Incompatibility of Pareto Eciency, Strategy-proofness and Mutual Best ∗

Umut Mert Dur

North Carolina State University July 9, 2015

Abstract While adopting a mechanism, a school board aims to have a reasonable balance between strategy-proofness, Pareto eciency, and fairness. In the school choice environment, Pareto eciency and fairness are incompatible, i.e., there does not exist a Pareto ecient and fair mechanism. Although there exists a Pareto ecient and strategy-proof mechanism (e.g. Top Trading Cycles mechanism), we show that any Pareto ecient and strategy-proof mechanism fails to satisfy a minimal fairness requirement: there does not exist a Pareto ecient, strategy-proof and mutual best. JEL Classication: C78, I28 Key Words: Matching Theory, Market Design, School Choice Problem, Top Trading Cycles

1

Introduction

School choice problem is introduced by Abdulkadiro§lu and Sönmez [2003]. In their seminal paper, two competing mechanisms, namely Deferred Acceptance (DA) mechanism [Gale and Shapley, 1962] and Top Trading Cycles (TTC) mechanism, have been suggested to the policy makers to replace Boston mechanism (BM) which is not immune to preference manipulation. The DA mechanism is strategy-proof1 , individually rational, non-wasteful, fair2 and it Pareto dominates any other fair mechanism [Gale and Shapley, ∗

Address: 2801 Founders Drive 4102 Nelson Hall, Raleigh, NC, 27695; e-mail: [email protected]; web

page: https://sites.google.com/site/umutdur/ 1 A mechanism is strategy-proof if it is a (weakly) dominant strategy for each student to state preferences truthfully. 2 A mechanism is fair if under its allocation there does not exist a student who prefers another school to his assignment and that school admitted a student with lower priority.

1

1962, Dubins and Freedman, 1981, Roth, 1982, Balinski and Sönmez, 1999]. Moreover, it is the unique strategy-proof mechanism which is also individually rational, non-wasteful and fair [Alcalde and Barbera, 1994]. However, it is not Pareto ecient. On the other hand, the TTC mechanism, which is based on Gale's top trading cycles idea [Shapley and Scarf, 1974], is Pareto ecient, strategy-proof, individually rational and non-wasteful. However, it is not fair. In particular, there does not exist a Pareto ecient and fair mechanism in school choice problem [Roth, 1982, Balinski and Sönmez, 1999]. Given the incompatibility between Pareto eciency and fairness, and the importance of strategy-proofness in the school choice problem, one may wonder to what extent priorities will be respected by strategy-proof and Pareto ecient mechanisms. In this paper, we use a very weak fairness notion called mutual best [Morrill, 2013]. In particular, we say a mechanism is mutual best if a student is always assigned to her top choice whenever she has the highest priority for her top choice among the students nding her top choice acceptable. We show that any strategy-proof and Pareto ecient mechanism does not satisfy mutual best. Our impossibility result implies that any mechanism belonging to the class of trading cycles mechanisms [Pycia and Ünver, 2011a], and therefore hierarchical mechanisms [Pápai, 2000], is not mutual best.

2

Model

A school choice problem is a 5-tuple (I, S, q, P, ) where

• I = {i1 , i2 , ..., in } is the nite set of students, • S = {s1 , s2 , ..., sm } is the nite set of schools, • q = (qs )s∈S is the quota vector where qs is the number of available seats in school s, • P = (Pi )i∈I is the preference prole where Pi is the strict preference of student i over the schools and being unassigned option denoted by ∅,

• = (s )s∈S is the priority prole where s is the strict priority relation of school s over I .

2

Let q∅ = ∞. In the rest of the paper we x I ,S , q , and , and represent a problem with

P . Let Ri be the at-least-as-good-as relation associated with Pi for all i ∈ I . A

matching µ : I → S is a function such that µ(i) ∈ S ∪ {∅}, |µ(i)| ≤ 1 and

|µ−1 (s)|

≤ qs for all i ∈ I and s ∈ S . We denote the set of matchings with M. A

mechanism Φ is a procedure which selects a matching for each problem. The matching selected by mechanism Φ in problem P is denoted by Φ(P ) and the assignment of each student i ∈ I is denoted by Φi (P ).

