Anti-bullying School Choice Mechanism Design Yusuke Kasuya∗ MEDS, Kellogg School of Management Northwestern University November 14, 2012 Preliminary Draft

Abstract The implications of peer effects on school choice mechanism design are of important, yet underexplored educational concerns. This paper addresses school bullying incidents as documented sources of significant negative peer effects among students. I consider school choice that requires that bullies and victims be assigned to different schools, and redefine the concepts of stability and efficiency accordingly. I then show that variants of the Gale-Shapley mechanism and the top trading cycles mechanism achieve these social goals respectively. Moreover, the mechanisms can potentially help prevent bullying in, for example, elementary schools since they assign less preferred middle schools to bullies thereby punishing them. I also discuss collaborative interaction between anti-bullying programs and school choice.

JEL Classification: C78; I21; I28 Keywords: School Choice; Mechanism Design; Peer Effects; Anti-bullying Program

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Email: [email protected] (any comments are welcomed). I gratefully acknowledge Jeff Ely, Fuhito Kojima, Jim Schummer, Peter K. Smith, Fran Thompson, Yuichi Toda, and Rakesh Vohra for enlightening discussion and their heartening encouragement. I also thank numerous seminar participants, Ryota Gamo, Daisuke Hirata, Kohei Kawaguchi, Mayumi Kuroda, Yusuke Narita, Ken Rigby, Tayfun S¨onmez, and Kentaro Tomoeda for incredibly helpful comments and conversations. All remaining errors are my own responsibility.

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1

INTRODUCTION

The design of rigorous and transparent school choice mechanisms has provoked recent heated policy debate in many countries such as the United States, the United Kingdom, and Japan. At the requests of politicians and academics, Abdulkadiro˘glu, and S¨onmez (2003) initiated the mechanism design approach to school choice, having brought economists to address and resolve various issues in real life.1 Nonetheless, much of the approaches advocated in the literature have ignored the implications of peer effects on school choice.2 For one thing, analytical models have ruled out forms of peer effects in preferences by assuming that composition of classmates does not influence a student’s evaluation of a match. However, as reported in an array of articles (Minei and Nakagawa 2005; Rothstein 2006; Woods et al. 1998), peer quality is an object of concern, and may matter more than school quality for students and their parents. For another thing, the welfare and fairness criteria employed in the literature do not incorporate a social concern for educational achievement which in turn is the product of peer effects among students. That the characteristics of peer group influence own educational outcome is well documented in the literature (Hanushek 1986), though the precise mechanisms behind these peer effects are still in question (Zimmerman 2003; Hoxby and Weingarth 2006). The purpose of this paper is to take a step in the direction of incorporating peer effects into the mechanism design approach to school choice by examining the particular case related to school bullying. There are at least four reasons that legitimate my pursuit. First, there is considerable evidence suggesting that the experience of being a victim adversely affects academic achievement and school adjustment (Kochenderfer and Ladd 1996; Hawker and Bouton 2000), and may cause mental illness in adolescence (Arseneault et al. 2009; Bond et al. 2001).3 As a result, bully-victim relationships underlie most childhood assaults, suicides, and 1

Economists have contributed to school admissions reform in New York City (Abdulkadiro˘glu et al. 2005a, 2009), Boston (Abdulkadiro˘glu et al. 2005b, 2006) and San Francisco. They have also discovered flaws in real-life mechanisms used in other districts and countries (Yasuda 2010; Pathak and S¨onmez 2011), and kept trying to convince policy makers to improve their admission systems. 2 Epple and Romano (2011) defined peer effects as follows: “For given educational resources provided to student A, if having student B as a classmate or schoolmate affects the educational outcome of A, then we regard this as a peer effect.” (P.1054) I adopt this definition while including psychological distress, social adjustment and school engagement in the denotation of “the educational outcome”. 3 Citation policy: whenever I refer to evidence on a nature of bullying or anti-bullying program for the first time, I specify its sources which investigate bullying incidents in the United States or European countries. If there are any documented conflicting views within these countries or between them and other countries such as Japan, I will describe them in due course.

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homicides (Anderson et al. 2001).4 Bullying in schools can also have serious negative consequences (e.g., lower academic achievement and distraction) for the perpetrators and bystanders as well (Fonagy et al. 2005; Gini and Pozzoli 2009; Nishina and Juvonen 2005).5 As such, in contrast to other sources of peer effects which have not found unanimous empirical support in the literature6 , bullying undoubtedly involves long-standing negative externalities among students which in particular can be fatal for the victims. Second, longitudinal research repeatedly confirms that the bully-victim relation is stable and persistent from primary to middle school and even afterwards (Olweus 1979, 1993; Pellegrini and Long 2002; Sourander et al. 2000). Hence for a school admission process, let alone a school transfer process where a victim needs to escape from a bullying episode by changing her school, each bullying incident in a previous school deserves due regard. And thus, it is important to separate bullies from their victims in the assignment. Third, the school choice program in Japan has been considered as a means to buffer victims against bullies.7 Traditionally, students in Japan are assigned to one of the school attendance areas based on their residence, and expected to enroll in designated primary and secondary public schools. Initially, they could rarely change the schools for important reasons such as bullying and disease. In 1997, the Ministry of Education announced that the municipal boards of education must more flexibly allow students to change their designated schools before or after the enrollment for specific reasons including bullying, absenteeism, and commuting distance.8 It then promoted the school choice program to enjoy the upside of it while dealing with these issues. Consequently, numerous municipalities have introduced variants of the open enrollment school admission/transfer process for primary and secondary public schools. 4

School bullying triggered the shooting rampage incidents at Columbine high school in 1999, Virginia Tech in 2007, Jokela high school in Tuusula, Finland, in 2007, and Tasso da Silveira Municipal School in Rio de Janeiro, Brazil, in 2011. See Wikipedia for details of these massacres. 5 Lazear (2001) builds on this fact, and the bad apple principle in general, to study class size effects. 6 For example, evidence on ability-based peer effects (e.g., whether and to what extent the average GPA of classmates raises one’s academic achievement) is mixed and disparate. Epple and Romano (2011) reviews the body of evidence that suggests that peer ability has positive impacts on one’s academic achievement. On the other hand, Abdulkadiro˘glu et al. (2011), Clarke (2008), and Cullen et al. (2006) show little evidence on the effects in the US and the UK. Racial, income, and ability-diversity in classroom have complex and equivocal consequences (Angrist and Lang 2004; Hoxby and Weingarth 2006). Duflo et al. (2011) found gains from tracking in Kenya while Pekkarinen et al. (2009) suggest the opposite effects in Finland. As discussed by Duflo et al. (2011) and Hoxby and Weingarth (2006), good strategies for administrative assignment of students to improve educational outcomes must have nuanced structural models of peer effects and incorporate other factors such as teacher incentives and gender/racial composition. I believe that in a “reduced-form” approach without specifying these elements in a model, the current analysis focusing on school bullying is a natural and inevitable first step. 7 I owe the following description to Yasuda (2010) and Yoshida et al. (2009). 8 http://www.mext.go.jp/a menu/shotou/gakko-sentaku/06041014/008/003.htm

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However the extant school choice mechanisms in Japan are not tailored to the needs of the victims having been historically at the center of the policy debate on school change/choice.910 Indeed, in 2008, the Ministry of Education called for careful treatment of those students during a school admission/transfer process.11 However, existing mechanisms are yet not designed to systematically separate bullies from their victims, and consequently they cannot accommodate the original motivation to afford students an escape from a toxic environment as well as the desire for respecting the rights for all families to choose the education. Fourth, since the initial success of a bullying prevention program in Norway (Olweus 1993)12 , numerous anti-bullying policies have been investigated and implemented in countries such as those in Scandinavia, the United Kingdom, France, Germany, the United States, Canada, Australia, and Japan (Smith et al. 1999; Smith et al. 2004). Probably due to the very recent developments in its formal analysis, however, the role of school choice in anti-bullying policy has escaped due notice both in the education and the market design literature. I believe that school choice mechanisms have a potential to play a significant role in antibullying programs. This is done by crafting the mechanisms in which the assignment for a student becomes less preferred if she bullies other student. Then the mechanisms may place some check on school bullying incidents. In this paper, motivated by these observations discussed at length above, I propose variants of canonical school choice mechanisms advocated in the literature: i.e., the Gale-Shapley (studentoptimal stable) mechanism, and the top trading cycles mechanism. The main theorems show that these mechanisms have anti-bullying properties (i.e., they separate bullies from their victims, and assign a less preferred school seat to a bully) together with other standard desiderata. I also demonstrate the generalizability and flexibility of the basic framework by taking up some particular issues in real life and showing how it could adapt to each of them. 9 For example, the school admission mechanism used in Tokyo is a variant of the serial dictatorship mechanism where each student can submit only one school but she is guaranteed to enter her designated school. 10 The municipal boards of education in Tokyo indeed respond to the needs of victims after they run school choice mechanisms conditional on the explicit requests (I thank Yusuke Narita who raised this point). There are several reasons to believe that this policy is fraught with difficulties. First, many students hesitate to submit requests because they fear their parents to know about their bullying episodes. Second, a student may not know if their assailants are going to be her classmates. Third, the ad hoc adjustment of student assignment is incongruent with the desiderata for mechanisms and outcomes. The framework in this paper is free from these difficulties. 11 http://www.mext.go.jp/a menu/shotou/gakko-sentaku/08042801.htm 12 An overview of the prototype of the Olweus Bullying Prevention Programme is found in Olweus (2004). See also http://www.Olweus.org.

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1.1

RELATED LITERATURE

To the best of my knowledge, there is no research in the whole literature of economics and education suggesting the relevance of school bullying to school admission process (or vice versa). However, besides its obvious relation to the literature on school choice mechanism design, this paper shares its motivations and analytical principles with some other prior work as well.13 Models of two-sided matching with peer effects have run the full spectrum. The first strand of literature studies stability of many-to-one matching when students (workers) have preferences over both schools (firms) and their schoolmates (colleagues). Dutta and Mass¨o (1997) introduced the canonical model which was further explicated by Ravilla (2007) and Echenique and Yenmez (2007). However, as Echenique and Yenmez (2007) lamented, these approaches to forms of peer effects in preferences faced severe challenges in their efforts to characterize the situations when the set of core (stable) matchings is nonempty.14 In contrast to these papers, I will simplify the model by assuming that students have preferences over schools but not over other students, deal with significant negative peer effects concerned with school bullying, and provide a series of positive results and school choice mechanisms that are practically implementable. The second strand of literature closely related to the first one incorporates a new element, such as a (bargaining) game after a match (Jackson and Watts 2010; Pycia 2012), or a social/friendship network structure behind students (Bodine-Baron et al. 2011), into the standard two-sided matching model thereby deal with peer effects. Their models are based on agents’ cardinal utility functions besides other detailed information about the environment (i.e., bargaining protocols, game forms, or friendship graphs), thus not tailored to (school choice) mechanism design. The third strand of literature departs further from mechanism design and adopts a general equilibrium model to study educational policies. Epple and Romano (1998) analyzed the implications of peer effects and school competition on various issues such as student sorting, school sorting, and educational outcomes.15 Other theoretical literature on peer group effects include Caucutt (2001), de Bartolom´e (1990), and Nechyba (2000). Although this paper does not address school competition especially among private schools, future research may want to investigate the intersection of their frameworks and mine. ¨ See S¨onmez and Unver (2011) and Pathak (2011) for up-to-date surveys on the mechanism design approach to school choice. Roth and Sotomayor (1990) is a classic for the two-sided matching theory, establishing a foundation for the whole school choice mechanism design literature. 14 “Our approach is motivated by a certain pessimism. It seems that general conditions for nonemptiness of the core are difficult to obtain, and that the few that are known are very strong.” (Echenique and Yenmez 2007, p.47) 15 Epple and Romano (2011) survey the literature on peer effects in education. 13

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The theoretical investigation in this paper targets centralized matching markets with constraints. There is an array of research having taken up such markets in real life, e.g., Japan residency matching program with regional caps (Kamada and Kojima 2011a,b), school choice (college admission) with affirmative action (Abdulkadiro˘glu 2005), and a residency matching problem when a couple of doctors needs to go to the same hospital (Dean et al. 2006). This paper joins this strain of research by dealing with both mathematically and conceptually new constraint (i.e., an agent (e.g., student, doctor, worker) is prohibited to affiliate an organization (e.g., school, hospital, firm) with some other agents) and demonstrating its relevance to the educational concern for school choice. The rest of the paper is organized as follows. Section 2 presents a model and the criteria for school choice mechanisms. Section 3 discusses stability, a variant of the Gale-Shapley mechanism, and a variant of the top trading cycles mechanism. Section 4 discusses the related topics and future directions. Section 5 concludes. Proofs are in the appendix unless otherwise noted.

