Elsevier

Journal of Economic Theory

Volume 166, November 2016, Pages 186-211
Journal of Economic Theory

Improving schools through school choice: A market design approach

https://doi.org/10.1016/j.jet.2016.07.001Get rights and content

Abstract

We study the effect of different centralized public school choice mechanisms on schools' incentives for quality improvement. To do so, we introduce the following criterion: A mechanism respects improvements of school quality if each school becomes weakly better off whenever that school improves, i.e., becomes more preferred by students. We first show that neither any stable mechanism nor mechanism that is Pareto efficient for students (such as the Boston and top trading cycles mechanisms) respects improvements of school quality. Nevertheless, for large school districts, we demonstrate that any stable mechanism approximately respects improvements of school quality; by contrast, the Boston and top trading cycles mechanisms fail to do so. Thus, a stable mechanism may provide better incentives for schools to improve themselves than the Boston and top trading cycles mechanisms.

Introduction

School choice has grown rapidly in the United States and many other countries such as Japan and the United Kingdom. In contrast to traditional neighborhood-based placement, school districts with school choice programs allow children and their parents to express preferences over public schools and use these preferences to determine student placement. Many politicians, school reformers, and academics have embraced school choice as a policy that will substantially improve educational outcomes; for instance, in their influential book Politics, Markets, and America's Schools, scholars John E. Chubb and Terry M. Moe (1990) argued that school choice is “the most promising and innovative reform” available to improve the quality of public schooling.

A large body of research in the market design literature now investigates how to assign school seats to students efficiently and fairly, recommending specific school choice mechanisms. In particular, beginning with the seminal paper by Abdulkadiroğlu and Sönmez (2003), it has been demonstrated that an extensively used school choice mechanism called the “Boston mechanism” provides strong incentives for students to misreport their preferences. Given this, two strategy-proof mechanisms have been proposed: the “student-optimal stable mechanism” (or “deferred acceptance algorithm”) and the “top trading cycles mechanism”. In fact, prompted by this research, the former has been adopted in New York City and Boston, while the latter has been adopted in New Orleans.2 Indeed, the successful implementation of school choice mechanisms in these cities has prompted other cities, such as Denver and Chicago, to implement new school choice mechanisms as well.

However, prior work on centralized public school choice in the market design literature has not analyzed the effect of different school choice mechanisms on overall school quality, but rather has always assumed that school quality is given and fixed. This is a serious omission, given that the major impetus for the introduction of school choice has been the argument, advanced by both academics and policymakers, that school choice will improve the quality of the public educational system as a whole by incentivizing schools to improve. Indeed, Hoxby (2003b) emphasized that “advocates of school choice…rely…on the idea that school productivity would increase.” Moe (2008) argued that school choice will induce schools “to educate, to be responsive, to be efficient, and to innovate.” Similarly, the 1991 National Governors' Association Report argues that the nation can “increase excellence by increasing choice,” that is, school choice will induce schools to improve, as expressed in the epigraph. The idea of using school choice mechanisms to incentivize schools to improve is becoming increasingly relevant as centralized school choice systems in the U.S. have started to include charter schools,3 which have greater discretion and incentives to improve. Nevertheless, formal analysis of the effects of different public school choice mechanisms on schools' incentives to improve has heretofore been absent.

This paper is a first step toward analyzing the question of how centralized public school choice mechanisms influence the incentives of schools to improve.4 In particular, we study how the design of a school choice mechanism affects the incentives for a school to improve holding the quality of other schools fixed. We consider multiple criteria of whether a school choice mechanism induces schools to improve: Our primary criterion, respecting improvements of school quality, requires that the set of students assigned to a school is never worse for that school while that school is more preferred by students. If a school's effort to improve its quality makes it more attractive to students, then requiring a school choice mechanism to assign a (weakly) better set of students to that school is a natural and mild condition in order for that school choice mechanism to incentivize that school to improve.

Despite the mildness of our criterion, our benchmark results demonstrate that no stable mechanism (such as the student-optimal stable mechanism) or mechanism that is Pareto efficient for students (such as the Boston and top trading cycles mechanisms) respects improvements of school quality.5 That is, for any such mechanism, there exist schools' rankings over students and student preferences over schools such that the outcome for a school becomes strictly worse as the school rises in the preference orderings of the students. Given this impossibility result, we consider domain restrictions on school preference profiles that ensure that the school choice mechanisms discussed above respect improvements of school quality. We show that the necessary and sufficient condition is that school preferences are virtually homogeneous, that is, all schools have essentially identical rankings over students;6 this result implies that no standard mechanism is guaranteed to respect improvements of school quality under reasonable school preferences.7

Even though the above results show that none of the standard school choice mechanisms respects improvements of school quality perfectly, it is possible that instances where a school benefits from discouraging student interest are rare for some mechanisms. If so, then those mechanisms may provide schools with incentives to improve in practice. To investigate this possibility, we next consider large market environments, with many schools and students, where we find clear differences in the incentives for quality improvement provided by different school choice mechanisms.

