Assigning more students to their top choices: A comparison of tie-breaking rules

Transcription

1 Assigning more students to their top choices: A comparison of tie-breaking rules Itai Ashlagi Afshin Nikzad Assaf Romm August 2016 Abstract School districts that implement stable matchings face various decisions that affect students assignments to schools. We study properties of the rank distribution of students with random preferences when schools use different tie-breaking rules to rank equivalent students. Under a single tie-breaking rule, where all schools use the same ranking, a constant fraction of students are assigned to one of their top choices. In contrast, under a multiple tie-breaking rule, where each school independently ranks students, a vanishing fraction of students are matched to one of their top choices. However, by restricting the students to submitting relatively short preference lists under a multiple tie-breaking rule, a constant fraction of students will be matched with one of their top choices, whereas only a small fraction of students will remain unmatched. Keywords. school choice; tie-breaking rule; deferred acceptance; stable matching Ashlagi: Sloan, MIT, iashlagi@mit.edu. Nikzad: Economics and MS&E departments, Stanford, nikzad@stanford.com. Romm: Economics, Hebrew University of Jerusalem, assaf.romm@mail.huji.ac.il. We thank Sergiu Hart, Avinatan Hassidim, Déborah Marciano-Romm, Parag Pathak, and Alvin Roth for helpful discussions. Ashlagi acknowledges the research support of National Science Foundation grant SES Romm acknowledges the research support of the Maurice Falk Institute and Israel Science Foundation grant 1780/16. 1

2 1 Introduction Two-sided matching markets have been an exciting research area ever since they were introduced by Gale and Shapley (1962). In these markets, each agent has ordinal preferences over agents from the other side of the market. Despite the vast literature following Gale and Shapley s work, little is known about whom agents should expect to be matched with. One area in which the rank distribution plays a crucial role is school choice, as school districts make policy decisions that directly affect students assignments. Many school districts have adopted student assignment mechanisms based on the studentproposing deferred acceptance (DA) algorithm (see Abdulkadiroğlu and Sönmez (2003)). 1 This algorithm selects a stable matching with respect to schools and students preferences. Often, however, schools are indifferent between large groups of students, and the manner in which ties are resolved has consequences for students welfare (Abdulkadiroğlu and Sönmez (2003); Erdil and Ergin (2008b)). Schools districts further decide how many schools students can rank, and the length of students lists of preferences also affects student assignments. This paper provides some insights into the impact of such choices on the students rank distribution. 2 School districts typically consider have mostly considered two natural tie-breaking rules: a multiple tie-breaking rule (MTB), under which every school independently assigns each applicant a random lottery number to break ties, and the single tie-breaking rule (STB), under which each student receives a single lottery number to be used for tie-breaking by all schools. A separate lottery for each school seems fairer, as students with bad draws at some schools may still have good chances at other schools, but such a system may be unnecessarily inefficient (Abdulkadiroğlu and Sönmez (2003)). Abdulkadiroğlu et al. (2009) and De Haan et al. (2015) use numerical experiments to empirically compare these tie-breaking rules using New York City and Amsterdam school choice data, respectively. They both find that STB results in more students receiving their top choices, but at the same time more students receive bad choices or remain unassigned. These trade-offs led to different implementations in each school district s first year of using DA, where New York City used STB and Amsterdam used MTB (Abdulkadiroğlu et al. (2009) document that even in New York City policymakers first leaned towards implementing 1 See Abdulkadiroğlu et al. (2005) and Abdulkadiroğlu et al. (2009) for implementation details about Boston and New York City, respectively. 2 See Ashlagi and Shi (2015), who study the design of school choice menus under public constraints. 2

3 MTB for reasons of equity). This paper takes a first analytical approach to explain these qualitative insights. We consider a two-sided matching model with students and schools, and students have random preferences over schools. To keep things simple, all students belong to a single priority class, implying that all students have a priori equal access to every school. Schools have a fixed capacity, and we investigate properties of the students rank distribution resulting from the student-proposing DA under STB and under MTB as the numbers of students and schools grow large. First we assume students rank all schools and ask how many students are likely to be assigned to one of their top choices under STB and MTB. Unless otherwise specified, by a student s top choice we refer to one of her k most preferred schools for some constant k that is independent of the number of schools (which in turn grows large) and is clearly known from the context. Before stating the results, we discuss the choice of this measure. In our model, when the number of students and the number of schools grow at the same rate, both tie-breaking rules will result in a student being assigned on average to one of her top O(ln n) choices (Pittel (1989); Knuth (1976)). Moreover, under MTB, she is assigned with high probability to one of her top O(ln 2 n) choices. Both of these quantities are well within her top 1% of schools. Finally, even in a large school district such as that of New York City (with approximately 700 programs), it may be significant for a student to be assigned to one of her very top choices (say, her top 3) rather than to a school within her top 1% (top 7 schools). We show that under STB, a constant fraction of students are assigned to their first choice. 3 This is intuitive, since the DA algorithm with STB is equivalent to random serial dictatorship, under which each student chooses a school in some random order. In sharp contrast, under MTB students are unlikely to get one of their top choices. Formally, for any constant k, with high probability a vanishing fraction of students are matched to one of their k most preferred schools as the market grows large. The following offers some intuition for this contrast. Consider a student s who has been temporarily assigned to her top choice by the end of the first round of DA. Since s was assigned to her top choice, say school c, she is likely to have a good lottery number at c. Under STB, future contenders for seats in c are likely to have worse lottery numbers than s, since they were rejected from some other schools. Under MTB, however, being rejected from a different school provides less 3 This further emphasizes the relevance of of our top choice measure as defined above. 3

