Reforms meet fairness concerns in school and college admissions
RREFORMS MEET FAIRNESS CONCERNS IN SCHOOLAND COLLEGE ADMISSIONS
SOMOUAOGA BONKOUNGOU AND ALEXANDER NESTEROV
Abstract.
Recently, many matching systems around the world have been re-formed. These reforms responded to objections that the matching mechanismsin use were unfair and manipulable. Surprisingly, the mechanisms remained unfaireven after the reforms: the new mechanisms may induce an outcome with a blockingstudent who desires and deserves a school which she did not receive.However, as we show in this paper, the reforms introduced matching mechanismswhich are more fair compared to the counterfactuals. First, most of the reformsintroduced mechanisms that are more fair by stability : whenever the old mechanismdoes not have a blocking student, the new mechanism does not have a blockingstudent either. Second, some reforms introduced mechanisms that are more fair bycounting : the old mechanism always has at least as many blocking students as thenew mechanism.These findings give a novel rationale to the reforms and complement the recentliterature showing that the same reforms have introduced less manipulable matchingmechanisms. We further show that the fairness and manipulability of the mecha-nisms are strongly logically related.
Keywords : market design, school choice, college admission, fairness, stability
JEL Classification : C78, D47, D78, D82 Introduction
In the last decades, there has been a wave of reforms of matching systems aroundthe world, ranging from college admissions systems in Chinese provinces, secondary
Higher School of Economics, St.Petersburg
E-mail addresses : [email protected], [email protected] . Date : July 29, 2020.We are grateful to Battal Do˘gan and Rustamdjan Hakimov for their suggestions. We thank BettinaKlaus, Camille Terrier, Inacio Bo, Madhav Raghavan, and the other participants of the onlineseminar of the Lausanne Market Design group for their comments, as well as the participants of theonline Conference on Mechanism and Institution Design 2020.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 grants 20-01-00687, 19-01-00762. a r X i v : . [ ec on . T H ] S e p EFORMS AND FAIRNESS 2 public school admissions systems in multiple districts in Ghana, to public schooladmissions systems in multiple cities in the US and in the UK.The old matching systems were criticized because they were vulnerable to gamingand were unfair. The most vivid example is, perhaps, the 2007 major reform in Eng-land, which covers 146 local school admissions systems. According to the Secretary ofState, Alan Johnson, the aim of the reform was to “ensure that admission authorities– whether local authorities or schools – operate in a fair way” (School AdmissionsCode, 2007). The reform prohibited the practice of giving “priority to children ac-cording to the order of other schools named as preference by their parents,” known asthe first-preference-first principle. According to this principle, a student who ranks aschool higher in her list, receives a higher admission priority at this school comparedto the students who rank this school lower. Prior to the reform, as many as one thirdof the schools in England used this principle.In 2009, the Chicago authorities replaced the so-called Boston mechanism thatused the same first-preference-first principle for each selective high school, arguingthat, due to this principle “high-scoring kids were being rejected simply because ofthe order in which they listed their college prep preferences” (Pathak and S¨onmez,2013). The same mechanism has been in use for college admission in all provincesin China. It raised similar complaints. For example, one parent said: “My child hasbeen among the best students in his school and school district. He achieved a scoreof 632 in the college entrance exam last year. Unfortunately, he was not accepted byhis first choice. After his first choice rejected him, his second and third choices werealready full. My child had no choice but to repeat his senior year” (Chen and Kesten,2017; Nie, 2007).The two complaints above illustrate an unfairness issue with the first-preference-first principle. This principle can induce a matching with a so-called blocking student ,that is, a student who prefers a school over her matching while at least one seat ofthis school has been assigned to a student with a lower priority (or even left empty).The blocking student desires and deserves this seat, yet she has not been assigned toit. A matching with no blocking student is called stable . We use these two conceptsto compare the mechanisms before and after each reform.The first fairness criterion is stability. We compare the mechanisms at each in-stance, taking the preferences as the reports. A mechanism is more fair by sta-bility than a second mechanism if it induces a stable matching whenever the secondmechanism induces a stable matching, and the reverse is not true. Namely, for some
EFORMS AND FAIRNESS 3
Reforms From To more fairby stability? more fairby counting?Arbitrarypriority Commonpriority Arbitrarypriority CommonpriorityUK(54), 2007/11
F P F k GS k notcomparable more notcomparable notcomparableChicago, 2009UK(4), 2007 β k GS k more more notcomparable notcomparableChicago, 2010Ghana, 2007/08UK(2), 2010 GS k GS k +1 more more more moreChina(13), 2001/12 β Ch ( e ) more more notcomparable notcomparable Table 1.
Comparison of the matching mechanisms by fairness criteria.
Notes : Each row compares the mechanism in the third column to the mechanism in the secondcolumn with respect to fairness by stability and fairness by counting. Common priority is a specialcase of arbitrary priority. The complete list of the the UK local matching systems and Chineseprovinces that underwent the reforms can be found in Pathak and S¨onmez (2013) and,respectively, in Chen and Kesten (2017). markets this mechanism induces a stable matching, while the second mechanism doesnot.Our main results using this criterion support that most of the reforms have adoptedmatching mechanisms that are more fair by stability (see Table 1). For example, inChina, this is true for half of its provinces (Chen and Kesten, 2017). In Chicago,the mechanism adopted after the 2009 reform is more fair by stability than the onepreviously used (Theorem 1); the one adopted after the 2010 reform is also more fairby stability than the mechanism adopted in 2009 (Theorem 2).The only exception is the 2007 reform in England in the districts where only someof the schools used the first-preference-first principle. For each of these districts, thereare instances where the matching was stable under the old mechanism but yet is notunder the new mechanism (Example 1). However, we restored the result when schoolsin such a district have a common priority order (e.g., based on students’ grades ora single lottery), that is, the mechanism adopted after the reform is more fair bystability than the one previously used (Proposition 1).We formulated a second fairness criterion based on counting the number of blockingstudents. A mechanism is more fair by counting (the number of blocking students)
EFORMS AND FAIRNESS 4 than a second mechanism if for each instance the second mechanism has at least asmany blocking students as the first mechanism. This criterion implies the previousone.Our main result for this criterion supports few reforms showing that the mechanismsadopted after these reforms are more fair by counting than the ones used before (seeTable 1). Broadly, these reforms involve extending ranking constraints in the Gale-Shapley mechanism. The Gale-Shapley mechanism with a shorter ranking constrainthas more, or an equal number of, blocking students than the Gale-Shapley mechanismwith a longer ranking constraint (Theorem 4). This reform took place in Chicago(2010), in Ghana (2007, 2008), in Newcastle (2010), and Surrey (2010) (Pathak andS¨onmez, 2013). But the criterion is too demanding for the other reforms. We providea counterexample showing that after these reforms the number of blocking studentsmay increase (examples 3,4).Overall, our results provide a new justification for the reforms, complementing theexisting ones. Pathak and S¨onmez (2013) were the first to observe these reformsand proposed a way to explain them using a notion of manipulability that comparesmechanisms according to the inclusion of instances where they are not vulnerableto gaming. These results were further strengthened for other mechanisms and othervulnerability criteria (Chen and Kesten, 2017; Decerf and Van der Linden, 2018; Duret al., 2018; Bonkoungou and Nesterov, 2020).We also find a logical relationship between stability and manipulability. Under theconstrained Gale-Shapley mechanism, when its outcome is stable the mechanism isnot manipulable; while for the constrained Boston mechanism, when the mechanismis not manipulable, its outcome is stable (Corollary 1 and Figure 1). For the serialdictatorship mechanism used in Chicago after 2009, the two concepts are equivalent:its outcome at an instance is stable if and only if the mechanism is not manipulableat this instance (Proposition 2).Another interesting example of the relationship between stability and manipula-bility is the reform in England. After this reform, the mechanisms in most schooldistricts did not become less manipulable (Bonkoungou and Nesterov, 2020); theyalso did not become more fair by stability either (Example 1 below). However, foreach instance, the reform was successful according to at least one of the two criteria(Proposition 3): if the reform disrupted fairness — by producing an unstable matchingwhile it was stable before the reform — the new matching is not vulnerable to gaming.Thus, at these instances, the mechanisms were not vulnerable after the reform.
EFORMS AND FAIRNESS 5
The rest of the paper is organized as follows. Next, we briefly review the relatedliterature not mentioned earlier. We then describe the model in section 2. We presentthe results on the comparisons in section 3 and on the relationship between stabilityand manipulability in section 4. We present all the proofs in the appendix.
Related literature.
Apart from the papers studying the reforms mentioned earlier(Pathak and S¨onmez, 2013; Chen and Kesten, 2017; Decerf and Van der Linden, 2018;Bonkoungou and Nesterov, 2020) the recent literature has been interested in otherways to compare mechanism by fairness and stability.Among the strategy-proof and Pareto efficient mechanisms, the Gale’s Top TradingCycles mechanism (Shapley and Scarf, 1974) is the most fair by stability when eachschool has one seat (Abdulkadiroglu et al., 2019). This result also holds for otherfairness comparisons, such as the set of blocking students (Dogan and Ehlers, 2020b)and the set of blocking triplets ( i, j, s ) – student i blocking the matching of school s and student j (Kwon and Shorrer, 2019). In fact, the result holds for each stabilitycomparison that satisfies few basic properties (Dogan and Ehlers, 2020b).Among the Pareto efficient mechanisms, the most fair by stability is the Efficiency-adjusted Deferred Acceptance mechanism (EADA) due to Kesten (2010) – both interms of blocking pairs and blocking triplets (Dogan and Ehlers, 2020a; Tang andZhang, 2020; Kwon and Shorrer, 2019). Independent from the present work, Doganand Ehlers (2020a) also use the fairness by counting criterion to show that amongefficient mechanisms EADA is not the most fair by counting, unless the priority profilesatisfies few acyclicity conditions. To our knowledge, this is the only paper that usesthe number of blocking students as a measure of fairness.The first paper that studied the constrained mechanisms is Haeringer and Klijn(2009). They study the stability of the Nash equilibrium outcomes of the game in-duced by these mechanisms. The most important insight is that the Nash equilibriumoutcomes of the constrained Boston mechanism are all stable, while the Nash equilib-rium outcomes of constrained Gale-Shapley may not be all stable. In addition, theNash equilibrium outcomes of a constrained Gale-Shapley are subset of the Nash equi-librium outcomes of any constrained Gale-Shapley with longer list. Therefore, whenthe Nash equilibrium outcomes the constrained Gale-Shapley with longer list are allstable, the Nash equilibrium outcomes of the constrained Gale-Shapley with shorterlist are also stable. Our results do not contradict the spirit of the previous results Ergin and S¨onmez (2006) also showed that the Nash equilibrium outcomes of the Boston mechanismare stable.
