RSD characterization for n=3 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
agent 2: a > b > c
agent 3: a > b > c
Application of Anonymity & No-waste Constraint gives 
         a   b   c
agent 1: 2/6 2/6 2/6 
agent 2: 2/6 2/6 2/6 
agent 3: 2/6 2/6 2/6 

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

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

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

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


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

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

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

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


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

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

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

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


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

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

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: b > c > a
agent 3: c > b > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: 0/6 */6 */6 
agent 3: */6 */6 */6 

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


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

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

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: b > c > a
agent 3: c > b > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: 0/6 */6 */6 
agent 3: 0/6 */6 */6 

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


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

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: a > c > b
agent 3: c > a > b
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: */6 2/6 */6 
agent 3: */6 1/6 */6 

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


Consider the profile
agent 1: a > b > c
agent 2: a > c > b
agent 3: c > a > b
Application of Ex-post Efficiency gives 
         a   b   c
agent 1: */6 */6 0/6 
agent 2: */6 2/6 */6 
agent 3: 0/6 1/6 */6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: */6 */6 0/6 
agent 2: */6 2/6 */6 
agent 3: 0/6 1/6 5/6 
Application of Anonymity & No-waste Constraint gives 
         a   b   c
agent 1: */6 3/6 0/6 
agent 2: */6 2/6 1/6 
agent 3: 0/6 1/6 5/6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: 3/6 3/6 0/6 
agent 2: 3/6 2/6 1/6 
agent 3: 0/6 1/6 5/6 

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

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

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

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c
agent 2: a > c > b
agent 3: c > b > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: */6 */6 */6 
agent 3: */6 */6 5/6 


Consider the profile
agent 1: a > b > c
agent 2: b > c > a
agent 3: c > b > a
Application of Ex-post Efficiency gives 
         a   b   c
agent 1: */6 0/6 0/6 
agent 2: 0/6 */6 0/6 
agent 3: 0/6 0/6 */6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: 6/6 0/6 0/6 
agent 2: 0/6 6/6 0/6 
agent 3: 0/6 0/6 6/6 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c
agent 2: b > a > c
agent 3: c > b > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: */6 6/6 */6 
agent 3: */6 */6 */6 

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


Consider the profile
agent 1: a > b > c
agent 2: b > a > c
agent 3: c > b > a
Application of Ex-post Efficiency gives 
         a   b   c
agent 1: */6 0/6 0/6 
agent 2: 0/6 6/6 0/6 
agent 3: 0/6 0/6 */6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: 6/6 0/6 0/6 
agent 2: 0/6 6/6 0/6 
agent 3: 0/6 0/6 6/6 

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

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

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: a > b > c
agent 3: c > b > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: */6 */6 0/6 
agent 3: */6 */6 */6 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: b > a > c
agent 3: b > c > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: */6 3/6 */6 
agent 3: 0/6 3/6 */6 

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


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

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

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: b > a > c
agent 3: c > a > b
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: */6 */6 0/6 
agent 3: */6 */6 6/6 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: a > b > c
agent 3: a > c > b
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: 2/6 */6 */6 
agent 3: 2/6 0/6 */6 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c
agent 2: a > b > c
agent 3: c > b > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: */6 */6 0/6 
agent 3: */6 */6 6/6 


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

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: b > a > c
agent 3: c > a > b
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: */6 */6 0/6 
agent 3: */6 0/6 6/6 

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


Consider the profile
agent 1: a > b > c
agent 2: b > a > c
agent 3: c > a > b
Application of Ex-post Efficiency gives 
         a   b   c
agent 1: */6 0/6 0/6 
agent 2: 0/6 */6 0/6 
agent 3: 0/6 0/6 6/6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: 6/6 0/6 0/6 
agent 2: 0/6 6/6 0/6 
agent 3: 0/6 0/6 6/6 

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

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


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

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

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c
agent 2: a > c > b
agent 3: c > b > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: 3/6 */6 */6 
agent 3: */6 */6 5/6 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: a > b > c
agent 3: b > c > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: */6 */6 */6 
agent 3: 0/6 */6 */6 


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

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: b > c > a
agent 3: c > a > b
The allocation takes the form
         a   b   c
agent 1: 6/6 */6 */6 
agent 2: */6 6/6 */6 
agent 3: */6 0/6 6/6 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c
agent 2: a > b > c
agent 3: b > c > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: 3/6 */6 */6 
agent 3: 0/6 */6 */6 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: a > c > b
agent 3: c > b > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: 3/6 */6 */6 
agent 3: 0/6 */6 5/6 


Consider the profile
agent 1: a > b > c
agent 2: b > c > a
agent 3: c > a > b
Application of Ex-post Efficiency gives 
         a   b   c
agent 1: 6/6 0/6 0/6 
agent 2: 0/6 6/6 0/6 
agent 3: 0/6 0/6 6/6 


