Leveing the Paying Fied: Sincere and Strategic Payers in the Boston Mechanism Parag A. Pathak and Tayfun Sönmez This version: August 2006 Abstract Motivated by empirica evidence on different eves of sophistication among students in the Boston Pubic Schoo (BPS) student assignment pan, we anayze the Nash equiibria of the preference reveation game induced by the Boston mechanism when there are two types of payers. Sincere payers are restricted to report their true preferences, whie strategic payers pay a best response. The set of Nash equiibrium outcomes is characterized in terms of the set of stabe matchings of an economy with a modified priority structure, where sincere students ose their priority to sophisticated students. Whie there are mutipe equiibrium outcomes, a sincere student receives the same assignment in a equiibria. Finay, the assignment of any strategic student under the Pareto-dominant Nash equiibrium of the Boston mechanism is weaky preferred to her assignment under the student-optima stabe mechanism, which was recenty adopted by BPS for use in 2006. Sti preiminary. For financia support, Pathak thanks the Nationa Science Foundation, the Spencer Foundation, and the Division of Research at Harvard Business Schoo and Sönmez thanks the Nationa Science Foundation. Fuhito Kojima and Avin E. Roth provided hepfu discussions. We are gratefu to seminar participants at the Harvard theory unch for comments. Harvard University, e-mai: ppathak@fas.harvard.edu Boston Coege, e-mai: sonmezt@bc.edu 1
1 Introduction In May 2005, Thomas Payzant, the Superintendent of Boston Pubic Schoos, recommended to the pubic that the existing schoo choice mechanism in Boston (henceforth the Boston mechanism) shoud be repaced with an aternative mechanism that removes the incentives to game the system that handicapped the Boston mechanism. Foowing his recommendation, the Boston Schoo Committee voted to repace the mechanism in Juy 2005 and adopt a new mechanism for the 2005-06 schoo year. Over 75,000 students have been assigned to pubic schoos in Boston under the mechanism before it was abandoned. The Boston mechanism is aso widey used to aocate seats at pubic schoos at severa schoo districts throughout the US incuding Cambridge MA, Charotte- Meckenburg, Denver, Miami-Dade, Minneapois, Rochester NY, Tampa-St. Petersburg, and White Pains NY. The major difficuty with the Boston mechanism is that students may find it in their interest to submit a rank order ist that is different from their true underying preferences. Loosey speaking, the Boston mechanism attempts to assign as many students as possibe to their first choice schoo, and ony after a such assignments have been made does it consider assignments of students to their second choices, etc. The probem with this is that if a student is not admitted to her first choice schoo, her second choice may aready be fied with students who isted it as their first choice. That is, a student may fai to get a pace in her second choice schoo that woud have been avaiabe had she isted that schoo as her first choice. If a student is wiing to take a risk with her first choice, then she shoud be carefu to rank a second choice that she has a chance of obtaining. Some parents and famiies understand these features of the Boston mechanism and have deveoped rues of thumb for how to submit preferences. For instance, the West Zone Parents Group (WZPG), a we-informed group of approximatey 180 members who meet reguary prior to admissions time to discuss Boston schoo choice at the K2 eve, recommends two types of strategies to its members. Their introductory meeting minutes on 10/27/2003 state: One schoo strategy is to find a schoo you ike that is undersubscribed and put it as a top choice, OR, find a schoo that you ike that is popuar and put it as a first choice and find a schoo that is ess popuar for a safe second choice. Using data on stated choices from Boston Pubic Schoos from 2000-2004, Abdukadiroğu, Pathak, Roth and Sönmez (2006) describe severa empirica patterns which suggest that there are different eves of sophistication among participants in the mechanism. Some fraction of 2
parents behave as the WZPG suggest, and avoid ranking two overdemanded schoos, especiay when they do not receive priority at their top choice schoo. On the other hand, neary 20% of students ist two overdemanded schoos, and 27% of these students end up unassigned. This empirica evidence, together with the theoretica evidence in Abdukadiroğu and Sönmez (2003) and the experimenta evidence in Chen and Sönmez (2006) was instrumenta in the decision to repace the Boston mechanism with the student-optima stabe mechanism (Gae and Shapey 1962). One of the remarkabe properties of the student-optima stabe mechanism is that it is strategy-proof: truth-teing is a dominant strategy for each student under this mechanism (Dubins and Freedman 1981, Roth 1982). Whether a famiy has access to advice on how to strategicay modify their rank order ist, either through the WZPG or through a famiy resource center, does not matter under the new mechanism. This feature was an important factor in the Superintendent Payzant s recommendation to change the mechanism. The BPS Strategic Panning team, in their 05/11/2005 dated recommendation to impement a new BPS assignment agorithm, emphasized: 1 A strategy-proof agorithm eves the paying fied by diminishing the harm done to parents who do not strategize or do not strategize we. In this paper, our aim is to formaize the intuitive idea that repacing the highy manipuabe Boston mechanism with the strategy-proof student-optima stabe mechanism eves the paying fied. To do so we consider a mode with both sincere and strategic