EFORMS AND FAIRNESS 6 because in Nash equilibrium students are best responding such that the equilibriumbehavior alters the fairness properties of the mechanisms.2.
Model
In a school choice model (Balinski and S¨onmez, 1999; Abdulkadiro˘glu and S¨onmez,2003), there is a finite and non-empty set I of students with a generic element i anda finite and non-empty set S of schools with a generic element s .Each student i has a strict preference relation P i over S ∪ {∅} , where ∅ representsthe outside option for this student. For each student i , let R i denote the “at least asgood as” relation associated with P i . School s is acceptable to student i if s P i ∅ ; andit is unacceptable to student i if ∅ P i s . The list P = ( P i ) i ∈ I is a preference profile.Given a proper subset I (cid:48) (cid:40) I of students, we will often write a preference profile as P = ( P I (cid:48) , P − I (cid:48) ) to emphasize the components for the students in I (cid:48) .Each school s has a strict priority order (cid:31) s over the set I of students, and acapacity q s (a natural number indicating the number of its available seats). The list (cid:31) = ( (cid:31) s ) s ∈ S is a priority profile and q = ( q s ) s ∈ S is a capacity vector. We extend eachpriority order (cid:31) s of school s to the set I of subsets of students and assume that thisextension is responsive to the priority order (cid:31) s over I as follows. The priority order (cid:31) s of school s is responsive (Roth, 1986) if • for each i, j ∈ I and each I (cid:48) ⊂ I \ { i, j } such that | I (cid:48) | < q s − , we have, (i) I (cid:48) ∪ { i } (cid:31) s I (cid:48) , and (ii) I (cid:48) ∪ { i } (cid:31) s I (cid:48) ∪ { j } if and only if i (cid:31) s j and • for each I (cid:48) ⊂ I such that | I (cid:48) | > q s , we have ∅ (cid:31) s I (cid:48) .The tuple ( I, S, P, (cid:31) , q ) is a school choice problem. We assume that there are morestudents than schools, that is, | I | > | S | . The set of students and the set of schools arefixed throughout the paper, and we denote the school choice problem by the triple ( P, (cid:31) , q ) , or even by the preference profile P only.A matching µ is a function µ : I → S ∪{∅} such that for each school s , | µ − ( s ) |≤ q s .If µ ( i ) (cid:54) = ∅ , then we say that student i is matched under µ . If µ ( i ) = ∅ , then we saythat she is unmatched under µ .Let ( P, (cid:31) , q ) be a problem. A matching µ is individually rational under P if foreach student i , µ ( i ) R i ∅ . A pair ( i, s ) of a student and a school blocks the matching µ under ( P, (cid:31) , q ) if s P i µ ( i ) and either there is a student j such that µ ( j ) = s and i (cid:31) s j or | µ − ( s ) | < q s . Student i is a blocking student for the matching µ under That is, for each s, s (cid:48) ∈ S ∪ {∅} , s R i s (cid:48) if and only s P i s (cid:48) or s = s (cid:48) . EFORMS AND FAIRNESS 7 ( P, (cid:31) , q ) if there is a school s such that the pair ( i, s ) blocks µ under ( P, (cid:31) , q ) . Amatching µ is stable under ( P, (cid:31) , q ) if it is individually rational under P and no pairof a student and a school blocks it.A mechanism ϕ is a function which maps each school choice problem to a match-ing. For each problem ( P, (cid:31) , q ) , let ϕ i ( P, (cid:31) , q ) denote the component for student i . A mechanism is individually rational if for each problem ( P, (cid:31) , q ) , ϕ ( P, (cid:31) , q ) isindividually rational under P . Given a mechanism ϕ and a problem ( P, (cid:31) , q ) , we saythat ϕ ( P, (cid:31) , q ) is stable whenever ϕ ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) .2.1. Mechanisms.
We are interested in the mechanisms that were used either beforeor after the reforms. We first describe the unconstrained versions.
Gale-Shapley.
Gale and Shapley (1962) showed that for each problem, there exists astable matching. In addition, there is a student-optimal stable matching, which is amatching that each student finds at least as good as any other stable matching. Foreach problem ( P, (cid:31) , q ) , this matching can be found via the Gale and Shapley (1962)student-proposing deferred acceptance algorithm. • Step 1:
Each student applies to her most-preferred acceptable school (if any).If a student did not rank any school acceptable, then she remains unmatched.Each school s considers its applicants I s at this step and tentatively accepts min( q s , | I s | ) of the (cid:31) s -highest priority applicants and rejects the remainingones. Let A s denote the set of students whom school s has tentatively acceptedat this step. • Step t ( t > ): Each student, who is rejected at step t − , applies to hermost-preferred acceptable school among those which have not yet rejected her(if any). If a student does not have any remaining acceptable school, then sheremains unmatched. Each school s considers the set A t − s ∪ I ts , where I ts are itsnew applicants at this step, and tentatively accepts min( q s , | A t − s ∪ I ts | ) of the (cid:31) s -highest priority applicants and rejects the remaining ones. Let A ts denotethe set of students whom school s has tentatively accepted at this step.The algorithm stops when every student is either accepted at some step or has appliedto all of her acceptable schools. The tentative acceptances become final at this step.Let GS ( P, (cid:31) , q ) denote the matching obtained. EFORMS AND FAIRNESS 8
Serial Dictatorship.
When schools have the same priority order, we call the Gale-Shapley mechanism the serial dictatorship mechanism. Let SD ( P, (cid:31) , q ) denote thematching assigned by the serial dictatorship mechanism to the problem ( P, (cid:31) , q ) . First-Preference-First.
The schools are exogenously divided into two subsets S fpf ⊂ S and S ep ⊂ S such that they are disjoint and S fpf ∪ S ep = S . The set S eq is a setof equal-preference schools and S fpf is a set of first-preference-first schools.The First-Preference-First mechanism (FPF) assigns to each problem ( P, (cid:31) , q ) , thematching GS ( P, ˆ (cid:31) , q ) , where the priority order of each equal-preference school ismaintained intact while the priority order of each first-preference-first school is ad-justed according to the rank that students have assigned to this school. Formally, thepriority profile ˆ (cid:31) is obtained as follows:1. for each equal-preference school s ∈ S ep , ˆ (cid:31) s = (cid:31) s and2. for each first-preference-first school s ∈ S fpf , ˆ (cid:31) s is defined as follows. Let I ( s ) be the set of students who have ranked school s first under P , I ( s ) the set of studentswho have ranked school s second under P , and so on. Note that we count the rankingof ∅ as well. • for each (cid:96), k ∈ { , . . . , | S | +1 } such that (cid:96) > k and each students i, j such that i ∈ I k ( s ) and j ∈ I (cid:96) ( s ) , i ˆ (cid:31) s j . • for each k ∈ { , . . . , | S | +1 } and each i, j ∈ I k ( s ) , i ˆ (cid:31) s j if and only if i (cid:31) s j .Let F P F ( P, (cid:31) , q ) denote the matching assigned to the problem ( P, (cid:31) , q ) . Boston.
Until 2005, the Boston public school system was using an immediate accep-tance mechanism called the Boston mechanism (Abdulkadiro˘glu and S¨onmez, 2003).This mechanism assigns to each problem ( P, (cid:31) , q ) , the matching as described in thefollowing algorithm. • Step 1:
Each student applies to her most-preferred acceptable school (if any).Each school s , considers its applicants I s at this step and immediately accepts min( q s , | I s | ) of the (cid:31) s -highest priority applicants and rejects the remainingones. For each school s , let q s = q s − min( q s , | I s | ) denote its remaining capacityafter this step. • Step t: ( t > Each student who is rejected at step t − , applies to hermost-preferred acceptable school among those which have not yet rejectedher (if any). Each school s considers its new applicants I ts at this step and According to our definition, a mechanism has as a domain the set of all problems — includingproblems where schools have different priorities.
EFORMS AND FAIRNESS 9 immediately accepts min( q t − s , | I ts | ) of the (cid:31) s -highest priority applicants andrejects the remaining ones. For each school s , let q ts = q t − − min( q t − s , | I ts | ) denote its remaining capacity after this step.The algorithm stops when every student is either accepted at some step or hasapplied to all of her acceptable schools. Let β ( P, (cid:31) , q ) denote the matching assignedby the Boston mechanism to the problem ( P, (cid:31) , q ) . Remark.
In the (algorithm of the) Boston mechanism, students applying to the sameschool at each step have assigned the same rank to it. Therefore, students applying to aschool at a given step of the algorithm rank this school higher than those applying to itat any step after. In particular, no student could be rejected by a school while anotherstudent, who has assigned a lower rank to it, is accepted. Thus, the Boston mecha-nism is a first-preference-first mechanism where every school is a first-preference-firstschool. This result follows from the Proposition 2 of Pathak and S¨onmez (2008).Constrained mechanisms.