Our first main result demonstrates that any stable mechanism (such as the student-optimal stable mechanism) approximately respects improvements of school quality in large markets. In other words, for nearly all preference profiles, a school is made weakly better off whenever students rank that school more highly.8

By contrast, surprisingly, our second main result shows that the Boston and top trading cycles mechanisms do not even approximately respect improvements in large markets. In particular, a school can obtain a better set of students by inducing students who the school finds undesirable to find the school unattractive. These “undesirable” students are often children with special needs,9 and the Boston and the top trading cycles mechanisms may induce schools to intentionally make themselves unattractive for these students.

Our results are the first to differentiate among centralized public school choice mechanisms based on their effects on schools' incentives to improve; they suggest that the student-optimal stable mechanism may be better at incentivizing schools to improve than other competing mechanisms, particularly the Boston and the top trading cycles mechanisms. Hence, the student-optimal stable mechanism may not only benefit students through its static efficiency and fairness properties, but also induce schools to improve quality in the long run.

Our model assumes that schools have intrinsic preferences over students; this assumption is motivated by school districts such as that in New York City where many public schools, commonly referred to as “screened schools”, explicitly rank students based on their academic record, attendance, test scores, and other similar criteria (Abdulkadiroğlu, Pathak, and Roth, 2005). Selective exam schools in many districts (e.g., New York City and Boston) rank students based on test scores and/or GPA and these rankings are widely believed to be largely consistent with the schools' own preferences. Similarly, in the Turkish higher educational system, schools rank students based on the students' scores on the national exam.10 In European countries such as Ireland and Hungary, many school districts let schools express their own preferences (Chen, 2012, Biro, 2012).

We also consider settings where each school does not have any intrinsic preferences over individual students, but instead is concerned solely with its enrollment. A mechanism respects improvements of school quality in terms of enrollment if the number of students attending a school weakly increases whenever that school is ranked more highly by students. Any stable mechanism, as well as the Boston mechanism, satisfies this criterion in general markets, while the top trading cycles mechanism does not. These results suggest an additional sense in which the student-optimal stable mechanism provides schools with better incentives to improve quality than the competing top trading cycles mechanism.

Another natural question is whether the mechanisms discussed here respect improvements of student quality, that is, whether a student is always weakly better off when schools rank that student more highly. We show that not only the student-optimal stable mechanism, but also the Boston and the top trading cycles mechanisms, satisfy this property.11

Theoretical analyses such as those by Abdulkadiroğlu and Sönmez (2003) and Ergin and Sönmez (2006) have argued for the student-optimal stable mechanism and the top trading cycles mechanism based on their incentive, fairness, and efficiency properties.12 Their research has led to several school choice reforms, which were organized and reported by Abdulkadiroğlu, Pathak, and Roth, 2005, Abdulkadiroğlu et al., 2009 and Abdulkadiroğlu, Pathak, Roth, and Sönmez, 2005, Abdulkadiroğlu et al., 2006. This line of studies is extensively surveyed by Roth (2008), Sönmez and Ünver (2011), and Pathak (2011). As we have already emphasized, all of these papers focus on the evaluation of mechanisms in terms of the efficiency and fairness of allocations, assuming (implicitly) that the quality of every school is fixed. While drawing extensively on this literature, we offer a new perspective for distinguishing desirable school choice mechanisms from undesirable ones by analyzing their effect on schools' incentives for improving their educational quality.

The closest work to ours is the pioneering study by Balinski and Sönmez (1999), who introduce the concept of respecting improvements of student quality. Our definition is a natural adaptation of their concept to the case in which a school improves in students' preference rankings.13 However, the results of Balinski and Sönmez (1999) cannot be directly applied, as the model of school choice is asymmetric between schools and students since schools have multiple seats while each student can attend only one school. In fact, while Balinski and Sönmez (1999) showed that the student-optimal stable mechanism respects improvements of student quality, we show that no stable mechanism, not even the school-optimal stable mechanism, respects improvements of school quality.14

From the methodological point of view, the current paper uses two types of analytical methods from the market design literature. First, our paper uses the large market approach used by, among others, Roth and Peranson (1999), Immorlica and Mahdian (2005), and Kojima and Pathak (2009).15 As these studies point out, large market analysis can often provide a positive result in cases where more traditional approaches cannot. An additional distinct feature of our work is that the large market approach is used to make a clear distinction between good mechanisms and bad ones; in this paper, we use the large market approach to provide guidance to policymakers regarding which school choice mechanisms will incentivize schools to improve. Second, we show impossibility results on the compatibility of some desirable properties and then find domain restrictions on the class of preferences such that the desirable properties hold simultaneously. In the context of school choice, previous studies such as those by Ergin (2002), Kesten (2006), and Haeringer and Klijn (2009) found domain restrictions for the student-optimal stable mechanism and the top trading cycles mechanism to satisfy several desirable properties. Similarly to these studies, we find new domain restrictions for a stable or Pareto efficient mechanism to respect improvements.

