The Modified Boston Mechanism∗ Umut Mert Dur† North Carolina State University April 7, 2015

Abstract Many school districts in the U.S. assign students to schools via the Boston mechanism. The Boston mechanism is not strategy-proof, and it is easy to manipulate. We slightly modify the Boston mechanism and show that the modified version outperforms the Boston mechanism in terms of strategy-proofness. In particular, the Boston mechanism is manipulable whenever the modified version is, but the modified version is not necessarily manipulable whenever the Boston mechanism is. We define a weaker form of consistency and characterize the modified Boston mechanism by this weaker form consistency and a new axiom called respect of priority of the top ranking students. JEL Classification: C78, D61, H75, I28 Key Words: Matching Theory, Boston Mechanism, School Choice

1

Introduction

Starting from Abdulkadiro˘ glu and S¨ onmez (2003), economists have pointed out the major deficiency of the Boston mechanism (BM): BM creates incentives for students to misreport their true preferences.1 Abdulkadiro˘glu and S¨onmez (2003) propose two strategy-proof mechanisms2 and suggest that school districts adopt one of these mechanisms. Thanks to the following experimental and theoretical papers focusing on this drawback of the BM, it was replaced with a strategy-proof mechanism in Boston and Chicago. However instead of abandoning BM, many school districts in US still assign students to the schools via BM.3 Moreover, BM was first replaced with a strategy-proof mechanism in Seattle but then BM was again adopted in ¨ Seattle in 2011 (Kojima and Unver, 2013). Given the tendency of school districts towards using BM, a natural question is whether we can improve the performance of the BM with minor changes. In this paper, we show that with a slight modification it ¨ I owe very special thanks to Thayer Morrill and Utku Unver for comments and suggestions. I would like to also thank Ay¸se Kabuk¸cuo˘ glu. † e-mail: [email protected]; web page: https://sites.google.com/site/umutdur/ 1 Pathak (2011) reviews the school choice literature including the works focusing on BM. 2 A mechanism is strategy-proof if any student cannot be better off when he misreports his true preference over the schools. These mechanisms are deferred acceptance (DA) and top trading cycles (TTC) mechanism. 3 Cambridge, Denver, Lee County, Seattle and Minneapolis are just a few districts currently using BM mechanism. ∗

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is possible to decrease the level of gaming under BM. In round k of BM the unassigned students apply to their k th choice school in their preference list. This may cause students to apply to a school which has filled all of its available seats in the previous rounds or a school that considers them unacceptable. Hence, these students waste their opportunity to be assigned in that round by applying to a school that will not accept them independent of the students applying to that school in that round. To fix this point in each round we allow a student to apply to her most preferred school among the schools that consider her acceptable and have available seats. The modified version of BM inherits desired features of the BM. It is constrained Pareto efficient and satisfies a weaker form of consistency4 (Theorem 1). This slight modification does not eliminate all the possible gaming. That is, the modified version of the BM is not strategy-proof. However, by using the notion introduced by Pathak and S¨ onmez (2013) we show that the modified version performs better than the BM in terms of vulnerability to manipulation (Theorem 2). In particular, whenever a student can manipulate the modified BM then BM can be manipulated by some students. Moreover, there exist problems in which the modified BM cannot be manipulated but BM can. In addition, any Nash equilibrium outcome of the preference revelation game associated with the modified BM leads to a stable matching under the true preferences (Theorem 3). We demonstrate except for the DA mechanism, all other members of the application rejection mechanisms introduced by Chen and Kesten (2013) do not perform better than the modified BM in terms of immunity to manipulation (Proposition 1). To our knowledge, there is no known use of the modified version. This modification can be thought of as a design that improves upon BM. We also show that the modified BM may outperform the BM if we consider the number of students assigned to their top k ≥ 3 choices. Moreover, the modified BM performs better than the DA mechanism based on the number of students assigned to their reported first choice under naive submission (Proposition 2). In addition to the standard school choice model, we study an environment composed of sincere and sophisticated student (Section 4.1). We show that all sophisticated students (weakly) prefer the Pareto dominant Nash equilibrium outcome of the preference revelation game induced by BM to the one induced by modified BM (Theorem 4). We provide a characterization of the modified version of the BM. We characterize the modified BM by using two axioms: weak consistency and a new axiom that we introduce in this paper (Theorem 5). We weaken the consistency axiom by only requiring the remaining students to be assigned to the same school whenever all the students assigned to their most preferred schools among the ones with available seats and considering them acceptable are removed along with their assignment. The weak consistency is satisfied by any consistent mechanism. Moreover, the DA mechanism, which does not satisfy consistency, is weak consistent. The axiom that we introduce is similar to the main axioms used in the characterization of BM ¨ ¨ by Kojima and Unver (2013) and Afacan (2013). Kojima and Unver (2013) and Afacan (2013) introduce 4

A mechanism is consistent if whenever a set of students are removed along with their assignment, then the assignments of the remaining students will be the same when we rerun the mechanism considering the remaining students and school seats.

2

respect of preference rankings 5 and respect of both preference rankings and priorities 6 , respectively. We say that a mechanism respects the priority of the top ranking students if whenever student i is one of the top q ranked students in the priority order of school s among the acceptable students considering s as the most preferred school in the set of schools with at least one available seat, then i is assigned to s by the mechanism.7 Respect of the priority of top ranking students is neither weaker nor stronger than the axioms ¨ introduced by Kojima and Unver (2013) and Afacan (2013). For instance, BM respects both preference rankings and priorities however it does not respect the priority of top ranking students (Example 2). On the other hand, the modified BM respects the priority of top ranking students (Theorem 1) but it does not respect both preference rankings and priorities (Example 1). Alcalde (1996) introduces a version of BM called Now-or-Never mechanism in marriage market. Similar to modified BM introduced here, under the Now-or-Never mechanism in each round each agent applies to the most preferred partner among the remaining ones. However, agents might still apply to partners that consider them unacceptable. He showed that Now-or-Never mechanism can implement all the stable allocations in undominated Nash equilibria. Different from Alcalde (1996), this paper focuses on the many to one matching problems and compares two mechanism based on their vulnerability to manipulation and gives a characterization of the modified BM. In the school choice environment, Miralles (2008) mentions that if students do not apply to schools without remaining seats then sincere students will be protected more against to the strategic students and they will be better off compared to the BM. This version of the BM is named as Corrected Boston mechanism. Miralles (2008) uses simulations to illustrate this point. In this paper, by using an environment composed of sincere and strategic agents we show that strategic students weakly prefer the Pareto dominant Nash equilibrium outcome of the BM to the Pareto dominant Nash equilibrium outcome of the modified BM (Theorem 4). Mennle and Seuken (2014) generalize the Now-orNever mechanism in a many to one matching market settings without priorities and call it Adaptive Boston mechanism. They compare its properties with the BM. Different from this paper, they do not provide a characterization for the Now-or-Never mechanism in their environment and an equilibrium analysis is not given in their paper. All these mechanisms were developed independently but it is worth mentioning that the modified BM is less manipulable than the others (Proposition 1). This paper is related to the literature comparing mechanisms based on their vulnerability to the manipulations. The notion used to compare two mechanisms which are not strategy-proof is introduced by Pathak and S¨onmez (2013). In their paper, they show that applications of this notion are related to the some of the school choice reforms and to the changes in the auction mechanism for US Treasury bonds. Chen and Kesten (2013) use this notion to compare the vulnerability to manipulation of a class of mechanisms including BM, DA and Shanghai mechanism. This paper is related also to the works on the axiomatic characterizations of 5

