Comparing School Choice and College Admission Mechanisms By Their Immunity to Strategic Admissions
aa r X i v : . [ ec on . T H ] J a n COMPARING SCHOOL CHOICE AND COLLEGE ADMISSIONMECHANISMS BY THEIR IMMUNITY TO STRATEGIC ADMISSIONS
SOMOUAOGA BONKOUNGOU AND ALEXANDER NESTEROV
Abstract.
Recently dozens of school districts and college admissions systems around theworld have reformed their admission rules. As a main motivation for these reforms thepolicymakers cited strategic flaws of the rules: students had strong incentives to game thesystem, which caused dramatic consequences for non-strategic students. However, almostnone of the new rules were strategy-proof. We explain this puzzle. We show that after thereforms the rules became more immune to strategic admissions: each student received asmaller set of schools that he can get in using a strategy, weakening incentives to manipu-late. Simultaneously, the admission to each school became strategy-proof to a larger set ofstudents, making the schools more available for non-strategic students. We also show thatthe existing explanation of the puzzle due to Pathak and S¨onmez (2013) is incomplete.
Keywords : matching market design, school choice, college admission, manipulability
JEL Classification : C78, D47, D78, D82 Introduction
In recent years, dozens of school districts around the world have reformed their school ad-missions systems. Examples include education policy reforms for K-9 Boston Public Schools(BPS) in 2005, Chicago Selective High Schools (SHS) in 2009 and 2010, Denver PublicSchools in 2012, Seattle Public Schools in 1999, Ghanaian Secondary Public Schools in 2007and Primary Public Schools in more than 50 cities and provinces in England and Wales in2005-2011. Like school admissions, many college admissions systems have also been reformed;well-known examples include college admissions in China and Taiwan.Sometimes the reforms were a pressing issue. The Chicago SHS, for example, called for areform in a midstream of their admissions process. What were the policymakers concernedabout, and what was it at stake for such a sudden midstream change? There are signs indi-cating that the public and the policymakers were concerned about the high vulnerability of
Higher School of Economics, St.Petersburg
E-mail addresses : [email protected], [email protected] . Date : January 16, 2019.We thank Parag Pathak and Tayfun S¨onmez for their comments and continuous communication. We alsothank Anna Bogomolnaia, Herve Moulin, Lars Ehlers, Sean Horan, Onur Kesten, Rustamdjan Hakimov,Szilvia Papai, Yan Chen, Fuhito Kojima, Alvin Roth, Fedor Sandomirskiy and Mikhail Panov for theirfeedback. Support from the Basic Research Program of the National Research University Higher School ofEconomics is gratefully acknowledged. The research is partially financed by the RFBR grant 20-01-00687.
MMUNITY TO STRATEGIC ADMISSIONS 2 the admissions mechanisms to strategic manipulations. For example, the former superinten-dent of the Boston BPS said that their mechanism “should be replaced with an alternative[...] that removes the incentives to game the system” (Pathak and S¨onmez, 2008).Indeed, this high vulnerability made strategy an essential decision for students and ledto serious mismatches between schools and students. Strategic rankings were playing anunbalanced role in admissions versus priorities/grades, and that was perceived undesir-able. For example, the education secretary in England remarked that their admissionssystem were forcing “many parents to play an admissions game with their children’s future ”(Pathak and S¨onmez, 2013). Prior to the reform in China, “a good score in the college en-trance exam is worth less than a good strategy in the ranking of colleges” (Chen and Kesten,2017; Nie, 2007).These issues also compromise the perceived fairness of the system as the consequences totruthful students can be disastrous. The Chicago SHS called for reform after they observedthat “High-scoring kids were being rejected simply because of the order in which they listedtheir college prep preferences” (Pathak and S¨onmez, 2013). As one parent in China reports: “My child has been among the best students in his school and school district. [...] Unfortu-nately, he was not accepted by his first choice. After his first choice rejected him, his secondand third choices were already full. My child had no choice but to repeat his senior year.” (Chen and Kesten, 2017; Nie, 2007). Reportedly, similar concerns resulted in protests inTaiwan (Dur et al., 2018).Did the reforms make the admissions mechanisms fully immune to manipulation? Theanswer is no. Except for Boston, each reform replaced one vulnerable mechanism withanother vulnerable mechanism. This is a puzzle, given what motivated these reforms. Butcould it be that the new mechanisms are more immune to manipulations than the old ones?Did the reforms weakened the incentives to manipulate and made the consequences thereofless harmful to others? To address this question we develop a criterion to rank mechanismsby their level of what we call immunity to strategic admissions.To explain this criterion, let us begin with the most immune mechanisms. A mechanismis strategy-proof when no student can ever gain from manipulating his preferences. That is,for such a mechanism the admission of each student i to each school s is strategy-proof. Letus generalize this definition to any, possibly not strategy-proof, mechanism. We say thatthe admission to school s is strategy-proof to student i via mechanism A if noneof i ’s profitable manipulations gives him an admission at school s . In other words, all themisreports that result in student i ’s admission to school s via mechanism A are not profitable:in each of those instances, he is weakly better-off reporting his preferences truthfully. Thisstudent has no reason to misreport his preferences to mechanism A when aiming at school MMUNITY TO STRATEGIC ADMISSIONS 3
Table 1.
Chicago selective high schools (SHS): rankings and cutoff grades.
Notes s . Student i may still profitably manipulate mechanism A and get an admission at otherschools — but not at s .We measure the level of immunity of a mechanism by how strategy-proof admission toeach school is. Formally, mechanism A is more immune (to strategic admissions) than B if for each student i the set of schools whose admission is strategy-proof to i via B is asubset of the set of schools whose admission is strategy-proof to i via A , while the converseis not true. Thus, with a more immune mechanism, each student faces more schools that hecannot be admitted to via a profitable strategy.We find that each reform made the mechanisms more immune. For each student we com-pare by inclusion the two sets — before and after the reforms — of schools whose admission isstrategy-proof. Each of the reforms enlarged this set. Simultaneously, following the reforms,the admission to each school became strategy-proof to a larger set of students.Roughly, the reforms made the mechanism more immune by using one or both of thefollowing features: they allowed students to submit longer lists of acceptable schools andmade admission to every school less sensitive to its rank in the list. Intuitively, a longer listallows students to be less strategic about selecting which schools to include in the list, whilelower sensitivity about ranking facilitates truthful ranking of the selected schools.We illustrate our concept and the result in the following example. Illustrative Example.
Let us consider the two reforms of the Chicago SHS in 2009 and2010. Each school uses a common priority based on students’ composite scores. The admis-sion to each of these schools is very competitive. To give you an idea, only 4 000 from morethan 10 000 participants were admitted in the 2018 admission session. In 2009, the ChicagoSHS replaced the Boston mechanism where students can rank only 4 schools ( β ) with a MMUNITY TO STRATEGIC ADMISSIONS 4 serial dictatorship with the same ranking constraint ( SD ). In 2010 the mechanism stayedthe same but the constrained was increased to 6 ( SD ).Among the 10 schools, 5 are elite, being the top 5 schools in the state of Illinois and amongtop 100 in the US (see Table 1). These schools are preferred by most, if not all, studentsover each other school. For simplicity, let us suppose that students have tier preferences:each student prefers each top 5 school over each non top 5 school, but may differ on rankingschools in each tier. Let each school have 400 seats.Under the mechanism β , each of the 400-highest priority students is guaranteed a seatin his most preferred school, while each other student may potentially get each school by aprofitable manipulation. The admission to every school is thus strategy-proof only for the400-highest priority students via β .However, under the mechanism SD , each of the 1600-highest priority students is guar-anteed one of his 4 most preferred schools (Lemma 3). The admission to every school isstrategy-proof for each of them, while each other student can potentially get every schoolby a profitable manipulation. Under SD , this is the 2400-highest priority students; whileonly the 2000-highest priority students can be admitted to the top 5 schools. Following thereforms, the immunity of the mechanism increased significantly as the share of students forwhom the admission to the top 5 schools is strategy-proof increased from 4% to 24% in 2009and further to 100% in 2010. End of the example.Our results rationalize all the reforms in the light of each of the cited concerns. The reformsdecreased each student’s incentives to manipulate as measured by the range of schools he orshe could get by manipulation. Intuitively, this range decreases due to the more competitiveschools as it was in the Chicago example, which makes the incentives to manipulate evenweaker. At the same time, the remaining manipulations harmed the truthful students lessas the admission to each school became more strategy-proof. Again, in the Chicago exampleall high-scoring kids were safe to submit their preferences truthfully.But the reforms could also be rationalized by that the instances without manipulations be-came more frequent. The state-of-the-art notion is manipulability due to Pathak and S¨onmez(2013): mechanism A is less manipulable than mechanism B if A is manipulable by at leastone student in a subset of preference profiles where B is manipulable.We found that this notion has limited applications for important reforms. First, contraryto what was claimed, it only partially explains a major reform in England and Wales thatwas followed by more than 50 local cities (see section 4 and Table 2). The serial dictatorship is a mechanism where students follow the common priority order and choosetheir most preferred schools among those that remain. The definitions of the mechanisms are given inthe next section. Constrained matching mechanisms were first studied in Haeringer and Klijn (2009) andCalsamiglia et al. (2010).
