Dynamic Matching in School Choice: Efficient Seat ...

MANAGEMENT SCIENCEArticles in Advance, pp. 1–21

http://pubsonline.informs.org/journal/mnsc ISSN 0025-1909 (print), ISSN 1526-5501 (online)

Dynamic Matching in School Choice: Efficient Seat ReassignmentAfter Late CancellationsItai Feigenbaum,a Yash Kanoria,b Irene Lo,c Jay Sethuramand

aLehmanCollege and the Graduate Center, City University of NewYork, Bronx, NewYork 10468; bDecision, Risk, andOperations Division,Columbia Business School, New York, New York 10027; cDepartment of Management Science and Engineering, Stanford University,Stanford, California 94305; dDepartment of Industrial Engineering and Operations Research, Columbia University, New York,New York 10027Contact: [email protected] (IF); [email protected], http://orcid.org/0000-0002-7221-357X (YK); [email protected],

http://orcid.org/0000-0002-0678-3494 (IL); [email protected], http://orcid.org/0000-0002-9985-0683 (JS)

Received: May 1, 2017Revised: July 28, 2018; February 7, 2019;July 15, 2019Accepted: July 27, 2019Published Online in Articles in Advance:May 8, 2020

https://doi.org/10.1287/mnsc.2019.3469

Copyright: © 2020 INFORMS

Abstract. In the school choice market, where scarce public school seats are assigned tostudents, a key operational issue is how to reassign seats that are vacated after an initialround of centralized assignment. Practical solutions to the reassignment problem must besimple to implement, truthful, and efficient while also alleviating costly student movementbetween schools. We propose and axiomatically justify a class of reassignment mecha-nisms, the permuted lottery deferred acceptance (PLDA) mechanisms. Our mechanismsgeneralize the commonly used deferred acceptance (DA) school choice mechanism to atwo-round setting and retain its desirable incentive and efficiency properties. School choicesystems typically run DA with a lottery number assigned to each student to break ties inschool priorities. We show that under natural conditions on demand, the second-round tie-breaking lottery can be correlated arbitrarily with that of the first round without affectingallocative welfare and that reversing the lottery order between rounds minimizes reas-signment among all PLDAmechanisms. Empirical investigations based on data fromNewYork City high school admissions support our theoretical findings.

Funding: Y. Kanoria was supported by the National Science Foundation’s Division of Civil, Mechanical,and Manufacturing Innovation [Grant CMMI-1201045]. J. Sethuraman was supported by the Na-tional Science Foundation [Grant CMMI-1201045].

History: Accepted by Gad Allon, operations management.Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2019.3469.

Keywords: dynamic matching • matching markets • school choice • deferred acceptance • tie breaking • cancellations • reassignments

1. IntroductionIn public school systems throughout the United States,students submit preferences over the schools forwhichthey are eligible for admission. Because this occursfairly early in the school year, students typically do notknow their options outside the public school systemwhen submitting their preferences. Consequently, asignificant fraction of students who are allotted a seatin a public school eventually do not use it, leading toconsiderable inefficiency. In the New York City (NYC)public high school system, over 80,000 students areassigned to a public school each year in March, andabout 10% of these students choose to not attend apublic school in September, possibly opting instead toattend a private or charter school.1 Schools find outabout many of the vacated seats only after classes be-gin, when students do not show up to class; such seatsare reassigned in an ad hoc manner by the schoolsusing decentralized procedures that can run monthsinto the school year. A well-designed reassignmentprocess, run after students learn about their outsideoptions, could lead to significant gains in overallwelfare.

