# Allocating students to schools: theory, methods, and empirical insights

Yeon-Koo Che, Julien Grenet, Yinghua He

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PARIS SCHOOL OF ECONOMICS
ECOLE D'ECONOMIE DE PARIS

WORKING PAPER N° 2025-48

Allocating Students to Schools:
Theory, Methods, and Empirical Insights

Yeon-Koo Che
Julien Grenet
YingHua He

JEL Codes: C18, C78, D47, I21, I28

Keywords: School Choice, Matching Theory, Deferred Acceptance, Immediate Acceptance, Top Trading Cycles, Preference Estimation
Allocating Students to Schools:
Theory, Methods, and Empirical Insights*

Yeon-Koo Che<fnref n="1" />

Julien Grenet<fnref n="2" />

YingHua He<fnref n="3" />

Chapter 4 of the *Handbook of the Economics of Matching*, Volume 2 (Elsevier)
edited by Yeon-Koo Che, Pierre-André Chiappori, and Bernard Salanié

DOI: 10.1016/bs.hesmat.2025.10.004

This version: August 2025

**Abstract**

This chapter surveys the application of matching theory to school choice, motivated by the shift from neighborhood assignment systems to choice-based models. Since educational choice is not mediated by price, the design of allocation mechanisms is critical. The chapter first reviews theoretical contributions, exploring the fundamental trade-offs between efficiency, stability, and strategy-proofness, and covers design challenges such as tie-breaking, cardinal welfare, and affirmative action. It then transitions to the empirical landscape, focusing on the central challenge of inferring student preferences from application data, especially under strategic mechanisms. We review various estimation approaches and discuss key insights on parental preferences, market design trade-offs, and the effectiveness of school choice policies.

**JEL Classification Numbers:** C18, C78, D47, I21, I28

**Keywords:** School Choice, Matching Theory, Deferred Acceptance, Immediate Acceptance, Top Trading Cycles, Preference Estimation

<footnote>*We are grateful to Christopher Campos, Aram Grigoryan, Dongwoo Hahm, Adam Kapor, Tomás Larroucau, Michael Lee, Minseon Park, and Olivier Tercieux for their comments. Our dear friend and coauthor, YingHua He, passed away in July 2024. We dedicate this chapter to his memory.</footnote>

<footnote n="1">¹Department of Economics, Columbia University, USA. Email: yeonkooche@gmail.com.</footnote>

<footnote n="2">²CNRS and Paris School of Economics, France. Email: julien.grenet@psemail.eu.</footnote>

<footnote n="3">³Department of Economics, Rice University, USA. Email: yinghua.he@rice.edu.</footnote>
# Contents

Introduction 3

1 Theories of School Choice 5

1.1 A Simple Economics of School Choice 5

1.2 Model Framework 8

1.3 School Assignment Methods before the Reforms 9

1.3.1 New York City School Assignment: Decentralized Matching 9

1.3.2 Boston School Assignment: Immediate Acceptance 11

1.4 Desirable Mechanisms: Efficiency vs. Stability 13

1.5 Coarse Priorities and Indifferences 20

1.6 Cardinal Welfare in School Choice 24

1.7 Affirmative Actions: Quotas and Reserves 28

1.7.1 Maximum Quotas 29

1.7.2 Minority Reserves 30

2 Methods for Preference Estimation 34

2.1 Preference Estimation under Immediate Acceptance and Other Strategic Mechanisms 36

2.2 Preference Estimation under the Deferred Acceptance Mechanism 41

2.2.1 The Truth-Telling Approach 41

2.2.2 The Optimal Portfolio Choice Approach 45

2.2.3 The Stability-Based Approach 48

2.2.4 Undominated Strategies 53

2.3 Identification and Estimation of Preferences 55

2.3.1 Specifying Student Preferences 55

2.3.2 Estimation under the Portfolio Choice Approach 60

2.3.3 Estimation under Truth-telling 64

2.3.4 Estimation under Stability 66

2.3.5 Using Undominated Strategies 71

2.3.6 Selecting Among Alternative Preference Inference Hypotheses 73

3 Empirical Insights from Centralized School Choice 75

3.1 Preferences and Application Behavior 76

3.1.1 Drivers of Preferences and Preference Heterogeneity 76

3.1.2 Information Frictions 78

3.1.3 Bounded Rationality 81

3.2 Market Design Trade-offs 82

3.2.1 Centralized vs. Decentralized Choice 82

3.2.2 Trade-offs between Assignment Mechanisms 84

3.2.3 Choice Support 88

3.3 School Choice Effectiveness 90

3.3.1 Market-Level Effects of School Choice and Supply-Side Responses 91

3.3.2 Distributional Effects 92

3.3.3 Dynamic Effects of School Choice and Interactions with Other Markets 95

References 117
# Introduction

The landscape of primary and secondary education has been significantly reshaped by the rise of school choice, a movement born out of concerns with traditional geographically based school assignment systems. For decades, the prevailing model automatically enrolled students in the public school nearest to their home. While convenient, this neighborhood assignment system frequently led to significant issues, including a lack of parental choice, particularly for families in areas with underperforming schools, and inequities stemming from socioeconomic disparities mirrored in school resources. Furthermore, the strong link between housing markets and school allocation under neighborhood assignment often resulted in allocational inefficiencies, as housing choices heavily dictated school placements. These concerns fueled advocacy for policies empowering parents with greater control over their children's educational paths.

Early school choice initiatives often centered on magnet schools and open enrollment programs, which allowed students to attend public schools outside their designated districts.<fnref n="1" /> Minnesota, for instance, enacted the first statewide open enrollment law in 1988, a model subsequently adopted by other states. Major urban centers like New York, Boston, and Seattle have also embraced such programs, with a growing focus on designing allocation mechanisms that are both efficient and fair. From a classical economics perspective, expanding choice in education is anticipated to enhance efficiency, much like broadening the scope of markets. However, a critical distinction arises: school choice is not typically mediated by price. Consequently, whether school choice fulfills its promise of improved allocation and other benefits hinges crucially on how these systems are organized and implemented.

It is precisely at this juncture that matching theory has become indispensable, offering powerful tools for both the theoretical design and empirical evaluation of school choice mechanisms. Theoretically, matching theory has been instrumental in analyzing the inherent trade-offs between competing desiderata such as fairness, efficiency, and strategy-proofness—the property that individuals have no incentive to misrepresent their preferences. It has also provided frameworks for optimizing tie-breaking procedures in situations of coarse priorities (where many students may have the same priority level), achieving desirable cardinal welfare outcomes that consider the intensity of preferences, and implementing affirmative action policies

<footnote n="1">¹Throughout, we identify school choice with open enrollment, thus excluding other initiatives such as charter schools and voucher programs, which some authors include as part of the school choice movement.</footnote>
effectively.

Empirically, the application of matching theory has enabled researchers to infer student preferences from school choice data and to evaluate the welfare implications of various assignment systems. The increasing availability of administrative data from school choice programs has spurred a rich body of empirical work focused on estimating student preferences and simulating the effects of counterfactual policies. A central challenge in this empirical agenda is inferring true preferences from observed ranking submissions, especially under mechanisms where strategic reporting might be prevalent.

While this chapter is primarily focused on school choice, we also draw selectively on insights from the closely related literature on centralized college admissions, particularly in settings where applicants are ranked according to exogenous priority criteria that closely parallel the school choice problem. The goal is not to provide a comprehensive review of all education-related matching markets—such as child-care assignment, foster-care placements, teacher assignment, or other emerging applications—but rather to highlight methodological contributions relevant for the analysis of school choice and to incorporate empirical evidence that informs its broader understanding. Within this scope, the chapter will navigate the multifaceted contributions of matching theory to the study of school choice. It is organized into three main parts.

Section 1 explores the theoretical underpinnings, beginning with the simple economics of school choice to illustrate the shortcomings of geographic assignment and the potential benefits of choice-based systems. It then establishes the formal model framework for school choice, discusses assignment methods used before major reforms, and explores desirable mechanisms like Deferred Acceptance and Top Trading Cycles, paying close attention to the trade-offs between efficiency and stability, the implications of coarse priorities, the pursuit of cardinal welfare, and the design of affirmative action policies.

Section 2 transitions to the empirical landscape, examining methods for estimating student preferences under both strategic and non-strategic (strategy-proof) mechanisms. This section will cover various approaches, including those that model preference reporting as a choice over lotteries, those grounded in the stability of matching outcomes, and those dealing with incomplete models of behavior, alongside a discussion of how to select among alternative preference hypotheses.

Building on these foundations, Section 3 synthesizes empirical lessons from the implementation of centralized school choice systems around the world, organizing the discussion around three central themes. It first examines how families form
preferences and make application decisions, with particular attention to information frictions and behavioral deviations from full rationality. It then turns to the design trade-offs in school choice systems, including the degree of centralization, the choice of assignment mechanisms, and the use of decision-support tools. Finally, it assesses the broader impact of school choice reforms on market-level outcomes, equity, and related behaviors such as residential mobility. Together, these empirical insights illustrate how the tools of matching theory have deepened our understanding of real-world school choice and informed the development and evaluation of assignment policies across diverse institutional contexts.

# 1 Theories of School Choice

## 1.1 A Simple Economics of School Choice

To illustrate the potential shortcomings of geographic assignment and the potential benefits of school choice, consider a simplified model. We have a unit mass of families, each with one child seeking to attend a public school. There are two districts, A and B, each capable of accommodating half the population, and each district has one public school: *a* in A and *b* in B.

We represent the preferences of each family with a point $(x, y)$ within $[-1/2, 1/2]^2$, as shown in Figure 1. The horizontal axis ($x$) captures the family's preference for residing in district A versus B, with higher values of $x$ indicating a stronger preference for A. Similarly, the vertical axis ($y$) represents the family's preference for school *a* over *b*, with higher $y$ values signifying a stronger preference for *a*. Assume for now that family preferences $(x, y)$ are uniformly distributed on the square $[-1/2, 1/2]^2$.

In an ideal allocation, a family would reside in district A if and only if $x > 0$ (they prefer A), and their child would attend school *a* if and only if $y > 0$ (they prefer *a*). The diagram visually depicts these ideal choices, with the light red area representing where school *a* is preferred and the light blue area showing where district A is preferred. The overlap, shaded in light purple, highlights the families who ideally desire both to live in district A and to have their child attend school *a*.

Under geographic assignment, families choose their child's school indirectly by choosing their residence. Essentially, a family chooses a bundle of a residential district and a school, facing housing rental prices $r_A$ and $r_B$, respectively, in districts A and B. Assuming a model where the utility difference between the bundles $(A, a)$ and $(B, b)$ is $x + y$, the housing market will reach equilibrium when the price difference between the two districts ($\delta = r_A - r_B$) is 0. The resulting assignment is
<figure><img src="image_3.png" /><figcaption>Figure 1 – Ideal Residential and School Choices</figcaption></figure>

depicted in Figure 2.

Geographic assignment, by tethering school choice to residence, compels the housing market to reconcile two distinct preferences: where families wish to live and where they want their children to attend school. This forced coupling generates distortions, visually represented in the figure by the four colored regions. Families in the blue region, for instance, would ideally reside in district *B* but choose *A* solely to access school *a*. This might stem from a stronger preference for *B*'s amenities or a parent's workplace location. Under geographic assignment, they sacrifice their residential preference to secure their educational one. Similarly, families in the red region, who ideally desire residence in *B* and school *a*, are forced to compromise their educational choice. The green and orange regions illustrate analogous inefficiencies, where families sacrifice either their residential or school preferences due to the constraints of geographic assignment. These inefficiencies underscore how this system can lead to suboptimal outcomes, forcing families to make trade-offs between their ideal living situations and their children's education.

School choice can mitigate these inefficiencies by decoupling families' residential choices from their school choices. In our example, if families are allowed to send
<figure><img src="image_4.png" /><figcaption>Figure 2 – Choices under Geographic School Assignment</figcaption></figure>

their children to either school, regardless of their district, the two choices become independent. Families can now choose their residence based solely on amenities and housing preferences, and select the ideal school for their children independently. Under this system, families will choose to reside in district A if and only if $x \ge \delta$, where the price difference $\delta$ adjusts to clear the housing market. They will also (try to) send their children to school a if and only if $y \ge 0$. Again, assuming a uniform distribution of $(x, y)$, the equilibrium price will be $\delta = 0$, and children will be assigned to their preferred schools. In short, school choice, in this stylized example, can achieve the first-best allocation.

While the previous example highlights the potential benefits of school choice in mitigating inefficiencies caused by geographic assignment, real-world implementations face additional challenges. Consider a scenario where the preference for schools is not uniformly distributed; instead, school a is more popular, leading to oversubscription. If tuitions could be adjusted to clear the market, an efficient allocation might emerge. However, this contradicts the principle of free public education. Therefore, alternative methods are necessary to address oversubscription.

One common approach is a lottery system, where seats at school a are assigned
randomly among those who prefer it ($y > 0$). However, this allocation ignores the intensity of preference, potentially leading to students with weak preferences for $a$ being admitted while those with strong preferences are excluded. Alternatively, granting geographic priority to district A residents for school $a$ essentially recreates a smaller-scale geographic assignment system, inheriting its associated inefficiencies, albeit to a lesser extent.

In reality, school choice mechanisms often fall short of achieving the theoretical first-best allocation. The lack of monetary transfers in public education necessitates alternative assignment methods based on elicited preferences. The design of these mechanisms becomes crucial in balancing efficiency, fairness, and the practical constraints of public schooling. In the following sections, we will explore how various school choice mechanisms attempt to achieve desirable allocations by strategically linking assignment to reported preferences.

## 1.2 Model Framework

A school choice model is very close to the many-to-one matching model introduced in Chapter 1. While there is an important difference (highlighted later), the model primitives follow from many-to-one matching, also known as the College Admissions problem.

A **school choice problem** consists of a set of **students**, $\mathcal{I} = \{i_1, \dots, i_I\}$, and a set of **schools**, $\mathcal{J} = \{s_1, \dots, s_J\}$. Each school $s \in \mathcal{J}$ has **capacity** $q_s$, the maximum number of students the school can enroll. Each student $i \in \mathcal{I}$ has a strict **preference ordering** $P_i$ over schools and the option to be unassigned. Each school $j \in \mathcal{J}$ has a strict **priority ordering** $\pi_j$ over the students. In the context of the public school system, one may not view priorities as schools' preferences; rather, they can be viewed as reflecting the policy objectives of the school system and other societal considerations. Examples include priority rankings based on sibling attendance, the proximity of residence to schools (e.g., "walk zone"), and/or test scores and grade point averages of the preceding schools. While priorities are often coarse, with many students falling within the same priority class (e.g., walk zone), it is convenient to begin with strict priorities as our baseline model; we will address the implications of coarse priorities in Section 1.5.

An **assignment** is a mapping $\mu : \mathcal{I} \cup \mathcal{J} \to \mathcal{J} \cup 2^{\mathcal{I}}$ such that for each $i \in \mathcal{I}$, $\mu(i) \in \mathcal{J} \cup \{i\}$ and, for each $s \in \mathcal{J}$, $\mu(s) \subseteq \mathcal{I}$; i.e., each student is assigned a school or himself/herself (not assigned to any school), and each school is assigned to a subset of students. Further, $\mu(i) = s$ if and only if $i \in \mu(s)$, and $|\mu(s)| \le q_s$ for each
so that the capacity constraint is satisfied.

There are several normative objectives we consider. First, we say $\mu$ is **efficient** if no other assignment $\mu'$ Pareto dominates $\mu$, i.e., making all students weakly better off and some strictly better off than $\mu$. Note that schools' priority orderings do not factor into this notion, in keeping with the idea that the schools are the resources to be consumed by students, rather than being welfare-relevant entities in their own right, as would be the case in two-sided matching.

At the same time, schools' priorities reflect policy and social considerations. There is a familiar way to incorporate them if one views the school assignment as a two-sided matching. An assignment is **stable** if it is (i) **individually rational**: for each student $i \in I$, $\mu(i)$ is weakly preferred to the option of being unassigned; it is (ii) **non-wasteful**: for each student $i \in I$, $sP_i\mu(i) \Rightarrow |\mu(s)| = q_s$; and there is (iii) **no justified envy**: if a student $i$ prefers a school $s$ to his/her assignment, then all students matched to school $s$ must have a higher priority than student $i$. While stability is justified in the two-sided matching context as an equilibrium notion, here it is justified as some basic efficiency desiderata (namely, (i) and (ii)) and a fairness desideratum (namely, (iii)).

In practice, students' preferences are unobserved by the school system, so the above normative properties cannot be satisfied unless the assignment process also provides students with incentives to reveal their preferences. Define an **assignment mechanism** to be a mapping from all possible profiles of student preferences to an assignment. We say that an assignment mechanism is **strategy-proof** if each student has an incentive to report his/her preference truthfully, for any preference profile reported by the other students. Strategy-proofness makes an assignment mechanism straightforward for participants, and enables other normative properties, such as efficiency and stability, to be achievable reliably and robustly.

## 1.3 School Assignment Methods before the Reforms

Before considering desirable school assignment methods, it is instructive to examine a couple of assignment methods used prior to the reforms.

### 1.3.1 New York City School Assignment: Decentralized Matching

Before the 2004 redesign, the New York City public high school assignment system was highly decentralized and complex. Abdulkadiroğlu et al. (2005) and Abdulkadiroğlu et al. (2017a) describe the system as follows: Rising NYC high school
students applied to five out of more than 600 school programs; they could receive multiple offers and be placed on waitlists. Students were allowed to accept only one school and one waitlist offer, and the cycle of offers and acceptances was repeated twice. Most students not assigned in these rounds went through an administrative process that manually placed them at schools close to their residences. The lack of centralized coordination meant that the number of offers and acceptances was limited, a problem known as *congestion* by Roth and Xing (1997).

How the congestion problem adversely impacts the outcome can be illustrated by a simple example from Che and Koh (2016). Suppose there are only two students, 1 and 2, applying to schools *a* and *b*. Each school has one seat to fill (and faces a prohibitively high cost of enrolling two students). Student *i* has priority measured by score $t_i$, $i = 1, 2$, where $0 < t_2 < t_1 < 2t_2$, for both schools. The priority score $t_i$ is a school's payoff from admitting student *i*. Each student has an equal probability of preferring either school, which is private information (unknown to the other student and to the schools). The applications are free of cost, and the timing of the decentralized process is as follows. First, students apply to schools simultaneously. Then, the two schools simultaneously offer admissions to sets of students. Finally, students decide which offer (if any) to accept. A school's seat remains empty if no applicant accepts its offer. While extremely simple, this model captures the lack of centralized coordination as well as the congestion (i.e., each school has only one round of offers).<fnref n="2" />

Given the large cost of over-enrollment, each school admits only one student. Their payoffs are described as follows:

<table><thead><tr><th>a's strategy \ b's strategy</th><th>b admits 1</th><th>b admits 2</th></tr></thead><tbody><tr><td>a admits 1</td><td>$\frac{1}{2}t_1, \frac{1}{2}t_1$</td><td>$t_1, t_2$</td></tr><tr><td>a admits 2</td><td>$t_2, t_1$</td><td>$\frac{1}{2}t_2, \frac{1}{2}t_2$</td></tr></tbody></table>

This game has a battle of the sexes structure (with asymmetric payoffs), so there are two different types of equilibria. First, there are two asymmetric pure-strategy equilibria in which one school admits student 1 and the other admits student 2. There is also a mixed-strategy equilibrium in which each school admits 1 with probability $\gamma := \frac{2t_1-t_2}{t_1+t_2} > 1/2$ and admits 2 with probability $1-\gamma$, where $\gamma$ is chosen such that the other school is indifferent over whom to admit. Both types of

<footnote n="2">²While the NYC mechanism allowed for more flexibility in terms of waitlist management, the coordination challenges are higher due to the large number of students and school programs. One can thus view the simple model as a fair stylization of the system.</footnote>
equilibria show the pattern of strategic targeting. In the pure-strategy equilibria, schools manage to avoid competition by targeting different students. The mixed-strategy equilibrium may involve schools using some "spurious" measure of student merit as a randomization device; schools following these strategies will appear to value such measures despite there being no intrinsic value to them.

This example, while extremely simple, reveals problems with decentralized matching in terms of welfare and fairness. First, the student with the highest score (student 1) may attend a less preferred school (in both types of equilibria), even though both schools prefer that student; that is, justified envy arises. Second, it could be the case that student 1 prefers *a* and student 2 prefers *b*, but the former is assigned *b* and the latter is assigned *a*, showing that the equilibrium outcome is inefficient among students. Lastly, the mixed-strategy equilibrium is wasteful because both schools may admit the same student, thus wasting the seat of the unmatched school.

### 1.3.2 Boston School Assignment: Immediate Acceptance

Unlike NYC, school systems in many cities such as Boston, Cambridge, Minnesota, and Seattle were using a centralized assignment process; but the assignment algorithm used in these districts, known as the **Immediate Acceptance (IA) mechanism** or simply **Boston mechanism**, was criticized on the normative ground mentioned above.

IA takes as input the rank-order list of schools submitted by students and the strict priority ordering of students for each school, based on sibling attendance and walk zone criteria, with ties broken by lotteries. Then, the assignment proceeds in multiple rounds. In the first round, students are assigned to their top-ranked school based on their priority order, starting with the highest priority down to the second highest, and so on. If a school's capacity is filled, it rejects the remaining students. In round two, rejected students are considered for the next highest-ranked school if seats are available at these schools; otherwise, they are rejected. This continues until all students have been assigned to a school or have exhausted their acceptable options.

