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Ex-post favoring ranks: a fairness notion for the random assignment problem

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Abstract

We introduce two notions of ex-post fairness, namely ex-post favoring ranks (EFR) and robust ex-post favoring ranks, which consider whether objects are received by those agents who have the highest rank for them. We examine their compatibility with standard properties of random assignments and state some impossibility theorems. We also propose and formalize a revised version of the Boston mechanism and prove that it provides an EFR random assignment.

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Notes

  1. A bistochastic matrix is a square matrix with nonnegative real numbers that the summation of elements in each of its rows and columns equals to 1.

  2. The trading cycle is a sequence of agent/object pairs, where each agent prefers (the fraction of) the object of the next pair to her, in a (probabilistic) deterministic setting. Therefore, each agent is willing to trade (a probability of getting) her object, in a (probabilistic) deterministic setting, with the agent in the successor pair. For the very last pair, the next pair is defined to be the very first one.

  3. Harless (2018) named it Respect for Rank However, to make the connection with Kojima and Ünver (2014) clearer, we rename this concept to interim favoring rank.

  4. The Boston mechanism has different variations, which are different, especially in terms of strategy-proofness (Mennle and Seuken, 2018; Dur, 2019). In its standard version, first, each school reports its strict priority ordering of students and each student submits his preference ranking over schools. Then, at Step 1, each student applies to his first best school while schools admit students, following their priority order, until there is left neither a seat nor a student who has listed that school as his first choice. At Step k, each of remaining student who has been rejected at step \(k-1\), applies to his kth best school even though it might not have any seat left. Each school considers students who have listed it as their kth choice and assigns remaining seats to these students, one at a time following their priority order, until there is left neither a seat nor a student who has listed that school as his kth choice. The algorithm terminates when no student applies to a school.

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Correspondence to Mehdi Feizi.

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Appendix: omitted proofs

Appendix: omitted proofs

Proof of Proposition 1

We prove that every decomposition of an IFR random assignment is a lottery over deterministic assignments that favor higher ranks. An IFR random assignment P could be decomposed into deterministic assignments, \(P=\sum \nolimits _{l=1}^{k}\lambda _{l}\Pi _{l}\) where \( \sum \nolimits _{l=1}^{k}\lambda _{l}=1\) . Suppose for some l , where \(\lambda _{l}\ne 0\) , \(\Pi _{l}\) does not favor higher ranks. Then, in this deterministic assignment, there are agents i and j , such that i could object j in the sense that for some object b and c , \((\Pi _{l})_{jb}=1\) and \(rk(b,\succ _{i})<rk(b,\succ _{j})\) , while \((\Pi _{l})_{ic}=1\) where \(rk(c,\succ _{i})>rk(b,\succ _{i}) \) . Now, consider the random assignment P . Since \( (\Pi _{l})_{jb}=1\) , we have \(p_{jb}\ge \lambda _{l}(\Pi _{l})_{jb}=\lambda _{l}>0\) , and since P is IFR, we must have \(p_{ic}=0\) . However, as \((\Pi _{l})_{ic}=1\) , we have \(p_{ic}\ge \lambda _{l}(\Pi _{l})_{ic}=\lambda _{l}>0\) , which is a contradiction.

We show the other direction by a counterexample: Take a random assignment (3) for the preference profile (2). The support of P has only four following deterministic assignments:

$$\begin{aligned} \Pi _{1}= & {} \left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) \text {, }\Pi _{2}=\left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right) \text {, } \\ \Pi _{3}= & {} \left( \begin{array}{cccc} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) \text {, }\Pi _{4}=\left( \begin{array}{cccc} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right) \text {,} \end{aligned}$$

where all favor higher ranks, and therefore P is REFR. However, P is not IFR since while \(rk(b,\succ _{1})<rk(b,,\succ _{2})\) and \(p_{2b}>0\), we have \(p_{1c}>0\) where \(rk(c,\succ _{1})>rk(b,,\succ _{1})\). QED.