famiies, 2 anayze the Nash equiibria of the preference reveation game induced by the Boston mechanism (or simpy the the Nash equiibria of the Boston game), and compare the equiibrium outcomes with the dominant-strategy outcome of the student-optima stabe mechanism. In Proposition 1, we characterize the equiibrium outcomes of the Boston game in terms of the set of stabe matchings of a modified probem where sincere students ose their priorities to strategic students. This resut impies that there exists a Nash equiibrium outcome where each student weaky prefers her assignment to any other equiibrium assignment. Hence, the Boston game is a coordination game among strategic students. We next examine properties of equiibrium. Sincere students may gain priority at a schoo at the expense of another sincere student by ranking the schoo higher on their preference ist 1 See Recommendation to Impement a New BPS Agorithm - May 11, 2005 at http://boston.k12.ma.us/assignment/. 2 This is consistent with the experimenta findings of Chen and Sönmez (2006) who have shown that about 20% of the subjects in the ab utiize the suboptima strategy of truth-teing under the Boston mechanism. 3
than other sincere students. This wi aow for some sincere students to benefit from the Boston mechanism, at the expense of other sincere students. In Proposition 2, we show that a sincere student receives the same assignment at a equiibria of the Boston game. Finay in Proposition 3, we compare the Boston game to the dominant strategy outcome of the student-optima stabe mechanism. We show that any strategic student weaky prefers her assignment under the Pareto-dominant Nash equiibrium outcome of the Boston game over the dominant-strategy outcome of the student-optima stabe mechanism. When ony some of the students are strategic, the Boston mechanism gives a cear advantage to strategic students provided that they can coordinate their strategies at a favorabe equiibrium. This resut might expain why, in testimony from the community about the Boston mechanism on 06/08/2005, the eader of the WZPG opposed changing the mechanism: There are obviousy issues with the current system. If you get a ow ottery number and don t strategize or don t do it we, then you are penaized. But this can be easiy fixed. When you go to register after you show you are a resident, you go to tabe B and the person ooks at your choices and ets you know if you are choosing a risky strategy or how to re-order it. Don t change the agorithm, but give us more resources so that parents can make an informed choice. Moreover, this resut stands in contrast to Ergin and Sönmez (2006), who show that the set of Nash equiibrium outcomes of the Boston game is equivaent to the set of stabe matchings of the underying probem when a students are strategic. 3 Their resut impies that a repacement of the Boston mechanism with the student-optima stabe mechanism shoud not be opposed since it is in a students best interest. The ayout of the paper is as foows. Section 2 defines the mode and Section 3 characterizes the set of equiibrium. Section 4 identifies comparative statics and Section 5 concudes. 2 The Mode In a schoo choice probem (Abdukadiroğu and Sönmez 2003) there are a number of students each of whom shoud be assigned a seat at one of a number of schoos. Each student has a strict preference ordering over a schoos as we as remaining unassigned and each schoo has a strict priority ranking of a students. Each schoo has a maximum capacity. 3 Kojima (2006) extends this resut to a mode with substitutabe (Keso and Crawford 1981) priorities. 4
Formay, a schoo choice probem consists of: 1. a set of students I = {i 1,..., i n }, 2. a set of schoos S = {s 1,..., s m }, 3. a capacity vector q = (q s1,..., q sm ), 4. a ist of strict student preferences P I = (P i1,..., P in ), and 5. a ist of strict schoo priorities π = (π s1,..., π sm ). For any student i, P i is a strict preference reation over S {i} where sp i i means student i stricty prefers a seat at schoo s to remaining unassigned. For any student i, et R i donote the at east as good as reation induced by P i. For any schoo s, the function π s : {1,...,n} {i 1,...,i n } is the priority ordering at schoo s where π s (1) indicates the student with highest priority, π s (2) indicates the student with second highest priority, and so on. Priority rankings are determined by the schoo district and schoos have no contro over them. We fix the set of students, the set of schoos and the capacity vector throughout the paper; hence the pair (P, π) denotes a schoo choice probem (or simpy an economy). The schoo choice probem is cosey reated to the we-known coege admissions probem (Gae and Shapey 1962). The main difference is that in coege admissions each schoo is a (possiby strategic) agent whose wefare matters, whereas in schoo choice each schoo is a coection of indivisibe goods to be aocated and ony the wefare of students is considered. The outcome of a schoo choice probem, as in coege admissions, is a matching. Formay a matching µ : I S I is a function such that 1. µ(i) S µ(i) = i for any student i, and 2. µ 1 (s) q s for any schoo s. We refer µ(i) as the assignment of student i under matching µ. A matching µ Pareto dominates (or is a Pareto improvement over) a matching ν, if µ(i)r i ν(i) for a i I and µ(i)p i ν(i) for some i I. A matching is Pareto efficient if it is not Pareto dominated by any other matching. A mechanism is a systematic procedure that seects a matching for each economy. 5