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MODEL

A school choice problem consists of a finite set I of students and a finite set S of schools. Each student i ∈ I has a strict preference ordering i over S ∪ {i} where i in its domain represents an outside option for the student.16 i is a weak preference ordering associated with i , i.e., s i s0 (s, s0 ∈ S ∪ {i}) means either s i s0 or s = s0 . For any student i ∈ I, s ∈ S is acceptable for her if s i i. For any J ⊆ I, the preference profile of the set J of students is denoted by  J ≡ (i )i∈J . A set of bullying incidents B is a subset of I × I where (i, j) ∈ B means “i has bullied j”.17 Formally, B satisfies the following condition: (i, i) < B for all i ∈ I (no student bullies herself). Each (i, j) can represent either an ongoing, or a passed bullying episode. This definition allows for the possibility that (i, j) ∈ B (i bullied j) and ( j, i) ∈ B ( j took revenge on i) at the same time. I will assume throughout that B is exogenously given to a social planner. This premise requires that at least the school bullying incidents of vital importance be well defined and detected. Antibullying school choice does not provide a cure-all for elicitation of the information (but see Section 3.3). Hence B must be obtained by means of an anti-bullying program (e.g., asking students to nominate bullies and victims), or other auxiliary policy. See Section 4.5.1 for further discussion. 16

An outside option would be a private school, an exam school, or homeschooling among others. See Section 4.3 for further discussion on the role of this concept in school choice. 17 A student is being bullied or victimized when he or she is exposed, repeatedly and over time, to negative actions on the part of one or more other students (Olweus 1991).

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Each school s ∈ S has q s available seats. Moreover, each school s ∈ S is endowed with a strict priority ordering  s over I. School priorities are determined by law and differ from school preferences.18 The priority profile of the set of schools is denoted by S ≡ ( s ) s∈S . Throughout the paper, I assume that no school rejects a student due to reasons other than a lack of school seats.19 A matching is a correspondence µ : I ∪ S → I ∪ S ∪ {∅} such that each student is assigned to either only one school or herself. If µ(s) = ∅, there is no student in s, thus |µ(s)| = 0. I denote by µ(µ(i)) the set of i’s colleagues including i herself (i.e., if i is single, it is herself; otherwise, it is the set of students who are assigned to µ(i)). A matching µ is feasible if |µ(s)| ≤ q s for every s ∈ S . A matching µ is individually rational if µ(i) i i for every i ∈ I. A matching µ satisfies the separation principle if there is no (i, j) ∈ B such that µ(i) = µ( j). In words, if i has bullied j, they are not assigned to the same school. This criterion reflects my motivation to escape victims from bullies. A matching µ dominates another matching ν if µ(i) i ν(i) for all i ∈ I, and µ(i) i ν(i) for some i ∈ I. A matching is constrained Pareto efficient if it is feasible, individually rational, satisfies the separation principle, and there is no other matching that satisfies these three conditions and dominates it. The concept of stability in the current framework which embodies a notion of fairness has two building blocks: “justified envy” and one’s “wish for a vacant seat”. Given a matching µ, a student i has justified envy toward student j if i prefers j’s assignment µ( j) to her own, she has a higher priority than j for µ( j), and there is no k ∈ µ(µ( j)) \ { j} such that (i, k) ∈ B or (k, i) ∈ B. To illustrate the meaning of justified envy, suppose i µ( j) j and there is k ∈ µ(µ( j)) \ { j} who has bullied i. Since j’s school admits k besides j, it must become less attractive to i. Hence i may not have envy toward j in the first place. Even if i still prefers to being enrolled in µ( j) in place of j (probably because µ( j) is an excellent school), this request is not justified since the recurring bullying incident would have negative externalities over other enrollees. Next suppose i µ( j) j and i’s victim k is also in j’s school. In this case, i’s complaint against j is not justified since replacing i with j clashes with the anti-bullying policy to buffer the victims against the bullies. The second notion is defined analogously to justified envy. Given a matching µ, a student i wishes for a vacant seat at a school s if s i µ(i), |µ(s)| < q s , and there is no j ∈ µ(s) such that (i, j) ∈ B or ( j, i) ∈ B. In other words, i wishes for a vacant seat at s if she has a “justified request” for the seat. Based on the two notions, now I formalize the stability concept. 18

See Section 4.4 for the implications of school preferences on anti-bullying school choice. I impose this assumption for convenience. Without it, all the results in this paper hold as they are with minor modifications to the framework. 19

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Definition 1. A matching is (constrained-)stable if (i) it is feasible, (ii) individually rational, (iii) satisfies the separation principle, (iv) there is no student who wishes for a vacant seat at some school, and (v) there is no student who has justified envy toward another student. When there are no bullying incidents (i.e., B = ∅), this concept reduces to the stability concept of Gale and Shapley (1962). As in the standard framework, stability in the current framework is intended to eliminate justified envy among students to consequently achieve fairness.20 In what follows, I fix (q s ) s∈S and S . A mechanism is a partial function that maps every (I , B) in its domain to a matching. For any student i, her assignment under a mechanism ϕ and (I , B) in its domain is denoted by ϕi (I , B). A mechanism ϕ is strategy-proof if there does not exist (I , B), 0

a student i ∈ I, and her stated preferences i such that 0

ϕi (i , −i , B) i ϕi (I , B), where −i ≡I\{i} (−J is substituted for I\J for any J ⊆ I). A mechanism ϕ is group strategy-proof 0 if there does not exist (I , B), a group of students I˜ ⊆ I, and their stated preferences  ˜ such that I

0 ˜ ϕi (I˜, −I˜, B) i ϕi (I , B) ∀i ∈ I,

˜ Group strategy-proofness may be important with strict preference holds for at least one student in I. since a group of bullies or victims may want to coordinate their preference revelations. FInally, I introduce a new criterion for evaluating mechanisms which will eventually connect school choice with anti-bullying programs. Definition 2. A mechanism ϕ is bullying-resistant if for any pair of contiguous inputs (I , B) and (I , B ∪ {(i, j)}) in its domain, ϕi (I , B) i ϕi (I , B ∪ {(i, j)}). There are two interrelated interpretations of this criterion. First, under a bullying-resistant mechanism, no student has an incentive (in terms of her preferences over schools) to bully others. In other words, if a mechanism violates the criterion, it itself might stimulate students to initiate or promote bullying problems in schools, which contradicts any anti-bullying policy. Second, it can be imposed as a disciplinary policy to pit a mechanism itself against bullying problems. If a mechanism is bullying-resistant, and this fact is widely recognized by students and 20

In Conclusion, I revisit the concept of stability and discuss its relation to privacy.

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their parents, it may work as a reactive and proactive deterrent against bullying incidents. I will elaborate on this point in Section 4.5.2.

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TWO COMPETING MECHANISMS

Since the seminal paper by Abdulkadiro˘glu and S¨onmez (2003), the debate in the literature has surrounded two mechanisms: the Gale-Shapley (GS) (student-optimal stable matching) mechanism (Gale and Shapley 1962) and the top trading cycles (TTC) mechanism (Shapley and Scarf 1974; Abdulkadiro˘glu and S¨onmez 2003). In this paper, I follow this tradition by considering variants of these two mechanisms.

3.1

STABILITY REVISITED: IMPOSSIBILITY RESULTS

Before introducing a variant of the GS mechanism, I reconsider stability. Although the stability concept introduced in the preceding section seems to fit naturally in the current framework, I will not pursue it in the rest of this paper. There are two reasons for this. First, as the following example demonstrates, a stable matching does not exist in general. Example 1. There are two schools {s1 , s2 } and three students {i1 , i2 , i3 }. The school capacities and a set of bullying incidents are given by q s1 = 2, q s2 = 1, B = {(i1 , i2 ), (i1 , i3 )}. The student preferences and the school priorities are: i1 : s1 , ∅

 s 1 : i3 , i1 , i2 , ∅

i2 : s1 , s2 , ∅

 s 2 : i2 , i1 , i3 , ∅

i3 : s2 , s1 , ∅ where “a : b, c, ∅” means b a c a ∅, and ∅ denotes a’s outside option if it is a student, and otherwise a vacant seat. If i1 enters s1 , i2 who has the highest priority for s2 must enter the school in order to achieve stability. Now consider the following matching:    s1 s2 ∅    i ∅ i i  . 1 2 3 A matrix denotes a matching: i1 is matched to s1 while s1 has its one of the available seats vacant, i2 is matched to s2 , and i3 is assigned her outside option. In this matching, i3 has justified envy 9

toward i1 , hence it is not stable. Next, suppose that i1 is assigned her outside option. Then, to achieve stability, i2 and i3 must enter their favorite schools. Now consider the following matching:    s1 s2 ∅    i ∅ i i  . 2 3 1 However, this matching is not stable because i1 has justified envy toward i2 .

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The second reason I no longer aim at the stability concept concerns its incompatibility with strategy-proofness. As the next example shows, there is no strategy-proof mechanism that selects a stable matching whenever there exists one. Example 2. There are two schools {s1 , s2 } and three students {i1 , i2 , i3 }. The school capacities and a set of bullying incidents are given by q s1 = 2, q s2 = 1, B = {(i1 , i3 ), (i2 , i3 )}. The student preferences and the school priorities are: i1 : s1 , ∅

 s 1 : i2 , i3 , i1 , ∅

i2 : s2 , s1 , ∅

 s 2 : i3 , i1 , i2 , ∅

i3 : s1 , s2 , ∅ In this problem, there are two stable matchings,    s1 s2   , µ =  i1 i2 i3 

   s1 s2 ∅   . and µ˜ =  i3 ∅ i2 i1 

Suppose that a mechanism chooses µ under the above preference profile I . Consider the following 0

reported preference ordering i3 of i3 , 0

i3 : s1 , ∅. Then, µ˜ is the unique stable matching under (i3 , {i1 ,i2 } ). Since i3 prefers µ(i ˜ 3 ) to µ(i3 ), she can 0

profitably misreport her preferences under I . Suppose on the other hand that a mechanism chooses µ˜ under I . Now consider the following 0

reported preference ordering i1 of i1 , 0

i1 : s1 , s2 , ∅. Then, µ is the unique stable matching under (i1 , {i2 ,i3 } ). Since i1 prefers s1 to her outside option 0

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according to her true preferences i1 , she prefers µ(i1 ) to µ(i ˜ 1 ), meaning that she can profitably misreport her preferences under I .

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Examples 1 and 2 lead me to seek a weaker, but still compelling stability concept. To do so, I impose more structure on each set of bullying incidents: it is said to satisfy the regularity condition if for all i, j ∈ I, (i, j) ∈ B implies that there is no k ∈ I such that ( j, k) ∈ B. One natural interpretation would be that B represents the ongoing bullying incidents at the time a social planner uses the information to determine the outcome.21 Under this condition, B separates students into three categories: a set Bu(B) of bullies (i.e., i ∈ Bu(B) if and only if (i, j) ∈ B for some j ∈ I), a set Vi(B) of victims (i.e., i ∈ Vi(B) if and only if ( j, i) ∈ B for some j ∈ I), and the rest. Note that Examples 1 and 2 have already demonstrated that, even under the regularity condition, the standard stability concept retains the drawbacks. This is exactly the reason why I pursue quasi stability in the rest of this paper. Definition 3. A matching is quasi stable if (i) it is feasible, (ii) individually rational, (iii) satisfies the separation principle, (iv) there is no student who wishes for a vacant seat at some school, and (vq ) whenever i has justified envy toward j, i ∈ Bu(B) and j < Bu(B). Under this criterion, whenever a bully has justified envy toward a non-bully, a social planner ignores it, by implicitly or explicitly bestowing a higher social priority on the latter. Besides, it requests an equal treatment of equals among non-bullies and bullies.22 In the appendix, I show that other stability concepts that meet (i) feasibility, (ii) individual rationality, (iii) the separation principle, and other reasonable conditions weaker than (iv) and (v) in Definition 1, turn out to wreck mechanism design embracing anti-bullying policy. Hence the quasi stability concept is, among several of the possibilities, the only refinement upon the standard stability concept that fits in the mechanism design analysis. A quasi stable matching is student-optimal if there does not exist any other quasi stable matching that dominates it. Student-optimality works as a normative refinement of quasi stability. While 21

The regularity condition features centrally only in the discussion of stability. As I will argue in Section 3.3, a variant of the top trading cycles mechanism can dispense with this condition, thus this interpretation as well. See Section 4.2 for further discussion on this condition. 22 Note that this partial elimination of justified envy is not equivalent to obtaining the complete elimination of it after lowering the school priorities for bullies below those for non-bullies. The partial elimination accommodates the possibility that a bully enters a school which is preferred by a non-bully to her own assignment. I regard the school priorities, which usually respect whether a student has siblings in a school and commuting time, for granted while considering a social priority. These two notions are conceptually different.