Yet no systematic way of reassigning students to unusedseats has been proposed in the literature. Our goal is todesign an explicit reassignment mechanism run at a latestage of the matching process that efficiently reassignsstudents to vacated seats.During the past 15 years, insights from matching

theory have informed the design of school choiceprograms in cities around the world. The formal studyof this mechanism design approach to school choiceoriginated in a paper by Abdulkadiroglu and Sonmez(2003). They formulated a model in which studentshave strict preferences over a finite set of schools, eachwith a given capacity, and each school partitions theset of students into priority groups. There is now avast and growing literature that explores many as-pects of school choice systems and informs how theyare designed in practice. However, most models con-sidered in this literature are essentially static. Incor-porating dynamic considerations in designing as-signment mechanisms, such as students learning newinformation at an intermediate time, is an importantaspect that has only recently started to be addressed.

1

http://pubsonline.informs.org/journal/mnsc

mailto:[email protected]


http://orcid.org/0000-0002-7221-357X

http://orcid.org/0000-0002-7221-357X


http://orcid.org/0000-0002-0678-3494

http://orcid.org/0000-0002-0678-3494


http://orcid.org/0000-0002-9985-0683

http://orcid.org/0000-0002-9985-0683



Our work provides some initial theoretical results inthis area and suggests that simple adaptations of one-shot mechanisms can work well in a more general setting.

We consider a two-round model of school assign-ment with finitely many schools. Students learn theiroutside option after the first-round assignment andvacate seats that can be reassigned. In the first round,schools have weak priorities over students, and stu-dents submit strict ordinal preferences over schools.Students receive a first-round assignment based onthese preferences via deferred acceptance with sin-gle tie breaking (DA-STB), a variant of the standarddeferred acceptance (DA) mechanism, where ties inschool preferences are broken via a single lottery or-dering across all schools. Afterward, students learntheir outside options (such as admission to a privateschool) and may no longer be interested in the seatallotted to them. In the second round, students areinvited to submit new ordinal preferences over schools,reflecting changes in their preferences induced by learn-ing their outside options. The goal is to reassign stu-dents so that the resulting assignment is efficient and thetwo-round mechanism is strategy-proof and does notpenalize students for participating in the second round.Because a significant fraction of seats available forreassignment are vacated only after the start of theschool year, a key additional goal is to ensure that thereassignment process minimizes the number of stu-dents who need to be reassigned.

We introduce a class of reassignment mechanismswith desirable properties: the permuted lottery deferredacceptance (PLDA) mechanisms. PLDA mechanismscompute a first-round assignment by running DA-STB and then a second-round assignment by run-ning DA-STB with a permuted lottery. In the secondround, each school first prioritizes students whowere

assigned to it in the first round, which guaranteeseach student a second-round assignment that he orshe prefers to his or her first-round assignment, thenprioritizes students according to their initial priori-ties at the school, and finally breaks ties at all schoolsvia a permutation of the (first-round) lottery num-bers. Our proposed PLDA mechanisms are based onschool choice mechanisms currently implemented inthe main round of assignments and can be imple-mented either as centralized PLDAs, which run a cen-tralized second round with updated preferences, oras decentralized PLDAs, which run a decentralizedsecond round via a waitlist system that closely mir-rors current reassignment processes.Our key insight is that the mechanism designer can

design the correlation between tie-breaking lotteriesto achieve operational goals. In particular, revers-ing the lottery between rounds minimizes reassign-ment without sacrificing student welfare. Our maintheoretical result is that under an intuitive order condi-tion, all PLDAs produce the same distribution overthe final assignment, and reversing tie-breaking lot-teries between rounds implements the centralized re-verse lottery DA (RLDA), which minimizes the num-ber of reassigned students. We axiomatically justifyPLDA mechanisms: absent school priorities, PLDAsare equivalent to the class of mechanisms that are two-round strategy-proofwhile satisfying natural efficiencyrequirements and symmetry properties.In a setting where all students agree on a ranking of

schools and there are no priorities, our results are veryintuitive. By reversing the lottery, we move a few stu-dents many schools up their preference list rather thanmany students a few schools up, thereby eliminatingunnecessary cascades of reassignment (see Figure 1).Surprisingly, however, our theoretical result holds in

Figure 1. Running DA with a Reversed Lottery Eliminates the Cascade of Reassignments

Notes. There are six studentswith identical preferences over schools and six schools eachwith a single-priority group. All students prefer schoolsin the order s1 � s2 � · · · � s6. The student assigned to school s1 in the first round leaves after the first round; otherwise, all students find allschools acceptable in both rounds. Running DAwith the same tie-breaking lottery reassigns each student to the school that is one better on his orher preference list, whereas reversing the tie-breaking lottery reassigns only the student initially assigned to s6.