4 information about a future contender s chances of taking s s place in school c. 4 Next we analyze the effect of shortening preference lists on student assignments. Using preference lists of bounded length is common practice in many school districts for a variety of reasons. In some school districts, administrators prefer to keep the length of preference lists to a small number to simplify the task of ranking schools (as supposedly was the case in New York City (Abdulkadiroğlu et al., 2009)). Moreover, some administrators may benefit indirectly from the fact that strategic considerations lead students to rank schools that are in less demand higher in their preference lists (Haeringer and Klijn, 2009), thus improving the expected rank distribution and lowering the number of unassigned students. 5 While shortening preference lists under STB changes the rank distribution only mildly, 6 the effect under MTB is much more pronounced. Furthermore, it is interesting to investigate the change in the distribution under MTB due to its fairness properties (see discussion in Section 5.3). Complementing our first result, we find that under MTB it is possible to select the length of the preference lists such that not only are a large fraction of students assigned to a top choice, but also only a small fraction of the students remain unassigned. An interesting case where shortening preference lists has a substantial impact on students rankings occurs when there is a shortage of seats, which reflects situations in which some schools are in significantly greater demanded than others. At one extreme where the shortage of seats is a constant fraction of the number of students, with high probability the rank distribution of students can be significantly improved without increasing the number of unassigned students. Intuitively, under MTB, allowing long preference lists creates a thick market, whereas truncating preference lists reduces competition and produces better outcomes for matched students. Thus our results and the intuition they convey represent a different (non strategic) argument for requiring short preference lists under MTB. From a technical standpoint, one novelty in the proofs is the explicit use of (non-trivial) capacities. Most papers that study random matching markets assume either explicitly or implicitly that the market contains large imbalances, which leads to an analysis similar to 4 See Remark 1, which describes a somewhat counterintuitive example of a market with asymmetric preferences, in which the expected number of students that receive their first choice is larger under MTB than under STB. 5 One of the authors, who is involved in redesigning school choice in Israel, had to suffer through this argument again and again as it was explicitly and proudly presented to him by school choice administrators in several Israeli cities. 6 This can be easily verified using computer simulations. Exact calculations can also be performed using methods similar to those in the proof of Proposition

5 that of one-to-one matching markets. 7 Our method allows us to extend the arguments to non-constant capacities. The main idea in the analysis is to simplify the random process that corresponds to the deferred acceptance algorithm by coupling it with a simpler random process. 1.1 Related Work Closely related are papers that study the trade-offs between STB and MTB. Abdulkadiroğlu et al. (2009) and De Haan et al. (2015) run numerical experiments using NYC and Amsterdam school choice data and find that STB results in many more students being assigned to one of their top choices and further the students rank distributions under STB and MTB have a single crossing point. Independent of this work, Arnosti (2015) explains the single crossing point in a cardinal utility model in which the length of preference lists is held constant. He further quantifies the number of students that remain unassigned under the different tie-breaking rules. Our work differs in at least two important ways. First, we also analyze markets in which the length of preference lists grows as the number of schools grows large. It is worth noting that our result that only a vanishing fraction of students receive one of their top choices under MTB holds even when students preference lists grow at a slow rate as the market grows large. Second, Arnosti s model assumes the imbalance between seats and students is of the order of the number of agents, whereas our model allows also for a small additive imbalance. Abdulkadiroğlu et al. (2015) analyze a model with a continuum of students and a finite number of schools, each with a mass of seats, in order to compare the two tie-breaking rules. They establish the existence of a unique set of cutoffs that clear the market under each tie-breaking rule and find that STB is ordinally efficient. 8 The trade-off between incentives and efficiency when preferences contain indifferences has led to several novel tie-breaking approaches, among which are stable improvement cycles (Erdil and Ergin (2008a)), efficiency-adjusted DA (Kesten (2011)), and choice-augmented DA (Abdulkadiroğlu et al. (2015)). Another closely related paper is Che and Tercieux (2015), who propose a DA-like mechanism based on a circuit breaker. Their algorithm does not 7 This is not to say that capacities do not significantly affect these papers approaches and analysis (see, for example, Kojima and Pathak (2009)). 8 See also Che and Kojima (2010), Liu and Pycia (2012), and Ashlagi and Shi (2014), who calculate cutoffs under STB in markets with a continuum of students. 5

