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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 docu- mented 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 mecha- nism 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 col- laborative interaction between anti-bullying programs and school choice.

JEL Classification: C78; I21; I28

Keywords: School Choice; Mechanism Design; Peer Effects; Anti-bullying Program

∗Email: yusuke.asia@gmail.com (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.

1 I

The design of rigorous and transparent school choice mechanisms has provoked recent heated pol- icy 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 mecha- nism 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

1Economists 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 mecha- nisms 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.

2Epple 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”.

3Citation 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.

homicides (Anderson et al. 2001).⁴ Bullying in schools can also have serious negative conse- quences (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 litera- ture⁶, 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.

4School 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.

5Lazear (2001) builds on this fact, and the bad apple principle in general, to study class size effects.

6For example, evidence on ability-based peer effects (e.g., whether and to what extent the average GPA of class- mates 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, Ab- dulkadiro˘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 struc- tural 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.

7I owe the following description to Yasuda (2010) and Yoshida et al. (2009).

8http://www.mext.go.jp/a menu/shotou/gakko-sentaku/06041014/008/003.htm

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 anti- bullying 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 (student- optimal 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 demon- strate 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.

9For 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.

10The 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.

11http://www.mext.go.jp/a menu/shotou/gakko-sentaku/08042801.htm

12An 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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To the best of my knowledge, there is no research in the whole literature of economics and ed- ucation 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 match- ing model thereby deal with peer effects. Their models are based on agents’ cardinal utility func- tions 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 equi- librium 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 competi- tion especially among private schools, future research may want to investigate the intersection of their frameworks and mine.

13See 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)

15Epple and Romano (2011) survey the literature on peer effects in education.

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 match- ing 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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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 s^′ (s, s^′ ∈ S ∪ {i}) means either s ≻_i s^′ or s = s^′. 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. Anti- bullying 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.

16An 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.

17A 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).

Each school s ∈ S has q_s available seats. Moreover, each school s ∈ S is endowed with a strict priority ordering ≻_sover 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)| ≤ qs for every s ∈ S . A matching µ is individually rational if µ(i) _i ifor 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 envytoward 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)} jand 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)} jand i’s victim k is also in j’s school. In this case, i’s complaint against j is not justified since replacing iwith 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.

18See Section 4.4 for the implications of school preferences on anti-bullying school choice.

19I impose this assumption for convenience. Without it, all the results in this paper hold as they are with minor modifications to the framework.

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 ϕ_i(≻_I, B). A mechanism ϕ is strategy-proof if there does not exist (≻I, B), a student i ∈ I, and her stated preferences ≻^′_i such that

ϕ_i(≻^′_i, ≻_−i, B) ≻i ^ϕi^(≻I, B),

where ≻−i^≡≻I\{i}(≻−Jis substituted for ≻I\Jfor any J ⊆ I). A mechanism ϕ is group strategy-proof if there does not exist (≻_I, B), a group of students ˜I ⊆ I, and their stated preferences ≻^′_˜Isuch that

ϕ_i(≻^′_˜I, ≻_{− ˜I}, B) i ^ϕi^(≻I, B) ∀i ∈ ˜I,

with strict preference holds for at least one student in ˜I. Group strategy-proofness may be important 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

20In Conclusion, I revisit the concept of stability and discuss its relation to privacy.

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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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.

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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^{, i}2^{, i}3}. 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, ∅ ≻_s1 : i₃, i₁, i₂, ∅

≻_i2 : s₁, s₂, ∅ ≻_s2 : 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 s₂must enter the school in order to achieve stability. Now consider the following matching:











s₁ s₂ ∅ i₁∅ i₂ i₃









 .

A matrix denotes a matching: i₁is matched to s₁ while s₁has 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

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 ∅ i₂∅ i₃ i₁









 .

However, this matching is not stable because i₁has justified envy toward i₂. ^{^} 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^{, ∅ ≻}s1 ^{: i}2^{, i}3^{, i}1^{, ∅}

≻_i2 : s₂, s₁, ∅ ≻_s2 : i₃, i₁, i₂, ∅

≻_i3 : s₁, s₂, ∅

In this problem, there are two stable matchings,

µ =











s₁ s₂ i₁i₂ i₃











, and ˜µ =











s₁ s₂ ∅ i₃∅ i₂ i₁









 .

Suppose that a mechanism chooses µ under the above preference profile ≻_I. Consider the following reported preference ordering ≻^′_i

3 ^{of i3,}

≻^′_i

3^{: s1, ∅.}

Then, ˜µ is the unique stable matching under (≻^′_i

3^{, ≻{i1,i2}). Since i3} prefers ˜µ(i3) to µ(i3), she can profitably misreport her preferences under ≻_I.