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a student-optimal stable matching in a model without any bullying incidents is unanimously preferred by the students to other stable matchings, this uniqueness property no longer obtains in the current model. Next example demonstrates this point. Example 3. There is one school {s} and two students {i1 , i2 }. The school capacity and a set of bullying incidents are given by q s = 1, B = {(i1 , i2 )}. The student preferences and the school priorities are:  s : i1 , i2 , ∅

i1 : s, ∅ i2 : s, ∅ In this problem, there are two quasi stable matchings:    s ∅   , µ =  i1 i2 

   s ∅   . and µ˜ =  i2 i1 

Since the school capacity is one, they are both student-optimal quasi stable matchings.

3.2

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TWO-ROUND GALE-SHAPLEY MECHANISM

Now I present a new mechanism that, for any given input, results in a student-optimal quasi stable matching. To do so, I first describe the deferred acceptance (DA) algorithm of Gale and Shapley (1962) and review its properties. Step 1: Let every student apply to her most preferred choice. If she chooses her outside option, she is tentatively accepted. Each school s ∈ S tentatively accepts the q s highest applicants following its priority order. The remaining students are rejected. Step t (> 1):

Every student who was rejected in the previous step applies to her next choice.

Each school s ∈ S tentatively accepts the q s highest students among the new applicants and those tentatively accepted. The remaining students are rejected. The algorithm terminates when every student is tentatively accepted by a school or her outside option. It produces the student-optimal stable matching when B = ∅. Moreover, the Gale-Shapley (GS) mechanism (also often called the student-optimal stable mechanism) induced by the DA al-

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gorithm is strategy-proof (Dubins and Freedman 1981; Roth 1982). The outcome, however, is not (constrained) Pareto efficient in general and thus the GS mechanism is not group strategy-proof. The GS mechanism or its variants have been implemented in many school districts such as New York City, Boston, Chicago, and England (Pathak and S¨onmez 2011). It is also now used for the medical interns match in Japan (Kamada and Kojima 2011a), and the United States (Roth 1984).23 The two-round deferred acceptance (TDA) algorithm reiterates the DA algorithm twice and hierarchically in the following way: Round 1:

Exclude all the bullies and run the DA algorithm. When it terminates, each school

accepts all the applicants and reduces its capacity by the number it has accepted. Round 2: Every bully puts every school where her victims were accepted in Round 1 below her outside option. Then run the DA algorithm for the schools and bullies with the modified preference profile and the remaining school seats. I refer to the induced preference revelation direct mechanism as the two-round Gale-Shapley (TGS) mechanism, denoted by ϕTGS . It allocates the slots to the non-bullys preferentially, then lets the bullies in the remaining slots so as not to violate the separation principle. The next theorem shows that the TGS mechanism inherits the plausible properties from the GS mechanism and still satisfies the bullying-resistance property. Theorem 1. The two-round Gale-Shapley mechanism produces a student-optimal quasi stable matching for any input in its domain, and is strategy-proof and bullying-resistant. Let me note that the TGS mechanism is not the unique mechanism satisfying all the desiderata. Consider a mechanism that produces µ in the problem of Example 3, and determines the allocation according to the TDA algorithm for all other inputs. While this mechanism differs from the TGS mechanism which produces µ˜ in the same problem, it also satisfies these desiderata. This observation opens the door for new mechanisms that may satisfy additional desiderata about, for example, the number of students admitted by schools, or the welfare of the victims.

3.3

TOP TRADING CYCLES MECHANISM

Here I present a variant of the top trading cycles (TTC) mechanism denoted by ϕT TC . Unlike the TGS mechanism, this mechanism works without the regularity condition. The original mechanism 23

Roth (2008) surveys the use of the GS mechanism in practice.

13

has been considered for use in several school districts in the United States, and it was finally adopted by the San Francisco Unified School District in 2010, and the New Orleans Recovery School District in 2012.24 The TTC mechanism for bullying incidents finds a matching via the following top trading cycles (TTC) algorithm: Step 1:

Let every student point to her favorite school. If there is no available and acceptable

school seat for her, she is assigned her outside option and removed. Each school s ∈ S points to the student with the highest priority among the remaining students. As there are a finite number of schools and students, there is at least one cycle, i.e., a sequence of distinct schools and students (i1 , s1 , i2 , s2 , ..., ik , sk ) where i1 points to s1 , s1 points to i2 , ..., ik points to sk , and sk points to i1 . In every such cycle, every student is assigned a seat at the school she points to and removed. Step t (> 1):

Let every remaining student i point to her favorite school among those who still

have available seats, and have not accepted in the previous steps any student j who victimizes or is victimized by i (i.e., (i, j) ∈ B or ( j, i) ∈ B). If there is no available and acceptable school seat for her, she is assigned her outside option and removed. Each school points to the student with the highest priority among the remaining students. There is at least one cycle. Every student in every cycle is assigned to the school she points to and removed. The algorithm terminates when there are no remaining students. It asks students endowed with the highest/top priorities to trade their rights repeatedly. A trade among a group of students occurs when each member points to her favorite school whose highest priority is bestowed on another student in the same group. Moreover, a bully is allowed to wish for a school only if it has not accepted her victims, and a victim examines a school only if it has not accepted her bullies. Abdulkadiro˘glu and S¨onmez (2003) introduced the original TTC mechanism for school choice which was an adaptation of Gale’s TTC mechanism (Shapley and Scarf 1974). They showed that the mechanism is strategy-proof and produces a Pareto efficient matching for any input. The next theorem summarizes similar properties of the TTC mechanism for bullying incidents. Theorem 2. The top trading cycles mechanism for bullying incidents produces a constrained Pareto efficient matching for any input, and is group strategy-proof and bullying-resistant.25 24

See http://www.sfusd.edu/en/assets/sfusd-staff/enroll/files/board-of-eduation-student-assignment-policy.pdf (San Francisco); http://www.nola.com/education/index.ssf/2012/04/centralized enrollment in reco.html (New Orleans). 25 Papai (2000) introduced a wide class of mechanisms called hierarchical exchange mechanisms, uniquely char-

14

Beyond the conflict between stability and efficiency, there are at least two advantageous natures in the TTC mechanism over the TGS mechanism. First, the mechanism can deal with any form of bullying that can be reasonably reduced to binary relations. As I will discuss in Section 4.2, some types of bullying would likely to fall outside of the scope created by the regularity condition. In contrast, the TTC mechanism may be capable of dealing with them thanks to its highly abstract construction. Second, there is a widespread sense that some types of bullying incidents, including indirect bullying and ijime (see Section 4.2), are difficult to detect (Morita et al. 1999; Olweus 1993). Thus B must be more or less constructed by students’ self-reports. Since the TTC mechanism treats a bully-role and a victim-role symmetrically, bullying-resistance implies that if a student reported her false bullying episode (involved either as a bully or a victim), her assignment would never become better off in terms of her preferences over schools. Meanwhile, a student may not report her insignificant bullying episode. She may weigh utility gained from participating in some excellent school against disutility caused by her assailants, and spontaneously abstain from reporting her bullying episode. In this way, the mechanism can potentially strike a balance between one’s preference for an excellent school and desire to eschew her assailants. Hence the mechanism may be equipped with a reasonable “incentive structure” surrounding a bully reporting system, which makes it useful in practice. Remark 1. Although the TTC algorithm for bullying incidents removes several cycles and students simultaneously, the outcome does not change if it instead removes a cycle and a student one at a time in random order. The proof is analogous to the one for the original TTC algorithm (see Carroll 2010; Lemma 1), which I omit. Remark 2. The serial dictatorship mechanism for bullying incidents can be defined in the same manner as the TTC mechanism for bullying incidents.26 Theorem 2 reveals, as its corollary, that this mechanism also satisfies the same desiderata in its statement. acterized by Pareto efficiency, reallocation-proofness, and group strategy-proofness. In contrast to the standard TTC mechanism (Abdulkadiro˘glu and S¨onmez 2003), however, the TTC mechanism for bullying incidents is not a hierarchical exchange mechanism since the separation principle is imposed on constrained Pareto efficiency. In the appendix, I give the independent proof for Theorem 2 which does not rely on the previous results in the literature. 26 The standard (random) serial dictatorship mechanism is exclusively used in real-life indivisible goods allocation problems such as the on-campus housing assignment in the US universities (Abdulkadiro˘glu and S¨onmez 1999), the allocation of dormitory rooms to students (Hylland and Zeckhauser 1979), and many school transfer processes (e.g., the Austin School “non-priority transfers” (http://archive.austinisd.org/academics/parentsinfo/transfer/)). School admission processes used in New York, Eugine (http://www.4j.lane.edu/choice]aboutschoolchoice), and Tokyo (Yasuda 2010) also rely on this mechanism.

15

4

DISCUSSION

4.1

BULLYING AS A GROUP PROCESS

It is often argued that bullying is a “group process” (O’Connell et al. 1999; Salmivalli et al. 1996; for a recent survey, see Salmivalli 2010). The “group” refers to either (i) a mob of students ganging up against one or a few students, or (ii) a peer cluster comprising a number of bystanders. Although these two points bear upon each other, and the existing bullying intervention policy is often tailored to the latter (Salmivalli et al. 2009a,b), I first address the former. Namely, is it possible to design a school choice mechanism that curtails the energy behind a group bullying (mobbing)? Here I give an initial answer to this question. To do so, I generalize the notion of bullyingresistance to study the resistibility of a mechanism to the impulse for mobbing. The next definition formulates the idea: Definition 4. A mechanism ϕ is group bullying-resistant if for any pair of contiguous inputs ˜ in its domain, ϕi (I , B) i ϕi (I , B ∪ {(i, j)|i ∈ I}) ˜ for all i ∈ I. ˜ (I , B) and (I , B ∪ {(i, j)|i ∈ I}) This criterion requires that for any member of a mob, her assignment gets weakly worse off if her bullying episode is revealed to a social planner. In other words, no one ever expects to benefit from initiating bullying even if other students would join her and thus induce mobbing. Given the formal definition of group resistance, the initial question is answered negatively by the following two impossibility theorems. Theorem 3. There is no mechanism for a domain satisfying the regularity condition that produces a constrained Pareto efficient matching for any input, and is group bullying-resistant. Theorem 4. There is no mechanism for a domain satisfying the regularity condition that produces a quasi stable matching for any input, and is group bullying-resistant. These negative answers may give an obstacle to progress in some cases, but not in others. There is typically a ringleader in a mob, supported by follower bullies (who join the mob after the leader), and reinforcers (who directly or indirectly encourage the bullying) (Salmivalli et al., 1996). It is by no means clear whether the set of bullying incidents B should include every student in every role in the mob. At present, there is not enough evidence regarding the peer-group socialization effect on bullying behaviors (but see Espelage et al. (2003) for an initial attempt in this line). To take a

16

step in this direction, economists need to wait for the accumulation of scientific evidence in both developmental and educational research. If the group bullying-resistance property turns out to be essential for anti-bullying school choice, one solution would be to dispense with the second round of the TDA algorithm. Namely, the mechanism that assigns outside options for bullies and determines a match for the rest by the standard GS mechanism is group bullying-resistant. While maintaining fairness among bullies and among non-bullies, it punishes bullies more severely than the TGS mechanism. I formulate this idea as a new stability concept which is weaker than quasi stability. Definition 5. A matching is weakly quasi stable if (i) it is feasible, (ii) individually rational, (iii) satisfies the separation principle, (ivw ) whenever i wishes for a vacant seat at some school, i ∈ Bu(B), and (vq ) whenever i has justified envy toward j, i ∈ Bu(B) and j < Bu(B). Proposition 1. A mechanism induced by a variant of the two-round deferred acceptance algorithm that stops at the end of the first round and assigns outside options to bullies produces a weakly quasi stable matching for any input in its domain, and is strategy-proof and group bullying-resistant. The proposition is true since the GS mechanism is strategy-proof. Condition (ivw ) in the definition of weak quasi stability would be justifiable if the systematic sanction on bullies is socially endorsed and the schools recognize that increasing the number of these students create more problems than benefits. Although this is merely one solution to the impossibility result set together by Theorem 3 and 4, other possible solutions are not scrutinized here and left for future study.

