Elsevier

Games and Economic Behavior

Volume 129, September 2021, Pages 198-237
Games and Economic Behavior

How lotteries in school choice help to level the playing field

https://doi.org/10.1016/j.geb.2021.05.010Get rights and content

Abstract

School authorities in the UK and the US advocate the use of lotteries to desegregate schools. We study a school choice mechanism employed in Berlin where a lottery quota is embedded in the immediate acceptance (IA) mechanism, and compare it to the deferred acceptance mechanism (DA) with a lottery quota. In both mechanisms, some seats are allocated based on academic achievement (e.g., grades), while seats in the lottery quota are allocated randomly. We find that, in theory, a lottery quota strengthens truth-telling in DA by eliminating non-truth-telling equilibria. Furthermore, the equilibrium outcome is stable for DA with a lottery but not for IA with a lottery. These predictions are borne out in the experiment. Moreover, the lottery quota leads to more diverse school populations in the experiment, as predicted. Students with the lowest grades profit more from the introduction of the lottery under IA than under DA.

Introduction

School choice has become common practice in urban areas in the US and in Europe. Students are not automatically assigned to a school, but can choose a school that best fulfills their needs and matches their interests. School choice is supposed to permit fair access to good schools and to avoid unjust segregation with respect to socio-demographic characteristics that can arise if there is residential segregation and students have to attend their district school.

However, some of the high hopes in school choice have been dashed by reality. Allowing parents and students to choose among schools has not led to a desegregation of schools, as documented by Allen et al. (2014) for the UK. For Sweden, Bohlmark et al. (2016) even document an increase in segregation following the introduction of school choice procedures. Three distinct reasons have been put forward:

(i) The persistence of segregation could be due to the preferences of students and parents. Homophily may lead parents to choose schools with students of a similar socio-economic background. Also, school segregation can result when there is residential segregation and parents prefer schools in their proximity.1 Relatedly, school choice has not always led to a higher demand for schools with a strong academic record than for poorly performing schools.2

(ii) School choice mechanisms themselves may inadvertently further segregation. If school choice procedures are complex and it is hard to find the optimal application strategies, then strategically sophisticated, better-informed parents have an advantage when trying to get their children into desirable schools.

(iii) The selection of students based on academic achievements can lead to a segregation of schools along this criterion, i.e., desirable schools admit the high-achieving students. This segregation along the lines of academic achievement may be considered problematic in itself. It is also possible that it leads to segregation according to socio-economic characteristics where they are correlated with academic achievement.

In this paper, we present mechanisms that deal with the second and third reason. In particular, we vary the manipulability and hence the strategic complexity of the admissions procedure. Moreover, we study to what extent a lottery can reduce the segregation of schools along the lines of academic achievement. We take an existing mechanism involving a lottery, examine it in light of its desegregation properties, and compare it to a natural alternative mechanism.

Segregation along the lines of academic achievement has been at the heart of a long-standing public debate in England. English grammar schools base their admission decisions on students' exam scores. In regions where these selective grammar schools are numerous and attract the highest achieving students, their counterparts, the so-called comprehensive schools, are left with mostly lower achieving students.3 To mitigate segregation, the official School Admissions Code 2007 in the UK proposed using lotteries alongside other admission criteria at oversubscribed schools, a proposal supported by Coldron et al. (2008) in their report on secondary school admissions in England. Noden et al. (2014) report that a small but growing number of English schools use lotteries as the main criterion to determine the student priorities. Similarly, in New York City, Educational Option schools use a combination of priorities based on academic performance and a lottery in an attempt to ensure a diverse student composition, but find it increasingly hard to attract high-achieving applicants who instead flock to selective exam schools.4

To reduce segregation and equalize educational opportunities, the city of Berlin introduced a new admission procedure in the academic year 2010/2011. Schools are no longer allowed to use geographic proximity as an admission criterion if the number of applicants exceeds the available seats. Instead, a school can assign at most 60% of the seats based on applicants' academic achievement and has to assign 30% via a lottery, with the remaining 10% reserved for cases of hardship.5

The introduction of the lottery quota was highly controversial. Left-leaning politicians who favored a less differentiated student composition across schools (and hence a more diverse student composition within schools) called for a larger lottery quota. Steffen Zillich (member of the Berlin parliament for Die Linke, the Left Party) argued that using a lottery opened up highly demanded schools to children from educationally deprived social groups and that lotteries counteract a further differentiation of schools.6 Right-leaning politicians criticized a lottery as arbitrary and favored academic achievement as the principal determinant of priorities; according to Mieke Senftleben (member of the Berlin parliament for the FDP, Free Democratic Party) a lottery undermines the principle of merit, as talent and effort become secondary.7 A lottery may also weaken incentives for quality improvements by the schools to attract the most able students, as considered, for example, by Hatfield et al. (2016).