MMUNITY TO STRATEGIC ADMISSIONS 5
Second, under realistic assumption on students preferences, the notion is not satisfactoryfor rationalizing the 2009 reform of the Chicago SHS. Indeed, for each preference profile, itsuffices that one student has a profitable manipulation to declare the profile as manipulable.However, with tier preference structure in the Chicago SHS — strong competition for theelite schools — at least one such student always exists. By truthfully ranking the top 5schools the mechanisms that allow students to submit at most 5 schools will leave the seatsat the other schools unassigned. Therefore, many students will be unmatched and at leastone student will have a profitable manipulation. Before and after the 2009 reform, bothmechanisms are manipulable at any tier preference profile. More generally, we show thatconstrained SD is not manipulable if and only if the constraint is not binding (Proposition2). However, immunity to strategic admissions changed significantly after both the 2009and 2010 reform, and the fact that admission to the elite schools is now strategy-proof mayrationalize why SD has been used in Chicago ever since.Very recently Decerf and Van der Linden (2018) used another frequency notion stemmingfrom Arribillaga and Mass´o (2016): they compare mechanisms by inclusions of vulnerableindividual preference relations, i.e., the preference relations for which truth is not always adominant strategy. This notion explains reforms where the constrained list of the Bostonmechanism were replaced by a constrained list of Gale-Shapley mechanism and where thelist in Gale-Shapley were extended. The major difference with our notion is that we quan-titatively measure what students can get or cannot get by manipulation. We particularlymeasure how schools are protected from strategic admissions.The paper is organized as follows. Section 2 presents the model and the main definitions.Section 3 presents the main results for immunity to strategic admissions and section 4 com-pares them to the results for manipulability. Section 5 develops an equilibrium refinementof immunity to strategic admissions. Section 6 concludes.2. Model
The school choice model originated in Balinski and S¨onmez (1999) and Abdulkadiro˘glu and S¨onmez(2003); constrained school choice was first studied in Haeringer and Klijn (2009).There is a finite set I of students with a generic element i and a finite set S of schoolswith a generic element s . Each student i has a strict preference relation P i over S ∪ {∅} (where ∅ stands for being unmatched). Each school s has a strict priority order ≻ s overthe set I of students and a capacity q s (a natural number indicating the number of seatsavailable at school s ). For each student i , let R i denote the “at least as good as” relation A strict preference relation is a complete, transitive and asymmetric binary relation.
MMUNITY TO STRATEGIC ADMISSIONS 6 associated with P i . The list P = ( P i ) i ∈ I is a preference profile, ≻ = ( ≻ s ) s ∈ S is a priorityprofile and q = ( q s ) s ∈ S is a capacity vector. We often write a preference profile P = ( P i , P − i ) to emphasize the preference relation of student i . The tuple ( I, S, P, ≻ , q ) is a school choiceproblem. We assume that there are more students than schools and at least two schoolsto reflect real-life school choice context. We fix the set of students and the set of schoolsthroughout the paper. For short, we call the pair ( ≻ , q ) a school choice environment andthe triple ( P, ≻ , q ) a school choice problem . School s is acceptable to student i if s P i ∅ .Otherwise, it is unacceptable. We will often specify a preference relation and a priority orderas follows P i ≻ s s is ′ j ∅ k to indicate that student i prefers school s to school s ′ and finds no other school acceptable;and, when there are three students, that student i has the highest priority at school s ,student j next and student k last.A matching µ is a function µ : I → S ∪ {∅} such that no school is assigned more studentsthan its capacity. Let ( P, ≻ , q ) be a problem. A matching µ is individually rational under P if for each student i , µ ( i ) R i ∅ . Student i has justified envy over student j in thematching µ if he prefers the school assigned to student j to his assignment and has higherpriority than j at that school. A matching µ is non-wasteful if no student prefers a schoolthat has an empty seat. A matching is stable if • it is individually rational, • no student has a justified envy over another, and • it is non-wasteful.A mechanism ϕ is a function that maps school choice problems ( P, ≻ , q ) to matchings. Let ϕ i ( P, ≻ , q ) denote the outcome for student i . We present real-life mechanisms next.2.1. Mechanisms.
Most real-life mechanisms can be described using the Gale-Shapley de-ferred acceptance mechanism. That is, for each s, s ′ ∈ S ∪ {∅} , s R i s ′ if and only s P i s ′ or s = s ′ . More generally, we write a preference profile P = ( P I ′ , P − I ′ ) to emphasize the components of a subset I ′ ofstudents. That is, | I | > | S |≥ . We assume that schools are not strategic. In practice, the priorities are determined by law or by students’performances, and known to students before they submit their preferences. That is for each school s , | µ − ( s ) |≤ q s . That is, for some school s , µ ( j ) = s , s P i µ ( i ) and i ≻ s j . That is, there is no student i and a school s such that | µ − ( s ) | < q s and s P i µ ( i ) . MMUNITY TO STRATEGIC ADMISSIONS 7
Gale-Shapley.
Gale and Shapley (1962) showed that for each problem, there is a stablematching. In addition, there is a student-optimal stable matching, that each student findsat least as good as any other stable matching. For each problem ( P, ≻ , q ) , let GS ( P, ≻ , q ) denote the student-optimal stable matching. Serial Dictatorship.
In environments where schools have the same priority order, we abuselanguage and call the Gale-Shapley mechanism, serial dictatorship. Let SD ( P, ≻ , q ) denotethe matching assigned by this mechanism to the problem ( P, ≻ , q ) where schools have thesame priority under ≻ . First-Preference-First.
The set of schools are partitioned into equal preference schools and first-preference-first schools. For each problem ( P, ≻ , q ) , the mechanism assigns thematching GS ( P, ≻ ′ , q ) where the priority profile ≻ ′ is obtained as follows:1. for each equal preference school s , ≻ ′ s = ≻ s and2. for each first-preference-first school s , ≻ ′ s is an adjustment of ≻ s with respect to P : • students who rank school s at a given position have higher priority under ≻ ′ s thanstudents who rank it lower than this position, and • students who ranked school s at the same position are ordered according to ≻ s .Let F P F ( P, ≻ , q ) denote the matching assigned by this mechanism to ( P, ≻ , q ) . Briefly,the original priority of each equal preference school remains unchanged, while the originalpriority of each first-preference-first school is adjusted by favoring higher ranks. Boston.
The Boston mechanism is a first-preference-first mechanism where every school is afirst-preference-first school. Let β ( P, ≻ , q ) denote the matching assigned to ( P, ≻ , q ) . Constrained versions.
In practice, students are allowed to report a limited number of schools.This means that schools that are listed below a certain position are not considered. For eachstudent i , each preference relation P i and each natural number k ≤ | S | , let P ki denote thetruncation of P i after the k ’th acceptable choice (if any). That is, every school ranked belowthe k ’th position under P i is unacceptable under P ki ; otherwise, the ranking is as in P i . Let P k = ( P ki ) i ∈ I . The k -constrained version ϕ k of the mechanism ϕ is the mechanism thatassigns to each problem ( P, ≻ , q ) the matching ϕ ( P k , ≻ , q ) . Chinese parallel.
This mechanism is determined by a parameter e ≥ (a natural number).For each problem ( P, ≻ , q ) , the outcome is a sequential application of constrained GS . Inthe first round, students are matched according to GS e . The matching is final for matchedstudents, while unmatched students proceed to the next round. In the next round, eachschool reduces its capacity by the number of students assigned to it the last round and the Gale and Shapley (1962) described an algorithm for finding this matching. According to our definition, a mechanism has as a domain the set of all problems — including problemswhere schools have different priorities.