The Boston Student Assignment Mechanism For any economy, the outcome of the Boston mechanism is determined in severa rounds with the foowing procedure: Round 1: In Round 1, ony the first choices of students are considered. For each schoo, consider the students who have isted it as their first choice and assign seats of the schoo to these students one at a time foowing their priority order unti there are no seats eft or there is no student eft who has isted it as their first choice. In genera, at Round k: Consider the remaining students. In Round k, ony the k th choices of these students are considered. For each schoo with sti avaiabe seats, consider the students who have isted it as their k th choice and assign the remaining seats to these students one at a time foowing their priority order unti there are no seats eft or there is no student eft who has isted it as his k th choice. The procedure terminates when each student is assigned a seat at a schoo. As any mechanism, the Boston mechanism induces a preference reveation game among students. We refer this game as the Boston game. Sincere and Strategic Students Motivated by the empirica observations in Abdukadiroğu, Roth, Pathak and Sönmez (2006), we assume that there are two types of students: sincere and strategic. Let N, M denote sets of sincere and strategic students, respectivey. We have N M = I and N M =. Sincere students are unaware about the strategic aspects of the student assignment process and they simpy revea their preferences truthfuy. The strategy space of each sincere student is a singeton under the Boston game and it consists of truth-teing. Each strategic student, on the other hand, recognizes the strategic aspects of the student assignment process, her strategy space is a strict preferences over the set of schoos pus remaining unassigned. Each strategic student seects a best response to the other students. We focus on the Nash equiibria of the Boston game where ony strategic students are active payers. 6
Stabiity The foowing concept that pays a centra roe in the anaysis of two-sided matching markets wi be usefu to characterize the Nash equiibria of the Boston game. A matching µ is stabe (Gae and Shapey 1962) if 1. it is individuay rationa in the sense that there is no student i who prefers remaining unassigned to her assignments µ(i), and 2. there is no student-schoo pair (i, s) such that, (a) student i prefers s to her assignment µ(i), and (b) either schoo s has a vacant seat under µ or there is a ower priority student j who nonetheess received a seat at schoo s under µ. Gae and Shapey (1962) show that the set of stabe matchings is non-empty and there exists a stabe matching, the student-optima stabe matching, that each student weaky prefers to any other stabe matching. We refer the mechanism that seects this stabe matching for each probem as the student-optima stabe mechanism. Dubins and Freedman (1981) and Roth (1982) show that for each student, truth teing is a dominant strategy. An Iustrative Exampe Since a student oses her priority to students who rank a schoo higher in their preferences, the outcome of the Boston mechanism is not necessariy stabe. However, Ergin and Sönmez (2006) have shown that any Nash equiibrium outcome of the Boston game is stabe when a students are strategic. Based on this resut they have argued that a change from the Boston mechanism to the student-optima stabe mechanism sha be embraced by a students for it wi resut in a Pareto improvement. This is not what happened in summer 2005 when Boston Pubic Schoos gave up the Boston mechanism and adopted the student-optima stabe mechanism. The eader of an organized parents group in Boston, the West Zone Parents Group (WZPG), pubicy opposed the change in the mechanism. A simpe exampe can iustrate how Ergin and Sönmez (2006) resut is affected when ony part of the students are strategic and provide an expanation to the resistance of the WZPG to the change of the mechanism. Exampe 1. There are three schoos a, b, c each with one seat and three students i 1, i 2, i 3. The priority ist π = (π a, π b, π c ) and student utiities representing their preferences P = (P i1, P i2, P i3 ) are as foows: 7
a b c u i1 1 2 0 u i2 0 2 1 u i3 2 1 0 π a : i 2 i 1 i 3 π b : i 3 i 2 i 1 π c : i 2 i 3 i 1 Students i 1 and i 2 are strategic whereas student i 3 is sincere. Hence the strategy space of each of students i 1, i 2 is {abc, acb, bac, bca, cab, cba} whereas the strategy space of student i 3 is the singeton {abc}. Hence we have the foowing 6 6 1 Boston game for this simpe exampe: abc acb bac bca cab cba abc (0,0,1) (0,0,1) (1,2,0) (1,2,0) (1,1,1) (1,1,1) acb (0,0,1) (0,0,1) (1,2,0) (1,2,0) (1,1,1) (1,1,1) bac (2,0,0) (2,0,0) (0,2,2) (0,2,2) (2,1,2) (2,1,2) bca (2,0,0) (2,0,0) (0,2,2) (0,2,2) (2,1,2) (2,1,2) cab (0,0,1) (0,0,1) (0,2,2) (0,2,2) (0,2,2) (0,2,2) cba (0,0,1) (0,0,1) (0,2,2) (0,2,2) (0,2,2) (0,2,2) where the row payer is student i 1 and the coumn payer is student i 2. There are four Nash equiibrium profies of the Boston game (indicated in bodface) each with a Nash equiibrium payoff of (1,2,0) and a Nash equiibrium outcome of µ = ( i 1 i 2 i 3 a b c We have the foowing usefu observations about the equiibria: 1. Truth-teing is not a Nash equiibrium of the Boston game. 2. Unike in Ergin and Sönmez (2006), the Nash equiibrium outcome µ is not a stabe matching of the economy (P, π). The sincere student i 3 not ony prefers schoo b to her assignment µ(i 3 ) = c but aso she has the highest priority there. Nevertheess, by being truthfu and ranking b second, she has ost her priority to student i 2 at equiibria. 3. The unique stabe matching of the economy (P, π) is ( ) i 1 i 2 i 3 ν =. a c b Matchings µ and ν are not Pareto ranked. Whie the strategic student i 1 is indifferent between the two matchings, the strategic student i 2 is better off under matching µ and 8 ).