Feigenbaum et al.: Dynamic Matching in School Choice2 Management Science, Articles in Advance, pp. 1–21, © 2020 INFORMS

a general setting with heterogeneous student pref-erences and arbitrary priorities at schools. The ordercondition can be interpreted as aggregate studentpreferences resulting in the same order of popularityof schools in the two rounds. Our results show that ifstudent preferences and school priorities produce suchagreement in aggregate demand across the two rounds,then reversing the lottery between rounds preserves exante allocative efficiency and minimizes reassignment.

We empirically assess the performance of RLDAusing data from the NYC public high school system.We first investigate a class of centralized PLDAs thatincludes RLDA, rerunning DA using the original lot-tery order (termed forward lottery deferred acceptance(FLDA)), and rerunning DA using an independentrandom lottery. We find that all these mechanismsprovide similar allocative efficiency, but RLDA re-duces the number of reassigned students significantly.For instance, in the NYC data set from 2004–2005, wefind that FLDA results in about 7,800 reassignmentsand RLDA results in about 3,400 reassignments outof a total of about 74,000 students who remained inthe public school system—that is, fewer than half thenumber of reassignments under FLDA. The gainsbecome even more marked if we compare with cur-rent practice: RLDA results in fewer than 40% of the8,600 reassignments under decentralized FLDA withwaitlists.2

To better evaluate the currently used waitlist sys-tems, we also empirically explore the performance ofdecentralized FLDA and RLDA as a function of thetime available to clear the market. We find that thetiming of information revelation can greatly impactboth allocative efficiency and congestion. If congestionis caused by students taking time to vacate previouslyassigned seats (see Figure 1), then reversing the lotteryincreases allocative efficiency during the early stagesof reassignment and decreases congestion. However,if congestion is caused by students taking time todecide on waitlist offers, these findings are reversed.In both cases, for reasonable timescales, the welfaregains from centralizing the system and reducing con-gestion can be substantial.

The rest of this paper is organized as follows. We out-line current practice in school admissions in Section 1.1and related literature in Section 1.2. In Section 2, wedescribe our model, our proposed PLDA mechanisms,and their properties. Section 3 presents our main re-sults, and Section 4 provides intuition and a flavorof our analysis via a special case of our model. Weprovide empirical results in Section 5 and concludein Section 6.

1.1. Current PracticeSchool systems in cities across the United States usesimilar centralized processes for admissions to public

schools. Students seeking admission to a schoolsubmit their preferences over schools to a centralauthority byDecember throughMarch, for admissionstarting the subsequent fall. Each school may havepriority classes of students, such as priority for stu-dents who live in the neighborhood, siblings of stu-dents already enrolled at that school, or students fromlow-income families. An assignment of seats to stu-dents is produced using the student-proposing DAalgorithm with single tie breaking. Students mustregister in their assigned school by April or early May.In March and April, students are also admitted to

private and charter schools via processes run con-currently with the public school assignment process.This results in an attrition rate of about 8%–10% of theseats assigned in the main round of public schooladmissions. Some schools account for this attrition byaccepting more students in the first round than theyhave seats for. However, such oversubscription ofstudents is usually conservative because of hard con-straints on space and teacher capacity.3 As a result,most schools have unused seats at the end of the firstround that can be reassigned. In addition, most publicschools find out aboutmany of these vacant seats onlyafter the start of the school year because they cannotrequire deposits or other forms of commitment fromstudents before the start of the school year.Reassignments in most school choice systems are