6 allow students who are preferred by a certain school to push out other students who rank that school much higher than the preferred students do, thus improving the rank distribution at the expense of a few students priorities. Their approach, which can be viewed as a dynamic truncation of preference lists (and is reminiscent of the Chinese Parallel, described by Chen and Kesten (2013)), aims to stem the inefficiency resulting from a lot of competition. The effect of requiring students to submit short preference lists has been described and studied by Haeringer and Klijn (2009) and experimentally tested by Calsamiglia et al. (2010). However, the idea of using truncation and dropping strategies dates back even further; see, for example, Roth and Rothblum (1999) and Kojima and Pathak (2009). 9 We are not the first to study properties of the rank distribution in two-sided random matching markets. Pittel (1989, 1992) studied the men s average rank in a balanced stable marriage model with random preferences and showed that under men-optimal stable matching, the men s average rank is approximately ln n and, with high probability, every man is assigned to one of his top ln 2 n preferred women. Ashlagi et al. (2016) showed that if there are fewer men than women, men are matched on average to at most their ln n-th choice under any stable matching. Our main result here, which shows that few students obtain one of their top preferences under MTB, is not implied by Ashlagi et al. (2016) (and thus even surprising when students are on the short side). Finally, a few studies compare tie-breaking rules under the top trading cycles (TTC) algorithm, which find Pareto efficient assignments. Both Pathak and Sethuraman (2011) and Carroll (2014) extend the results of Abdulkadiroğlu and Sönmez (1998) and show that there is no difference between a single tie-breaking (i.e., using random serial dictatorship) and a multiple tie-breaking rule (TTC with random endowments). 2 The model and preliminary findings In a school choice problem there is a set of n students, each of whom can be assigned to one seat at one of m schools. Denote the set of students by S and the set of schools by C, each of which has a fixed capacity. Every student has a strict preference ranking over all schools. The rank of a school c in a preference list of student s is the number of schools that s weakly prefers to c (so s s most 9 See also Coles and Shorrer (2014), who study truncation in random markets, and Gonczarowski (2014) for an algebraic approach. 6

7 preferred school has rank 1). Unless otherwise specified, students preferences over schools are drawn independently and uniformly at random. Each school c C has a strict priority ranking over students that is used to break ties between students. A matching is a mapping from students to schools such that each student is assigned at most one school and the number of students assigned to any given school is at most the capacity of that school. A matching is unstable if there is a student s and a school c such that s prefers to be assigned to c over her current assignment, and, c either has a vacant seat or an assigned student with a lower priority ranking than s. A matching is stable if it is not unstable. We consider two common tie-breaking rules that school districts use to determine schools priority rankings over students. Under the multiple tie-breaking rule (MTB) each school independently selects a priority ranking over all students uniformly at random. Under the single tie-breaking rule (STB) all schools use the same priority ranking order, which is selected a priori uniformly at random. 10 We study properties of the rank distribution of students in the assignments generated by the student-proposing deferred acceptance (DA) algorithm when schools priority rankings are generated by each of these tie-breaking rules. For brevity we refer to the assignments as the outcomes under MTB and STB. 3 Average rankings and top choices We study properties of the rank distribution in school choice problems with many schools, each of which has a small capacity. Each school has the same capacity q, which is a constant independent of n and m, and we further denote by q = qm the total number of seats. Unless mentioned otherwise we use d to denote a constant parameter that is independent of n and q. Section 3.1 examines properties of the rank distribution under MTB, and Section 3.2 studies the rank distribution under STB. The main results in these sections are summarized in the following statement. Main Theorem. Suppose n q d for some constant d 0. Then for any constant k, as n approaches infinity, the expected fraction of students who are matched to one of their top k 10 One way to implement STB is to assign each student a lottery number drawn independently and uniformly at random from [0, 1]. MTB can be implemented in a similar fashion by using a different draw (and thus different lottery numbers) for each school. 7

8 choices under student-optimal stable matching approaches zero under MTB, but approaches a positive constant under STB. The Main Theorem holds true under weaker assumptions that do not necessarily require students to rank all schools and may allow k and q to grow large; this is discussed in the following sections. A few more words are appropriate regarding the assumption that preferences are drawn uniformly at random. Our analysis still holds when there is limited alignment in students preferences, allowing for schools to differ slightly in their popularity, resulting in the same qualitative insights. As students preferences become more (positively) correlated, the rank distributions under STB and MTB become more similar; at the extreme of fully aligned preferences, the rank distributions are identical. 3.1 Properties of the rank distribution under MTB We begin with studying the average rank. For this purpose we apply a result by Ashlagi et al. (2016), who find that there is a significant advantage to being on the short side in a two-sided random matching market. Denote by Avg(π) the matched students average rank of schools under the student-optimal stable matching when the tie-breaking rule π is used. Proposition 3.1 (Ashlagi et al. (2016)). Suppose q = 1, n = q + d for some (possibly negative) constant d, and fix any ɛ > 0. If d > 0, the probability that Avg(MT B) is at least (1 ɛ) n converges to 1 as n grows large. If d 0, the probability that Avg(MT B) is at ln n most (1 + ɛ) ln n converges to 1 as n grows large. So with a shortage of seats students are, on average, assigned to much worse choices than they are when there is a surplus of seats. Next we study the fraction of students who get one of their top choices. The next result shows that very few students receive one of their top choices under MTB as the market grows large even when students are on the short side of the market. Denote by R k (π) the expected fraction of students who get one of their top k choices under student-optimal stable matching when the tie-breaking rule π is used. Theorem 3.2. Suppose n q d for some constant d Then for any constant k, R k (MT B) approaches 0 as n approaches infinity. 11 In fact, the result holds true even when d = n b, where b is a constant less than 1. The same proof goes through, with minor modifications (merely different choices of constants). 8