Haeringer and Klijn (2009) first observed that in practice,students are allowed to report a limited number of schools. This means that schoolsthat are listed below a certain position are not considered. Let k ∈ { , . . . , | S |} . Foreach student i , the truncation after the k ’th acceptable school (if any) of the preferencerelation P i with x acceptable schools is the preference relation P ki with min( x, k ) acceptable schools such that all schools are ordered as in P i . Let P k = ( P ki ) i ∈ I . Theconstrained version ϕ k of the mechanism ϕ is the mechanism that assigns to eachproblem ( P, (cid:31) , q ) the matching ϕ ( P k , (cid:31) , q ) . That is, ϕ k ( P, (cid:31) , q ) = ϕ ( P k , (cid:31) , q ) . Chinese parallel.
Chen and Kesten (2017) describe a parametric mechanism thatmany Chinese provinces have been using. The parameter e ≥ is a natural number.For each problem ( P, (cid:31) , q ) , the outcome is a sequential application of constrained GS . In the first round, the matching is final for matched students under GS e ( P, (cid:31) , q ) , while unmatched students proceed to the next round. In the next round, eachschool reduces its capacity by the number of students assigned to it in the last round,each matched student replaces her preferences with a preference relation where shefinds no school acceptable and the unmatched students (in the previous round) arematched according to GS e for the reduced capacities and the new preference profile.The process continues until either no school has a remaining seat or no unmatched EFORMS AND FAIRNESS 10 student finds a school with a remaining seat acceptable. Let Ch ( e ) ( P, (cid:31) , q ) denotethe matching assigned by the mechanism to ( P, (cid:31) , q ) . Results
Fairness by stability.
Our starting point is a comparison according to the setinclusion of the problems where mechanisms are stable.
Definition 1 (Chen and Kesten, 2017) . Mechanism ϕ (cid:48) is more fair by stability than ϕ if(i) at each problem where ϕ is stable, ϕ (cid:48) is also stable and(ii) there exists a problem where ϕ (cid:48) is stable but ϕ is not. Although this criterion is less demanding, in the sense that it does not take intoaccount the problems where mechanisms produce unstable outcomes, it does notexplain all reforms. Indeed, it does not explain many changes that followed the 2007reform in the UK. Indeed, the constrained First-Preference-First mechanism is notcomparable to the constrained Gale-Shapley mechanism according to this concept.We demonstrate this in the following example.
Example 1.
Let I = { i , . . . , i } and S = { s , . . . , s } . Let school s be a (the only)first-preference-first school. Let ( P, (cid:31) , q ) be a problem where each school has one seatand the remaining components are specified as follows. The sign ... indicates that theremaining part is arbitrary. P i P i P i P i P i P i P i (cid:31) s (cid:31) s (cid:31) s (cid:31) s (cid:31) s s s s s s s s i i i i i s s s s s s s ... ... i i ... s ∅ ∅ s s s s i i s ∅ ∅ s ∅ ... ... ∅ s ∅ The outcomes of the constrained First-Preference-First
F P F and the constrainedGale-Shapley GS at ( P, (cid:31) , q ) are specified as follows: F P F ( P, (cid:31) , q ) = (cid:32) i i i i i i i s ∅ s s s ∅ s (cid:33) , This definition of the Chinese parallel mechanisms is given only for the symmetric version whereeach round has the same length e . See Chen and Kesten (2017) for details. EFORMS AND FAIRNESS 11 GS ( P, (cid:31) , q ) = (cid:32) i i i i i i i s ∅ s s s ∅ s (cid:33) . The matching
F P F ( P, (cid:31) , q ) is stable. However, the matching GS ( P, (cid:31) , q ) is notstable. Indeed, the pair ( i , s ) blocks this matching because student i is unmatchedand finds school s acceptable, but student i is matched to s while i (cid:31) s i .The intuition is that the constraint in GS shortened the chains of the rejectionsneeded to reach a stable matching in the Gale-Shapley algorithm. For example, student i is temporarily matched to school s at some step of the algorithm. At the student-optimal stable matching for ( P, (cid:31) , q ) , school s is assigned to student i . However, weneed an application of student i at that school to displace student i from s . Thisdoes not occur under GS because no student initiates the rejection chain. However,under F P F , the application of student i at school s causes the rejection of student i at s (student i has ranked it higher than i and school s is a first-preference-firstschool). This is the rejection needed to reach the student-optimal stable matching. In this example, we illustrate how the constrained GS mechanism has shortenedthe chains needed to reach a stable matching. It is well known that this type ofchains cause welfare losses of the unconstrained GS (Kesten, 2010). However, underthe Boston mechanism, (where all schools are first-preference-first schools) there isno such chain. The following result is an implication of this fact.
Theorem 1.
Suppose that there are at least two schools and let k > . The con-strained Gale-Shapley mechanism GS k is more fair by stability than the constrainedBoston mechanism β k . See the appendix for the proof. Similarly, when schools have a common priorityorder, there is no such chain in the Gale-Shapley mechanism. We restore the resultfor this case.
Proposition 1.
Suppose that there are at least two schools and at least one first-preference-first school, and let k > . The constrained serial dictatorship mechanism SD k is more fair by stability than the constrained First-Preference-First mechanism F P F k . Note that this matching is the student (and school)-optimal stable matching. These chains are initiated by the so-called interrupters. These are students who initiate chains ofrejections which return to them (Kesten, 2010).
EFORMS AND FAIRNESS 12
See the appendix for the proof. The constrained GS with shorter and longer listsmechanisms can also be compared with this criterion. However, the intuition for thisresult is different. When the constrained Gale-Shapley with shorter lists is stable, therestriction has no effect on the outcome. Lemma 1.
Let ( P, (cid:31) , q ) be a problem and k > . Then GS k ( P, (cid:31) , q ) is stable if andonly if GS k ( P, (cid:31) , q ) = GS ( P, (cid:31) , q ) . See the appendix for the proof. Then, when the constraint in GS k does not affectthe outcome, the longer constraint in GS k +1 will not affect the outcome either. Theorem 2.
Suppose that there are at least three schools and let k > (cid:96) where k is lessthan the number of schools and (cid:96) ≥ . Then, the constrained Gale-Shapley mechanism GS k is more fair by stability than GS (cid:96) . See the appendix for the proof. Finally, we consider the Chinese mechanisms.These mechanisms are known to be comparable in terms of fairness by stability, butonly in case one tier is a multiple of another (Chen and Kesten, 2017). We presentthis result for completeness.
Theorem 3 (Chen and Kesten, 2017) . For each e ≥ , m > , the Chinese mechanism Ch ( me ) is more fair by stability than Ch ( e ) . Fairness by counting.
In this section we present the results for a strongercomparison criterion. Unlike the previous notion, we compare the number of blockingstudents. Therefore, the mechanisms are compared for all problems (even where theyinduce unstable outcomes).
Definition 2.
An individually rational mechanism ϕ (cid:48) is more fair by counting (the blocking students) than an individually rational mechanism ϕ if (i) for each problem, there are at least as many blocking students of the outcomeof ϕ as there are of the outcome of ϕ (cid:48) , and (ii) there is a problem where there are more blocking students of the outcome of ϕ than the outcome of ϕ (cid:48) . Fairness by counting is a stronger notion than stability considered earlier. If amechanism ϕ (cid:48) is more fair by counting than ϕ , then for each problem where ϕ inducesa stable matching, i.e., there is no blocking student, ϕ (cid:48) also necessarily induces a stable EFORMS AND FAIRNESS 13 matching. Our main result with this concept is a strengthening of the comparisonbetween different constraints of the Gale-Shapley mechanism.We illustrate the intuition using the example below.
Example 2.
Let I = { i , . . . , i } and S = { s , . . . , s } . Let ( P, (cid:31) , q ) be a problemwhere each school has one seat, and the remaining components are specified as follows. P i P i P i P i P i (cid:31) s (cid:31) s (cid:31) s (cid:31) s s s s s s i i i i s s s s s i i i ... s s s s ... ... ... ...Let us compare the mechanisms GS and GS . We have GS ( P, (cid:31) , q ) = (cid:32) i i i i i ∅ s s ∅ s (cid:33) where student i is the unique blocking student for the matching under ( P, (cid:31) , q ) . In-deed, she is unmatched, finds s acceptable while she has a higher priority at s than i . Let us shorten the reported list only for student i . Then, GS ( P i , P − i , (cid:31) , q ) = (cid:32) i i i i i s ∅ s ∅ s (cid:33) . As a result of this replacement, there are three types of students, given their statusin the previous matching. First, student i — who was matched — became a blockingstudent. Second, student i — who was a blocking student — is not a blocking studentfor the new matching. Finally, student i is a new blocking student.The intuition of this result is that by shortening the schools listed by student i ,she is worse off while the other students are weakly better off. First, she is a blockingstudent for the new matching. Second, student i is not a blocking student for thenew matching, though she was a blocking student for the old matching. But a newblocking student appears so that there are two blocking students in total. This turns out to be true in general. When a student shortens the list, the set ofblocking students changes, but the size of this set never decreases (and sometimesincreases). And when all students shorten their lists, we get the following result.
EFORMS AND FAIRNESS 14
Theorem 4.