performed using a decentralized waitlist system.4

Students are put on waitlists for all schools that theyranked above their first-round assignment and or-dered by first-round priorities (after tie breaking).Students who do not register by the deadline arepresumed to be uninterested, and their seats are of-fered to waitlisted students in sequence, with moreseats becoming available over time as students receivenew offers from outside the system or are reassignedvia waitlists to other public school seats. Studentsoffered seats by the waitlist system usually have justunder a week to make a decision and are only boundby the final offer they choose to accept.5 Overall, thistypically results in a drawn-out reassignment processthat continues all summer until after classes begin andin some cities (e.g., NYC kindergarten, Boston, andWashington, DC) up to several months after the startof the school year.Our proposed class of mechanisms generalizes

these waitlist systems as follows. Waitlists are PLDAmechanisms where (1) the second round is imple-mented in a decentralized fashion as informationabout vacated seats propagates through the system,and (2) the tie-breaking lotteries used in the tworounds are the same. We show that permuting the tie-breaking lottery numbers before creating waitlistsprovides a class of reassignment mechanisms that,given sufficient time, results in similar allocative

Feigenbaum et al.: Dynamic Matching in School ChoiceManagement Science, Articles in Advance, pp. 1–21, © 2020 INFORMS 3

efficiency while providing flexibility for optimizingother objectives.

1.2. Related WorkThemechanism design approach to school choice wasfirst formulated by Balinski and Sonmez (1999) andAbdulkadiroglu and Sonmez (2003). Since then, ac-ademics have worked closely with school authoritiesto redesign school choice systems to increase stu-dent welfare.6 A significant portion of the literaturehas focused on providing solutions for a single roundof centralized school assignment—see, for example,Pathak (2011) andAbdulkadiroglu and Sonmez (2011)for recent surveys. Many of these works provide axi-omatic justifications for two canonical mechanisms—DA (Gale and Shapley 1962) and top trading cycles(TTCs)—and their variants, in terms of their desirableproperties. We provide a similar framework for thereassignment problem by proposing and character-izing PLDA mechanisms by their incentive and effi-ciency properties.

There is a growing operations literature on de-signing the school choice process to optimize quan-titative objectives. Ashlagi and Shi (2014) considerhow to improve community cohesion in school choiceby correlating the lotteries of students in the samecommunity, and Ashlagi and Shi (2015) show howto maximize welfare given busing cost constraints.Several papers also explore how school districts canuse rules for breaking ties in school priorities as policylevers. Arnosti (2015), Ashlagi and Nikzad (2016),and Ashlagi et al. (2015) show that DA-STB assignsmore students to one of their top k schools (for small k)compared with DA using independent lotteries atdifferent schools, and Abdulkadiroglu et al. (2009)empirically compare these tie-breaking rules. Erdiland Ergin (2008) also exploit indifferences to improveallocative efficiency. We explore the design of tie-breaking rules in the reassignment setting and cor-relate tie breaking across rounds.

There is also a vast literature on dynamic matchingand reassignments. The reassignment of donated or-gans has been extensively studied in work on kidneyexchange (see, e.g., Roth et al. 2004; Anderson et al.2015, 2017; Ashlagi et al. 2019). Reassignments re-sulting from cancellations also arise in online as-signment settings such as kidney transplantation (see,e.g., Zenios 1999, Su and Zenios 2006) and publichousing allocation (see, e.g., Kaplan 1987, Arnostiand Shi 2017). An important difference is that theseare online settings where agents and objects arriveover time and arematched on an ongoing basis. In suchsettings, matches are typically irrevocable, so optimalassignment policies account for typical cancellationand arrival statistics and optimize for agents arrivingin the future (see, e.g., Dickerson and Sandholm