9 A few comments are appropriate: 1. Theorem 3.2 holds true also when preference lists are not complete, as long as they grow at a non zero rate. That is, preference lists are of length f(m) where f approaches infinity as m approaches infinity. 2. Theorem 3.2 is true also when k = k(n) is not held constant but grows slower than ln n (recall from Theorem 3.1 that students are assigned on average to at most their ln n rank). 3. Theorem 3.2 still holds when q grows slower than ln n as long as k is held constant. 4. Ashlagi et al. (2016) compute the average rank of students in a one-to-one large random matching market. They find that when students are on the short side (n q), they obtain on average roughly their ln n rank and when students are on the long side n (n > q), they obtain on average at least their rank. Neither of these results imply ln n Theorem 3.2, which shows that a vanishing fraction of students obtain one of their top k choices for any constant k (k is independent of n). Hence, our result is especially surprising when students are on the short side of the market. Moreover, our analysis extends to many-to-one markets with constant capacities. 5. Theorem 3.2 applies when there is not a large surplus of seats. Indeed, when the surplus of seats is sufficiently large, many students will one of their top choices: Ashlagi et al. (2016) analyze a one-to-one matching market, in which the number of students is a fraction of the number of seats (n = λ qm for some λ < 1 and q = 1) and find that students will obtain, on average, a constant rank (so many students will be assigned to one of their top choices). The proof, given in Appendix A, is based on the following steps. Fix some arbitrary school c. By symmetry and linearity of expectation observe that R k (MT B) equals the expected number of students who are admitted to c, and c is among their top k preferences. Denote the latter amount by E [X c ]. Next, we show that school c receives many more applications than the number of students who list c as one of their top k choices. Formally, c receives at least Ω(ln n) proposals with high probability. Moreover, with high probability the number of students who list c as one of their top k choices, denoted by ψ(k), is very small (sub-logarithmic in n). Therefore, since each student who applies to c is admitted with a 9

10 probability of at most O(q/ ln n), E [X c ] O 0 as n approaches infinity. ( ) q ψ(k). This implies that E [X ln n c ] approaches 3.2 Properties of the rank distribution under STB Next we discuss properties of the the rank distribution under STB (proofs are deferred to Appendix E). Proposition 3.3. Under STB, students average rank of schools is O(ln min{n, q}). The intuition for the result follows from observing that the DA algorithm under STB is equivalent to the random serial dictatorship algorithm, in which students pick in random order their favorite school from schools that still have vacancies. When it is the turn of the (i + 1)-th student to choose a school, the rank she will obtain is, roughly speaking, a geometric random variable with a probability of success of at least these random variables gives the result. i m i and summing up When preferences are random in the random serial dictatorship algorithm, many students (among those who pick a school early) are likely to obtain one of their top choices, in contrast to MTB. Proposition 3.4. Let t = min{n, q} be the total number of assigned students under STB. Then: (i) The expected number of students assigned to their first choice under STB is at least t/2. Moreover, (ii) the total number of students assigned to their first choice under STB is w.h.p. at least (1 ε)(t/2) for any ε > It follows directly from Proposition 3.4 that when the number of students is not more than the number of seats, at least half of the students to be assigned to their top choice in expectation. Remark 1. The following example shows that the expected number of students assigned to their first choice can be larger under MTB than under STB. There are 5 schools with capacities q 1 = 40, q 2 = 10, q 3 = 500, q 4 = 5000, and q 5 = 20, 000. There are 4 types of students. There are 50 students of type 1 with preference c 1 > c 2 > c 3 > c 4 > c 5, 10 students of type 2 with preference c 2 > c 5, 500 students of type 3 with preference c 3 > c 4 > c 5, 12 Given a sequence of events {E n }, we say that this sequence occurs with high probability (w.h.p.) if lim n 1 P[E n] n θ = 0, for some constant θ > 0. 10