To illustrate IA, consider the following example: Let $\mathcal{I} = \{1,2,3,4\}$ be the set of students and $\mathcal{J} = \{a,b,c\}$ be the set of schools, with school capacities $q_a = 2$ and $q_b = q_c = 1$. The student preferences and school priorities are described in Figure 3.

The outcome of IA in this example is $\mu = (1a, 2a, 3b, 4c)$ if the students report their preferences truthfully.
<table><caption>**Figure 3 – Immediate Acceptance**</caption><thead><tr><th>P<sub>1</sub></th><th>P<sub>2</sub></th><th>P<sub>3</sub></th><th>P<sub>4</sub></th><th>&gt;<sub>a</sub></th><th>&gt;<sub>b</sub></th><th>&gt;<sub>c</sub></th></tr></thead><tbody><tr><td>a</td><td>a</td><td>b</td><td>b</td><td>4</td><td>3</td><td>1</td></tr><tr><td>b</td><td>b</td><td>a</td><td>a</td><td>1</td><td>4</td><td>2</td></tr><tr><td></td><td></td><td></td><td>c</td><td>2</td><td>1</td><td>3</td></tr><tr><td></td><td></td><td></td><td></td><td>3</td><td>2</td><td>4</td></tr></tbody></table>

The example illustrates several problems with IA. First, the matching is not stable, assuming that students report their preferences truthfully. For instance, student 1 is assigned school *a*, but school *a* prefers student 4, who also prefers school *a* to her assigned school *c*. The failure of stability could lead to outcomes that violate school priorities. This means that a student may be assigned to a school even if another student with a higher priority for that school remains unassigned. Such priority violations raise concerns about fairness and can even result in legal challenges from dissatisfied students and parents.

The instability in turn implies that the mechanism is not strategy-proof: student 4 could obtain her second most-preferred school *a* by misrepresenting her preferences to (*a*,*b*,*c*). This failure of strategy-proofness is intuitive. If a student's most preferred school is highly popular and thus difficult to get into, but her second most-preferred is not much worse and is less competitive, then strategically ranking the latter for her top spot is quite rational.<fnref n="3" /> A lab experiment conducted by Chen and Sönmez (2006) shows that about 80% of the subjects misreport their preferences under IA. The failure of strategy-proofness raises multiple issues. First, how participants strategize is not straightforward, requiring sophistication in strategic thinking. Second, participants may differ in the sophistication of their strategic thinking and may put the more sophisticated at an advantage.<fnref n="4" /> Third, the strategic nature of

<footnote n="3">³At a conference organized by the Federal Reserve Bank of Chicago in 1994 ("Midwest approaches to school reform"), Meyer and Glazerman report:</footnote>

<footnote>> It may be optimal for some families to be strategic in listing their school choices. For example, if a parent thinks that their favorite school is oversubscribed and they have a close second favorite, they may try to avoid "wasting" their first choice on a very popular school and instead list their number two school first.</footnote>

<footnote>In a meeting of the West Zone Parents Group of the city of Boston, it was said:</footnote>

<footnote>> One school choice strategy is to find a school you like that is undersubscribed and put it as a top choice, or, second a school that you like that is popular and put it as a first choice and find a school that is less popular for a "safe" second choice.</footnote>

<footnote n="4">⁴ Pathak and Sönmez (2008) apply the full-information Nash equilibrium analysis of Ergin and</footnote>
the mechanism makes it more challenging for one to decipher the data to estimate preferences accurately. We turn to these issues in the empirical part of this survey.

How can we predict the outcome of IA and its welfare implications? Predicting the outcome of a strategic mechanism is challenging since one must rely on some notion of equilibrium; it is often unclear whether participants play the chosen notion of equilibrium. A case in point is Ergin and Sönmez (2006), which studies Nash equilibria of the IA mechanism under the assumption that each student observes the preferences and priorities of all other students; they show that the set of Nash equilibria coincides with that of stable matching outcomes.<fnref n="5" /> Such an analysis sheds some light on the difference between DA and IA, yet the full-information Nash equilibrium concept is very restrictive and unlikely to hold in practice.

## 1.4 Desirable Mechanisms: Efficiency vs. Stability

The challenges encountered in both NYC and Boston school assignment systems highlight the need to have an assignment mechanism that is straightforward for the participants to play; namely, the assignment mechanism should be strategy-proof. Beyond the strategy-proofness, one could aim for two desiderata: stability (less ambitiously, no justified envy) and efficiency.

**Deferred Acceptance and Top Trading Cycles.** As it turns out, there are strategy-proof mechanisms that attain these objectives. Stability can be attained by the Student-Proposing Deferred Acceptance (DA) algorithm which runs in multiple rounds (see Gale and Shapley, 1962; Abdulkadiroğlu and Sönmez, 2003): in round 1, students apply to their most-preferred acceptable schools, and the schools accept tentatively the students up to their capacities based on their priority orderings and reject the rest; in the subsequent rounds, rejected students apply to the most-preferred schools that have not rejected them, and the schools again tentatively accept the students according to their priority orderings up to the capacity, and reject the rest, possibly including those whom they may have accepted before; this

<footnote>Sönmez (2006) to study how sincere applicants who reveal their preferences truthfully fare against sophisticated ones who play a best response, and formalize the sense in which naivety disadvantages the former.</footnote>

<footnote n="5">⁵The argument for this is rather simple. Take any stable matching; say $\mu = (1a, 2c, 3b, 4a)$. It is an equilibrium for all students to report only the assigned school under $\mu$ to be acceptable; a unilateral deviation can never enable a deviator to obtain a better school (since there is no blocking pair). Meanwhile, any unstable matching can never be an equilibrium, since one can unilaterally deviate to obtain a block (either by herself or with a school).</footnote>
process continues until there are no more rejected students.<fnref n="6" /> The DA mechanism is strategy-proof: namely, students have no incentive to misrepresent their preference orderings when they are used to implement DA.

Meanwhile, the Top Trading Cycles (TTC) mechanism guarantees efficient assignment among students. TTC operates by recursively forming and clearing cycles of students and schools (Abdulkadiroğlu and Sönmez, 2003). Initially, each school is assigned a counter to track the number of remaining seats. In step 1, each student points to her most preferred school, and each school points to the student with the highest priority. This creates at least one cycle. Students in a cycle are assigned their pointed-to schools, removed from the process, and the school's counter is decremented. In the subsequent steps, the process repeats with the remaining students and schools. Students point to their most preferred school among those with available seats, and schools point to their highest-priority remaining student. Cycles are formed, students are assigned, and counters are updated. The algorithm ends when all students are assigned or all submitted choices have been considered. TTC is strategy-proof, meaning students have no incentive to misrepresent their preference orderings when they are used to implement TTC.

Unfortunately, stability and efficiency may not be both satisfied. To see this, suppose there are three students $\mathcal{I} = \{1, 2, 3\}$ and three schools $\mathcal{J} = \{a, b, c\}$, each with unit quota. The students' preferences and priorities are described in Figure 4.

<table><caption>**Figure 4 – DA versus TTC**</caption><thead><tr><th>P<sub>1</sub></th><th>P<sub>2</sub></th><th>P<sub>3</sub></th><th>&gt;_a</th><th>&gt;_b</th><th>&gt;_c</th></tr></thead><tbody><tr><td>b</td><td>a</td><td>a</td><td>1</td><td>2</td><td>1</td></tr><tr><td>a</td><td>b</td><td>b</td><td>3</td><td>1</td><td>2</td></tr><tr><td>c</td><td>c</td><td>c</td><td>2</td><td>3</td><td>3</td></tr></tbody></table>

DA produces an assignment $\mu_{DA} = (1a, 2b, 3c)$. This assignment is not Pareto efficient since 1 and 2 would be strictly better off, with 3 remaining the same, if 1 and 2 swapped their assignments. Indeed, the resulting “efficient” assignment $\mu_{TTC} = (1b, 2a, 3c)$ is attained by TTC, where a cycle $1 \to b \to 2 \to a \to 1$ is formed and cleared in the first round. Note that this assignment is unstable: 3 has justified envy toward 2.

This example illustrates the trade-off between stability and efficiency. While the objectives may conflict with each other, if a school district seeks either objective,

<footnote n="6">⁶The reader is referred to the formal description of DA in Chapter 1.</footnote>
there is a case for choosing between DA and TTC. The next result shows that student-proposing DA implements a stable matching that is most efficient among all stable matchings in the following sense:

**Theorem 1** (Gale and Shapley, 1962). DA assignment is student-optimal in the sense that it Pareto dominates all other stable assignments for the students.<fnref n="7" />

This means that DA achieves stability with minimal efficiency loss. The intuition behind this result is explored in Chapter 1, but roughly, it is related to the fact that the set of stable matchings forms a complete lattice in terms of relative welfare of students versus school priority orderings, and the student-optimal DA selects the extremal stable matching in favor of students.

How well does TTC perform in minimizing the stability loss? The next result establishes an analogous result to Theorem 1 for TTC in this regard, but in a much weaker and nuanced sense:<fnref n="8" />

**Theorem 2** (Abdulkadiroğlu et al., 2020a). (i) In the one-to-one matching setting (i.e., schools have unit quota), TTC is “justified envy minimal:” no other efficient and strategy-proof mechanism entails “fewer” incidences of justified envy, in the set inclusion sense.<fnref n="9" />

(ii) In the general setting with multiple school quotas, when students' preferences and priorities are generated in the i.i.d. fashion, TTC is less likely to create justified envy than random serial dictatorship (RSD).<fnref n="10" />

*Proof*. We only provide the proof for (i). Let $\varphi$ be a Pareto efficient and strategy-proof mechanism. We show that if $\varphi$ admits less justified envy (in the sense of admitting a subset of blocking pairs) than TTC at any priority orderings $>$, then $\varphi(P) = \text{TTC}(P)$ for all $P$, where $\text{TTC}(P)$ is the TTC assignment given preferences $P$ and priorities $>$. (We suppress the dependence of the matching on $>$..) Suppose, to the contrary, there exists $P$ such that $\varphi(P) \neq \text{TTC}(P)$. Let $I_k(P)$ be the set of agents who are matched in step $k$ of $\text{TTC}(P)$, and let $\ell$ be the smallest $k$ such that $I_k(P)$ contains an agent who is assigned differently between TTC and $\varphi$. By definition, for some $i \in I_\ell(P)$, $\varphi(P)(i) \neq \text{TTC}(P)(i)$. Let $c = \{s_k, i_k\}_{k=1,...,K}$ be the cycle such

<footnote n="7">⁷See the proof in Chapter 1.</footnote>

<footnote n="8">⁸Doğan and Ehlers (2022) show the robustness of (i) to several measures of relative stability.</footnote>

<footnote n="9">⁹One may be interested in dropping strategy-proofness, namely a mechanism that is efficient and justified envy undominated (in the set containment sense). Abdulkadiroğlu et al. (2020a) show that such a mechanism is NP-hard to find.</footnote>

<footnote n="10">¹⁰See Chapter 1 for the exact definition.</footnote>
that $s_k$ points to $i_k$, and $i_k$ points to $s_{k+1}$, with $s_{K+1} \equiv s_1$. Suppose $i$ is matched with TTC($P$)($i$) and $i = i_K$.

Consider the preference relation: $P'_{i_K} : s_1, s_K, \dots$. Since we have only altered the preferences of $i_K$ in $c$ and TTC($P$)($i_K$) = $s_1$ has become her first choice, the TTC matching remains the same: $\text{TTC}(P'_{i_K}, P_{-i_K}) = \text{TTC}(P)$. Since $\varphi(P)(i_K) \neq \text{TTC}(P)(i_K) = s_1$, and $\varphi(P)(i_K)$ is still available at step $\ell$, we obtain

$$s_1 P_{i_K} \varphi(P)(i_K). \qquad (1)$$

But since $\text{TTC}(P'_{i_K}, P_{-i_K})(i_K) = s_1 P'_{i_K} s_K$, $(i_K, s_K)$ does not block $\text{TTC}(P'_{i_K}, P_{-i_K})$.

Since $\varphi$ has less justified envy than TTC at $\succ$, $(i_K, s_K)$ should not block $\varphi(P'_{i_K}, P_{-i_K})$. Since $s_K$ points to $i_K$ in cycle $c$, $i_K \succ_{s_K} j$ for all $j \neq i_K$ who are still unassigned at step $\ell$. Given that $s_K$ will be assigned one of the agents still unassigned at step $\ell$, by construction, we must then have $\varphi(P'_{i_K}, P_{-i_K})(i_K) \in \{s_1, s_K\}$. Given (1), strategy-proofness of $\varphi$ implies $\varphi(P'_{i_K}, P_{-i_K})(i_K) = s_K$; otherwise $i_K$ would be able to manipulate $\varphi$ in economy ($P$) by submitting $P'_{i_K}$ to obtain $s_1$. Hence, $i_K$ obtains her second choice under $P'_{i_K}$. Now, we have $\varphi(P'_{i_K}, P_{-i_K})(i_{K-1}) \neq s_K = \text{TTC}(P'_{i_K}, P_{-i_K})(i_{K-1})$, and so we can apply the same argument for $i_{K-1}$.

By iterating the argument for $i_{K-1}, \dots, i_1$, and for every agent in the cycle $c$, we obtain that $\varphi(P'_c, P_{-c})(i_k) = s_k$, where $P'_c = \{P'_{i_k}\}_{i_k \in c}$ and $P'_{i_k} : s_{k+1}, s_k, \dots$. Applying the argument iteratively, we find $\varphi(P'_c, P_{-c})(i_k) = s_k$ for all $k$. But this contradicts the Pareto efficiency of $\varphi$ because every agent in the cycle will be better off if every $i_k$ is matched with $s_{k+1}$ (modulo $k$) without changing the matching of other agents, establishing the claim. $\square$

The sense in which TTC performs also “well” in stability is weak. First, the constrained-stability result holds only in the one-to-one matching setting. Even in this case, DA’s constrained efficiency holds more powerfully. Second, even for the general matching setting, TTC’s stability performance could be shown only relatively better than RSD. Still, these results are meaningful since RSD has been the standard method for implementing an efficient assignment.

The logic of why TTC may do better in terms of stability than other efficient mechanisms, such as RSD, is that TTC prioritizes the assignment of students based on their priority standing: those with high priorities at schools that are highly demanded are more likely to be part of the cycles at the early rounds, enabling them to claim their preferred schools. If their preferred schools happen to be those they have high priorities for, then no other students will have justifiable envy of those students. By contrast, RSD ignores students’ priorities in its assignment, so it
will entail justifiable envy in this case.

To illustrate, consider Figure 5, which has the same priorities as Figure 4 but different student preferences.

<table><caption>**Figure 5 – TTC Admits Short Cycles**</caption><thead><tr><th>P<sub>1</sub></th><th>P<sub>2</sub></th><th>P<sub>3</sub></th><th>&gt;<sub>a</sub></th><th>&gt;<sub>b</sub></th><th>&gt;<sub>c</sub></th></tr></thead><tbody><tr><td>a</td><td>b</td><td>a</td><td>1</td><td>2</td><td>1</td></tr><tr><td>b</td><td>a</td><td>b</td><td>3</td><td>1</td><td>2</td></tr><tr><td>c</td><td>c</td><td>c</td><td>2</td><td>3</td><td>3</td></tr></tbody></table>

In this case, TTC yields $(1a, 2b, 3c)$, since 1 and $a$ form a cycle of length 1, or a short cycle, and likewise 2 and $b$ form a short cycle. This means that the assignment is not only efficient but also stable, admitting no justified envy.

However, suppose the preferences are as in Figure 4. Then, recall that the assignment $\mu_{TTC} = (1b, 2a, 3c)$ resulted from clearing a cycle of length 2, or a long cycle, $1 \to b \to 2 \to a \to 1$, in the first round. In this case, 3 has justifiable envy for 2. In fact, whenever a student obtains her assignment via a long cycle, she, a priori, has no reason to have a higher priority at the assigned school than any other student, including those who may envy her, just as in the case of RSD. The logic of Abdulkadiroğlu et al. (2020a) is that TTC may reduce justified envy relative to the RSD insofar as students are assigned via short rather than long cycles.

Still, Theorem 2 provides some justification for the use of TTC in case the policymaker cares about efficiency. In practice, however, when centralized school choice assignment has been adopted worldwide, most of the school systems have adopted DA. The only exception is the New Orleans School System, which had used TTC for a year before switching to DA.

One reason for the infrequent use of TTC is that it is difficult for school systems to explain to parents.<fnref n="11" /> The other issue is that parents may not tolerate violations of their priorities that occur under TTC. Despite Theorem 2, there are two reasons why the latter loss is significant.

Che and Tercieux (2018) suggests that the stability benefit of TTC may be small as the market size grows large. Consider the one-to-one matching in which students' preferences and priorities are drawn identically and independently. Figure 6, from Che and Tercieux, shows that as the market size $n$ (i.e., the number of students, which equals the number of schools) increases, the proportion of incidences of

<footnote n="11">¹¹See Leshno and Lo (2021) for some recent attempts to make the mechanism explainable.</footnote>
<figure><img src="image_5.png" /><figcaption>**Figure 6** – The Fraction of Incidences of Justified Envy out of all Incidences of Envy</figcaption></figure>

Source: Che and Tercieux (2018, Figure 5).

justified envy among the incidences of envy under TTC converges to that under RSD.<fnref n="12" /> The reason for this is that as the market size *n* grows, the fraction of the students assigned via long cycles converges to 1.

Second, Che and Tercieux (2019) consider whether the trade-off between DA and TTC vanishes in a large market if one considers approximate versions of stability and efficiency. They consider an environment in which students have cardinal preferences over schools that consist of common and idiosyncratic terms of payoffs for schools, and schools have cardinal payoffs corresponding to their priorities. They then define an assignment to be **asymptotically efficient** if, for any $\epsilon > 0$, the proportion of students who may benefit more than $\epsilon$ from a Pareto-improving reassignment vanishes in probability as $n \to \infty$, and **asymptotically stable** if the proportion of “blocks” that benefit the deviating student and school by more than $\epsilon$, out of all possible blocks, vanishes in probability as $n \to \infty$. When common payoff terms are important, more precisely in the sense that some schools are uniformly

<footnote n="12">¹²Incidences of envy are the (random) set of students and the (random) set of schools for each student in the set such that each student prefers the set of schools over her assigned school. Note that the incidences of envy are the same between TTC and RSD, due to the equivalence result by Abdulkadiroğlu and Sönmez (1998) and Knuth (1996). The normalization serves as a scaling device that keeps the equivalence nontrivial.</footnote>
better than others, regardless of the idiosyncratic payoffs, they show that the TTC fails to be asymptotically stable, suggesting that the stability loss remains significant (see Theorem 8 of Chapter 2, and Theorem 2 of Che and Tercieux, 2019).

Combined, these two results suggest that as long as respecting priorities remains an important policy objective, the TTC remains problematic, possibly explaining the infrequent use of the TTC (or RSD).

**Compromise between Efficiency and Stability.** Che and Tercieux (2019) also show that when common payoff terms are important in the sense mentioned above, DA fails to be asymptotically efficient, meaning the efficiency loss of DA does not vanish as the market grows large. Combining with the asymptotic instability of TTC, this suggests that if the policymaker cares about both efficiency and stability/no justified envy, the two well-known mechanisms may entail significant drawbacks. It is also not clear why one should satisfy one desideratum exactly with the minimal loss of the other. In principle, one may prefer to achieve both desiderata approximately. Che and Tercieux (2019) show that such a compromise is indeed possible.

To explain their proposed solution, it is helpful to understand why DA fails to achieve asymptotic efficiency. Whenever students compete for limited seats at a school, they are rationed based on their priorities at the school, independently of their relative preferences about the school. For example, it is plausible that a student who ranks the school at the bottom of her preference orderings may win against the student for whom the school is most preferred. In other words, whenever students have conflicting preferences and compete for a popular school, the conflicts are resolved in a way that is likely to undermine their welfare.

Their proposal, called Deferred Acceptance with Circuit Breaker (DACB), modifies DA so that the tentative assignments are finalized periodically whenever some student reaches a pre-specified cutoff number of applications for the first time. The process then repeats with the application counter refreshed after each assignment. See Che and Tercieux (2019) and Chapter 2 for the precise definition of DACB. They show that with the cutoff number of applications suitably calibrated, DACB can achieve asymptotic efficiency and asymptotic stability. While the mechanism is not strategy-proof, it is asymptotically incentive compatible—namely, truth-telling is an $\epsilon$-Bayesian Nash equilibrium: for each $\epsilon > 0$, truth-telling gives a student within $\epsilon$ of the highest possible payoff given all others report truthfully if the number $n$ of students is sufficiently large. A more detailed presentation of this result can be found in Chapter 2.
1.5 Coarse Priorities and Indifferences

In real-world school choice scenarios, school priorities are often coarse, meaning
they do not strictly rank all students. For example, in the Boston Public Schools
(BPS) system, priorities are based on factors like sibling attendance, walk zone, and
whether the student currently attends the school. This coarseness leads to many
ties in priority rankings. Formally, we extend the priority rankings ≻ to ≲, which
allows for possible ties.