the sincere student i 3 is better off under matching ν. That is, the strategic student i 2 is better off under the Nash equiibria of the Boston game at the expense of the sincere student i 3. We next characterize the Nash equiibrium outcomes of the Boston game which wi be usefu to generaize the above observations. 3 Characterization of Nash Equiibrium Outcomes Given an economy (P, π), we wi construct an augmented economy that wi be instrumenta in describing the set of Nash equiibrium outcomes of the Boston game. Given an economy (P, π) and a schoo s, partition the set of students I into m sets as foows: I 1 : Strategic students and sincere students who rank s as their first choices under P, I 2 : sincere students who rank s as their second choices under P, I 3 : sincere students who rank s as their third choices under P,.. I m : sincere students who rank s as their ast choices under P. Given an economy (P, π) and a schoo s, construct an augmented priority ordering π s as foows: each student in I 1 has higher priority than each student in I 2, each student in I 2 has higher priority than each student in I 3,..., each student in I m 1 has higher priority than each student in I m, and for any k m, priority among students in I k is based on π s. Define π = ( π s ) s S. We refer to the economy (P, π s ) as the augmented economy. Exampe 1 continued. Let us construct the augmented economy for Exampe 1. Since ony student i 3 is sincere, π is constructed from π by pushing student i 3 to the end of the priority 9
ordering at each schoo except her top choice a (where she has the owest priority to begin with): π a : i 2 i 1 i 3 π a : i 2 i 1 i 3 π b : i 3 i 2 i 1 π b : i 2 i 1 i 3 π c : i 1 i 3 i 2 π c : i 1 i 2 i 3 The key observation here is that the unique Nash equiibrium outcome µ of the Boston game is the unique stabe matching for the augmented economy (P, π). Whie the uniqueness is specific to the above exampe, the equivaence is genera. We are ready to present our first resut. Proposition 1: The set of Nash equiibrium outcomes of the Boston game under (P, π) is equivaent to the set of stabe matchings under (P, π). Proof: : Fix an economy (P, π) and et µ be stabe under (P, π). Let preference profie Q be such that Q i = P i for a i N and µ(i) is the first choice under Q i for a i M. Matching µ is stabe under (Q, π) as we. Let ν be the outcome of the Boston mechanism under (Q, π). We first show, by induction, that ν = µ. Consider any student j who does not receive her first choice s 1 j under Q at matching µ. By construction of Q, student j is sincere. Since µ is stabe under (Q, π) and since student j does not ose priority to any student at schoo s 1 j when priorities change from π to π, she has ower priority under π s 1 j than any student who has received a seat at s 1 j under µ. Each of these students rank s 1 j as their first choices under Q and schoo s 1 j does not have empty seats under µ for otherwise (j, s) woud bock µ under (Q, π). Therefore ν(j) s 1 j. So a student can receive her first choice under Q at matching ν ony if she receives her first choice under Q at matching µ. But then, since the Boston mechanism is Pareto efficient, matching ν is Pareto efficient under (Q, π) which in turn impies that ν(i) = µ(i) for any student i who receives her first choice under Q at matching µ. Next given k > 1, suppose 1. any student who does not receive one of her top k choices under Q at matching µ does not receive one of her top k choices under Q at matching ν either, and 2. for any student i who receives one of her top k choices under Q at matching µ, ν(i) = µ(i). 10
We wi show that the same hods for (k + 1) and this wi estabish that ν = µ. Consider any student j who does not receive one of her top k + 1 choices under Q at matching µ. By construction of Q, student j is sincere and by assumption she does not receive one of her top k choices under Q at matching ν. Consider (k + 1) th choice s k+1 j of student j under Q j. Since µ is stabe under (Q, π), there is no empty seat at schoo s k+1 j for otherwise pair (j, s k+1 j ) woud bock matching µ under (Q, π). Moreover since µ is stabe under (Q, π), for any student i with µ(i) = s k+1 j one of the foowing three cases shoud hod: 1. i M and by construction s k+1 j is her first choice under Q i, 2. i N and s k+1 j is one of her top k choices under Q i, 3. i N, she has ranked s k+1 j than j under π s k+1. j as her (k + 1) th choice under Q i, and she has higher priority If either of the first two cases hods, then ν(i) = s k+1 j by inductive assumption. If Case 3 hods, then student i has not received one of her top k choices under Q i at matching ν by the inductive assumption and furthermore she has ranked schoo s k+1 j under Q i. Since she has higher priority than j under π s k+1 j as her (k + 1) th choice, ν(j) = s k+1 j impies ν(i) = s k+1 j. Therefore considering a three cases, ν(j) = s k+1 j impies ν(i) = s k+1 j for any student i with µ(i) = sj k+1 and since schoo s k+1 j does not have empty seats under µ, ν(j) s k+1 j. So a student can receive one of her top k + 1 choices under Q at matching ν ony if she receives