11 and 5000 students of type 4 with preferences c 4 > c 5. Computer simulations show that the expected fractions of students who obtain their first choice under STB and under MTB are and , respectively. 13 The main idea behind this example is that the expected length of rejection chains is shorter under MTB than under STB. 14 Nevertheless, this example is carefully crafted and the gap in favor of MTB is very small. We believe, however, that more students are assigned to their first choice under STB than under MTB under more general assumptions about preferences (for example, when preferences are generated from a symmetric multinomial logit model). 4 Short preference lists School districts often limit the number of schools students can rank. Since students do not apply to all schools they find acceptable, competition is artificially limited, which is a way to resolve the issues that Theorem 3.2 exposes. Of course, shortening the lists may also increase the number of unassigned students. We quantify this trade-off in our model (the proof appears in Appendix A.2). Theorem 4.1. Let U k denote the number of unassigned students given that students submit only their k most preferred schools under the MTB rule. Also, let q = min(q, n). 1. If n = (1 + ε)q for some constant ε (ε can be negative). Then, there exists a random variable such that U k max(εq, 0) +, where (a) E [ ] e ε k q, (b) is not much larger than µ = e ε k q, in the following sense (Chernoff bounds): P [ > µ(1 + δ)] e δ2 µ 3 0 < δ < 1 ( ) e δ 1 µ P [ > δµ] δ > 1 13 The averages are taken over i iterations. For any large enough i (e.g., for i 10 4 ), the averages coincide with the two numbers mentioned above, regardless of the random seed initializer. 14 To see this, first notice that after the first round of the student-proposing DA algorithm, all students of types 2 and 3 and 4 are accepted to their first choice. Also observe that after the first round, 10 students of type 1 will be rejected from their first choice, c 1. These rejected students are less likely to be accepted to their second choice, c 2, under STB than under MTB. Furthermore, a student who is accepted to c 2 in the second round triggers a rejection chain that goes immediately to school c 5, which has enough capacity for all students. Finally, roughly speaking, a student of type 1 who is accepted to c 3 is likely to trigger a chain of length 2 till it reaches c δ δ

12 2. If n = q + d with d = o(q) (d can be negative), then there exists a random variable such that U k < max(d, 0) +, where: (a) E [ ] 2q/k for all k m. (b) W.v.h.p. (2 + ε )q/k, for all ε, ε > 0 and k q 1/3 ε. When preference lists are long, very few students are assigned to one of their top choices under MTB. Theorem 4.1 suggests a remedy by shortening the lists. When there is an excess of seats, shortening the lists can significantly improve the average rank of assigned students, while ensuring that the number of assigned students does not decrease significantly or does not decrease at all, as shown in the following corollary. Corollary 4.2. Suppose n = (1+ε)q, and let k = 1+δ ε Then, w.h.p. = 0. ln q, where ε, δ are positive constants. Proof. By Theorem 4.1 we have E [ ] e (1+δ) ln q q = q δ. So, P [ > 1] q δ. Observe that when preference lists are short, DA is not strategyproof. However, since preferences are drawn uniformly at random in our model, it is safe to abstract away from strategic decisions since in equilibrium each student ranks her top k choices. Our results depend on the assumption that students preferences are drawn uniformly at random. However, one can apply Theorem 4.1, for example, to a model with two tiers of schools. In this model students prefer first-tier schools to second-tier schools but there is a shortage of seats in first-tier schools. Theorem 4.1 provides an upper bound on the number of unassigned students when preference lists are short. In particular, the fraction of first-tier seats remaining vacant decreases exponentially with the length of the preference lists. In Section 5.2 we provide further simulation results that quantify the number of unassigned students under a more involved preference structure. 5 Simulations This section presents simulation results that complement our theoretical results. We first consider markets with complete preference lists for students while varying parameters of school choice problems. Then we consider school choice problems in which students have short preference lists. Finally we investigate how the length of the preference lists affects the standard deviation of a student s rank as well as the number of unassigned students. 12

13 5.1 Long preference lists The first computational experiment illustrates the effect of the market size on the cumulative rank distributions under STB and MTB. For each set of parameters considered, we sample realizations by drawing complete preference lists uniformly at random an independently for each student. In addition, under MTB, for each market realization we draw a complete order over students for each school, independently and uniformly at random. Under STB, for each market realization we draw a single order over students uniformly at random. Then, we compute the student optimal stable matching. The statistics presented are generated by taking average over at least 1000 samples for each set of parameters. Figure 1 plots the (average) students cumulative rank distribution under MTB (left panel) and under STB (right panel) in three different balanced markets and where each school has unit capacity. The left panel (under MTB) shows that as the number of students (and seats) grows large, the percentage of students who get their top choices under MTB becomes smaller. Observe that the difference between the percentage of those students when n = 10 4 and n = 10 3 is roughly similar to the difference between n = 10 5 and n = 10 4 ; this follows from the logarithmic rate of decay as shown in the proof of Theorem 3.2. Under MTB, as the number of students increases, the rank distribution becomes flatter. The corresponding graphs for STB almost coincide with each other for all simulated markets n=10^3 n=10^4 n=10^ n=10^3 n=10^4 n=10^ Figure 1: Students cumulative rank distribution in three different balanced markets (n = 10 3, 10 4, 10 5 ) in which each school has capacity q = 1. The left panel plots the distributions under MTB, and the right panel plots the distributions under STB. 15 Under STB, the larger the capacity in each school, the higher the rank distribution since more students are assigned to their top choices. 13