MMUNITY TO STRATEGIC ADMISSIONS 8 unmatched students are matched according to GS e , and so on. Let Ch ( e ) ( P, ≻ , q ) denotethe matching assigned by the mechanism to ( P, ≻ , q ) . Results
We first introduce our immunity notion. We would like to test how immune is a mechanismto strategic admissions, focusing on individual students. We build a definition from thefollowing question. When can we say that an admission to school s by student i is not dueto strategic manipulation? Obviously, when no profitable manipulation by i gives him anadmission at s , then none of this admissions is due to strategy. Definition 1.
Let ϕ be a mechanism and ( ≻ , q ) an environment. • The admission to school s is strategy-proof to student i via mechanism ϕ if there is no preference profile P and a profitable manipulation P ′ i that gives him s : s = ϕ i ( P ′ i , P − i , ≻ , q ) P i ϕ i ( P, ≻ , q ) . • Mechanism ϕ is strategy-proof to student i if the admission to every school isstrategy-proof to student i via ϕ . We qualify these admissions as non-strategic and the rest as strategic. A closer definitionin the literature is strategy-proofness saying that mechanism ϕ is strategy-proof if for eachenvironment ( ≻ , q ) and each student i , there is no preference profile P and P ′ i such that ϕ i ( P ′ i , P − i , ≻ , q ) P i ϕ i ( P, ≻ , q ) . If a mechanism ϕ is strategy-proof, then for each environment, the admission to every schoolis strategy-proof to every student via ϕ . The unconstrained GS , for example, is strategy-proof (Roth, 1982; Dubins and Freedman, 1981). Next, for each student, we find the set ofschools for which the admission is strategy-proof to him and rank mechanisms as follows. Definition 2.
Mechanism ϕ is more immune to strategic admissions than mechanism ψ if • for each environment ( ≻ , q ) , if the admission to school s is strategy-proof to student i via ψ , then the admission to school s is also strategy-proof to i via ϕ and, This definition of the Chinese parallel mechanisms is given only for the symmetric version where eachround has the same length e . Our results hold also for the asymmetric mechanisms with different lengths ofthe rounds. See Chen and Kesten (2017) for details. MMUNITY TO STRATEGIC ADMISSIONS 9 • there is an environment ( ≻ , q ) where the admission to some school s is strategy-proofto some student i via ϕ , but not via ψ . Observe that the immunity relation is transitive as it is based on set inclusion.In the following section, we apply this definition to the mechanisms before and after thereforms. Subsequently, we discuss its explanatory power compared to the state-of-the-artnotion.3.1.
Reforms and immunity to strategic admissions.
In England and Wales.
According to the field observation (see Table 2) more than 50 areasin England and Wales have replaced different constrained versions of
F P F with constrained GS . In the following theorem, we show that they have replaced mechanisms with moreimmune alternatives. Theorem 1.
Suppose that there are at least k schools where k > and at least one first-preference-first school. Then GS k is more immune to strategic admissions than F P F k . See the appendix for the proof. Kingston upon Thames has replaced a constrained
F P F with a constrained GS but with a longer list. This replacement also resulted in a moreimmune mechanism: Corollary 1.
Let k > ℓ and suppose that there are at least ℓ schools. Then GS k is moreimmune to strategic admissions than F P F ℓ . See the appendix for the proof.
In Chicago and Denver.
In 2009, the Chicago Selective High Schools moved from a con-strained Boston to a constrained serial dictatorship. A similar replacement has been ob-served in Denver and 4 other cities in England. As the Boston mechanism is a special caseof
F P F , and SD is a special case of GS , we also explain this replacement. Corollary 2.
Let k ≥ ℓ and suppose that there are at least k schools. Then GS k is moreimmune to strategic admissions than β ℓ .In Chicago and Ghana. In 2010, the Chicago Selective High Schools again replaced its con-strained Serial Dictatorship with a version with longer list. In 2007, the Ghanaian SecondarySchools undertook a similar change, from a constrained GS to a version with longer list andextending the list again in 2008. These type of changes have also been observed in Newcastleand Surrey in England. Theorem 2.
Let k > ℓ and suppose that there are at least k schools. Then GS k is moreimmune to strategic admissions than GS ℓ . MMUNITY TO STRATEGIC ADMISSIONS 10
See the appendix for the proof. In the following table, we list all reforms in school choicesystems. We also feature those that are comparable ´a la Pathak and S¨onmez (2013) andthose that are not (see section 4 for the results).
In college admission in China.
Starting from 2001, most of the Chinese provinces changedtheir mechanisms from the Boston mechanism to various other parallel mechanisms. Considertwo Chinese mechanisms: one with parameter e and the other with e > e ′ . For these twomechanisms we obtain the following comparison. Theorem 3.
Let e > e ′ and suppose that there is at least e schools. Then Ch ( e ) is moreimmune to strategic admissions than Ch ( e ′ ) . See the appendix for the proof.4.
Comparison with Manipulability
In this section, we introduce another notion — manipulability due to Pathak and S¨onmez(2013), and compare it with our notion of immunity to strategic admissions.
Definition 3. (Pathak and S¨onmez, 2013). Let ϕ and ψ be two mechanisms. Then mecha-nism ϕ is less manipulable than mechanism ψ if • in each environment ( ≻ , q ) , each preference profile P that is vulnerable under ϕ , i.e.,there exists some student i and some misreport P ′ i such that, ϕ i ( P ′ i , P − i , ≻ , q ) P i ϕ i ( P, ≻ , q ) , is also vulnerable under ψ and, • there is an environment where a preference profile is vulnerable under ψ but not under ϕ . Broadly, whenever a student has a profitable manipulation at a preference profile P under ϕ , then some — possibly other — student has a profitable manipulation under ψ , while thereverse is not true in some environment. It is important to note that a profitable manipulationof one student is enough to declare a preference profile as vulnerable under a mechanism.Comparing mechanisms with respect to certain property profile by profile is common in theliterature (a notable example is Kesten (2006)), but it has two important limitations whenapplied to our case.4.1. Limitation in England and Wales.
First, we prove that the notion of manipulabilitydoes not explain the reforms in England and Wales. Recall that the education officials havereplaced the constrained First-Preference-First with a constrained Gale-Shapley. Indeed, weprovide a counterexample to Proposition 3 in Pathak and S¨onmez (2013).
MMUNITY TO STRATEGIC ADMISSIONS 11
Table 2.
School Admissions Reforms (documented in Pathak and S¨onmez (2013))Allocation system Year From To Manipulable?(More or less?) Immune?(More or less?)Boston Public School (K, 6, 9) 2005 Boston GS Less MoreChicago Selective High Schools 2009 Boston SD Less More2010 SD SD Less MoreGhana—Secondary schools 2007 GS GS Less More2008 GS GS Less MoreDenver Public Schools 2012 Boston GS Less MoreSeattle Public Schools 1999 Boston GS Less More2009 GS Boston More LessEnglandBath and North East Somerset 2007 FPF GS Not comparable MoreBedford and Bedfordshire 2007 FPF GS Not comparable MoreBlackburn with Darwen 2007 FPF GS Not comparable MoreBlackpool 2007 FPF GS Not comparable MoreBolton 2007 FPF GS Not comparable MoreBradford 2007 FPF GS Not comparable MoreBrighton and Hove 2007 Boston GS Less MoreCalderdale 2006 FPF GS Not comparable MoreCornwall 2007 FPF GS Not comparable MoreCumbria 2007 FPF GS Not comparable MoreDarlington 2007 FPF GS Not comparable MoreDerby 2005 FPF GS Not comparable MoreDevon 2006 FPF GS Not comparable MoreDurham 2007 FPF GS Not comparable MoreEaling 2006 FPF GS Not comparable MoreEast Sussex 2007 Boston GS Less MoreGateshead 2007 FPF GS Not comparable MoreHalton 2007 FPF GS Not comparable MoreHampshire 2007 FPF GS Not comparable MoreHartlepool 2007 FPF GS Not comparable MoreIsle of Wright 2007 FPF GS Not comparable MoreKent 2007 Boston GS Less MoreKingston upon Thames 2007 FPF GS Not comparable MoreKnowsley 2007 FPF GS Not comparable MoreLancashire 2007 FPF GS Not comparable MoreLincolnshire 2007 FPF GS Not comparable MoreLuton 2007 FPF GS Not comparable MoreManchester 2007 FPF GS Not comparable MoreMerton 2006 FPF GS Not comparable MoreNewcastle 2005 Boston GS Less More2010 GS GS Less MoreNorth Lincolnshire 2007 FPF GS Not comparable MoreNorth Somerset 2007 FPF GS Not comparable MoreNorth Tyneside 2007 FPF GS Not comparable MoreOldham 2007 FPF GS Not comparable MorePeterborough 2007 FPF GS Not comparable MorePlymouth 2007 FPF GS Not comparable MorePoole 2007 FPF GS Not comparable MorePortsmouth 2007 FPF GS Not comparable MoreRichmond 2005 FPF GS Not comparable More (Continued)
MMUNITY TO STRATEGIC ADMISSIONS 12
Table 1.