A mechanism respects preference rankings if whenever student i prefers school s to his assignment under the mechanism, then all the seats of s are filled by students who rank s at least as high as student i. 6 A mechanism respects both preference rankings and priorities if whenever student i prefers school s to his assignment under the mechanism, then all the seats of s are filled by students who rank s at least as high as him, and in the case of equal ranking, each of students has higher priority for s than i. 7 Here, q is the number of available seats in s.

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the matching mechanisms. By using respecting preference rankings, consistency, resource monotonicity8 and ¨ an invariance property Kojima and Unver (2013) characterize the set of mechanisms coinciding with BM for some priority order. Afacan (2013) shows that BM is the unique individually rational mechanism which also respects both preference rankings and priorities. There are also papers characterizing other school choice mechanisms. Kojima and Manea (2010) and Morrill (2013) characterize the DA mechanism. Abdulkadiro˘ glu and Che (2010), Morrill (2012) and Dur (2012) provide the characterization of TTC mechanism.

2

Model

A school choice problem is a list [I, S, q, P, ] where • I = {i1 , i2 , ..., in } is the set of students, • S = {s1 , s2 , ..., sm } is the set of schools, • q = (qs )s∈S is the quota vector where qs is the number of available seats in school s, • P = (Pi )i∈I is the preference profile where Pi is the strict preference of student i over the schools including no-school option, • = (s )s∈S is the priority profile where s is the priority relation of school s over I and vacant position denoted by ∅.9 The no-school option is denoted with s∅ and qs∅ = |I|. Let Ri be the at-least-as-good-as relation associated with the strict preference order Pi and for all s, s0 ∈ S ∪ {s∅ } sRi s0 if and only if s = s0 or sPi s0 . We assume that there are no ties in the priority profiles of schools. We say a student i is acceptable for s if i s ∅. Let S+ [I, S, q, P, ] be the set of schools with at least one available seat in [I, S, q, P, ]. That is, S+ [I, S, q, P, ] = {s ∈ S ∪ {s∅ }|qs ≥ 1}. In the rest of the paper we fix the set of students, schools and priority orders and define a problem with [q, P ]. A matching is a function µ : I → S ∪ {s∅ } such that µ(i) = s and µ(i) = s0 if only if s = s0 . If µ(i) = s∅ then student i is unassigned. In a matching µ, the number of students assigned to a school s cannot exceed qs . Let M be the set of all possible matchings. A mechanism is a procedure which selects a matching for each problem. The matching selected by mechanism φ in problem [q, P ] is denoted by φ [q, P ]. Let φi [q, P ] denote the assignment of student i ∈ I by mechanism φ in problem [q, P ]. A matching µ is individually rational if µ(i)Ri s∅ for all i ∈ I and j s ∅ for all j ∈ µ−1 (s) and s ∈ S. A matching µ is non-wasteful if there does not exist any student-school pair (i, s) such that sPi µ(i), i s ∅ and |µ−1 (s)| ≤ qs . A matching µ is fair if there does not exist any student-school pair (i, s) such that sPi µ(i) 8

A mechanism is resource monotonic if any student is not assigned to a worse school when the number of available seats in each school weakly increases. It is easy to show that modified version of BM satisfies resource monotonicity but since we do not use resource monotonicity in our other results its proof is not included to the text. 9 ¨ Similarly, Kojima and Unver (2013) and Afacan (2013) define priority order over the set of students and vacant position.

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and i s j for some j ∈ µ−1 (s). A matching µ is stable if it is individually rational, non-wasteful and fair. A mechanism φ is stable if for all problems it selects a stable matching. Student i strictly prefers matching µ to matching µ0 if he strictly prefers µ(i) to µ0 (i), µ(i)Pi µ0 (i). A matching µ Pareto dominates matching ν if µ(i)Ri ν(i) for all i ∈ I and µ(i)Pi ν(i) for some i ∈ I. A matching µ is constrained Pareto efficient if there does not exist a matching ν ∈ M such that ν(i)Ri µ(i) for all i ∈ I, ν(i)Pi µ(i) for some i ∈ I and j s ∅ for all j ∈ ν −1 (s) and s ∈ S.10 A mechanism φ is constrained Pareto efficient if for all problems it selects a constrained Pareto efficient matching. A mechanism φ is strategy-proof if it is (weakly) dominant strategy for all students to tell their preferences truthfully. Formally, a mechanism φ is strategy-proof if for every preference profile P and Pi0 φi [q, P ] Ri φi [q, (Pi0 , P−i )] for all student i ∈ I. Here, P−i represents the true preference profile of students in I \ {i}. In addition to these frequently used axioms we define two more axioms that we use in Section 5. Before giving the formal definition of these axioms we introduce some notions. We say a school s is available for student i if qs > 0 and i s ∅. Let bi [q, P ] be the most preferred available school for i in problem [q, P ]. We denote the set of students whose most preferred available school is s in problem [q, P ] by Ts [q, P ]. Let ms : N → Ts [q, P ] be a function and ms (k) be the student who has the k th highest priority for s among the students in Ts [q, P ] according to the priority order s . That is, ms (1) has the highest priority for school s among the students in Ts [q, P ], i.e. ms (1) s x for all x ∈ Ts [q, P ] \ {ms (1)}. Let I˜ be a subset of the students such that each i ∈ I˜ is assigned to bi [q, P ] by mechanism φ. We say mechanism φ is weakly consistent if whenever we remove all agents in I˜ along with their assignments then φ assigns the remaining agents to the same school in the reduced problem obtained after the removal of the agents in I˜ with their assignments. For any i ∈ I let P ∅ be the set of preference relations that rank s∅ as the first choice. A mechanism φ is weakly consistent if φj [q 0 , (PI˜∅ , P−I˜)] = φj [q, P ] for all j ∈ / I˜ and ˜ P ∈ P |I| and P ∅ ∈ P ∅ where I˜ = {i ∈ I : φi [q, P ] = bi [q, P ]} and φi [q 0 , (PI˜∅ , P−I˜)] = s∅ for all i ∈ I, qs0 = qs − |{i ∈ I˜ : φi [q, P ] = s}|. Weak consistency is a weaker form of the consistency axiom used in the ¨ ¨ characterization of the BM by Kojima and Unver (2013). In particular, Kojima and Unver (2013) do not ˜ have any restriction on set I. Hence, any consistent mechanism (i.e., BM and serial dictatorship mechanism) satisfies weak consistency. Moreover, DA mechanism which is not consistent satisfies weak consistency. s A matching µ respects priority of top-ranking students if µ(i) = s for all i ∈ ∪qk=1 ms (k). A mechanism φ respects priority of top-ranking students if φ[q, P ] respects priority of top ranking students for any problem [q, P ]. Note that since we are only considering the most preferred schools among the ones with available seats, respecting priority of top-ranking students is not stricter condition than either respecting preference ¨ rankings (Kojima and Unver, 2013) or respecting both preference rankings and priorities (Afacan, 2013). 10