one of her top k + 1 choices under Q at matching µ. Moreover matching ν is Pareto efficient under (Q, π) and therefore ν(i) = µ(i) for any student i who receives her (k + 1) th choice under Q i at µ competing the induction and estabishing ν = µ. Next we show that Q is a Nash equiibrium profie and hence ν is a Nash equiibrium outcome. Consider any strategic student i M and suppose sp i ν(i) = µ(i) for some schoo s S. Since ν = µ is stabe under (Q, π) and since student i gains priority under π s over ony students who rank s second or worse under Q, not ony any student j I with ν(j) = s ranks schoo s as her first choice under Q i but she aso has higher priority under π s. Therefore regardess of what preferences student i submits, each student j I with ν(j) = s wi receive a seat at schoo s. Moreover by stabiity of ν = µ under (Q, π) there are no empty seats at schoo s and hence student i cannot receive a seat at s regardess of her submitted preferences. Therefore matching ν is a Nash equiibrium outcome. : 11
Suppose matching µ is not stabe under (P, π). Let Q be any preference profie where Q i = P i for any sincere student i and where µ is the outcome of the Boston mechanism under (Q, π). We wi show that Q is not a Nash equiibrium strategy profie of the Boston game under (P, π). First suppose µ is not individuay rationa under (P, π). Then there is a student i I with ip i µ(i). Since the Boston mechanism is individuay rationa, student i shoud be a sophisticated student who has ranked the unacceptabe schoo µ(i) as acceptabe. Let Pi 0 be a preference reation where there is no acceptabe schoo. Upon submitting Pi 0, student i wi profit by getting unassigned. Hence Q cannot be an equiibrium profie in this case. Next suppose there is a pair (i, s) that bocks µ under (P, π). Since µ is the outcome of the Boston mechanism under (Q, π), student i cannot be a sincere student. Let Pi s be a preference reation where schoo s is the first choice. We have two cases to consider: Case 1: Schoo s has an empty seat at µ. Reca that by assumption µ is the outcome of the Boston mechanism under (Q, π). Since s has an empty seat at µ, there are fewer students who rank s as their first choice under Q than the capacity of schoo s. Therefore upon submitting the preference reation Pi s, student i wi profit by getting assigned a seat at schoo s. Hence Q cannot be an equiibrium profie. Case 2: Schoo s does not have an empty seat at µ. By assumption µ is the outcome of the Boston mechanism under (Q, π) and there is a student j with µ(j) = s athough i has higher priority than j under π s. If schoo s is not j s first choice under Q j then there are fewer students who rank s as their first choice under Q than the capacity of schoo s, and upon submitting the preference reation Pi s, student i wi profit by getting assigned a seat at schoo s contradicting Q being an equiibrium profie. If on the other hand schoo s is j s first choice under Q j, then either j is sophisticated or j is sincere and s is her first choice under P j. In either case i having higher priority than j under π s impies i having higher priority than j under π s. Moreover since µ(j) = s, the capacity of schoo s is stricty arger than the number of students who both rank it as their first choice under Q and aso has higher priority than j under π s. Therefore the capacity of schoo s is stricty arger than the number of students who both rank it as their first choice under Q and aso has higher priority than i under π s. Hence upon submitting the preference reation Pi s, student i wi profit by getting assigned a seat at schoo s contradicting Q being an equiibrium profie. Since there is no Nash equiibrium profie Q for which µ is the outcome of the Boston mechanism under (Q, π), µ cannot be a Nash equiibrium outcome of the Boston game under (P, π). 12
Therefore at Nash equiibria strategic students gain priority at the expense of sincere students. Another impication of Proposition 1 is that the set of equiibrium outcomes inherits some of the properties of the set of stabe matchings. In particuar there is a Nash equiibrium outcome of the Boston game that is weaky preferred to any other Nash equiibrium outcome by any student. We refer this outcome as the Pareto-dominant Nash equiibrium outcome. Equiibrium Assignments of Sincere Students The student-optima stabe mechanism repaced the Boston mechanism in Boston in 2005. In the foowing section we wi compare the equiibrium outcomes of the Boston game with the dominant-strategy equiibrium outcome of the student-optima stabe mechanism. One of the difficuties in such comparative static anaysis is that the Boston game has mutipe equiibria in genera. Nevertheess, as we present next, mutipicity is not an issue for sincere students. Proposition 2: Let µ, ν be both Nash equiibrium outcomes of the preference reveation game induced by the Boston mechanism. For any sincere student i N, µ(i) = ν(i). Proof: Fix an economy (P, π). Let µ, ν be both Nash equiibrium outcomes of the preference reveation game induced by the Boston mechanism. By Proposition 1, µ, ν are stabe matchings under (P, π). Let µ = µ ν and µ = µ ν be the join and meet of the stabe matching attice. That is, µ, µ are such that, for a i I, { { µ(i) if µ(i)r i ν(i) ν(i) if µ(i)r i ν(i) µ(i) = µ(i) = ν(i) if ν(i)r i µ(i) µ(i) if ν(i)r i µ(i) Since the set of stabe matchings is attice (attributed to John Conway by Knuth 1976), µ and µ are both stabe matchings under (P, π). Let T = {i I : µ(i) µ(i)}. That is, T is the set of students who receive a different assignment under µ and µ. If T M, then we are done. So suppose there exists i T N. We wi show that this eads to a contradiction. Let s = µ(i), s = µ(i), and j µ 1 (s ) T. Such a student j I exists because by the rura hospitas theorem of Roth (1985) the same set of students and the same set of seats are assigned under any pair of stabe matchings. Note that j µ 1 (µ(i)). Caim: j N. Proof of the Caim: By construction of µ and µ, sp i s and therefore schoo s is not i s first choice. Moreover by Roth and Sotomayor (1989) each student in µ(s )\µ(s ) has higher priority 13
under π s than each student in µ(s ) \ µ(s ), and hence i has higher priority than j under π s. But since i is sincere by assumption and since s is not her first choice, student j has to be sincere as we for otherwise she woud have higher priority under π s. Next construct the foowing directed graph: Each student i T N is a node and there is a directed ink from i T N to j T N if j µ 1 (µ(i)). By the above Caim there is at east one directed ink emanating from each node. Therefore, since there are finite number of nodes, there is at east one cyce in this graph. Pick any such cyce. Let T 1 T N be the set of students in the cyce, and et T 1 = k. Reabe students in T 1 and their assignments under µ, µ so that the restriction of matchings µ and µ to students in T 1 is as foows: ( ) i 1 i 2... i k µ T1 = s 1 s 2... s k ( ) i 1 i 2... i k 1 i k µ T1 = s 2 s 3... s k s 1 Note that a schoo may appear more than once in a cyce so that schoos s t, s u does not need to be distinct for t u (athough they woud have if the cyce we pick is minima). This has no reevance for the contradiction we present next. Let r i,s be the ranking of schoo s in P i (so r i,s = means that s is i s th choice). By Roth and Sotomayor (1989) i k has higher priority at schoo s 1 than i 1 under π s 1, and since i 1, i k are both sincere, r i k,s 1 r i 1,s 1 Simiary r i 1,s 2 r i 2,s 2. r i k 1,s r k i k,s k Moreover since µ(i)p i µ(i) for each i T, s 1 P i 1s 2 r i 1,s 1 < r i 1,s 2 s 2 P i 2s 3 r i 2,s 2 < r i 2,s 3. s k 1 P i k 1s k r i k 1,s < r k 1 i k 1,s k s k P i ks 1 r i k,s < r k i k,s 1 14
Combining the inequaities, we obtain r i k,s 1 r i 1,s 1 < r i 1,s 2 r i 2,s 2 < r i 2,s 3... r i k 1,s k 1 < r i k 1,s k r i k,s k < r i k,s 1 estabishing the desired contradiction. Hence there exists no i N with µ(i) µ(i). But that means there exists no i N with µ(i) ν(i) either, competing the proof. 4 Comparative Statics: Comparing Mechanisms The outcome of the student-optima stabe mechanism can be obtained with the foowing student-proposing deferred acceptance agorithm (Gae and Shapey 1962): Step 1: Each student proposes to her first choice. Each schoo tentativey assigns its seats to its proposers one at a time foowing their priority order. Any remaining proposers are rejected. In genera, at: Step k: Each student who was rejected in the previous step proposes to her next choice. Each schoo considers the students it has been hoding together with its new proposers and tentativey assigns its seats to these students one at a time foowing their priority order. Any remaining proposers are rejected. The agorithm terminates when no student proposa is rejected and each student is assigned her fina tentative assignment. Any student who is not hoding a tentative assignment remains unassigned. Comparing Mechanisms for Sincere Students Sincere students ose priority to strategic students under the Boston mechanism. They may aso be affected by other sincere students, so that some sincere students may benefit at the expense of other sincere students under the Boston mechanism. More precisey, a sincere student may prefer the Boston mechanism to the student-optima stabe mechanism since: she gains priority at her first choice schoo over sincere students who rank it second or ower, and in genera 15
she gains priority at her k th choice schoo over sincere students who rank it (k + 1) th or ower, etc. Exampe 2. There are three schoos a, b, c each with one seat and three sincere students i 1, i 2, i 3. Preferences and priorities are as foows: P i1 : P i2 : P i3 : a b c a b c b a c π a : i 1 i 2 i 3 π b : i 1 i 2 i 3 π c : i 1 i 2 i 3 Outcomes of the Boston mechanism and the student-optima stabe mechanism are ( ) ( ) i 1 i 2 i 3 i 1 i 2 i 3 and a c b a b c respectivey. Under the Boston mechanism the sincere student i 3 gains priority at her top choice schoo b over the sincere student i 2. Hence student i 3 prefers her assignment under the Boston mechanism whereas student i 2 prefers her assignment under the student-optima stabe mechanism. Comparing Mechanisms for Strategic Students Unike a sincere student, a strategic student may be assigned seats at different schoos at different equiibrium outcomes of the Boston game. Hence we wi concentrate on the Paretodominant Nash equiibrium outcome of the Boston game. Proposition 3: The schoo a strategic student receives in the Pareto-dominant equiibrium of the Boston mechanism is weaky better than her dominant-strategy outcome under the student-optima stabe mechanism. Proof: Let µ I and ν I be the student-optima stabe matching for economies (P, π), (P, π) respectivey. Define matching ν 0 as foows: ν 0 (i) = µ I (i) for a i M, ν 0 (i) = i for a i N. If ν 0 is stabe under (P, π), then each sophisticated student i M weaky prefers ν I (i) to ν 0 (i) = µ I (i) and we are done. So w..o.g. assume ν 0 is not stabe under (P, π). We wi 16