2015). In our setting, the matching for the entire sys-tem is coordinated in time, and we improve welfareby controlling both the initial assignment and sub-sequent reassignment of objects among the same setof agents.Another relevant strand in the reassignment liter-

ature is the work of Abdulkadiroglu and Sonmez(1999) on house allocation models with housing en-dowments. Our second round can be thought of asschool seat allocation where some agents already owna seat and we wish to reassign seats to reach an effi-cient assignment. There are also a growing numberof papers that consider a dynamic model for schooladmissions (see e.g., Compte and Jehiel 2008, Combeet al. 2016). A critical distinction between these worksand ours is that in our model, the initial endowmentis determined endogenously by preferences, so wepropose reassignment mechanisms that are imper-vious to students manipulating their first-round en-dowment to improve their final assignment.A number of recent papers, such as those by Dur

(2012), Pereyra (2013), and Kadam and Kotowski(2014), focus on the strategic issues in dynamic re-assignment. These works develop solution conceptsin finite markets with specific cross-period con-straints and propose DA-like mechanisms that im-plement them. In recent complementary work, Narita(2016) analyzes preference data from NYC schoolchoice, observes that a significant fraction of prefer-ences is permuted after the initial match, and proposesa modified version of DA with desirable propertiesin this setting. We similarly propose PLDA mecha-nisms for their desirable incentive and efficiencyproperties. In addition, our large market and consis-tency assumptions allow us to uncover considerablestructure in the problem and provide conditions un-der which we can optimize over the entire class ofPLDA mechanisms.Our work also has some connections to the queu-

ing literature. The class of mechanisms that emergesin our setting involves choosing a permutation of theinitial lottery order, and we find that the reverselottery minimizes reassignment within this class. Thisis similar to choosing a service policy in a queuingsystem (e.g., first in first out, last in first out, shortestremaining time first, etc.) in order to minimize costfunctions (see, e.g., Lee and Srinivasan 1989). “Work-conserving” service policies can result in identicalthroughput but different expected waiting times, andwe similarly find that PLDA mechanisms may haveidentical allocative efficiency but different numbersof reassignments. Our continuum model parallelsfluid limits and deterministic models employed inqueuing (Whitt 2002), revenue management (TalluriandVanRyzin 2006), and other contexts in operationsmanagement.


2. Model2.1. Definitions and NotationWe consider the problem of assigning a set of studentsΛ to seats in a finite set of schools S � {s1, . . . , sN}. Eachstudent can attend at most one school. There is acontinuum7 of students with an associated measure η:for any (measurable) subset A ⊆ Λ, we use η(A) todenote themass of students inA. The outside option issN+1 /∈ S. The capacities of the schools are q1, . . . , qN ∈R+ and qN+1 � ∞. A set of students of η-measure atmost qi can be assigned to school si.

Each student submits a strict preference orderingover his or her acceptable schools, and each schoolpartitions eligible students into priority groups. Eachstudent has a type θ � (�θ, �θ, pθ) that encapsulatesboth his or her preferences and school priorities. Thestudent’s first- and second-round preferences, re-spectively, �θ and �θ are strict ordinal preferencesover S ∪ {sN+1}, and schools before (after) sN+1 in theordering are acceptable (unacceptable). We think ofsN+1 as the best guaranteed outside option available tothe student, with the understanding that it can “im-prove” from the first to the second round—for ex-ample, because a new private school offer comes in.The student’s priority class pθ encodes his or her pri-ority pθi at each school si. Each school si has ni prioritygroups. We assume that schools prefer higher pri-ority groups, that students ineligible for school sihave priority pi � −1, and that pi ∈ {−1, 0, 1, . . . ,ni − 1}. Eligibility and priority groups are exoge-nously determined and publicly known. Each studentλ � (θλ,L(λ)) ∈ Λ also has afirst-round lottery numberL(λ) ∈ [0, 1]. We sometimes use the notation (�λ, �λ,pλ) as a less cumbersome alternative to (�θλ

, �θλ, pθ

λ).Let Θ be the set of all student types so that Λ � Θ ×[0, 1] denotes the set of students. For each θ ∈ Θ, letζ(θ) � η({θ} × [0, 1]) be the measure of all studentswith type θ.

We assume that all students have consistent pref-erences, defined as follows.