Suppose on the other hand that a mechanism chooses ˜µ under ≻_I. Now consider the following reported preference ordering ≻^′_i

1 ^{of i1,}

≻^′_i

1^{: s1, s2, ∅.}

Then, µ is the unique stable matching under (≻^′_i , ≻_{i2,i3}). Since i₁ prefers s₁ to her outside option

according to her true preferences ≻_i1, she prefers µ(i₁) to ˜µ(i₁), meaning that she can profitably

misreport her preferences under ≻_I. ^

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 condi- tion, the standard stability concept retains the drawbacks. This is exactly the reason why I pursue quasi stabilityin 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 (v_q) 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 match- ing that dominates it. Student-optimality works as a normative refinement of quasi stability. While

21The 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.

22Note 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.

a student-optimal stable matching in a model without any bullying incidents is unanimously pre- ferred 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^{, i}2}. 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:

≻_i1 : s, ∅ ≻_s: i₁, i₂, ∅

≻_i2 : s, ∅

In this problem, there are two quasi stable matchings:

µ =









 s ∅ i₁ i₂











, and ˜µ =









 s ∅ i₂ i₁









 .

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

3.2 T

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M

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_shighest 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-

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 ϕ^{T GS}. 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 observa- tion 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 T

T

C

M

Here I present a variant of the top trading cycles (TTC) mechanism denoted by ϕ^{T T C}. Unlike the TGS mechanism, this mechanism works without the regularity condition. The original mechanism

23Roth (2008) surveys the use of the GS mechanism in practice.

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, ..., i_k, s_k) where i1points to s1, s1points to i2, ..., i_k points to s_k, and s_k 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

24See 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).

25Papai (2000) introduced a wide class of mechanisms called hierarchical exchange mechanisms, uniquely char-

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 Bmust 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 util- ity 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 hierar- chical 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.

26The 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 ad- mission processes used in New York, Eugine (http://www.4j.lane.edu/choice♯aboutschoolchoice), and Tokyo (Yasuda 2010) also rely on this mechanism.

4 D

4.1 B

G

P

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 bullying- resistance 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 (≻_I, B) and (≻I, B ∪ {(i, j)|i ∈ ˜I}) in its domain, ϕi^(≻I, B) i ^ϕi^(≻I, B ∪ {(i, j)|i ∈ ˜I}) for all 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

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, (iv_w) whenever i wishes for a vacant seat at some school, i ∈ Bu(B), and (v_q) 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 (iv_w) in the def- inition 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 prob- lems 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 I

B

, B

/V

,

I

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 I

B

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

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) ∈ Bcan 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 B

/V

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.

27Girls 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).

28http://www.cbc.ca/news/background/bullying/cyber bullying.html

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 longitudi- nal analysis on bully/victims such as Pollastri et al. (2010) may help disentangle any confusion.

4.2.3 I

(J

B

)

As I stated in the “citation policy” (footnote 3), much of the cited evidence on the nature of bul- lying 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 be- tween 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

29See Morita et al. (1999) for a rather formal definition of ijime and the history of socially acknowledged ijime problems in Japan.

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 O

O

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

30http://www.seattleschools.org/ See “school choice form instructions”.

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 vic- tims. 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 I

S

P

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 nu- anced 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 ≻sover 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 ≻_s ∅ with i, j < ˜I, ˜I ∪ {i} ≻_s ˜I ∪ { j} ⇔ {i} ≻_s { j}, and (ii) for any i ∈ I and ˜I ≻_s ∅ with i < ˜I,

˜I ∪ {i} ≻_s ˜I ⇔ {i} ≻_s ∅ (Roth 1985).³¹ A matching µ is individually rational if µ(i) _i ifor every i ∈ I, and {i} ≻_s ∅ for all i ∈ µ(s) for every s ∈ S . A matching µ dominates another matching ν

31Note 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.

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 ⊆ µ(s) such that (i) |(µ(s) \ ˜I) ∪ {i}| ≤ q_s, (ii) (µ(s) \ ˜I) ∪ {i} ≻_s µ(s), and (iii) there is no j ∈ 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 condi- tions 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 unjusti- fiable, 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 hy- pothetical 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^{, ..., i}5}. The school capacities and a

32School 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.

33See the appendix (section G) for a discussion of strong quasi stability which will eventually turn out to be incom- patible with anti-bullying school choice.