¨ Kojima and Unver (2013) use the same efficiency notion.

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3

Boston Mechanism vs. Modified Boston Mechanism

In this section we give the outline of BM (Abdulkadiro˘glu and S¨onmez, 2003) and introduce the modified version of BM. BM is a direct mechanism and for any given problem [q, P ] it works iteratively in a number of rounds:11 Boston Mechanism (BM): Round 1: Each student applies to his top ranked school in S ∪ {s∅ }. For each school s, consider the acceptable students who have applied to it in this round and assign seats of the school s to these students one at a time following their priority order s until either there is no seat left or there is no student left who has applied to it in this round. Remove all assigned students. In general, Round k: Each remaining student applies to his k th choice school in S ∪ {s∅ }. For each school s, consider the acceptable students who have applied to it in this round and assign seats of the school s to these students one at a time following their priority order s until either there is no seat left or there is no student left who has applied to it in this round. Remove all assigned students. The algorithm terminates when all students are assigned to schools in S ∪ {s∅ }. In many school districts the assignment of students to the schools are done through BM. Deficiencies of BM have been extensively analyzed in the literature (see Abdulkadiro˘glu and S¨onmez (2003), Chen and S¨onmez (2006), Ergin and S¨ onmez (2006) and Pathak and S¨onmez (2008)). The major deficiency of BM is that it can be easily manipulated. Thanks to the earlier papers, in 2005 the Boston school district replaced BM with student proposing DA mechanism. However, many districts are still using BM and not replacing it with a strategy-proof mechanism. Given the resistance toward adopting a strategy-proof mechanism, our goal in this paper is by minimal modifications on BM getting a new mechanism which performs better than BM in terms of vulnerability to manipulation while achieving other desired features of BM.12 Under BM in Round k only the k th choices of the remaining students are considered. It is highly possible that either all the seats at the k th choice of a student may have been filled in the previous rounds or that student might be considered unacceptable by his k th choice. That is, that student is not actively participating the Round k. This may cause that student to miss an opportunity to get a seat in his next best choice. Motivated by this observation we modify BM and introduce the modified Boston mechanism (mBM) and for any given problem [q, P ] it works iteratively in a number of rounds: Modified Boston Mechanism (mBM): Round 0: Set c1s = qs for all s ∈ S and c1s∅ = |I|. Round 1: Let Q1 = {s ∈ S ∪ {s∅ }|c1s > 0}. Each student applies to his top ranked school in Q1 considering him acceptable. For each school s consider the students who have applied to it in this round and assign seats of the school s to these students one at a time following their priority order s until either there is no seat left or there is no student left who has applied to it in this round. Remove all assigned students. Let a1s be the number of students assigned to school s in this round. Set c2s = c1s − a1s . 11 12

¨ Kojima and Unver (2013) defines the BM in the same way. ¨ BM is constrained Pareto efficient, consistent, resource monotonic and population monotonic (see Kojima and Unver (2013)).

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In general, Round k: Let Qk = {s ∈ S ∪ {s∅ }|cks > 0}. Each student applies to his top ranked school in Qk considering him acceptable. For each school s consider the students who have applied to it in this round and assign seats of the school s to these students one at a time following the priority order s until either there is no seat left or there is no student left who has applied to it in this round. Remove all assigned students. Let aks be the number of students assigned to school s in this round. Set ck+1 = cks − aks . s The mechanism terminates when all students are assigned. Now-or-Never mechanism, and therefore Corrected BM and Adaptive BM, is very similar to mBM but differs by allowing students to apply to the schools considering them unacceptable as long as those schools have available seats. We illustrate the difference between BM and mBM in the following example. Example 1 There are 4 schools S = {s1 , s2 , s3 , s4 } with q = (1, 1, 1, 1) and 4 students I = {i1 , i2 , i3 , i4 }. The preference profile and priorities are: i1 :s1 Pi1 s2 Pi1 s3 Pi1 s4 Pi1 s∅ s1 :i1 s1 i3 s1 i4 s1 i2 s1 ∅ i2 :s1 Pi2 s2 Pi2 s4 Pi2 s3 Pi2 s∅ s2 :i2 s2 i3 s2 i4 s2 i1 s2 ∅ i3 :s2 Pi3 s3 Pi3 s1 Pi3 s4 Pi3 s∅ s3 :i3 s3 i2 s3 i4 s3 i1 s3 ∅ i4 :s2 Pi4 s3 Pi4 s1 Pi4 s4 Pi4 s∅ s4 :i3 s4 i4 s4 i1 s4 ∅ s4 i2 BM Round 1 Round 2

s1 ∗ i1 , i2

s2 ∗ i3 , i4

s3

i2

i∗4

Round 3 Round 4

mBM s4

s∅

s1 ∗ i1 , i2

s2 ∗ i3 , i4

s3

s4

s∅

i∗2 , i4 i∗4

i2 i2 i∗2

Round 5

We represent the rounds of each mechanism above. In each round the student with star is assigned to the school in which he applies to. In this example BM and mBM selects different outcomes. Example 1 also points out the difference between Now-or-Never and mBM. If we allow students to apply to the schools considering them unacceptable as in Now-or-Never mechanism, then the modified BM and BM will select the same outcome. That is, in Example 1 Now-or-Never and BM mechanisms select the same outcomes.

