Assignment Maximization
aa r X i v : . [ ec on . T H ] D ec ASSIGNMENT MAXIMIZATION
MUSTAFA O ˘GUZ AFACAN, IN ´ACIO B ´O, AND BERTAN TURHAN
Abstract.
We evaluate the goal of maximizing the number of individuals matched to acceptableoutcomes. We show that it implies incentive, fairness, and implementation impossibilities. Despitethat, we present two classes of mechanisms that maximize assignments. The first are Pareto efficient,and undominated — in terms of number of assignments — in equilibrium. The second are fair forunassigned students and assign weakly more students than stable mechanisms in equilibrium.
JEL classification : D47, C78, D63.
Keywords : Market Design, Matching, Maximal Matching, Fairness, Object Allocation, School Choice.1.
Introduction
In this paper, we consider the economic problems faced by a market designer who wants to producestudent matchings (or object allocations) that are responsive to agents’ preferences and leave thesmallest number of them unmatched (that is, have maximum cardinality among individually rationalmatchings). One of the main motivations for studying this problem is the fact that, in practice, market de-signers often make adaptations to standard procedures with the objective of preventing agents frombeing left unmatched. The real life use of allocation mechanisms in school choice procedures, forexample, often consists of using a standard mechanism, such as the Gale-Shapley deferred acceptance(Gale and Shapley, 1962), followed by some additional procedure to assign the students who wereleft unmatched into some school. These secondary steps or other ad-hoc methods for filling up theremaining seats, however, result in the loss of the properties of the mechanism that was used in thefirst place, such as fairness and strategy-proofness (Dur and Kesten, 2019). In this paper we startinstead from the presumption that the market designer has the objective of leaving the minimumnumber of students unmatched. While this objective is not attainable via a strategy-proof mecha-nism, we propose mechanisms that produce maximum matchings and are efficient or satisfy a novelfairness criterion, when students are non-strategic. We also show that they satisfy desirable char-acteristics in equilibrium, and increase the cardinality of the matching as the proportion of truthfulagents increases.
Mustafa O˘guz Afacan : Sabancı University, Faculty of Art and Social Sciences, Orhanli, 34956, Istanbul, Turkey.e-mail: [email protected].
In´acio B´o : University of York, Department of Economics and Related Studies. website: ;e-mail: [email protected].
Bertan Turhan : Iowa State University, Department of Economics, 260 Heady Hall, Ames, IA, 50011, USA. e-mail:[email protected] thank Ahmet Alkan, Orhan Ayg¨un, Mehmet Barlo, Umut Dur, Andrei Gomberg, Isa Hafalır, Onur Kesten,Vikram Manjunath, Tridib Sharma, Tayfun S¨onmez, Alex Teytelboym, William Thomson, and Utku ¨Unver for helpfulcomments. Afacan acknowledges the Marie Curie International Reintegration Grant. B´o acknowledges financialsupport by the Deutsche Forschungsgemeinschaft (KU 1971/3-1). Even though our entire analysis translates naturally to most unit-demand discrete assignment problems, we will usethe framing of school allocation throughout the paper. Related Literature
While algorithms for finding maximum matchings are well-known (Kuhn, 1955; Berge, 1957), theresearch on the incentives induced by the use of these procedures is limited, and typically rely onrandom mechanisms. One exception is Afacan and Dur (2018), which follows-up to this paper andshows that no strategy-proof and individually rational mechanism systematically matches more stu-dents than either of Boston, Gale-Shapley deferred acceptance, and serial dictatorship mechanisms.Krysta et al. (2014) consider the problem of producing maximal matchings in a house allocationproblem. They show that there is no mechanism that is deterministic, maximal, and strategy-proof, and provide instead a random mechanism that is strategy-proof and yields approximately-maximal outcomes. Bogomolnaia and Moulin (2015) evaluate the trade-off between maximalityand envy-freeness, a notion of fairness that is stronger than the ones we consider in this paper.Bogomolnaia and Moulin (2004) consider the random assignment when agents have dichotomouspreferences. When that is the case, Pareto efficiency is equivalent to maximality of the matching,and moreover, since agents are indifferent between all “acceptable” allocations, maximality doesn’tresult in incentive problems even in deterministic mechanisms. Noda (2018) studies the matchingsize achieved by strategy-proof mechanisms in a general model of matching with constraints.Finding the matching with maximum cardinality subject to some constraints is also a problemthat is explored in the literature. Irving and Manlove (2010) consider the problem of finding stablematchings with maximum cardinality when priorities have ties, which is known to be an NP-hardproblem, and present heuristics for finding them. Ashlagi et al. (2020) also consider