Proof of Proposition 2

(i) For the preference profile

$$\begin{aligned} \begin{array}{cccccccc} 1: &{} a &{} \succ _{1} &{} b &{} \succ _{1} &{} c &{} \succ _{1} &{} d \\ 2: &{} a &{} \succ _{2} &{} c &{} \succ _{2} &{} b &{} \succ _{2} &{} d \\ 3: &{} b &{} \succ _{3} &{} c &{} \succ _{3} &{} d &{} \succ _{3} &{} a \\ 4: &{} b &{} \succ _{4} &{} d &{} \succ _{4} &{} a &{} \succ _{4} &{} c \end{array} \text {,} \end{aligned}$$
(6)

the random assignment

$$\begin{aligned} P=\left( \begin{array}{cccc} 0 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1/2 &{} 0 &{} 1/2 \\ 0 &{} 1/2 &{} 0 &{} 1/2 \end{array} \right) =\frac{1}{2}\overset{\text {FHR}}{\overbrace{\left( \begin{array}{cccc} 0 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right) }}+\frac{1}{2}\overset{\text {Not FHR}}{\overbrace{\left( \begin{array}{cccc} 0 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) }}\text {,} \end{aligned}$$
(7)

is not REFR since it has a decomposition with the deterministic assignment which is not FHR: the second deterministic matrix is not FHR: agent 1 gets object c which agent 3 prefers it more while the latter has been given a less preferable object d .

However, P is OE since it is acyclic: we only have \(a\tau \left( P,\succ _{1}\right) c\) (since \(a\succ _{1}c\) and \(p_{1c}>0\)), \(b\tau \left( P,\succ _{1}\right) c\) (since \(b\succ _{1}c\) and \(p_{1c}>0\)), \(b\tau \left( P,\succ _{3}\right) d\) (since \(b\succ _{3}d\) and \(p_{3d}>0\)), \(c\tau \left( P,\succ _{3}\right) d\) (since \(c\succ _{3}d\) and \(p_{3d}>0\)), and \(\tau \left( P,\succ _{4}\right) \) (since \(b\succ _{4}d\) and \(p_{4d}>0\)) which do not make a cycle.

For the preference profile

$$\begin{aligned} \begin{array}{cccccccc} 1: &{} a &{} \succ _{1} &{} b &{} \succ _{1} &{} c &{} \succ _{1} &{} d \\ 2: &{} c &{} \succ _{2} &{} b &{} \succ _{2} &{} d &{} \succ _{2} &{} a \\ 3: &{} c &{} \succ _{3} &{} d &{} \succ _{3} &{} b &{} \succ _{3} &{} a \\ 4: &{} a &{} \succ _{4} &{} d &{} \succ _{4} &{} b &{} \succ _{4} &{} c \end{array} \text {,} \end{aligned}$$
(8)

the random assignment

$$\begin{aligned} P=\left( \begin{array}{cccc} 1/2 &{} 1/2 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1/2 &{} 1/2 \\ 0 &{} 1/2 &{} 1/2 &{} 0 \\ 1/2 &{} 0 &{} 0 &{} 1/2 \end{array} \right) =\frac{1}{2}\overset{\text {FHR}}{\overbrace{\left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right) }}+\frac{1}{2}\overset{\text {FHR}}{\overbrace{\left( \begin{array}{cccc} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \end{array} \right) }} \end{aligned}$$
(9)

is not OE since P is not acyclic: we have \(b\tau \left( P,\succ _{2}\right) d\) (since \(b\succ _{2}d\) and \( p_{2d}>0\) ), \(d\tau \left( P,\succ _{3}\right) b\) (since \( d\succ _{3}b\) and \(p_{3b}>0\) ), which make a cycle.

Yet, P is REFR since it has a unique decomposition (9) into FHR deterministic assignments: We must give a to either to agent 1 or agent 4, since if we give it to agent 2 or agent 3, a trade happens as both agents 2 and 3 know a as their worst choice, while agents 1 and 4 know a as their first best choice. On the one hand, once we give a to agent 1 , as \(p_{2b}=p_{4b}=0\), we have to give b to agent c, and as \(p_{4c}=0\) we only have one choice to give c to agent 2 and finally d to agent 4. On the other hand, once we give a to agent 4 , as \(p_{1d}=p_{3d}=0\) we have to give d to agent 2, and since \(p_{1c}=0\) we only have one choice to give c to agent 3 and finally give b to agent 1.