4

Results

We first analyze the mBM and show which axioms it satisfies. Then, we compare BM and mBM in terms of their vulnerability to manipulation. In Subsection 4.1 we focus on the equilibrium analysis when some students are sincere. 7

Theorem 1 mBM is constrained Pareto efficient, weakly consistent and respects priority of top-ranking students. Proof. Constrained Pareto Efficiency: Consider any problem [q, P ]. Denote the students assigned to a school including s∅ in Round k with Ck and mBM [q, P ] with µ. Students in C1 are assigned to their most preferred schools among the ones with available seats and considering them acceptable. If there exists i ∈ C1 who prefers another school s to µ(i) then qs = 0 or ∅ s i. If qs = 0 then assigning i to s will either violate the feasibility of the matching. If ∅ s i then any matching in which i is assigned to s cannot Pareto dominates µ. Now consider the students in Ck>1 . The students in Ck are assigned to their most preferred schools among the ones with available seats in round k. If there exists i ∈ Ck who prefers school s to µ(i) then either qs = 0 or all seats of s are filled with students in ∪k−1 t=1 Ct or ∅ s i. If qs = 0 or all seats of k−1 s are filled with students in ∪t=1 Ct then assigning i to s will either violate the feasibility of the matching or make at least one student in ∪k−1 t=1 Ct worse off. If ∅ s i then any matching in which i is assigned to s cannot Pareto dominates µ. Weak Consistency: For any problem [q, P ], by the definition of mBM students in C1 are the only students assigned to their most preferred available school, C1 = {i ∈ I : µ(i)=bi [q, P ]}. Let qs0 = qs − |{i ∈ C1 : µ(i) = s}|. Let P 0 = (PC∅1 , P−C1 ). First consider Round 1 of mBM for [q 0 , P 0 ]. For all i ∈ I \ C1 , bi [q 0 , P 0 ] is the most preferred schools among the ones with available seats and considering i as acceptable in Round 2 of mBM in [q, P ]. In the first round of mBM for [q 0 , P 0 ], all students in C1 apply to s∅ and all students in I \ C1 apply to the schools that they have applied in Round 2 of mBM for [q, P ]. Therefore, students in C2 are assigned to the same schools that they have been assigned in µ in the first round of mBM for [q 0 , P 0 ] and students in C1 are assigned to s∅ . As in the second round of mBM for [q, P ], all the available seats of the schools that students in I \ (C1 ∪ C2 ) apply in the first round of mBM for [q 0 , P 0 ] are filled by students in C2 . Then only students in C1 ∪ C2 are removed. Therefore, in the second round of mBM for [q 0 , P 0 ], the set of remaining schools and students are the same as the ones in the third round of mBM for [q, P ]. By repeating this procedure we can show that students Ck>1 are assigned to their assignment in µ in the Round k − 1 of mBM when we apply it to [q 0 , P 0 ]. Respect of Priority of Top-Ranking Students: For any problem [q, P ] in the first round of mBM any student i such that m−1 s (i) ≤ qs is assigned to s. Therefore, mBM respects priority of top-ranking students. It is worth mentioning that a version of the mBM in which students apply to the most preferred school considering them acceptable in each round is consistent.13 In the following examples we show that BM does not respect priority of top-ranking students. Example 2 Let S = {s1 , s2 }, I = {i1 , i2 }, q = (1, 1). The preference profile and priorities are: s2 Pi1 s1 Pi1 s∅ and s1 Pi2 s∅ Pi2 s2 , i1 s1 i2 s1 ∅ and i2 s2 ∅ s2 i1 . Any mechanism which respects priority of top-ranking students assign i1 to s1 because s1 is i1 ’s most preferred school among the ones with available seats and 13

¨ We refer Kojima and Unver (2013) for this result.

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considering her and she has the highest priority for s1 . On the other hand, BM assigns i1 to s∅ and i2 to s1 . Therefore it does not respect priority of top-ranking students. Example 3 Let S = {s1 , s2 }, I = {i1 , i2 }, q = (0, 1). The preference profile and priorities are: s2 Pi1 s1 Pi1 s∅ and s1 Pi2 s∅ Pi2 s2 , i1 s i2 s ∅ for all s ∈ S. As in Example 2, any mechanism which respects priority of top-ranking students assign i1 to s1 because s1 is i1 ’s most preferred school among the ones with available seats and considering her and she has the highest priority for s1 . On the other hand, BM assigns i1 to s∅ and i2 to s1 . Therefore it does not respect priority of top-ranking students. An important drawback of the BM is its vulnerability to manipulation (Abdulkadiro˘glu and S¨ onmez, 2003). Boston school district replaced it with DA mechanism which is strategy-proof. mBM also fails to satisfy strategy-proofness.14 Pathak and S¨ onmez (2013) introduce a notion which enables us to compare two mechanisms which are not strategy-proof in terms of their vulnerability to manipulation. Based on their notion, mechanism ψ is more manipulable than mechanism φ if whenever φ can be manipulated by student i ∈ I in problem [q, P ] then there is at least one student j ∈ I who can manipulate ψ in the same problem and there exists at least one problem in which none of the students can manipulate φ but at least one student can manipulate ψ. In this paper, we compare BM and mBM based on this notion and show that BM is more manipulable than mBM. Theorem 2 BM is more manipulable than mBM. Proof. We first show that whenever mBM can be manipulated by a student then BM can be also manipulated. Consider a problem [q, P ] in which student i ∈ I can manipulate mBM, i.e. mBMi [q, (P 0 , P−i )]Pi mBMi [q, P ]. We consider two cases. We take mBM [q, P ] 6= BM [q, P ] in the first one and mBM [q, P ] = BM [q, P ] in the second one. Case 1: Both BM and mBM are constrained Pareto efficient and individually rational. Then, there exists a student j ∈ I such that mBMj [q, P ]Pj BMj [q, P ]. Let mBMj [q, P ] = s1 and BMj [q, P ] = s2 . Let ABM and AmBM be the set of acceptable students for s1 who applied to s1 in round 1 under BM and mBM , respectively. First note that qs1 > 0 and j s1 ∅ since j is assigned to it under mBM . By the definitions of BM and mBM , ABM ⊆ AmBM . If j is assigned to s1 in Round 1 then the number of students in AmBM with higher priority for s1 than j is less than qs1 . Then, the number of students in ABM with higher priority for s1 than j is less than qs1 . If j submits P 0 : s1 P 0 x for all x ∈ (S ∪ s∅ ) \ {s1 } then BM will assign j to s1 . That is, j can manipulate BM by submitting P 0 , i.e. BMj [q, (P 0 , P−j )]Pj BMj [q, P ]. If j is assigned to s1 in Round k > 1 then the number of students in AmBM and ABM are less than qs1 . P 0 is also a profitable deviation for j under BM, i.e. BMj [q, (P 0 , P−j )]Pj BMj [q, P ]. That is, whenever mBM [q, P ] 6= BM [q, P ] there exists at least one student who can manipulate BM . Case 2: Let s˜ be the school that j is assigned when he manipulates under mBM . Let P˜ be the preference list that i submits to manipulate mBM , i.e. mBM [q, (P˜ , P−i )] = s˜Pi mBM [q, P ]. Let A˜BM 14

To see this consider Example 1. Student i2 can get s2 and be better off by ranking s2 at the top of the preference list.