4.2

INDIRECT BULLYING, BULLY/VICTIMS, AND IJIME

In this section, I describe three types of bullying, i.e., indirect bullying, bully/victims, and ijime (Japanese bullying), that may fall outside of the scope created by the regularity condition. I also discuss how the analytical framework in this paper, and especially the TTC mechanism for bullying incidents are adaptable to these problems.

4.2.1

INDIRECT BULLYING

The analytical framework standing on the regularity condition is tailored to direct bullying. In direct bullying, a victim confronts a bully face to face (examples cover physical aggression such

17

as shoving, slapping, choking, and verbal abuse such as calling a derogatory nickname to make a victim feel mocked or humiliated). In indirect bullying, a victim is undermined her reputation by gossip and shunning of any sort, but does not know who bullies her. This type of bullying is difficult to stop and it is hard to detect the culprit if any.27 Cyberbullying (Li et al. 2012) is a well known example of indirect bullying. It occurs through the Internet, using electronic forms of contact. The Canadian Broadcasting Company, for example, reported in March 2005:28 David Knight’s life at school has been hell. He was teased, taunted and punched for years. But the final blow was the humiliation he suffered every time he logged onto the internet. Someone had set up an abusive website about him that made life unbearable. “Rather than just some people, say 30 in a cafeteria, hearing them all yell insults at you, it’s up there for 6 billion people to see. Anyone with a computer can see it,” says David. “And you can’t get away from it. It doesn’t go away when you come home from school. It made me feel even more trapped.” He felt so trapped he decided to leave school and finish his final year of studies at home. One defensible way to wrestle with indirect bullying would be to handle incidents where (i, j) ∈ B can imply “ j has been a victim of an indirect bullying episode and she is in desperate need not to go to the same school with i since i may be a latent culprit or i may disturb her new life”. To the extent that finding a culprit is a troubling issue, a victim can also be classified as a latent bully. As such, indirect bullying seems at odds with the regularity condition.

4.2.2

BULLY/VICTIMS

Bully/Victims are those who “both bully and have been victimized” (Haynie et al. 2001). Although many reports build on different methods to extract the information about bully/victims, and thus it is difficult to compare and pool these results about the prevalence of this kind, the general conclusion is that some students are indeed bullying others and victimized by others at the same time (Veenstra et al. 2005). Hence the regularity condition may restrict the scope of the benchmark framework. 27 Girls are apt to engage in indirect or relational bullying which includes social exclusion. On the other hand, boys’ bullying tends to be direct (Archer and Cote 2005). 28 http://www.cbc.ca/news/background/bullying/cyber bullying.html

18

The literature on bullying has confirmed that bully/victims have distinct characteristics, both behaviorally and psychologically, in comparison to other bullies and victims (Berger 2007; Haynie et al. 2001; Kumpulainen et al. 1998). As such, it is not clear whether one should treat bullying episodes involving bully/victims and those involving just bullies in the same manner. A longitudinal analysis on bully/victims such as Pollastri et al. (2010) may help disentangle any confusion.

4.2.3

IJIME (JAPANESE BULLYING)

As I stated in the “citation policy” (footnote 3), much of the cited evidence on the nature of bullying and anti-bullying programs concern bullying incidents in the western countries. However, a number of researchers have reported differences in the nature and extent of school bullying between countries (Smith et al. 1999). The definitions of bullying (Smith et al. 2002) and children’s perceptions of these problems (Kanetsuna et al. 2006) also vary across countries. In this section, I focus on Japanese bullying as one example of non-western bullying and examine its fit for the analytical framework in this paper. Ijime, which is the Japanese term closest in meaning to the word bullying, has been a pressing social issue in Japan for the last 30 years.29 Although bullying is generally characterized by direct forms of aggression which often occurs in the playground (Craig et al. 2000), a major type of ijime is “characterized by more indirect forms of aggressions, often conducted by a group of pupils who were a victim’s classmates or even “friends” to the victim, mainly in the classroom” (Kanetsuna et al. 2006). Verbal forms (e.g., teasing, verbal threats) are more frequent than physical forms such as hitting and kicking (Morita et al. 1999). Hence, a major type of ijime causes mental rather than physical suffering to victims. Taki (2001) argues that personal factors (i.e., rearing conditions) do not meaningfully characterize bullies and victims in Japan. Thus ijime “can happen at any time, at any school and among any children” (p.2). His empirical data also suggests that from 1997 to 2000, over 30 percent of primary and junior high school students had bullied others and also experienced being bullied. Hence, ijime shares some of the features of indirect bullying and bully/victims-problems. Moreover, bullies can become victims and vice versa in a short period, which is quite opposite to the stable nature of the bully and victim roles in the west. Evidently, however, some students in Japan indeed think of themselves as victims and desire to escape from a toxic environment through 29

See Morita et al. (1999) for a rather formal definition of ijime and the history of socially acknowledged ijime problems in Japan.

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school choice. In this case, it would be unwise to assume the regularity condition to design a school choice mechanism. Finally, the implications of bullying-resistance on the TTC mechanism are worth highlighting. For both indirect bullying and ijime, it is often hard to detect the culprit if any. Imagine that a school choice mechanism is bullying-resistant, and some student is tormented by a slander. For those not being against her, it is of their interest to stick up for her so as not to be deemed as latent culprits. This interest may facilitate their positive and overt attitudes toward the victim. Hence bullying-resistance could induce a “group process” by converting “a peer cluster comprising a number of bystanders” into a group of defenders. This peer involvement on behalf of the victim would eventually put an end to a bullying (ijime) incident as it is well documented in the literature (Kanetsuna et al. 2006; O’Connell et al. 1999; Salmivalli et al. 1996).

4.3

OUTSIDE OPTION

In Section 2, I assumed that each student i has a preference ordering i over S ∪ {i} where “i” in the domain was interpreted as her “outside option”. In this section, I elaborate on the meaning of this concept and clarify the scope of the benchmark framework. Usually, the outside option refers to an opportunity (e.g., a private school, an exam school, or homeschooling) that is not under control of a public school choice. On the other hand, although labeled “outside option”, it should sometimes be interpreted as a public school preassigned to each student by a mechanism (i.e., “default option”). For instance, a mechanism used in Tokyo ensures seats in designated public schools for its participants (Yasuda 2010). School choice enrollment in Seattle also lets participants choose their initially assigned schools.30 Moreover, school transfer mechanisms used in many countries generally allow students to choose their current schools in the end. Since a bully and her victim may be ensured the same public school by a mechanism, or a bully (victim) may be assigned to a school that turns out to be the ensured option for her victim (bully), the current framework must be carefully applied in these practical situations. In the rest of this section, I describe two possible solutions that may help administrators carry out anti-bullying school choice in these situations. First, when a bully and her victim are supposed to be given the same school as their ensured options, it may be reasonable to change one or both of them to another neighborhood school, before collecting the information about their preferences 30

http://www.seattleschools.org/ See “school choice form instructions”.

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to determine the assignment. This approach may resonate with administrators in Tokyo who have historically allowed victims to change their designated schools. Second, it may also be reasonable to withdraw the ensured school options for bullies or victims. In fact, the New York City Department of Education and the Boston Public Schools do not preassign any schools to students. Hence if there are fairly many feasible schools for the students, it may be justifiable to run a mechanism without any ensured options for at least some of them.

4.4

INCORPORATING SCHOOL PREFERENCES

I have assumed thus far that school seats are objects to be consumed by students, and school priorities are different from school preferences. There are, however, at least three reasons to believe that the implications of school preferences on anti-bullying policy deserve investigation. First, although the literature on school choice has generally assumed that school priorities are ¨ set by law (S¨onmez and Unver 2011), in some practical cases, schools are active players who report their preferences over students (e.g., “screened schools” in New York City; universities in the Turkish higher educational system (Balinski and S¨onmez 1999)). Second, it has been widely believed and suggested (Hanushek 1986) that officials in public schools, and other institutions of higher education (Epple et al. 2006), do care the composition and academic performance of students. Since nobody can overemphasize the importance of human capital accumulation in schools, public school choice in the future may possibly employ more nuanced information, besides the simple priority rankings, to place students. Preference rankings of schools over the set of subsets of students may potentially constitute additional useful information for a social planner. Now I introduce a new model that incorporates schools preferences over students. Each school s ∈ S has a strict preference ordering  s over the set of subsets of students. I assume that a school preference ordering  s satisfies a responsiveness condition, i.e., (i) for any {i}, { j}  s ∅ and any ˜ I˜ ∪ {i}  s I˜ ∪ { j} ⇔ {i}  s { j}, and (ii) for any i ∈ I and I˜  s ∅ with i < I, ˜ I˜  s ∅ with i, j < I, I˜ ∪ {i}  s I˜ ⇔ {i}  s ∅ (Roth 1985).31 A matching µ is individually rational if µ(i) i i for every i ∈ I, and {i}  s ∅ for all i ∈ µ(s) for every s ∈ S . A matching µ dominates another matching ν 31

Note that by this condition, I assume that schools are not given any information about bullying incidents among students; otherwise they may eschew admitting a bully together with her victim. This assumption is feasible since this type of information is in general confidential. On the other hand, however, the implications of this section (Theorem 5, Proposition 3) hold as they are if schools know the identity of each bully and victim, and lower the preference ranking of those violating the separation principle.

21

if µ(i) i ν(i) for all i ∈ I, µ(s)  s ν(s) for all s ∈ S , and at least one of them holds with a strict relation. A matching is constrained Pareto efficient if it is feasible, individually rational, satisfies the separation principle, and there is no other matching that satisfies these three conditions and dominates it. Given a matching µ, (i, s) is called a blocking pair if s i µ(i) and there is a possibly empty set ˜ ∪ {i}| ≤ q s , (ii) (µ(s) \ I) ˜ ∪ {i}  s µ(s), and (iii) there is no j ∈ µ(s) \ I˜ I˜ ⊆ µ(s) such that (i) |(µ(s) \ I) such that (i, j) ∈ B or ( j, i) ∈ B. A matching is stable if it is feasible, individually rational, satisfies the separation principle, and there is no blocking pair. It is easy to show by Example 1 in Section 3.1, that a stable matching may not exist in this model. In the following, I fix (q s ) s∈S and S .32 A mechanism and strategy-proofness are defined in the same manner as the basic framework. Example 2 in Section 3.1 demonstrates that there is no mechanism that is strategy-proof and selects a stable matching whenever there exists one in this model (note that, in the example, {i2 }  s1 {i3 } and the responsiveness condition imply {i1 , i2 }  s1 {i3 }, thus the matching µ remains stable). As in the previous analysis, I study a weaker notion of stability. Given a matching µ, a blocking pair (i, s) is permissible if i ∈ Bu(B), s i µ(i), and for any set I˜ ⊆ µ(s) satisfying the three conditions in the definition of a blocking pair, I˜ \ Bu(B) , ∅. A matching is quasi stable if it is feasible, individually rational, satisfies the separation principle, and any blocking pair is permissible.33 To convince students and school principals that their deviation from the assignment is unjustifiable, it is important to provide a rationale for instability. Quasi stability in this framework only allows “justified envy” from a bully to a set of students involving non-bullies. In other words, a hypothetical replacement of a student i with a set of students I˜ ⊆ µ(s) helps a bully i at the sacrifice of non-bullies, which is incongruent with the differential social priorities for bullies and non-bullies. A student-optimal quasi stable matching is defined analogously to the one in the previous framework. The next example demonstrates that even a student-optimal quasi stable matching may not be constrained Pareto efficient in this framework. Example 4. There are two schools {s1 , s2 } and six students {i1 , ..., i5 }. The school capacities and a 32

School preferences are usually not known to a social planner, thus they should be revealed by the institutions in actuality. I fix them, and thus assume that they are exogenously given to a social planner, because the strategic behaviors of the institutions are not in my interest. In fact, when B = ∅, there is no mechanism that produces a stable matching for any input and is dominant strategy incentive compatible for students and schools (Roth 1982), or even only for schools (Roth 1985). These facts carry over to the current model and analysis. 33 See the appendix (section G) for a discussion of strong quasi stability which will eventually turn out to be incompatible with anti-bullying school choice.