In practice, the assignment algorithm requires strict rankings, which are obtained
by breaking ties in various ways. There are two common methods for tie-breaking:

* Single Tie-Breaking (STB): A single lottery is used to create a priority ranking of students that applies to all schools.

* Multiple Tie-Breaking (MTB): A separate lottery is used for each school, resulting in independent rankings across schools.

It is well known that, with indifference, a student-optimal stable matching—a stable matching that Pareto dominates all other stable matchings—may not be well defined. Hence, a more reasonable objective is to achieve a **constrained-efficient** stable matching, defined as a stable matching that is Pareto-undominated by other stable matchings. Unfortunately, this more reasonable goal may not be achieved by DA with standard tie-breaking rules such as MTB and STB.

To illustrate the inefficiency of MTB, consider the example depicted in Figure 4,
except that schools have no priorities, meaning schools are indifferent over all
students. Now, suppose the MTB were used to break ties. Then, the priorities ≻ in
Figure 4 could be realized with positive probability (more precisely, equal to 1/27).
In that case, a matching $\mu_{MTB} = (1a, 2b, 3c)$ would be realized, which we know is
Pareto-dominated by $\mu^* = (1b, 2a, 3c)$. Moreover, the latter matching is stable. In
other words, the MTB yields a stable matching that is not constrained-efficient, or
*Pareto-undominated by other stable matchings*.

Suppose instead that STB were used to break ties. Then, effectively, the mechanism becomes the random serial dictatorship, which we know would be efficient. Further, given that there are no priorities, the efficient assignment would be stable. Hence, in this case, DA-STB implements a student-optimal stable matching. Indeed, if schools have no priorities, then DA-STB reduces to RSD, so it produces an efficient and student-optimal stable assignment.

More generally, however, even DA-STB cannot consistently select a constrained-
efficient stable matching. To illustrate this, suppose the students have the same
preferences and priorities as in Figure 4, except that school *a* is indifferent between 3 and 2. In this case, a STB may yield the priorities orderings precisely as in Figure 4 with probability 1/2, so that the assignment $\mu_{STB} = (1a, 2b, 3c)$ arises, which again is Pareto-dominated by $\mu^* = (1b, 2a, 3c)$. Importantly, $\mu^*$ is stable, so again $\mu_{STB}$ fails to be constrained-efficient.

These examples illustrate the possibility that DA with a standard tie-breaking rule leaves room for Pareto improvement without sacrificing stability. We shall discuss how this can be achieved shortly. Before proceeding, however, it is important to note that such an improvement cannot be achieved without a price. The next theorem shows that any such Pareto improvement can only be achieved at the expense of strategy-proofness.

**Theorem 3** (Abdulkadiroğlu et al., 2009). Assume that all students are acceptable to schools. For any tie-breaking rule, there is no strategy-proof mechanism that Pareto dominates DA with that tie-breaking rule.

*Proof*. Consider any tie-breaking rule $\tau$. We fix schools' weak priority orderings $\asymp$ and an arbitrary tie-breaking rule $\tau$ that completes $\asymp$ to a strict order $\succ^\tau$. One can show, and we take as given, the fact: If $\nu$ Pareto dominates $\mu = \text{DA}^\tau(P)$ for a given tie-breaking rule $\tau$, then the same set of students is matched in both $\nu$ and $\mu$.

Suppose that there exists a strategy-proof mechanism $\varphi$ and tie-breaking rule $\tau$ such that $\varphi$ Pareto dominates $\text{DA}^\tau$. There exists a profile $P$ such that $\varphi_i(P)R_i\text{DA}_i^\tau(P)$ for all $i$, and $\varphi_j(P)P_j\text{DA}_j^\tau(P)$ for some $j$, where $aR_ib$ means either $aP_ib$ or $a = b$. We will say that the matching $\varphi(P)$ Pareto dominates the matching $\text{DA}^\tau(P)$, where $\text{DA}^\tau(P)$ denotes the student-optimal stable matching for $(P; \succ^\tau)$. Let $s_i = \text{DA}_i^\tau(P)$ and $\hat{s}_i = \varphi_i(P)$ denote $i$'s assignment under $\text{DA}^\tau(P)$ and $\varphi(P)$ respectively, where $\hat{s}_iP_i s_i$.

Consider profile $P' = (P'_i, P_{-i})$ where $P'_i$ ranks $\hat{s}_i$ as the only acceptable school. Since $\text{DA}^\tau$ is strategy-proof, $s_i = \text{DA}_i^\tau(P)R_i\text{DA}_i^\tau(P')$, and since $\text{DA}_i^\tau(P')$ is either $\hat{s}_i$ or $i$, we conclude that $\text{DA}_i^\tau(P') = i$. Then, since the set of assigned students is identical between the two mechanisms, we must have $\varphi_i(P') = i$. Suppose $P'$ are the actual preferences. Then, $i$ could state $P_i$ and be matched to $\varphi_i(P) = \hat{s}_i$ which under $P'_i$ she prefers to $\varphi_i(P') = i$. This shows that $\varphi$ is not strategy-proof, a contradiction. $\square$

The theorem implies that there is no strategy-proof mechanism that delivers a Pareto improvement over a DA assignment with any exogenous tie-breaking rule. Put differently, if a school choice mechanism aims to improve upon the DA
algorithm with a given tie-breaking rule by making some students better off without harming others, it cannot be strategy-proof. This highlights the inherent trade-off between strategy-proofness and potential welfare performance in school choice.

Indeed, the theorem applies broadly to any mechanism that may or may not yield a stable outcome. For example, recall that TTC and Serial Dictatorship are Pareto efficient and strategy-proof. The theorem implies that neither mechanism can Pareto dominate DA when all students are acceptable to all schools.

These findings underscore the importance of DA as a benchmark for evaluating other school choice mechanisms. While DA may not always produce the most efficient outcome, it offers a balance between strategy-proofness, fairness, and efficiency that is difficult to surpass without sacrificing one of these desirable properties.

**Stable Improvement Cycles Algorithm.** The discussion so far has focused on strategy-proof methods for breaking ties in priorities, where students have no incentive to misrepresent their preferences. However, in some contexts, ensuring truthful revelation may not be the primary concern. For instance, if there is a high degree of trust between students and the school district, or if students are well-informed about school priorities and the consequences of different choices, strategic behavior may be less prevalent.

In such scenarios, it may be possible to achieve improvements over the DA algorithm while maintaining stability. One approach is the **Stable Improvement Cycles (SIC) algorithm**.

The SIC algorithm starts with any stable matching $\mu$. For example, one can first run the student-proposing DA with some exogenous tie-breaking rule and find $\mu$. We say that student $i$ **desires** $s$ at $\mu$ if she prefers school $s$ to her assignment $\mu(i)$, and let $D_s$ be the set of highest $\succ_s$-priority students among those who desire $s$. We can then form a **stable improvement cycle** that consists of distinct students $i_1, \dots, i_n \equiv i_0$ ($n \ge 2$) such that each student $i_l$, for $l = 0, \dots, n-1$, is in $D_{\mu(i_{l+1})}$.

Given a stable improvement cycle, define a new matching $\mu'$ by:

$$ \mu'(j) = \begin{cases} \mu(j) & \text{if } j \notin \{i_1, \dots, i_n\} \\ \mu(i_{l+1}) & \text{if } j = i_l \end{cases} $$

Observe that matching $\mu'$ continues to be stable and it Pareto dominates $\mu$.

**Theorem 4** (Erdil and Ergin, 2008). Fix $(P, \succ, q)$ and let $\mu$ be a stable matching. If $\mu$ is Pareto-dominated by another stable matching, then $\mu$ admits a stable improvement
cycle.

This theorem implies a method for finding a constrained-efficient stable matching using the SIC algorithm. Start with a stable matching. Repeatedly search for and execute stable improvement cycles. Terminate when no further stable improvement cycles are possible. The algorithm guarantees that we will eventually reach a constrained-efficient matching. This means that starting from a stable matching, by successive execution of stable improvement cycles, we will reach a point where no further Pareto improvements are possible while maintaining stability.

In light of Theorem 3, one would expect that the SIC algorithm is not strategy-proof. This is indeed the case; a student may profitably misrepresent his/her preferences to receive a school that he/she may not like in the first-round DA in order to trade for a better school from the SIC round. Despite this problem, Erdil and Ergin (2008) argue that the gains from misrepresenting preferences are rather limited: for any school one may receive from a misreported preference, there is some probability that one could get an even better school by truth-telling.

**Alternative tie-breaking rules in the NYC school assignment.** Summarizing the results so far, we have shown that DA with standard tie-breaking rules such as STB and MTB does not yield a constrained-efficient stable matching. Nevertheless, there is a sense in which DA-STB delivers a better efficiency outcome than DA-MTB. Abdulkadiroğlu et al. (2015) and Allman et al. (2023) show that DA-STB allocates *popular* schools—popular in the sense that the numbers of students top-ranking them exceed their capacities—ex-ante efficiently, whereas DA-MTB does not, in a continuum economy with no priorities. We have also shown that the SIC algorithm can produce a constrained-efficient stable matching although at the expense of strategy-proofness.

Can we quantify the efficiency losses arising from DA-STB and DA-MTB in the field? The question is ultimately an empirical one. Table 1 in Abdulkadiroğlu et al. (2009) provides some ideas about the magnitude of these losses. In line with the intuition provided above, it shows that DA-STB tends to deliver better outcomes than DA-MTB: for each $k \le 4$, the number of students who receive k-th or better options is larger in the former than in the latter. Perhaps consistent with this outcome, the NYC school system adopted DA-STB rather than DA-MTB. Still, this comparison does not extend to those assigned to lower-ranked schools or unassigned.<fnref n="13" /> The same table also suggests that the improvement from SIC on

<footnote n="13">¹³Arnosti (2023) provides a formal result that matches the observed single-crossing property.</footnote>
DA-STB is rather modest, particularly for the price of sacrificing strategy-proofness.

## 1.6 Cardinal Welfare in School Choice

The previous sections focused on students' ordinal preferences for welfare evaluation, without considering their preference intensities. In practice, however, students may differ in their preference intensities even among schools they rank identically, and such differences may be relevant for evaluating the overall student welfare resulting from an assignment.

To illustrate this point, consider an example with three students $\mathcal{I} = \{1, 2, 3\}$ and three schools $\mathcal{J} = \{a, b, c\}$, each with one seat. The schools have no priorities in keeping with the feature mentioned in the previous section. All students have the same ordinal rankings with $aPbPc$. But their preference intensities described by the (normalized) von Neumann-Morgenstern (vNM) values associated with alternative assignments are:<fnref n="14" />

<table><thead><tr><th></th><th>$u_j^1$</th><th>$u_j^2$</th><th>$u_j^3$</th></tr></thead><tbody><tr><th>$j = a$</th><td>4</td><td>4</td><td>3</td></tr><tr><th>$j = b$</th><td>1</td><td>1</td><td>2</td></tr><tr><th>$j = c$</th><td>0</td><td>0</td><td>0</td></tr></tbody></table>

In this scenario, any assignment of students across the schools is ex-post Pareto efficient. However, there is a sense in which student 3 would suffer less than either student 1 or 2 would from being assigned $b$ rather than $a$. One way to operationalize such a cardinal welfare concept is by measuring students' preferences toward lotteries of alternative assignments. Indeed, lotteries are quite natural in the school choice context due to coarse priorities, as noted in the previous section.

**IA versus DA revisited.** To see how the ex-ante welfare concept helps us to operationalize cardinal preferences, recall the earlier discussion about Immediate Acceptance (IA) and Deferred Acceptance (DA) algorithms. Given identical ordinal preferences, one cannot compare the two based on either ex-post Pareto efficiency or stability. Both are indistinguishable based on these criteria.

To be specific, suppose first that DA is used to assign students, with ties broken by any lotteries, e.g., STB or MTB. Since it is strategy-proof, all three students

<footnote n="14">¹⁴The values are normalized to have the same total sum across students.</footnote>
submit true (ordinal) preferences, and they will be assigned to the schools with
equal probabilities. This leads to an expected utility of $EU_1 = EU_2 = EU_3 = \frac{5}{3}$ for
the three students.

Now, suppose IA is used to assign students with any lotteries as a tie-breaker. No matter how student 3 ranks the schools, students 1 and 2 prefer to rank truthfully, meaning truthful reporting is a dominant strategy for these students. Given this, if student 3 ranks truthfully, then all three students are assigned with equal probabilities across the schools just like under DA, so she will get the expected utility of 5/3. Suppose student 3 ranks $(b, a, c)$. Then, she will be assigned $b$ with probability one, and receive an expected utility of 2, so she will indeed "manipulate" her preference. More interestingly, such a behavior not only increases her ex-ante welfare, but it also benefits the first two students; they are now assigned between $a$ and $c$ with equal probabilities, thus enjoying the expected utility of 2 each. Remarkably, the IA assignment Pareto dominates the DA assignment in ex-ante welfare. This result is generalized for an arbitrary number of schools and capacities.<fnref n="15" />

**Theorem 5** (Abdulkadiroğlu et al., 2011b). Consider a school choice model ($\mathcal{I}$, $\mathcal{J}$, $q$), where students have identical ordinal preferences but have vNM utilities ($u_s$)$_{s \in \mathcal{J}}$ drawn as private information according to an arbitrary distribution with finite support, and the schools have no priorities and use symmetric lotteries to break ties. In any symmetric Bayes–Nash equilibrium of the IA algorithm, each type of student is (at least weakly) better off than she is under DA.

*Proof*. Let $\mathcal{J} = \{s_1, \dots, s_m\}$, with capacity $q_s$ for each $s \in \mathcal{J}$. Without loss, $\sum_s q_s \le |\mathcal{J}|$, the number of students; and assume that assignment at any school is better than non-assignment. Each student draws a vNM $\mathbf{u} = (u_1, \dots, u_m)$, $u_1 > \dots > u_m$, from a finite support according to probability mass function $\pi(\mathbf{u})$. Under DA, each student has a probability $q_s/|\mathcal{J}|$ of getting assigned $s$.

Now consider IA. Fix any symmetric Bayes–Nash equilibrium $(\sigma, \dots, \sigma)$, in which each player employs a mixed strategy $\sigma: \mathbf{u} \mapsto \Delta(\mathcal{R})$, where $\mathcal{R}$ is the set of all rank-order lists of $\mathcal{J}$. (Such an equilibrium is well-defined by Nash’s existence theorem.) Let $x_s(\mathbf{u})$ be the probability that a student of type $\mathbf{u}$ obtains $s \in \mathcal{J}$ in that equilibrium. In equilibrium, all seats are filled; otherwise, one can deviate to increase the assignment probability at the school with empty seats and reduce the chance of non-assignment, a contradiction. Consequently, for each $s \in \mathcal{J}$, $|\mathcal{J}| \sum_{\mathbf{u}} \pi(\mathbf{u})x_s(\mathbf{u}) = q_s$.

<footnote n="15">¹⁵See also Miralles (2012) who establishes a similar result in a large continuum economy model.</footnote>
Consider a symmetric Bayes-Nash equilibrium of IA, and a student of any type **u**.
Suppose the student deviates to randomizing over the strategies other students
employ in equilibrium according to their relative "frequencies" $\pi$. Such a deviation
will yield a random assignment $(x_s(\mathbf{u}'))_{s \in \mathcal{J}}$ with probability $\pi(\mathbf{u}')$, for each $\mathbf{u}'$.
Since such a deviation cannot be profitable in equilibrium,

$$
\begin{align*}
U^{\text{IA}}(\mathbf{u}) &:= \sum_{s \in \mathcal{J}} u_s x_s(\mathbf{u}) \ge \sum_{\mathbf{u}} \sum_{s \in \mathcal{J}} u_s (\pi(\mathbf{u}) x_s(\mathbf{u})) \\
&= \sum_{s \in \mathcal{J}} u_s \sum_{\mathbf{u}} (\pi(\mathbf{u}) x_s(\mathbf{u})) = \sum_{s \in \mathcal{J}} u_s \frac{q_s}{|\mathcal{I}|} =: U^{\text{DA}}(\mathbf{u}).
\end{align*}
$$

Since the argument applies to all **u**, the desired result is proven. □

This theorem suggests the potential for IA to attain a more desirable allocation
than DA in ex-ante welfare. In the above example, the first two students have a
higher willingness to pay for *a* than student 3, when the “payment” is the risk/the
probability of getting assigned *c*, the worst school. IA activates such a trading
possibility, while DA does not. The reason can be traced to the fact that the students
are given more “say” under IA than DA, about how they wish to be treated when
competing for a school. While the students, using their strategic ranking of schools,
can improve their odds of assignment at a school, DA does not allow for this ability;
any competition among students for a school is entirely resolved by how the school
ranks them, not how the students rank the school.<fnref n="16" />

Recall that the latter feature is precisely what allowed DA to be strategy-proof.
The theorem suggests that there may be a silver lining with a strategic mechanism
such as IA in eliciting cardinal preferences and using them to achieve a desirable
outcome.

There are several reasons, however, that one should not overstate the case for
IA based on the preceding argument. First, the preceding analysis is limited to the
special case of the identical ordinal preferences and no priorities. Troyan (2012)
shows that the interim Pareto dominance of IA over DA may not hold when schools
have coarse priorities, but that IA Pareto dominates all strategy-proof mechanisms
in the ex-ante sense, or utilitarian welfare sense, even in the presence of coarse
priorities. It remains unknown, however, to what extent a similar result holds when
students have heterogeneous ordinal preferences. Second, it is unclear whether

<footnote n="16">¹⁶This difference did not go unnoticed. In the wake of the redesign of the assignment system, a parent noted: “I’m troubled that you’re considering a system that takes away the little power that parents have to prioritize… what you call this strategizing as if strategizing is a dirty word…” (Boston Public School Hearing, 2005).</footnote>
parents or students may play equilibrium strategies; namely, we do not know
how closely the theoretical prediction based on equilibrium analysis describes the
outcome of a strategic mechanism such as IA. Quite possibly, individuals may not
understand the mechanism well and may not know how best to strategize in the
face of such a mechanism. It is also conceivable that some parents are better than
others at strategizing, and if this confers an advantage to the former, there may be
an issue of equity.<fnref n="17" /> Third, IA may create different kinds of inequity, conferring a
strategic advantage to applicants with high outside options (e.g., private schools).
To see this, consider the same example as above, except that student 3 now has an
outside option worth 1.5 + ε (whereas students 1 and 2 have zero outside options).
With the insurance of 1.5 + ε for an arbitrarily small ε > 0, student 3 now prefers
to take a risk and rank (a, b, c), and with probability 1/3 succeeds in taking a away
from those who arguably value it more. See Akbarpour et al. (2022), Shorrer (2019),
and Calsamiglia et al. (2021) for formalizing this intuition. All in all, there are
compelling reasons to weigh strategy-proofness more favorably than the potential
benefit from its violation.

Of course, this caveat raises an interesting research question: can one promote
cardinal welfare better than DA with a minimal sacrifice of its desirable strategy-
proofness property?

**Choice-Augmented Deferred Acceptance.** The Choice-Augmented Deferred Acceptance (CADA) mechanism, developed by Abdulkadiroğlu et al. (2015), offers a method for improving cardinal welfare with a modest sacrifice on the desirable property of strategy-proofness. It does so by modifying DA to allow students to influence how they are treated in ties while maintaining truthful revelation of ordinal preferences.

CADA consists of two main steps:

(i) Ordinal Preference Submission and Target School Declaration: Students submit their ordinal preferences over schools, just like in DA. In addition, they declare a "target" school.

(ii) Modified DA with Targeted Tie-Breaking: The mechanism then runs a
modified version of the DA algorithm. In this modified version, any ties
at each school are resolved by the STB, except that the realized lottery

<footnote n="17">¹⁷However, it is worth noting that strategically sophisticated applicants may not harm naive students. Suppose in the above example that all students are initially naive and rank (a, b, c), they all enjoy the expected utility of 5/3. Suppose now only applicant 3 becomes sophisticated and ranks (b, a, c); this benefits the two naive applicants by reducing the competition for a, which stands in contrast to Pathak and Sönmez (2008) (see footnote 4).</footnote>
order at each school is reordered to prioritize the students who declare that school over the others who did not.<fnref n="18" />

The DA algorithm is then run based on the ordinal preferences from step (i) and the priority lists from step (ii). It is easy to see that the CADA is ordinally strategy-proof, meaning it is a dominant strategy for each student to rank schools truthfully. The scope for the strategization is limited to the tie-breaking part of DA, which can be influenced by the “strategic” choice one makes about the target schools. Since the tie-breaking is subordinated to the “intrinsic” priorities, CADA, just like DA, respects the true priorities of students. In short, CADA produces a stable matching with no justified envy. In particular, when the priorities are strict, CADA reduces to DA. When the priorities are coarse (as is often the case), however, CADA can elicit cardinal preferences in a way that improves ex-ante welfare.

To see this, suppose CADA is used in the above example. All students report truthfully their ordinal preference $(a, b, c)$. For targeting, students 1 and 2 will name $a$ for the target school (in a dominant strategy); and given this, student 3 will name $b$ for the target school. Then, school $a$ will prioritize 1 and 2 over 3, so either of the first two will win (based on random tie-breaking). In the second round, when the two rejected students, who include 3, apply to school $c$, it will prioritize student 3, resulting in that student receiving $b$, and the other getting $c$. Hence, the ex-ante efficient assignment will be obtained.