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and A˜mBM be the set of acceptable students who applied to school s˜ in round 1 of [q, P ] under BM and mBM , respectively. By the definitions of BM and mBM , A˜BM ⊆ A˜mBM . Then the number of students in A˜mBM and A˜BM with higher priority for s˜ than i is less than qs˜. Otherwise i cannot be assigned to s˜ in mBM [q, (P˜ , P−i )]. If i submits P 00 : s˜P 00 x for all x ∈ (S ∪ s∅ ) \ {˜ s} then BM will assign him to s˜ in problem [q, (P 00 , P−i )]. Now we show that there exists a problem in which mBM cannot be manipulated but BM can be manipulated. Consider the problems in Example 2 and 3. mBM assigns i1 to s1 and i2 to s∅ in both problems. None of the students can be better off by deviating. BM assigns i1 to s∅ and i2 to s1 . In both problems, if i1 deviates and ranks s1 as his top choice then he will be assigned to s1 by BM. Theorem 2 shows that a simple modification may improve BM in terms of strategy-proofness. Note that BM is more manipulable than the versions of the mBM in which in each round students apply to the most preferred school among the ones either considering them acceptable or having available seats. Moreover, Now-or-Never mechanism, which corresponds to a version of mBM in which students can apply to the schools considering themselves unacceptable, is more manipulable than mBM.15 Proposition 1 Now-or-Never mechanism is more manipulable than mBM. Proof. One can show whenever mBM can be manipulated by a student then Now-or-Never mechanism can be also manipulated by following the same steps in the proof of Theorem 2. Moreover, in the problem illustrated in Example 2, Now-or-Never mechanism is manipulable but mBM is not. One might think that the modification will decrease the number of students assigned to one of their top k choices. In that case the modification may not be desirable since BM attracts school districts because it maximizes the number of students assigned to their most preferred school. If each school has at least one available seat then the first rounds of BM and mBM will be the same and they will assign the same number of students to their top choice. In the following example we show that mBM may outperform BM when the number of students assigned to top k ≥ 3 choices are considered. Example 4 There are m > 3 schools and students: S = {s1 , s2 , ..., sm } and I = {i1 , i2 , ..., im }. For each s ∈ S, qs = 1. Each student ik ∈ I \ {i1 , i2 } prefers sk most and has the highest priority for sk . Let s3 Pi1 s4 Pi1 s1 Pi1 ..., sPi1 s2 Pi1 s∅ for all s ∈ S, s3 Pi2 s1 Pi2 s2 Pi2 ... and i1 s1 i2 and all students are acceptable for all schools. BM and mBM assign all students in I \ {i1 , i2 } to their top choice. BM assigns i2 to his second choice and i1 to his last choice. mBM assigns both i1 and i2 to their third choices. The number of students assigned to their top k ∈ {3, ..., m − 1} choices under mBM is greater than the one under BM. If we define the utility of school district based on the weighted averages of the number of students assigned to top k choices where the weights are decreasing then we can find a weight profile under which mBM performs better than BM, i.e., the summation of the weights of top 3 and 4 choices are greater than the one of top 2. 15

Miralles (2008) and Mennle and Seuken (2014) introduce more general version Now-or-Never mechanism in the school choice environment independently. Therefore, the following result also hold for Corrected BM and Adaptive BM.

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An important reason for school districts using the BM instead of the DA mechanism is that it maximizes the number of students assigned to their top reported choice. One might think that due to the modification mBM will not perform better than DA when we compare the number of students assigned to top reported choice under naive submission. In the following proposition, we show this is not true. Proposition 2 In any problem [q, P ], the number of students assigned to their top choice under mBM is (weakly) more than the number of students assigned to their top choice under DA. Proof. One can easily verify that for any problem, if in the first round of DA student i applies to the most preferred school among the ones with available seats and considering her acceptable and in the following rounds i applies to the most preferred school which has not rejected her yet and is not preferred to the school she has applied in the first round, then DA will select the same outcome that it selects according to the usual definition. Hence, students apply to the same schools in the first round of both the mBM and this version of the DA mechanism. In each round of the mBM the acceptances are permanent. On the other hand, in each round of the DA the acceptances are tentative. Hence, a student who is assigned to her top choice in the first round of the mBM may be assigned to a lower ranked school by DA mechanism. Since both BM and mBM are vulnerable to manipulation, we investigate the equilibrium outcomes of preference revelation game induced by the mBM. We analyze the game under complete information of payoffs, available strategies and priorities. Agents are assumed to play simultaneously. Here, the set of actions is all the possible orders of schools and being unassigned option. Ergin and S¨onmez (2006) show that the set of Nash equilibrium outcome of preference revelation game induced by the BM is equal to the set of stable matching under the true preferences. In Theorem 3, we show this results holds under mBM. Theorem 3 The set of Nash equilibrium outcome of preference revelation game induced by the mBM is equal to the set of stable matching under the true preferences. Proof. Consider a problem [q, P ]. We first show that every Nash equilibrium outcome of the preference revelation game associated with mBM leads to a stable matching under the true preferences. Let Q = (Qi )i∈I be a Nash equilibrium profile and µ be the resulting outcome of the mBM, i.e. mBM [q, Q] = µ. By the definition of the mBM, i s ∅ for all i ∈ µ−1 (s) and s ∈ S. Suppose µ is not stable. Then there is a student–school (including s∅ ) pair (i, s) such that sPi µ(i), i s ∅, and |µ−1 (s)| < qs or i s j for some ˜ i be a profile in which s ∈ S ∪ {s∅ } is j ∈ µ−1 (s). This implies that s is not ranked at the top of Qi . Let Q ˜ i , Q−i ) and therefore Q is not a Nash equilibrium ranked at the top. mBM will assign i to s under profile (Q profile. Hence, µ is stable under [q, P ]. Conversely, let µ be a stable matching under true preferences, P . Let Q = (Qi )i∈I be a preference profile such that each student i ranks µ(i) as her top choice under Qi . Under problem [q, Q] mBM terminates at the first round and mBM [q, Q] = µ. We need to show that Q is a Nash equilibrium profile. Suppose Q is not a ˜ i such that mBMi [q, (Q ˜ i , Q−i )] = sPi µ(i). Nash equilibrium profile. Then there exists i ∈ I and a strategy Q First of all s 6= s∅ . Otherwise, µ is individually irrational. Since mBM assigns i to s, i s ∅. By nonwastefulness, |µ−1 (s)| = qs . Moreover, by our construction and fairness of µ, the number of student with 11

