Leveling the Playing Field : Sincere and Sophisticated Players in the Boston Mechanism ∗

Empirical and experimental evidence suggests different levels of sophistication among families in the Boston Public School student assignment plan. We analyze the preference revelation game induced by the Boston mechanism with sincere players who report their true preferences and sophisticated players who play a best response. We characterize the set of Nash equilibrium outcomes as the set of stable matchings of a modified economy, where sincere students lose priority to sophisticated students. Any sophisticated student weakly prefers her assignment under the Pareto-dominant Nash equilibrium of the Boston mechanism to her assignment under the recently adopted student-optimal stable mechanism. (JEL C78, D63, D78, I20) ∗For financial support, Pathak thanks the National Science Foundation, the Spencer Foundation, and the Division of Research at Harvard Business School and Sönmez thanks the National Science Foundation. John Friedman, Fuhito Kojima and Alvin E. Roth provided helpful discussions. We thank seminar participants at the Harvard theory lunch for comments. We are especially grateful to anonymous referees whose comments significantly improved the paper. †Society of Fellows, Harvard University, Cambridge MA 02138, e-mail: ppathak@fas.harvard.edu ‡Department of Economics, Boston College, Chestnut Hill, MA 02467, e-mail: sonmezt@bc.edu

In May 2005, Dr. Thomas Payzant, the Superintendent of the Boston Public Schools (BPS), recommended to the public that the existing school choice mechanism in Boston (henceforth the Boston mechanism) should be replaced with an alternative mechanism that removes the incentives to "game the system" that handicapped the Boston mechanism.The mechanism had been used to assign over 75,000 students to school from July 999 until July 2005.2Following Payzant's recommendation, the Boston School Committee voted to replace the mechanism in July 2005 and adopt a new mechanism for the 2005-2006 school year.
The major difficulty with the Boston mechanism is that students may benefit from misrepresenting their preferences over schools.Loosely speaking, the Boston mechanism attempts to assign as many students as possible to their first choice school, and only after all such assignments have been made does it consider assignments of students to their second choices, and so on.If a student is not admitted to her first choice school, her second choice may be filled with students who have listed it as their first choice.That is, a student may fail to get a place in her second choice school that would have been available had she listed that school as her first choice.If a student is willing to take a risk with her first choice, then she should be careful to rank a second choice that she has a chance of obtaining.Some families understand these features of the Boston mechanism and have developed rules of thumb for submitting preferences strategically.For instance, the West Zone Parents Group (WZPG), a well-informed group of approximately 80 members who meet regularly prior to admissions time to discuss Boston school choice for elementary school, recommends two types of strategies to its members.Their introductory meeting minutes on October 27, 2003, state: One school choice strategy is to find a school you like that is undersubscribed and put it as a top choice, OR, find a school that you like that is popular and put it as a first choice and find a school that is less popular for a "safe" second choice.
The Boston mechanism is also widely used throughout several US school districts, including Cambridge, MA, Charlotte-Mecklenburg, NC, Denver, CO, Miami-Dade, FL, Minneapolis, MN, and Tampa-St. Petersburg, FL.Using data on stated choices from the Boston Public Schools from 2000-2004, Atila Abdulkadirog ˘lu et al. (2006) describe several empirical patterns which suggest that there are different levels of sophistication among the families who participate in the mechanism.Some fraction of parents behave as the WZPG suggest and avoid ranking two overdemanded schools as their top two choices.On the other hand, nearly 20 percent of students list two overdemanded schools as their top two choices, and 27 percent of these students are unassigned by the mechanism. 3This empirical evidence, together with the theoretical arguments in Abdulkadirog ˘lu and Sönmez (2003) and the experimental study of Yan Chen and Sönmez (2006), was instrumental in the decision to replace the Boston mechanism with the student-optimal stable mechanism (David Gale and Lloyd Shapley 962).
One of the remarkable properties of the student-optimal stable mechanism is that it is strategy-proof: truth-telling is a dominant strategy for each student.If families have access to advice on how to strategically misrepresent their preferences from groups like the WZPG or through family resource centers, they can do no better than by submitting their true preferences to the mechanism.This feature was an important factor in Superintendent Payzant's recommendation to change the mechanism.The BPS Strategic Planning Team, in their May , 2005, recommendation to implement a new BPS assignment algorithm, emphasized:4 A strategy-proof algorithm "levels the playing field" by diminishing the harm done to parents who do not strategize or do not strategize well.
In this paper, we investigate the intuitive idea that replacing the Boston mechanism with the strategy-proof student-optimal stable mechanism "levels the playing field."To do so, we consider a model with both sincere and sophisticated families,5 analyze the Nash equilibria of the preference revelation game induced by the Boston mechanism (or simply the Nash equilibria of the Boston game), and compare the equilibrium outcomes with the dominant-strategy outcome of the student-optimal stable mechanism.In Proposition , we characterize the equilibrium outcomes of the Boston game as the set of stable matchings of a modified economy where sincere students lose their priorities to sophisticated students.This result implies that there exists a Nash equilibrium outcome where each student weakly prefers her assignment to any other equilibrium assignment.Hence, the Boston game is a coordination game among sophisticated students.
We next examine properties of equilibria.While no sophisticated student loses priority to any other student, some of the sincere students may gain priority at a school at the expense of other sincere students by ranking the school higher on their preference list.As a result, a sincere student may still benefit from the Boston mechanism.In Proposition 2, we show that a sincere student receives the same assignment in all equilibria of the Boston game.
In Proposition 3, we compare the equilibria of the Boston game to the dominant-strategy outcome of the student-optimal stable mechanism.We show that any sophisticated student weakly prefers her assignment under the Pareto-dominant Nash equilibrium outcome of the Boston game over the dominant-strategy outcome of the student-optimal stable mechanism.When only some of the students are sophisticated, the Boston mechanism gives a clear advantage to sophisticated students, provided they can coordinate their strategies at a favorable equilibrium. 6This result might explain why, in testimony from the community about the Boston mechanism, the leader of the WZPG opposed changing the mechanism. 7he position of the WZPG may be interpreted as a desire to maintain their strategic advantage over sincere students under the Boston mechanism.In a model where all students are sophisticated, the set of Nash equilibrium outcomes of the Boston game coincides with the set of stable matchings of the underlying economy (Haluk Ergin and Sönmez 2006). 8This theoretical result would suggest that in a school district with only sophisticated students, a transition to the student-optimal stable mechanism should be embraced by all student groups, for it would be in the best interest of all students.In contrast, Proposition 3 may explain why the WZPG did not embrace the transition.
Our last result, Proposition 4, examines what happens when a sincere student becomes sophisticated.Comparing the Pareto-dominant Nash equilibrium outcomes of the two scenarios, the student in question is weakly better off when she is sophisticated, although each other sophisticated student weakly prefers she remain sincere.
The layout of the paper is as follows.Section I defines the model and Section II characterizes the set of equilibria.Section III presents comparative statics, and Section IV concludes.Finally, the Appendix contains the proofs.