Abdulkadiroğlu et al. (2015) consider a model with a mass of students and a finite number of schools with masses of seats and with no priorities. They then measure the efficiency of a mechanism by the **scope of efficiency**—formally, the maximal set $\mathcal{J}' \subset \mathcal{J}$ of schools whose allocation shares among students cannot be reallocated in a way that yields a Pareto improvement. It is as if a classic market exists, and delivers Pareto efficiency, but only with shares of $\mathcal{J}'$ being allowed to be traded. They show that CADA supports a larger scope of efficiency than DA in terms of set-inclusion or set-cardinality.

## 1.7 Affirmative Actions: Quotas and Reserves

The concept of affirmative action in public school assignments has been a subject of extensive debate, reflecting societal goals to address historical inequalities and promote diversity within educational institutions. School districts often implement

<footnote n="18">¹⁸Specifically, two random priority lists, $T$ and $R$, of students are generated. For each school, the students who targeted that school are ranked first, according to $T$, and then those who did not are ranked next according to $R$.</footnote>
controlled choice programs to offer parental choice while striving to maintain a
balance across various student characteristics, such as race, ethnicity, or socioeco-
nomic status. Before the advent of school choice policies, neighborhood assignment
frequently led to socioeconomically segregated areas, as wealthier families relo-
cated to neighborhoods with their preferred schools. This left families without
such means constrained to neighborhood schools, regardless of their qualities or
match values. To counteract these shortcomings, controlled school choice programs
have become increasingly popular. This section will explore the common methods
of incorporating affirmative action—maximum quotas and reserves—into school
assignment mechanisms like Deferred Acceptance (DA).

1.7.1 Maximum Quotas

In practice, controlled school choice programs often enforce diversity objectives
by setting hard upper bounds (“ceilings”) and/or hard lower bounds (“floors”)
for different student types. An affirmative action policy may take the form of
a maximum quota against the advantaged group and a minimum quota for a
protected disadvantaged group. For example, the Jefferson County School District
in Kentucky requires elementary schools to allocate between 15% and 50% of
their seats to students from the geographic area with the highest concentration of
beneficiaries of affirmative action policies. Similarly, in New York City, “Educational
Option” (EdOpt) schools have to accept students across different ability ranges: 16%
must score above grade level on the standardized English Language Arts test, 68%
at grade level, and the remaining 16% below grade level.

Abdulkadiroğlu and Sönmez (2003) were among the first to formally incorporate
type-specific quotas (upper bounds) into their seminal work on school choice
mechanisms. They consider maximal quotas for non-overlapping types of students.
They then modify DA to respect the type-specific quotas: in each step, each school
accepts (tentatively) the students of each type at most up to its maximal quota and
rejects the others based on its priority, and once the quota is filled for a given type,
it only replaces the students in that type as better students apply.

This maximum-quota-modified DA, labeled DA<sub>MaQ</sub>, retains the desirable proper-
ties of DA: it is strategy-proof, meaning students have no incentive to misrepresent
their preferences, and it produces a student-optimal stable matching.

However, a critical question arises: Do maximum quotas always benefit the
minority groups they are intended to protect? Surprisingly, this is not necessarily
the case. As highlighted by Kojima (2012), affirmative action policies can have
"perverse consequences." To illustrate, consider an example from Kojima (2012): Suppose there are three students, 1, 2 and 3, and two schools, *a* and *b*, with capacities $q_a = 2$ and $q_b = 1$. Students 1 and 2 belong to a majority group, and 3 belongs to a minority group.

<table><caption>**Figure 7 – The Perverse Effects of Maximum Quotas**</caption><thead><tr><th>P<sub>1</sub></th><th>P<sub>2</sub></th><th>P<sub>3</sub></th><th>&gt;_a</th><th>&gt;_b</th></tr></thead><tbody><tr><td>a</td><td>a</td><td>b</td><td>1</td><td>2</td></tr><tr><td></td><td>b</td><td>a</td><td>3</td><td>1</td></tr><tr><td></td><td></td><td></td><td>2</td><td>3</td></tr></tbody></table>

Suppose first there are no maximum quotas against the majority group, then DA produces a matching $\mu = (1a, 2a, 3b)$ in which all students are assigned their first-best choices.

Next, school *a* introduces an affirmative action policy for the minority group via a maximum quota of one against the majority group {1, 2}. DA<sub>MaQ</sub> then produces a matching (1a, 2b, 3a). The reason is that the maximum quota on the majority group forces school *a* to reject 2 when both 1 and 2 apply. Student 2 then applies to *b*, and displaces 3 at school *b* (since 2 has a higher priority than 3 at *b*). Not only is the new matching Pareto inferior to the old matching, but the new matching is strictly worse for the minority student 3, as she prefers *b* over *a*.

### 1.7.2 Minority Reserves

To circumvent the inefficiencies and perverse effects caused by rigid majority quotas, Hafalir et al. (2013) propose a different interpretation of affirmative action policies based on “minority reserves.” The idea of minority reserves is that schools give higher priority to minority students up to a certain “reserve” number, but unlike a maximum quota against the majority group, it does not cap the seats allocated to the majority group. Minority reserves are extensively used in practice.<sup>19</sup>

The Deferred Acceptance (DA) algorithm adapted for minority reserves, labeled DA<sub>MiR</sub>, works as follows:

* **Step 1:** Start with the matching in which no student is matched. Each student *i* applies to her first-choice school. Each school *s* first accepts as many as $r_s^m$

<footnote n="19"><sup>19</sup>See https://www.schools.nyc.gov/enrollment/enrollment-help/meeting-student-needs/diversity-in-admissions and Dur et al. (2020) for more details of the use of minority reserves in NYC and Chicago exam schools.</footnote>
minority applicants with the highest priorities if there are enough minority applicants. Then it accepts applicants with the highest priorities from the remaining applicants until its capacity is filled or there are no applicants left. Any remaining applicants are rejected by s.

* **Step k:** Start with the tentative matching obtained at the end of Step $k-1$. Each student $i$ who got rejected at Step $k-1$ applies to her next-choice school. Each school $s$ considers the new applicants and students admitted tentatively at Step $k-1$. Among these students, school $s$ first accepts as many as $r_s^m$ minority students with the highest priorities if there are enough minority students. Then it accepts students with the highest priorities from the remaining students. The rest of the students, if any remain, are rejected by $s$. If there are no rejections, then stop.

The algorithm terminates when no rejection occurs and the tentative matching at that step is finalized. This mechanism is designed such that schools dynamically adjust their priorities: giving the highest priority to student types who have not filled their reserve, but no such priority once their reserves are filled. This dynamic priority structure ensures that the DA algorithm with minority reserves preserves its desirable properties, including being student-optimal, fair, non-wasteful, and group strategy-proof.

It is easy to see that $\text{DA}_{\text{MiR}}$ is more flexible than $\text{DA}_{\text{MaQ}}$. Suppose the maximum quota $q_s^M$ against the majority group is replaced by a minority reserve set at $r_s^m = q_s - q_s^M$. The minority group is prioritized over the $r_s^m$ reserved seats. However, if there are not enough minority students to fill the reserves, majority students can still be admitted to occupy those reserved seats, meaning that the number of majority students admitted can exceed what would be allowed under a strict quota if there are not enough minority students who prefer that school to fill the reserved slots.

Hafalir et al. (2013) demonstrate a crucial theoretical advantage of minority reserves:

**Theorem 6.** Consider majority quotas $q_s^M$ and minority reserves $r_s^m$ such that $r_s^m = q_s - q_s^M$, for each $s \in \mathcal{J}$. $\text{DA}_{\text{MiR}}$ with reserves ($r_s^m$) produces a matching that Pareto dominates $\text{DA}_{\text{MaQ}}$ matching with quotas ($q_s^M$).

This theorem implies that the student-proposing DA with minority reserves ($\text{DA}_{\text{MiR}}$) weakly Pareto dominates DA with majority quotas ($\text{DA}_{\text{MaQ}}$), suggesting that switching from majority quotas to minority reserves can make all students at least weakly better off.
This result can be illustrated easily in Figure 7 by Kojima (2012). Suppose school *a* replaces its maximum quota of 1 against students {1, 2} with a reserve of 1 for student 3. $DA_{MiR}$ then produces $(1a, 2a, 3b)$,<fnref n="20" /> which benefits both majority and minority students.

Even with minority reserves, however, some minorities may be made worse off, while others remain the same. Consider the following example from Hafalir et al. (2013), in which there are three students 1, 2, and 3, with {2, 3} being minorities, and three schools, *a*, *b*, and *c*, each with one seat. See Figure 8.

<table><caption>**Figure 8 – Minority Reserves Can Hurt Minorities**</caption><thead><tr><th>P<sub>1</sub></th><th>P<sub>2</sub></th><th>P<sub>3</sub></th><th>&gt;<sub>a</sub></th><th>&gt;<sub>b</sub></th><th>&gt;<sub>c</sub></th></tr></thead><tbody><tr><td>a</td><td>c</td><td>a</td><td>1</td><td>1</td><td>1</td></tr><tr><td>c</td><td>a</td><td>b</td><td>2</td><td>2</td><td>2</td></tr><tr><td>b</td><td>b</td><td>c</td><td>3</td><td>3</td><td>3</td></tr></tbody></table>

With no affirmative action, DA yields matching $(1a, 2c, 3b)$, which is Pareto efficient. Suppose (only) school *a* introduces a minority reserve of 1. Then, $DA_{MiR}$ yields $(1c, 2a, 3b)$, which is Pareto inferior to the DA matching without affirmative action. The intuition is that 3 now outcompetes 1 at *a* in step 1 based on her minority status, only to be displaced by 2 later, following 2's displacement by 1 at *c*.

Despite these specific examples, simulations by Hafalir et al. (2013) indicate that the positive welfare effects of minority reserves on minority students are “substantial,” with improvements for up to 30% of minority students on average. Even more significant benefits are achieved for majority students, with up to 50% being better off under $DA_{MiR}$ compared to $DA_{MaQ}$. These simulations, which account for correlations in student preferences and school priorities, demonstrate that minority reserves, on average, make minorities better off (while potentially making majorities worse off) than no affirmative action, and that $DA_{MiR}$ benefits both minorities and majorities significantly compared to $DA_{MaQ}$. Crucially, majority quota-based mechanisms are highly sensitive to quota size, especially for majority welfare, whereas minority reserve-based mechanisms temper the adverse effects on majorities.

Ehlers et al. (2014) further investigate the implications of “hard bounds” versus “soft bounds” in controlled school choice.<fnref n="21" /> They formally define “hard bounds” as

<footnote n="20">²⁰The algorithm ends in step 1 where both 1 and 2 are assigned school *a*, and 3 is assigned *b*. Since no minority student applies to *a*, accepting both majority students satisfies the policy.</footnote>

<footnote n="21">²¹Echenique and Yenmez (2015) provide axiomatic characterizations of choice rules based on</footnote>
feasibility constraints that must be met, such as fixed upper and lower bounds for student types at schools. With hard bounds, they demonstrate serious limitations: a key finding is that assignments that are both "fair across types" and "non-wasteful" may not even exist. This non-existence problem extends even to assignments that are "fair for the same types" (where envy is only considered between students of the same type) if non-wastefulness is also required. This impossibility arises because lower bounds can create complementarities among students for schools, a known risk factor for nonexistence.

Given these difficulties with hard bounds, the paper introduces "soft bounds" as a novel interpretation, where control constraints dynamically regulate school priorities. This is precisely the mechanism behind minority reserves discussed earlier, where priorities adapt to fill floors or address ceilings. Under this "soft bounds" view, an assignment that is fair across types and non-wasteful is guaranteed to exist. The student-proposing DA, when operating with these soft bounds, achieves a student-optimal, fair, and non-wasteful assignment, and is also group strategy-proof. While the student-proposing DA maximizes student welfare among fair and non-wasteful assignments, it may not necessarily respect the controlled choice constraints (i.e., the floors and ceilings) in some cases. In contrast, the school-proposing DA with soft bounds minimizes violations of controlled choice constraints among fair and non-wasteful assignments.

In summary, Ehlers et al. (2014) highlight a fundamental trade-off: hard bounds are problematic due to existence issues and lack of strategy-proofness, whereas soft bounds (like minority reserves) restore desirable properties such as fairness, non-wastefulness, and truthfulness. This implies that if policymakers are not overly paternalistic, soft-bound policies are the superior choice, even though desired diversity might not be completely guaranteed.

A crucial aspect, often implicit in the mechanisms discussed so far, is the *precedence order* in which different types of seats are filled when a student is eligible for more than one. So far, our discussion has implicitly assumed a certain precedence order for filling these seats: reserved seats are processed first for the protected group. However, recent research highlights that this seemingly minor implementation detail can profoundly alter the intended outcomes of affirmative action policies (Dur et al., 2018b, 2020; Sönmez and Yenmez, 2022).

Specifically, the order in which a school fills its seats, whether it is through open competition first or through reserved categories first, can determine who gets in

<footnote>reserves and on quotas.</footnote>
and from which group. For instance, in Boston, a policy designed to set aside 50% of seats for local "walk-zone" students effectively yielded almost no advantage for them in practice (Dur et al., 2018b). This was because the district's mechanism filled the reserved walk-zone seats first. While this sounds intuitive, it meant that the strongest walk-zone students took those reserved spots. When it came to filling the "open" seats (available to all), the remaining walk-zone students, generally having lower lottery numbers or merit among the overall applicant pool, were then outcompeted by non-walk-zone students for those general seats. Had the open seats been processed first, more walk-zone students overall would have gained admission (see Dur et al., 2018b). This revelation underscores that the sequencing of seat allocation is not a neutral technicality. It can create "statistical targeting" effects, where policies that appear neutral on paper (like equal reserve sizes) can implicitly favor or disadvantage certain groups based on their score distributions (see Dur et al., 2020). This highlights a fundamental trade-off: market designers have additional leverage beyond just setting quotas or reserves; they can use precedence orders to intentionally (or unintentionally) influence which groups are ultimately admitted, impacting both diversity and merit-based goals (see Dur et al., 2020; Sönmez and Yenmez, 2022; Abdulkadiroğlu and Grigoryan, 2025). In particular, Abdulkadiroğlu and Grigoryan (2025) provide an axiomatic foundation based on constrained efficiency for pinning down a precedence order.

## 2 Methods for Preference Estimation

The rapid expansion of centralized school choice systems across the globe (see Neilson, 2024) has opened up new empirical opportunities for researchers to analyze how these procedures perform, how participants behave, and how alternative assignment mechanisms affect educational outcomes. A central ingredient of this empirical agenda is the use of rich administrative data generated by centralized matching platforms—including submitted rank-order lists (ROLs), priorities, school capacities, and final assignments—to recover information about applicants’ underlying preferences. Such information is essential for evaluating the performance of different mechanisms, conducting counterfactual policy simulations, and assessing the welfare and distributional impacts of school choice reforms.

However, extracting reliable preference information from observed ROLs poses significant challenges. As highlighted in Section 1.3.2, in strategic mechanisms such as Immediate Acceptance, submitted rankings do not necessarily reflect applicants’
true preferences. Even in strategy-proof mechanisms like Deferred Acceptance, reported preferences may not be truthful: for instance, applicants might omit schools they view as unattainable. In addition, list-length constraints, which are common in many systems, further distort observed rankings when applicants strategically exclude schools not because of lack of preference but to avoid going unmatched. These features make the interpretation of reported preferences inherently non-trivial.

The stakes of misinterpreting these data are high. Preference estimates feed directly into welfare analyses and policy recommendations. Misreading why disadvantaged students avoid ranking selective schools, for example, may lead to erroneous conclusions about their preferences and the design of policies aimed at closing opportunity gaps.

This section reviews the methodological approaches that have been developed to estimate preferences from the data generated by school choice mechanisms, highlighting both foundational frameworks and more recent advances. A key feature of most centralized school choice systems is that schools do not submit preferences but instead rank students according to pre-specified priority criteria. The resulting priorities are policy parameters chosen by the designer and, crucially for empirical work, are known functions of observable characteristics. This implies that only students' preferences must be recovered, distinguishing these settings from matching markets with non-transferable utility where both sides' preferences need to be estimated (e.g., Agarwal, 2015; Diamond and Agarwal, 2017; He and Magnac, 2022; Friedrich et al., 2024; He et al., 2024; Klein et al., 2024; Ederer, 2025).<fnref n="22" />

Agarwal and Somaini (2020) provide an excellent overview of many of the methods discussed in this section. We build on their survey by integrating recent developments and extending the scope to new empirical applications. While our review is mostly centered on preference estimation in school choice settings, we also draw on studies of centralized college admissions where programs rank students using pre-specified rules, as the methodological approaches are largely common to both contexts. Our focus is on the identification and estimation of preferences, and we do not cover the complementary literature that exploits quasi-experimental variation embedded in centralized assignment mechanisms to evaluate school effectiveness (e.g., Deming, 2011; Pop-Eleches and Urquiola, 2013; Abdulkadiroğlu et al., 2011a, 2014, 2017b, 2022; Angrist et al., 2017, 2023a). Nonetheless, we return to some of the key insights from this literature in Section 3 of the chapter.

We begin by distinguishing between preference estimation approaches suited to

<footnote n="22">²²See Agarwal and Somaini (2023) for a recent survey of empirical models of non-transferable utility matching.</footnote>
IA and other inherently manipulable mechanisms (Section 2.1), and those suited to DA (Section 2.2), which is strategy-proof for applicants under certain conditions.<fnref n="23" /> Although there is some methodological overlap, each setting poses distinct identification challenges, which we address separately for expositional clarity. Beyond this distinction, we emphasize recent innovations that incorporate insights from behavioral market design, allowing for belief errors, strategic mistakes, and bounded rationality in estimation. We then turn to a more detailed discussion of identification and estimation techniques across empirical approaches (Section 2.3). For a summary of the applications discussed throughout this section, we refer the reader to Table 1, which outlines the main settings, mechanisms, and estimation approaches used in the literature.

## 2.1 Preference Estimation under Immediate Acceptance and Other Strategic Mechanisms

Early contributions in the empirical market design approach to school choice have focused on the challenges of recovering preferences under strategic mechanisms—particularly Immediate Acceptance (IA)—reflecting both their widespread use and the rich theoretical literature analyzing their properties.

**Evidence of strategic behavior under manipulable mechanisms.** A key milestone in the empirical analysis of manipulable school choice mechanisms has been the consistent finding, both in lab experiments and real-world settings, that they induce pervasive strategic behavior, often at the expense of less informed or less sophisticated agents. In a seminal study, Chen and Sönmez (2006) showed that participants in a school choice experiment were significantly more likely to misrepresent their preferences under IA than under either DA or TTC. In one treatment, fewer than 14% of participants reported their true preferences under IA, compared to 72% under DA and 50% under TTC. The finding that IA results in lower rates of truthful reporting compared to DA or TTC has consistently been confirmed by nearly all

<footnote n="23">²³We do not discuss preference estimation under the Top Trading Cycles mechanism. At present, there are no prominent empirical implementations of TTC, and most of the methods covered in this section are not applicable. In particular, neither the optimal portfolio approach nor the stability-based approach can be used, as the assumptions required for their validity do not hold under TTC. Although the truth-telling approach discussed in Section 2.2.1 could, in principle, apply in this context, it rests on strong assumptions that may not be satisfied in practice. If TTC sees wider adoption, developing preference estimation methods tailored to this mechanism would be a promising direction for future work.</footnote>
subsequent school choice experiments (for a comprehensive survey, see Hakimov and Kübler, 2021).

Field studies have corroborated these experimental findings, offering both direct and indirect evidence of preference manipulation under strategic assignment mechanisms. These include analyses of reported rankings under IA or closely related mechanisms (e.g., Abdulkadiroğlu et al., 2006a,b; Agarwal and Somaini, 2018) and survey data eliciting students’ true preferences (e.g., Budish and Cantillon, 2012; Kapor et al., 2020; De Haan et al., 2023). For example, Agarwal and Somaini (2018) exploit geographic discontinuities in proximity-based priorities under Cambridge Public Schools’ (CPS) Controlled Choice Plan, which uses a variant of IA, to show that students adjust their rankings strategically to retain priority, providing clear evidence of manipulation.

This growing body of evidence has been instrumental in challenging common misconceptions among school district officials, who often misinterpret the high proportion of students assigned to their top-listed school under IA as a sign of genuine satisfaction. In several cities, including Boston and Amsterdam, these insights were pivotal in prompting the adoption of DA (Abdulkadiroğlu et al., 2006a; De Haan et al., 2023). From a methodological perspective, this evidence has motivated economists to develop models that incorporate strategic behavior in estimating student preferences.

**Preference reporting as a choice over lotteries.** The now-dominant approach to preference estimation in strategic assignment mechanisms was developed by Agarwal and Somaini (2018). They introduce a general framework for inferring student preferences from a broad class of mechanisms and apply it empirically to the CPS Controlled Choice Plan in Cambridge. This class, referred to as *Report-Specific Priority + Cutoff* (RSP+C) mechanisms, is defined by the feature that an applicant's probability of admission to a given school depends on two elements: (i) the applicant's priority at the school (which, under IA, may depend on the submitted report), and (ii) the school's market-clearing cutoff, which reflects its capacity constraint. Most of the manipulable and non-manipulable mechanisms analyzed in the empirical literature, including IA and DA, fall within this class, with the notable exception of TTC.<fnref n="24" />

To model strategic behavior in RSP+C mechanisms, Agarwal and Somaini assume that applicants submit ROLs that maximize their expected utility, given their beliefs

<footnote n="24">²⁴Agarwal and Somaini (2018) note that although Leshno and Lo (2021) derive a cutoff representation for TTC, it does not belong to the class of RSP+C mechanisms.</footnote>
about their admission probabilities at the different schools. In this framework, applicants hold private information about their own preferences over schools, but do not observe the preferences of other participants in the market. Beliefs about assignment probabilities depend on the submitted ROL, the priority at each school, and expectations about the strategies of other applicants.