14 5.2 Short preference lists Next we simulate the effect of limiting the length of preference lists on the rank distribution under MTB. Figure 2 plots the (average) cumulative rank distributions for markets, in which schools have relatively large capacities. The markets simulated consist of m = 100 schools, each with capacity q = 100, and n = students. Note that when students submit their full preferences, the number of students getting their first choice is very small (recall that under STB almost half of the population gets their first choice). However, as lists become shorter, the rank distribution (for the listed positions) improves, although the number of unassigned students does not notably increase k=5 k=10 k= k=20 k=50 k= Figure 2: Students cumulative rank distribution under MTB with different lengths of preference lists. There are m = 100 schools, each school has capacity q = 100, and n = The left panel plots the distributions for preference lists with lengths 5, 10, and 20, and the right panel plots the distribution for preference lists of lengths 20, 50, and 100. Our finding that one may truncate preference lists while leaving few students unassigned depends on the assumption that students preferences are drawn uniformly at random. We next explore the effect of correlation in students preferences on the number of unassigned students under MTB. We simulated markets with 100 schools, each with capacity 100, and 10,000 students. Students preference rankings are drawn independently from a multinomial logit discrete choice model: we assume each school c has a weight w c = e c m λ representing its popularity and students rank schools on their list proportionally to these weights (the w chance that school c is a student first choice is m c ). In this model, the larger λ, the more c =1 w c students preferences are aligned (setting λ = 0 results in the uniform random distribution). For simplicity we assume students are sincere (and do not calculate equilibrium behavior); one can therefore interpret the findings as an upper bound on the fraction of unassigned 14

15 % unassigned students. Figure 3 reports the percentage of unassigned students for different choices of λ (degree of correlation) and varying lengths of the preference lists. Observe that a larger λ results in a larger fraction of unassigned students MTB STB 1/5 1/10 1/20 2/5 2/10 2/20 4/5 4/10 4/20 dergree of correlation (λ)/length of prefernece list Figure 3: Percentage of students remaining unassigned under MTB and STB with varying lengths of preference lists and varying demand over schools. 5.3 Fairness Our theory suggests that significantly more students are assigned to one of their top choices under STB than under MTB. MTB, however, may still be attractive from the standpoint of fairness, because students with bad lottery numbers are more likely to be assigned to lower choices under STB than under MTB (intuitively a student who receives a bad draw under STB is likely to be rejected in schools that other students have applied to). 16 Next we compare the tie-breaking rules with and without truncation from the standpoint of fairness and adopt the standard deviation of students ranks as a proxy for fairness. First we assume that students rank all schools. Let µ STB and µ MTB denote the studentoptimal matchings under STB and MTB, respectively. Table 1 reports the expected average rank and expected standard deviation of students ranks (denoted by Avg and Std ) under µ STB and µ MTB in simulated markets with different imbalances and capacities. Note that when there is an excess of seats (first two columns), the average rank for students is similar under both tie-breaking rules, whereas MTB has a lower standard deviation. When there is a shortage of seats, both the average and the standard deviation are smaller under STB than under MTB. We refer the reader to Ashlagi and Nikzad (2016) who analyze such differences in the variance of the rank in large random markets. 16 A (loose) argument to see this in markets with unit capacities and a surplus of seats is the following: students average rank is of the same order under STB and MTB (Ashlagi et al. (2016); Knuth (1976)) yet many more students obtain their top choices under STB than under MTB. 15

16 n qm m (q) (5) 1000 (10) Avg(µ STB )/Avg(µ MTB ) 1.77/ / / /234.9 Std(µ STB )/Std(µ MTB ) 2.73/ / / /210.6 Avg(µ STB )/Avg(µ MTB ) 1.57/ / / /206.8 Std(µ STB )/Std(µ MTB ) 2.5/ / / / Table 1: Average rank and the standard deviation of students ranks under STB and MTB in the student-optimal stable matching for different markets where students have full preference lists. A student s most preferred rank is 1, and larger rank indicates a less preferred school. Shortening preference lists allows more students to obtain top choices under MTB (Theorem 4.1). However, even in this case the rank distributions under MTB and STB have a single crossing point (Arnosti (2015)), implying that there is a real trade-off between the tie-breaking rules. Next we explore how the standard deviation of students ranks varies with the length of the preference list. When preference lists are truncated, some students may not be assigned even when there is a surplus of seats. Due to the symmetry in students preferences, we restrict attention only to assigned students. Intuitively, truncation should result in a lower standard deviation due to reduced competition. Table 2 reports the expected percentage of unassigned students (denoted by UA) and the sample standard deviation of a student rank in simulated markets with 1000 schools, where each school has 10 seats, an imbalance of 100 students, and varying lengths of preference lists. k n qm UA(µ STB )/UA(µ MTB ) 0.032/ / / / /0.01 Std(µ STB )/Std(µ MTB ) 0.72/ / / / /3.83 UA(µ STB )/UA(µ MTB ) 0.023/ / / / /0 Std(µ STB )/Std(µ MTB ) 0.96/ / / / /0.95 Table 2: Percentage of unassigned students and the standard deviation of a student rank under STB and MTB in student-optimal stable matching for different markets. The first two rows report results for markets with students and the last two rows report results for markets with 9900 students. Column headings represent the length of preference lists. A student s most preferred rank is 1, and a larger rank indicates a less preferred school. 16