School Admissions Reforms (
Continued )Allocation system Year From To Manipulable?(More or less?) Immune?(More or less?)Sefton primary 2007 Boston GS Less MoreSefton secondary 2007 FPF GS Not comparable MoreSlough 2006 FPF GS Not comparable MoreSomerset 2007 FPF GS Not comparable MoreSouth Gloucestershire 2007 FPF GS Not comparable MoreSouth Tyneside 2007 FPF GS Not comparable MoreSouthhampton 2007 FPF GS Not comparable MoreStockton 2007 FPF GS Not comparable MoreStoke-on-Trent 2007 FPF GS Not comparable MoreSuffolk 2007 FPF GS Not comparable MoreSunderland 2007 FPF GS Not comparable MoreSurrey 2007 FPF GS Not comparable More2010 GS GS Less MoreSutton 2006 FPF GS Not comparable MoreSwindon 2007 FPF GS Not comparable MoreTameside 2007 FPF GS Not comparable MoreTelford and Wrekin 2007 FPF GS Not comparable MoreTorbay 2007 FPF GS Not comparable MoreWarrington 2007 FPF GS Not comparable MoreWarwickshire 2007 FPF GS Not comparable MoreWilgan 2007 FPF GS Not comparable MoreWalesWrexham County Borough 2011 FPF GS Not comparable More
Claim 1 (Proposition 3 — Pathak and S¨onmez (2013)) . Suppose that there are at least k schools where k > . Then, GS k is less manipulable than F P F k . We provide a counterexample to this claim. We specify the relevant part of the prioritiessuch that the sign ... after student i indicates that the part below i is arbitrary and omitted. Example 1.
Counterexample.
Suppose that there are seven students and seven schools. Each school has one seat: q s = 1 for each school s . Let P and ≻ be as specified below. MMUNITY TO STRATEGIC ADMISSIONS 13 P P P P P P P ≻ s ≻ s ≻ s ≻ s ≻ s ≻ s ≻ s ≻ ′ s s s s s s s s s ∅ ∅ ∅ s s s ... ... ... ... s ∅ s ∅ ... ... ... s ∅ ... ∅ Suppose that school s is the only first-preference-first school. Then the matching is asfollows: F P F ( P, ≻ , q ) = GS ( P, ≻ , q ) = ∅ s s s s s s ! . Every student except student received his most preferred school. Therefore only student could potentially manipulate each of F P F and GS . Let P s denote student ’s preferencerelation where school s is the only acceptable school. Then GS ( P s , P − , ≻ , q ) = s s s s s s s ! . By reporting the preference relation P s to GS , student obtained an acceptable school s but is unmatched when he reports his true preference relation P . Therefore the profile P isvulnerable under GS . However, because school s is a first-preference-first school and thatstudent has ranked it higher than student , we have (where ≻ s is replaced by ≻ ′ s ) F P F ( P s , P − , ≻ , q ) = ∅ s s s s s s ! . By reporting P s to F P F student is unmatched, the same as when he reports his truepreferences. It can be verified that it is enough to check for manipulations by ranking schoolsfirst. In addition, student cannot misrepresent his preferences to obtain a seat at school s , s and s . Therefore the profile P is not vulnerable under F P F .The intuition is that when student claims school s he causes the rejection of student . Then student claims school s . Under GS student is rejected from school s and heapplies to school s , ending the process. However under F P F , school s is a first-preference-first school which student has ranked first. This time it is student who is rejected fromschool s . Then he claims school s and causes the rejection of student . Ultimately, student claims school s and takes it back from student . End of the example.Nevertheless, when each school is a first-preference-first school then the comparison —between the constrained Boston and the constrained Gale-Shapley — is valid. Proposition 1. (Pathak and S¨onmez, 2011) Suppose that there are at least k schools where k > . Then GS k is less manipulable than β k . MMUNITY TO STRATEGIC ADMISSIONS 14
Limitation in Chicago.
The other limitation of manipulability is that it is insensi-tive when applied to constrained versions of strategy-proof mechanisms: it only bites forthose profiles where the constraint becomes not binding. Indeed, if at a particular profilea constrained mechanism is not manipulable, then at this profile it must be very close toits unconstrained version. We formalize this intuition for the serial dictatorship mechanismused in Chicago.
Proposition 2.
A preference profile P is not vulnerable under SD k if, and only if SD k ( P ) = SD ( P ) .Proof. The if part is straightforward: if SD k ( P ) = SD ( P ) , i.e. if the constraint is notbinding, then at SD k ( P ) each student receives the best available school among remainingones and cannot profitably misreport his preferences.We prove the only if part by contraposition. Suppose that SD k ( P ) = SD ( P ) and considerthe highest priority student i for whom SD ki ( P ) = s = SD i ( P ) . Each student with higherpriority than i received under SD k ( P ) the same school as under SD ( P ) , therefore under SD k ( P ) student i had the same choice set of remaining schools as under SD ( P ) . The onlyway i missed school s under SD k ( P ) is if the constraint k was binding for him: each of histop k schools were already assigned, and school s was not listed. However, school s still hadavailable seats, and i could profitably manipulate SD k at P by listing school s as one of histop k schools. (cid:3) For k = 1 this result also applies to Boston (constrained and unconstrained) with acommon priority: at a given preference profile P , Boston is not manipulable if and only ifits outcome coincides with the outcome of the unconstrained serial dictatorship at P .For realistic profiles, however, the constraint is almost guaranteed to be binding at leastfor one student, and thus the constrained mechanism remains manipulable at this profile.This occurs, for instance, when the preferences of students are correlated, as often is thecase. Next we generalize the Chicago example presented in the introduction. We show thatwhen students have tier preferences, constrained versions of serial dictatorship are alwaysmanipulable. In general, the more tiers there are, the more binding the constraint will be. Example 2.
Serial dictatorship and tier preferences.
Consider a school choice problem with n students and m schools, each two schools s, s ′ ∈ S having the same capacity q s = q s ′ and a common priority ranking ≻ s = ≻ s ′ .Assume that students have tier preferences: the set of schools S is partitioned into t > sets S , S , ..., S t and each student i ∈ I prefers each school in S j from a higher tier j < t MMUNITY TO STRATEGIC ADMISSIONS 15 over each school in S j +1 from a lower tier. Assume also that each student finds each schoolacceptable and that there is shortage of seats, n > q × m .It is straightforward to show that whenever the number of schools in the upper tiers isat least as large as the constraint, | S | − | S t | ≥ k , the constrained serial dictatorship SD k isalways manipulable. Otherwise, if every student reports truthfully, then some students areunmatched while acceptable schools in the last tier S t are unassigned. By Proposition 2 this isnecessary and sufficient for manipulability of SD k . This is why for correlated preferences themanipulability criterion is not sensitive and the constrained serial dictatorship mechanismis as manipulable as the Boston mechanism.In contrast, our notion remains sensitive in this domain: as the constraint changes, theimmunity to strategic admissions changes as well. We formulate this as a proposition. Proposition 3.
Let k ≥ and ( ≻ , q ) an environment where schools have a common priority.Let the capacities of the schools be increasingly ordered q ≤ q ≤ . . . ≤ q | S | and ˆ q = q + . . . + q k . Then, the mechanism SD k is strategy-proof for the ˆ q -highest priority students. See the appendix for the proof. Proposition 3 is formulated for the entire domain ofpreferences, and it remains true in the domain of tier preferences. Therefore, in the example,the share of students for whom the admission to each school is strategy-proof is q × k/n .The schools in the upper tiers are more protected from strategic admissions. By switchingfrom β to SD in 2009, and to SD in 2010, the strategy-proof admissions to Chicago eliteschools, increased from to and eventually to , respectively.5. Strategic admissions in equilibrium
In this section, we develop a more refined concept of immunity to strategic admissions.Previously, we called admission of student i to school s strategic if there exists a profile anda profitable deviation for i that places i to s . But this deviation did not need to be optimal,same as the reports of other students did not need to be optimal, and thus not rationalizable.Now we require the strategies to be mutually optimal.Let us motivate this with an example. Example 3.