22

set of bullying incidents are given by q s1 = q s2 = 2, B = {(i1 , i4 ), (i1 , i5 ), (i2 , i4 ), (i3 , i5 )}. Let student and school preferences be such that: i1 : s2 , s1 , ∅

 s1 : {i3 , i4 }, {i1 , i2 }, {i1 }, {i5 }, {i3 }, {i4 }, {i2 }, ∅

i2 : s2 , s1 , ∅

 s2 : {i1 , i2 }, {i3 , i4 }, {i3 }, {i1 }, {i5 }, {i2 }, {i4 }, ∅

i3 : s1 , s2 , ∅ i4 : s1 , s2 , ∅ i5 : s1 , s2 , ∅ The preference lists for the schools are curtailed suitably. Consider the following two matchings,    s1 s2 ∅   , µ =  i1 i2 i3 i4 i5 

   s1 s2 ∅   . and µ˜ =  i3 i4 i1 i2 i5 

It is easy to show that µ is a student-optimal quasi stable matching, and is dominated by µ. ˜

^

The two-round deferred acceptance algorithm in this framework is characterized by letting every school accept or reject students in each step following its preference ordering over the set of singletons comprising one student. Note that the algorithm uses only the information about schools’ preference rankings over students, which makes it applicable to school choice in real life. The next theorem shows that the induced two-round Gale-Shapley mechanism satisfies the same desiderata as those for the same mechanism in priority-based school choice, and its outcome is indeed constrained Pareto efficient for any input. Theorem 5. The two-round Gale-Shapley mechanism produces a student-optimal quasi stable and constrained Pareto efficient (in terms of both student and school preferences) matching for any input and is strategy-proof and bullying-resistant. Finally, I briefly study a blocking group/coalition, comprising one school and several students. A school may desire to deviate with several students, and its principal must be convinced that the action is unjustifiable in light of anti-bullying policy. Given a matching µ, (I1 , s) (I1 ⊆ I, I1 , ∅) is called a blocking group if s i µ(i) ∀i ∈ I1 , and there is a possibly empty set I2 ⊆ µ(s) such that (i) |(µ(s) \ I2 ) ∪ I1 | ≤ q s , (ii) (µ(s) \ I2 ) ∪ I1  s µ(s), and (iii) there is no (i, j) ∈ B such that i, j ∈ (µ(s) \ I2 ) ∪ I1 . The next proposition describes properties of a blocking group for an output of the TGS mechanism. 23

Proposition 2. For any output of the two-round Gale-Shapley mechanism, if (I1 , s) is a blocking group and I2 satisfies the three conditions in its definition, I1 ∩ Bu(B) , ∅, and I2 \ Bu(B) , ∅. Moreover, (i) |I1 ∩ Bu(B)| > |I2 ∩ Bu(B)| or (ii) there is (i, j) ∈ B such that i ∈ I1 and j ∈ I2 . Hence any group deviation from the output of the TGS mechanism helps some bullies to the detriment of some non-bullies. Moreover, the deviation either (i) results in an increase in the number of bullies in a school, which may make the execution of its education program difficult, or (ii) admits a bully and discards her victim, which clearly clashes with anti-bullying policy. Although undeniably preliminary, I believe that this section has provided an initial successful attempt to elucidate the implications of school preferences on anti-bullying school choice. From a more general perspective, it is a promising enterprise to study whether a quasi stable matching or its variants can serve as an appropriate social goal in centralized two-sided matching markets when institutions have more complex preferences. I leave this analysis for future research.

4.5

ANTI-BULLYING PROGRAMS AND SCHOOL CHOICE

In this section, I discuss the complementary roles of anti-bullying programs and market design through their concomitant research agenda for school bullying and school choice. I also discuss possible future directions that would advance a fruitful collaboration between the two fields.

4.5.1

ANTI-BULLYING PROGRAM CAN INFORM SCHOOL CHOICE

Anti-bullying programs can inform school choice, above and beyond the market design perspective, through a channel of research on bullying. First, longitudinal studies and other close examination of social relationships among students and teachers would tell us nuanced implications about the goodness and badness of student assignment. Research has shown since its outset (Olweus 1993) that causes of bullying can not be attributed to personality of students alone, but should also be found in social environment such as teachers, principals, and education curricula. Hence future research on school choice, and probably other student assignment as well, must bear on these implications and go beyond a simple utilitarian perspective. Second, the expertise of bullying researchers can clarify whether and to what extent a social planner can extract and construct a set of bullying(∗) problems B(∗) . It is often said that school bullying remains undiscovered. Adults are generally not privy to the bullying episodes (Olweus 1991). And a bullying incident in western countries frequently occurs during the activity in the 24

playground (Craig et al. 2000), but it is difficult to detect each problem since usually a few adults supervise students outside the classrooms. At the same time, anti-bullying legislation in almost all the states in the United States and European countries stipulate that teachers and principals must be responsive to bullying incidents in their schools. On this background, researchers have utilized and tested many strategies to detect bullying incidents in schools. Examples include self-reports from students (questionnaires/interviews), teacher and/or parent ratings of individual students, asking students to nominate bullies and victims (Veenstra et al. 2005), and identification of possible victims and bullies (Olweus 1993). Hence I expect and hope that a social planner, with the help of school principals, teachers, and parents, will be able to uncover and construct the information B(∗) which is close enough to the “real” set of bullying incidents. Another possible approach that can be brought up by economists is concerned with agency problems. Contract theory and personnel economics have a long history of addressing employeremployee (principal-agent) problems especially when workers may game performance measures to their advantage (Lazear 1986; Lazear and Oyer 2009). A natural question is: Is there any effective scoring rule that prompts teachers to find out and report the bullying incidents that occurred before their eyes? A solution may be similar to the rules characterized in other areas (e.g., Fong 2009). On the other hand, the analytical framework for anti-bullying school choice will also need to be resilient enough to deal with the possibility of partial revelation of bullying incidents. It no doubt should be based on the accumulated knowledge in the literature on bullying as to which type of bullying episode is easily detected, and which should be focused. Above all these difficulties, I believe that anti-bullying school choice can more or less help students reduce there fear for bullying incidents at their new schools.

4.5.2

SCHOOL CHOICE CAN INFORM ANTI-BULLYING PROGRAMS

Dan Olweus, the pioneer of the anti-bullying programs, reported extraordinary high prevalence of the bullying incident in Norway (15 percent of 568,000 primary or junior high school students were involved in 1983-4), which then reduced by half in two years after he implemented the Olweus Bullying Intervention Programme (Olweus 1993). Inspired by this remarkable success, a plethora of programs have been marketed in many countries, although generally they have been less successful or even failed to show any significant improvement.34 With the history of success 34

However, as reported in Berger (2007), few school interventions have been scientifically evaluated.

25

and disappointment, there has been an ongoing debate as to what bullying intervention/prevention strategy is appropriate and effective.35 As a result, researchers have reached agreement on some pivotal aspects of the effective anti-bullying policy. One of the keys to successful intervention is a set of rules against bullying and consequences for breaking them (Smith et al. 2004). For example, Olweus (1994) claimed that “the key principles for preventing school bullying include the “nonhostile, nonphysical sanctions” against “violations of limits and rules”. The examples of the sanction strategies span the withdrawal of privileges, and the zero-tolerance policies. In this light, the bullying-resistance property of the anti-bullying school choice mechanisms is congruent with these premises. Moreover, anti-bullying school choice requires a city(community)-level intervention, and dispenses with any suspension strategies in schools.These are probably the defining characteristics of this new bullying prevention strategy, compared to other school-level intervention and a zerotolerance policy. I believe that this distinct and new deterrent to bullying can help increase the effectiveness of each anti-bullying program.

5

CONCLUSION

In this paper, I considered the design of school choice mechanisms in terms of their resistance against bullying incidents, motivated by an educational concern for peer effects, the historical background of school choice in Japan, and the widespread anti-bullying policies in the world. Under the constraint that a student has to be separated from her assailants, I formalized the criterion of bullying-resistance that requires that the assignment for a student be less preferred if she is involved in an additional bullying episode as a bully. Based on the observation that the standard stability concept is not appropriate as a social goal in the benchmark framework, I first defined the quasi stability concept, which puts lower social priority on bullies compared to a non-bulliies, thereby allows the former to have justified envy toward latter. Then I proposed the two-round Gale-Shapley mechanism, which produces a studentoptimal quasi stable matching for any input, and is strategy-proof and bullying-resistant. Finally, I showed that a variant of the Top Trading Cycles mechanism produces a constrained Pareto efficient matching for any input, and is group strategy-proof and bullying-resistant. An auxiliary policy is indispensable for each mechanism to stand as a part of the anti-bullying 35

Smith et al. (2004) and Merrell et al. (2008) survey both large-scale and small-scale anti-bullying programs conducted in the world.

26

program. The detection of the school bullying incidents of paramount gravity needs to be achieved by the program as a whole. Admittedly, a policy maker needs to determine the roles and details of each mechanism in the program depending on circumstances at hand. I demonstrated in the discussion sections that the benchmark framework proposed in this paper is adaptable to some contingent demands. For a victim, experience of bullying during the school days will loom large in her mind for the rest of her life. In the worst case, she commits suicide to alleviate the pain. There is every reason to believe that researchers together with a board of education should seek an effective policy to help the potential victims. In this course, both theoretical and empirical analysis of anti-bullying school choice need to be further investigated.

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APPENDIX [NOT FOR PUBLICATION] A PROOF OF THEOREM 1 Outputs of the TGS mechanism satisfy feasibility, individually rationality, and the separation principle by the flow of the TDA algorithm. To prove other properties, I first introduce the following lemma which follows from Erdil and Ergin (2008; Lemma 1): Lemma 1. Suppose B = ∅ and let µ be an arbitrary stable matching. For any µ˜ which dominates it and for any s ∈ S , |µ(s)| < q s if and only if |µ(s)| ˜ < q s , and in this case, µ(s) = µ(s). ˜ Proof of Theorem 1 Quasi Stability: Let µ ≡ ϕTGS (I , B). If there exist i ∈ I and s ∈ S such that s i µ(i) and |µ(s)| < q s , then i ∈ Bu(B) (otherwise i was rejected by s at Round 1 of the TDA algorithm), and s accepts her victim (otherwise i was rejected by s at Round 2). Hence, in µ, no student wishes for a vacant seat at some school. Suppose there exist i, j ∈ I such that µ( j) i µ(i) and i µ( j) j. By its construction of µ, i ∈ Bu(B) (otherwise i was rejected by µ( j) at Round 1). If j ∈ Bu(B), µ( j) accepts i’s victim at Round 1 (otherwise i was rejected by µ( j) at Round 2). Hence, if i has justified envy toward j, i ∈ Bu(B) and j < Bu(B). Student-optimality: Let µ ≡ ϕTGS (I , B). Suppose that there exists a quasi stable matching µ˜ that dominates µ. Let I˜ = {i ∈ I | µ(i) ˜ i µ(i)}. There are two cases to consider. Case 1: I˜ \ Bu(B) , ∅. Let µ1 denote a matching such that µ1 (i) = µ(i) for all i < Bu(B), and µ1 (i) = i for all i ∈ Bu(B). We define µ˜ 1 in the same way. Consider a preference profile of bullies such that i i s for all s ∈ S and for all i ∈ Bu(B). By assumption, µ˜ 1 dominates µ1 . Since µ1 is a student-optimal stable matching given the modified preferences, there is at least one student i < Bu(B) at µ˜ 1 who (i) wishes for a vacant seat at some school s ∈ S , or (ii) has justified envy toward another student j < Bu(B) (i.e., µ˜ 1 ( j) i µ˜ 1 (i) and i µ˜ 1 ( j) j). Suppose the first possibility holds. By Lemma 1, if s ∈ S has a vacant seat at µ˜ 1 , it also has a vacant seat at µ1 . Since s i µ˜ 1 (i) i µ1 (i), she wishes for the vacant seat at µ1 , which contradicts the stability of µ1 . Next, suppose the second possibility holds. Then |µ˜ 1 (µ˜ 1 ( j))| = qµ˜ 1 ( j) is true for the same reason we discussed above. Since |µ˜ 1 (µ˜ 1 ( j))| = qµ˜ 1 ( j) implies |µ( ˜ µ( ˜ j))| = qµ( ˜ µ( ˜ j)) ∩ Bu(B) = ∅. Hence, ˜ j) and µ( i < Bu(B) and she has justified envy toward j at µ, ˜ which contradicts the quasi stability of µ. ˜