object assign-ment problems under distributional constraints. The authors show that variants of serial dictatorshipand Probabilistic Serial (Bogomolnaia and Moulin, 2001) mechanisms assign at least as many agentsas one can match under the constraints, while the violations of the constraints are relatively small.Assignment maximization has been the primary objective in the organ exchange literature, asit means the maximum number of transplants. This literature was initiated by the seminal workon kidney exchange of Roth et al. (2004). In a subsequent study, in order to accommodate sev-eral physical and geographical restrictions in operating transplants, Roth et al. (2005) introducethe idea of pairwise kidney exchange where exchanges can only be made between two pairs. Theysuggest implementing the priority-based maximal matching algorithm from the combinatorial op-timization literature (Korte and Vygen, 2011). The first stages of both EAM and FAM are adap-tations of the priority-based maximal matching algorithm. Some other studies on organ exchangesinclude S¨onmez et al. (2018), Andersson and Kratz (2018), Chun et al. (2018), Ergin et al. (2017),Nicol´o and Rodr´ıguez-Alvarez (2017), and Ergin et al. (2018).Refugee reassignment is another real-world application in which maximality might be a primarydesign objective. Andersson and Ehlers (2018) study the problem of finding housing for refugees oncethey have been granted asylum. The authors propose an easy-to-implement mechanism that findsan efficient stable maximum matching. They show that such a matching guarantees that housing isefficiently provided to a maximum number of refugees and that no unmatched refugee-landlord pairprefers each other.Our “fairness for unassigned students” is a weakening of the usual stability of Gale and Shapley(1962), therefore, the current study is also related to the surging literature on the weakening ofstability in different ways. Among others, Dur et al. (2018), Afacan et al. (2017), Morrill and Ehlers(2018), and Troyan et al. (2018) are recent papers from that literature.3.
Model A school choice problem consists of a finite set of students I = { i , ..., i n } , a finite set of schools S = { s , ..., s m } , a strict priority structure for schools ≻ = ( ≻ s ) s ∈ S where ≻ s is a linear order over I ,a capacity vector q = ( q s , ..., q s m ), and a profile of strict preference of students P = ( P i ) i ∈ I , where P i is student i ’s preference relation over S ∪ {∅} and ∅ denotes the option of being unassigned. We SSIGNMENT MAXIMIZATION 3 denote the set of all possible preferences for a student by P . Let R i denote the at-least-as-good-as preference relation associated with P i , that is: sR i s ′ ⇔ sP i s ′ or s = s ′ . A school s is acceptable to i if sP i ∅ , and unacceptable otherwise. Let Ac ( P i ) = { c ∈ S : cP i ∅} .In the rest of the paper, we consider the tuple ( I, S, ≻ , q ) as the commonly known primitive of theproblem and refer to it as the market . We suppress all those from the problem notation and simplywrite P to denote the problem. A matching is a function µ : I → S ∪ {∅} such that for any s ∈ S , | µ − ( s ) | ≤ q s . A student i is assigned under µ if µ ( i ) = ∅ . For any k ∈ I ∪ S , we denote by µ k theassignment of k . Let | µ | be the total number of students assigned under µ .A matching µ is individually rational if, for any student i ∈ I , µ i R i ∅ . A matching µ is non-wasteful if for any school s such that sP i µ i for some student i ∈ I , | µ s | = q s . A matching µ is fair if there is no student-school pair ( i, s ) such that sP i µ i , and for some student j ∈ µ s , i ≻ s j . Amatching µ is stable if it is individually rational, non-wasteful, and fair.In the rest of the paper, we will consider only individually rational matchings. Therefore, wheneverwe refer to a matching, unless explicitly stated, we refer to an individually rational matching. Let M be the set of matchings.A matching µ dominates another matching µ ′ if, for any student i ∈ S , µ i R i µ ′ i , and for somestudent j , µ j P j µ ′ j . A matching µ is efficient if it is not dominated by any other matching. Wesay that a matching µ size-wise dominates another matching µ ′ if | µ | > | µ ′ | . A matching µ is maximal if it is not size-wise dominated. A mechanism ψ is a function from P | I | to M . A Mechanism ψ is strategy-proof if there existno problem P , and student i with a false preference P ′ i such that ψ i ( P ′ i , P − i ) P i ψ i ( P ).A mechanism ψ size-wise dominates another mechanism φ if, for any problem P , φ ( P ) doesnot size-wise dominate ψ ( P ), while, for some problem P ′ , ψ ( P ′ ) size-wise dominates φ ( P ′ ). Amechanism ψ is maximal if it is not size-wise dominated by any other mechanism.We start our analysis by first observing that none among four well-known mechanisms commonlyused and considered for the kind of allocation problems that we are considering — deferred-acceptance( DA ), top trading cycles ( T T C ), Boston ( BM ), and serial dictatorship ( SD ) — is maximal. Proposition 1.