I. The Model
In a school choice problem (Abdulkadirog ˘lu and Sönmez 2003), there are a number of students, each of whom should be assigned a seat at one of a number of schools.Each student has a strict preference ordering over all schools as well as remaining unassigned, and each school has a strict priority ranking of all students.Each school has a maximum capacity.
Formally, a school choice problem consists of: • A set of students i 5 5i , … , i n 6, • A set of schools s 5 5s , … , s m 6, • A capacity vector q 5 1q s , … , q s m 2 , • A list of strict student preferences p i 5 1p i , … , p in 2 , and • A list of strict school priorities p 5 1p s , … , p s m 2.
For any student i, p i is a strict preference relation over s < 5i 6 where sp i i means student i strictly prefers a seat at school s to being unassigned.For any student i, let R i donote the "at least as good as" relation induced by p i .For any school s, the function p s : 5, … , n 6 S 5i , … , i n 6 is the priority ordering at school s where p s 12 indicates the student with highest priority, p s 122 indicates the student with second highest priority, and so on.Priority rankings are determined by the school district and schools have no control over them.We fix the set of students, the set of schools, and the capacity vector throughout the paper; hence the pair 1p, p2 denotes a school choice problem (or simply an economy).
The school choice problem is closely related to the well-known college admissions problem (Gale and Shapley 962).The main difference is that in college admissions each school is a (possibly strategic) agent whose welfare matters, whereas in school choice each school is a collection of indivisible goods to be allocated and only the welfare of students is considered.
The outcome of a school choice problem, as in college admissions, is a matching.Formally, a matching m : i S s < i is a function such that • m 1i 2 o s 1 m 1i 2 5 i for any student i, and • Z m 2 1s 2 Z # q s for any school s.
We refer to m 1i 2 as the assignment of student i under matching m.
A matching m pareto dominates (or is a pareto improvement over) a matching n, if m 1i 2R i n 1i 2 for all i [ i and m 1i 2p i n 1i 2 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 selects a matching for each economy.