Formally, let $i \in \mathcal{I} := \{1, \dots, I\}$ index students and $j \in \mathcal{J} := \{1, \dots, J\}$ index schools. Let $u_{ij}$ denote student $i$'s indirect utility from being assigned to school $j$, and $t_{ij}$ represent her priority at school $j$. Priorities may be strict, as when they are determined by an exam score, or coarse, as when students fall into broad priority classes (e.g., based on walk zones). In the latter case, a random tie-breaker, denoted by $\tau_{ij}$, is used to rank students within the same priority class. Given a utility vector $\mathbf{u}_i := (u_{i1}, \dots, u_{ij})$ and priority profile $\mathbf{t}_i := (t_{i1}, \dots, t_{ij})$, student $i$'s expected utility from submitting ROL $R$ is $\mathbf{u}_i \cdot \mathbf{L}_{R,i}$, where $\mathbf{L}_{R,i}$ is the vector of the student's perceived assignment probabilities at each school when submitting $R$. Thus, an applicant's choice over ROLs can be viewed as a portfolio choice over lotteries: among all possible reports $R$, the student selects the one that yields the highest expected utility.

This framework implies a two-step estimation strategy: (1) for each priority profile $\mathbf{t}_i$ and potential ROL $R$, estimate the applicant's beliefs about the probability of admission at each school; (2) using the assignment probabilities estimated in the first step, recover the preference parameters that rationalize the reports submitted. Further details on the identification and estimation of assignment probabilities and preference parameters within this framework are provided in Section 2.3.2.

**Beliefs about assignment probabilities.** Agarwal and Somaini (2018) consider several alternative assumptions regarding how agents form beliefs about their assignment probabilities. In their baseline model, applicants are assumed to hold *rational expectations*: they form accurate beliefs about their admission probabilities based on their own priorities, the distribution of priorities and preferences in the applicant population, and the reporting strategies of others. Under this assumption, uncertainty in admission probabilities stems from two sources: (i) uncertainty about the reports submitted by other participants and (ii) randomness introduced by lottery tie-breakers when priorities are coarse.

In school choice environments, rational expectations may appear restrictive, given empirical evidence that many agents behave unsophisticatedly in strategic mechanisms. For instance, Abdulkadiroğlu et al. (2006a,b) document that under Boston's former manipulable assignment mechanism, 13% of applicants ranked two
overdemanded schools as their top two choices while having random priority at the first, and 23% of these applicants remained unassigned after the main admission round and were subsequently placed administratively in schools with unfilled seats. While listing one overdemanded school first can be a calculated risk under IA, listing two as the top choices is a clear mistake when a student ends up unassigned and does not withdraw from the public school system. This strategy effectively wastes the second slot, reducing the chances of securing a seat at the third or any subsequent choice.<fnref n="25" />

To relax the rational expectations assumption, Agarwal and Somaini (2018) explore alternative models of belief formation: (i) In the *adaptive expectations* model, agents form expectations based on their own priorities and past-year data on submitted reports; (ii) In the *coarse beliefs* model, they ignore their own priorities—either due to lack of information or misunderstanding of the mechanism—and rely only on aggregate information such as prior-year application numbers and school capacities; (iii) Drawing on the framework of Pathak and Sönmez (2008), the authors also introduce a model of *heterogeneous sophistication*, in which the applicant pool consists of a mix of sophisticated agents (who report optimally given correct beliefs) and naive agents (who report their preferences truthfully even when doing so is suboptimal).

An alternative and more direct way to recover preferences in IA settings, without relying on specific assumptions about belief formation, is to elicit applicants’ subjective expectations. Kapor et al. (2020) develop a framework that combines survey data on students’ stated preferences and beliefs about admission probabilities with observed application and enrollment decisions. This hybrid revealed-preference and stated-belief approach extends the methods of Agarwal and Somaini (2018) by replacing rational or adaptive expectations with elicited beliefs in the utility maximization framework. Importantly, the analysis assumes that choices are optimal given subjective beliefs, so deviations from predicted behavior are attributed to belief errors rather than optimization failures. Applied to high school choice in New Haven, where the assignment mechanism closely resembles IA, the authors find that applicants often make choices based on systematically mistaken beliefs, leading to substantial welfare losses.<fnref n="26" /> This approach underscores the value of

<footnote n="25">²⁵We caution the reader, however, that this is only suggestive evidence, since applicants may not have accurate beliefs about which schools are eventually overdemanded.</footnote>

<footnote n="26">²⁶While IA prioritizes how highly a student ranks a school, considering priority groups only afterward, the assignment mechanism used in New Haven reverses this order, giving precedence to a student’s priority group before considering the school’s position in her ROL. This feature makes the mechanism close in spirit to the Choice-Augmented Deferred Acceptance (CADA) mechanism</footnote>
combining administrative and survey data to recover preferences in school choice
settings where agents face strategic incentives and belief errors can significantly
distort observed behavior.

**Other approaches.** Contemporaneously with Agarwal and Somaini (2018), several alternative methods have been developed to estimate preferences using data from strategic mechanisms (e.g., Hwang, 2015; He, 2017; Calsamiglia et al., 2020; Bayraktar and Hwang, 2024). These approaches are generally more tailored to specific institutional settings and thus require some adaptation before being applied elsewhere.

Using data from Barcelona, where school assignment follows an IA mechanism
with a single tie-breaking lottery, Calsamiglia et al. (2020) develop a method suited
to contexts with limited uncertainty about admission cutoffs. This situation typically
arises when the number of students is large relative to the number of schools, so that
aggregate uncertainty about cutoffs effectively disappears. The model assumes two
types of applicants: strategic agents, who hold rational expectations and maximize
expected utility based on admission probabilities, and non-strategic ones, who
submit truthful rankings. Given fixed cutoffs, the optimal report for a strategic
agent is derived using backward induction. Instead of evaluating all possible ROLs,
the method sequentially builds the list from the bottom up, selecting at each rank *r*
the school that maximizes expected utility, conditional on rejection from higher-
ranked schools and the continuation value of subsequent choices. Preferences
and the type shares are estimated by testing the optimality of submitted ROLs for
strategic agents, while assuming sincere reporting for non-strategic ones.

Like Agarwal and Somaini (2018), this approach relies on a complete model in
which an applicant's report is uniquely determined by her beliefs about admission
probabilities. By contrast, Hwang (2015), He (2017), and Bayraktar and Hwang (2024)
propose partial identification strategies that relax these behavioral assumptions.

Hwang (2015) and Bayraktar and Hwang (2024) allow for belief errors under two
weaker conditions: (i) agents can correctly rank schools by competitiveness, i.e., they
correctly anticipate which schools will be oversubscribed or undersubscribed in each
round of the IA mechanism, and they know the relative ranking of admission cutoffs,
though not their exact values; (ii) they do not rank a more competitive school above
a less competitive one unless they strictly prefer it. Violating the latter condition

<footnote>described in Section 1.6, with the difference that applicants do not declare a single “target” school to be prioritized among students in the same priority group at that school. Instead, the school’s position in the submitted ROL is used to prioritize applicants within the same priority group.</footnote>
would reduce the chance of being assigned to a more preferred option, making
such a strategy suboptimal. Using data from Seoul, the authors derive moment
inequalities to partially identify preferences and construct confidence regions for
the model's parameters.

He (2017) develops a related approach in the context of middle school admissions
in Beijing. His method allows parents to make mistakes—such as playing “safe”
strategies too often—by allowing for heterogeneous levels of sophistication, moving
beyond the dichotomy between naive and fully strategic parents. The only rationality
requirement is that parents do not play dominated strategies in equilibrium, for
example, ranking an unacceptable school first. At the same time, the framework
allows for multiple best responses, such as ranking or excluding a school with
zero admission probability. These relaxed assumptions yield bounds on choice
probabilities, which in turn set-identify preference parameters. Importantly, He
shows that this bounded rationality framework substantially improves fit relative to
models that assume fully strategic, best-responding agents.

## 2.2 Preference Estimation under the Deferred Acceptance Mechanism

At first glance, estimating preferences under the Deferred Acceptance mechanism
may appear more straightforward than under IA. Because the canonical version of
DA is strategy-proof for applicants, submitted ROLs are often interpreted as truthful
reflections of preferences. However, a growing body of research has cautioned
against taking this literal interpretation in real-world applications. In practice, truth-
telling may be undermined by factors such as list-length restrictions, application
costs, or behavioral frictions. In response, recent work has developed empirical
methods that estimate preferences in DA settings while explicitly allowing for
deviations from truthful reporting.

### 2.2.1 The Truth-Telling Approach

A central rationale for adopting strategy-proof mechanisms like DA is to eliminate
the need for applicants to strategize over their rankings, thereby simplifying the
decision-making process and helping to level the playing field between strategically
sophisticated agents and those who are less informed or able to game the system
(Pathak and Sönmez, 2008). This property has led many empirical studies using data
from centralized school choice or college admissions systems operating under DA to
estimate preferences under the assumption that applicants submit truthful rankings (e.g., Hällsten, 2010; Kirkebøen, 2012; Abdulkadiroğlu et al., 2017a; Laverde, 2024; De Groote et al., 2025). However, the validity of this assumption warrants careful scrutiny, as the notion of truth-telling admits multiple interpretations that require clarification.

**Strict versus weak truth-telling.** The theoretical justification for truthful preference reporting under DA rests on relatively stringent conditions. The strategy-proofness of DA is only guaranteed if agents can rank all schools at no cost. In this case, *strict truth-telling* (STT), i.e., ranking *all* schools in the order of true preferences, as defined by Fack et al. (2019), is a dominant strategy. However, because STT is only a *weakly* dominant strategy, multiple equilibria can arise: an applicant might achieve the same outcome by submitting a non-truthful report, such as omitting schools where she anticipates no chance of admission. For STT to be the unique equilibrium, an additional requirement is that each student has a strictly positive probability of being admitted to every school. These conditions are more likely to be met when applicants face no restrictions on the number of schools they can rank and when there is substantial uncertainty about admission probabilities, such as in systems where priorities are determined by lotteries.

In practice, however, applicants rarely rank all available options, even when permitted to do so at no cost. This raises the question of how to interpret the omission of certain schools or programs from an applicant's ROL. One approach is to introduce an outside option and reformulate the STT assumption accordingly: students are assumed to rank all schools they find acceptable (i.e., those they prefer to the outside option) and omit those they deem unacceptable. This modified version of STT can be rationalized as an equilibrium outcome when applicants are fully rational, unconstrained in list length (as in the Boston pre-kindergarten setting studied by Laverde, 2024), and face substantial uncertainty about admission outcomes.<fnref n="27" />

A more commonly used alternative in the literature omits the outside option and assumes that applicants are *weakly truth-telling* (WTT): they rank their most preferred schools truthfully, and any unranked school is assumed to be less preferred than those included in the ROL. For instance, if there are four schools, {a, b, c, d}, and student *i* submits the ROL (*c*, *a*), WTT dictates that her true preferences are

<footnote n="27">²⁷This formulation, however, introduces a potential issue of multiple equilibria: if applicants can always decline an unacceptable assignment, they may be indifferent about whether to include such schools in their ROL.</footnote>
$c \succ a \succ b, d$. WTT can thus be viewed as a truncated version of STT. While it is widely adopted in empirical work, it lacks a solid theoretical foundation, as it assumes that the number of ranked options in any ROL is exogenous to applicants' preferences. Under this assumption, WTT can be used to identify preferences through standard discrete choice models, which we discuss in more detail in Section 2.3.3.

**Deviations from truth-telling in DA.** Despite its intuitive appeal, the WTT assumption in DA environments has been challenged in settings where applicants face list-length restrictions, incur application costs, have limited uncertainty about their admission chances, or are inattentive to mistakes that carry minor payoff consequences.

WTT is most clearly undermined in contexts where DA imposes a cap on the number of schools applicants can rank. This is a common feature in many centralized school choice and college admission systems worldwide. For example, in Paris, students applying to public high schools can rank up to 8 schools, even though the number of available options in a district typically ranges from 10 to 17. Similar list-length constraints exist in school choice systems implemented in Boston, Chicago, New York City, Finland, Ghana, Singapore, and Turkey, as well as in higher education admissions in countries like Australia, Chile, France, Norway, Spain, Tunisia, and Turkey (see Table 1 in Fack et al., 2019 and Figure 6 in Neilson, 2024).

When such constraints are imposed, DA ceases to be strategy-proof: students risk remaining unassigned if the mechanism exhausts the options in their submitted ROL (Haeringer and Klijn, 2009). In response, constrained applicants may rationally choose to “skip” schools with low admission probabilities in favor of safer, though less preferred, options. For instance, a student who can rank only three out of four acceptable schools, with preferences $a \succ b \succ c \succ d$, may submit the ROL $(a, c, d)$ if she anticipates a low admission probability at $b$, in order to reduce the risk of remaining unassigned in the event that $d$ is her only feasible option. Calsamiglia et al. (2010) provide experimental evidence of this behavior, showing that constrained DA leads to lower rates of WTT compliance than its unconstrained counterpart.

Beyond the case of list-length restrictions, WTT can also fail when students face low uncertainty about their admission chances. This is particularly relevant in environments where schools or colleges rank applicants using a priority index, such as a test score, which is known to students when they submit their ROL. If a student expects her score to be too low for a selective school (based on past admission data, for example), she may rationally choose to “skip the impossible” by omitting it
from her ROL, despite preferring it to some of the schools she does list. Even in the
absence of explicit list-length limits, the cognitive burden of ranking many schools
or the desire to avoid disappointment from likely rejections (Meisner and van
Wangenheim, 2023) may discourage applicants from listing all acceptable options,
thereby increasing the likelihood of WTT violations. Consistent with this pattern,
Chen and Pereyra (2019) combine survey and administrative data from Mexico
City's high school assignment system and find that 20% of students who declared
a selective school as their top choice did not list it first on their actual application.
Since fewer than 3% of these WTT-violating students listed the maximum 20 schools
allowed, list-length constraints alone cannot explain these omissions. The authors
provide suggestive evidence that applicants excluded preferred options because
they believed their chances of admission were close to zero.

In light of these findings, relying on the fact that most applicants do not list
the maximum number of allowed schools is not a strong argument in favor of
the WTT assumption. If listing additional options is perceived as costly, students
may still submit “short lists,” omitting programs they believe have low admission
probabilities, even when those programs are preferred to their outside option or to
others included in their ROL. Survey evidence supports this behavior. In Chile, for
example, Larroucau and Rios (2020) find that although 90% of college applicants list
fewer than the maximum number of allowed programs, only 43% report ranking
their true top choice. Similarly, Chrisander and Bjerre-Nielsen (2023) show that in
Denmark, 20% of surveyed college applicants agree with the statement that they
would change their ROL if they could be admitted to any program, with 60% of
these respondents omitting their most preferred option, even though only 2% submit
the maximum of eight choices.

Deviations from truth-telling have also been documented in unconstrained DA
settings, which are strategy-proof. Evidence comes from both lab experiments
(Chen and Sönmez, 2006; Li, 2017) and high-stakes matching environments such as
medical residency and university admissions (Larroucau and Rios, 2020; Hassidim
et al., 2021; Arteaga et al., 2022; Artemov et al., 2023; Shorrer and Sóvágó, 2023). For
instance, in the Israeli psychology graduate match, Hassidim et al. (2021) report that
19% of applicants submitted non-truthful ROLs by either failing to list a scholarship
position for a program or by ranking a non-scholarship position above its scholarship
counterpart, contrary to their interests. Similar patterns are observed in Australia
(Artemov et al., 2023), Hungary (Shorrer and Sóvágó, 2023), and in the U.S. National
Resident Matching Program, where Rees-Jones (2018) finds that 17% of applicants
acknowledged misrepresenting their preferences.
These widespread deviations from truthful reporting have motivated two strands of literature that seek to estimate preferences without relying on the WTT assumption: the optimal portfolio choice approach and the stability-based approach, which we discuss in turn.

### 2.2.2 The Optimal Portfolio Choice Approach

A first approach to addressing deviations from truthful reporting in DA settings is the framework proposed by Agarwal and Somaini (2018), presented in the previous section. This framework interprets each applicant's ROL as the outcome of an optimal portfolio choice over admission lotteries, conditional on preferences and beliefs about admission chances. Although originally applied to the IA mechanism, it extends to any mechanism that admits a cutoff representation, which includes DA.

**Implementing the portfolio choice approach in DA environments.** The optimal portfolio choice approach has been applied in DA settings where truth-telling may appear implausible because of list-length restrictions or application costs (e.g., Luflade, 2019; Larroucau and Rios, 2020, 2023; Idoux, 2023; Vrioni, 2024; Agte et al., 2024; Ajayi and Sidibé, 2024; Lee and Son, 2024; Wang et al., 2025). These settings are typically characterized by a large number of school options and relatively limited uncertainty about admission probabilities. For example, in New York City middle school admissions, applicants may rank up to 12 choices from among 450 programs (Idoux, 2023). In Chilean college admissions, students can apply to at most 10 out of 1,400 programs (Larroucau and Rios, 2020). In both cases, only a minority of applicants exhaust their lists (8% in NYC, 10% in Chile). However, these “short lists” cannot be readily interpreted as evidence of truthful reporting: constructing a ROL entails significant information acquisition costs, which may lead applicants to skip programs they believe are unattainable or to stop adding schools once they feel assured of admission to one of their ranked options. Such behavior weakens the plausibility of the WTT assumption.

When applying the portfolio choice approach to DA settings, researchers have developed various strategies for embedding application costs within the Agarwal and Somaini (2018) framework. Larroucau and Rios (2023) do not model application costs explicitly but assume that students include a program in their ROL only if it strictly increases their expected utility. This implies that students exclude programs with zero admission probabilities and do not rank less preferred programs below
those with guaranteed admission. Idoux (2023) specifies a linear application cost function, $C_i(R) = c_i|R|$, where $|R|$ denotes the number of programs listed on the applicant's ROL and $c_i$—the cost of adding a program—is applicant-specific and follows a truncated normal distribution with lower bound $c > 0$. Ajayi and Sidibé (2024) introduce search frictions: applicants have imperfect information about school characteristics and incur a search cost $c$ to learn about a school. In a related approach, Lee and Son (2024) model applicants as considering only a subset of schools, with consideration sets determined by a latent variable whose distribution depends on observables, some of which are excluded from the utility function.

A common computational difficulty across these implementations stems from the fact that, regardless of the specific modeling assumptions, applying the portfolio choice approach to large-scale settings with many alternatives leads to a curse of dimensionality. In what follows, we discuss methods that have been developed to address this issue.

**The curse of dimensionality: solutions.** A major challenge in applying the portfolio choice approach is that the number of possible ROLs increases exponentially with the number of available options. For instance, with just 10 options, the total number of possible ROLs approaches 9.9 million. Except in special cases, no polynomial-time algorithms or even qualitative characterizations of optimal behavior are available, making the problem computationally intractable. To address this, the literature has developed a range of solutions that either target specific admission processes or introduce behavioral assumptions about how applicants form their ROLs.

One influential approach originates with Chade and Smith (2006), who show that when applicants view admission probabilities as independent across programs and application costs depend only on the number of applications submitted, the optimal portfolio problem becomes downward recursive. Under these assumptions, it can be solved in polynomial time using the Marginal Improvement Algorithm (MIA), a greedy algorithm that constructs the optimal ROL by sequentially adding the program offering the highest marginal gain in expected utility, given the current portfolio.

Luflade (2019) implements this algorithm to construct optimal ROLs in the context of Tunisian college admissions, where applicants may rank up to ten programs among more than 600 alternatives. Applicants are assumed to form expectations about admission probabilities based on their own (known) priority score and programs' past cutoffs (i.e., the adaptive expectations model), with cutoffs assumed to be independently normally distributed. In a similar vein, Larroucau and Rios (2020)
and Larroucau and Rios (2023) apply the MIA to Chilean college admissions, where
applicants observe their priority scores before submitting ROLs and uncertainty
arises from stochastic cutoffs. They assume that students form rational expectations
about admission probabilities and take the distributions over cutoffs to be indepen-
dent across programs. Larroucau and Rios (2020) show that whenever beliefs on
admission probabilities can be estimated in a first stage and assumed to be inde-
pendent across programs, then checking optimality requires comparing a ROL to a
restricted set of “one-shot swaps,” which considerably reduces the dimensionality
of the problem.<fnref n="28" /> An alternative approach is proposed by Hernández-Chanto (2021)
for DA environments where applicants are ranked according to a known common
priority score and face list-length restrictions. Under the assumptions that prefer-
ences are independent of priority scores and that applicants’ behavior conforms to
a set of rationality axioms, he develops a concatenation algorithm to sequentially
infer students’ underlying ordinal preferences from their observed, constrained
reports.<fnref n="29" /> These recovered preferences are then used to condition the estimation of
cardinal utilities in a second step, substantially reducing the dimensionality of the
problem, as the optimality of each observed report only needs to be checked against
the subset of ROLs that are consistent with the inferred ordinal preferences.