17 6 Discussion In their seminal paper comparing school choice mechanisms, Abdulkadiroğlu and Sönmez write: Using a single tie-breaking lottery might be a better idea [...]. In this case, any inefficiency will be necessarily caused by a fundamental policy consideration and not by an unlucky lottery draw. In other words, the tie-breaking will not result in additional efficiency loss if it is carried out through a single lottery (while that is likely to happen if the tie-breaking is independently carried out across different schools). (Abdulkadiroğlu and Sönmez, 2003, footnote 14). The current paper provides a first analytical treatment for this type of comparison between STB and MTB rules using a model with random preferences. The results support empirical evidence in NYC and Amsterdam school choice data that STB outperforms MTB at the higher end of the rank distribution (Theorem 3.2). 17 MTB is often considered to be fairer than STB by school choice practitioners, since a bad draw for a student under STB is irreversible. In a balanced market it follows from Pittel (1992) and Knuth (1976) that with high probability every student is assigned to at most her O ( ln 2 n ) rank under both tie-breaking rules. However, under STB the last assigned student receives her n -th choice, in expectation. Simulations suggest that in general there is 2 a tipping point in the rank distribution, such that STB dominates until that point (whereby it assigns more people to their top choices), but MTB dominates beyond this point. Arnosti (2015) further establishes this single crossing result when students preference lists are short. It is also interesting to gain insights into the rank distribution when students are on the long side of the market. This case is relevant, for example, when some schools are considered much better than others and there are not enough seats for students in these better schools. In such markets the model predicts that STB significantly outperforms MTB both in terms of students average rank (Proposition 3.1 and Proposition 3.3) and in assigning more students to their top choices (Theorem 3.2 and Proposition 3.4). Length of preference lists. School districts, such as those in Boston and NYC, often allow students to rank only a few schools. One concern is that many students will remain 17 See Abdulkadiroğlu et al. (2009) and De Haan et al. (2015). 17

18 unassigned, which will result in an excessive administrative burden. 18 Our results provide a rationale for shortening lists in addition to some guidance for practitioners (Theorem 4.1). By limiting (appropriately) the length of preference lists under MTB, the social planner can bound the fraction of students who get assigned to a school they do not like, whereas allowing long lists may grant only a very small number of students their top choices (this effect is even stronger when the students are on the long side). 19 This observation also resonates with the logic behind the circuit-breaker mechanism of Che and Tercieux (2015). Indeed, offering a small number of choices and then later administratively assigning unmatched students can be viewed as a rudimentary way to implement a two-stage mechanism. Large capacities. Our model assumes schools have small capacities. One motivation is that in practice there are typically multiple priority classes, which affect students assignments. For example, schools may fill many of their seats with students that belong to top priority classes (due to their proximity to the school or siblings priority), which suggests that competition concerns only the remaining seats. Nevertheless, we will extend the discussion to markets with a continuum of students and a finite number of large schools. Consider a continuum of N students and m schools, each with mass capacity q and where students have uniformly random preferences over schools. 20 Using the cutoff characterization of stable matchings (Azevedo and Leshno (2016)), observe that when N mq, all (except possibly a measure 0 of) students will be assigned to their first choice under both tie-breaking rules. When N > mq, under STB all assigned students but a measure 0 are assigned to their first choice, 21 but under MTB a mass of students are assigned to their second or worse choice. Final remarks. The tie-breaking rule may have a significant effect on the resulting rank distribution. We believe that information about this distribution is not only crucial to guiding design, but also valuable for students who go through the (arguably) costly task of ranking schools. 22 This paper contributes to the study of tie-breaking rules by applying methods from a growing line of research on two-sided matching markets with random preferences 18 Abdulkadiroğlu et al. (2009) provide empirical evidence that many students in NYC remain unmatched after the first round. 19 We do not compare formally the number of unassigned people under both STB and MTB; however, we believe that MTB actually performs better in that it leaves fewer people unassigned. 20 This example is a special case of the continuum model in Abdulkadiroğlu et al. (2015) and Azevedo and Leshno (2016)). 21 The reason is that the chance to meet the cutoff at each school is mq N and each school is listed first by N m students, implying that exactly q students will be assigned to that school. 22 Abdulkadiroğlu et al. (2009) document that a significant fraction of students rank very few schools. 18

19 that aims to understand outcomes in typical markets. While it is clear that quantitative findings reached using these methods should be treated with caution, our results provide new qualitative insights that explain the different performance of STB and MTB. References Atila Abdulkadiroğlu and Tayfun Sönmez. Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica, 66(3): , Atila Abdulkadiroğlu and Tayfun Sönmez. School choice: A mechanism design approach. American Economic Review, 93(3): , ISSN Atila Abdulkadiroğlu, Parag A Pathak, Alvin E. Roth, and Tayfun Sönmez. The boston public school match. American Economic Review, 95(2): , Atila Abdulkadiroğlu, Parag A. Pathak, and Alvin E. Roth. Strategy-proofness versus efficiency in matching with indifferences: Redesigning the nyc high school match. American Economic Review, 99(5): , Atila Abdulkadiroğlu, Yeon-Koo Che, and Yosuke Yasuda. Expanding choice in school choice. American Economic Journal: Microeconomics, 7(1):1 42, Nick Arnosti. Short lists in centralized clearinghouses. Available at SSRN, Itai Ashlagi and Afshin Nikzad. What matters in tie-breaking rules? how competition guides design. Unpublished working paper, Itai Ashlagi and Peng Shi. Improving community cohesion in school choice via correlatedlottery implementation. Operations Research, 62(6): , Itai Ashlagi and Peng Shi. Optimal allocation without money: An engineering approach. Management Science, 62(4): , Itai Ashlagi, Yashodhan Kanoria, and Jacob D Leshno. Unbalanced random matching markets: the stark effect of competition. Journal of Political Economy, Forthcoming. Eduardo M Azevedo and Jacob D Leshno. A supply and demand framework for two-sided matching markets. Journal of Political Economy, 124(5): ,