For the preference profile

$$\begin{aligned} \begin{array}{cccccccc} 1: &{} a &{} \succ _{1} &{} d &{} \succ _{1} &{} c &{} \succ _{1} &{} b \\ 2: &{} c &{} \succ _{2} &{} a &{} \succ _{2} &{} d &{} \succ _{2} &{} b \\ 3: &{} c &{} \succ _{3} &{} d &{} \succ _{3} &{} b &{} \succ _{3} &{} a \\ 4: &{} a &{} \succ _{4} &{} c &{} \succ _{4} &{} b &{} \succ _{4} &{} d \end{array} \text {,} \end{aligned}$$

the random assignment

$$\begin{aligned} P=\left( \begin{array}{cccc} 1/2 &{} 0 &{} 0 &{} 1/2 \\ 0 &{} 0 &{} 1/2 &{} 1/2 \\ 0 &{} 1/2 &{} 1/2 &{} 0 \\ 1/2 &{} 1/2 &{} 0 &{} 0 \end{array} \right) =\frac{1}{2}\overset{\text {FHR}}{\overbrace{\left( \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \end{array} \right) }}+\frac{1}{2}\overset{\text {FHR}}{\overbrace{\left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) }} \end{aligned}$$

is OE since it is acyclic: we have \(a\tau \left( P,\succ _{1}\right) d\) (since \(a\succ _{1}d\) and \(p_{1d}>0\) ), \(c\tau \left( P,\succ _{2}\right) d\) and \(a\tau \left( P,\succ _{2}\right) d\)(since \(c\succ _{3}d\succ _{3}a\) and \(p_{2d}>0\)), \(c\tau \left( P,\succ _{3}\right) b\) and \(d\tau \left( P,\succ _{3}\right) b\)(since \( c\succ _{3}d\succ _{3}b\) and \(p_{3b}>0\)), and finally \( a\tau \left( P,\succ _{4}\right) b\) and \(c\tau \left( P,\succ _{4}\right) b\)(since \(a\succ _{4}c\succ _{4}b\) and \( p_{4b}>0\)), which do not make a cycle.

P is also REFR since it has a unique decomposition into FHR deterministic assignments: The decomposition of P is also unique: We must give a to either to agent 1 or agent 4, since if we give it to agent 2 or agent 3 , a trade happens. On the one hand, once we give a to agent 1, as \(p_{3d}=p_{4d}=0\), we have to give d to agent 2, and as \(p_{1c}=p_{4c}=0\) we only have one choice to give c to agent 3 and finally b to agent 2. On the other hand, once we give a to agent 4, as \(p_{1b}=p_{2b}=0\) we have to give b to agent 3, and since \(p_{1c}=0\) we only have one choice to give c to agent 3 and finally give d to agent 1.

However, P is not IFR since \(p_{2d}>0\) , and while agent 3 ranks d higher than agent 2 does, i.e., \(rk(d,\succ _{3})<rk(d,\succ _{2})\), P assigns an inferior object b, i.e., \(rk(b,\succ _{3})>rk(d,\succ _{3})\), to agent 3 with a positive probability.

ii) For the preference profile (6),the random assignment (7) is OE not EFR since in its only possible decomposition into deterministic assignments, there is a deterministic matrix which does not favor higher ranks. For the preference profile (8), the random assignment (9) is not OE since it is not acyclic, while it has a decomposition into FHR deterministic assignments.

We already showed in Example 1 that an assignment (3) for the preferences profile (2) is not IFR while it is EFR. We now show that it is also OE as it is acyclic: we only have \(a\tau \left( P,\succ _{1}\right) c\), \(b\tau \left( P,\succ _{1}\right) c\), \( d\tau \left( P,\succ _{1}\right) c\), (since \(a\succ _{1}b\succ _{1}d \) \(\succ _{1}c\) and \(p_{1c}>0\)), \(d\tau \left( P,\succ _{2}\right) b\), \(a\tau \left( P,\succ _{2}\right) b\) (since \(d\succ _{2}a\) \(\succ _{2}b\) and \( p_{2d}>0\)), \(a\tau \left( P,\succ _{3}\right) c\), \(d\tau \left( P,\succ _{3}\right) c\), and \(b\tau \left( P,\succ _{3}\right) c\), (since \(a\succ _{3}d\succ _{3}b\succ _{3}c\) and \(p_{3b}>0\)), \(d\tau \left( P,\succ _{4}\right) b\) (since \(d\succ _{4}b\) and \(p_{4b}>0\)) which do not make a cycle.

iii) We show both directions by counterexamples: For the preference profile (6), the random assignment (7) is robust ex-post Pareto efficient since both deterministic assignments in its only decomposition are Pareto efficient. However, it is not EFR as there exists a deterministic assignment in its decomposition, which is not FHR.