These methods, however, break down when admission probabilities are corre-
lated across programs and applicants recognize these correlations in forming their
expectations. This typically arises when programs share a lottery or rely on priority
scores based on similar criteria that applicants observe only imperfectly. In such
settings, rejection from one program conveys information about the likelihood of
admission at others, violating the independence assumption.

Ali and Shorrer (2025) propose a solution for environments where admissions are
based on a “common score,” such as a single exam score or tie-breaking lottery, and
where students must choose their portfolio before learning their own score.<fnref n="30" /> They
show that although Chade and Smith’s MIA fails in such environments, the optimal
portfolio problem remains recursively solvable, and they develop an algorithm that

<footnote n="28"><sup>28</sup>A one-shot swap from a ROL *R* is a ROL *R*′ that differs from *R* by a single school. For example, if there are three schools {*a*, *b*, *c*}, then the ROL *R* = (*a*, *b*) has four one-shot swaps: (*a*, *c*), (*b*, *c*), (*c*, *a*), and (*c*, *b*), whereas the number of possible reports is 15.</footnote>

<footnote n="29"><sup>29</sup>The algorithm exploits variation across “tiers” of applicants, defined by how a student’s priority score compares to the maximum historical cutoff in each program (e.g., tier 1 students can access all programs, tier 2 all but the most selective, and so on). By comparing reports across tiers, the algorithm constructs a set of recoverable ordinal preferences that are consistent with the specified rationality axioms.</footnote>

<footnote n="30"><sup>30</sup>While admission cutoffs are assumed to be fixed and known to applicants in their core model, the results extend to cases where applicants know the relative ranking of cutoffs but not their precise values.</footnote>
finds the optimal portfolio in polynomial time, drawing on a recursive approach similar to that introduced by Calsamiglia et al. (2020) for the IA mechanism (see Section 2.1). Ajayi and Sidibé (2024) apply this framework to Ghana's centralized high school admissions system, where students submit ROLs before sitting for a common entrance exam and assignments are made using DA.

Outside of these special cases, general polynomial-time solutions are not available. Moreover, even in settings where the algorithms of Chade and Smith (2006) and Ali and Shorrer (2025) are applicable, solving the portfolio problem remains computationally burdensome in large markets. This has led researchers to develop models of bounded rationality to simplify the decision problem.

Using data from New York City middle school admissions, Idoux (2023) develops a model in which applicants hold rational expectations over their admission probabilities but face application costs and do not optimize over the full set of ROLs. Instead, they are assumed to rely on a sequential heuristic that ignores the full impact of each choice on continuation values. This heuristic reduces the dimensionality of the problem by introducing bounds on the indirect utilities associated with schools. Similarly, Wang et al. (2025) model applicants' construction of their ROL as a cognitively limited process, drawing on the direct cognition framework of Gabaix et al. (2006), in which "at each decision point, agents act as if their next search of operations were their last opportunity for search." In their application, applicants construct their ROL step by step, myopically choosing at each stage the program offering the highest expected utility at that point, without considering how this choice affects the value of subsequent options. This behavior tends to produce overly conservative preference submissions. Taking a different perspective, Ajayi and Sidibé (2024) and Lee and Son (2024) explore models of sequential search under imperfect information about program characteristics or admission probabilities: applicants gradually expand their consideration set and stop searching when the expected value of additional information falls below its cost. While these various approaches greatly enhance tractability and facilitate estimation, they rely on context-specific behavioral assumptions that may not generalize across settings.

### 2.2.3 The Stability-Based Approach

An alternative to the truth-telling and optimal portfolio choice approaches is the stability-based approach, which offers a different framework for estimating preferences in DA settings. Rather than modeling agents' beliefs and application behavior, this approach leverages a key equilibrium property of the matching outcome under
DA: stability, or the absence of justified envy.

In the context of school choice, stability implies that each student is matched to her most preferred *feasible* school, where feasibility is determined by whether the student's priority exceeds the school's ex-post cutoff. For example, consider a district with four schools, {a, b, c, d}, all of which rank students based on a common priority index, such as an exam score. Suppose the cutoffs are P_a = 0.2, P_b = 0.4, P_c = 0.6, and P_d = 0.8. A student with a score of 0.7 is thus eligible for admission at schools a, b, and c. If she is observed to be assigned to school b, the stability assumption implies that she prefers b over both a and c, i.e., b is her most preferred feasible option.

Although stability is guaranteed when students truthfully rank all schools, it may still hold even when they do not. This makes it a weaker, and arguably more realistic, behavioral assumption in many settings. Importantly, because it applies to the matching outcome rather than to the ranking behavior itself, the stability assumption enables identification of student preferences from observed assignments and feasible choice sets, without relying on submitted ROLs.

**Stability in strict-priority DA settings.** Early applications of the stability-based approach to preference estimation in centralized admissions include Bordon and Fu (2015) for Chilean college admissions, Burgess et al. (2015) for school choice in England, and Akyol and Krishna (2017) for Turkish high school admissions. Its theoretical foundations were subsequently formalized by Fack et al. (2019) in the context of “strict-priority” DA settings. These are environments in which schools or colleges rank students based on a priority index, such as a test score, that is known to applicants at the time of ROL submission. In such settings, Fack et al. argue that stability is a more plausible and robust assumption than WTT, particularly when list-length limits or application costs are present.

The key insight from Fack et al. (2019) is that in such markets, stability is more likely to be satisfied than truth-telling as the economy grows large. Building on theoretical results in Azevedo and Leshno (2016), they show that a unique stable matching exists in the continuum economy, and that in a sequence of finite random economies with Bayes–Nash equilibrium strategies, the resulting matchings are *asymptotically stable*—meaning the fraction of students assigned to their most preferred feasible schools converges to one almost surely as the market size increases (see Chapter 2 for details). The intuition behind this result is that aggregate uncertainty about cutoffs diminishes with market size. As uncertainty declines, students can more accurately identify which schools are feasible given their priority scores,
and they tend to secure stable assignments by ranking and being matched to their
most preferred feasible option. At the same time, when list-length restrictions or
application costs are present, reduced uncertainty may increase the incentive to omit
low-probability options, leading to WTT violations and weakening its empirical
relevance.

A fundamental distinction between truth-telling and stability-based approaches
lies in their underlying choice structure. Under the truth-telling assumption, stu-
dents choose from a universal set of schools, leading to a standard discrete choice
model. In contrast, the stability-based approach implies a model with *personalized
choice sets*, where each student is assumed to “choose” from the subset of schools
that are feasible given her priority score and the school cutoffs. A related implication
is that estimation under stability relies on observed assignments, while truth-telling
models use submitted ROLs. The trade-off, then, is between an approach that ac-
commodates non-truthful behavior but uses only final matches and feasible sets, and
one that assumes truthful reporting in order to exploit the richer data on rankings.

Growing skepticism about the plausibility of truth-telling in strict-priority DA
settings has led empirical work on centralized school choice and college admissions
to rely increasingly on the stability assumption for preference estimation. Applica-
tions include school choice in England (Burgess et al., 2015), Ghana (Ajayi, 2024),
Mexico City (Bobba et al., 2023; Ngo and Dustan, 2024; Pariguana and Ortega-Hesles,
2025), New York City (Hahm and Park, 2024), Paris (Fack et al., 2019), Sweden (An-
dersson et al., 2024), and Turkey (Akyol and Krishna, 2017); and college admissions
in Brazil (Barahona et al., 2023), Chile (Bordon and Fu, 2015; Bucarey, 2018; Kapor
et al., 2024), Denmark (Chrisander and Bjerre-Nielsen, 2023; Gandil, 2025), Germany
(Grenet et al., 2022), and Turkey (Arslan, 2021; Akyol et al., 2024). Beyond student
assignment, the stability-based approach has also been applied to preference estima-
tion in other education-related matching markets, including childcare allocation in
Japan (Kamada et al., 2025) and centralized teacher assignment in France (Combe
et al., 2022, 2025).

**Stability: robustness to mistakes.** Fack et al. (2019) provide a rationale for the stability-based approach in strict-priority settings where agents are fully rational. In their framework, deviations from truth-telling arise when the probability of admission to a school is too low to justify the application cost of including it in a student's ROL. However, this framework does not account for broader forms of non-truthful behavior, such as those documented in the literature on strategic mistakes in strategy-proof environments (see Section 2.2.1). These mistakes typically involve
the use of weakly dominated strategies, even when they result in no actual payoff losses. Examples include “skipping” schools with positive admission probabilities or “flipping” the order of schools, e.g., ranking school *a* above school *b* despite preferring *b* to *a*.

A central empirical finding in this literature is that while mistakes are common, most have minimal payoff consequences in the sense that they rarely alter matching outcomes. This is because misranked or omitted options tend to have low admission probabilities. For example, Shorrer and Sóvágó (2023) show that although 17% of college applicants in Hungary ranked a program without a scholarship above an identical program with a scholarship (which is an identifiable mistake), only 4% of these applicants would have received a different assignment had their mistake been unilaterally corrected.

Building on this empirical pattern, Artemov et al. (2023) offer a broader theoretical foundation for the stability-based approach by allowing for mistakes. They study a strategy-proof DA environment with strict, known priorities and propose a solution concept called *robust equilibrium*, which relaxes Bayesian Nash equilibrium by allowing agents to make mistakes as long as the resulting payoff losses are arbitrarily small. In this framework, applicants may adopt dominated strategies, as long as the resulting payoff loss is negligible in a large economy. While universal truth-telling clearly constitutes a robust equilibrium, it is not the only one. In fact, all but a vanishing fraction of applicants may submit untruthful ROLs, such as omitting their most preferred school or misordering schools in their ROLs, as long as doing so does not affect their admission outcomes appreciably.

Despite these potentially widespread deviations from truth-telling, Artemov et al. (2023) show that the key predictions of stable matching under DA remain valid. Under mild regularity conditions, all robust equilibria yield a virtually unique matching outcome in a sufficiently large economy. This outcome is asymptotically stable, meaning that almost all students are matched to their most preferred feasible school, and it converges to the outcome that would arise under truth-telling. In other words, even in the presence of mistakes, the matching outcome closely approximates that generated by fully rational applicants.

These results carry important implications for preference estimation in DA settings. On the one hand, they cast doubt on the validity of approaches that rely on truth-telling as an identifying assumption, showing that such methods are vulnerable to seemingly minor strategic errors. On the other hand, they strengthen the case for the stability-based approach, which relies on the observed *outcome* of DA rather than on assumptions about applicant *behavior*, and is more robust to
small, payoff-irrelevant mistakes.

Although Artemov et al. caution against relying on truth-telling for identification, they do support its use for counterfactual simulations, provided the underlying preferences have been consistently estimated. Indeed, the asymptotic stability of robust equilibria justifies an approach in which preferences are estimated using the stability-based method, and truthful reporting is assumed only at the simulation stage. Despite the presence of strategic mistakes, the resulting counterfactual assignment is well approximated by the outcome that would arise if all applicants reported truthfully. By contrast, using observed ROLs directly across regimes is not theoretically justified. If a policy reform makes previously unattainable schools feasible, applicants’ ROLs would likely change in response. Artemov et al. illustrate this point using Monte Carlo simulations: assuming fixed ROLs across regimes leads to a misprediction rate of 40%, whereas the stability-based approach with counterfactual truth-telling yields an error rate below 5%.

**Leveraging uncertainties to infer preferences.** While Artemov et al. (2023) focus on strict-priority settings, Che et al. (2023) extend the stability-based approach to strategy-proof environments where applicants face uncertainty about their priorities, and show how such uncertainty can be leveraged to infer students’ preferences. These settings include cases where lotteries are used to break ties among applicants, or where priority scores are unknown or only imperfectly known at the time applicants submit their ROLs.<fnref n="31" />

Unlike in settings with known priorities, such as those studied by Fack et al. (2019) and Artemov et al. (2023), priority uncertainty implies that admission probabilities remain uncertain even in large markets. Without knowing which schools are feasible, applicants must be more cautious to avoid costly mistakes. Yet uncertainty does not make all forms of misranking consequential. For instance, consider three schools, {a, b, c}, that admit students based on a common tie-breaking lottery. Suppose the cutoffs are always ordered such that $P_a > P_b > P_c$, where $P_j$ denotes the cutoff for school j. A student with preferences $b \succ a \succ c$ would incur no payoff loss by submitting (b, c) or (b, c, a) instead of the fully truthful ranking (b, a, c), since a is irrelevant regardless of her lottery draw: whenever a is feasible, b is also feasible—and preferred. Such “mistakes” are payoff-irrelevant. Thus, truth-telling

<footnote n="31">³¹Examples include admissions systems in which applicants are ranked based on an entrance exam taken after submitting their ROL, such as New York City’s Specialized High Schools (Corcoran and Levin, 2011) and high school choice in Ghana (Ajayi and Sidibé, 2024), or where exam results are disclosed only after ROL submission, as in Chinese college admissions (Chen and Kesten, 2017).</footnote>
cannot be taken for granted even in uncertain environments, as agents may still
make harmless deviations.

At the same time, uncertainty renders some mistakes costly, thereby encouraging
more accurate preference revelation. In the same example, reversing the order of
b and c would be a payoff-relevant mistake, since there is a positive probability of
encountering a lottery draw where either all three schools are feasible or only b
and c are.<fnref n="32" /> In the latter case—when the draw falls between P_b and P_a—ranking c
above b would result in an avoidable welfare loss. To avoid such losses, the student
must carefully rank b ahead of c. Che et al. show how this logic can be used to infer
preferences from submitted ROLs. In the example, one can conclude that b ≻ a and
b ≻ c by carefully examining the structure of uncertainty the student faces.

This approach can reveal more information than what is obtainable by invoking
stability in environments without priority uncertainty, such as in Fack et al. (2019).
For example, suppose that schools rank applicants based on GPA, and a student
“knows” her GPA is above the cutoff for school c but below that of b. In this case,
c is her only feasible option. Whether she submits (b,c) or (b,c,a), she will be
assigned to c, and her ROL provides no information about preferences under the
stability assumption alone. By contrast, the methods developed by Che et al. (2023)
exploit the structure of uncertainty to recover as much information as possible on
participants’ preferences in DA environments. These methods are discussed in more
detail in Section 2.3.4.

2.2.4 Undominated Strategies

A drawback of the stability-based approach is that it relies only on comparisons
between ex-post feasible options, without exploiting the full content of the submitted
ROL. This leads in a loss of information when estimating preferences. A natural
question, then, is whether additional identifying restrictions can be derived from
ROL data in DA settings.

As noted above, Che et al. (2023) show that in strategy-proof DA environments
with priority uncertainty, ROLs can be used to infer preferences consistent with
robust equilibrium play. However, this approach is not readily applicable to DA
settings where strategy-proofness fails, such as when students face list-length
constraints or application costs.

In such cases, Fack et al. (2019) show that researchers can still extract useful

<footnote n="32"><sup>32</sup>The first case occurs when the student's lottery number is above P<sub>a</sub>, making all schools feasible; the second arises when the number lies between P<sub>b</sub> and P<sub>a</sub>, rendering only b and c feasible.</footnote>
identifying restrictions if they are willing to assume that applicants do not play weakly dominated strategies. Indeed, Haeringer and Klijn (2009, Proposition 4.2) demonstrate that under DA, even when list-length constraints or application costs are present, submitting a ROL in which schools are not ranked in true preference order is weakly dominated by submitting a ROL that truthfully ranks the same schools. For example, if a student's preferences are $a \succ b \succ c \succ d$, then submitting $(b,a,c)$ is weakly dominated by $(a,b,c)$, since ranking $b$ above $a$ can only reduce the chance of being admitted to the more preferred school without improving the chances at $b$.

The undominated-strategies assumption provides over-identifying restrictions that can complement those derived from stability.<fnref n="33" /> The benefits of combining the two assumptions can be illustrated with a stylized example. Consider a DA setting with strict priorities, where students may rank up to three schools out of four. Under stability, preferences over $a$ and $b$ are primarily estimated using students who have both schools in their feasible sets. But if all students rank both schools, the undominated-strategies assumption allows the use of the full sample of students, including those for whom $a$ or $b$ was not feasible ex-post. Section 2.3.5 details how the moment inequalities derived from undominated strategies can be combined with the moment equalities derived from stability to sharpen identification.

Undominated strategies can also be used on their own to recover preference information in non-strategy-proof DA environments, akin to the approaches developed by Hwang (2015), He (2017), and Bayraktar and Hwang (2024) in IA settings. For example, consider a DA market where assignments are based purely on lotteries and applicants face severe list-length limits. In such contexts, neither truth-telling nor stability is likely to hold, and researchers may be reluctant to impose the behavioral restrictions embedded in optimal portfolio models. Yet the undominated-strategies assumption yields inequality restrictions that support partial information of preferences. However, as the econometric structure is incomplete (Tamer, 2003, 2010), the assumption does not pin down a unique ROL given utilities $u_i$ and priorities $t_i$. Instead, it generates a set of inequalities consistent with a range of preference parameters. Section 2.3.5 explains how these inequalities can be used to achieve partial identification of student preferences. Still, the power of this approach may be limited when submitted ROLs are short relative to the total number of available options, in which case the identified bounds can be too wide to be informative.

<footnote n="33">³³Note that the undominated-strategies assumption does not add new identifying power in approaches based on truth-telling or optimal portfolio choice in DA settings, as these frameworks already rule out the submission of ROLs that are not true partial orders of applicants' preferences.</footnote>
Finally, it is important to note that undominated strategies do not nest the stability assumption. While they involve weaker behavioral assumptions than truth-telling or optimal portfolio choice, they are stronger than stability in some dimensions. In particular, they rule out certain types of payoff-irrelevant mistakes— such as arbitrarily ranking schools with negligible admission probabilities—that stability-based methods allow. The undominated strategies assumption is therefore best suited to settings without “safety” or “impossible” schools, where arbitrary rankings are less likely.

## 2.3 Identification and Estimation of Preferences

This section describes how the approaches discussed in the previous sections can be implemented to estimate student preferences in both strategic and non-strategic assignment mechanisms. We start by introducing the random utility framework for representing preferences and the conditions required for identification.

### 2.3.1 Specifying Student Preferences

**Random utility model.** Let $i \in \mathcal{I}$ index applicants and $j \in \mathcal{J}$ index schools or programs. A general specification of applicant $i$'s indirect utility from being assigned to school $j$ is given by:

$$u_{ij} = V(z_{ij}, \xi_j, \epsilon_i), \qquad (2)$$

where $z_{ij}$ is a vector of observable characteristics that may vary by student, school, or both; $\xi_j$ captures school-level attributes observed by students but unobserved by the econometrician; and $\epsilon_i$ denotes student-specific unobserved heterogeneity in preferences.

In school choice settings, $z_{ij}$ typically includes student-level characteristics (e.g., academic achievement, socioeconomic background), school-level attributes (e.g., average test score), and student-school-specific variables (e.g., commuting distance). The vector $\xi_j$ reflects school-level characteristics that are observed by students but unobserved by the econometrician, such as disciplinary climate, teacher quality, or school culture. The vector $\epsilon_i$ captures individual heterogeneity in how parents or students value both observed and unobserved school attributes.

Although fairly general, the specification in (2) embeds two simplifying assumptions. First, utility is assumed to depend only on the applicant's own assignment, ruling out preferences over peers at the individual level. However, preferences
for school-wide peer characteristics (e.g., racial or socioeconomic mix) can be accommodated by including lagged aggregates in $z_{ij}$. Second, the model assumes that preferences are fixed and known to students ex ante. Yet empirical evidence suggests that this assumption is unlikely to hold in contexts where participants face a large set of alternatives and must therefore learn their preferences through a costly discovery process (e.g., Narita 2018, Grenet et al. 2022). To address this limitation, an increasing number of studies adopt models that allow applicants' preferences to evolve over time through dynamic learning (e.g., Bordon and Fu, 2015; Narita, 2018; Grenet et al., 2022; Larroucau and Rios, 2023; Agte et al., 2024; Hahm and Park, 2024; Kapor, 2024; Arcidiacono et al., 2025; De Groote et al., 2025).

**Non-parametric identification of preferences.** Agarwal and Somaini (2018) provide general identification results for the random utility model described in (2). The goal is to identify and estimate the joint distribution of the utility vector $\mathbf{u}_i = (u_{i1}, \dots, u_{iJ})$ conditional on observed characteristics $\mathbf{z}_i = (\mathbf{z}_{i1}, \dots, \mathbf{z}_{iJ})$ and school-specific unobservables $\xi = (\xi_1, \dots, \xi_J)$, denoted by the conditional density $f_U(u_{i1}, \dots, u_{iJ} | \mathbf{z}_i, \xi)$.