Equilibrium in Boston.
Suppose there are three students i , j and k and three schools s , s and s with one seateach. The preferences and the priorities are as follows. Coles et al. (2013) observed that the academic job market has this structure and referred to it as blockcorrelated preferences.
MMUNITY TO STRATEGIC ADMISSIONS 16 P i P j P k ≻ s ≻ s ≻ s s s s j j js s s k k ks s s i i i ∅ ∅ ∅ Let us consider the Boston mechanism. Its outcome for this problem is specified as follows. β ( P, ≻ , q ) = i j ks s s ! . Suppose instead that student i reports the preference relation P s i where he ranks school s first. If student j and k report truthfully as in P , we have β ( P s i , P − i , ≻ , q ) = i j ks s s ! . According to the notion developed earlier, the admission to school s is not strategy-proofto student i via the Boston mechanism. However, it is not a best response for student k toreport truthfully P k , when student i reports P s i . Student i has the lowest priority at everyschool. The lack of strategy-proof admissions for student i stems from the fact that otherstudents report their preferences truthfully without best-responding. End of example.This example demonstrates that sometimes the admission to some schools is not strategy-proof to some students only because others are not best responding. This type of admissionsmay disappear when we require best responses. To take these best responses into account weintroduce an equilibrium concept. If we fix an environment ( ≻ , q ) , any mechanism ϕ inducesa normal form game such that the students are the players, the strategies are the preferencesand the outcome function is ϕ ( ., ≻ , q ) . Then, a strategy profile P ′ is a Nash equilibrium of the game ( I, P, ϕ ( ., ≻ , q )) if for each student i , P ′ i is a best response to P ′− i . When thereis no ambiguity on the environment, we denote the game as ( P, ϕ ) . Definition 4.
Let ϕ be a mechanism and ( ≻ , q ) an environment. The admission to school s is strategy-proof to student i via mechanism ϕ in equilibrium if there is no pref-erence profile P and a preference relation P ′ i such that (1) ( P ′ i , P − i ) is a Nash equilibrium of the game ( P, ϕ ) and (2) s = ϕ i ( P ′ i , P − i , ≻ , q ) P i ϕ i ( P, ≻ , q ) Note that when the admission to school s is strategy-proof to student i via ϕ , there is nopreference profile and a deviation that satisfy the condition (2). Therefore the admissionto school s is strategy-proof to student i via ϕ in equilibrium. Clearly, Definition 4 is more That is, for each student i , there is no strategy P ′′ i such that ϕ i ( P ′′ i , P ′− i , ≻ , q ) P i ϕ i ( P ′ , ≻ , q ) . MMUNITY TO STRATEGIC ADMISSIONS 17 stringent than Definition 1. Let us now use this notion to define a ranking criterion analogousto the one defined in Definition 2.
Definition 5.
Mechanism ϕ is strongly more immune to strategic admissions thanmechanism ψ if • for each environment ( ≻ , q ) , if the admission to school s is strategy-proof to student i via ψ in equilibrium, then the admission to school s is also strategy-proof to him via ϕ in equilibrium and, • there is an environment ( ≻ , q ) where the admission to some school s is strategy-proofto some student i via ϕ in equilibrium, but not via ψ . Despite this stringent notion, the main results for constrained Gale-Shapley mechanismand the First-Preference-First mechanism remain true.
Theorem 4.
Let k > ℓ and suppose that there are at least k schools and at least one first-preference-first school. Then • GS k is strongly more immune to strategic admissions than GS ℓ , • GS k is strongly more immune to strategic admissions than F P F k . See the appendix for the proof. In contrast, the prior ranking of the Chinese mechanismsdoes not hold anymore.
Proposition 4.
There is e > e ′ and at least e schools such that the the mechanism Ch ( e ) isnot strongly less immune to strategic admissions than Ch ( e ′ ) .Proof. We prove by the following example and the mechanisms Ch and Ch = β .Suppose that there are 4 students and 4 schools. Let ( P ∗ , ≻ , q ) be a problem where eachschool has one seat and P and ≻ are specified as follows. P ∗ i P ∗ j P ∗ k P ∗ m P ∗ t P ′ i ≻ s ≻ s ≻ s ≻ s s s s s s s k i j ts s s s ∅ s m k ... ... s ∅ ∅ s ∅ j ... s ∅ t ∅ i Then we have Ch (2) ( P ∗ , ≻ , q ) = i j k m t ∅ s s s s ! . MMUNITY TO STRATEGIC ADMISSIONS 18
Suppose that student i reports the preference relation P ′ i . We show that ( P ′ i , P ∗− i ) is a Nashequilibrium of the game ( P ∗ , Ch (2) ) . First,(1) Ch (2) ( P ′ i , P ∗− i , ≻ , q ) = i j k m ts s s ∅ s ! . In the matching in equation 1, every student, other than i and m , is matched to hisr mostpreferred school under P ∗ . Thus we need to check that it is a best response for student i and m . School s and s are assigned to the highest priority students. Therefore, student i cannot get a seat at each of these schools by reporting a preference relation other than P ′ i .Let us consider now student m . In any strategy where he did not include school s amongthe top two acceptable schools, the outcome is the matching in equation 1. Suppose that heuses a strategy P ′ m where he includes school s among the top two acceptable schools. Then, Ch (2) ( P ′ i , P ′ m , P ∗−{ i,m } , ≻ , q ) = i j k m ts s s ∅ s ! , where student m remains unmatched. Therefore, student i and m do not have a profitabledeviation and ( P ′ i , P ∗− i ) is a Nash equilibrium of the game ( P ∗ , Ch (2) ) . Since s = Ch (2) i ( P ′ i , P ∗− i , ≻ , q ) P ∗ i Ch (2) i ( P ∗ , ≻ , q ) , then the admission to school s is not strategy-proof to student i via Ch (2) in equilibrium.Next, we show that the admission to school s is strategy-proof to student i via Ch (1) = β in equilibrium. Suppose that for some preference profile P and P ′ i , we have(2) s = Ch (1) i ( P ′ i , P − i , ≻ , q ) P i Ch (1) i ( P, ≻ , q ) . We now show that ( P ′ i , P − i ) is not a Nash equilibrium of the game ( P, β ) . This will completethe proof showing that student i does not have a strategic admission to school s via β . TheBoston mechanism produces a Pareto optimal matching with respect to reported preferences.Therefore equation 2 implies that some student j ′ is worse-off under β ( P ′ i , P − i , ≻ , q ) comparedto β ( P, ≻ , q ) . We consider two cases depending on what j ′ gets:Case 1: Student j ′ is matched to his first choice school under β ( P, ≻ , q ) , denoted by s .Then student j ′ is the highest priority student among those who ranked school s first. Since j ′ is worse-off, and thus not matched to school s under β ( P ′ i , P − i , ≻ , q ) , then student i hasranked school s first under P ′ i and is matched to it. By equation 2, s = s which contradictsthe fact that student j ′ has higher priority than student i under ≻ s .Case 2: Student j ′ is not matched to his first choice school under β ( P, ≻ , q ) . Let s = β j ′ ( P, ≻ , q ) . Then no student ranked school s first under P . Let P sj ′ be a preference relationwhere he has ranked school s first. We claim that β j ′ ( P ′ i , P sj ′ , P −{ i,j ′ } , ≻ , q ) = s . Suppose, tothe contrary, that this is not the case. Then student has ranked school s first under P ′ i , and MMUNITY TO STRATEGIC ADMISSIONS 19 is the only student who has ranked it first under ( P ′ i , P − i ) . Thus s = β i ( P ′ i , P − i , ≻ , q ) = s .Since student i has lower priority than student j ′ under ≻ s , β j ′ ( P ′ i , P sj ′ , P −{ i,j ′ } , ≻ , q ) = s ,contradicting our assumption that student j ′ is not matched to school s .Therefore student j ′ has a profitable deviation from ( P ′ i , P − i ) , and ( P ′ i , P − i ) is not a Nashequilibrium of the game ( P, β ) . Under ( ≻ , q ) , the admission to school s is strategy-proof tostudent i via Ch (1) in equilibrium. (cid:3) Conclusion