For the preference profile

$$\begin{aligned} \begin{array}{cccccccc} 1: &{} a &{} \succ _{1} &{} d &{} \succ _{1} &{} b &{} \succ _{1} &{} c \\ 2: &{} c &{} \succ _{2} &{} b &{} \succ _{2} &{} d &{} \succ _{2} &{} a \\ 3: &{} c &{} \succ _{3} &{} d &{} \succ _{3} &{} b &{} \succ _{3} &{} a \\ 4: &{} a &{} \succ _{4} &{} b &{} \succ _{4} &{} c &{} \succ _{4} &{} d \end{array}, \end{aligned}$$

the random assignment

$$\begin{aligned} P= & {} \left( \begin{array}{cccc} 1/2 &{} 1/2 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1/2 &{} 1/2 \\ 0 &{} 0 &{} 1/2 &{} 1/2 \\ 1/2 &{} 1/2 &{} 0 &{} 0 \end{array} \right) =\frac{1}{2}\overset{\text {FHR}}{\overbrace{\left( \begin{array}{cccc} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 \end{array} \right) }}+\frac{1}{2}\overset{\text {FHR}}{\overbrace{\left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) }} \\= & {} \frac{1}{2}\overset{\text {Not Pareto}}{\overbrace{\left( \begin{array}{cccc} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \end{array} \right) }}+\frac{1}{2}\overset{\text {Pareto}}{\overbrace{\left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) }}\text {,} \end{aligned}$$

is EFR since there it has a decomposition into FHR deterministic assignments. However, P is not REP since there is a decomposition of it that admits a non-Pareto optimal deterministic assignment consistent with it.

For the preference profile (6), the random assignment

$$\begin{aligned} P= & {} \left( \begin{array}{cccc} 1/2 &{} 0 &{} 1/2 &{} 0 \\ 1/2 &{} 0 &{} 1/2 &{} 0 \\ 0 &{} 1/2 &{} 0 &{} 1/2 \\ 0 &{} 1/2 &{} 0 &{} 1/2 \end{array} \right) =\frac{1}{2}\overset{\text {Not FHR but Pareto}}{\overbrace{\left( \begin{array}{cccc} 0 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) }}+\frac{1}{2}\overset{\text {FHR and Pareto}}{\overbrace{\left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right) }} \\= & {} \frac{1}{2}\overset{\text {FHR and Pareto}}{\overbrace{\left( \begin{array}{cccc} 0 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right) }}+\frac{1}{2}\overset{\text {FHR and Pareto}}{\overbrace{\left( \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 \end{array} \right) }}\text {,} \end{aligned}$$

has only two possible decompositions where in both all deterministic assignments are Pareto while there is a deterministic assignment in the first decomposition that does not favor higher ranks. Therefore, P is REP and EFR, but not REFR. QED.

Proof of Proposition 3

Suppose for the preferences profile,

$$\begin{aligned} \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 &{} : &{} a &{} \succ _{1} &{} b &{} \succ _{1} &{} \ldots \\ 2 &{} : &{} a &{} \succ _{2} &{} b &{} \succ _{2} &{} \ldots \\ \ldots &{} : &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots \\ \mathbf {i} &{} \mathbf {:} &{} \mathbf {b} &{} \mathbf {\succ }_{i} &{} \mathbf {a} &{} \mathbf {\succ }_{i} &{} \mathbf {\ldots } \\ \ldots &{} : &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots \\ n &{} : &{} a &{} \succ _{n} &{} b &{} \succ _{n} &{} \ldots \end{array} \text {,} \end{aligned}$$

P is a random assignment where it is both EF and EFR with a decomposition \(P=\sum \nolimits _{l=1}^{k}\lambda _{l}\Pi _{l}\) where \(\Pi _{l}\) is FHR for all l.

On the one hand, since b is the first best object of only agent i, all deterministic FHR assignments, \(\Pi _{l}\) , should assign object b to agent i, which implies that \(p_{ib}=1\) (hence agent i gets all other objects with zero probability, particularly object a, i.e., \( p_{ia}=0\)). Moreover, any other agent \(j\ne i\) has no chance to get object b, i.e., \(p_{jb}=0\); particularly agent 1, \(p_{1b}=0\).