Suppose that there are at least two schools and let k > (cid:96) where k is lessthan the number of schools and (cid:96) ≥ . The constrained Gale-Shapley mechanism GS k is more fair by counting than GS (cid:96) . See the appendix for the proof. Next, we show that the other comparisons donot extend to this stronger criterion. The first example shows that the constrainedBoston mechanism is not comparable to the constrained GS . Example 3 (Constrained Boston and GS) . Let n ≥ , I = { i , ..., i n } and S = { s , . . . , s } . Let ( P, (cid:31) , q ) be a problem where each school has one seat and the re-maining components are specified as follows. P i P i P i P i P i . . . P i n − P i n (cid:31) s, s ∈ S s s s s s s s s i ... ... ... s s s s s i s s s s ... i ... s s s i ∅ ∅ ∅ i ... i n The outcomes of β and GS for this problem are specified as follows: β ( P, (cid:31) , q ) = (cid:32) i i i i i . . . i n − i n s s s s ∅ . . . ∅ s (cid:33) and GS ( P, (cid:31) , q ) = (cid:32) i i i i i . . . i n − i n s s s s ∅ . . . ∅ s (cid:33) . Let us compare the number of blocking students for the two outcomes. On onehand, student i is the only blocking student for β ( P, (cid:31) , q ) . Indeed, the pair ( i , s ) blocks β ( P, (cid:31) , q ) under ( P, (cid:31) , q ) . On the other hand, students i , . . . , i n − are allblocking students for GS ( P, (cid:31) , q ) because they are unmatched, each of them prefersschool s to being unmatched, and has higher priority than i n under (cid:31) s . Since n ≥ ,there are at least two blocking students for GS ( P, (cid:31) , q ) . Therefore, there are moreblocking students for GS ( P, (cid:31) , q ) than β ( P, (cid:31) , q ) . Under Theorem 1, there is aproblem where GS is stable but not β . Next, the symmetric Chinese parallel mechanisms are also not comparable in termsof fairness by counting.
EFORMS AND FAIRNESS 15
Example 4 (Chinese parallel) . We consider Example 3. Consider the Chinese mech-anisms Ch (1) = β and Ch (3) and note that for the problem ( P, (cid:31) , q ) specified in thatexample, Ch (1) ( P, (cid:31) , q ) = β ( P, (cid:31) , q ) and Ch (3) ( P, (cid:31) , q ) = GS ( P, (cid:31) , q ) . Accordingto the conclusion in Example 3, there are more blocking students for Ch (3) ( P, (cid:31) , q ) than Ch (1) ( P, (cid:31) , q ) . According to Chen and Kesten (2017), there is a problem where Ch (3) produces a stable outcome but Ch (1) does not. The overall ranking with respect to the two criteria are presented in Table 1. Inthe next section, we investigate the relationship between stability and manipulability.
Remark.
Dogan and Ehlers (2020a) introduced a criterion where they compare mech-anisms by the inclusion of the blocking pairs and blocking students. However, thesecriteria are stronger than fairness by counting and will lead to negative results for ourcomparisons. To see this, consider Example 3. In this example, ( i , s ) is a blockingpair of SD ( P, (cid:31) , q ) but not β ( P, (cid:31) , q ) . In addition, ( i , s ) is a blocking pair of β ( P, (cid:31) , q ) but not SD ( P, (cid:31) , q ) .For the comparison between different constrained Gale-Shapley, consider Example2. There, ( i , s ) is a blocking pair of GS ( P, (cid:31) , q ) but not GS ( P, (cid:31) , q ) . In addition, ( i , s ) is a blocking pair of GS ( P, (cid:31) , q ) but not GS ( P, (cid:31) , q ) . Stability and manipulability
In this section, we will elucidate the relation between blocking students and manip-ulating students, i.e., those who may benefit from misrepresenting their preferencesto the mechanisms. We define those students below as well as the definition of anon-manipulable mechanism.
Definition 3.
Let ϕ be a mechanism.(i) Student i is a manipulating student of ϕ at ( P, (cid:31) , q ) if there is a preferencerelation ˆ P i such that ϕ i ( ˆ P i , P − i , (cid:31) , q ) P i ϕ i ( P, (cid:31) , q ) . (ii) The mechanism ϕ is not manipulable at ( P, (cid:31) , q ) if there is no manipulatingstudent of ϕ at ( P, (cid:31) , q ) . It turns out that there is a strong relation between blocking students and manipulat-ing students for the constrained Boston mechanism and the constrained Gale-Shapleymechanism. Interestingly, these relations for the two mechanisms are reversed.
Theorem 5.
Let ( P, (cid:31) , q ) be a problem and k > . Then, EFORMS AND FAIRNESS 16 (i) every blocking student of the outcome β k ( P, (cid:31) , q ) of the constrained Bostonmechanism is a manipulating student of β k at ( P, (cid:31) , q ) and(ii) every manipulating student of the constrained Gale-Shapley mechanism GS k at ( P, (cid:31) , q ) is a blocking student of GS k ( P, (cid:31) , q ) . See the appendix for the proof. These results have important implications for therelation between manipulability and stability. To see this, suppose that there is nomanipulating student for the constrained Boston mechanism β k at ( P, (cid:31) , q ) . Then,under part (i) of this theorem, there is not a blocking student of β k ( P, (cid:31) , q ) . Since β k is individually rational, then β k ( P, (cid:31) , q ) is stable. Suppose now that there is noblocking student for GS k ( P, (cid:31) , q ) . Since GS k is individually rational, this means that GS k ( P, (cid:31) , q ) is stable. Then, there is no manipulating student for GS k at ( P, (cid:31) , q ) .That is, GS k is not manipulable at ( P, (cid:31) , q ) . We summarize these results in thefollowing corollary; see the appendix for the proof. β k notmanipulable GS k stable GS k not manipulable β k stable Figure 1.
The set inclusion relations of the problems where GS k and β k are stable or not manipulable. Corollary 1.
Let ( P, (cid:31) , q ) be a problem and k > . (i) Suppose that the constrainedBoston mechanism β k is not manipulable at ( P, (cid:31) , q ) . Then, β k ( P, (cid:31) , q ) is stable.(ii) Suppose that the constrained Gale-Shapley mechanism GS k ( P, (cid:31) , q ) is stable.Then, GS k is not manipulable at ( P, (cid:31) , q ) . EFORMS AND FAIRNESS 17 GS k notmanipulable GS k +1 stable GS k +1 notmanipulable GS k stable Figure 2.
The set inclusion relations of the problems where GS k and GS k +1 are stable or not manipulable. Note that there are problems where the reverse of each of these results does nothold. See Example 5 below for the constrained Gale-Shapley mechanism. To seea counterexample of the reverse of the case (i), consider a problem ( P, (cid:31) , q ) wherestudents have a common ranking of schools, have ranked k schools acceptable andwhere each school has one seat. Then, β k ( P, (cid:31) , q ) is stable. However, the studentwho has received her third ranked school is better off top ranking the school she hasranked second as her top choice.The diagrams illustrate the structure of the interplay between stability and manip-ulability for the constrained Boston and the constrained Gale-Shapley mechanisms.An implication of the latter results is a manipulability comparison introduced byPathak and S¨onmez (2013). Under part (i) of Corollary 1, when the constrainedBoston mechanism is not manipulable then it is stable. By Theorem 1, the constrainedGale-Shapley mechanism is also stable. By part (ii) of Corollary 1, the constrainedGale-Shapley mechanism is not manipulable. This is the comparison established byPathak and S¨onmez (2013). Corollary 2. (Pathak and S¨onmez, 2013). Let ( P, (cid:31) , q ) be a problem, k > andsuppose that the constrained Boston mechanism β k is not manipulable at ( P, (cid:31) , q ) .Then, the constrained Gale-Shapley mechanism GS k is not manipulable at ( P, (cid:31) , q ) . Another implication is for the serial dictatorship mechanism. The manipulationstrategy under the constrained GS is to include an acceptable school in the list. Butwhen the constrained GS is stable, all the seats of such a school are assigned to
EFORMS AND FAIRNESS 18 higher priority students, and such a manipulation does not help. This implies thatconstrained serial dictatorship mechanism is non-manipulable and stable for the sameset of problems.
Proposition 2.
Let ( P, (cid:31) , q ) be a problem and k > . The constrained serial dicta-torship mechanism SD k is stable if and only if it is not manipulable at ( P, (cid:31) , q ) . See the appendix for the proof. In general, the constrained Gale-Shapley mech-anism may be unstable while not manipulable. We illustrate this in the followingexample.
Example 5.
Let I = { i , . . . , i } and S = { s , . . . , s } . Let ( P, (cid:31) , q ) be a problemwhere each school has one seat and the remaining components are specified as follows. P i P i P i P i (cid:31) s (cid:31) s (cid:31) s (cid:31) s s s s s i i i ...... s s s ... i i s ... ... i i ∅ i i Let us consider the constrained Gale-Shapley mechanism GS . We have GS ( P, (cid:31) , q ) = (cid:32) i i i i s ∅ s s (cid:33) . This matching is not stable under ( P, (cid:31) , q ) because student i is unmatched, findsschool s acceptable while student i is matched to it and i (cid:31) s i . We claim that GS is not manipulable at ( P, (cid:31) , q ) . Only student i could benefit from misrepresenting herpreferences to the mechanism GS because each of the other students is matched toher most-preferred school. Let P s i be a preference relation where student i has rankedonly school s acceptable. Then, GS ( P s i , P − i , (cid:31) , q ) = (cid:32) i i i i s ∅ s s (cid:33) , that is, student i remains unmatched even by ranking school s first. (It is easyto verify that any other manipulation also leaves i unmatched.) Therefore, GS isnot manipulable at ( P, (cid:31) , q ) . The intuition is that this ranking initiates a chain ofrejections which returns to this student. Student i becomes a so-called “interrupter”when she ranks school s first (Kesten, 2010). EFORMS AND FAIRNESS 19
We also establish another direct corollary of Theorem 5 with two additional results.We show that when switching from constrained Boston to constrained GS, of fromthe constrained GS to a longer list, the mechanism becomes more fair by stabilityand less manipulable.
Corollary 3.