A. the Boston student Assignment mechanism
The Boston mechanism is by far the most popular mechanism used in school districts throughout the United States.For any economy, the outcome of the Boston mechanism is determined in several rounds with the following procedure: Round 1.-In Round , only the first choices of students are considered.For each school, consider the students who have listed it as their first choice and assign seats of the school to these students one at a time following their priority order until there are no seats left or there is no student left who has listed it as her first choice.
Round k.-In general, at Round k, consider the remaining students.Only the k th choices of these students are considered.For each school with seats still available, consider the students who have listed it as their k th choice and assign the remaining seats to these students one at a time following their priority order, until there are no seats left or there is no student left who has listed it as her k th choice.
The procedure terminates when each student is assigned a seat at a school.The Boston mechanism induces a preference revelation game among students.We refer to this game as the Boston game.

B. sincere and sophisticated students
We assume that there are two types of students: sincere and sophisticated.Let N, m denote sets of sincere and sophisticated, respectively.We have N < m 5 i and N > m 5 ~.Sincere students simply reveal their preferences truthfully.The strategy space of each sincere student is a singleton under the Boston game.Each sophisticated student, on the other hand, recognizes the strategic aspects of the student assignment process, and the support of her strategy space is all strict preferences over the set of schools, plus remaining unassigned.We focus on the Nash equilibria of the Boston game where only sophisticated students are active players.Each sophisticated student selects a best response to the other students.

C. stability
The following concept, which plays a central role in the analysis of two-sided matching markets, will be useful to characterize the Nash equilibria of the Boston game.
A matching m is stable if: • It is individually rational in the sense that there is no student i who prefers remaining unassigned to her assignment m 1i 2 , and • There is no student-school pair 1i, s 2 such that: S Student i prefers s to her assignment m 1i 2 , and S Either school s has a vacant seat under m or there is a lower priority student j who nonetheless received a seat at school s under m.
Gale and Shapley (962) show that the set of stable matchings is nonempty and there exists a stable matching, the student-optimal stable matching, that each student weakly prefers to any other stable matching.We refer to the mechanism that selects this stable matching for each problem as the student-optimal stable mechanism.Lester E. Dubins and David Freedman (98) and Roth (982) show that truth-telling is a dominant strategy for each student under this mechanism.

D. An illustrative Example
Since a student "loses" her priority to students who rank a school higher in their preferences, the outcome of the Boston mechanism is not necessarily stable.However, Ergin and Sönmez (2006) show that any Nash equilibrium outcome of the Boston game is stable when all students are sophisticated.Based on this result, they have argued that a change from the Boston mechanism to the student-optimal stable mechanism should be embraced by all students, for it will result in a Pareto improvement.This is not what happened in summer 2005 when the Boston Public Schools gave up the Boston mechanism and adopted the student-optimal stable mechanism.A simple example provides some insight into the resistance of sophisticated players to the change of the mechanism.
There are four Nash equilibrium profiles of the Boston game (indicated in boldface), each with a Nash equilibrium payoff of (, 2, 0) and a Nash equilibrium outcome of We have the following useful observations about the equilibria: • Truth-telling, i.e., the profile 1bac, bca, abc 2 , is not a Nash equilibrium of the Boston game.
• Unlike in Ergin and Sönmez (2006), the Nash equilibrium outcome m is not a stable matching of the economy 1p, p2.The sincere student i 3 not only prefers school b to her assignment m 1i 3 2 5 c but also has the highest priority there.Nevertheless, by being truthful and ranking b second, she has lost her priority to student i 2 at equilibrium.
• The unique stable matching of the economy 1p, p2 is Matchings m and n are not Pareto ranked.While the sophisticated student i is indifferent between the two matchings, the sophisticated student i 2 is better off under matching m and the sincere student i 3 is better off under matching n.That is, the sophisticated student i 2 is better off under the Nash equilibria of the Boston game at the expense of the sincere student i 3 .
We next characterize the Nash equilibrium outcomes of the Boston game, which will be useful to generalize the observations above.