On the other hand, since P is EF and object a is the first best of all agents \(j\ne i\), the chance of receiving abject a must be divided equally among all these \(n-1\) agents, i.e., \(p_{ja}=p_{1a}=\frac{1}{n-1}\). Now, since

$$\begin{aligned} p_{1a}+p_{1b}=\frac{1}{n-1}<p_{ia}+p_{ib}=1\text {,} \end{aligned}$$

the allocation of agent 1, regarding her preference, does not stochastically dominate the allocation of agent i, agent 1 envies agent i, and P is not EF, which is a contradiction. QED.

Proof of Theorem 1

(i) Suppose \( n=|A|=|N|=4\), and mechanism \(\mu \) is EFR and P is the output of \(\mu \) on preference profile:

$$\begin{aligned} \begin{array}{cccccccc} 1: &{} a &{} \succ _{1} &{} b &{} \succ _{1} &{} c &{} \succ _{1} &{} d \\ 2: &{} a &{} \succ _{2} &{} b &{} \succ _{2} &{} c &{} \succ _{2} &{} d \\ 3: &{} b &{} \succ _{3} &{} a &{} \succ _{3} &{} c &{} \succ _{3} &{} d \\ 4: &{} b &{} \succ _{4} &{} a &{} \succ _{4} &{} c &{} \succ _{4} &{} d \end{array} \text {.} \end{aligned}$$

We should have \(p_{ic}\le 1/4\) for \(i=1,\ldots 4\). Without loss of generality, suppose \(p_{1c}\le 1/4\). As \(\mu \) is EFR and SETE, we have \(p_{1a}=p_{2a}=1/2\), and \(p_{1b}=0\). Now, if agent 1 misreports to \(\succ _{1}^{\prime }:a\succ _{1}^{\prime }c\succ _{1}^{\prime }b\succ _{1}^{\prime }d\), since \( \mu \) is EFR and SETE, we have \(p_{1a}^{\prime }=p_{2a}^{\prime }=1/2\), \(p_{1b}^{\prime }=0\), and \(p_{1c}^{\prime }=1/2\) . (Since agent 1 prefers c more than any other agent, in any deterministic FHR assignments, she is assigned either her first best object a, or her second best object c, i.e., \(p_{1a}^{\prime }+p_{1c}^{\prime }=1\).) Now, we have \( p_{1a}^{\prime }=1/2\ge p_{1a}=1/2\), \(p_{1a}^{\prime }+p_{1b}^{\prime }=1/2\ge p_{1a}+p_{1b}=1/2\) , and \( p_{1a}^{\prime }+p_{1b}^{\prime }+p_{1c}^{\prime }=1>p_{1a}+p_{1b}+p_{1c}=3/4 \). Therefore, agent 1 is better off misreporting.

Now, suppose \(n>4\), and we have additional agents 5 ,6, \(\ldots \), n and objects \(o_{5}\) , \(o_{6}\), ..., \(o_{n}\), where \(o_{i}\) is the first-best object of each new agent i. Since mechanism \( \mu \) is EFR and P is the output of \(\mu \) on preference profile

$$\begin{aligned} \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1: &{} a &{} \succ _{1} &{} b &{} \succ _{1} &{} c &{} \succ _{1} &{} d &{} \succ _{1} &{} o_{5} &{} \succ _{1} &{} o_{6} &{} \succ _{1} &{} \ldots &{} \succ _{1} &{} o_{n} \\ 2: &{} a &{} \succ _{2} &{} b &{} \succ _{2} &{} c &{} \succ _{2} &{} d &{} \succ _{2} &{} o_{5} &{} \succ _{2} &{} o_{6} &{} \succ _{2} &{} \ldots &{} \succ _{2} &{} o_{n} \\ 3: &{} b &{} \succ _{3} &{} a &{} \succ _{3} &{} c &{} \succ _{3} &{} d &{} \succ _{2} &{} o_{5} &{} \succ _{2} &{} o_{6} &{} \succ _{2} &{} \ldots &{} \succ _{2} &{} o_{n} \\ 4: &{} b &{} \succ _{4} &{} a &{} \succ _{4} &{} c &{} \succ _{4} &{} d &{} \succ _{2} &{} o_{5} &{} \succ _{2} &{} o_{6} &{} \succ _{2} &{} \ldots &{} \succ _{2} &{} o_{n} \\ 5: &{} o_{5} &{} \succ _{6} &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots \\ 6: &{} o_{6} &{} \succ _{6} &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots \\ \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots \\ n: &{} o_{n} &{} \succ _{n} &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots \end{array} \text {,} \end{aligned}$$