35

Case 2: I˜ ⊆ Bu(B). Let µ2 denote a matching such that µ2 (i) = µ(i) for all i ∈ Bu(B) and µ2 (i) = i for all i < Bu(B). We define µ˜ 2 in the same way. Consider the following modification of the preference profile of students and school capacities: for all i ∈ Bu(B), if s ∈ S accepts her victim at µ1 (= µ˜ 1 ), she lowers the preference ranking of s below her outside option; for each i < Bu(B), i i s for all s ∈ S ; each school s ∈ S reduces its capacity by |µ1 (s)| (=|µ˜ 1 (s)|). Then µ2 is a student-optimal stable matching and µ˜ 2 dominates µ2 . Suppose i ∈ Bu(B) wishes for a vacant seat of s ∈ S at µ˜ 2 . By Lemma 1, s also leaves the seat vacant and does not accept any victims of i at µ. ˜ This contradicts the quasi stability of µ. ˜ Next, suppose i ∈ Bu(B) has justified envy toward j ∈ Bu(B) at µ˜ 2 . Since µ˜ 2 ( j) does not accept any victims of i at µ, ˜ this assumption also contradicts the quasi stability of µ. ˜ Strategy-proofness: Since the GS mechanism is strategy-proof, no one has any incentives to misrepresent her preferences at Round 1 and 2. Thus, the TGS mechanism is also strategy-proof. Bullying-resistance: Suppose that B and B0 = B ∪ {(i, j)} are both sets of bullying incidents. There are two cases to consider. Case 1: i ∈ Bu(B). Let 0i denote i’s preference ordering that is used for the DA algorithm at Round 2 when the set of bullying incidents is B. When the set of bullying incidents is B0 , i (I , B0 ) below her outside option. Since the GS may further lower the preference ranking of ϕTGS j mechanism is strategy-proof, ϕTGS (I , B) 0i ϕTGS (I , B0 ). By the construction of preferences, i i (I , B0 ) 0i i. Note that for all s, s0 ∈ S ∪ {i} such that s, s0 0i i, s 0i s0 implies s i s0 . Hence ϕTGS i we have ϕTGS (I , B) i ϕTGS (I , B0 ). i i Case 2: i < Bu(B). Consider the following variations of the bullies’ preferences: for each (˜ Bu(B) , −Bu(B) , B0 ) k ∈ Bu(B), k ˜ k s for all s ∈ S , and ˜ i places only her assignment ϕTGS i above her outside option (if ϕTGS (˜ Bu(B) , −Bu(B) , B0 ) = i, then all the schools in S move bei low her outside option). Then ϕTGS (˜ Bu(B) , −Bu(B) , B0 ) = ϕTGS (˜ Bu(B0 ) , −Bu(B0 ) , B). Since the GS i i TGS ˜ TGS ˜ mechanism is strategy-proof, ϕi (Bu(B) , −Bu(B) , B) i ϕi (Bu(B0 ) , −Bu(B0 ) , B), which implies ϕTGS (˜ Bu(B) , −Bu(B) , B) i ϕTGS (˜ Bu(B) , −Bu(B) , B0 ). By the assumption i < Bu(B), ϕTGS (˜ Bu(B) , −Bu(B) i i i , B) = ϕTGS (I , B). Since ϕTGS (˜ Bu(B) , −Bu(B) , B0 ) i ϕTGS (I , B0 ), we conclude that ϕTGS (I , B) i i i i i ϕTGS (I , B0 ). This completes the proof. i

Q.E.D.

B PROOF OF THEOREM 2 Proof. Outputs of the TTC mechanism are always feasible, individually rational, and satisfy the separation principle by the construction of the TTC algorithm. Let µ ≡ ϕT TC (I , B). 36

Constrained Pareto Efficiency: Suppose that µ˜ is a feasible and individually rational matching that dominates µ. Let i ∈ I be a student who prefers µ(i) ˜ to µ(i), and suppose that there is no other student who prefers the assignment under µ˜ to the one under µ while being removed earlier in the TTC algorithm compared to i. Thus, for any student j who is removed earlier than i , µ( ˜ j) = µ( j). By the construction of the TTC algorithm, µ(i) ˜ i µ(i) implies that µ˜ does not satisfy the separation principle. Hence, µ is constrained Pareto efficient. Group Strategy-proofness : For an arbitrary matching µ, ˜ let µi˜ denote i’s preferences such that s µi˜ i if and only if s = µ(i) ˜ and s ∈ S . Then, µ is equivalent to ϕT TC (µi , −i , B) for any i ∈ I. (∵ Let C 1 , C 2 , ..., C n , ... be a sequence of one-by-one cycle removals given the preference profile I where we stretch the interpretation of a “cycle” to include the singletons comprising only one student. Suppose i is in C n . Hence whatever preferences i reports, we can remove C 1 , C 2 , ..., C n−1 in the first n − 1 steps. Now suppose i reports µi . After removing C 1 , ..., C n−1 , we can immediately remove C n , followed by C n+1 , C n+2 , ... which results in µ.) ˜ Let µ˜ denote ϕT TC (˜ I˜, −I˜, B). By the Suppose ϕTi TC (˜ I˜, −I˜, B) i µ(i) holds for all i ∈ I. previous discussion, ϕT TC (µI˜˜ , −I˜, B) = µ. ˜ Let C 1 , C 2 , ... be a sequence of one-by-one cycle removals in the TTC algorithm given an input (I , B). Suppose that it is possible to remove the first t − 1 cycles C 1 , ..., C t−1 in the same order under (µI˜˜ , −I˜, B) and there is a student in C t who is a member of I˜ (indeed, if C t is the first cycle where a nonempty subset of I˜ is involved, it is possible to remove C 1 , C 2 , ..., C t−1 in the first t − 1 steps no matter what profile of preferences I˜ reports). There are two cases to consider. Case 1: C t is a singleton comprising one student i. By the construction of the TTC algorithm, after removing C 1 , ..., C t−1 there is no remaining school that is preferable for i in terms of i to her outside option and has not accepted any student who victimizes her or is victimized by her. Hence i µi˜ s holds for all s ∈ S , meaning that C t can be removed immediately after C 1 , ..., C t−1 even under the preference profile (µI˜˜ , −I˜). Case 2: C t comprises a school. Each student in the cycle points to her best remaining school among those who have not accepted any student who victimizes her or is victimized by her. Hence ˜ µ(i) µ˜ i and i µ˜ s for all s ∈ S \ {µ(i)}. for any student i who is in it and also a member of I, i

i

As in the previous case, therefore, C can be removed immediately after C 1 , ..., C t−1 even under the t

preference profile (µI˜˜ , −I˜).

To summarize, it is possible to remove C 1 , C 2 , ... in the same order under ((µI˜˜ , −I˜), B), which

results in µ. Hence µ˜ = µ, which implies that the TTC mechanism is group strategy-proof.

37

Bullying-resistance: Suppose that B and B0 = B ∪ {(i, j)} are both sets of bullying incidents. Let µ0 ≡ ϕT TC (I , B0 ). Consider the TTC algorithm for (I , B). If i and j are removed at the same step, they are also removed simultaneously under (I , B0 ) and assigned to the same schools across µ and µ0 . If i is removed earlier than j, µ(i) = µ0 (i). Finally, if j is removed earlier than i, the allocation to those removed earlier than i is the same across µ and µ0 . Hence, µ(i) is different from µ0 (i) only if µ(i) = µ( j), and in this case µ(i) i µ0 (i). This completes the proof.

Q.E.D.

C PROOF OF THEOREM 3 Proof. Consider a problem where I = {i1 , ..., i6 }, S = {s1 , s2 , s3 , s4 }, q s1 = q s2 = q s3 = 1, q s4 = 2, B = ∅, and preferences are given by i1 , i2 : s4 , s1 , ∅ i3 , i4 : s4 , s2 , ∅ i5 , i6 : s4 , s3 , ∅. Suppose in the way of contradiction that there exists a mechanism that produces a feasible, individually rational, and constrained Pareto efficient matching for any input and is group bullyingresistant. Consider the following possible outcome of a mechanism:    s1 s2 s3 s4 ∅ ∅   . µ =  ∅ i3 i5 i1 i2 i4 i6 

(1)

If B is replaced by B0 = {(i2 , i1 ), (i4 , i1 ), (i6 , i1 )}, i1 or i2 must leave s4 . By the group bullyingresistance property, neither i4 nor i6 enters s4 . Then by constrained Pareto efficiency, either i3 or i5 enters s4 , which induces a transfer of i4 or i6 to s2 or s3 . This contradicts the group bullyingresistance property. Consider the following possible outcome of a mechanism    s1 s2 s3 s4 ∅   . µ0 =  i2 i4 i5 i1 i3 i6  If B is replaced by B00 = {(i2 , i1 ), (i3 , i1 ), (i4 , i1 ), (i6 , i1 )}, i1 or i3 must leave s4 . By the group bullyingresistance property, s4 does not accept i2 , i4 , nor i6 . By constrained Pareto efficiency, i5 and i6 enter s4 and s3 respectively, which contradicts the group bullying-resistance property. This completes the proof.

Q.E.D. 38

D PROOF OF THEOREM 4 Proof. Suppose that there exists a mechanism that produces a quasi stable matching and is group bullying-resistant. Consider the problem in the previous proof and the following school priorities:  s1 ,  s2 ,  s3 ,  s4 : i1 , i2 , i3 , i4 , i5 , i6 . The matching µ (see (1)) is the unique (quasi) stable matching when B = ∅. Following the argument in the proof of Theorem 3, we observe that if B is replaced by B0 = {(i2 , i1 ), (i4 , i1 ), (i6 , i1 )}, there is at least one student in {i2 , i4 , i6 } who gets better off. This is a contradiction.

Q.E.D.