A. An Augmented Economy
Given an economy 1p, p2 , we will construct an augmented economy that will be instrumental in describing the set of Nash equilibrium outcomes of the Boston game.Given an economy 1p, p2 and a school s, partition the set of students i into m sets as follows: Given an economy 1p, p2 and a school s, construct an augmented priority ordering p ˜ s as follows: • Each student in i s has higher priority than each student in i 2 s , each student in i 2 s has higher priority than each student in i 3 s , … , each student in i s m2 has higher priority than each student in i s m , and • For any k # m, priority among students in i k s is based on p s .
Define p ˜ 5 1p ˜ s 2 s[s .We refer the economy 1p, p ˜ 2 as the augmented economy.
In the augmented economy, the priorities at each school are adjusted so that sincere students who rank a school as their second choice or lower are ordered following all sophisticated students.The augmented priorities will reflect the fact that in the Boston mechanism these students will receive a lower priority because they do not rank the school as their top choice.
Example 1 (continued).Let us construct the augmented economy for Example .Since only student i 3 is sincere, p ˜ is constructed from p by pushing student i 3 to the end of the priority ordering at each school except her top choice a (where she has the lowest priority to begin with): The key observation is that the unique Nash equilibrium outcome m of the Boston game is the unique stable matching for the augmented economy 1p, p ˜ 2.
While the uniqueness is specific to the example above, the equivalence is general.We are ready to present our first result.
PROPOSITION : the set of Nash equilibrium outcomes of the Boston game under 1p, p2 is equivalent to the set of stable matchings under 1p, p ˜ 2 .This is a generalization of the main result of Ergin and Sönmez (2006), who show that the set of Nash equilibrium outcomes of the Boston game is equal to the set of stable matchings of the original economy when all students are sophisticated.In this case, any stable matching m can be sustained at equilibrium when each student strategically ranks m 1i 2 as her top choice.Moreover, any equilibrium outcome is stable for otherwise student j of a blocking pair 1 j, s 2 can have a profitable deviation by ranking s as her top choice.When only a subset of students are sophisticated, a sincere student i loses the ability to rank her assignment m 1i 2 at a stable matching m as her first choice.She also loses the ability to engage in a profitable deviation when she is in a blocking pair of an unstable matching.Hence, a version of Ergin and Sönmez (2006) holds where sincere students lose priority to sophisticated students.
A key implication of Proposition is that sophisticated students gain priority at the expense of sincere students at Nash equilibria.Another implication is that the set of equilibrium outcomes inherits some of the properties of the set of stable matchings.In particular, there is a Nash equilibrium outcome of the Boston game that is weakly preferred to any other Nash equilibrium outcome by all students.We refer to this as the pareto-dominant Nash equilibrium outcome.Therefore, the Boston game is a coordination game among sophisticated students.

B. Equilibrium Assignments of sincere students
The student-optimal stable mechanism replaced the Boston mechanism in Boston in 2005.In the following section we will compare the equilibrium outcomes of the Boston game with the dominant-strategy equilibrium outcome of the student-optimal stable mechanism.One of the difficulties in such comparative static analysis is that the Boston game has multiple equilibria in general.Nevertheless, as we now discuss, multiplicity is not an issue for sincere students.9PROPOSITION 2: Let m, n be both Nash equilibrium outcomes of the preference revelation game induced by the Boston mechanism.for any sincere student i [ N, m 1i 2 5 n 1i 2 .What makes multiple stable matchings possible in school choice is a possible "conflict" between school priorities and student preferences (e.g., student i has higher priority at school a than student j, student j has higher priority at school b than student i, but student i prefers b to a and student j prefers a to b.)Under the augmented priorities, sincere students are never involved in such conflicts: they have lower priority than sophisticated students and among sincere students a school gives higher priority to the sincere student who ranks it higher in her preferences.That is why a sincere student always receives the same assignment in any Nash equilibrium.

A. Comparing mechanisms
The outcome of the student-optimal stable mechanism can be obtained with the following student-proposing deferred acceptance algorithm (Gale and Shapley 962): Step 1: Each student proposes her first choice.Each school tentatively assigns its seats to its proposers one at a time following their priority order.Any remaining proposers are rejected.
Step k: In general, at this step, each student who was rejected in the previous step proposes to her next choice.Each school considers the students it has been holding along with its new proposers and tentatively assigns its seats to these students one at a time following their priority order.Any remaining proposers are rejected.
The algorithm terminates when no student proposal is rejected and each student is assigned her final tentative assignment.Any student who is not holding a tentative assignment remains unassigned.
Sincere students lose priority to sophisticated students under the Boston mechanism.They may also 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 precisely, a sincere student may prefer the Boston mechanism to the student-optimal stable mechanism since: • She gains priority at her first choice school over sincere students who rank it second or lower, and, in general, • She gains priority at her k th choice school over sincere students who rank it 1k 1 2 th or lower, etc.
Example 2. There are three schools, a, b, c, each with one seat and three sincere students, i , i 2 , i 3 .Preferences and priorities are as follows: Outcomes of the Boston mechanism and the student-optimal stable mechanism are respectively.Under the Boston mechanism, the sincere student i 3 gains priority at her top choice school 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-optimal stable mechanism.Unlike a sincere student, a sophisticated student may be assigned seats at different schools at different equilibrium outcomes of the Boston game.We first concentrate on the Pareto-dominant Nash equilibrium outcome of the Boston game.Since this equilibrium outcome is the studentoptimal stable matching of the augmented economy, it is not surprising that a sophisticated student weakly prefers it to the student-optimal stable matching of the original economy.This is what is shown in our next result.