Our work here is also related to the pre-existing literature on matching and investment, which considers settings in which firms and workers, both of whom have utility functions that are quasilinear in a numeraire, make (non-contractable) investments before engaging in trade. For instance, Cole et al., 2001a, Cole et al., 2001b considered such a setting and showed that there exists an equilibrium that achieves a fully efficient outcome.16 However, these works rely on agents having utility functions that are quasilinear in a transferable numeraire, and so their results can not be directly applied to our setting, as our setting (which considers public schools) does not admit the possibility of transfers.

Finally, this paper is part of the vast literature on school choice (in a broad sense), including educational vouchers, charters, and pilot schools. A large number of papers analyze, both theoretically and empirically, the competitive effects of school choice on educational outcomes: see the work of Belfield and Levin (2002), Hoxby (2003a) and Macleod and Urquiola (2013) for recent overviews. However, the number of papers on the competitive effects of public school choice, the focus of our paper, is relatively small; one exception is the work of Hastings et al. (2008), who analyze how public school choice provides different types of schools with different incentives for improvements. Our paper poses several open questions, both empirical and theoretical, to this literature on the economics of school choice: see the conclusion for details.

The remainder of this paper is organized as follows. In Section 2, we present our model and formally define the student-optimal stable mechanism, the Boston mechanism, and the top trading cycles mechanism. In Section 3, we formally define respecting improvements of school quality and present our benchmark impossibility results. In Section 4, we present our large market results. Section 5 analyzes an alternative criterion (respecting improvements of school quality in terms of enrollment). We conclude in Section 6. The online appendix discusses several additional alternative criteria as well as a number of related topics.

Section snippets

Model

There is a finite set S of students and a finite set C of schools.17 Each student sS has a strict preference relation s over C{}, where ∅ denotes the outside option of the student.18 The weak preference relation associated with s is denoted by s and so we write csc¯ (where c,c¯C{}) if either csc¯ or c=c¯

Respecting improvements of school quality

The main goal of this paper is to analyze how the design of a school choice mechanism affects the incentives of schools to improve themselves. To do this, we now define a criterion for evaluating school choice mechanisms in terms of the incentives they provide for school improvement. We first formally specify the concept of an improvement of school quality in our model.

Definition 1

A preference relation s is an improvement for school c over the preference relation s if

  • (1)

    for all c¯C{}, if csc¯, then cs

Respecting improvements in large markets

While the results of Section 3 show that no standard school choice mechanism always respects improvements of school quality, violations of this condition may be rare for some school choice mechanisms. In this section, we investigate this possibility by considering large, random markets which contain many students and schools; in that environment, we study the probability that each mechanism respects improvements.

Alternative criterion: respecting improvements of school quality in terms of enrollment

In the preceding discussion on respecting improvements of school quality, whether a mechanism respects improvements is judged in terms of schools' preferences. This means that we implicitly assume that school preferences in the model are the preferences by which schools evaluate matchings. However, in several real-life school choice systems, school preferences do not necessarily reflect schools' true preferences. Rather, they are often priorities set by law, as is the case for schools in

Conclusion

School choice has become a widespread and successful education policy in recent years. In this work, we considered how the design of a centralized public school choice mechanism affects the incentives of schools to improve their educational quality. We first defined the concept of respecting improvements of school quality, which requires that the outcome of a mechanism becomes weakly better for a school whenever that school becomes more preferred by students.49

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    A previous version of this paper appeared under the title “Promoting School Competition Through School Choice: A Market Design Approach.” We are grateful to Parag Pathak for continuous discussions on related issues. Michihiro Kandori, Yusuke Kasuya, and Dan Sasaki provided comments that motivated the analysis in Section 1.1 of the online appendix. We also thank Atı̇la Abdulkadı̇roğlu, Daron Acemoglu, Mustafa Oguz Afacan, Joshua Angrist, Itai Ashlagi, Eduardo Azevedo, Eric Budish, Gabriel Carroll, Yeon-Koo Che, Glenn Ellison, Dan Fragiadakis, Benjamin Golub, Guillaume Haeringer, Tadashi Hashimoto, James Heckman, Eun Jeong Heo, Brent Hickman, Bengt Holmstrom, Nicole Immorlica, Navin Kartik, Onur Kesten, Scott Duke Kominers, Jonathan Levin, Mihai Manea, Bentley MacLeod, Paul Milgrom, Terry Moe, Herve Moulin, Muriel Niederle, Derek Neal, Phil Reny, Michael Riordan, Alvin E. Roth, Kyoungwon Seo, Tayfun Sönmez, Yuki Takagi, William Thomson, Jean Tirole, Juuso Toikka, Peter Troyan, Xin Wei, Muhamet Yildiz and participants at numerous seminars and conferences for useful comments. We gratefully acknowledge comments by the Editor and three referees of the journal. Much of this work was completed while Hatfield was at the Stanford University Graduate School of Business. Kojima gratefully acknowledges financial support from the Sloan Foundation (FG-BR2013-005), as well as financial support from the National Research Foundation through its Global Research Network Grant (NRF-2013S1A2A2035408). Stephen Nei, Bobak Pakzad-Hurson, Fanqi Shi, and Qihang Wu provided excellent research assistance.

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