Pareto dominates another matching ν if µ(i)R ν(i) for each student i ∈ I and µ(j)P ν(j) for at least one student j ∈ I . A matching µ is Pareto ecient A matching µ

i

j

if there does not exist another matching ν ∈ M which Pareto dominates µ. A matching µ is

fair if there does not exist a student school pair (i, s) where s P

i

and i s j for some j ∈ A mechanism Φ is a preference relation

strategy-proof

P0

µi

µ−1 (s).

such that

if there does not exist a student i ∈ I and

Φi (P 0 , P−i )

Pi Φi (Pi ). A mechanism Φ is

Pareto

ecient (fair) if for any problem P its outcome Φ(P ) is Pareto ecient (fair). 3

Results In a problem P , let As (P ) be the set of students ranking s above ∅: As (P ) = {i ∈

I|sPi ∅}. Let iTs be the student who has the highest priority in school s among the students in As (P ). A matching µ is

mutual best if for any school s, whenever i

T s

top of her preference list, i.e., sPiTs x for all x ∈ S ∪ {∅}, then is

µ(iTs )

ranks s at the

= s. A mechanism Φ

mutual best if Φ(P ) is mutual best for all P . In words, a mechanism satises mutual

best if a student is always assigned to her most preferred school whenever she has the highest priority for it among the students considering it acceptable. It is easy to see that any fair mechanism is mutual best, i.e., DA mechanism is mutual best. Hence, there exists a strategy-proof and mutual best mechanism. Moreover, BM, which is not a fair mechanism, is mutual best. Since BM is Pareto ecient, the existence of Pareto ecient and mutual best mechanism is guaranteed. However, any Pareto ecient and strategy-proof mechanism fails to satisfy mutual best.

Theorem 1 There does not exist a mechanism that is Pareto ecient, strategy-proof

and mutual best.

Proof. Suppose Φ is a Pareto-ecient, strategy-proof and mutual best mechanism.

Consider a problem where S = {s1 , s2 , s3 }, I = {i1 , i2 , i3 , i4 }. Each school has unit capacity. Preferences and priorities are given by 3

Pi1

P i2

P i3

P i4

s1

s2

s3

s2

s1

s3

s1

i1

i2

i4

∅

s2

s1

s3

i3

i3

i1

∅

∅

∅

i2

i1

i2

i4 i4 i3 Denote this problem with P . Since i2 has the highest priority at s2 among the students considering it acceptable, any mutual best and strategy-proof mechanism assigns

i2 a school at least as good as s2 . Otherwise, due to mutual best i2 can get s2 by reporting it at the top of his preference list and this violates strategy-proofness. Similarly, any mutual best and strategy-proof mechanism assigns i3 and i4 a school at least as good as

s1 and s3 , respectively. If i2 is assigned to s1 by Φ, either i3 or i4 will be unassigned. This violates mutual best. Hence i2 needs to be assigned to be assigned to s2 and Φ selects the following matching: Φi1 (P ) = ∅, Φi2 (P ) = s2 , Φi3 (P ) = s3 and Φi4 (P ) = s1 . Now consider the preference prole P˜i1 : s2 P˜i1 s1 P˜i1 ∅. Let P˜ = (P˜i1 , Pi2 , Pi3 , Pi4 ). In problem P˜ , since i1 has the highest priority at s1 among the students considering it acceptable, any mutual best and strategy-proof mechanism assigns i1 a school at least as good as s1 . Similarly, any mutual best and strategy-proof mechanism assigns i2 and i4 a school at least as good as s2 and s3 , respectively. If i4 is assigned to s1 by Φ in problem P˜ , either i1 or i4 will be unassigned. This violates mutual best. Hence i4 needs to be assigned to be assigned to s3 and Φ selects the following matching:Hence, Φi (P˜ ) = s2 , 1

Φi2 (P˜ ) = s1 , Φi3 (P˜ ) = ∅ and Φi4 (P˜ ) = s3 . Since Φi1 (P˜ )Pi Φi1 (P ), Φ is manipulated by i1 and it cannot be strategy-proof.

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American

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Journal of Mathematical