PROPOSITION 3: the school a sophisticated student receives in the pareto-dominant equilibrium of the Boston mechanism is weakly better than her dominant-strategy outcome under the student-optimal stable mechanism.
A theoretical setback for Proposition 3 is that it does not extend to all Nash equilibria.Our next example makes this point.
Example 3.There are two schools, a, b, each with one seat and two sophisticated students, i , i 2 .Preferences and priorities are as follows: pAthAk ANd sÖNmEz: LEVELiNg thE pLAyiNg fiELd Since both students are sophisticated, the augmented economy 1p, p ˜ 2 is the same as the original economy 1p, p2.Hence, 5m, n6 is the set of stable matchings for both economies where Therefore, while m is the dominant strategy outcome of the student-optimal stable mechanism, both m and n are Nash equilibrium outcomes of the Boston game.Hence, both students strictly prefer the dominant strategy outcome of the student-optimal stable mechanism (i.e., matching m) to one of the Nash equilibria of the Boston game (i.e., matching n ).
The case for Proposition 3 would have been stronger if the Boston game had a unique equilibrium.As we show in Example 3, this is not necessarily the case.There is, however, evidence in the literature that suggests that the size of the set of stable matchings may be very small in real-life applications of college admissions problems.Using data for years 99-994 and 996 for the thoracic surgery market, Roth and Elliot Peranson (999) have shown that there are two stable matchings each for years 992 and 993, and one stable matching each for 99, 994, and 996.One caveat of these computational experiments is that the thoracic surgery market used the hospital-optimal stable mechanism in these years, and truth-telling is not a dominant strategy for interns or for hospitals under this mechanism.So it is theoretically possible that the small number of stable matchings is an implication of preference manipulation.The same computational exercise is on firmer ground for school years 2005-2006 and 2006-2007 for Boston Public School student admissions, where the student-optimal stable mechanism was used (which is strategy-proof in the context of school choice).The results of these computational experiments are very similar to those of Roth and Peranson: at grade K2, the main entry to elementary school, for school years 2005-2006 and 2006-2007 (each with more than 2,800 students), there is only one stable matching for either year.At grade 6, the situation is not very different.For school year 2005-2006 there are only two stable matchings, and among more than 3,200 students only two are affected by the choice of a stable matching.For school year 2006-2007 there are also two stable matchings, and among more than 2,900 students only three are affected by the choice of a stable matching.The reason this is happening is that, for most students, the factors that give a student higher priority at a Boston school (i.e., proximity and the presence of a sibling) also make that school more desirable for the student.
These computational experiments suggest that while the existence of multiple equilibria is a theoretical possibility under the Boston game, it likely affects a very small minority of students.As we show in Proposition , the set of Nash equilibrium outcomes is equal to the set of stable matchings of an augmented economy where sincere students lose priority to sophisticated students.Using data for school years 2005-2006 and 2006-0207 and admission to grade K2 and grade 6, we ran computational experiments by randomly setting 20 percent of students to be sincere and the rest to be sophisticated.We calculated the student-optimal stable matching and the school-optimal stable matching for the resulting augmented economy and repeated the same exercise ,000 times to calculate how many students are affected on average by the multiplicity of the Nash equilibria.We repeated the same experiment for the cases where 40 percent, 60 percent, and 80 percent of the students are sincere, respectively.Table summarizes the results of our computational experiment.Most of the time the augmented economy has a unique stable matching and, more specifically, no more than 0.38 students (less than 0.03 percent of students) are affected on average by the multiplicity of the Nash equilibria in each of the treatments.Hence, while Proposition 3 does not theoretically extend to all equilibria, the computational experiments suggest that multiplicity is not a significant problem in our application.