All strategy-proof mechanisms are alike, each vulnerable mechanism is vulnerable in itsown way. Compared to another, a mechanism can be vulnerable at a larger set of profiles as inPathak and S¨onmez (2013), larger set of preference relations as in Decerf and Van der Linden(2018). A mechanism can also be manipulated by a larger set of agents, giving them strongerincentives to manipulate and causing worse consequences for others — as measured by therange of outcomes that these manipulations induce. This is the focus of our paper.We compare vulnerable mechanisms by what we call immunity: that is how many agentscan manipulate them and to what extent. This notion is first introduced by Bonkoungou(2019) to study the relation between favoring higher ranks and incentives in school choice.Here we argued that this metric represents the concerns of the policy-makers and the publicthat accompanied recent reforms in admissions systems around the world; and showed thateach of these reforms made the mechanisms more immune. Namely, after each reform, ineach environment (set of students, schools, priorities and capacities), for each student thereare fewer schools that he can access by profitable manipulation, and each school is moreprotected from these manipulations.Immunity also comes in an equilibrium version: when student i manipulates the mechanismto get to a school at preference profile P , we require P and i ’s deviation to form an equilibriumin the game induced by the mechanism. This concept is arguably less realistic for marketswhere best responses is hard to expect, e.g., when the market is large, but it is a standardrefining criterion for smaller problems. Our main results carry over to this equilibriumversion of immunity for each mechanism except for the Chinese (Theorem 4, Proposition 4).Pathak and S¨onmez (2013)’s seminal paper was one of the first attempts to compare vul-nerable mechanisms and it has generated a literature on other applications and extensions.Arribillaga and Mass´o (2016) ranked voting rules by inclusion of the vulnerable preferencerelations of each agent i , that is the relations for which there exist preferences of others suchthat i can manipulate. This notion was recently used by Decerf and Van der Linden (2018)to rank constrained Boston and GS mechanisms. Andersson et al. (2014a) ranked budget MMUNITY TO STRATEGIC ADMISSIONS 20 balanced and fair rules by counting, for each preference profile, the number of agents whohave profitable manipulations. They find rules that minimize (by inclusion) the number ofagents and coalitions that can manipulate. In the same problem Andersson et al. (2014b)find the rule that minimizes the maximal gain that an agent can get by manipulation. Next,Chen and Kesten (2017) and Dur et al. (2018) used manipulability to compare mechanismsused in China and Taiwan, respectively.Another related notion is manipulability ranking criterion due to Chen et al. (2016): com-pared to mechanism A, they define mechanism B to be more manipulable if for each agent,at each profile where he can manipulate and get a particular outcome under mechanism A,he can do the same under mechanism B. This notion is useful for ranking all stable mech-anisms in an intuitive sense. However, unlike the results in Pathak and S¨onmez (2013) andin the current paper, these results cannot be attained using the equilibrium version of theirmanipulability criterion: in equilibrium each stable mechanism is as manipulable as another.As this notion also relies on the preference by preference and agent by agent comparison, itdoes not explain any of the reforms studied here (except the Chicago SHS).We should note that immunity to manipulation, regardless of the precise metric used tomeasure it, is not the final criterion in selecting a mechanism. Perhaps, the ultimate concernof the policy-makers and the parents is not the vulnerability itself, but rather the complexityof finding an optimal strategy. This complexity results in drawbacks such as higher number ofmismatches, wastes, justified envy and overall dissatisfaction with the system. Surprisingly,the mechanism designers around the world are ready to tolerate certain levels of these draw-backs, but why that ever do it when a strategy-proof mechanism is readily available — forexample, the unconstrained version of the student-proposing deferred acceptance mechanismis strategy-proof (Roth, 1982; Dubins and Freedman, 1981) — remains obscure.
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MMUNITY TO STRATEGIC ADMISSIONS 22
Appendix: Proofs
Theorem 1: Suppose that there are at least k schools where k > and at least one first-preference-first school. Then GS k is more immune to strategic admissions than F P F k .Proof of Theorem 1. We prove the theorem by contraposition. Suppose that there are atleast k schools and at least one first-preference-first school (in each environment). Let anenvironment ( ≻ , q ) be given and suppose that the admission to school s is not strategy-proofto student i via GS k . There is a preference profile P and a preference relation P ′ i such that(3) s = GS ki ( P ′ i , P − i , ≻ , q ) P i GS ki ( P, ≻ , q ) . We also show that the admission to school s is not strategy-proof to i via F P F k . We aregoing to prove two facts and draw a lemma. Fact 1: GS ki ( P, ≻ , q ) = ∅ If GS ki ( P, ≻ , q ) = s ′ ∈ S then school s ′ is one of the schools that student i has rankedamong his top k schools under P i . Since s P i GS ki ( P, ≻ , q ) , we have s P i s ′ . Thus school s isone of the schools that student i has ranked among his top k schools under P i . Then school s and school s ′ are acceptable under P ki . By definition GS k ( P ′ i , P − i , ≻ , q ) = GS ( P ′ ki , P k − i , ≻ , q ) and GS k ( P, ≻ , q ) = GS ( P k , ≻ , q ) . Therefore(4) GS i ( P ′ ki , P k − i , ≻ , q ) = s P ki s ′ = GS i ( P k , ≻ , q ) . This means, contrary to the fact that GS is strategy-proof (Roth, 1982; Dubins and Freedman,1981), that student i manipulates GS at P k . Therefore, GS ki ( P, ≻ , q ) = ∅ . Fact 2:
Student i did not rank school s among the top k schools under P i .Otherwise, school s is acceptable under P ki and the fact that GS i ( P k , ≻ , q ) = ∅ , by Fact1 , we would have GS i ( P ′ ki , P k − i , ≻ , q ) = s P ki ∅ = GS i ( P k , ≻ , q ) , also contradicting the fact that GS is strategy-proof. Because GS k is individually rational,equation 3 implies that school s is acceptable to student i under P i . Now because school s isacceptable under P i but not under P ki , student i has ranked more than k schools acceptableunder P . Let us state these results in a lemma that we use later. Lemma 1.
Let us suppose that for some preference profile P , a student i and P ′ i , we have s = GS ki ( P ′ i , P − i , ≻ , q ) P i GS ki ( P, ≻ , q ) . Then (i) student i has ranked more than k schools acceptable under P i , and (ii) school s isacceptable under P i but not ranked among the top k schools under P i . We now prove that the admission to school s is not strategy-proof to student i via F P F k .For each ℓ ≤ k , let s ℓ denote the school that student i has ranked at position ℓ under P i . Let MMUNITY TO STRATEGIC ADMISSIONS 23 µ = GS k ( P, ≻ , q ) . Because µ is stable under ( P k , ≻ , q ) and that for each ℓ ≤ k , s ℓ P ki µ ( i ) due to Fact 1 (that is, µ ( i ) = ∅ ) and Lemma 1, for each ℓ ≤ k , we have | µ − ( s ℓ ) | = q s ℓ andfor each student j ∈ µ − ( s ℓ ) , j ≻ s ℓ i . Construct a preference profile P ∗ as follows:(5) P ∗ i P ∗ j = i s µ ( j ) s ∅ ... s k s ∅ Each of the students in µ − ( s ) have higher priority than student i under ≻ s and haveranked it first under P ∗ . Thus F P F ki ( P ∗ , ≻ , q ) = s . More generally, for each ℓ ≤ k , eachstudent in µ − ( s ℓ ) has higher priority than student i under ≻ s ℓ and has ranked school s ℓ higher than student i . Therefore F P F k ( P ∗ , ≻ , q ) = µ where student i is unmatched. Let P si be a preference relation where student i finds only school s acceptable. We claim that F P F ki ( P si , P ∗− i , ≻ ) = s . We consider two cases:Case 1: | µ − ( s ) | < q s . In that case, clearly F P F ki ( P si , P ∗− i , ≻ , q ) = s because no more than q s students finds school s acceptable under ( P si , P ∗ k − i ) .Case 2: | µ − ( s ) | = q s . In this case, we claim that there is at least one student in µ − ( s ) who has lower priority than student i under ≻ s . Suppose, to the contrary, that each studentin µ − ( s ) has higher priority than student i under ≻ s .By definition, GS k ( P, ≻ , q ) = GS ( P k , ≻ , q ) . Then µ is stable under ( P k , ≻ , q ) . It isalso stable under ( P si , P k − i , ≻ , q ) . This is because, any student other than i does not havea justified envy and