Let ( P, (cid:31) , q ) be a problem.(i) Let k > and suppose that the constrained Boston mechanism β k is stableat ( P, (cid:31) , q ) . Then, the constrained Gale-Shapley mechanism GS k is stable and notmanipulable at ( P, (cid:31) , q ) .(ii) Let k > (cid:96) > and suppose that the constrained Gale-Shapley mechanism GS (cid:96) is stable at ( P, (cid:31) , q ) . Then, the mechanism GS k is stable and not manipulable at ( P, (cid:31) , q ) . Finally, we partially restore the comparisons for the First-Preference-First mech-anism. Although the constrained First-Preference-First mechanism and the con-strained Gale-Shapley mechanism are not comparable by manipulability (Bonkoungouand Nesterov, 2020) and by fairness by stability (Example 1), there is a surprisinginterplay between the two concepts.
Proposition 3.
Let ( P, (cid:31) , q ) be a problem, k > and suppose that the constrainedFirst-Preference-First mechanism F P F k is stable under ( P, (cid:31) , q ) . Then, the con-strained Gale-Shapley mechanism GS k is not manipulable at ( P, (cid:31) , q ) . See the appendix for the proof.This result helps to evaluate the reforms in England, where
F P F k was replaced by GS k . Even though for some problems the reform was unsuccessful in one of the twodimensions — decreasing fairness by stability (Example 1) or increase manipulability(Pathak and S¨onmez, 2013; Bonkoungou and Nesterov, 2020) — the reform could notbe unsuccessful in both dimensions.To sum up the results of this section, stability and manipulability are logicallyrelated, and the relationship depends on the mechanism.5. Conclusions
In response to objections, many school districts around the world have recentlyreformed there admissions systems. The main reason for these objections was thatthe mechanisms were unfair and manipulable. Yet, the mechanisms remained unfairand manipulable even after the reforms. We used two criteria and showed that many
EFORMS AND FAIRNESS 20 reforms resulted in more fair matching mechanisms, first by relying on stability andsecond by counting and comparing the number of the blocking students.The reforms concern essentially two major changes. First, they kept the constrainton the number of schools that each student is allowed to report but replaced theBoston mechanism (or a hybrid between Gale-Shapley and Boston mechanism) withthe Gale-Shapley’s student-proposing deferred acceptance mechanism. Second, someschool districts extended the number of schools that each student is allowed to reportbut kept the Gale-Shapley mechanism. Our findings support these reforms, as wellas such changes in the future.
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Appendix: Proofs
The following result from the literature will be useful throughout the appendix.This result is known as the rural hospital theorem . Lemma 2 (Rural hospital theorem) . (Roth, 1986). Let ( P, (cid:31) , q ) be a problem, ν and µ two stable matchings. Then, EFORMS AND FAIRNESS 22 (i) the same set of students are matched under ν and µ and(ii) each school is matched to the same number of students under ν and µ .Proposition 3: Let ( P, (cid:31) , q ) be a problem, k > and suppose that the constrainedFirst-Preference-First mechanism F P F k is stable at ( P, (cid:31) , q ) . Then, the constrainedGale-Shapley mechanism GS k is not manipulable ( P, (cid:31) , q ) .Proof. We first establish two claims.
Claim 1: Suppose that student i is matched to school s under GS k ( P, (cid:31) , q ) and let P si be a preference relation where she has ranked only school s as an acceptable choice.Then, she is matched to school s under GS k ( P si , P − i , (cid:31) , q ) . Suppose that GS ki ( P, (cid:31) , q ) = s . As shown by Roth (1982), GS i ( P k , (cid:31) , q ) = s implies that GS i ( P si , P k − i , (cid:31) , q ) = s . Since k > , the truncation of P si after the k ’thacceptable school is nothing but P si . Thus, GS ki ( P si , P − i , (cid:31) , q ) = s . Claim 2: Suppose that student i can manipulate GS k at the problem ( P, (cid:31) , q ) . Thenshe is unmatched under GS k ( P, (cid:31) , q ) . This result follows from Pathak and S¨onmez (2013).We are now ready to prove the proposition. Let ( P, (cid:31) , q ) be a problem and supposethat µ = F P F k ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) . Under Claim 2, it is enough to showthat an unmatched student under GS k ( P, (cid:31) , q ) has no profitable misrepresentation.Because GS k is individually rational, under Claim 1, we need to restrict ourselves tomanipulation by top ranking a school first. Since µ is stable under ( P, (cid:31) , q ) , then µ is also stable under ( P k , (cid:31) , q ) . Since GS k is individually rational, we need to checkthat there is no blocking pair. Suppose, to the contrary, that a pair ( i, s ) is a blockingpair for µ under ( P k , (cid:31) , q ) . Then, s P ki µ ( i ) and either (i) school s has an empty seatunder µ or (ii) there is a student j such that µ ( j ) = s and i (cid:31) s j . Note that s P ki µ ( i ) implies that s P i µ ( i ) . Therefore, ( i, s ) is also a blocking pair for µ under ( P, (cid:31) , q ) .This conclusion contradicts our assumption that µ is stable under ( P, (cid:31) , q ) .Therefore, µ is stable under ( P k , (cid:31) , q ) . Since GS ( P k , (cid:31) , q ) is the student-optimalstable matching under ( P k , (cid:31) , q ) ,(1) for each student i, GS i ( P k , (cid:31) , q ) R ki µ ( i ) . In line with Lemma 2, the same number of students are matched under µ and GS ( P k , (cid:31) , q ) . Let i be a student and s a school and suppose that i is unmatched EFORMS AND FAIRNESS 23 under GS ( P k , (cid:31) , q ) and that s P i GS i ( P k , (cid:31) , q ) . Then, student i is also unmatchedunder µ . Thus, s P i µ ( i ) = ∅ . Because µ is stable under ( P, (cid:31) , q ) , this impliesthat every student in µ − ( s ) has higher priority than i under (cid:31) s . Let P si denote apreference relation where i has ranked only school s acceptable. Since µ is stableunder ( P k , (cid:31) , q ) , it is also stable under ( P si , P k − i , (cid:31) , q ) . Under Lemma 2, the set ofmatched students is the same at all stable matchings. Thus, student i is also un-matched under GS ( P si , P k − i , (cid:31) , q ) . Then, under Claim 1, there is no strategy P (cid:48) i suchthat GS ki ( P (cid:48) i , P − i ) = s . Thus, the mechanism GS k is not manipulable at ( P, (cid:31) , q ) . (cid:3) Theorem 1: Suppose that there are at least three schools and let k > . The constrainedGale-Shapley mechanism GS k is more fair by stability than the constrained Bostonmechanism β k .Proof. The Boston mechanism is a special case of the First-Preference-First mecha-nism when every school is a first-preference-first school. Let ( P, (cid:31) , q ) be a problemand suppose that β k ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) . As stated in equation 1, eachstudent finds the outcome GS k ( P, (cid:31) , q ) at least as good as β k ( P, (cid:31) , q ) under P k . Wealso know that the Boston mechanism is Pareto efficient, that is, for each problemthere is no other matching that each student finds at least as good as its outcome (Ab-dulkadiro˘glu and S¨onmez, 2003). Therefore, the matching β k ( P, (cid:31) , q ) = β ( P k , (cid:31) , q ) is Pareto efficient under P k . Thus, GS k ( P, (cid:31) , q ) = β k ( P, (cid:31) , q ) and consequently, GS k ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) .We construct a problem where GS k is stable but not β k . Since there are at leasttwo schools and more students than schools, let s , s be two distinct schools and i , i and i three students. Let ( P, (cid:31) , q ) be a problem where each school has one seat andthe remaining components are specified as follows. P i (cid:54) =3 P (cid:31) s ∈ S s s i s s i ∅ ∅ i ...Since k ≥ , GS k ( P, (cid:31) , q ) = GS ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) . However, thematching β k ( P, (cid:31) , q ) = (cid:32) i i i (cid:54) = 1 , s s ∅ (cid:33) is not stable because the pair ( i , s ) blocks it under ( P, (cid:31) , q ) . EFORMS AND FAIRNESS 24 (cid:3)