construct a sequence of matchings ν 0, ν 1,...,ν k where ν k is stabe under (P, π), and ν (i)r i ν 1 (i) for a i M and 1. Consider matching ν 0. Since ν 0 is not stabe but individuay rationa under (P, π), there is a bocking pair. Pick any schoo s 1 in a bocking pair and et i 1 be the highest priority student (sincere or sophisticated) under π s 1 who stricty prefers s 1 to her assignment under ν 0. Caim 1: Schoo s 1 has an empty seat under ν 0. Proof of Caim 1: We have three cases to consider. Case 1: i 1 M. By construction ν0 1 (s 1 ) = (µ I ) 1 (s 1 ) M and (i 1, s 1 ) does not bock µ I under (P, π). When priorities change from π to π, no sophisticated student oses priority to any other student and therefore (i 1, s 1 ) can bock ν 0 under (P, π) ony if schoo s 1 has an empty seat under ν 0. Case 2: i 1 N and schoo s 1 is not student i 1 s first choice. Student i 1 has ower priority under π s 1 than any sophisticated student. Since ν0 1 (s1 ) M, pair (i 1, s 1 ) can bock ν 0 under (P, π) ony if schoo s 1 has an empty seat under ν 0. Case 3: i 1 N and schoo s 1 is student i 1 s first choice. If µ I (i 1 ) = s 1, then by construction the seat i 1 occupies at s 1 under µ I is empty under ν 0. If µ I (i 1 ) s 1, then a seats at s 1 are occupied under µ I by higher priority students under π. Since (i 1, s 1 ) bocks ν 0 under (P, π), at east one of these students must be a sincere student who ranks s 1 second or worse in her preferences. That is because student i 1 gains priority over ony these student under π. Therefore at east one seat at s 1 must be empty under ν 0. This competes the proof of Caim 1. Construct matching ν 1 by satisfying pair (i 1, s 1 ) at matching ν 0 : ν 1 (i) = ν 0 (i) for a i I \ {i 1 }, ν 1 (i 1 ) = s 1 By construction ν 1 (i)r i ν 0 (i) for a i I. If ν 1 is stabe under (P, π), then for a i M, ν I (i)r i ν 1 (i)r i ν 0 (i) }{{} =µ I (i) and we are done. If not, we proceed with the construction of matching ν 2. 17
In genera for any > 0, if ν is not stabe under (P, π) construct ν +1 as foows: Pick any schoo s +1 in a bocking pair for ν and et i +1 be the highest priority student (sincere or sophisticated) under π s +1 who stricty prefers s +1 to her assignment under ν. As we prove next, s +1 has an empty seat under ν. Construct matching ν +1 by satisfying pair (i +1, s +1 ) at matching ν : ν +1 (i) = ν (i) for a i I \ {i +1 }, ν +1 (i +1 ) = s +1 Caim 2: Schoo s +1 has an empty seat under ν. Proof of Caim 2: We have three cases to consider. Case 1: i +1 M. By construction ν 1 (s +1 ) [(µ I ) 1 (s +1 ) M] {i 1,...,i }. Since (i +1, s +1 ) does not bock µ I under (P, π) and since student i +1 does not gain priority over any sophisticated student when priorities change from π to π, any student in (µ I ) 1 (s +1 ) M has higher priority than student i +1 under π s +1. Moreover for any i m {i 1,...,i } with i m ν 1 (s +1 ), we must have s +1 = s m and thus student i m has higher priority than student i +1 at schoo s +1 = s m under π by the choice of bocking pairs. Therefore student i +1 has ower priority under π s +1 than any student in ν 1 (s +1 ) and hence pair (i +1, s +1 ) can bock ν under (P, π) ony if schoo s +1 has an empty seat at ν. Case 2: i +1 N and schoo s +1 is not student i +1 s first choice. By construction ν 1 (s +1 ) M {i 1,..., i }. Student i +1 has ower priority under π s +1 than any sophisticated student. Moreover for any i m {i 1,..., i } with i m ν 1 (s +1 ), we must have s +1 = s m and thus student i m has higher priority than student i +1 at schoo s +1 = s m under π by the choice of bocking pairs. Therefore student i +1 has ower priority under π s +1 than any student in ν 1 (s +1 ) and hence pair (i +1, s +1 ) can bock ν under (P, π) ony if schoo s +1 has an empty seat at ν. Case 3: i +1 N and schoo s +1 is student i +1 s first choice. Reca that ν 1 (s +1 ) [(µ I ) 1 (s +1 ) M] {i 1,...,i }. First suppose µ I (i +1 ) s +1. For any for any i m {i 1,...,i } with i m ν 1 (s +1 ), we must have s +1 = s m and thus student i m has higher priority than student i +1 at schoo s +1 = s m under π by the choice of bocking pairs. Moreover s +1 is i +1 s first choice and yet µ I is stabe under (P, π). Therefore a students in (µ I ) 1 (s +1 ) has higher priority under π s +1 than i +1 does. But student i +1 does not gain priority at schoo s +1 over any sophisticated student when priorities change from 18