None of DA , T T C , BM , and SD is maximal.Proof. Let I = { i , i } and S = { a, b } , each with unit quota. Let P i : a, b, ∅ and P i : a, ∅ . Thepriorities are such that agent i has the top priority at object a . Then, the DA , T T C , and BM outcomes are the same. If we write µ for their outcome, then µ i = a and µ i = ∅ . Likewise, for SD ,let us consider the ordering where agent i comes first. Then, the SD outcome is the same as µ .This shows that none of these mechanisms is maximal because the matching µ ′ where µ ′ i = b and µ ′ i = a is individually rational and matches more agents than µ . (cid:3) Given the lack of maximality of the well-known mechanisms, in the rest of the paper, we introducetwo maximal mechanisms and study their properties.3.1.
A Class of Efficient Maximal Mechanisms.
Given a problem P and an enumeration of thestudents in I ( i , ..i n ), Step 0.
Let ξ = M . Step 1.Sub-step 1.1.
Define the set ξ ⊆ ξ as follows: Notice that the notions of size domination and maximality we use is in th set of agents (or nodes) involved in amatching. In most of the literature in graph theory, the cardinality of a matching is measured in the set of edges of thegraph that are part of the matching. While when considering the set of edges there is a difference between maximaland maximum cardinality matchings, in our setup these are equivalent: maximal matchings are always maximum. For the description of these mechanisms, the reader could refer to Abdulkadiroglu and S¨onmez (2003).
SSIGNMENT MAXIMIZATION 4 ξ = (cid:26) { µ ∈ ξ : µ i = ∅} If ∃ µ ∈ ξ such that µ i = ∅ ξ otherwiseIn general, for every k ≤ n , Sub-step 1.k.
Define the set ξ k ⊆ ξ k − as follows: ξ k = (cid:26) { µ ∈ ξ k − : µ i k = ∅} If ∃ µ ∈ ξ k − such that µ i k = ∅ ξ k − otherwiseStep 1 ends with the selection of a matching µ ∈ ξ n . Step 2.
In general:
Sub-step 2.k.
Let ˜ µ be the matching obtained in the previous step of the procedure. If ˜ µ does notadmit an improving chain or cycle then the algorithm ends with the final outcome of ˜ µ . Otherwise,pick such a chain or cycle, and obtain a new matching by assigning each student in the chosen chain(cycle) to the school he prefers in the chain (cycle), and move to the next sub-step. Theorem 1.
Every
EAM mechanism is maximal and efficient.
Notice, however, that while EAM mechanisms are maximal, they are not fair.3.2.
A Class of Maximal and Fair for Unassigned Students Mechanisms.
We say that amatching µ is fair for unassigned students if there is no student-school pair ( i, s ) where µ i = ∅ and i ≻ s j for some j ∈ µ s . A mechanism ψ is fair for unassigned students if, for any problem P , ψ ( P ) is fair for unassigned students.Below is a description of how each mechanism in this class works. Given a problem P , Step 1.
Pick an
EAM mechanism ψ , and let ψ ( P ) = µ . Step 2.
In general,
Sub-step 2.k.
Let ˜ µ be the matching obtained in the previous step. If ˜ µ is fair for unassignedstudents, the algorithm terminates with the outcome ˜ µ . Otherwise, pick a student-school pair ( i, s )such that sP i ∅ , ˜ µ i = ∅ , and i ≻ s j for some j ∈ ˜ µ s . Place student i at school s , and let the lowestpriority student in ˜ µ s be unassigned, while keeping everyone else’s assignment the same. Note thatas in each sub-step the number of assigned students is preserved, ˜ µ is maximal. Hence, we have | ˜ µ s | = q s . Let ˆ µ be the obtained matching, and move to the next sub-step.As, in every sub-step, a higher priority student is placed at a school while a lower priority one isdisplaced from the school, and both the students and schools are finite, the algorithm terminates infinitely many rounds. The above procedure defines a class of mechanisms, each of which is associatedwith different selections of the first stage EAM mechanism as well as the student-school pairs in thecourse of Step 2. We refer them as “Fair Assignment Maximizing” (
F AM ) mechanisms.
Theorem 2.
Every
F AM mechanism is fair for unassigned students and maximal.Proof.
Let ψ be a F AM mechanism, and µ be the outcome of its first step. As µ is the outcome ofan EAM mechanism, and in Step 2 of ψ , no student is assigned to one of his unacceptable choices, ψ is individually rational. Because µ is maximal and the number of assigned students is preservedas | µ | in the course of Step 2, ψ is maximal. Moreover, as ψ does not stop until no student-schoolpair violates fairness for unassigned students, ψ is fair for unassigned students as well. (cid:3) Incentives and Equilibrium Analysis
In this section we show that the mechanisms in the classes
EAM and
F AM have surprisinglyregular properties in terms of equilibrium outcomes. Consider the preference reporting game induced
SSIGNMENT MAXIMIZATION 5 by a mechanism ψ . At problem P , a preference submission P ′ = ( P ′ i ) i ∈ I is a (Nash) equilibrium of ψ if for every student i , ψ i ( P ′ ) R i ψ i ( P ′′ i , P ′− i ) for any P ′′ i ∈ P . Let Ω be the set of mechanismsthat admit an equilibrium in any problem P ∈ P | I | . In the rest of this section, we consider only themechanisms in Ω. Proposition 2.
Every
EAM and
F AM mechanism is in Ω . Moreover, for any problem, an EAM mechanism has a unique equilibrium outcome that is equivalent to the outcome of the serial dictator-ship where the student ordering is the same as that used in that
EAM mechanism.
Proposition 2 shows, therefore, that equilibrium outcomes of
EAM are not only Pareto efficient,but will match as many students as a commonly used strategy-proof mechanism.Our next question is how mechanisms compare, in terms of the number of assignments, in equilib-rium. For that, we define the concept of size-wise domination in equilibrium.Definition 1.
For a given market (
I, S, ≻ , q ), a mechanism ψ size-wise dominates another mech-anism φ in equilibrium if, for any problem P and for every equilibria P ′ , P ′′ under ψ and φ ,respectively | ψ ( P ′ ) | ≥ | φ ( P ′′ ) | , and there exists a problem P ∗ such that for every equilibria ˆ P , ˜ P under ψ and φ , respectively | ψ ( ˆ P ) | > | φ ( ˜ P ) | . Theorem 3.
In any market ( I, S, ≻ , q ) , no EAM mechanism is size-wise dominated by an individ-ually rational mechanism in equilibrium.
While we do not have a similar result to above for the
F AM mechanisms, we are able to comparethe number of assigned students under the
F AM in equilibrium and the weakly dominant strategyequilibrium of the DA , which is truth-telling. Theorem 4.
Regarding the
F AM mechanisms:(i) For any problem P and any stable matching for P µ ∗ , for every equilibrium P ′ of a F AM mechanism ψ , | ψ ( P ′ ) | ≥ | µ ∗ | .(ii) There exist a F AM mechanism ψ , problem P , and an equilibrium profile P ′ of ψ at P suchthat | ψ ( P ′ ) | > | µ ∗∗ | , where µ ∗∗ is any stable matching for P . One may interpret the results in this section as an indication that there isn’t much gain in usingmaximal mechanisms such as EAM and FAM, since when agents respond to their incentives, outcomesare similar to those produced by other non-maximal mechanisms. Below we show, however, that thereare improvements in terms of the cardinality of the matching, as long as some fraction of the studentsare sincere.
Proposition 3.
For any maximal mechanism ψ , problem P , and student i with false preferences P ′ i such that ψ i ( P ′ i , P − i ) P i ψ i ( P ) , we have | ψ ( P ) | ≥ | ψ ( P ′ i , P − i ) | . Moreover, there exist a problem ˜ P andstudent i with false preferences ¯ P i such that ψ i ( ¯ P i , ˜ P − i ) ˜ P i ψ i ( ˜ P ) and | ψ ( ˜ P ) | > | ψ ( ¯ P i , ˜ P i ) | . In a preference-reporting game induced by a maximal mechanism where the only active playersare strategic students in the sense that the rest is always sincere, Proposition 3 leads to the followingcorollary.
Corollary 1.
Under any maximal mechanism, as the set of sincere students increases, in any prob-lem, the number of students matched in equilibrium either stays the same or increases.
SSIGNMENT MAXIMIZATION 6
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SSIGNMENT MAXIMIZATION 8
Appendix
Proofs.
Theorem 1.
We will use the following Lemma:
Lemma.