The empirical approaches outlined in the previous sections yield revealed preference relations, derived either from submitted ROLs (as in the portfolio choice and truth-telling approaches) or from observed assignments (as in the stability-based approach). Under complete models of behavior, a given utility vector $\mathbf{u}_i$ maps to a unique ROL or assignment, allowing the likelihood of the observed outcome to be written as a function of the underlying distribution of preferences. For instance, under the portfolio choice approach, the probability that student $i$ submits her observed ROL $R_i$ is:

$$ \mathbb{P}(i \text{ submits } R_i | \mathbf{z}_i, \xi) = \mathbb{P}(R_i = \arg \max_{R \in \mathcal{R}_i} \mathbf{u}_i \cdot \mathbf{L}_{R,i} | \mathbf{z}_i, \xi; f_U), $$

where $\mathbf{L}_{R,i}$ denotes student $i$'s vector of (perceived) assignment probabilities under ROL $R$ and $\mathcal{R}_i$ is the set of possible ROLs that student $i$ can submit. Submitting $R_i$ reveals that her utility vector $\mathbf{u}_i$ lies within the convex set $C_{R_i}$:

$$ C_{R_i} = \{ \mathbf{u}_i \in \mathbb{R}^J : \mathbf{u}_i \cdot (\mathbf{L}_{R_i,i} - \mathbf{L}_{R,i}) \ge 0 \text{ for all } \mathbf{L}_{R,i} \in \mathcal{L}_i \}, \quad (3) $$

where $\mathcal{L}_i$ denotes the set of all assignment lotteries faced by $i$. The subset $C_{R_i}$ is defined by linear inequalities on $\mathbf{u}_i$ and characterizes the set of utilities for which $R_i$ is optimal given the applicant's beliefs. Analogous restrictions on $\mathbf{u}_i$ arise under the
truth-telling and stability approaches, as discussed in Sections 2.3.3 and 2.3.4.

Agarwal and Somaini (2018) analyze the conditions required for non-parametric identification of the joint distribution of preferences $f_U$ in mechanisms that admit a cutoff representation (RSP+C). They show that point identification can be achieved by "tracing out" the distribution of utilities using two types of exogenous variation: (i) variation in the choice environment (e.g., different priorities or cutoffs faced by otherwise similar applicants), and (ii) variation generated by a preference shifter, i.e., a student-school-specific covariate $z_{ij}^0$ that enters utility additively and is orthogonal to unobserved heterogeneity.

The first source of variation applies, for example, when components of applicant priority (such as sibling or walk-zone priority) are plausibly exogenous to preferences, or when the data span multiple years with fixed school sets but varying capacities or cutoffs.<fnref n="34" /> However, Agarwal et al. note that such variation is often limited in practice, particularly in settings with few priority types or few cohorts, making full non-parametric identification difficult.

Alternatively, preferences can be non-parametrically identified if the covariates $z_{ij}$ include a “special regressor,” as defined in the econometrics literature (see Lewbel, 2014, for a review). In the context of school choice, such a variable must vary at the student-school level, enter additively in the utility function, and be orthogonal to unobserved preference heterogeneity ($\epsilon_i$). A commonly used special regressor in empirical applications is the distance between a student and a school. When this assumption is satisfied, (2) can be rewritten as:

$$u_{ij} = V(\tilde{z}_{ij}, \xi_j, \epsilon_i) - d_{ij}, \qquad (4)$$

where $d_{ij}$ is the distance from student $i$ to school $j$, and $\tilde{z}_{ij} := z_{i,j} \setminus d_{ij}$. The critical assumption is that distance is orthogonal to unobserved preference heterogeneity. This may be violated if families sort into neighborhoods based on unobserved school preferences (see Park and Hahm, 2023). The credibility of this assumption is thus context-specific and becomes more plausible when the data include rich controls to adjust for potential confounders.

Agarwal and Somaini (2018) show that when the revealed preference sets (like $C_{R_i}$) are defined by systems of linear inequalities, variation in $d_{ij}$ non-parametrically identifies $f_U$.<fnref n="35" />

<footnote n="34">³⁴See Carvalho et al. (2019) for a discussion of how exogenous variation in the choice environment can be used to non-parametrically identify preferences in a college admissions context.</footnote>

<footnote n="35">³⁵See Agarwal and Somaini (2020, Section 3.2), for a graphical illustration of the identification argument.</footnote>
Note that preference distributions in random utility models are only identified up to an affine transformation, requiring both location and scale normalization. Location is typically normalized by setting the utility of a reference alternative—often the outside option (denoted school 0)—to zero, i.e. $u_{i0} = 0$, anchoring all utility comparisons to that baseline. If the model does not include an outside option, the mean utility of an arbitrary school may be normalized to zero instead. For scale, identification is generally achieved by normalizing the variance of $\epsilon_i$ or by fixing the coefficient of one observed variable. In (4), the scale normalization is ensured by setting the coefficient on distance to -1, allowing preferences to be interpreted in terms of students' willingness to travel.<fnref n="36" />

**Common parametrizations.** Although the identification results in Agarwal and Somaini (2018) do not rely on parametric assumptions about the random utility function $V(\cdot)$, such assumptions are typically necessary in empirical applications to ensure computational tractability.

*Multinomial logit.* A widely used parametrization in the literature (e.g., Burgess et al., 2015; Fack et al., 2019; Nguyen, 2021; Oosterbeek et al., 2021; Grenet et al., 2022; Ngo and Dustan, 2024) is to specify preferences using a multinomial logit model of the form:

$$u_{ij} = V(\mathbf{x}_j, \mathbf{w}_{ij}, \xi_j, \alpha, \beta) = \underbrace{\mathbf{x}_j'\alpha + \xi_j + \mathbf{w}_{ij}'\beta}_{\delta_j} + \epsilon_{ij}, \quad (5)$$

where $u_{i0} = \epsilon_{i0}$ and $\epsilon_{ij}$ follows an extreme-value type I (EVT1) or Gumbel distribution with location parameter 0 and scale parameter 1.<fnref n="37" />

In this formulation, $\mathbf{x}_j$ includes observed school-specific characteristics, while $\mathbf{w}_{ij}$ captures observed characteristics that vary across both students and schools, potentially through interactions between student and school attributes. Unobserved school-specific characteristics are summarized by a single dimension, $\xi_j$, which reflects the overall influence of unobserved attributes that make a school more or

<footnote n="36">³⁶While this willingness-to-travel representation aids interpretation, it does not allow for interpersonal utility comparisons. Because distance is not a transferable unit of measurement, utilitarian welfare metrics based on the Kaldor-Hicks criterion (such as total or average utility) cannot be justified. Still, ordinal comparisons (e.g., the share of students who prefer one assignment over another) remain valid. See Agarwal and Somaini (2020) for further discussion.</footnote>

<footnote n="37">³⁷When the model includes distance to school, the scale normalization can instead be achieved by fixing the coefficient on distance to -1, as in (4). In this case, the model parameters are interpreted in terms of willingness-to-travel, and the scale of the EVT1 distribution becomes a parameter $\sigma$ to be estimated.</footnote>
less desirable to all applicants. Observed and unobserved school characteristics are often grouped into a single school fixed effect $\delta_j := x'_j\alpha + \xi_j$, which flexibly captures all systematic variation in average school desirability.<fnref n="38" />

A major advantage of the logit model is computational: as discussed in Sections 2.3.3 and 2.3.4, it yields closed-form expressions for the likelihood of observed ROLs or assignments under both the truth-telling and stability-based approaches, enabling estimation via maximum likelihood. However, a key limitation is that it assumes fixed coefficients $\alpha$ and $\beta$ across individuals, preventing unobserved heterogeneity in how students value observed school characteristics. This leads to restrictive substitution patterns due to the independence of irrelevant alternatives (IIA) property.<fnref n="39" />

To introduce greater flexibility while preserving tractability, a growing number of studies (e.g., Abdulkadiroğlu et al., 2020b; Barahona et al., 2023; Campos and Kearns, 2024; Corradini, 2024; Laverde, 2024; Campos et al., 2025) estimate (5) separately across subsamples defined by combinations of observable student characteristics such as gender, race/ethnicity, or academic performance. This stratified specification allows model parameters to vary freely across subgroups, thereby capturing observed heterogeneity in preferences.

*Multinomial mixed logit.* To allow for unobserved heterogeneity in how students value observed characteristics, the multinomial logit model can be extended by introducing applicant-specific random coefficients $\beta_i$ on $w_{ij}$, resulting in a mixed logit specification:

$$u_{ij} = \delta_j + w'_{ij}\beta_i + \epsilon_{ij}, \quad (6)$$

where $\beta_i$ is a vector of individual-specific taste coefficients drawn from a multivariate normal distribution, i.e. $\beta_i \sim N(\beta, \Sigma_\beta)$, with $\beta$ and $\Sigma_\beta$ as parameters to be estimated, and $\epsilon_{ij}$ follows an EVT1 distribution.

This model introduces flexible substitution patterns and richer heterogeneity in preferences, but at the cost of losing closed-form expressions for choice probabilities. As a result, estimation requires multi-dimensional numerical integration. Common simulation-based methods include maximum simulated likelihood, the method of simulated moments, and simulated scores (see, e.g., Hastings et al., 2009; Luflade,

<footnote n="38">³⁸Without such fixed effects, $\xi_j$ would be absorbed into the error term, potentially biasing estimates of $\alpha$ and $\beta$ when $x_j$ and $w_{ij}$ are correlated with $\xi_j$.</footnote>

<footnote n="39">³⁹For example, IIA implies that improving a positively valued attribute of one school reduces the choice probabilities of all other schools proportionally—a substitution pattern known as “proportionate shifting” (see Train, 2009).</footnote>
2019; Larroucau and Rios, 2023). These methods can be computationally demanding, especially as the dimension of $\beta_i$ increases.<fnref n="40" />

*Multinomial probit.* To circumvent the computational burden associated with evaluating integrals in mixed logit models, an increasingly popular alternative is to specify student preferences using a multinomial probit model estimated via Bayesian methods (e.g., Abdulkadiroğlu et al., 2017a; Agarwal and Somaini, 2018; Kapor et al., 2020; Larroucau and Rios, 2020; Che et al., 2023; Agte et al., 2024; Campos and Kearns, 2024; Kapor et al., 2024). In this setup, both the random coefficients and the idiosyncratic preferences in (6) are assumed to follow normal distributions, i.e., $\beta_i \sim N(\beta, \Sigma_\beta)$ and $\epsilon_{ij} \sim N(0, \sigma_\epsilon^2)$.

Probit models offer two main advantages. First, under both the truth-telling and stability assumptions, Bayesian methods provide a tractable framework for estimating random-coefficient models without requiring the simulation of choice probabilities, a feature that facilitates preference estimation in large-scale markets such as the Chilean college admissions system analyzed by Kapor et al. (2024). Second, the probit specification is particularly well-suited for the portfolio choice approach in strategic settings, where even logit models without random coefficients yield likelihood functions that lack closed-form expressions. It is also recommended for stability-based inference in DA environments with priority uncertainty (Che et al., 2023), where the likelihood is typically non-analytic (see Section 2.3.4).

The next section illustrates how Bayesian methods can be implemented to estimate probit-based models under the portfolio choice approach.

### 2.3.2 Estimation under the Portfolio Choice Approach

In the portfolio choice approach, each applicant is assumed to submit the ROL $R_i$ that maximizes her expected utility, given her beliefs about the assignment probabilities associated with each possible report:

$$ \mathbb{P}(R_i = \arg \max_{R \in \mathcal{R}} \mathbf{u}_i \cdot \mathbf{L}_{R,i} | \mathbf{z}_i; \boldsymbol{\theta}), \quad (7) $$

where $\boldsymbol{\theta}$ denotes the preference parameters to be estimated, and $\mathbf{L}_{R,i}$ is the vector of student $i$'s perceived assignment probabilities across schools when she submits ROL $R$.

<footnote n="40">⁴⁰This is particularly true for the method of maximum simulated likelihood, which may exhibit substantial asymptotic bias if the number of simulation draws per observation does not grow fast enough relative to the sample size (Lee, 1995).</footnote>
Agarwal and Somaini (2018) propose a two-step estimator for $\theta$. In the first step, assignment probabilities $L_R$ are estimated for each student-ROL pair. In the second step, the preference parameters $\theta$ are estimated, treating the first-stage estimates $\hat{L}_R$ as known.<fnref n="41" />

**Step 1: Estimating assignment probabilities.** Under the rational expectations assumption, applicants have correct beliefs about the probability of assignment to each school, conditional on their submitted ROL. This implies that each component $L_{R,i}^{(j)}$ of the assignment probability vector $L_{R,i}$ corresponds to the probability that student $i$ is matched to school $j$, given she submits ROL $R$.

In mechanisms that admit a cutoff representation (RSP+C), these probabilities depend on the student's effective priority at each school and the school's admission cutoff. The student's effective priority can be summarized by an ex-post eligibility score, denoted by $e_{ij}$, which aggregates the relevant inputs of the assignment mechanism. Under DA, $e_{ij}$ can be constructed as a lexicographic function of the student's priority $t_{ij}$ and, in case of coarse priorities, a random tie-breaker $\tau_{ij}$. Under IA, the eligibility score $e_{ij}$ also depends on the position of school $j$ in the student's ROL, making it endogenous to the submitted ranking. The admission cutoff at school $j$, denoted by $p_j$, is defined as the lowest eligibility score among students assigned to the school when its capacity is exhausted, and as the lowest possible score otherwise.

Given these definitions, the probability that student $i$ is matched to $j$ conditional on submitting ROL $R$ is given by:

$$L_{R,i}^{(j)} = \mathbb{P} (e_{ij} \ge p_j \text{ and } e_{ij'} < p_{j'} \text{ if } j' \text{ is ranked above } j), \quad (8)$$

that is, the probability that student $i$'s eligibility score exceeds the cutoff at school $j$ and falls below the cutoffs at all schools she ranks higher.

From the applicant's perspective, assignment uncertainty stems from two sources: (i) the ROLs submitted by other applicants, and (ii) random tie-breakers, when priorities are coarse. To approximate both, Agarwal and Somaini (2018) propose a resampling procedure. This involves repeatedly drawing (with replacement) samples of students, their submitted ROLs, and lottery draws when relevant. The assignment mechanism is then simulated on each resampled dataset to compute market-clearing cutoffs. The empirical distribution of these cutoffs is used to estimate $\hat{L}_R$ for each

<footnote n="41">⁴¹Agarwal and Somaini (2018) note that while joint estimation of $L_R$ and $\theta$ is theoretically feasible, the two-step approach is more computationally tractable, though it may entail some loss in efficiency.</footnote>
student and ROL, by averaging over simulations—across both cutoff realizations and, in the presence of lotteries, multiple tie-breaker draws. Agarwal and Somaini (2018) show that this estimator is consistent and asymptotically normal.

This procedure can be extended to accommodate alternative belief formation assumptions, such as the adaptive or coarse expectations models discussed in Agarwal and Somaini (2018). It is also compatible with models allowing for heterogeneous degrees of sophistication in application behavior (e.g., He, 2017; Agarwal and Somaini, 2018; Calsamiglia et al., 2020). Alternatively, perceived assignment probabilities $\hat{L}_R$ can be constructed using data on applicants’ stated beliefs, when available from surveys (Kapor et al., 2020).

**Step 2: Estimating preference parameters.** Given the estimated assignment probabilities $\hat{L}_R$ from Step 1, the second step estimates preference parameters using (7) and (8). The corresponding log-likelihood function is:

$$ \ln \mathcal{L}_{\text{Portfolio}}(\theta | \mathbf{z}) = \sum_{i=1}^{I} \ln \mathbb{P}(R_i = \arg \max_{R \in \mathcal{R}} \mathbf{u}_i \cdot \hat{\mathbf{L}}_{R,i} | \mathbf{z}_i; \theta). \quad (9) $$

Because this likelihood lacks a closed-form expression, estimation of $\theta$ typically relies on simulation-based methods. Common approaches include maximum simulated likelihood (e.g., Calsamiglia et al., 2020; Vrioni, 2024) and the method of simulated moments (e.g., Ajayi and Sidibé, 2024). An alternative, increasingly used in recent work, is to adopt a probit specification and estimate $\theta$ via Bayesian methods (e.g., Agarwal and Somaini, 2018; Kapor et al., 2020; Larroucau and Rios, 2020; Hernández-Chanto, 2021; Idoux, 2023; Agte et al., 2024; Campos and Kearns, 2024; Xu and Hammond, 2024).

As outlined in Section 2.3.1, the probit model assumes that both the idiosyncratic taste parameters $\beta_i$ and the error terms $\epsilon_{ij}$ follow normal distributions: $\beta_i \sim N(\beta, \Sigma_\beta)$ and $\epsilon_{ij} \sim N(0, \sigma_\epsilon^2)$. Abdulkadiroğlu et al. (2017a) and Agarwal and Somaini (2018) adapt the Gibbs sampling procedure of McCulloch and Rossi (1994) and Rossi et al. (1996) to estimate such models in the school choice context. The algorithm specifies conjugate priors for $\theta := (\delta, \beta, \Sigma_\beta, \sigma_\epsilon^2)$ and proceeds iteratively, starting from initial values $\theta_0$ and a set of utilities consistent with the observed ROLs. Each iteration consists of two steps: (i) drawing each student’s latent utility vector $\mathbf{u}_i$ from its posterior distribution, conditional on the parameters and the revealed preference inequalities from (3); and (ii) drawing the parameters $\theta$ from their posterior distribution, conditional on the sampled utilities. After discarding an
initial set of “burn-in” draws, posterior means (as point estimates) and standard
deviations (as standard errors) of the parameters can be obtained by sampling from
their posterior distributions.<fnref n="42" /> By the Bernstein-von Mises theorem (Van der Vaart,
2000, Section 10.2), these posterior means converge asymptotically to the same
distribution as maximum likelihood estimators.<fnref n="43" />

The key advantage of the Gibbs sampler is that it sidesteps the need to compute
the choice probabilities in (9) directly. Instead, it generates samples that approximate
the posterior distribution of the parameters of interest. In school choice applications,
the crucial insight behind step (i) is that conditional draws of $u_{ij}$—given current
values for $\mathbf{u}_{i,-j}$ and $\theta$—are obtained from a (possibly two-sided) truncated normal
distribution. This follows from the joint normality assumption and the fact that the
revealed preference conditions impose the constraint $\mathbf{u}_i \cdot (\mathbf{L}_{R_i,i} - \mathbf{L}_{R,i}) \ge 0$ for all
$R \in \mathcal{R}_i$, which bounds the support of each $u_{ij}$ conditional on the others.

The main computational challenge in implementing the portfolio choice ap-
proach lies in the exponential growth of the set of ROLs that must be considered for
each student as the number of options increases. Whether using simulation-based
or Bayesian methods, the utility vector of each student must be evaluated against
all possible alternative ROLs, a task that becomes computationally prohibitive in
large markets. While this challenge is manageable in small-scale settings—such as
those studied by Agarwal and Somaini (2018) and Kapor et al. (2020)—applications
to more complex environments often require simplifying assumptions or approxi-
mations. Several strategies have been proposed to address this dimensionality issue
(see Section 2.2.2). These include assuming that applicants know their priorities
and perceive admission probabilities as independent across schools or programs
(Luflade, 2019; Larroucau and Rios, 2020, 2023; Vrioni, 2024); that applicants face
limited uncertainty over admission cutoffs when priorities are based on a “common
score,” such as a single tie-breaking lottery (Calsamiglia et al., 2020) or exam score
(Ajayi and Sidibé, 2024); or that applicants rely on decision heuristics to simplify
the application process due to bounded rationality (Idoux, 2023; Lee and Son, 2024;
Wang et al., 2025). The validity of these modeling choices should be carefully
assessed in light of the institutional and informational context of each study.

<footnote n="42">⁴²Priors for $\delta$ and $\beta$ are typically specified as Gaussian, while those for $\Sigma_{\beta}$ and $\sigma_{\epsilon}^2$ are generally chosen to follow inverse-Wishart distributions.</footnote>

<footnote n="43">⁴³Since the assignment probabilities $L_R$ are estimated in a first step, standard errors must account for the resulting estimation error. Agarwal and Somaini (2018) address this using a bootstrap procedure described in Appendix E.2 of their paper.</footnote>
### 2.3.3 Estimation under Truth-telling

Empirical studies in DA environments have commonly relied on the weak truth-telling assumption to recover student preferences. WTT can be viewed as a truncated version of STT, and rests on two assumptions: (a) the length of each submitted ROL is exogenous to student preferences, and (b) students rank their most preferred schools truthfully while omitting less-preferred options beyond a certain point.

As an illustration, suppose a student faces four schools, {$a, b, c, d$}, and submits the ROL $R_i = (b, a)$. Under WTT, the student is assumed to have preferences $b \succ a \succ c, d$, while the choice to rank only two schools is treated as independent of her preferences over the full set of options.

To fix ideas, suppose that student preferences can be represented using a multinomial logit model without an outside option, specified as:

$$u_{ij} = V_{ij} + \epsilon_{ij} = V(\mathbf{z}_{ij}, \boldsymbol{\theta}) + \epsilon_{ij},$$

where $V(\cdot, \cdot)$ is a known function of $\mathbf{z}_{ij}$, a vector of observable student-school characteristics, and $\boldsymbol{\theta}$, a vector of parameters to be estimated. The idiosyncratic shocks $\epsilon_{ij}$ are assumed to be i.i.d. across students and schools, and to follow an EVT1 distribution with location parameter 0 and scale parameter 1.

Let $\sigma_i(\mathbf{u}_i, \mathbf{t}_i)$ denote student $i$'s strategy, mapping her utility vector $\mathbf{u}_i$ and priority type $\mathbf{t}_i$ into a submitted ROL. The observed ROL is written as $R_i = (r^1, \dots, r^{|R_i|})$, where $r^k$ denotes the school ranked in position $k$, and $|R_i|$ is the number of schools ranked.