student i ’s envy is not justified — every student in µ − ( s ) has higherpriority than i under ≻ s .By Roth (1986) the set of students who are matched is the same at all stable matchings.Since student i is unmatched under µ , we have GS i ( P si , P k − i , ≻ , q ) = ∅ . In addition, usinga result by Roth (1982), GS i ( P ′ ki , P k − i , ≻ , q ) = s implies that GS i ( P si , P k − i , ≻ , q ) = s . Thiscontradicts the conclusion that GS i ( P si , P k − i , ≻ , q ) = ∅ . Therefore there is at least one studentin µ − ( s ) who has lower priority than student i under ≻ s .Thus F P F ki ( P si , P ∗− i , ≻ , q ) = s . Finally, because F P F ki ( P ∗ , ≻ , q ) = ∅ and school s isacceptable under P ∗ i by construction, s = F P F ki ( P si , P ∗− i , ≻ , q ) P ∗ i F P F ki ( P ∗ , ≻ , q ) , MMUNITY TO STRATEGIC ADMISSIONS 24 proving that the admission to school s is not strategy-proof to student i via F P F k . Finally,we construct an environment where the admission to some school is strategy-proof to somestudent via GS k but not via F P F k .Let ( ≻ , q ) be an environment such that schools have the same priority order and have oneseat each. Since there is at least one first-preference-first school, let school s be one sucha school. Let student i be the student who is ranked first, j second and m third, in thecommon priroity. Since k ≥ , the admission to school s is strategy-proof to student j via GS k . Let us now show that the admission to school s is not strategy-proof to student j via F P F k . This follows from Lemma 1 and because he is always matched to one of his top twoschools when he ranks at least two schools acceptable. Let P be a preference profile suchthat the components for i , j and m are specified as below. P i P j P m s ′ s ′ ss s s ′ ∅ ∅ ∅ Then
F P F kj ( P, ≻ , q ) = ∅ because school s is a first-preference-first school for which student j did not rank as high as student m . Let P si be a preference relation where s is the onlyacceptable school for student j . Then F P F kj ( P sj , P − j , ≻ , q ) = s . Therefore the admission toschool s is not strategy-proof to student j via F P F k . (cid:3) Theorem 2: Let k > ℓ and suppose that there are at least k schools. Then GS k is moreimmune to strategic admissions than GS ℓ .Proof of Theorem 2. We prove the theorem by contraposition. Let k > ℓ and suppose thatthere is at least k schools. We show that GS k is more immune to strategic admissions than GS ℓ . Fix an environment ( ≻ , q ) and suppose that the admission to school s is not strategy-proof to student i via GS k . Then there is a preference profile P and a preference relation P ′ i such that s = GS ki ( P ′ i , P − i , ≻ , q ) P i GS ki ( P, ≻ , q ) . We show that the admission to school s is not strategy-proof to student i via GS ℓ .By Lemma 1, student i has ranked more than k schools acceptable under P i and school s is acceptable under P i but not ranked among the top k schools. For each ℓ ′ ≤ k , let s ℓ ′ denote the school that student i has ranked at position ℓ ′ under P i .By definition GS k ( P, ≻ , q ) = GS ( P k , ≻ , q ) . Let µ = GS ( P k , ≻ , q ) . Because µ is stableunder ( P k , ≻ , q ) and student i is unmatched under µ by Fact 1 , for each ℓ ′ ≤ k , s ℓ ′ P ki µ ( i ) implies that for each student j ∈ µ − ( s ℓ ′ ) , j ≻ s ℓ ′ i . Let P ∗ be a preference profile defined asfollows: MMUNITY TO STRATEGIC ADMISSIONS 25 (6) P ∗ i P ∗ j = i s µ ( j ) s ∅ ... s ℓ s ∅ Then GS ℓ ( P ∗ , ≻ , q ) = µ , where student i is unmatched. Let P si be a preference relationwhere s is the only acceptable school for student i . If | µ − ( s ) | < q s , then GS ℓi ( P si , P ∗− i , ≻ , q ) = s . If | µ − ( s ) | = q s , then by Case 2 above, student i has higher priority than somestudent in µ − ( s ) . Therefore, GS ℓi ( P si , P ∗− i , ≻ , q ) = s . Therefore the admission to school s isnot strategy-proof to student i via GS ℓ .We provide an environment where the admission to some school is strategy-proof to astudent via GS ℓ but not via GS k . Fix an environment ( ≻ , q ) where schools have the samepriority and where each has one seat. Then GS ℓ is strategy-proof to the top ℓ students underthe common priority but not to the ( ℓ + 1) ’s priority student. Since k ≥ ℓ + 1 , then GS k isstrategy-proof to the ( ℓ + 1) ’s priority student. (cid:3) Proof of Corollary 1.
Fix an environment ( ≻ , q ) and suppose that the admission to school s is not strategy-proof to student i via GS k . By Theorem 2 the admission to school s is alsonot strategy-proof to student i via GS ℓ . By Theorem 1 the admission to school s is also notstrategy-proof to student i via F P F ℓ .It remains to provide an environment where the converse is not true. We provided suchan environment in the proof of Theorem 1: there the admission to some school s is strategy-proof to student i via GS ℓ but not via F P F ℓ . By Theorem 2 the admission to school s isstrategy-proof to student i via GS k . (cid:3) Theorem 3: Let e > e ′ and suppose that there is at least e schools. Then Ch ( e ) is moreimmune to strategic admissions than Ch ( e ′ ) .Proof of Theorem 3. First, we collect some basic results that are proven in Chen and Kesten(2017) needed to prove Theorem 3.
Lemma 2. (Chen and Kesten, 2017). Let e be given. Let P be a preference profile, i astudent, s ′ a school and P s ′ i a preference relation where i has ranked school s ′ first. (i) Suppose that student i is matched to school s ′ under Ch ( e ) ( P, ≻ , q ) . Then he is alsomatched to school s ′ under Ch ( e ) ( P s ′ i , P − i , ≻ , q ) . MMUNITY TO STRATEGIC ADMISSIONS 26 (ii)
Suppose that student i prefers school s ′ to his matching under Ch ( e ) ( P, ≻ , q ) and hasranked it among his top e schools under P . Then he cannot obtain a seat at school s ′ by misrepresenting his preferences. We prove the theorem by contraposition. Fix an environment ( ≻ , q ) and suppose that theadmission to school s is not strategy-proof to student i via Ch ( e ) . Then there is a preferenceprofile P and a preference relation P ′ i such that(7) s = Ch ( e ) i ( P ′ i , P − i , ≻ , q ) P i Ch ( e ) i ( P, ≻ , q ) . Let P si be a preference relation where student i has ranked school s first. By Lemma 2 (i), Ch ( e ) i ( P si , P − i , ≻ , q ) = s . Then student i is matched in the first round of the mechanism.Thus,(8) GS i ( P si , P e − i , ≻ , q ) = s. Since s P i Ch ( e ) i ( P, ≻ , q ) , then student i has been rejected by school s at some round. Henceall the seats of school s have been assigned under µ = Ch ( e ) ( P, ≻ , q ) . That is | µ − ( s ) | = q s .Let N = µ − ( s ) denote the set of students who are matched to school s under Ch ( e ) ( P, ≻ , q ) .We now prove that the admission to school s is not strategy-proof to student i via Ch ( e ′ ) .By Lemma 2 (ii), student i did not rank school s among his top e schools under P . Byequation 7, if µ ( i ) is a school, then it is ranked lower that the position e under P i . Weconsider two cases: Case 1:
At least one student in N has lower priority than student i under ≻ s . Since e ′ < e ,student i has ranked more than e ′ schools above school s under P i . For each ℓ = 1 , . . . , e ′ ,let s ℓ be the ℓ ′ th ranked school under P i . Let P ∗ denote the following preference profile: P ∗ i P ∗ j = i s µ ( j ) ... ∅ s e ′ sµ ( i ) ∅ Note that student i is not matched in the first round Ch ( e ) . Thus GS i ( P e , ≻ , q ) = ∅ .Then for each ℓ = 1 , . . . , e ′ , each student matched to school s ℓ in µ has higher priority thanstudent i under ≻ s ℓ . Furthermore, under P ∗ , each student in N has ranked school s firstand student i did not rank it among the top e ′ . Therefore, This is because for each ℓ = 1 , . . . , e ′ and each student j such that µ ( j ) = s ℓ , s ℓ = GS j ( P e , ≻ , q ) and s ℓ P ei µ ( i ) . MMUNITY TO STRATEGIC ADMISSIONS 27 Ch ( e ′ ) ( P, ≻ , q ) = µ. Since at least one student in N has lower priroity than student i under ≻ s we have, Ch ( e ′ ) ( P si , P ∗− i , ≻ , q ) = s, proving that the admission to school s is not strategy-proof to student i via Ch ( e ′ ) . Case 2:
Every student in N has higher priority than student i under ≻ s . We claim that Claim: at least one student in N has ranked school s below position e under P . Proof of the claim.