E PROOF OF THEOREM 5 Before proceeding to the proof of the theorem, I introduce a lemma without proof, which will be used subsequently. Lemma 2. Suppose B = ∅. Then any stable matching is (constrained) Pareto efficient in terms of student and school preferences. This follows from the fact that when B = ∅ and school preferences are strict, the set of stable matchings is equivalent to the weak core. For specific details, the reader is referred to Roth and Sotomayor (1990; Proposition 5.36). Proof of Theorem 5 Quasi Stability: Let µ denote an outcome of the TGS mechanism and suppose that (i, s) is a blocking pair at µ. Let I˜ denote an arbitrary subset of µ(s) satisfying the three conditions in the definition of a blocking pair. It is easy to show that if i < Bu(B) and either I˜ ∩ Bu(B) , ∅ or I˜ = ∅, the outcome of Round 1 is not stable among non-bullies. This is a contradiction. Next if i < Bu(B), I˜ , ∅, and I2 ∩ Bu(B) = ∅, then by the responsiveness condition, for any student j in I˜ \ Bu(B), {i}  s { j}. This again contradicts stability. Finally if i ∈ Bu(B) and I˜ ⊆ Bu(B), then by the construction of Round 2, there must be j ∈ µ(s) \ I˜ such that (i, j) ∈ B. Then (i, s) is by no means a blocking pair. Hence i ∈ Bu(B) and I˜ \ Bu(B) , ∅, meaning µ is quasi stable. Proofs for other properties going along with the TGS mechanism except constrained Pareto efficiency are the same as those in the proof of Theorem 1. Note that Lemma 1 holds as it is under 39

the responsive school preferences (domination in the statement of the lemma is defined in terms of student preferences). Constrained Pareto Efficiency: Let µ denote an outcome of the TGS mechanism. Suppose in the way of contradiction that µ is dominated by another matching µ˜ that is feasible and satisfies the separation principle. There are two cases to consider. Case 1: ∀i ∈ I \ Bu(B), µ(i) ˜ = µ(i). This means that in the hypothetical Round 2 of the TDA algorithm, after assigning µ(i) = µ(i) ˜ to all i < Bu(B), under the modified preferences of bullies and reduced numbers of the seats of schools, µ˜ gives a description of a matching that dominates a stable matching in the residual problem. This contradicts Lemma 2. Case 2: ∃i ∈ I \ Bu(B), µ(i) ˜ , µ(i). For any i < Bu(B), µ(i) ˜ i µ(i), and at least one of them holds with a strict relation. By Lemma 1, for any s ∈ S , |µ(s) ˜ \ Bu(B)| < q s ⇔ |µ(s) \ Bu(B)| < q s and in this case, µ(s) ˜ \ Bu(B) = µ(s) \ Bu(B). Hence by Lemma 2, there is at least one school s such that |µ(s) ˜ \ Bu(B)| = |µ(s) \ Bu(B)| = q s and µ(s)  s µ(s). ˜ This contradicts the assumption that µ˜ dominates µ.

Q.E.D.

F PROOF OF PROPOSITION 2 Proof. Given an output µ of the TGS mechanism, suppose that (I1 , s) is a blocking group and I2 ⊆ µ(s) is a set of students satisfying the three conditions in the definition of a blocking group. There are two cases to consider. Case 1: µ(s) ∩ Bu(B) = ∅. If |µ(s)| < q s , there is (i, j) ∈ B such that i ∈ I1 and j ∈ I2 , confirming the statement of the proposition. Suppose instead that |µ(s)| = q s and I2 , ∅. If I1 ∩ Bu(B) = ∅, by the responsiveness condition, there is at least one student i in I1 and one student j in I2 such that i, j < Bu(B), s i µ(i), and {i}  s { j}  s ∅. This contradicts stability among non-bullies. Hence I1 ∩ Bu(B) , ∅. Since I2 ∩ Bu(B) = ∅ holds by assumption, I2 \ Bu(B) , ∅ and |I1 ∩ Bu(B)| > 0 = |I2 ∩ Bu(B)|. Case 2: µ(s) ∩ Bu(B) , ∅. If I1 \ Bu(B) , ∅, there are i ∈ I1 \ Bu(B) such that {i}  s ∅ and a vacant seat at s at the end of Round 1 of the TDA algorithm. This contradicts the stability of the outcome of Round 1 among non-bullies. Hence I1 ⊆ Bu(B). Suppose there is no (i, j) ∈ B such that i ∈ I1 and j ∈ I2 (otherwise, the statement of the proposition is immediately confirmed). By the definition of a blocking group, there is no (i, j) ∈ B such that i ∈ I1 and j ∈ µ(s). Hence for all i ∈ I1 , s is placed above her outside option in her preference ordering during Round 2 of the TDA algorithm. By the construction of the DA 40

algorithm, |µ(s)| = q s and for all i ∈ I1 and all j ∈ µ(s) ∩ Bu(B), { j}  s {i}. Hence, I2 \ Bu(B) , ∅ since otherwise |I1 | ≤ |I2 | (by |µ(s)| = q s ), { j}  s {i} for all j ∈ I2 and i ∈ I1 , and the responsiveness condition together imply µ(s)  s (µ(s) \ I2 ) ∪ I1 , which contradicts the definition of a blocking group. Moreover, if |I2 ∩ Bu(B)| ≥ |I1 ∩ Bu(B)| = |I1 |, then µ(s)  s µ(s) \ (I2 \ Bu(B))  s (((µ(s) \ (I2 \ Bu(B))) \ (I2 ∩ Bu(B))) ∪ I1 = (µ(s) \ I2 ) ∪ I1 . Thus µ(s)  s (µ(s) \ I2 ) ∪ I1 , a contradiction. Hence |I1 ∩ Bu(B)| > |I2 ∩ Bu(B)|.

Q.E.D.

G STRONG QUASI STABILITY In this section, I discuss the notion of strong quasi stability for school choice when schools have preferences/priorities over the set of subsets of students. Given a matching µ, a blocking pair (i, s) is strongly permissible if i ∈ Bu(B) and for any set I˜ ⊆ µ(s) that satisfies the three conditions in the definition of a blocking pair, I˜ , ∅ and I˜ ∩ Bu(B) = ∅. A matching is strongly quasi stable if it is feasible, individually rational, satisfies the separation principle, and any blocking pair is strongly permissible. Strong quasi stability only allows “justified envy” from a bully to non-bullies. One may thus argue that it fits in the notion of social priority manipulation more than quasi stability. However, the following two examples demonstrate that this stability concept has the same drawbacks as the standard stability concept in the priority-based school choice. Example 5 (Strongly quasi stable matching may not exist). There are two schools {s1 , s2 } and four students {i1 , i2 , i3 , i4 }. The school capacities and a set of bullying incidents are given by q s1 = 2, q s2 = 1, B = {(i1 , i2 ), (i1 , i3 ), (i4 , i3 )}. The student and school preferences are such that: i1 : s1 , ∅

 s1 : {i3 }, {i1 , i4 }, {i1 }, {i2 , i4 }, {i2 }, {i4 }, ∅

i2 : s1 , s2 , ∅

 s2 : {i2 }, {i3 }, ∅

i3 : s2 , s1 , ∅ i4 : s1 , ∅ The preference lists for the schools are curtailed suitably. This problem is obtained by adding i4 to the problem in Example 1 and modifying the school preferences accordingly. If i4 is not 41

assigned any school seat, the only candidate for a strongly quasi stable matching is described by µ1 in Example 1 since it is the unique quasi stable matching in the previous framework. Hence we only need to consider the following four matchings:          s1 s2 ∅   s1 s2 ∅   s1 s2 ∅   s1 s2 ∅   , µ2 =       µ1 =  i i i i  , µ3 = i i i i  , µ4 = i i i i  . i2 ∅ i3 i1 i4  1 4 2 3 1 4 3 2 2 4 3 1 However, every matching accepts a blocking pair that is not strongly permissible (µ1 : (i4 , s1 ), µ2 : (i3 , s1 ), µ3 : (i2 , s2 ), µ4 : (i1 , s1 )).

^

Example 6 (No mechanism that is strategy-proof selects a strongly quasi stable matching whenever there exists one). There are two schools {s1 , s2 } and five students {i1 , ..., i5 }. The school capacities and the set of bullying incidents are given by q s1 = 2, q s2 = 1, B = {(i1 , i3 ), (i2 , i3 ), (i4 , i5 )}. The student and school preferences are: i1 : s1 , ∅

 s1 : {i2 }, {i3 , i4 }, {i3 }, {i1 }, {i4 }, ∅

i2 : s2 , s1 , ∅

 s2 : {i3 }, {i1 }, {i2 }, ∅

i3 : s1 , s2 , ∅ i4 : s1 , ∅ i5 : ∅ The preference lists for the schools are curtailed suitably. This problem is obtained by adding i4 and i5 to the problem in Example 2 and modifying the school preferences accordingly. There are two strongly quasi stable matchings:    s1 s2 ∅   . and µ˜ =  i3 i4 i2 i1 i5 

   s1 s2 ∅   , µ =  i1 i2 i3 i4 i5 

We consider a mechanism that chooses a strongly quasi stable matching whenever there exists one. Suppose that it chooses µ under the above preference profile I . Consider the following reported 0

preference ordering i3 of i3 : 0

i3 : s1 , ∅. Then, µ˜ is the unique strongly quasi stable matching under (i3 , {i1 ,i2 } ). Since i3 is better off at µ˜ 0

than at µ, she can profitably misreport her preferences under I . 42

Suppose on the other hand that the mechanism chooses µ˜ under I . Now consider the following 0

reported preference ordering i1 of i1 : 0

i1 : s1 , s2 , ∅. Then, µ is the unique strongly quasi stable matching under (i1 , {i2 ,i3 } ). Note that if i2 is replaced 0

with i1 at µ, ˜ the resulting matching is quasi stable in the corresponding problem of the previous framework, but it is not strongly quasi stable since (i2 , s1 ) is a blocking pair that is not strongly permissible. Since i1 prefers s1 to her outside option according to her true preferences i1 , she prefers µ(i1 ) to µ(i ˜ 1 ), meaning that she can profitably misreport her preferences under I .

^

H OTHER STABILITY CONCEPTS In this section, I discuss some refinements of the stability concept other than quasi stability, and show that they are all incompatible with anti-bullying school choice mechanism design. Hence in the framework of this paper, the anti-bullying perspective is linked inexorably to quasi stability. There are three stability concepts, α, β, and γ stability, and their variants. Definition 6. A matching is α stable if (i) it is feasible, (ii) individually rational, (iii) satisfies the separation principle, (iv) there is no student who wishes for a vacant seat at some school, and (vα ) whenever i has justified envy toward j, (i, j) ∈ B. Definition 7. A matching is β stable if (i) it is feasible, (ii) individually rational, (iii) satisfies the separation principle, (iv) there is no student who wishes for a vacant seat at some school, and (vβ ) whenever i has justified envy toward j, i ∈ Bu(B) and j ∈ Vi(B). Definition 8. A matching is γ stable if (i) it is feasible, (ii) individually rational, (iii) satisfies the separation principle, (iv) there is no student who wishes for a vacant seat at some school, and (vγ ) whenever i has justified envy toward j, i < Vi(B) and j ∈ Vi(B). At each variant of stability, bullies, victims, and bystanders are equally treated among themselves respectively. The α stability concept espouses the principle that the difference in social priorities for two students should be grounded in their own bullying episode. In other words, if two students are in no bully-victim relation, there should be no discord between them. The β stability concept expands the range of applications of the social priority manipulation. It intends to prioritize victims at the expense of bullies, but not of others who play no role in the bullying episodes. 43

Finally, the γ stability concept postulates that the victims should be given priority and rights to enroll in their preferred schools above all requests of non-victims. This may be practically desirable if, besides the policy to separate certain pairs of students, victims call for careful treatments so that they can go to safe schools. Analogously to weak quasi stability (Definition 5), weak stability, weak α stability, weak β stability, and weak γ stability are defined by replacing the condition (iv) in Definitions 1, 6, 7, and 8 with (ivw ) (whenever i wishes for a vacant seat at some school, i ∈ Bu(B)) respectively. Claim 1 and 2 below show that the weak stability concept and the β (or α) stability concept, either weak or not, have the same drawbacks as those for the basic stability concept. Claim 1. A weakly β stable matching does not exist in general. Proof. Consider a problem where I = {i1 , ..., i8 }, S = {s1 , ..., s5 }, q s1 = 2, q s2 = q s3 = q s4 = q s5 = 1, B = {(i2 , i3 ), (i5 , i3 ), (i6 , i3 ), (i6 , i7 )} (Bu(B) = {i2 , i5 , i6 } and Vi(B) = {i3 , i7 }), and preferences and school priorities are such that: i1 : s1 , s2 , ∅

 s 1 : i7 , i6 , i3 , i2 , i5 , i1 , i4 , ∅

i2 : s1 , s3 , ∅

 s 2 : i1 , i3 , i4 , ∅

i3 : s2 , s1 , s5 , ∅

 s3 : i2 , i6 , i8 , ∅

i4 : s1 , s2 , s4 , ∅

 s4 : i4 , i5 , i8 , ∅

i5 : s4 , s1 , ∅

 s5 : i3 , i7 , ∅

i6 : s3 , s1 , ∅ i7 : s5 , s1 , ∅ i8 : s3 , s4 , ∅ The priority lists for the schools are curtailed suitably. Suppose that i7 is matched with s5 . Then the unique candidate for a weakly β stable matching is the one marked with ♣ in Table 1. In this matching, i3 has justified envy toward i7 and i3 , i7 ∈ Vi(B), meaning that it is not weakly β stable. Next, suppose that i7 is enrolled in s1 . Then, i3 needs to enter s5 since otherwise i7 wishes for a vacant seat at s5 . To avoid the case where i3 wishes for a vacant seat at s1 or has justified envy toward i7 ’s buddy, s1 needs to admit i6 . However, (i6 , i7 ) ∈ B requests the separation of i7 from i6 . Hence, if i7 is in s1 , the resulting matching is by no means weakly β stable. Finally, if i7 is not enrolled in any school, she would wish for a vacant seat at s1 , or have 44

justified envy toward a student in it (note that i6 is the only student who victimizes her). This completes the proof.