Under the WTT assumption, the probability that student $i$ submits ROL $R_i$ can be decomposed as:

$$\mathbb{P}(i \text{ submits } R_i \mid \mathbf{z}_i; \boldsymbol{\theta}) = \mathbb{P}(\sigma_i = R_i \mid \mathbf{z}_i; \boldsymbol{\theta}; |\sigma_i| = |R_i|) \times \mathbb{P}(|\sigma_i| = |R_i| \mid \mathbf{z}_i; \boldsymbol{\theta}),$$

where the second term—the probability that the student submits a list of length $|R_i|$—is treated as exogenous and unrelated to the student's preferences $u_{ij}$ for all $j$. Estimation therefore focuses on the first term. When the preferences shocks $\epsilon_{ij}$ follow an EVT1 distribution, this conditional probability admits a closed-form expression:

$$\begin{aligned} \mathbb{P}(\sigma_i = R_i | \mathbf{z}_i; \boldsymbol{\theta}; |\sigma_i| = |R_i|) &= \mathbb{P} \left( u_{i,r^1} > \dots > u_{i,r^{|R_i|}} > u_{i,j'} \quad \forall j' \in \mathcal{J} \setminus R_i \mid \mathbf{z}_i; \boldsymbol{\theta}; |\sigma_i| = |R_i| \right) \\ &= \prod_{j \in R_i} \frac{\exp(V_{ij})}{\sum_{j' \neq R_i} \exp(V_{ij'})} \end{aligned}$$
where $j' \not\in R_i$ $j$ denotes schools that are not ranked above $j$ in $R_i$, including $j$ itself and all unranked schools. This formulation corresponds to the rank-ordered (or “exploded”) logit, which can be interpreted as a sequence of conditional logits: the first for the top-ranked school being preferred to all others, the second for the next-ranked school being preferred to all remaining options, and so on.<fnref n="44" />

With a location normalization (e.g., setting $V_{i,1} = 0$ for all $i$), the model can be estimated via maximum likelihood using the following log-likelihood function:<fnref n="45" />

$$ \ln \mathcal{L}_{\text{WTT}} (\boldsymbol{\theta} \mid \mathbf{z}, \{ |R_i| \}_{i=1,...,I}) = \sum_{i=1}^{I} \sum_{j \in R_i} V_{ij} - \sum_{i=1}^{I} \sum_{j \in R_i} \ln \left( \sum_{j' \neq j} \exp(V_{ij'}) \right), $$

where the first term sums the deterministic utilities of ranked schools, and the second accounts for the normalizing denominators in the rank-ordered logit. The resulting estimator is denoted by $\hat{\theta}_{\text{WTT}}$.

The truth-telling framework can be extended to accommodate unobserved tastes for school characteristics by introducing random coefficients, using either a mixed logit or a probit specification as in (6). Although the likelihood function no longer admits a closed-form expression under these more flexible models, preference parameters can still be estimated using simulation-based methods (e.g., Hastings et al., 2009) or Bayesian approaches such as Gibbs sampling (e.g., Abdulkadiroğlu et al., 2017a; Pathak and Shi, 2021). Compared to the portfolio choice approach, the computational burden of estimating random-coefficient models under WTT is substantially lower. This is because estimation under WTT relies solely on the internal rank-ordering within each submitted list and assumes that unranked schools are strictly less preferred than the lowest-ranked option, thereby avoiding the need to evaluate expected utility across all possible ROLs.

$$ \mathbb{P}(\sigma_i = (b, a) \mid \mathbf{z}_i; \boldsymbol{\theta}; |\sigma_i| = 2) = \frac{\exp(V_{ib})}{\sum_{j=a,b,c,d} \exp(V_{ij})} \frac{\exp(V_{ia})}{\sum_{j=a,c,d} \exp(V_{ij})}, $$

i.e., the product of two terms: the probability that the student prefers school $b$ to all others, and the probability that she prefers school $a$ to the remaining unranked schools ($c$ and $d$).

<footnote n="44">⁴⁴In the earlier example with four schools, the WTT assumption implies that the probability that student $i$ submits $R_i = (b, a)$ is:</footnote>

<footnote n="45">⁴⁵The rank-ordered logit model can be implemented using standard software tools such as the Rologit command in Stata or the ROlogit package in R.</footnote>
### 2.3.4 Estimation under Stability

Under the assumption that the matching outcome is stable, each student is matched to her most preferred school among those that are feasible ex-post. This section outlines how the stability-based approach can be used to estimate preferences in DA environments. We begin with settings in which students are ranked by strict, known priority scores, and then extend the analysis to cases where applicants face uncertainty about their priorities.

**Strict priorities.** In settings with strict priorities, stable matching can be formulated as the outcome of a discrete choice model with personalized choice sets.

Let $\mu$ denote the matching and $\mathbf{P}(\mu)$ the associated vector of admission cutoffs, which are random variables determined by the unobserved utility shocks $\epsilon$. A school $j$ is ex-post feasible to student $i$ if $i$'s priority score $t_{i,j}$ at that school meets or exceeds the cutoff, i.e., $t_{i,j} \ge P_j$. Denote by $S(\mathbf{t}_i, \mathbf{P})$ the set of schools that are feasible for student $i$ under the realized cutoffs.

For example, consider a district with four schools, $\{a, b, c, d\}$, admission cutoffs $P_a = 0, P_b = 0.4, P_c = 0.6$, and $P_d = 0.8$, and a student with priority scores $t_{i,a} = 0.5, t_{i,b} = 0.2, t_{i,c} = 0.8$, and $t_{i,d} = 0.7$. The student's feasible set is then $S(\mathbf{t}_i, \mathbf{P}) = \{a, c\}$.

Under the stability assumption, the observed match $j^* = \mu(i)$ corresponds to the utility-maximizing option within the feasible set. That is, the probability that student $i$ is matched to school $j^*$ is given by:

$$ \mathbb{P} \left( j^* = \mu(i) = \arg \max_{j \in S(\mathbf{t}_i, \mathbf{P})} u_{ij} \mid \mathbf{z}_i, \mathbf{t}_i, S(\mathbf{t}_i, \mathbf{P}); \theta \right). $$

This characterization transforms the assignment problem into a discrete choice model, where each student “chooses” a school from her personalized feasible set. Standard methods from the discrete choice literature can then be applied to estimate preferences.

The identification of preferences in this framework relies on two exogeneity assumptions:

* EXO 1 (Exogeneity of priority scores). $\mathbf{t}_i \perp \epsilon_i \mid \mathbf{z}_i$: Conditional on observed covariates, a student's priority scores are independent of her unobserved preference shocks.

* EXO 2 (Exogeneity of feasible set). $S(\mathbf{t}_i, \mathbf{P}) \perp \epsilon_i \mid \mathbf{z}_i$: Conditional on $\mathbf{z}_i$, a student's set of feasible schools is independent of her unobserved preferences.
Assumption EXO1 is necessary because preferences are only identified over the schools that fall within a student's personalized choice set. In particular, the preferences of students with limited access—due to low priority scores—must be inferred from those of students with broader access, i.e., higher priority scores. This assumption may fail if, conditional on observables, priority scores are correlated with unobserved determinants of preference such as ability. For instance, when priorities are based on test scores, the stability assumption does not reveal information about low-scoring students' preferences for popular schools because such schools are generally infeasible to them. This may lead to a failure to identify how test scores determine student preferences. This issue can be mitigated if alternative measures of ability are available, as in Fack et al. (2019)'s study of high school choice in Paris. Moreover, if priority scores have full support within each observable ability group, e.g., because they combine multiple priority criteria, then even students with relatively weak academic records may occasionally have access to the full set of schools, restoring non-parametric identification.

Assumption EXO2 requires that the personalized choice set $S(t_i, \mathbf{P})$ be exogenous to a student's unobserved preferences. Importantly, this does not imply that the cutoffs $\mathbf{P}$ are independent of preference shocks, but only that each student's feasible set is conditionally independent of her own idiosyncratic tastes. This condition may be violated in finite markets, for example, if a student can affect the cutoff of a school by choosing whether or not to apply, thereby changing which schools are feasible to her. However, such endogeneity becomes increasingly implausible in large markets, where the influence of any single student's application decision on admission cutoffs is negligible.

When student preferences follow a logit specification, and under assumptions EXO1 and EXO2, the probability that student $i$ is matched with her assigned school $j^*$ can be written as:

$$ \mathbb{P} \left( j^* = \mu(i) = \arg \max_{j' \in S(t_i, \mathbf{P})} u_{ij'} \mid \mathbf{z}_i; \theta \right) = \frac{\exp(V_{ij^*})}{\sum_{j' \in S(t_i, \mathbf{P})} \exp(V_{ij'})} $$

This is the standard conditional logit expression, with the key distinction that each student's choice set is individualized, restricted to schools that are feasible given her priority scores.<fnref n="46" />

This leads to the following log-likelihood function, which can be estimated by

<footnote n="46">⁴⁶In the earlier example with four schools, student $i$ is matched with school $c$ and her feasible set is $\{a, c\}$. Then, under stability, $c$ must be her most preferred school in that set, and the match</footnote>
maximum likelihood:

$$ \ln \mathcal{L}_{\text{Stability}}(\theta | \mathbf{z}) = \sum_{i=1}^{J} V_{i,j^*} - \sum_{i=1}^{I} \ln \left( \sum_{j' \in S(\mathbf{t}_i, \mathbf{P})} \exp(V_{ij'}) \right), \quad (10) $$

with the resulting estimator denoted by $\hat{\theta}_{\text{Stability}}$.

As with WTT, the stability-based approach can be extended to accommodate random coefficients via mixed logit or probit models. However, it relies on more limited information: while WTT leverages the full ranking of submitted preferences, the stability assumption restricts attention to the matching outcome only. As a result, the stability-based framework may entail a loss of information, particularly regarding substitution patterns between schools. This limitation becomes more pronounced when estimating flexible models that allow for rich heterogeneity in preferences over school attributes.<fnref n="47" />

**Priority uncertainty.** The stability-based approach just described does not directly extend to DA environments where students face uncertainty about their priorities, such as when assignments depend on lottery-based tie-breakers.

In such cases, one might consider mechanically applying the stability condition by defining each student's feasible schools based on ex-post cutoffs derived from an arbitrary lottery draw. However, this method would be problematic for two reasons. First, it lacks a strong theoretical foundation, especially in light of results from Che et al. (2023) on the asymptotic stability of robust equilibria in DA markets with payoff-insignificant mistakes. Specifically, the ex-post cutoffs from a single lottery realization could represent a low-probability event from the applicant's perspective. Hence, there is no guarantee that the preferences inferred by this

probability is:

$$ \mathbb{P}(\mu(i) = c | \mathbf{z}_i; \theta) = \frac{\exp(V_{ic})}{\exp(V_{ia}) + \exp(V_{ic})}. $$

<footnote n="47">⁴⁷This limitation of the stability-based approach has implications not only for preference estimation and welfare analysis but also for causal inference. Preference estimates often feed into control-function approaches that correct for selection when estimating treatment effects by capturing idiosyncratic preference heterogeneity (e.g., Abdulkadiroğlu et al., 2020b; Barahona et al., 2023; Campos et al., 2025; Gandil, 2025). If stability-based estimates systematically underrepresent certain types of heterogeneity (e.g., among low-achieving students), this shortcoming may carry over into the causal analysis. By contrast, WTT-based estimates exploit the full information in applicants' ROLs and, although potentially biased in demand estimation, may yield less biased causal forecasts. Developing a deeper understanding of these trade-offs and their implications for causal inference represents an important avenue for future research.</footnote>
method reflect the applicant's true preferences. Second, this approach requires
observing the realized lottery numbers, which are often unavailable in practice.
In their absence, researchers might resort to simulating lotteries, but this raises
concerns about sensitivity to the simulated draw. Without a principled method for
selecting among simulations, the inference becomes vulnerable to cherry-picking.

To address these challenges, Che et al. (2023) develop a procedure known as the *Transitive Extension of Preferences from Stability* (TEPS), which adapts the stability-based approach to environments with priority uncertainty. Rather than relying on a single cutoff realization, TEPS constructs a transitive preference relation that is consistent with all possible stable assignments under the observed ROL. The authors apply this method to school choice data from Staten Island in New York City.

The TEPS method proceeds in three steps to infer preference relations from each applicant's ROL. To illustrate, consider a setting with six schools {a, b, c, d, e, f} that use tie-breaking lotteries (possibly in conjunction with coarse priorities), and suppose a student submits the ROL (e, d, c, b).

*Step 1: Simulating uncertainty and compiling choice data.* The first step simulates the uncertainty faced by applicants due to tie-breaking lotteries. This involves repeatedly executing the DA algorithm under different realizations of the lottery and recording two key pieces of information for each draw: (i) the set of schools that are feasible (i.e., schools the applicant could have been admitted to given her lottery position), and (ii) the school to which she is assigned.

For example, suppose the following four scenarios arise with positive probability:

• the feasible set is {d,e} with probability 0.4, and the student is matched to e;

• the feasible set is {a,b} with probability 0.3, and the student is matched to b;

• the feasible set is {a,b,c} with probability 0.25, and the student is matched to c;

• the feasible set is {b,e} with probability 0.05, and the student is matched to b.

These events are illustrated in Panel (a) of Figure 9.

*Step 2: Inferring preference relations within each realization.* Assuming that students are matched to their most preferred school among those that are feasible in each lottery realization (as implied by robust equilibrium in large markets), one can recover a set of ordered preference relations from each assignment. In the example: assignment to *e* when the feasible set is {d,e} or {b,e} implies *e* ≻ *d* and *e* ≻ *b*;
<figure><img src="image_6.png" /></figure>

(a) Step 1

<figure><img src="image_7.png" /></figure>

(b) Step 2

<figure><img src="image_8.png" /></figure>

(c) Step 3

<caption>**Figure 9** – Transitive Extension of Preferences from Stability (TEPS): Example</caption>

Source: Che et al. (2023).

assignment to $c$ when the feasible set is $\{a, b, c\}$ implies $c \succ a$ and $c \succ b$; and
assignment to $b$ when the feasible set is $\{a, b\}$ implies $b \succ a$. These inferred relations
are visualized in Panel (b) of Figure 9 as directed trees, with each tree rooted at the
school to which the student is assigned.

*Step 3: Extending preference relations by transitivity.* The final step aggregates all preference relations recovered in Step 2 by applying the transitivity axiom. In the example, this yields the extended set of preference relations $\{(b,a), (c,b), (c,a), (e,b), (e,d), (e,a)\}$, which are depicted as a transitive closure in Panel (c) of Figure 9.

The TEPS procedure recovers all preference relations that are consistent with the stability condition across all possible realizations of uncertainty and the transitivity of preferences. Importantly, while these assumptions allow researchers to infer rich ordinal information, they do not uniquely determine the ROL a student submits. In fact, given other participants’ strategies, a student may have multiple best-response ROLs. Despite this multiplicity, Che et al. show that TEPS does not suffer from incompleteness or incoherence in the sense of Tamer (2003). The key reason is that this procedure relies solely on the distribution of possible assignment outcomes for each student, which is unique in a large market, regardless of which best-responding ROL is observed.

Once preference relations have been inferred, one can estimate a parametric utility function that best fits these preferences. Che et al. recommend a multinomial probit specification estimated via Gibbs sampling, as the sampler can flexibly draw cardinal utilities that satisfy the observed ordinal constraints.<fnref n="48" />

<footnote n="48">⁴⁸In this framework, constructing the likelihood function under a logit specification is generally infeasible, even with i.i.d. EVT1 errors, e.g., if the inferred preferences are $b \succ c$ and $c \succ d \succ e$. By contrast, these ordinal constraints can be directly translated into inequality bounds on the utilities, which are readily handled in a Gibbs sampler.</footnote>
### 2.3.5 Using Undominated Strategies

The assumption that applicants do not play weakly dominated strategies does not generally imply a unique ROL for each student (see Sections 2.1 and 2.2.4). Instead, it yields a set of inequality restrictions that can be transformed into moment inequalities. These can be used to partially identify preferences (e.g., Hwang, 2015; He, 2017; Bayraktar and Hwang, 2024) or serve as over-identifying restrictions in estimation (e.g., He, 2017; Fack et al., 2019).

**Estimation with moment inequalities.** Consider a DA environment under the assumption that students do not play weakly dominated strategies (see Section 2.2.4). In such settings, submitted ROLs provide partial information about students' preferences, which can be exploited to derive inequality restrictions. These restrictions, in turn, can be translated into conditional and unconditional moment inequalities for estimation.

To illustrate, consider two schools, a and b. Because students do not necessarily rank all schools, only some ROLs reveal a preference between a and b. Under the assumption of undominated strategies, the probability that student i ranks a above b in her ROL $R_i$ provides a lower bound on the probability that she prefers a to b:

$$ \begin{aligned} \mathbb{P}(a \succ_{R_i} b | \mathbf{z}_i; \boldsymbol{\theta}) &= \mathbb{P}(\{u_{i,a} > u_{i,b}\} \cap \{a, b \in R_i\} | \mathbf{z}_i; \boldsymbol{\theta}) \\ &\le \mathbb{P}(u_{i,a} > u_{i,b} | \mathbf{z}_i; \boldsymbol{\theta}). \end{aligned} $$

Similarly, the probability that b is ranked above a implies an upper bound:

$$ \mathbb{P}(u_{i,a} > u_{i,b} | \mathbf{z}_i; \boldsymbol{\theta}) \le 1 - \mathbb{P}(b \succ_{R_i} a | \mathbf{z}_i; \boldsymbol{\theta}). $$

Together, these bounds yield two conditional moment inequalities:

$$ \begin{aligned} \mathbb{P}(u_{i,a} > u_{i,b} | \mathbf{z}_i; \boldsymbol{\theta}) - \mathbb{E}[1(a \succ_{R_i} b) | \mathbf{z}_i; \boldsymbol{\theta}] &\ge 0; \\ 1 - \mathbb{E}[1(b \succ_{R_i} a) | \mathbf{z}_i; \boldsymbol{\theta}] - \mathbb{P}(u_{i,a} > u_{i,b} | \mathbf{z}_i; \boldsymbol{\beta}) &\ge 0. \end{aligned} $$

Such inequalities can be derived for any pair or subset of schools and then interacted with functions of observable covariates $\mathbf{z}_i$ to construct a vector of $M_1$ unconditional moment inequalities, denoted by $(m_1, \dots, m_{M_1})$.

Estimation proceeds by testing whether these inequalities hold in the data. Andrews and Shi (2013) propose a Cramér-von Mises-type statistic for this purpose,
based on the modified methods of moments (or sum function):

$$ T_{\text{MI}}(\boldsymbol{\theta}) = \sum_{k=1}^{M_1} \left[ \frac{\bar{m}_k(\boldsymbol{\theta})}{\hat{\sigma}_k(\boldsymbol{\theta})} \right]_{-}^{2}, \qquad (11) $$

where $\bar{m}_k(\boldsymbol{\theta})$ is the sample mean, $\hat{\sigma}_k(\boldsymbol{\theta})$ is the sample standard deviation of the kth moment, and $[x]_{-} := \min\{0, x\}$ ensures that only violations of the inequality contribute to the test statistic. Bugni et al. (2017) develop methods to construct marginal confidence intervals using this statistic. For each parameter component $\theta_k$, they provide a test for the null hypothesis $H_0: \theta_k = \theta_0$ for each candidate value $\theta_0 \in \mathbb{R}$. The confidence interval for $\theta_k$'s true value is then the convex hull of all $\theta_0$ values for which the null is not rejected.

**Combining moment equalities and moment inequalities.** Approaches based solely on moment inequalities in school choice settings often yield wide confidence intervals of the parameters of interest, constrained by the limitations of current econometric tools. As a result, such inference may be uninformative in practice. A more effective strategy is to treat inequality restrictions as over-identifying information within a complete model framework (Moon and Schorfheide, 2009). For example, one can combine the restrictions implied by undominated strategies (expressed as moment inequalities) with the moment equalities implied by the stability assumption. Specifically, the likelihood function in (10) can be rewritten as a system of moment equalities by equating predicted and observed school-level matching frequencies:

$$ \sum_{i=1}^{I} \mathbb{P} \left( j = \arg \max_{j' \in S(t_i, \mathbf{p})} u_{ij'} \mid \mathbf{z}_i; \boldsymbol{\theta} \right) - \mathbb{E} \left( \sum_{i=1}^{I} 1(\mu(i) = j) \right) = 0, \quad \forall j \in \mathcal{J}, $$

where $1(\mu(i) = j)$ is an indicator equal to one if student $i$ is matched to school $j$. These moment equalities can be further interacted with covariates $\mathbf{z}_i$ to construct $M_2$ unconditional moment equalities ($m_{M_1+1}, \dots, m_{M_1+M_2}$).

To jointly incorporate the full set of $M_1$ moment inequalities and $M_2$ moment equalities, the test statistic in (11) can be extended to:

$$ T_{\text{MEI}}(\boldsymbol{\theta}) = \sum_{k=1}^{M_1} \left[ \frac{\bar{m}_k(\boldsymbol{\theta})}{\hat{\sigma}_k(\boldsymbol{\theta})} \right]_{-}^{2} + \sum_{k=M_1+1}^{M_1+M_2} \left[ \frac{\bar{m}_k(\boldsymbol{\theta})}{\hat{\sigma}_k(\boldsymbol{\theta})} \right]_{-}^{2}. \qquad (12) $$

Here, the first term penalizes violations of inequality constraints (as before), while