for \(i>4\), we should have \(p_{io_{i}}=1\) and \( p_{jo_{i}}=0\) for \(j\ne i\). Otherwise there exits a deterministic assignment in any decomposition of P that is not favoring higher ranks. For \(n\ge 4\), agents 1 to 4 get the very same allocation as when we had \(n=4\) agents and objects. Therefore, with the same argument, agent 1 has an incentive to misreport. Thus, \(P_{1}^{\prime }\succ _{1}^{sd}P_{1}\) , and \(\mu \) is not weak-strategy proof.

ii) Suppose that the number of agents and objects is at least three, i.e., \(n=|A|=|N|\ge 3\), and for all agents, object a is the first best and object b is the second best. For every EFR mechanism \(\mu (\succ )=P\), there exists some agent i such that, \(p_{ia}+p_{ib}<1\). (Since if for all agents i, we have \(p_{ia}+p_{ib}=1\), as the number of agents is at least three, then \(\sum \nolimits _{i\in N}p_{ia}+\sum \nolimits _{i\in N}p_{ib}>2\), which contradicts with the fact that matrix P is bistochastic.) Let us define \(\varepsilon =1-(p_{ia}+p_{ia})>0\) . For a utility function, \(u_{i}\), which respects the preference \( \succ _{i}\), let us also assume \(u_{i}(a)=10+\varepsilon \), \(u_{i}(b)=10\), and \(u_{i}(c)<\varepsilon /n\) for all \( c\in A/\left\{ a,b\right\} \). We prove that given \(u_{i}\), agent i has the incentive to misreport her preference as \( i:b\succ _{i}^{^{\prime }}a\succ _{i}^{^{\prime }}\ldots \).

Since \(P^{^{\prime }}=\mu (\succ _{i}^{^{\prime }},\succ _{-i})\) satisfies EFR and b is the first best object of i in \(\succ _{i}^{^{\prime }}\), and not the first best object of anyone else, she must get it for sure, i.e., \(p_{ib}^{^{\prime }}=1\), and \(p_{id}^{^{\prime }}=0\) for \(d\in A/\left\{ b\right\} \). Therefore, the (expected) utility of agent i in the assignment \(P^{^{\prime }}\) is \(U_{i}(P^{^{\prime }})=p_{ib}^{^{\prime }}u_{i}(b)+\sum \nolimits _{d\in A/\left\{ b\right\} }p_{id}^{^{\prime }}u_{i}(d)=10\). However,

$$\begin{aligned} U_{i}(P)= & {} p_{ia}u_{i}(a)+p_{ib}u_{i}(b)+\sum \nolimits _{c\in A/\left\{ a,b\right\} }p_{ic}u_{i}(c) \\= & {} p_{ia}\left( 10+\varepsilon \right) +10p_{ib}+\sum \nolimits _{c\in A/\left\{ a,b\right\} }p_{ic}u_{i}(c) \\= & {} 10\left( p_{ia}+p_{ib}\right) +p_{ia}\varepsilon +\sum \nolimits _{c\in A/\left\{ a,b\right\} }p_{ic}u_{i}(c) \\< & {} 10\left( p_{ia}+p_{ib}\right) +p_{ia}\varepsilon +\sum \nolimits _{c\in A/\left\{ a,b\right\} }p_{ic}(\frac{\varepsilon }{n}) \\< & {} 10\left( p_{ia}+p_{ib}\right) +p_{ia}\varepsilon +(n-2)(\frac{\varepsilon }{n}) \\< & {} 10\left( 1-\varepsilon \right) +\varepsilon +\varepsilon <10=U_{i}(P^{^{\prime }})\text {.} \end{aligned}$$

1-Hence, given the utility function \(u_{i}\), agent i could gain more via misreporting her preference, and thus it is not possible for the mechanism \(\mu \) to be both SP and EFR. QED.

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Ramezanian, R., Feizi, M. Ex-post favoring ranks: a fairness notion for the random assignment problem. Rev Econ Design 25, 157–176 (2021). https://doi.org/10.1007/s10058-021-00246-7

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