higher priority than i applying to s is qs . Hence, all students in µ−1 (s) will be assigned to s by mBM under ˜ i , Q−i )] and Q is a Nash equilibrium profile. problem [q, (Q Chen and Kesten (2013) introduce a parametric family of application-rejection mechanisms including BM and DA mechanisms as two extremes. In particular, they characterize each member by some positive number e ∈ {1, 2, .., ∞} of parallel and periodic choices through which the application and rejection procedure continues before the final outcome is reached.16 They showed that BM (e = 1) is more manipulable than any other application-rejection mechanism (e > 1). One can wonder how mBM performs compared to the application rejection mechanisms based on vulnerability to manipulation. In the following proposition we show that one cannot compare mBM and any application rejection mechanism except BM and DA in terms of vulnerability to manipulation. Proposition 3 mBM is not more manipulable than any application rejection mechanism except DA and any application-rejection mechanism except BM is not more manipulable than mBM. Proof. We first show that mBM is not more manipulable than any application rejection mechanism except DA. Let ϕe be the application-rejection mechanism with e parallel and periodic choices through with the application and rejection procedure continues until the final outcome. Since ϕ∞ corresponds to DA mechanism, it suffices to show that there always exists a problem in which mBM cannot be manipulated but ϕe can be manipulated where e is finite positive natural number. Let I = {i1 , i2 }, S = {s1 , s2 , ..., se , se+1 } and qs = 1 for all s ∈ S. The preference and priority profiles are: sx Pi1 sx+1 Pi1 s∅ , se+1 Pi2 s∅ Pi2 sx , i2 sx ∅ sx i1 and i1 se+1 i2 se+1 ∅ for all x ∈ {1, 2, ..., e}. ϕe will assign i1 to s∅ and i2 to se+1 , whereas mBM will assign i1 to se+1 and i2 to s∅ . Either i1 or i2 cannot be assigned to a better schools by mBM if one of them deviates from true preference. On the other hand, if i1 ranks se+1 at the top of his preference list he will be assigned to se+1 by ϕe . Now we show that any application-rejection mechanism except BM is not more manipulable than mBM by using an example. Let I = {i1 , i2 , i3 }, S = {s1 , s2 } and qs = 1 for all s ∈ S. The preference and priority profiles are: s1 Pi s2 Pi s∅ for all i ∈ I and i1 s i2 s i3 for all s ∈ S. Any application-rejection mechanism ϕe with e > 1 and mBM will assign i1 to s1 , i2 to s2 and i3 to s∅ under true preferences. None of the students can get a better outcome by misreporting under ϕe . However, if i3 ranks s2 at the top of her submitted preference list, she will be assigned to s2 by mBM.

4.1

Sincere and Sophisticated Students

In this subsection we consider an environment composed of two types of students: sincere and sophisticated.17 We denote the set of sincere and sophisticated students with N and M , respectively. Here, we investigate the equilibrium outcomes of the preference revelation game induced by the mBM in which only sophisticated 16

For formal definition of the family of application-rejection mechanisms see Chen and Kesten (2013). A school choice model with sincere and sophisticated students was first introduced by Pathak and S¨ onmez (2008). In this subsection we use their notation. Miralles (2008) compares the welfare effects of gaming under BM on the sincere and sophisticated players by using simulations. 17

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students are active players. In particular, we focus on a game in which the strategy space of each sincere student is composed of her true preference and the strategy space of each sophisticated player is all strict preferences over the set of schools. Pathak and S¨onmez (2008) consider the same environment under BM. Our goal in this subsection is to compare the equilibrium assignments of sophisticated and sincere students under the preference revelation games induced by BM and mBM.18 We start our analysis by first focusing on the assignments of sincere students under mBM game. As one can verify easily, mBM game has multiple equilibria. In the following proposition, we show that the assignment of each sincere student is the same in each equilibrium outcome of mBM game. Proposition 4 For a given problem [q, P ], let µ and ν be both Nash equilibrium outcomes of the mBM game. Then, µ(i) = ν(i) for any sincere student i ∈ N . We prove the proposition by using an augmented problem of [q, P ]. Let [q, P˜ ] be the augmented problem of [q, P ] where P˜i = Pi for all i ∈ M and P˜i : bi [q, P ]P˜i s∅ P˜i x for all x ∈ S \ {bi [q, P ]} for all i ∈ N . Proof. Let Q = (Qi )i∈I be a Nash equilibrium profile and µ be the resulting outcome of mBM, i.e. mBM [q, Q] = µ. By our construction, Qi = Pi for all i ∈ N . Construct a new matching µ ˜ as follows: (1) for all i ∈ M , µ ˜(i) = µ(i), and (2) for all i ∈ N, µ ˜(i) = µ(i) if µ(i) = bi [q, P ] and µ ˜(i) = s∅ otherwise. We first show that µ ˜ is a stable matching in problem [q, P˜ ]. Claim: µ ˜ is a stable matching in problem [q, P˜ ]. Proof. Individual Rationality: By the definition of the mBM, i s ∅ for all i ∈ µ ˜−1 (s) and s ∈ S. By ˜ i ∅ for all i ∈ N . Suppose there our construction of preference profile P˜ and the definition of mBM, µ ˜(i)R ˜ i s for all s ∈ S will be a exists i ∈ M such that s∅ P˜i µ ˜(i). Hence, s∅ Pi µ(i). Then, preference profile s∅ Q profitable deviation for i and Q cannot be a Nash equilibrium profile. Non-wastefulness: Suppose that there exists a student school pair (i, s) such that i s ∅, |˜ µ−1 (s)| < qs , and sP˜i µ ˜(i). If |˜ µ−1 (s)| < qs then |µ−1 (s)| < qs or there exists j ∈ N ∩ µ−1 (s) such that bj [q, Q] 6= s. In both cases, the number of applicants for school s in the first round of mBM in problem [q, Q] is less than qs . If i ∈ N then s = bi [q, Q], i.e., i applies to s in the first round of mBM. Hence, i ∈ / N . If i ∈ M , then ˜ preference profile sQi x for all x ∈ S ∪ {s∅ } will be a profitable deviation for i and Q cannot be a Nash equilibrium profile. Fairness: Suppose that there exists a student–school pair (i, s) such that sP˜i µ ˜(i) and i s j for some −1 j ∈ µ ˜ (s). If i ∈ N then s = bi [q, Q] 6= µ(i). Note i applies to s in the first round of mBM in [q, Q]. Hence, i will be assigned to s whenever a student with lower priority is assigned. Then, i ∈ / N . If i ∈ M , ˜ i x for all x ∈ S ∪ {s∅ } will be a profitable deviation for i and Q cannot be a Nash then preference profile sQ equilibrium profile. By the Rural Hospital Theorem (Roth, 1986), the sets of unassigned and assigned students in any stable outcome of [q, P˜ ] are the same. Moreover, in any stable matching of [q, P˜ ] the number of students assigned to each school is the same. If µ ˜(i) = s then i is assigned to the same school s in any stable outcome of [q, P˜ ]. 18

In the rest of our analysis, we use mBM game and BM game instead of the preference revelation game induced by mBM and BM, respectively.