Besides the use of a lottery, the Berlin mechanism is controversial in that it applies the immediate acceptance algorithm. This algorithm has been widely used in many cities, notably in Boston where it first attracted the interest of economists. Following protests from parents, and after the involvement of economists who helped design a new mechanism, Boston abandoned the immediate acceptance mechanism in 2005. The main criticism was that under the immediate acceptance mechanism parents had to manipulate their rank-order lists over schools to achieve a good outcome.8 Such manipulations required strategic sophistication and information about the demand for the schools. Thus, the mechanism favored strategically sophisticated and better-informed parents over others.9 Under the new mechanism in Boston that is based on the deferred acceptance algorithm, parents cannot gain from misrepresenting their true preferences. This property, called strategy-proofness, levels the playing field among parents. Moreover, truthful reports can serve as a valuable feedback to school authorities on the quality of and the demand for particular schools.

In this paper, we use theory and experiments to investigate the existing mechanism in Berlin, and more generally, to understand the influence of a lottery quota on the two most frequently applied matching mechanisms, the immediate and the deferred acceptance mechanism. Following the controversies that accompanied the introduction of a lottery and taking into account the criticism of the immediate acceptance mechanism, we seek to understand whether the mechanism achieves the political goals of a more diverse student composition of schools and hence less segregation along the lines of academic achievement. In addition, we investigate an alternative mechanism with a lottery that is based on the deferred acceptance algorithm. We show how a lottery quota combined with the deferred acceptance mechanism levels the playing field in two of the three dimensions mentioned above: First, it gives students with lower academic achievements a chance to get a seat at their preferred school, thereby reducing segregation according to academic achievement. Second, it reinforces the strategy-proofness of the deferred acceptance mechanism by making it a strict best response for more students to report their true preferences, thereby reducing the complexity of the admission game.

Previous evidence from the field and the laboratory shows that participants in the deferred acceptance mechanism often fail to understand its incentive properties (Chen and Sönmez, 2006; Hassidim et al., 2018). Instead, they manipulate their rank-order lists in a systematic fashion (Echenique et al., 2016; Ding and Schotter, 2017; Klijn et al., 2013). A number of recent studies of matching mechanisms are motivated by the question of how to help participants make the right choices (see for example Bó and Hakimov, 2020). Our study contributes to this discussion by showing that a lottery quota in the deferred acceptance mechanism strengthens truth-telling by giving each applicant a small positive probability to get a seat at the most preferred school(s). In particular, this makes misreporting the most preferred school strictly dominated by truth-telling. In equilibrium, a lottery can also make it a strict (rather than a weak) best response to truthfully reveal the complete preferences.

In the context of school choice, incomplete information about other players, e.g., their preferences, has been studied as a source of uncertainty. Ehlers and Massó (2015) analyze ordinal Bayes-Nash equilibria for stable mechanisms in two-sided, many-to-one matching markets with incomplete information.10 In particular, they show that the set of equilibria shrinks as the support of the common prior expands (Corollary 1b) and that truth-telling is an equilibrium if schools' priorities are perfectly correlated (Proposition 1a). For one-sided many-to-one matching markets (i.e., non-strategic schools) with serial dictatorship, Chen and Pereyra (2019) show that when each student's prior about other students' preferences and school priorities has full support, then truth-telling is the unique ordinal Bayes-Nash equilibrium. Featherstone and Niederle (2016) provide an example where truth-telling is the unique ordinal Bayes-Nash equilibrium in anonymous strategies under immediate acceptance with incomplete information about student preferences. In all of these papers, the uncertainty due to incomplete information about others' preferences can strengthen the incentives to tell the truth.

In contrast to the work on incomplete information about other players, participants in our setting are uncertain about their lottery number, which determines their priority for some seats. This is closely related to tie-breaking, see Abdulkadiroğlu et al. (2009), which also creates uncertainty about the priorities. In our setting schools have two types of seats – those where priority is based on academic achievement, and lottery seats. We find that, just like uncertainty from incomplete information about others' preferences, the uncertainty introduced by the lottery quota reduces the set of equilibria and strengthens truth-telling incentives under the stable deferred acceptance mechanism. It also tends to increase truth-telling under immediate acceptance in the aggregate, though not for all students.

The issue of segregation resulting from school choice has been studied in depth by Calsamiglia et al. (2021). They consider a school-choice model with multiple communities and schools as local public goods, thereby enriching the framework of Epple and Romano, 1998, Epple and Romano, 2008 with different school choice mechanisms, namely immediate and deferred acceptance. Families decide where to live, and the quality of a school is affected by the ability of its students. Even in the absence of priorities based on residence or academic achievement, segregation by student ability may arise under immediate acceptance if high ability applicants care more about peer quality than applicants of lower ability. In contrast, deferred acceptance does not give rise to segregation, as long as the ordinal preferences over schools are identical for applicants of different abilities. In our setup preferences over schools are exogenous and peer effects do not affect school quality. Since schools prioritize students based on academic achievement, sorting along this dimension arises under both mechanisms, but is mitigated by the lottery.