Proposition 1: Suppose that there are at least two schools and at least one first-preference-first school, and let k > . The constrained serial dictatorship mechanism SD k is more fair by stability than the constrained First-Preference-First mechanism F P F k .Proof. Let ( P, (cid:31) , q ) be a problem where schools have a common priority order andsuppose that F P F k ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) . Under equation 1, each studentfinds the outcome SD k ( P, (cid:31) , q ) at least as good as F P F k ( P, (cid:31) , q ) under P k . With acommon priority order, there is a unique stable matching under ( P, (cid:31) , q ) which is alsoPareto efficient under P . Therefore, because F P F k ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) ,we have F P F k ( P, (cid:31) , q ) = SD ( P, (cid:31) , q ) .Next, every student who is matched under SD ( P, (cid:31) , q ) is matched to one of her top k -ranked acceptable schools. Therefore, SD ( P, (cid:31) , q ) = F P F k ( P, (cid:31) , q ) is also Paretoefficient under P k . Thus, equation 1 implies that SD k ( P, (cid:31) , q ) = F P F k ( P, (cid:31) , q ) andconsequently, SD k ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) .We can adapt the example provided in the proof of Theorem 1 to show that thereis a problem where SD k is stable but not F P F k . (cid:3) Lemma 1: Let ( P, (cid:31) , q ) be a problem and k > . Then GS k ( P, (cid:31) , q ) is stable if andonly if GS k ( P, (cid:31) , q ) = GS ( P, (cid:31) , q ) .Proof. The “if” part is straightforward because GS ( P, (cid:31) , q ) = GS k ( P, (cid:31) , q ) is thestudent-optimal stable matching under ( P, (cid:31) , q ) .The “only if” part. Suppose that GS k ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) . Let N = { i ∈ I | GS i ( P, (cid:31) , q ) = ∅} denote the set of students who are unmatched un-der GS ( P, (cid:31) , q ) . Step 1 : For each i ∈ N , GS i ( P, (cid:31) , q ) = GS i ( P k , (cid:31) , q ) .This follows from the assumption that GS ( P k , (cid:31) , q ) is stable and Lemma 2. Step 2 : For each i ∈ I \ N , GS i ( P, (cid:31) , q ) = GS i ( P k , (cid:31) , q ) .Let i ∈ I \ N . Because GS ( P, (cid:31) , q ) is the student-optimal stable matching under ( P, (cid:31) , q ) ,(2) GS i ( P, (cid:31) , q ) R i GS ki ( P, (cid:31) , q ) . EFORMS AND FAIRNESS 25
Note that for each student j ∈ N , the preference relation P j can be interpreted asif she has extended her list of acceptable schools from P kj . As shown by Gale andSotomayor (1985), when a subset of students extend their list of acceptable schools,none of the remaining students are better off. Therefore,(3) for each student j ∈ I \ N, GS j ( P kN , P − N , (cid:31) , q ) R j GS j ( P, (cid:31) , q ) . Because GS is individually rational under P , under equation 3, student i is alsomatched under GS ( P kN , P − N , (cid:31) , q ) . Next, since GS ( P k , (cid:31) , q ) is stable under ( P, (cid:31) , q ) ,by assumption, Lemma 2 implies that the same set of students are matched underboth GS ( P, (cid:31) , q ) and GS ( P k , (cid:31) , q ) . Therefore, i is also matched under GS ( P k , (cid:31) , q ) .Next, note that the students in I \ N have extended their list of acceptable schoolsunder ( P kN , P − N ) from P k . Then, at the end of the Gale-Shapley algorithm for theproblem ( P k , (cid:31) , q ) , each of the students in I \ N is accepted by a school. The schoolthat each of them has listed below the school that has accepted her at this step of thealgorithm and how she has ranked them do not affect the outcome of the algorithm.Thus, GS ( P k , (cid:31) , q ) = GS ( P kN , P − N , (cid:31) , q ) . This equation and equation 3 imply that GS i ( P k , (cid:31) , q ) R i GS i ( P, (cid:31) , q ) . Since thepreference relation P i is strict, this relation and equation 2 imply that GS i ( P k , (cid:31) , q ) = GS i ( P, (cid:31) , q ) . Finally, by
Step 1 and
Step 2 , the matching is the same for each student under GS k ( P, (cid:31) , q ) and GS ( P, (cid:31) , q ) , the desired conclusion. (cid:3) Corollary 1: Let ( P, (cid:31) , q ) be a problem and k > . (i) Suppose that the constrained Boston mechanism β k is not manipulable at theproblem ( P, (cid:31) , q ) . Then, β k ( P, (cid:31) , q ) is stable. (ii) Suppose that the constrained Gale-Shapley mechanism GS k is stable under ( P, (cid:31) , q ) . Then, GS k is not manipulable at ( P, (cid:31) , q ) .Proof. We prove (i) by the contraposition. Suppose that β k ( P, (cid:31) , q ) is not stableunder ( P, (cid:31) , q ) . Since β k is individually rational, there is a pair ( i, s ) of a student EFORMS AND FAIRNESS 26 and a school which blocks β k ( P, (cid:31) , q ) under ( P, (cid:31) , q ) . In assent with (i) of Theorem5, student i is a manipulating student of β k at ( P, (cid:31) , q ) . Thus, β k is manipulable at ( P, (cid:31) , q ) .We now prove part (ii). Suppose that GS k ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) . Asshown in Lemma 1, we have GS k ( P, (cid:31) , q ) = GS ( P, (cid:31) , q ) . As shown by Pathak andS¨onmez (2013), only unmatched students could benefit from misrepresenting theirpreferences to GS k . Let i be an unmatched student under µ = GS k ( P, (cid:31) , q ) , s a schoolsuch that s P i µ ( i ) and P si a preference relation where she has ranked only school s as an acceptable school. Because µ is stable under ( P, (cid:31) , q ) , each student in µ − ( s ) has a higher priority than i under (cid:31) s . Therefore, µ is also stable under ( P si , P k − i , (cid:31) , q ) . Under Lemma 2, student i is unmatched under both µ and GS ( P si , P k − i , (cid:31) , q ) .Corresponding to Claim 1, there is no strategy P (cid:48) i such that student i is matched toschool s under GS k ( P (cid:48) i , P − i , (cid:31) , q ) . Therefore, student i cannot manipulate GS k at ( P, (cid:31) , q ) . Since the choice of i is arbitrary, GS k is not manipulable at ( P, (cid:31) , q ) . (cid:3) Theorem 4: Let k > (cid:96) and suppose that there are at least k schools. The constrainedGale-Shapley mechanism GS k is more fair by counting than GS (cid:96) .Proof. Let ( P, (cid:31) , q ) be a problem. The strategy is to start from GS ( P k , (cid:31) , q ) andreplace the preference relations in P k one at a time, with a preference relation in P (cid:96) for the corresponding student until we get GS ( P (cid:96) , (cid:31) , q ) . We prove the theorem byshowing that the number of blocking students is not decreasing after each replacement.First, we prove two lemmas. The first lemma will be used at some steps of thesecond. The second will be the main part to proving the theorem. Lemma 3.
Let N be a subset of students and µ = GS ( P (cid:96)N , P k − N , (cid:31) , q ) . Any blockingstudent for µ under ( P, (cid:31) , q ) is unmatched.Proof. We prove the lemma by the contradiction. Suppose, to the contrary, that somestudent i is a blocking student for µ = GS ( P (cid:96)N , P k − N , (cid:31) , q ) under ( P, (cid:31) , q ) such that µ ( i ) = s for some school s . Then, there is a school s (cid:48) such that s (cid:48) P i µ ( i ) and either(i) | µ − ( s (cid:48) ) | < q s (cid:48) or (ii) there is a student j such that µ ( j ) = s (cid:48) and i (cid:31) s (cid:48) j . Let x ∈ { (cid:96), k } be such that x = (cid:96) if i ∈ N and x = k if i / ∈ N . Since µ ( i ) = s , school s EFORMS AND FAIRNESS 27 is one of the top x acceptable schools under P i . Thus s (cid:48) P xi µ ( i ) = s . This relation,together with the case (i) or (ii) imply that the pair ( i, s (cid:48) ) blocks the matching µ under ( P (cid:96)N , P k − N , (cid:31) , q ) . This conclusion contradicts the fact that µ is stable under ( P (cid:96)N , P k − N , (cid:31) , q ) . (cid:3) The next lemma is the main part for proving the theorem. For simplicity, let ˆ P = ( P (cid:96)N , P k − N ) , i / ∈ N and denote µ =: GS ( ˆ P , (cid:31) , q ) and, ν = GS ( P (cid:96)i , ˆ P − i , (cid:31) , q ) . Lemma 4.