A maximal matching µ is efficient if and only if it does not admit an improving chain orcycle.Proof. “Only If ” Part. Let µ be an efficient matching. If it admits an improving chain { i , ..i n , c , .., c n +1 } ,then we can define a new matching by assigning each agent i k to c k +1 while keeping the assignmentsof the others the same. By the improving chain definition, that new matching dominates µ , con-tradicting our starting supposition that µ is efficient. The same argument shows for the case ofimproving cycle. “If ” Part. Let µ be a maximal matching that does not admit improving chains or cycles. Assumefor a contradiction that there exists a matching µ ′ that dominates µ .Let W = { i ∈ I : µ ′ i P i µ i } . By the supposition, W = ∅ . Note that for any student i with µ i = ∅ ,we have µ ′ i = ∅ . This, along with the maximality of µ , implies that | µ ′ | = | µ | . Hence, for any student i with µ i = ∅ , µ ′ i = ∅ .Enumerate the students in W = { i , .., i n } and write µ ′ i k = c k for any k = 1 , .., n . If | µ c k | < q c k forsome k , then the pair { i k , c k } constitutes an improving chain, a contradiction.Suppose that | µ c k | = q c k for any k = 1 , .., n . As c does not have excess capacity at µ , and µ ′ i = c ,we have another student in W , say i , such that µ i = c . Then, consider student i , and as c doesnot have excess capacity at µ and µ ′ i = c , we have another student in W , say i , such that µ i = c .If we continue to apply the same arguments to the other students in W , as W is finite, we wouldeventually obtain an improving cycle, a contradiction. (cid:3) Let now ψ be an EAM mechanism, and µ and µ ′ be its first stage and final outcome, respectively.As students are not assigned to one of their unacceptable schools in Step 1 of ψ , µ is individuallyrational.Assume for a contradiction that µ is not maximal and there exists µ ′′ = µ such that | µ ′′ | > | µ | .Let { i , .., i n } be the agent-enumeration that is used under ψ .As | µ ′′ | > | µ | , there exists some agent i k ∈ I such that µ ′′ i k = ∅ and µ i k = ∅ . Let i k ′ be the firstagent according to the above enumeration such that µ ′′ i k ′ = ∅ and µ i k ′ = ∅ . This means that for each k < k ′ , either µ i k = ∅ or µ i k = ∅ and µ ′′ i k = ∅ . Let B ( µ, k ′ ) = { i ∈ N : µ i k = ∅ for any k < k ′ } . Thatis, it is set of agents who come before agent i k ′ in the above enumeration and are assigned undermatching µ .Now consider agent i k ′ . By the definition of ψ , µ i k ′ = ∅ because it is not possible to match agent i k ′ to some of his acceptable objects while keeping all the agents in B ( µ, k ′ ) assigned to one of theiracceptable objects. This means that in order for agent i k ′ to receive one of his acceptable objects,one of the assigned agents under µ from B ( µ, k ′ ) has to be unassigned. This arguments holds foreach other agent who is assigned under µ ′′ , but not under µ . This implies that µ is maximal.In Step 2 of ψ , new matchings are obtained by implementing improving chains and cycles (if any).By their definitions no student receives a worse school than his assignment µ . This, along with theindividual rationality of µ , implies that µ ′ is maximal. The efficiency of µ ′ directly comes from theLemma above. Proposition 2.
Let ψ be an EAM mechanism. The first student in Step 0 of the
EAM obtainshis top choice by reporting it as the only acceptable choice. By the same reasoning, the secondstudent can obtain his top choice among the remaining schools with seats after considering the firststudent’s assignment by reporting that school as his only acceptable choice. Once we repeat thesame arguments for every other student, we not only find an equilibrium of ψ , but also conclude that SSIGNMENT MAXIMIZATION 9 it is the unique equilibrium outcome, which coincides with the outcome of serial dictatorship withthe ordering being the same as that in Step 0 of ψ .Let φ be a F AM mechanism. Let µ be a stable matching at P . Consider the preferences submission P ′ under which for any student i , the only acceptable school is µ i . Any unassigned student at µ reports no school acceptable at P ′ . It is easy to verify that φ ( P ′ ) = µ .Next, we claim that P ′ is an equilibrium submission under φ . Suppose for a contradiction thatthere exist a student i and P ′′ i such that φ i ( P ′′ i , P ′− i ) P i φ i ( P ′ ). For ease of writing, let φ i ( P ′′ i , P ′− i ) = s and φ i ( P ′ ) = s ′ . As µ is stable, | µ s | = q s . This, along with the definition of P ′ and φ i ( P ′′ i , P ′− i ) = s ,implies that there exists a student j = i such that µ j = s and φ j ( P ′′ i , P ′− i ) = ∅ . Moreover, from thestability of µ , we also have j ≻ s i . These altogether contradict the fairness for unassigned studentsof φ , showing that P ′ is equilibrium of φ . Theorem 3.
We will use the following.
Lemma.