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Therefore, the number of sophisticated students assigned to each school s ∈ S is the same under all stable outcomes of [q, P˜ ]. Claim implies that the number of sophisticated students assigned to each school s ∈ S is the same under all Nash equilibrium outcome of the mBM game under problem [q, P ]. Given the strategy space of each sincere student is composed of her true preference, mBM will assign them to the same school in all Nash equilibrium outcome. Pathak and S¨ onmez (2008) show the same result under the equivalent BM game. In the proof of Proposition 4, we use a different way. Moreover, we can show that sincere students are assigned to the same schools under all Nash equilibria outcome of the BM game by following the steps of the proof of Proposition 4.19 Proposition 5 For a given problem [q, P ], let µ and ν be both Nash equilibrium outcomes of the BM game. Then, µ(i) = ν(i) for any sincere student i ∈ N . Different from the proof of Proposition 4, we create the augmented economy as follows: Let [q, Pˆ ] be the augmented problem of [q, P ] where Pˆi = Pi for all i ∈ M and Pˆ : sPˆi s∅ Pˆi x for all x ∈ S \ {s} for all i ∈ N where s is the top ranked school in Pi . Proof. We can prove this proposition by following the same steps in the proof of Proposition 4. In the following proposition, we show that in both the BM and the mBM game there exists a unique Pareto dominant equilibrium and all sophisticated students are weakly better off under the Pareto dominant equilibrium of the BM game. Theorem 4 Each sophisticated student weakly prefers the school she receives in the Pareto dominant equilibrium of the BM game to the school she receives in the Pareto dominant equilibrium of the mBM game. Proof. Proposition 4 implies that the equilibrium outcomes of the mBM game are Pareto ranked based on the assignments of the sophisticated students. Similarly, Proposition 5 implies that the equilibrium outcomes of the BM game are Pareto ranked based on the assignments of the sophisticated students. We first consider the mBM game. Let µ ˜ be the student optimal stable matching in problem [q, P˜ ]. We show that there exists a unique Pareto dominant equilibrium outcome of mBM game and in that outcome each sophisticated student i is assigned to µ ˜(i). By the proof of Proposition 4, it suffices to show that there exists a Nash equilibrium outcome of the mBM game such that each i ∈ M is assigned to µ ˜(i). Consider preference profile Q = (Qi )i∈I such that (1) for each i ∈ M µ ˜(i)Qi s∅ Qi s for all s ∈ S \ {˜ µ(i)} and (2) for each i ∈ N Qi = Pi . By the definition of the mBM, mBM [q, Q](i) = µ ˜(i) for all i ∈ M . Suppose that there exists i ∈ M such that sPi µ ˜(i) and i s ∅. By the proof of Proposition 4 for each student i ∈ M the number of students with higher priority than i applying to s in the first round mBM in problem [q, Q] is not less than qs . Hence, i cannot get s by changing his strategy and Q is a Nash equilibrium. In the proof of Proposition 4, we show that in any Nash equilibrium outcome sophisticated students are assigned to the school they get in a stable outcome of [q, P˜ ]. Hence, there exists a unique Pareto dominant equilibrium 19

If s∅ is the top ranked school then s∅ Pˆi s for all s ∈ S.

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outcome of mBM and in that equilibrium each sophisticated students gets their assignment under µ ˜. Let µ be the unique Pareto dominant equilibrium outcome of the mBM game. By following the same arguments and using Proposition 5, we can show that there exists a unique Pareto dominant equilibrium outcome of the BM and in that equilibrium outcome each sophisticated student gets her assignment under the student optimal stable matching of [q, Pˆ ] denoted by νˆ. Let ν be the unique Pareto dominant equilibrium outcome of mBM. Since we are only comparing the assignments of the sophisticated players and µ(i) = µ ˜(i) and ν(i) = νˆ(i) ˜ for all i ∈ M , we look at the outcome of the DA mechanism in problems [q, P ] and [q, Pˆ ]. In particular, we apply the sequential DA mechanism introduced by McVitie and Wilson (1971). By our construction, P˜i = Pˆi = Pi for all i ∈ M . Each i ∈ N ranks only one school acceptable in Pˆi and if bi [q, P ] is that school ¯ be the subset of the sincere students such that bi [q, P ]Pˆi s∅ for then it is also top ranked school in P˜i . Let N ¯ . Note that any student i ∈ N \ N ¯ ranks a school at the top of Pˆi which considers him unacceptable all i ∈ N or does not have available seat. When we apply sequential DA mechanism in both problems we first allow ¯ to apply to their top ranked schools. Since the top ranked school of i ∈ N ¯ is the same in each student i ∈ N ¯ applies to her first choice in both both P˜i and Pˆi , we will have the same tentative matching when each i ∈ N problem. Then we allow all sophisticated students to apply until sophisticated students are not rejected any more. Since Pˆi = P˜i for all i ∈ M , we will have the same tentative matching in both problem. Denote this tentative matching with µ1 . Since all the students who are not assigned to a school or s∅ are the sincere students, one can easily verify that µ1 is the final outcome of sequential DA mechanism in problem [q, Pˆ ]. That is, µ1 (i) = ν(i) for all i ∈ M . Moreover, any further application in problem [q, P˜ ] will weakly worsen the assignment of sophisticated students. That is, µ1 (i)Ri µ ˜(i) = µ(i) for all i ∈ M . Hence, ν(i)Ri µ(i) for all i ∈ M . In the following example, we illustrate a case in which all sophisticated and some sincere students strictly prefer the Pareto dominant equilibrium outcome of BM game and the remaining sincere students strictly prefer the Pareto dominant equilibrium outcome of the mBM game. Example 5 There are 3 schools S = {s1 , s2 , s3 } with q = (1, 1, 1) and 3 students I = {i1 , i2 , i3 }. The sets of sophisticated and sincere students are as follows: M = {i1 } and N = {i2 , i3 }. The preference profile and priorities are: i1 :s1 Pi1 s2 Pi1 s3 Pi1 s∅ s1 :i2 s1 i1 s1 i3 s1 ∅ i2 :s3 Pi2 s1 Pi2 s2 Pi2 s∅ s2 :i1 s2 i3 s2 i2 s2 ∅ i3 :s2 Pi3 s3 Pi3 s1 Pi3 s∅ s3 :i3 s3 i2 s3 ∅ s3 i1 ! i1 i2 i3 The Pareto dominant equilibrium outcome of the BM game is ν = . The Pareto dominant s1 s∅ s2 ! i1 i2 i3 equilibrium outcome of the mBM game is µ = . i1 and i3 prefer ν to µ and i2 prefers µ to ν. s2 s1 s3 As can be seen from Example 5, we cannot have a conclusion for the sincere students similar to the Theorem 4. However, Miralles (2008), by using simulations, shows that sincere students are better off under the Corrected BM compared to the BM. 15