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

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c
agent 2: a > c > b
agent 3: c > b > a
The allocation takes the form
         a   b   c
agent 1: 3/6 */6 */6 
agent 2: 3/6 */6 */6 
agent 3: 0/6 */6 5/6 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c
agent 2: a > c > b
agent 3: c > b > a
Application of Ex-post Efficiency gives 
         a   b   c
agent 1: 3/6 */6 0/6 
agent 2: 3/6 */6 */6 
agent 3: 0/6 */6 5/6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: 3/6 3/6 0/6 
agent 2: 3/6 */6 */6 
agent 3: 0/6 1/6 5/6 
Application of Anonymity & No-waste Constraint gives 
         a   b   c
agent 1: 3/6 3/6 0/6 
agent 2: 3/6 2/6 1/6 
agent 3: 0/6 1/6 5/6 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: c > a > b
agent 3: c > b > a
The allocation takes the form
         a   b   c
agent 1: */6 */6 */6 
agent 2: */6 2/6 3/6 
agent 3: */6 */6 3/6 


Consider the profile
agent 1: a > b > c
agent 2: c > a > b
agent 3: c > b > a
Application of Ex-post Efficiency gives 
         a   b   c
agent 1: */6 */6 0/6 
agent 2: */6 2/6 3/6 
agent 3: 0/6 */6 3/6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: */6 */6 0/6 
agent 2: 1/6 2/6 3/6 
agent 3: 0/6 3/6 3/6 
Application of Anonymity & No-waste Constraint gives 
         a   b   c
agent 1: 5/6 1/6 0/6 
agent 2: 1/6 2/6 3/6 
agent 3: 0/6 3/6 3/6 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c
agent 2: b > a > c
agent 3: b > c > a
The allocation takes the form
         a   b   c
agent 1: 5/6 */6 */6 
agent 2: */6 3/6 */6 
agent 3: 0/6 3/6 */6 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c
agent 2: b > a > c
agent 3: b > c > a
Application of Ex-post Efficiency gives 
         a   b   c
agent 1: 5/6 0/6 */6 
agent 2: */6 3/6 */6 
agent 3: 0/6 3/6 */6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: 5/6 0/6 1/6 
agent 2: */6 3/6 */6 
agent 3: 0/6 3/6 3/6 
Application of Anonymity & No-waste Constraint gives 
         a   b   c
agent 1: 5/6 0/6 1/6 
agent 2: 1/6 3/6 2/6 
agent 3: 0/6 3/6 3/6 

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

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: a > b > c
agent 3: b > c > a
The allocation takes the form
         a   b   c
agent 1: 3/6 */6 */6 
agent 2: 3/6 */6 2/6 
agent 3: 0/6 */6 */6 


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

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: a > c > b
agent 3: b > a > c
The allocation takes the form
         a   b   c
agent 1: 3/6 */6 */6 
agent 2: 3/6 0/6 3/6 
agent 3: 0/6 */6 1/6 


Consider the profile
agent 1: a > b > c
agent 2: a > c > b
agent 3: b > a > c
We already know that the allocation has the form
         a   b   c
agent 1: 3/6 */6 */6 
agent 2: 3/6 0/6 3/6 
agent 3: 0/6 */6 1/6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: 3/6 */6 */6 
agent 2: 3/6 0/6 3/6 
agent 3: 0/6 5/6 1/6 
Application of Anonymity & No-waste Constraint gives 
         a   b   c
agent 1: 3/6 1/6 2/6 
agent 2: 3/6 0/6 3/6 
agent 3: 0/6 5/6 1/6 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c
agent 2: a > b > c
agent 3: b > c > a
The allocation takes the form
         a   b   c
agent 1: 3/6 */6 2/6 
agent 2: 3/6 */6 2/6 
agent 3: 0/6 */6 */6 
Note that 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
agent 2: a > c > b
agent 3: b > c > a
The allocation takes the form
         a   b   c
agent 1: 3/6 */6 */6 
agent 2: 3/6 0/6 3/6 
agent 3: 0/6 5/6 */6 


Consider the profile
agent 1: a > b > c
agent 2: a > b > c
agent 3: b > c > a
We already know that the allocation has the form
         a   b   c
agent 1: 3/6 */6 2/6 
agent 2: 3/6 */6 2/6 
agent 3: 0/6 */6 */6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: 3/6 1/6 2/6 
agent 2: 3/6 1/6 2/6 
agent 3: 0/6 */6 */6 
Application of Anonymity & No-waste Constraint gives 
         a   b   c
agent 1: 3/6 1/6 2/6 
agent 2: 3/6 1/6 2/6 
agent 3: 0/6 4/6 2/6 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c
agent 2: a > b > c
agent 3: b > a > c
The allocation takes the form
         a   b   c
agent 1: 3/6 */6 */6 
agent 2: 3/6 */6 2/6 
agent 3: */6 4/6 2/6 


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


Consider the profile
agent 1: a > b > c
agent 2: a > c > b
agent 3: b > c > a
We already know that the allocation has the form
         a   b   c
agent 1: 3/6 */6 */6 
agent 2: 3/6 0/6 3/6 
agent 3: 0/6 5/6 */6 
Application of Lottery Constraint gives 
         a   b   c
agent 1: 3/6 */6 */6 
agent 2: 3/6 0/6 3/6 
agent 3: 0/6 5/6 1/6 
Application of Anonymity & No-waste Constraint gives 
         a   b   c
agent 1: 3/6 1/6 2/6 
agent 2: 3/6 0/6 3/6 
agent 3: 0/6 5/6 1/6 


For all the preferences profiles the outcome of the mechanism was deduced from the axioms.
The conjecture is proved for N=3.
Hurah!!