Let ψ be an EAM and φ be an individually rational mechanism. In any market ( I, S, ≻ , q ) and problem P , if | ψ ( P ′ ) | < | φ ( P ′′ ) | where P ′ and P ′′ are equilibria under ψ and φ , respectively,then there exists a student i such that ψ i ( P ′ ) P i φ i ( P ′′ ) P i ∅ .Proof. In a market (
I, S, ≻ , q ) and problem P , let | ψ ( P ′ ) | < | φ ( P ′′ ) | where P ′ and P ′′ are equilibriaunder ψ and φ , respectively. This implies that for some school s , | ψ s ( P ′ ) | < | φ s ( P ′′ ) | ≤ q s . Hence,let i ∈ φ s ( P ′′ ) \ ψ s ( P ′ ). By the individual rationality of φ and P ′′ being equilibrium under φ ,we have sP i ∅ , where φ i ( P ′′ ) = s . As the unique equilibrium outcome of ψ coincides with the(truthtelling) outcome of a SD mechanism (Proposition 5), we have ψ ( P ′ ) = SD ( P ). Hence, school s has an excess capacity under SD ( P ). Moreover, from above, ψ i ( P ′ ) = SD i ( P ) = s . Hence,by the non-wastefulness of SD , i must be matched to a school strictly better than s and therefore ψ i ( P ′ ) = SD i ( P ) P i φ i ( P ′′ ) P i ∅ , which finishes the proof. (cid:3) Let now (
I, S, ≻ , q ) be a market and ψ be an EAM mechanism. Assume for a contradictionthat an individually rational mechanism φ size-wise dominates ψ in equilibrium. This in particularimplies that for some problem P , | ψ ( P ′ ) | < | φ ( P ′′ ) | for every equilibria P ′ and P ′′ under ψ and φ ,respectively. In what follows, we will fix one such pair P ′ , P ′′ . We prove the result in two steps. Step 1.
By the Lemma above, there exists a student i such that ψ i ( P ′ ) P i φ i ( P ′′ ) P i ∅ . Let ¯ P i bethe preference relation that keeps the relative rankings of the schools the same as under P i , whilereporting any school that is worse than ψ i ( P ′ ) as unacceptable. In other words, ¯ P i truncates P i below ψ i ( P ′ ). Let us write ¯ P = ( ¯ P i , P − i ). Recall that the unique equilibrium outcome of ψ always coincideswith the truthtelling outcome of a SD mechanism (Proposition 5). Moreover, by the construction of¯ P , SD ( P ) = SD ( ¯ P ). This in turn implies that ψ ( P ′ ) = ψ (cid:0) ¯ P ′ (cid:1) for every equilibrium ¯ P ′ under ψ inproblem ¯ P .We next consider problem ¯ P . If there exists no student j such that ψ j (cid:0) ¯ P ′ (cid:1) ¯ P j φ j (cid:0) ¯ P ′′ (cid:1) ¯ P j ∅ forsome equilibria ¯ P ′ and ¯ P ′′ under ψ and φ , respectively, then we move to Step 2. Otherwise, we picksuch student j . Note that because of the definition of ¯ P i states that any outcome below ψ i (cid:0) ¯ P ′ (cid:1) is unacceptable for i and φ is individually rational, ψ j (cid:0) ¯ P ′ (cid:1) ¯ P j φ j (cid:0) ¯ P ′′ (cid:1) ¯ P j ∅ cannot hold for j = i ,therefore j = i . Then, as the same as above, let ¯ P j be the preference list that truncates P j below ψ j (cid:0) ¯ P ′ (cid:1) . Let us write ˜ P = ( ¯ P i , ¯ P j , P −{ i,j } ). By the same reason as above, ψ ( P ′ ) = ψ (cid:16) ˜ P ′ (cid:17) for anyequilibrium ˜ P ′ under ψ in problem ˜ P .We next consider problem ˜ P . If there exists no student k such that ψ k (cid:16) ˜ P ′ (cid:17) ˜ P k φ k (cid:16) ˜ P ′′ (cid:17) ˜ P k ∅ forsome equilibria ˜ P ′ and ˜ P ′′ under ψ and φ , respectively, then we move to Step 2. Otherwise, we picksuch a student k . By the same reason as above, student k is different than both i and j . Then, wefollow the same arguments above and obtain a new preference profile. In each iteration, we have toconsider a different student. But then, since there are finitely many students, this case cannot holdforever. Hence, we eventually obtain a problem, say ˆ P , in which there exists no student h such that SSIGNMENT MAXIMIZATION 10 ψ h ( ˆ P ′ ) ˆ P h φ h (cid:16) ˆ P ′′ (cid:17) ˆ P h ∅ for some equilibria ˆ P ′ and ˆ P ′′ under ψ and φ , respectively, and move to Step2. We also have ψ ( P ′ ) = ψ ( ˆ P ′ ) for any equilibrium ˆ P ′ under ψ in problem ˆ P . Step . By the Lemma above, in problem ˆ P , we have (cid:12)(cid:12)(cid:12) ψ (cid:16) ˆ P ′ (cid:17)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) φ (cid:16) ˆ P ′′ (cid:17)(cid:12)(cid:12)(cid:12) for any equilibriaˆ P ′ and ˆ P ′′ under ψ and φ , respectively. If it holds strictly for some equilibria, then we reach acontradiction. Suppose (cid:12)(cid:12)(cid:12) ψ ( ˆ P ′ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) φ (cid:16) ˆ P ′′ (cid:17)(cid:12)(cid:12)(cid:12) for any equilibria ˆ P ′ and ˆ P ′′ .We now claim that ˆ P ′′ is an equilibrium under φ in problem P . Suppose it is not, and letstudent k have a profitable deviation, say ¨ P k , from ˆ P ′′ k . This means that φ k (cid:16) ¨ P k , ˆ P ′′− k (cid:17) P k φ k (cid:16) ˆ P ′′ (cid:17) .But then, by construction above, ˆ P k preserves the relative rankings under P k . This implies that φ k (cid:16) ¨ P k , ˆ P ′′− k (cid:17) ˆ P k φ k (cid:16) ˆ P ′′ (cid:17) , contradicting ˆ P ′′ being an equilibrium under φ in problem ˆ P .Recall that ψ ( P ′ ) = ψ ( ˆ P ′ ). Hence, this, along with (cid:12)(cid:12)(cid:12) ψ ( ˆ P ′ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) φ (cid:16) ˆ P ′′ (cid:17)(cid:12)(cid:12)(cid:12) and our above finding,implies that in problem P , | ψ ( P ′ ) | = (cid:12)(cid:12)(cid:12) φ (cid:16) ˆ P ′′ (cid:17)(cid:12)(cid:12)(cid:12) where P ′ and ˆ P ′′ are equilibria under ψ and φ ,respectively. Therefore, we constructed an equilibrium pair for problem P where ψ matches as manystudents as φ , contradicting our assumption that this does not hold in problem P . Theorem 4. ( i ). First, by the rural hospital theorem (Roth, 1984), the number of assignments in anystable matching is the same as that of DA. Let ψ be a F AM mechanism. Assume for a contradictionthat there exist a problem P and an equilibrium profile P ′ under ψ such that | ψ ( P ′ ) | < | DA ( P ) | .For ease of writing, let DA ( P ) = µ and ψ ( P ′ ) = µ ′ .We now claim that for some student i , µ i = s for some school s whereas µ ′ i = ∅ and, moreover, | µ ′ s | < q s . To prove this claim, let us define W = { i ∈ I : µ i = s and µ ′ i = ∅} . By our suppositionthat | DA ( P ) | > | ψ ( P ′ ) | , we have W = ∅ . Suppose that for each i ∈ W with µ i = s , | µ ′ s | = q s . Butthen this implies that | µ ′ | ≥ | µ | , contradicting our initial supposition, which finishes the proof of theclaim.Let i ∈ I such that µ i = s , µ ′ i = ∅ , and | µ ′ s | < q s . Now, consider the following preferences P ′′ : P ′′ k = (cid:26) P ′ k If k = is, ∅ If k = i First, observe that there exists a (individually rational) matching at P ′′ that assigns | µ ′ | + 1 manystudents (to see this, keep the assignment of everyone except student i the same as at µ ′ , and placestudent i at school s ). Therefore, due to the maximality of ψ , we have | ψ ( P ′′ ) | ≥ | µ ′ | + 1. If student i is assigned to school s at ψ ( P ′′ ) then this contradicts P ′ being equilibrium under ψ . Hence, ψ i ( P ′′ ) = ∅ . But then, by the definition of P ′′ , ψ ( P ′′ ) is individually rational at P ′ . This, alongwith the maximality of ψ , implies that | ψ ( P ′ ) | ≥ | ψ ( P ′′ ) | , contradicting our previous finding that | ψ ( P ′′ ) | ≥ | ψ ( P ′ ) | + 1, which finishes the proof of the first part.( ii ). Let us consider I = { i, j, k, h } and S = { a, b, c } , each with unit capacity. The preferencesand the priorities are given below. P i : a, b, ∅ ; P j : c, a, ∅ ; P k : c, a, ∅ ; P h : c, ∅ . ≻ a : k, i, j, h ; ≻ b : k, h, j, i ; ≻ c : k, h, i, j .Let ψ be a F AM mechanism with the student ordering k, j, i, h . Mechanism ψ is such that itproduces matching µ at P where µ i = b , µ j = a , µ k = c , and µ h = ∅ . For any P ′ i ∈ P with bP ′ i ∅ ,let ψ ( P ′ i , P − i ) = µ ′ where µ ′ i = b , µ ′ j = ∅ , µ ′ k = a , and µ ′ h = c . Moreover, for any P ′ i ∈ P with ∅ P ′ i b , ψ ( P ′ i , P − i ) = µ ′′ where µ ′′ i = ∅ , µ ′′ j = ∅ , µ ′′ k = a , and µ ′′ h = c . And, for any P ′ h ∈ P , let ψ ( P − h , P ′ h ) = µ .Note that student j can never get school c under ψ by misreporting because otherwise student h would be unassigned, and he has higher priority at school c . It is immediate to see that the abovematchings can be obtained in the course of F AM through particular selection. All of these show
SSIGNMENT MAXIMIZATION 11 that under ψ , truth-telling is an equilibrium at P , and | ψ ( P ) | = 3. On the other hand, DA ( P ) issuch that DA i ( P ) = a , DA k ( P ) = c , and DA h ( P ) = DA j ( P ) = ∅ . Hence, | ψ ( P ) | > | DA ( P ) | ,finishing the proof of the second part. Proposition 3.