B. Becoming sophisticated
Our final result concerns a sincere student i who becomes sophisticated.While student i weakly benefits from this transition under the Pareto-dominant Nash equilibrium of the Boston game, students who have been sophisticated weakly suffer.
To state this result, we must define additional notation.First, fix an economy 1p, p2.Let m , i be the set of sophisticated students and N be the set of sincere students.Next, consider an initially sincere student i [ N and suppose she becomes sophisticated.Let m 2 5 m < 5i 6 be the set of sophisticated students including i, and let N 2 5 N \ 5i 6 be the set of remaining sincere students.
Let n be the Pareto-dominant Nash equilibrium of the Boston game where m and N are the sophisticated and sincere players, respectively.Let m be the Pareto-dominant Nash equilibrium of the Boston game where m 2 and N 2 are the sophisticated and sincere players, respectively.PROPOSITION 4: Let i, m , n, m be as described above.student i weakly benefits from becoming sophisticated in the pareto-dominant Nash equilibrium of the Boston game, whereas all other sophisticated students weakly suffer.that is, This proposition suggests that groups such as the West Zone Parents Group do not exist only to share information on how to become strategic, because educating a sincere player will not benefit an existing sophisticated player.Rather, this proposition suggests that the theoretical function of the West Zone Parents Group may be to share information and coordinate behavior among the sophisticated players.