Definition 1. Preferences (�, �) are consistent if thesecond-round preferences � are obtained from � viatruncation—that is, (a) schools do not become ac-ceptable only in the second round, ∀si ∈ S si�sN+1implies si � sN+1, and (b) the relative ranking of schoolsis unchanged across rounds, ∀si, sj ∈ S if si�sN+1 andsi�sj, then si � sj. Type θ is consistent if (�θ, �θ) areconsistent.

Assumption 1 (Consistent Preferences). If ζ(θ) > 0, thenthe type θ is consistent.

In otherwords,we assume that the only change thata student makes to his or her preferences between thetwo rounds is truncating his or her preference list. Theconsistency assumption allows us to study the effects

of students leaving the public school system but pre-cludes the possibility of students revising their relativeranking of schools between rounds—for example, asa result of making mistakes in the first round—orobtaining more information about the schools. It alsoallows us to propose strategy-proofmechanisms8 andto optimize for allocative efficiency and reassign-ment. In settings where the consistency assumptiondoes not hold, we note that the mechanisms we willpropose are still well defined, and we will show thatthey retain many of their desirable properties.We also assume that consistent types have full

support. We use this only to characterize our proposedmechanisms (Theorem 3) and do not need it for ourpositive results (Theorems 1 and 2).

Assumption 2 (Full Support). For all consistent typesθ ∈ Θ, it holds that ζ(θ) > 0.

Finally, we assume that the first-round lottery num-bers are drawn independently and uniformly from[0, 1] and do not depend on preferences: η {θ} ×((a, b)) � (b − a)ζ(θ) ∀θ ∈ Θ, 0 ≤ a ≤ b ≤ 1.9

An assignment μ : Λ → S specifies the school towhich each student is assigned. For any assignmentμ,we let μ(λ) denote the school to which student λ isassigned, and in a slight abuse of notation, we let μ(si)denote the set of students assigned to school si. Weassume that μ(si) is η-measurable and that the as-signment is feasible—that is, η(μ(si)) ≤ qi for all si ∈ S,and if μ(λ) � si, then pλi ≥ 0. We let μ and μ denote thefirst- and second-round assignments, respectively.

2.1.1. Timeline. Students report first-round prefer-ences�. Themechanismdesigner obtains a first-roundassignment μ by running DA-STB with lottery L andannounces μ and L. Students then observe their out-side options and update their preferences to �. Finally,students report their second-round preference �,and the mechanism designer obtains a second-roundassignment μ by running a reassignment mecha-nism M and announces μ. We illustrate the timelinein Figure 2.

2.1.2. Informational Assumptions. Eligibility and pri-orities are exogenously determined and publiclyknown. The mechanism is publicly announced beforepreferences are submitted. Before first-round report-ing, each student knows his or her first-round pref-erences and that his or her second-round preferenceswill be obtained from these preferences via truncation.Each student has imperfect information regarding hisor her own second-round preferences (i.e., the point oftruncation) at that stage and believes with positiveprobability that his or her preferences in both roundswill be identical.10 We assume that students know thedistribution η over student types and lotteries (an


assumption we need only for our characterizationresult, Theorem 3). Each student is assumed to learnhis or her lottery number after the first round because,in practice, students are often permitted to inquireabout their position on each school’s waitlist; ourresults hold even if students do not learn their lotterynumbers.

Definition 2. A student λ ∈ Λ is a reassigned student if heor she is assigned to a different school in S in the secondround than in the first round. That is, λ is a reassignedstudent under reassignment μ if μ(λ) �� μ(λ) and11

μ(λ) �� sN+1, μ(λ) �� sN+1.Most reassignments happen around the start of the

school year, a time when they are costly for schools andstudents alike. Hence, in addition to providing an efficientfinal assignment, we also want to reduce congestion byminimizing the number of reassigned students.