5

Characterization

In this section we characterize mBM based on weak consistency and the respect of priority of top-ranking students. Theorem 5 mBM is the unique mechanism which is weakly consistent and respects priority of top-ranking students. Proof. Suppose not. There exists mechanism φ which is weakly consistent and respects priority of top-ranking students. There exists at least one problem [q, P ] such that φ[q, P ] 6= mBM [q, P ]. Consider any problem [q, P ] such that there exists j ∗ ∈ I φj ∗ [q, P ] 6= mBMj ∗ [q, P ]. Let J[q, P ] be the set of students assigned to a school in Round 1 of mBM in [q, P ]. Claim 1: φi [q, P ] = mBMi [q, P ] for all i ∈ J[q, P ]. Proof: Suppose not. There exists k ∈ J[q, P ] and φk [q, P ] 6= mBMk [q, P ]. Let mBMk [q, P ] = s0 . Student k applies to his most preferred school available school in the first round of mBM. Since he is assigned to s0 then there do not exist qs0 students whose most preferred available school is s0 and has higher priority for s0 than k. Hence, φ does not respect priority of top-ranking students. Claim 2: Only students in J[q, P ] are assigned to their most preferred available school in φ[q, P ] and mBM [q, P ]. Proof: By Claim 1 and the definition of mBM students in J[q, P ] are assigned to their most preferred available school. For any k ∈ I \ J[q, P ], by the definition of mBM and Claim 1 all the available seats of k’s most preferred available schools are filled by students in J[q, P ] in mBM [q, P ] and φ[q, P ]. ∅ ∅ 1 1 ∗ Claim 3: φj ∗ [q 1 , (PJ[q,P ] , P−J[q,P ] )] 6= mBMj [q , (PJ[q,P ] , P−J[q,P ] )] where qs = qs − |{i ∈ J[q, P ] : mBMi [q, P ] = s}|. Proof: Recall our assumption φj ∗ [q, P ] 6= mBMj ∗ [q, P ]. By Claim 1, j ∗ ∈ / J[q, P ]. By Claim 2 and ∅ ∅ 1 1 weak consistency, φj ∗ [q , (PJ[q,P ] , P−J[q,P ] )] = φj ∗ [q, P ] and mBMj ∗ [q , (PJ[q,P ] , P−J[q,P ] )] = mBMj ∗ [q, P ]. ∅ ∅ 1 1 ∗ Then φj ∗ [q 1 , (PJ[q,P ] , P−J[q,P ] )] 6= mBMj [q , (q , (PJ[q,P ] , P−J[q,P ] ))]. ∅ ∗ Let P 1 = (PJ[q,P ] , P−J[q,P ] ). Note that Claim 1-3 are true for any problem [q, P ] such that φj [q, P ] 6= mBMj ∗ [q, P ] for some j ∗ ∈ I . Therefore, we can apply these claims to problem [q 1 , P 1 ] and get another problem where j ∗ is assigned to different schools by φ and mBM . Since there are finite number of students and there always exist a set of students assigned to a school in the first round of mBM for any problem we can apply these claims to the reduced problems finite number of times. At some point we will reach a problem [˜ q , P˜ ] where all students are assigned in the first round of mBM. This contradicts the fact that j ∗ is assigned to different schools by mBM and φ. These two axioms are independent. For instance, a mechanism which is only composed of the first round of mBM and assigns all the remaining students to s∅ satisfies respect of priority of top-ranking students but not consistency. On the other hand BM is weakly consistent but does not respect priority of top-ranking students.

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6

Simulations

In Section 4, we compare the BM and the mBM mechanisms based on their vulnerability to manipulation and show that mBM is less manipulable than BM. In this section, by using simulations we compare the the fraction of students who can manipulate each mechanism in various scenarios. We consider environments with 5, 10, 20 and 50 schools. Each school has 5 seats and the number of students is equal to the total number of seats. We construct the preference profile of each student i by incorporating the possible correlation among students’ preferences. In particular, we calculate student i’s utility from school s as follows: Ui,s = α × Z(s) + (1 − α) × Z(i, s)) Here, Z(s) is an i.i.d standard uniformly distributed random variable and represents the common tastes of students on school s. Z(i, s) is also an i.i.d standard uniformly distributed random variable and represents the tastes of student i on school s. The correlation in the students preferences is captured by α ∈ [0, 1]. As α increases the student preferences over the schools become more similar. By using the utilities students getting from each school we construct the ordinal preferences of students over schools. In order to construct schools priorities over students, we draw a priority number for each school student pair. In particular, W (i, s) is an i.i.d standard uniformly distributed random variable between [0, 1] and represents the priority number of student i for school s. By using the priority numbers we construct the strict priority order of schools over students. We use a threshold value to determine the set of unacceptable students for each school. We say student i is unacceptable for school s if W (i, s) is less than the threshold value. We run our simulations for different values of threshold: 0.1, 0.2, 0.3 and 0.4. For each α and threshold, we run the BM and mBM 10,000 times by using different random orderings of the students and calculate the fraction of students who can manipulate these mechanisms. Threshold=0.2 1

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Figure 3: Fractions of students who can manipulate mBM and BM (20 schools with 5 seats) Threshold=0.2 1

0.8

0.8

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Fraction

Threshold=0.1 1

0.4 0.2 0

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1

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Threshold=0.4

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Figure 4: Fractions of students who can manipulate mBM and BM (50 schools with 5 seats)

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The figures illustrate that our theoretical results are supported by the simulations. In all cases, the fraction of students who can manipulate BM is more than the fraction of students who can manipulate BM. Under both mechanisms, as the preferences become more correlated the fraction of students who can manipulate increases. As the number of students who are unacceptable for some schools increases, i.e., threshold value increases, the difference between the fractions of students who can manipulate BM and mBM increases. Finally as the number of schools increases, under both mechanisms, the fraction of students who can manipulate increases.

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