IV. Conclusion
Boston Public Schools stated that one of their main rationales for changing their student assignment system is that it levels the playing field.They identified a fairness rationale for a strategy-proof system.In this paper, we examined this intuitive notion and showed that the Boston mechanism favors sophisticated parents at Pareto-dominant Nash equilibrium, providing formal support for BPS's position.
Despite its theoretical weaknesses, performance in laboratory experiments, and empirical evidence of confused play, the Boston mechanism is the most widely used school choice mechanism in the United States.It is remarkable that such a flawed mechanism is so widely used.John E. Chubb and Terry M. Moe (999) argue that important stakeholders often control the mechanisms of reform in education policy.In the context of student assignment mechanisms, the important stakeholders may be sophisticated parents who have invested energy in learning about the mechanism, and the choice of the Boston mechanism may reflect their preferences.
Appendix: Proofs PROOF OF PROPOSITION : B (Any stable matching under 1p, p ˜ 2 is an equilibrium outcome of the Boston game under 1p, p2 2: Fix an economy 1p, p2 and let m be stable under 1p, p ˜ 2. Let preference profile Q be such that Q i 5 p i for all i [ N and m 1i 2 is the first choice under Q i for all i [ m. Matching m is stable under 1Q, p ˜ 2 as well.Let n be the outcome of the Boston mechanism under 1Q, p2.We first show, by induction, that n 5 m.
Consider any student j who does not receive her first choice s j under Q at matching m.By construction of Q, student j is sincere.Since m is stable under 1Q, p ˜ 2 and since student j does not lose priority to any student at school s j when priorities change from p to p ˜ , she has lower priority under p s j than any student who has received a seat at s j under m.Each of these students ranks s j as their first choice under Q and school s j does not have empty seats under m, for otherwise 1 j, s 2 would block m under 1Q, p ˜ 2. Therefore, n 1 j 2 Z s j .So a student can receive her first choice under Q at matching n only if she receives her first choice under Q at matching m.But then, since the Boston mechanism is Pareto efficient, matching n is Pareto efficient under 1Q, p2 , which in turn implies that n 1i 2 5 m 1i 2 for any student i who receives her first choice under Q at matching m.
Next, given k ., suppose: (i) Any student who does not receive one of her top k choices under Q at matching m does not receive one of her top k choices under Q at matching n either, and (ii) For any student i who receives one of her top k choices under Q at matching m, n 1i 2 5 m 1i 2.
We will show that the same holds for 1k 1 2 , and this will establish that n 5 m.Consider any student j who does not receive one of her top k 1 choices under Q at matching m.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 n.Consider 1k 1 2 th choice s j k1 of student j under Q j .Since m is stable under 1Q, p ˜ 2 , there is no empty seat at school s j k1 , for otherwise pair 1 j, s j k1 2 would block matching m under 1Q, p ˜ 2.Moreover, since m is stable under 1Q, p ˜ 2 , for any student i with m 1i 2 5 s j k1 one of the following three cases should hold: . i [ m and by construction s j k1 is her first choice under Q i , 2. i [ N and s j k1 is one of her top k choices under Q i , 3. i [ N, she has ranked s j k1 as her 1k 1 2 th choice under Q i , and she has higher priority than j under p s j k1 .
If either of the first two cases holds, then n 1i 2 5 s j k1 by the inductive assumption.If Case 3 holds, then student i has not received one of her top k choices under Q i at matching n by the inductive assumption, and furthermore she has ranked school s j k1 as her 1k 1 2 th choice under Q i .Since she has higher priority than j under p s j k1 , n 1 j 2 5 s j k1 implies n 1i 2 5 s j k1 . Therefore, considering all three cases, n 1 j 2 5 s j k1 implies n 1i 2 5 s j k1 for any student i with m 1i 2 5 s j k1 , and since school s j k1 does not have empty seats under m, we must have n 1 j 2 Z s j k1 . So a student can receive one of her top k 1 choices under Q at matching n only if she receives one of her top k 1 choices under Q at matching m.Moreover, matching n is Pareto efficient under 1Q, p2 and therefore n 1i 2 5 m 1i 2 for any student i who receives her 1k 1 2 th choice under Q i at m completing the induction and establishing n 5 m.
Next, we show that Q is a Nash equilibrium profile and hence n is a Nash equilibrium outcome.Consider any sophisticated student i [ m and suppose sp i n 1i 2 5 m 1i 2 for some school s [ s.
Since n 5 m is stable under 1Q, p ˜ 2 and since student i gains priority under p ˜ s over only students who rank s second or worse under Q, not only does any student j [ i with n 1 j 2 5 s rank school s as her first choice under Q i , but she also has higher priority under p s .Therefore, regardless of what preferences student i submits, each student j [ i with n 1 j 2 5 s will receive a seat at school s.Moreover by stability of n 5 m under 1Q, p ˜ 2 , there are no empty seats at school s and hence student i cannot receive a seat at s regardless of her submitted preferences.Therefore, matching n is a Nash equilibrium outcome.
1 (Any equilibrium outcome of the Boston game under 1p, p2 is a stable matching under 1p, p ˜ 2 2: Suppose matching m is not stable under 1p, p ˜ 2. Let Q be any preference profile where Q i 5 p i for any sincere student i and where m is the outcome of the Boston mechanism under 1Q, p2.We will show that Q is not a Nash equilibrium strategy profile of the Boston game under 1p, p2.
First, suppose m is not individually rational under 1p, p ˜ 2.Then, there is a student i [ i with ip i m 1i 2. Since the Boston mechanism is individually rational, student i should be a sophisticated student who has ranked the unacceptable school m 1i 2 as acceptable.Let p i 0 be a preference relation where there is no acceptable school.Upon submitting p i 0 , student i will profit by getting unassigned.Hence, Q cannot be an equilibrium profile in this case.
Next, suppose there is a pair 1i, s 2 that blocks m under 1p, p ˜ 2. Since m is the outcome of the Boston mechanism under 1Q, p2 , student i cannot be a sincere student.Let p i s be a preference relation where school s is the first choice.We have two cases to consider: Case 1: School s has an empty seat at m.
Recall that by assumption m is the outcome of the Boston mechanism under 1Q, p2.Since s has an empty seat at m, there are fewer students who rank s as their first choice under Q than the capacity of school s.Therefore, upon submitting the preference relation p i s , student i will profit by getting assigned a seat at school s.Hence, Q cannot be an equilibrium profile.
Case 2: School s does not have an empty seat at m.
By assumption m is the outcome of the Boston mechanism under 1Q, p2 and there is a student j with m 1 j 2 5 s, although i has higher priority than j under p ˜ s .If school 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 school s, and upon submitting the preference relation p i s , student i will profit by getting assigned a seat at school s contradicting Q being an equilibrium profile.If, on the other hand, school 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 p ˜ s implies i having higher priority than j under p s .Moreover, since m 1 j 2 5 s, the capacity of school s is strictly larger than the number of students who rank it as their first choice under Q while having higher priority than j under p s .Therefore, the capacity of school s is strictly larger than the number of students who Similarly, and, similarly, Combining the inequalities, we obtain establishing the desired contradiction.Hence, there exists no i [ N with m -1i 2 Z m -1i 2. But that means there exists no i [ N with m 1i 2 Z n 1i 2 , completing the proof.
The following lemma will be useful to prove Proposition 3 and Proposition 4. We need the following piece of notation to present this lemma.Given a preference profile p and a school s, let f s 1p2 denote the set of students who rank school s as their first choice under p.
LEMMA : fix a preference profile p, a list of priorities p, and a set of students J , i. Let priorities s be such that, for any school s: (i) Any student in J < f s 1p2 has higher priority under s s than any student in i \ 1J < f s 1p2 2 , and (ii) for any student j [ J < f s 1p2 and any student i [ i, if j has higher priority than i under p s , then j also has higher priority than i under s s .
Let m, n be the student-optimal stable matching for economies 1p, p2 , 1p, s2 , respectively.then: PROOF OF LEMMA : Let m, n be the outcomes of the student-proposing deferred acceptance algorithm for economies 1p, p2 , 1p, s2 , respectively.Assume by way of contradiction that there exists a student j [ J such that m 1 j 2p j n 1 j 2.Then, in the execution of the student-proposing deferred acceptance algorithm for economy 1p, s2 , there exists a student who is rejected by her assignment under matching m.Let j * be the first such student in set J and suppose he is rejected by school m 1 j * 2 at Step k * of the algorithm.But this rejection means that there exists a student j9 who has higher priority at school m 1 j * 2 under s than student j * , and does not propose to school m 1 j * 2 in the algorithm for economy 1p, p2 , but has proposed to school m 1 j * 2 in the algorithm for economy 1p, s2.This implies that m 1 j92p j9 m 1 j * 2R j9 n 1 j92 and so student j9 has been rejected from school m 1 j92 before Step k * in the execution of the student-proposing deferred acceptance algorithm for economy pAthAk ANd sÖNmEz: LEVELiNg thE pLAyiNg fiELd 1p, s2.Recall that we assumed j * is the first such student in set J. Hence, all that remains to obtain the desired contradiction is to show j9 [ J.
Since j9 has higher priority at school m 1 j * 2 under s than student j * (who is a member of set J), we must have j9 [ J < f m 1 j* 2 1p2.But since j9 does not propose to school m 1 j * 2 in the algorithm for economy 1p, p2 , it cannot be her first choice 1i.e., j9 o f m 1 j* 2 1p2.2Hence, j9 [ J, resulting in the desired contradiction.