There are at least as many blocking students for ν than µ under ( P, (cid:31) , q ) .Proof. Let n be the number of blocking students for µ under ( P, (cid:31) , q ) . We show thatthere are at least n blocking students for ν under ( P, (cid:31) , q ) .Let us first show that every student other than i finds ν at least as good as µ under ˆ P . To see this, note that student i has extended her list of acceptable schools under ˆ P i = P ki from P (cid:96)i . As shown by Gale and Sotomayor (1985), after this extension nostudent, other than i , is better off. That is,(4) for each student j (cid:54) = i, ν ( j ) ˆ R j µ ( j ) . We divide the rest of the proof into two cases. In the first case, student i is unmatchedunder µ . For this case, we will show that any blocking student for µ under ( P, (cid:31) , q ) isalso a blocking student for ν under ( P, (cid:31) , q ) . In the second case, student i is matchedunder µ . We will show that either µ = ν (in case i is also matched under ν ), or i isa blocking student for ν . Case I: Suppose that student i is unmatched under µ , that is, µ ( i ) = ∅ . EFORMS AND FAIRNESS 28
We first show that ν ( i ) = ∅ . Suppose, to the contrary, that ν ( i ) = s , for someschool s . Then s is one of the top (cid:96) acceptable schools of student i under P i . Since k > (cid:96) , school s is also one of the top k acceptable schools under ˆ P i = P ki . Therefore, GS i ( P (cid:96)i , ˆ P − i , (cid:31) , q ) = s ˆ P i µ ( i ) = GS i ( ˆ P , (cid:31) , q ) = ∅ . This relation shows that student i is better off misrepresenting her preference to theGale-Shapley mechanism, contradicting the fact that this mechanism is not manip-ulable (Dubins and Freedman, 1981; Roth, 1982). Therefore, ν ( i ) = µ ( i ) = ∅ . Thisequality together with equation 4 imply that each student finds the matching ν atleast as good as µ under ˆ P . Because µ = GS ( ˆ P , (cid:31) , q ) is stable under ( ˆ P , (cid:31) , q ) , it isalso stable under ( P (cid:96)i , ˆ P − i , (cid:31) , q ) . To see this, note that student i is unmatched under µ and that for each school s such that s P (cid:96)i µ ( i ) , school s does not have an emptyseat under µ and every student in µ − ( s ) has higher priority than i under (cid:31) s . Since ν = GS ( P (cid:96) − i , ˆ P − i , (cid:31) , q ) is also stable under ( P (cid:96) − i , ˆ P − i , (cid:31) , q ) , under Lemma 2, we havethe following conclusion. Conclusion: (a) the same set of students are matched (unmatched) under ν and µ and (b) every school is matched to the same number of students under both ν and µ . Let us now prove that every blocking student for µ under ( P, (cid:31) , q ) is also a blockingstudent for ν under ( P, (cid:31) , q ) . Let j be a blocking student for µ under ( P, (cid:31) , q ) . Thereare two cases. Case I.1: j = i Then, there is a school s such that s P i µ ( i ) and either (i) school s has an emptyseat under µ or (ii) there is a student j (cid:48) such that µ ( j (cid:48) ) = s and i (cid:31) s j (cid:48) .Consider the case (i) where school s has an empty seat under µ . Then, under part(b) of the previous conclusion, s has an empty seat under ν . Since ν ( i ) = ∅ , i is ablocking student for ν under ( P, (cid:31) , q ) .Consider the case (ii) where there is a student j (cid:48) such that µ ( j (cid:48) ) = s and i (cid:31) s j (cid:48) .Without loss of generality, suppose that school s does not have an empty seat under µ . Then, under part (b) of the previous conclusion, school s does not have an emptyseat under ν . Suppose that ν ( j (cid:48) ) = s . Since ν ( i ) = ∅ and i (cid:31) s j (cid:48) , then the pair ( i, s ) blocks ν under ( P, (cid:31) , q ) and i is a blocking student for ν under ( P, (cid:31) , q ) . Supposethat ν ( j (cid:48) ) (cid:54) = s . Since | ν − ( s ) | = q s , there is j (cid:48)(cid:48) ∈ ν − ( s ) \ µ − ( s ) . Under equation 4, EFORMS AND FAIRNESS 29 s = ν ( j (cid:48)(cid:48) ) ˆ P j (cid:48)(cid:48) µ ( j (cid:48)(cid:48) ) . Since µ = GS ( ˆ P , (cid:31) , q ) is stable under ( ˆ P , (cid:31) , q ) and µ ( j (cid:48) ) = s , then j (cid:48) (cid:31) s j (cid:48)(cid:48) . Because (cid:31) s is transitive, i (cid:31) s j (cid:48) and j (cid:48) (cid:31) s j (cid:48)(cid:48) imply that i (cid:31) s j (cid:48)(cid:48) . Since s P i ν ( i ) = ∅ , the pair ( i, s ) blocks ν under ( P, (cid:31) , q ) and i is a blocking student for ν under ( P, (cid:31) , q ) . Case I.2 : j (cid:54) = i There is a school s such that s P j µ ( j ) and either (i) school s has an empty seatunder µ or (ii) there is a student j (cid:48) such that µ ( j (cid:48) ) = s and j (cid:31) s j (cid:48) . As shownin Lemma 3, because student j is a blocking student for µ under ( P, (cid:31) , q ) , we have µ ( j ) = ∅ .Let us consider the case (i). Under Lemma 2, student j is also unmatched under ν and school s has an empty seat under ν . Thus, j is a blocking student for ν under ( P, (cid:31) , q ) .Second, consider (ii) where there is a student j (cid:48) such that µ ( j (cid:48) ) = s and j (cid:31) s j (cid:48) .If ν ( j (cid:48) ) = s , then ( j, s ) is a blocking pair of ν under ( P, (cid:31) , q ) and j is a blockingstudent for ν under ( P, (cid:31) , q ) . Without loss of generality, suppose that | µ − ( s ) | = q s and ν ( j (cid:48) ) (cid:54) = s . According to part (b) of the above conclusion, | ν − ( s ) | = q s . Then,there is a student j (cid:48)(cid:48) ∈ ν − ( s ) \ µ − ( s ) . Because µ ( i ) = ∅ , we have j (cid:48)(cid:48) (cid:54) = i , and byequation 4, s = ν ( j (cid:48)(cid:48) ) ˆ P j (cid:48)(cid:48) µ ( j (cid:48)(cid:48) ) . Since µ is stable under ( ˆ P , (cid:31) , q ) and µ ( j (cid:48) ) = s , then this equation implies that j (cid:48) (cid:31) s j (cid:48)(cid:48) . Because (cid:31) s is transitive, j (cid:31) s j (cid:48) and j (cid:48) (cid:31) s j (cid:48)(cid:48) imply that j (cid:31) s j (cid:48)(cid:48) . Since s P j ν ( j ) = ∅ , then the pair ( j, s ) blocks ν under ( P, (cid:31) , q ) and j is a blocking studentfor ν under ( P, (cid:31) , q ) .In conclusion, every blocking student for µ under ( P, (cid:31) , q ) is also a blocking studentfor ν under ( P, (cid:31) , q ) . There are n blocking students for ν under ( P, (cid:31) , q ) . Case II: Student i is matched under µ .If student i is also matched under ν = DA ( P (cid:96)i , ˆ P − i , (cid:31) , q ) , then ν = µ . To see this,let ν ( i ) = s for some school s . School s is one of the top (cid:96) acceptable schools ofstudent i under P i . The Gale-Shapley mechanism is invariant to the modification of EFORMS AND FAIRNESS 30 the preferences of the students for the part below their outcomes. We know that P ki is one such modification of P (cid:96) below school s . Thus, ν = µ . We now consider thecase where i is not matched under ν .Suppose that student i is unmatched under ν . The strategy of the proof is to showthat i is a blocking student for ν under ( P, (cid:31) , q ) and that there are also at least n − other blocking students for ν under ( P, (cid:31) , q ) . We depict the flow of these students inFigure 3. Step 1 : Student i is a blocking student for ν under ( P, (cid:31) , q ) .Recall that we assumed that student i is matched under µ = GS ( ˆ P , (cid:31) , q ) , where ˆ P = ( P (cid:96)N , P k − N ) and i / ∈ N . Let s = µ ( i ) . School s is one of the top k acceptableschools under P i . Since ν ( i ) = ∅ , if school s has an empty seat under ν , thenclearly the pair ( i, s ) blocks ν under ( P, (cid:31) , q ) and i is a blocking student for ν under ( P, (cid:31) , q ) . Suppose that | ν − ( s ) | = q s . Since µ ( i ) = s and ν ( i ) = ∅ , there is a student j ∈ ν − ( s ) \ µ − ( s ) . By equation 4, i ≤ Unmatched MatchedBlockingstudents
Figure 3.
Case II: flow of students across matched, unmatched, and block-ing status, from µ to ν : at most one student can leave the blocking status tothe matched status; student i left the matched status to the blocking status,and no student can leave the blocking status and remain unmatched. EFORMS AND FAIRNESS 31 s = ν ( j ) ˆ P j µ ( j ) . Since µ is stable under ( ˆ P , (cid:31) , q ) and µ ( i ) = s , we have i (cid:31) s j . Therefore, the pair ( i, s ) blocks ν under ( P, (cid:31) , q ) and i is a blocking student for ν under ( P, (cid:31) , q ) . Step 2 : Every blocking student for µ under ( P, (cid:31) , q ) who is unmatched under ν isalso a blocking student for ν under ( P, (cid:31) , q ) .Let j be a blocking student for µ under ( P, (cid:31) , q ) and suppose that she is unmatchedunder ν . There is a school s such that s P j µ ( j ) and either (i) school s has an emptyseat under µ or (ii) there is a student j (cid:48) such that µ ( j (cid:48) ) = s and j (cid:31) s j (cid:48) . In addition,because j is the blocking student of µ under ( P, (cid:31) , q ) , by Lemma 3, we have µ ( j ) = ∅ .Let us consider the case (i) where school s has an empty seat under µ . We alsoshow that s has an empty seat under ν . Assume otherwise. Then, there is j (cid:48) ∈ ν − ( s ) \ µ − ( s ) . We know that student i is unmatched under ν . Thus, j (cid:48) (cid:54) = i . Underequation 4, s = ν ( j (cid:48) ) ˆ P j (cid:48) µ ( j (cid:48) ) . This contradicts the fact that µ is stable under ( ˆ P , (cid:31) , q ) because s has an empty seat under µ . Therefore, s has an empty seat under ν . Then the pair ( j, s ) blocks ν under ( P, (cid:31) , q ) and j is a blocking student for ν under ( P, (cid:31) , q ) .Let us now consider the case (ii) where there is a student j (cid:48) such that µ ( j (cid:48) ) = s and j (cid:31) s j (cid:48) . If school s has an empty seat under ν , then because student j is unmatchedunder ν , she is a blocking student for ν under ( P, (cid:31) , q ) . Suppose that school s doesnot have an empty seat under ν . If ν ( j (cid:48) ) = s then the pair ( j, s ) blocks ν under ( P, (cid:31) , q ) and j is a blocking student for ν under ( P, (cid:31) , q ) because ν ( j ) = ∅ and j (cid:31) s j (cid:48) . Suppose that ν ( j (cid:48) ) (cid:54) = s . Because school s does not have an empty seat under ν , there is j (cid:48)(cid:48) ∈ ν − ( s ) \ µ − ( s ) . Since student i is unmatched under ν , we have j (cid:48)(cid:48) (cid:54) = i .By equation 4, we have s = ν ( j (cid:48)(cid:48) ) ˆ P j (cid:48)(cid:48) µ ( j (cid:48)(cid:48) ) . Since µ is stable under ( ˆ P , (cid:31) , q ) , theequation and the fact that µ ( j (cid:48) ) = s imply that j (cid:48) (cid:31) s j (cid:48)(cid:48) . Because (cid:31) s is transitive, j (cid:31) s j (cid:48) and j (cid:48) (cid:31) s j (cid:48)(cid:48) imply that j (cid:31) s j (cid:48)(cid:48) . Since s P j ν ( j ) = ∅ , then the pair ( j, s ) blocks ν under ( P, (cid:31) , q ) and j is a blocking student for ν under ( P, (cid:31) , q ) . Step 3 : Every student but i who is matched under µ is also matched under ν .Suppose that for some student j (cid:54) = i and some school s , µ ( j ) = s . Under equation4, we have ν ( j ) ˆ R j µ ( j ) = s . Since µ is individually rational under ˆ P , ν ( j ) (cid:54) = ∅ . EFORMS AND FAIRNESS 32