Let P ′ = ( P ′ i , P − i ), ψ ( P ) = µ , and ψ ( P ′ i , P − i ) = µ ′ . Assume that | µ ′ | > | µ | . By oursupposition, µ ′ i P i µ i . This, along with the fact that P j = P ′ j for each j = i , µ ′ is individually rationalin problem P . But then, | µ ′ | > | µ | contradicts the fact that µ is maximal in problem P .Consider a problem where { i, j } ⊆ N , { a, b } ⊆ S , each with unit capacity. Let P i : a, ∅ , P j : a, ∅ ,and each other student (if any) finds every school unacceptable. Without loss of generality, assumethat the outcome of ψ in that problem, say µ , is such that µ i = a , and each other student isunassigned.Consider a problem where P ′ i : a, b, ∅ , while each other student’s preferences are the same asabove. Under the true preferences, ψ produces µ ′ where µ ′ i = b , µ ′ j = a , and each other student isunassigned. However, student i can misreport his preferences by submitting P i above as, under thisfalse profile, ψ produces matching µ above. Finally, note that ||
Let P ′ = ( P ′ i , P − i ), ψ ( P ) = µ , and ψ ( P ′ i , P − i ) = µ ′ . Assume that | µ ′ | > | µ | . By oursupposition, µ ′ i P i µ i . This, along with the fact that P j = P ′ j for each j = i , µ ′ is individually rationalin problem P . But then, | µ ′ | > | µ | contradicts the fact that µ is maximal in problem P .Consider a problem where { i, j } ⊆ N , { a, b } ⊆ S , each with unit capacity. Let P i : a, ∅ , P j : a, ∅ ,and each other student (if any) finds every school unacceptable. Without loss of generality, assumethat the outcome of ψ in that problem, say µ , is such that µ i = a , and each other student isunassigned.Consider a problem where P ′ i : a, b, ∅ , while each other student’s preferences are the same asabove. Under the true preferences, ψ produces µ ′ where µ ′ i = b , µ ′ j = a , and each other student isunassigned. However, student i can misreport his preferences by submitting P i above as, under thisfalse profile, ψ produces matching µ above. Finally, note that || µ ′ ||
Let P ′ = ( P ′ i , P − i ), ψ ( P ) = µ , and ψ ( P ′ i , P − i ) = µ ′ . Assume that | µ ′ | > | µ | . By oursupposition, µ ′ i P i µ i . This, along with the fact that P j = P ′ j for each j = i , µ ′ is individually rationalin problem P . But then, | µ ′ | > | µ | contradicts the fact that µ is maximal in problem P .Consider a problem where { i, j } ⊆ N , { a, b } ⊆ S , each with unit capacity. Let P i : a, ∅ , P j : a, ∅ ,and each other student (if any) finds every school unacceptable. Without loss of generality, assumethat the outcome of ψ in that problem, say µ , is such that µ i = a , and each other student isunassigned.Consider a problem where P ′ i : a, b, ∅ , while each other student’s preferences are the same asabove. Under the true preferences, ψ produces µ ′ where µ ′ i = b , µ ′ j = a , and each other student isunassigned. However, student i can misreport his preferences by submitting P i above as, under thisfalse profile, ψ produces matching µ above. Finally, note that || µ ′ || > ||
Let P ′ = ( P ′ i , P − i ), ψ ( P ) = µ , and ψ ( P ′ i , P − i ) = µ ′ . Assume that | µ ′ | > | µ | . By oursupposition, µ ′ i P i µ i . This, along with the fact that P j = P ′ j for each j = i , µ ′ is individually rationalin problem P . But then, | µ ′ | > | µ | contradicts the fact that µ is maximal in problem P .Consider a problem where { i, j } ⊆ N , { a, b } ⊆ S , each with unit capacity. Let P i : a, ∅ , P j : a, ∅ ,and each other student (if any) finds every school unacceptable. Without loss of generality, assumethat the outcome of ψ in that problem, say µ , is such that µ i = a , and each other student isunassigned.Consider a problem where P ′ i : a, b, ∅ , while each other student’s preferences are the same asabove. Under the true preferences, ψ produces µ ′ where µ ′ i = b , µ ′ j = a , and each other student isunassigned. However, student i can misreport his preferences by submitting P i above as, under thisfalse profile, ψ produces matching µ above. Finally, note that || µ ′ || > || µ ||