Q.E.D. s1 ∅∅ i∅ i4 ∅ i1 i2 i1 i2 i1 i3 i1 i4 i1 i5 i1 i6 i2 i4 i2 i5 i2 i5 i2 i5 i2 i5 i2 i5 i2 i6 i2 i6 i3 i4 i4 i5 i4 i6 i5 i6

s2 i3 i4 ∅ i3 i1 -

s3 i∗6 i∗6 i∗6 i∗6 i∗6 ∅ i8 -

s4 i∗5 i∗5 ∅ i8 i4 i4 i4 -

Remark i1 → s1 i4 → s1 (i , i4 ) i1 → s1 i4 → s4 (i5 ) i3 → s2 (i4 ) i3 → s2 i2 → s1 (i1 ) i2 → s1 (i1 ) i2 → s1 (i1 ) i1 → s1 (i4 ) i8 → s4 i5 → s4 (i8 ) i4 → s2 i1 → s2 (i3 ) ♠ ♣ i8 → s3 i6 → s3 (i8 ) i1 → s1 (i4 ) i1 → s1 (i4 ) i1 → s1 (i4 ) i2 → s1 (i5 )

Table 1: In the Remark column, “i → s” means “i wishes for a vacant seat at s” or “i has justified envy toward another student in s”. Especially, “i → s( j)” means “i has justified envy toward j who belongs to s”. An asterisk at the upper right of a student’s identity means that she needs to enroll in a school specified above so as not to have justified envy toward another student in s1 . Claim 2. 1. There is no mechanism that is strategy-proof and selects a weakly stable matching whenever there exists one. 2. There is no mechanism that is strategy-proof and selects an α stable matching whenever there exists one. 3. There is no mechanism that is strategy-proof and selects a weakly α stable matching whenever there exists one. 45

4. There is no mechanism that is strategy-proof and selects a β stable matching whenever there exists one. 5. There is no mechanism that is strategy-proof and selects a weakly β stable matching whenever there exists one. Proof. I first prove the first statement. Consider a problem where I = {i1 , i2 , i3 , i4 }, S = {s1 , s2 }, q s1 = 2, q s2 = 1, B = {(i1 , i3 ), (i1 , i4 ), (i2 , i3 )} (Bu(B) = {i1 , i2 }), and preferences and school priorities are such that: i1 : s1 , ∅

 s1 : i2 , i3 , i1 , i4 , ∅

i2 : s2 , s1 , ∅

 s2 : i3 , i1 , i2 , i4 , ∅

i3 : s1 , s2 , ∅ i4 : s1 , ∅ Note that this problem is similar to the one in Example 2: differences appear in i4 ,  s1 , and  s2 . There are two weakly stable matchings in this problem:    s1 s2 ∅   , µ =  i1 i2 i3 i4 

   s1 s2 ∅   . and µ˜ =  i3 i4 i2 i1 

Suppose that a mechanism chooses µ under the above preference profile I . Consider the following 0

reported preference ordering i3 of i3 , 0

i3 : s1 , ∅. Then, µ˜ is the unique weakly stable matching under (i3 , {i1 ,i2 ,i4 } ). Since i3 prefers µ(i ˜ 3 ) to µ(i3 ), 0

she can profitably misreport her preferences under I . Suppose on the other hand that a mechanism chooses µ˜ under I . Now consider the following 0

reported preference ordering i1 of i1 , 0

i1 : s1 , s2 , ∅. Then, µ is the unique stable matching under (i1 , {i2 ,i3 ,i4 } ). Since i1 prefers s1 to her outside op0

tion according to her true preferences i1 , she prefers µ(i1 ) to µ(i ˜ 1 ), meaning she can profitably misreport her preferences under I . This completes the proof of the first statement. Next, I prove the last statement. Consider a problem where I = {i1 , ..., i9 }, S = {s1 , ..., s6 }, 46

q s1 = 2, q s2 = q s3 = q s4 = q s5 = q s6 = 1, B = {(i2 , i3 ), (i5 , i3 ), (i6 , i3 ), (i6 , i7 )} (Bu(B) = {i2 , i5 , i6 } and Vi(B) = {i3 , i7 }), and preferences and school priorities are such that: i1 : s1 , s2 , ∅

 s1 : i7 , i6 , i3 , i2 , i5 , i1 , i4 , ∅

i2 : s1 , s3 , ∅

 s 2 : i1 , i3 , i4 , ∅

i3 : s2 , s1 , s3 , ∅

 s 3 : i2 , i6 , i9 , i3 , i8 , ∅

i4 : s1 , s2 , s4 , ∅

 s4 : i4 , i5 , i8 , ∅

i5 : s4 , s1 , ∅

 s5 : i3 , i7 , ∅

i6 : s3 , s1 , ∅

 s6 : i5 , i9 , ∅

i7 : s5 , s1 , ∅ i8 : s3 , s4 , ∅ i9 : s6 , s3 , ∅ The priority lists for the schools are curtailed suitably. Note that this problem is similar to the one in the proof of Claim 1: differences appear in i3 , i9 ,  s3 , and  s6 . Since i3 i3 s5 and i5 i5 s6 , i7 and i9 enter s5 and s6 respectively in all weakly β stable matchings. Hence the only matching in Table 1 that also gives a description of a weakly β stable matching in this problem is the one marked with ♣. Moreover, since s3 i3 i3 , there is another weakly β stable matching in which i3 enters s3 . In sum, there are two weakly β stable matchings in this problem:    s1 s2 s3 s4 s5 s6 ∅   , µ =  i2 i5 i1 i6 i4 i7 i9 i3 i8 

   s1 s2 s3 s4 s5 s6 ∅   . and µ˜ =  i2 i6 i1 i3 i4 i7 i9 i5 i8 

We consider a mechanism that chooses a weakly β stable matching whenever there exists one. Suppose that it chooses µ under the above preference profile I . Consider the following reported 0

preference ordering i3 of i3 , 0

i3 : s2 , s1 , s3 , s5 , ∅. Then, µ˜ is the unique weakly β stable matching under (i3 , I\{i3 } ). To see this is true, suppose that 0

i3 enters s5 . Then, i7 is in s1 and, to avoid the case where i3 wishes for a vacant seat at s1 or she has justified envy toward i7 ’s buddy, i6 has to be in s1 . But this is incompatible with (i6 , i7 ) ∈ B. Hence at any weakly β stable matching, i3 does not enter s5 , meaning that µ˜ is the unique one. As a result, i3 can misrepresent her preferences to her advantage.

47

Suppose on the other hand that the mechanism chooses µ˜ under I . Now consider the following 0

reported preference ordering i5 of i5 : 0

i5 : s4 , s1 , s6 , ∅. Then, µ is the unique weakly β stable matching under (i5 , I\{i5 } ). To see this is true, suppose that 0

i5 enters s6 . Then, the resulting matching is by no means weakly β stable as it is shown by Table 2. As a result, i5 can beneficially misrepresent her preferences. To summarize, any mechanism that selects a weakly β stable matching whenever there exists one is by no means strategy-proof. Note that in this problem, the two weakly β stable matchings µ and µ˜ are both α stable, thus they are weakly α stable and β stable (µ is stable; i6 has justified envy toward i3 at µ, ˜ but (i6 , i3 ) ∈ B). Moreover, µ˜ is α stable under (i3 , I\{i3 } ) and µ is α stable under (i5 , I\{i5 } ). Thus, we have already 0

0

verified the statements 2-4. This completes the proof. s1 ∅∅ i∅ i1 i2 i1 i3 i1 i6 i2 i6 i2 i6 i2 i6 i2 i6 i2 i7 i3 i7

s2 -

s3 ∅ i3 i8 i9 -

s4 i4 i4 i4 i4 i4 i4 i4 i4 i4 i4 i4

s5 -

s6 i5 i5 i5 i5 i5 i5 i5 i5 i5 i5 i5

Q.E.D.

Remark i4 → s1 i4 → s1 (i: arbitrary) i5 → s1 (i1 ) i3 → s2 i2 → s1 (i1 ) i3 → s3 i9 → s3 (i3 ) i9 → s3 (i8 ) i6 → s3 (i9 ) i7 → s5 i7 → s5

Table 2: The description follows the same rule as the one for Table 1. Note that if i4 is not in s4 , i5 wishes for a vacant at that school. Hence if i5 enters s6 at some weakly β stable matching, i4 must enter s4 at the same time. Contrary to the previous two stability concepts, a γ stable matching always exists and is implementable via a strategy-proof mechanism. To see this is true, consider an algorithm, similar to the TDA algorithm, where only the victims participate in the first round, assignment for them is finalized after the first run of the DA algorithm, then each bully puts every school where her victims were accepted in the previous round below her outside option, and finally the second DA algorithm is run for the rest of the students and the school seats. It is easy to show that the output 48

of this algorithm is γ stable and the induced mechanism is strategy-proof. However, Claim 3 below shows that not only this mechanism, but also any mechanism that produces a (weakly) γ stable matching for any input is by no means bullying-resistant. Therefore, if this stability concept is pursued along a school choice mechanism despite its incongruence with the desideratum, a punitive deterrent against school bullying, if any is desired, must be supplemented by other auxiliary policies in the whole anti-bullying program. Given its plausibility and potential usability, I believe that the further examination of the virtues of the γ stability concept must be undertaken as a theoretical enterprise in the future work. Claim 3. There is no mechanism that produces a weakly γ stable matching for any input and is bullying-resistant. Proof. Consider a problem where I = {i1 , ..., i9 }, S = {s1 , ..., s6 }, q s1 = 2, q s2 = q s3 = q s4 = q s5 = q s6 = 1, B = {(i2 , i3 ), (i5 , i3 ), (i6 , i3 ), (i6 , i7 )} (Vi(B) = {i3 , i7 }), and preferences and school priorities are such that: i1 : s1 , s2 , s6 , ∅

 s1 : i7 , i6 , i3 , i2 , i5 , i1 , i4 , ∅

i2 : s1 , s3 , ∅

 s2 : i1 , i3 , i4 , ∅

i3 : s2 , s1 , s5 , ∅

 s3 : i2 , i6 , i8 , ∅

i4 : s1 , s2 , s4 , ∅

 s4 : i4 , i5 , i8 , ∅

i5 : s4 , s1 , ∅

 s 5 : i3 , i7 , ∅

i6 : s3 , s1 , ∅

 s 6 : i1 , i9 , ∅

i7 : s5 , s1 , ∅ i8 : s3 , s4 , ∅ i9 : s6 , ∅ The priority lists for the schools are curtailed suitably. Note that this problem is similar to the one in the proof of Claim 1: differences appear in i1 , i9 , and  s6 . Since i7 ∈ Vi(B), she must be enrolled in s5 at any weakly γ stable matching for the same reason I described previously. Then in Table 1, according to the Remark column, the only candidate for a weakly γ stable matching is the one marked with ♠ because i3 ∈ Vi(B) and i1 < Vi(B). There is thus a unique weakly γ stable

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matching:    s1 s2 s3 s4 s5 s6 ∅   . µ =  i2 i5 i3 i6 i4 i7 i1 i8 i9  In µ, i1 has justified envy toward i3 . Now suppose that the set of bullying incidents B changes to B0 = B ∪ {(i9 , i1 )}. Since i1 ∈ Vi(B0 ), µ is no longer weakly γ stable. In any weakly γ stable matching under B0 , i9 enters s6 instead of i1 , violating the bullying-resistance criterion.

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Q.E.D.