π to π. Therefore any student in (µ I ) 1 (s +1 ) M has higher priority under π s +1 than student i +1 does, which in turn impies any student in ν 1 (s +1 ) has higher priority under π s +1 than student i +1 does. Hence pair (i +1, s +1 ) can bock ν under (P, π) ony if schoo s +1 has an empty seat at ν. Next suppose µ I (i +1 ) = s +1. Let i m {i 1,...,i } be such that i m ν 1 (s +1 ). We have s +1 = s m and since student i m has higher priority than student i +1 at schoo s +1 under π, the same shoud be true under π as we. That is because, student i +1 does not ose priority to any student at schoo s +1 when priorities change from π to π. Since pair (i m, s m ) = (i m, s +1 ) bocks matching ν m 1 under (P, π) by definition, and since assignments ony improve as we proceed by the sequence ν 0, ν 1,...,ν k, s +1 P i m ν m 1 (i m ) R i m ν 0 (i m ). So on one hand µ I (i +1 ) = s +1 where i +1 has ower priority at s +1 than i m under π, and on the other hand i m stricty prefers s +1 to its assignment under ν 0. That means not ony student i m is sincere, but aso µ I (i m ) = s +1 for otherwise µ I cannot be stabe under (P, π). So for each i m ν 1 (s +1 ), there is one empty seat at s +1 under ν 0. In addition there is at east one more empty seat, namey the seat sincere student i +1 occupies at schoo s +1 at matching µ I. Therefore under matching ν there must sti be at east one empty seat at schoo s +1. This covers a three cases and competes the proof of Caim 2. We are now ready to compete the proof. Since each student weaky prefers matching ν to matching ν 1 for any 0, eventuay the sequence terminates which means no pair bocks the fina matching ν k in the sequence and thus ν k is stabe under (P, π). Therefore for any strategic student i M, ν I (i) R i ν k (i) R i ν 0 (i) }{{} =µ I (i) where the first reation hods by definition of the student-optima stabe matching. This competes the proof. Whie the Boston mechanism is easy to describe, it induces a compicated coordination game among strategic students. Therefore, it is important to be cautious in interpreting Proposition 3. When strategic students and their famiies (such as the members of the WZPG) are abe to reach the Pareto-dominant Nash equiibrium assignment, they may prefer keeping the Boston mechanism for they wi reap the benefit of their strategic advantage. 19
5 Concusion Boston Pubic Schoos stated that their main rationae for changing their student assignment system is that it eves the paying fied. They identified a fairness rationae for a strategyproof system. In this paper, we examined this intuitive notion and showed the assignment of a strategic students in the Pareto-dominant Nash equiibrium of the Boston mechanism is weaky better than their assignment under the student-optima stabe mechanism, providing forma support for BPS s caim. Despite its theoretica weaknesses, performance in aboratory experiments, and empirica evidence of confused pay, the Boston mechanism is the most widey used schoo choice mechanism in the United States. This paper proposes another theoretica rationae for abandoning the mechanism based on fairness or equa access, which was centra in Boston s decision. The reason why the mechanism continues to be used is a puzze, which might be reated to our anaysis here. Chubb and Moe (1999) argue that important stakehoders often contro the mechanisms of reform in education poicy. In the context of student assignment mechanisms, the important stakehoders may be strategic parents who have invested energy in earning about the mechanism, and the choice of the Boston mechanism may refect their preferences. References [1] A. Abdukadiroğu, P. A. Pathak, A. E. Roth, and T. Sönmez (2006), Changing the Boston Schoo Choice Mechanism: Strategyproofness as Equa Access, Boston Coege, Coumbia, Harvard working paper. [2] A. Abdukadiroğu and Tayfun Sönmez (2003), Schoo Choice: A Mechanism Design Approach, American Economic Review, 93: 729-747. [3] J. Chubb and T. Moe (1990), Poitics, Markets and America s Schoos. Washington D.C.: Brookings Institution Press. [4] Y. Chen and T. Sönmez (2006), Schoo Choice: An Experimenta Study, Journa of Economic Theory, 127: 202-231. [5] L. E. Dubins and D. A. Freedman (1981), Machiavei and the Gae-Shapey Agorithm, American Mathematica Monthy, 88: 485-494. 20
[6] H. Ergin and T. Sönmez (2006), Games of Schoo Choice Under the Boston Mechanism, Journa of Pubic Economics, 90: 215-237. [7] D. Gae and L. Shapey (1962), Coege Admissions and the Stabiity of Marriage, American Mathematica Monthy, 69: 9-15. [8] A. Keso and V. Crawford (1981), Job Matching, Coaition Formation, and Gross Substitutes, Econometrica, 50: 1483-1504. [9] D. Knuth (1976), Marriage Stabes, Les Press de Universitie de Montrea, Montrea. [10] F. Kojima (2006), Games of Schoo Choice under the Boston Mechanism with Genera Priority Structures, mimeo, Harvard University. [11] A. E. Roth (1982), The Economics of Matching: Stabiity and Incentives, Mathematics of Operations Research, 7: 617-628. [12] A. E Roth and M. Sotomayor(1989), The Coege Admissions Probem Revisited, Econometrica, 57, 1989, 559-570. 21