20 Caterina Calsamiglia, Guillaume Haeringer, and Flip Klijn. Constrained school choice: An experimental study. American Economic Review, 100(4): , Gabriel Carroll. A general equivalence theorem for allocation of indivisible objects. Journal of Mathematical Economics, 51: , Yeon-Koo Che and Fuhito Kojima. Asymptotic equivalence of probabilistic serial and random priority mechanisms. Econometrica, 78(5): , Yeon-Koo Che and Olivier Tercieux. Mimeo, Efficiency and stability in large matching markets. Yan Chen and Onur Kesten. From boston to chinese parallel to deferred acceptance: Theory and experiments on a family of school choice mechanisms. Discussion Paper, Social Science Research Center Berlin (WZB), Research Area Markets and Politics, Research Unit Market Behavior, Peter Coles and Ran Shorrer. Optimal truncation in matching markets. Games and Economic Behavior, 87: , Monique De Haan, Pieter A Gautier, Hessel Oosterbeek, and Bas Van der Klaauw. The performance of school assignment mechanisms in practice. CEPR Discussion Paper No. DP10656, Aytek Erdil and Haluk Ergin. What s the matter with tie-breaking? improving efficiency in school choice. American Economic Review, 98(3): , 2008a. Aytek Erdil and Haluk Ergin. What s the matter with tie-breaking? improving efficiency in school choice. American Economic Review, 98(3): , 2008b. David Gale and Lloyd S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69(1):9 15, ISSN Yannai A Gonczarowski. Manipulation of stable matchings using minimal blacklists. In Proceedings of the fifteenth ACM conference on Economics and computation, pages ACM, Guillaume Haeringer and Flip Klijn. Constrained school choice. Journal of Economic Theory, 144(5): ,

21 Onur Kesten. On two kinds of manipulation for school choice problems. Economic Theory, pages 1 17, ISSN /s Donald Ervin Knuth. Mariages stables et leurs relations avec d&autres problèmes combinatoires. Presses de l Université de Montréal, Fuhito Kojima and Parag A. Pathak. Incentives and stability in large two-sided matching markets. American Economic Review, 99(3):608 27, doi: /aer Qingmin Liu and Marek Pycia. Ordinal efficiency, fairness, and incentives in large markets. Working paper, Parag A Pathak and Jay Sethuraman. Lotteries in student assignment: An equivalence result. Theoretical Economics, 6(1):1 17, Boris Pittel. The average number of stable matchings. SIAM Journal on Discrete Mathematics, 2(4): , Boris Pittel. On likely solutions of a stable marriage problem. The Annals of Applied Probability, 2(2): , Alvin E. Roth and Uriel G. Rothblum. Truncation strategies in matching markets-in search of advice for participants. Econometrica, 67(1):21 43, ISSN A Analysis We fix some notation and then proceed to the proof. We use a version of the deferred acceptance (DA) algorithm that accepts an infinite stream of integers, representing a source of randomness from which students preferences are drawn. We use square brackets to denote a range, i.e., [n] = {1,..., n}. Let S be a sequence of integers in [m], such that each integer in [m] appears an infinite number of times in S. Let S[j] denote the prefix of S up to the j-th place, and let S h denote the h-th integer in S. The student-proposing DA algorithm with k rounds based on S (denoted by DA (k)) works as described in Algorithm 1 by letting all proposers in a specific round draw schools one by one, while skipping any school they have already proposed to. Note that when a prefix of size j is read, it is possible that fewer than j proposals were made, because some integers were read by students who already proposed to 21

22 those schools. The regular DA algorithm (without limit on the number of rounds) is denoted by DA (= DA ( )). Algorithm 1: DA (k) based on S Input: k, S, schools preferences Output: Matching µ P [n] h 0 c C : µ[c] for round 1 to k do for s in P do if s already proposed to all schools then continue end while s already proposed to S h do h h + 1 end µ[s h ] µ[s h ] {s} h h + 1 end P for c in C do T top q students in µ[c] according to c s preference P P (µ[c] \ T ) µ[c] T end sort P end We almost always omit the dependency of DA (k) and DA on the schools preferences, assuming they are drawn at random as described above. Whenever we refer to DA (k) or DA without specifying a stream S, it means that we refer to the operation of the DA algorithm on a random stream in which every integer is drawn uniformly at random from [m]. At the same time we ignore the (measure 0) event of having a stream in which some integer appears only finitely many times. Some of our results are related to the expectation of the rank distribution and others are concentration results. For the latter we often use the expression with high probability (also denoted by w.h.p.) to mean with probability at least 1 n λ, and with very high probability (w.v.h.p.) to mean with probability at least 1 e nλ, for some λ > 0. 22