PROOF OF PROPOSITION 3:
Let m, n be the student-optimal stable matching for economies 1p, p2 , 1p, p ˜ 2 , respectively.We have to show that n 1 j 2R j m 1 j 2 for any j [ m.For any school s, the priority order p ˜ s is such that: (i) Any student in m < f s 1p2 has higher priority under p ˜ s than any student in i \ 1m < f s 1p2 2 , and (ii) For any student j [ m < f s 1p2 and any student i [ i, if j has higher priority than i under p s , then j also has higher priority than i under p ˜ s .
Therefore, n 1 j 2R j m 1 j 2 for any j [ m by Lemma .
PROOF OF PROPOSITION 4: Fix an economy 1p, p2 , and let m , i be the set of sophisticated students and N be the set of sincere students.Next consider an initially sincere student i [ N and suppose she becomes sophisticated.Let m 2 5 m < 5i 6 be the set of sophisticated students including i, and let N 2 5 N \ 5i 6 be the set of remaining sincere students.Let n be the Pareto-dominant Nash equilibrium of the Boston game where m and N are the sophisticated and sincere players, respectively.Let m be the Pareto-dominant Nash equilibrium of the Boston game where m 2 and N 2 are the sophisticated and sincere players, respectively.Let p ˜ be the augmented priority ordering when m is the set of sophisticated students, and let p ˜ 2 be the augmented priority ordering when m 2 is the set of sophisticated students.By Proposition , n is the student-optimal stable matching for economy 1p, p ˜ 2 and m is the student-optimal stable matching for economy 1p, p ˜ 2 2.
By the construction of the augmented priorities, student i does not lose priority to any student when priorities change from p ˜ to p ˜ 2 , while priorities between other students remain the same between p ˜ and p ˜ 2 .Therefore, m 1i 2 R i n 1i 2 immediately follows from Michel Balinski and Sönmez (999).Moreover, by construction of the augmented priorities, for any school s: (i) Any student in m < f s 1p2 has higher priority under p ˜ s than any student in i \ 1m < f s 1p2 2 , and (ii) For any student j [ m < f s 1p2 and any other student h [ i, if j has higher priority than h under p ˜ s 2 , then j also has higher priority than h under p ˜ s .
Therefore, n 1 j 2 R j m 1 j 2 for any j [ m by Lemma .

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Sincere students who rank s as their first choices under p and all sophisticated students, thE AmERiCAN ECONOmiC REViEW i 2 s : Sincere students who rank s as their second choices under p, i 3 s : Sincere students who rank s as their third choices under p, ( ( i m s : Sincere students who rank s as their last choices under p.

Table - Average
Number of Students Receiving Different Schools in Student-Optimal versus School-Optimal Matching Note: This table is based on data provided by Boston Public Schools for Round of their admissions process in 2005-2006 and 2006-2007.pAthAk ANd sÖNmEz: LEVELiNg thE pLAyiNg fiELd