Suppose, to the contrary, that every student in N has ranked school s among the top e schools under P . Let η = GS ( P e , ≻ , q ) . Student i is not matched to histop e schools under Ch ( e ) ( P, ≻ , q ) . Therefore, η ( i ) = ∅ . Every student in N is matched toschool s under µ and has ranked it among the top e schools under P . Then for each j ∈ N , η ( j ) = s . Because every student in N has higher priority than student i under ≻ s , η is alsostable under ( P si , P e − i , ≻ , q ) . By Roth (1984) the set of matched students is the same at allstable matchings. Hence GS i ( P si , P e − i , ≻ , q ) = ∅ . This contracts equation 8. (cid:3) Since there is at least one student in N who ranked school s below position e under P andthat e ′ < e , there is at least one student in N who ranked school s below position e ′ under P . Let j be one such student. For each ℓ = 1 , . . . , e ′ , let s iℓ and s jℓ denote the ℓ ′ th rankedschool of student i and j , respectively, under P . Let P ∗ be the following profile. P ∗ i P ∗ j P ∗ k = i,j s i s j µ ( k ) ... ... ∅ s ie ′ s je ′ s sµ ( i ) ∅∅ All seats of each of the schools s j , . . . , s je ′ are assigned in the first round of Ch ( e ) ( P, ≻ , q ) .Since student i is matched (if any) in a round later than the first round of Ch ( e ) ( P, ≻ , q ) , µ ( i ) is not one of the schools s j . . . , s je ′ . Let ℓ = 1 . . . , e ′ . Because student j has rankedschool s jℓ among the top e schools under P and has been rejected, all its seats have beenassigned at the first round of Ch ( e ) ( P, ≻ , q ) to students who have higher priority than himunder ≻ s jℓ . Thus each student in µ − ( s jℓ ) has higher priority than student j under ≻ s jℓ . Then Ch ( e ′ ) ( P ∗ , ≻ , q ) is completed in two rounds, and because student j has higher priority than MMUNITY TO STRATEGIC ADMISSIONS 28 student i under ≻ s , we have Ch ( e ′ ) ( P ∗ , ≻ , q ) = µ. Now under ( P si , P ∗− i ) , there is q s students (including student i ) who have ranked school s among the top e ′ schools. Therefore Ch ( e ′ ) i ( P si , P ∗− i , ≻ , q ) = s and s = Ch ( e ′ ) i ( P si , P ∗− i , ≻ , q ) P ∗ i Ch ( e ′ ) i ( P ∗ , ≻ , q ) = µ ( i ) , proving that the admission to school s is not strategy-proof to student i via Ch ( e ′ ) . (cid:3) Proposition 3: Let k ≥ and ( ≻ , q ) an environment where schools have a common priority.Let the capacities of the schools be increasingly ordered q ≤ q ≤ . . . ≤ q | S | and ˆ q = q + . . . + q k . Then, the mechanism SD k is strategy-proof for the ˆ q -highest priority students.Proof of Proposition 3. The mechanism SD is strategy-proof. Let P be a preference profileand suppose that student i is matched under SD k ( P, ≻ , q ) or has ranked less than or k acceptable schools. By Lemma 1, student i cannot manipulate SD k at P . Let i be one ofthe ˆ q -highest priority students. We show that he never misses one of his k most preferredschools whenever he ranks at least k acceptable schools. This will complete the proof.Suppose, to the contrary, that student i ranks at least k acceptable schools and ends upunmatched under SD k ( P, ≻ , q ) . Let S ′ denote the set of his k most preferred schools. Then,at his turn, all the seats of the schools in S ′ have been selected. Then at least q ′ = P s ∈ S ′ q s students have moved before student i . Then student i is not one of the q ′ -highest prioritystudents. This contradicts the fact that student i is one of the ˆ q -highest priority studentsbecause ˆ q ≤ q ′ . (cid:3) Theorem 4: Let k > ℓ and suppose that there are at least k schools and at least onefirst-preference-first school. Then • GS k is strongly more immune to strategic admissions than GS ℓ , • GS k is strongly more immune to strategic admissions than F P F k .Proof of Theorem 4. We prove the theorem by contraposition. Suppose that there are atleast k schools and at least one first-preference-first school. Let an environment ( ≻ , q ) begiven and suppose that the admission to school s is not strategy-proof to student i via GS k in equilibrium. There is a preference profile P and a preference relation P ′ i such that • ( P ′ i , P − i ) is a Nash equilibrium of the ( P, GS k ) and • s = GS ki ( P ′ i , P − i , ≻ , q ) P i GS ki ( P, ≻ , q ) .We show that the admission to school s is not strategy-proof to student i via F P F k and GS ℓ in equilibrium. The difference with the proof of Theorem 1 and Theorem 2 is that MMUNITY TO STRATEGIC ADMISSIONS 29 we further assumed that ( P ′ i , P − i ) is a Nash equilibrium of the ( P, GS k ) . For each of thepreference profiles that we constructed in equations 5 and 6, we have s = GS ki ( P si , P ∗− i , ≻ , q ) P ∗ i GS ki ( P ∗ , ≻ , q ) , where student i has ranked school s first under P si . Note that there is a student in µ − ( s ) who has lower priority than student i under ≻ s . Let j be the lowest priority students amongthem. Now, under F P F k ( P si , P ∗ , ≻ , q ) student j is unmatched, student i is matched toschool s and each of the remaining students is matched to their first choice school. Thestrategy ( P si , P ∗− i ) is a Nash equilibrium of ( P, F P F k ) . Indeed, student i cannot get a seatat a school s ′ that he prefers to s because each student in µ − ( s ′ ) has ranked s ′ first andhas higher priority than him under ≻ s ′ . Similarly, each student matched to school s under F P F k ( P si , P ∗− i , ≻ , q ) has higher priority than student j under ≻ s and has ranked it firstunder ( P si , P ∗− i ) . Thus student j cannot be matched to school s by reporting a preferencerelation other than P ∗ j . The prove that the admission to school s is not strategy-proof tostudent i via F P F k in equilibrium.The argument can be used to prove that ( P si , P ∗− i ) is a Nash equilibrium of the game ( P, GS ℓ ) where P ∗ is the preference profile in equation 6.We provide an environment where the admission to some school is strategy-proof to somestudent via GS k in equilibrium but not via F P F k . Let ( ≻ , q ) be an environment where theschools have a common priority and where each school has one seat. By assumption thereare at least k ≥ schools and students. Let students be ordered from , the highest prioritystudent, to | I | , the lowest priority student. By Proposition 3, GS k = SD k is strategy-prooffor student . Without loss of generality, let us assume that school s is a first-preference-firstschool. Let P be the following preference profile: P P P P −{ , , } s s s ∅∅ s ∅∅ Since k ≥ , F P F k ( P, ≻ , q ) = ∅ . Let P s be a preference relation where student ranksschool s first. Clearly, ( P s , P − ) is a Nash equilibrium of the game ( P, F P F k ) and s = F P F k ( P s , P − , ≻ , q ) P F P F k ( P, ≻ , q ) . Therefore, the admission to school s is not strategy-proof to student via F P F k in equi-librium.We consider the same environment to show that the admission to some school is strategy-proof to some student via GS k in equilibrium but not via GS ℓ . Since k > ℓ , k ≥ ℓ + 1 and bythe Proposition 3, GS k = SD k is strategy-proof to the student ℓ + 1 . Let P be a preference MMUNITY TO STRATEGIC ADMISSIONS 30 profile such that for each student i , P i is specified as follows (recall that there is at least k schools). P i s ... s ℓ s ℓ +1 ∅ Then GS ℓℓ +1 ( P, ≻ , q ) = ∅ . Let P s ℓ +1 ℓ +1 be a preference relation where student ℓ + 1 rankedschool s ℓ +1 first. Clearly, ( P s ℓ +1 ℓ +1 , P − ( ℓ +1) ) is a Nash equilibrium of the game ( P, GS ℓ ) , and s ℓ +1 = GS ℓℓ +1 ( P s ℓ ℓ +1 , P − ( ℓ +1) , ≻ , q ) P ℓ +1 GS ℓℓ +1 ( P, ≻ , q ) . Therefore, the admission to school s ℓ +1 is not strategy-proof to student student ℓ + 1 via GS ℓ in equilibrium.in equilibrium.