We proceed as follows: First, we describe the existing Berlin school choice mechanism and a version of the deferred acceptance mechanism that incorporates a lottery. We show that the deferred acceptance lottery mechanism preserves the desirable properties of the deferred acceptance algorithm, including stability and strategy-proofness. Furthermore, we show that if students' priorities are the same across schools, then adding a lottery to the deferred acceptance mechanism always increases truth-telling in equilibrium (Theorem 1). For general priorities, adding a lottery to the deferred acceptance mechanism makes it a strictly dominant strategy to truthfully rank the most preferred school. While adding a lottery to the immediate acceptance mechanism gives some students an incentive to truthfully rank their most preferred school, some students might demote their most preferred school to avoid competition for a lottery seat. Finally, a lottery quota embedded in immediate acceptance leads to unstable outcomes.

To provide an empirical test of the predicted properties of the mechanisms involving lotteries, we conducted an experiment. We consider a setup where preferences over schools are correlated and academic achievement determines the students' priorities at schools. We compare the immediate acceptance mechanism with a lottery quota (which is a stylized version of the mechanism used in Berlin) to the immediate acceptance mechanism without a lottery and to the deferred acceptance mechanism with and without a lottery. This allows us to test whether, on average, the lottery quota leads to more truthful revelation of preferences under both mechanisms, as predicted by theory for the school choice problems we consider. Moreover, we test the predictions on stability by counting the number of blocking pairs. Finally, we consider which students benefit most from the introduction of a lottery in the two mechanisms and what the effects of the lottery are on the distribution of payoffs across students and on the composition of schools.

The experimental findings support the main theoretical predictions concerning comparisons of the mechanisms. The results show that lotteries increase truth-telling and lead to a more diverse student body at schools with respect to academic achievement. In particular, lotteries harm good-but-not-excellent students who are displaced by students with lower academic achievements but more luck in the lottery. While these findings hold for both school choice mechanisms, the immediate acceptance mechanism leads to more diverse schools than the deferred acceptance mechanism for the same lottery quota, at the cost of being manipulable and less stable. We conclude that where a lottery quota is used to strike a compromise between meritocratic and egalitarian principles, the size of the quota should take into account the mechanism used. At the same time, the fairness aspect of strategy-proofness and stability may be considered.

The paper is organized as follows. Section 2 introduces four school choice mechanisms, namely Immediate Acceptance (IA), Deferred Acceptance (DA), Immediate Acceptance with a lottery quota for one third of the seats (IA33), and Deferred Acceptance with the same lottery quota (DA33). Then, we analyze the effect of a lottery on stability and truth-telling under DA and IA in equilibrium. Section 3 presents the experimental design and the equilibrium analysis for the four mechanisms, including the hypotheses. The experimental results are found in Section 4, and Section 5 concludes.

Section snippets

School choice mechanisms

A school choice problem is a many-to-one matching problem between a set of students and a set of schools that have a limited number of seats to allocate. Each student has strict preferences over schools and each school has strict priorities over students. Schools having priorities as opposed to preferences imply that they do not act strategically, e.g., because priorities are mandated by the local education authorities. Note that in the remainder of the paper, the shorter term “priorities” can

Design of the experiment

The experiment is the first to investigate behavior under the two most important school choice mechanisms, IA and DA, in combination with a lottery quota. We study the effect of a lottery on truth-telling, stability, the utility of students, and the student composition at schools.

Experimental results

We first analyze the observed preference manipulations across the four mechanisms (IA, IA33, DA, and DA33). Then, we investigate the payoffs of students, the distributional effects of the lottery quota, and the composition of schools.

Conclusions

The paper is the first to study how to embed lotteries in school choice mechanisms with the aim to reduce disparities in the student composition between schools. We show that there are important differences between the effect of a lottery quota on the immediate acceptance mechanism (IA33) and the deferred acceptance mechanism (DA33).

Our experiment replicates previous evidence that truth-telling in the strategy-proof DA mechanism is significantly lower than predicted (see the survey by Hakimov

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    The authors would like to thank Bistra Solakova for her contribution to this research in its early phase (see Solakova, 2011) and to Renke Schmacker and Felix Bönisch for excellent research assistance. We are grateful to Nina Bonge who helped us run the experiments and to Jennifer Rontganger for copy editing. We received valuable comments from participants of the Matching-in-Practice workshop in Cologne, the Market Design Workshop at ZEW Mannheim, the German Society of Operations Research Conference at HU Berlin, the seminar at the MPI for Collective Goods in Cologne, as well as from Bob Hammond. We would like to thank two reviewers, the advisory editor, and the editor for their constructive comments and thoughtful handling of the paper. Bettina Klaus and Dorothea Kübler gratefully acknowledge the hospitality of Stanford University where part of this paper was written. Financial support by the Deutsche Forschungsgemeinschaft through CRC TRR 190 (project number 280092119), from the Swiss National Science Foundation (SNSF) through Project 100018_162606, and from the Fonds de la Recherche Scientifique (FRS-FNRS) through Grant 1.B.222.17F is gratefully acknowledged.

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