Step 4 : There are at least n blocking students for ν under ( P, (cid:31) , q ) .Let j be a blocking student for µ under ( P, (cid:31) , q ) who is not a blocking student for ν under ( P, (cid:31) , q ) . Then, j is matched under ν . Otherwise, according to step 2, sheis also a blocking student for ν under ( P, (cid:31) , q ) . We prove, more generally, that thereare at most one student who is unmatched under µ but matched under ν . To do that,we compare for each school the number of students matched to it under µ and ν .Let s be a school. Suppose that it does not have an empty seat under µ . Then,we have | ν − ( s ) |≤ | µ − ( s ) | = q s . Suppose now that s has an empty seat under µ .Suppose that there is j (cid:48) ∈ ν − ( s ) \ µ − ( s ) . Then, because student i is unmatchedunder ν , j (cid:48) (cid:54) = i . By equation 4, s = ν ( j (cid:48) ) ˆ P j (cid:48) µ ( j (cid:48) ) . This contradicts the fact that µ is stable under ( ˆ P , (cid:31) , q ) because school s has anempty seat under µ . Thus, there is no student matched to school s under ν but notunder µ . Therefore, | ν − ( s ) |≤ | µ − ( s ) | . We conclude that no school is matched tomore students under ν than under µ . Thus,(5) (cid:88) s ∈ S | ν − ( s ) |≤ (cid:88) s ∈ S | µ − ( s ) | . By step 3, all students, but student i , who are matched under µ are also matchedunder ν . Therefore, the set of students who are matched under ν consists of thefollowing students: • the students who are matched under µ , except student i and • the students who are unmatched under µ but matched under ν .Let x denote the number of the students who are unmatched under µ but matchedunder ν . Then, we have (cid:88) s ∈ S | ν − ( s ) | = (cid:88) s ∈ S | µ − ( s ) |− (cid:124) (cid:123)(cid:122) (cid:125) number of studentsmatched under µ and ν + x, EFORMS AND FAIRNESS 33 where the first two expressions on the right-hand side indicate that we subtractedstudent i from those who are matched under µ . By rearranging this equation, we get (cid:88) s ∈ S | ν − ( s ) |− (cid:88) s ∈ S | µ − ( s ) | = x − ≤ , where the inequality follows from equation 5. Thus, there is at most one student whois unmatched under µ but matched under ν . According to Lemma 3, all blockingstudents for µ under ( P, (cid:31) , q ) are unmatched under µ . Then, there is at most oneblocking student for µ under ( P, (cid:31) , q ) who is matched under ν . In assent with thisresult together with step 2, there is at most one blocking student for µ under ( P, (cid:31) , q ) who is not a blocking student for ν under ( P, (cid:31) , q ) . Among the n blocking studentsfor µ under ( P, (cid:31) , q ) , at most one of them is not a blocking student for ν under ( P, (cid:31) , q ) . Therefore, excluding student i , there are at least n − blocking studentsfor ν under ( P, (cid:31) , q ) . Since student i is also a blocking student for ν under ( P, (cid:31) , q ) ,there are at least n blocking students for ν under ( P, (cid:31) , q ) . (cid:3) We complete the proof of the theorem by applying the lemma sequentially. Let n bethe number of the blocking students for GS ( P k , (cid:31) , q ) under ( P, (cid:31) , q ) . For simplicity,let I = { , , . . . , | I |} . Under Lemma 4, there are at least n blocking students for µ = GS ( P (cid:96) , P k − , (cid:31) , q ) under ( P, (cid:31) , q ) . By the same lemma, compared to µ , there are at least n blockingstudents of the matching µ = GS ( P (cid:96) { , } , P k −{ , } , (cid:31) , q ) under ( P, (cid:31) , q ) . With a repeated replacement of the remaining components of P k withtheir counterparts in P (cid:96) , we draw the conclusion that there are at least n blockingstudents for GS ( P (cid:96) , (cid:31) , q ) under ( P, (cid:31) , q ) .Finally, we describe a problem where the outcome of GS (cid:96) has more blocking stu-dents than the outcome of GS k . Let ( P, (cid:31) , q ) be a problem where each school has oneseat, each student has k acceptable schools and such that students have a commonranking of schools. Then, GS k ( P, (cid:31) , q ) = GS ( P, (cid:31) , q ) . Thus GS k ( P, (cid:31) , q ) is stableunder ( P, (cid:31) , q ) . Let s be the school that students have ranked at the k ’th positionstarting from the top. Since there are more students than schools and k > (cid:96) , at leastone student is not matched under GS (cid:96) ( P, (cid:31) , q ) and no student is matched to school EFORMS AND FAIRNESS 34 s even though every student prefers it to being unmatched. Then, there is at leastone blocking student for GS (cid:96) ( P, (cid:31) , q ) . Therefore, there are more blocking studentsfor GS (cid:96) ( P, (cid:31) , q ) than GS k ( P, (cid:31) , q ) under ( P, (cid:31) , q ) . (cid:3) Proposition 2: Let ( P, (cid:31) , q ) be a problem and k > . The constrained serial dictator-ship mechanism SD k is stable if and only if it is not manipulable at ( P, (cid:31) , q ) .Proof. As shown by Bonkoungou and Nesterov (2020), SD k is not manipulable at ( P, (cid:31) , q ) if and only if SD k ( P, (cid:31) , q ) = SD ( P, (cid:31) , q ) . Suppose that SD k ( P, (cid:31) , q ) isstable. Then, according to Lemma 1, SD k ( P, (cid:31) , q ) = SD ( P, (cid:31) , q ) and thus SD k is not manipulable at ( P, (cid:31) , q ) . Suppose that SD k is not manipulable at ( P, (cid:31) , q ) .Then, SD k ( P, (cid:31) , q ) = SD ( P, (cid:31) , q ) and thus stable. (cid:3) Theorem 5: Let ( P, (cid:31) , q ) be a problem and k > . (i) Every blocking student of β k ( P, (cid:31) , q ) is a manipulating student of the con-strained Boston mechanism β k at ( P, (cid:31) , q ) . (ii) Every manipulating student of the constrained Gale-Shapley mechanism GS k at ( P, (cid:31) , q ) is a blocking student of GS k ( P, (cid:31) , q ) .Proof. Proof of (i) . Let i be a student and suppose that she is a blocking studentof µ = β ( P k , (cid:31) , q ) . There is a school s such that the pair ( i, s ) blocks µ under ( P, (cid:31) , q ) . Then, we have s P i µ ( i ) and either (a) school s has an empty seat under µ or (b) there is a student j such that µ ( j ) = s and i (cid:31) s j . We claim that student i did not rank school s first under P i . Otherwise, school s has rejected student i atthe first step of the Boston algorithm under ( P k , (cid:31) , q ) . This is because k > andthe top ranked schools are considered under β k . This contradicts the assumptionthat school s has an empty seat or has accepted student j and i (cid:31) s j . Let P si bea preference relation where i has ranked school s first. Since s has an empty seatunder β k ( P, (cid:31) , q ) or has accepted student j and i (cid:31) s j , there are less than q s studentswho have ranked school s first under P k and have a higher priority than i under (cid:31) s .Therefore, β i ( P si , P k − i , (cid:31) , q ) = s . Since s P i µ ( i ) , then i is a manipulating student of β k at ( P, (cid:31) , q ) . Proof of (ii) . We prove this part by contradiction. Suppose that student i isa manipulating student of GS k at ( P, (cid:31) , q ) but is not a blocking student for µ = GS k ( P, (cid:31) , q ) under ( P, (cid:31) , q ) . By Claim 2 above, i is unmatched under GS k ( P, (cid:31) , q ) . EFORMS AND FAIRNESS 35
Let s be a school such that s P i µ ( i ) . Then, | µ − ( s ) | = q s and every student in µ − ( s ) has higher priority than i under (cid:31) s . Let P si be a preference relation where i has rankedonly school s as an acceptable school. Since µ is stable under ( P k , (cid:31) , q ) , it is alsostable under ( P si , P k − i , (cid:31) , q ) . This follows from the fact that µ ( i ) = ∅ and that everystudent in µ − ( s ) has a higher priority than i under (cid:31) s . According to Lemma 2, theset of unmatched students is the same under µ and GS k ( P si , P k − i , (cid:31) , q ) . Thus, i is alsounmatched under GS k ( P si , P k − i , (cid:31) , q ) . According to Claim 1, there is no misreport bywhich i is matched to s . Since s has been chosen arbitrarily, i is not a manipulatingstudent of GS k at ( P, (cid:31) , q ) . This conclusion contradicts our assumption that student i is a manipulating student of GS k at ( P, (cid:31) , q ) . (cid:3) Theorem 2: Suppose that there are at least three schools and let k > (cid:96) where k is lessthan the number of schools and (cid:96) ≥ . Then, the constrained Gale-Shapley mechanism GS k is more fair by stability than GS (cid:96) .Proof. Suppose that GS (cid:96) ( P, (cid:31) , q ) is stable under ( P, (cid:31) , q ) . Then, there is no blockingstudent for it under ( P, (cid:31) , q ) . According to Theorem 4, there is no blocking studentfor GS k ( P, (cid:31) , q ) under ( P, (cid:31) , q ) . Since GS k ( P, (cid:31) , q ) is individually rational under P ,then it is stable under ( P, (cid:31) , q ) .We described an example in the proof of Theorem 4 where there is a blockingstudent (pair) for GS (cid:96) but not GS k . Since GS k and GS (cid:96) are individually rational, atthis problem GS k is stable but not GS (cid:96) ..