2.2. MechanismsA mechanism is a function that maps the realization offirst-round lotteries L, school priorities p, and students’first-round preference reports � into an assignment μ.A reassignment mechanism is a function that maps therealization of first-round lotteries L, first-round as-signment μ, school priorities p, and students’ second-round reports � into a second-round assignment12 μ.A two-round mechanism obtained from a reassignmentmechanism M is a two-round mechanism where thefirst-round mechanism is DA-STB (see Definition 3)and the second-round mechanism is M.

In the first round, seats are assigned according tothe student-optimal DA algorithm with single tiebreaking (STB) as follows. A single lottery ordering ofthe students L is used to resolve ties in the prioritygroups at all schools, resulting in an instance of thetwo-sided matching problem with strict preferencesand priorities. In each step of DA, unassigned stu-dents apply to their most-preferred school that hasnot yet rejected them. A school with a capacity of qtentatively accepts the q highest-ranked eligible ap-plicants (according to its priority ranking of stu-dents after breaking all ties) and rejects any remainingapplicants. The algorithm runs until there are no new

student applications, at which point it terminates andassigns each student to his or her tentatively assignedschool seat. The strict student preferences,weak schoolpriorities, anduse ofDA-STBmirror current practice inmany school choice systems, such as those in NYC,Chicago, and Denver (see, e.g., Abdulkadiroglu andSonmez 2003).DA can also be formally defined in terms of ad-

missions scores and cutoffs.

Definition 3 (Deferred Acceptance; Azevedo and Leshno2016). The DA mechanism with single tie breaking(DA-STB) is a function DAη((�λ, pλ)λ∈Λ,L) mappingstudent preferences, priorities, and lottery numbersinto an assignment μ, defined by a vector of cutoffsC ∈ RN+ as follows. Each student λ is given a score rλi �pλi + L(λ) at school si and is assigned to his or hermost-preferred school as per his or her preferences, amongthose where his or her score exceeds the cutoff:

μ(λ) � max�λ

si ∈ S : rλi ≥ Ci

{ }∪ sN+1{ }

( ). (1)

Moreover, C is market clearing—namely,

η μ si( )( ) ≤ qi, for all si ∈ S,with equality if Ci > 0.

(2)Azevedo and Leshno (2016) showed that the set of

assignments satisfying Equations (1) and (2) forms anonempty complete lattice and typically consists of asingle uniquely determined assignment.13 This uniqueassignment in the continuum further corresponds tothe scaling limit of the set of stable matches obtainedin finite markets as the number of students grows(with school capacities growing proportionally).Throughout this paper, in the (knife-edge) case wherethere are multiple assignments satisfying Definition 3,we pick the student-optimal matching.Given cutoffs {Ci}Ni�1, we will also find it helpful to

define for each priority class π the cutoffs within thepriority class at each school Cπ,i ∈ [0, 1] by Cπ,i � 0 ifCi ≤ πi, Cπ,i � 1 if Ci ≥ πi + 1, and Cπ,i � Ci − πi oth-erwise. Thus, Cπ,i is the lowest lottery number astudent θ with priority pθ � π can have and still beable to attend school si.

Figure 2. Timeline of the Two-Round Mechanism Design Problem


We now turn to the mechanism design problem.We emphasize that we consider only two-round mech-anisms whose first-round mechanism is the currentlyused DA-STB—that is, the only freedom afforded theplanner is the design of the reassignment mechanism.We propose the following class of two-round mech-anisms. Intuitively, these mechanisms run DA-STBtwice, once in each round. They explicitly correlatethe lotteries used in the two rounds via a permutationP and in the second round give each student toppriority in the school to which he or she was assignedin the first round to guarantee that each student re-ceives a (weakly) better assignment.

Definition 4 (Permuted Lottery Deferred Acceptance(PLDA)Mechanisms). LetP : [0, 1] → [0, 1] be ameasure-preserving bijection, let L be the realization of first-round lottery numbers, and let μ be the first-roundassignment obtained by running DA with lottery L.Define a new economy η, where to each student λ ∈ Λwith priority vector pλ and first-round lottery andassignment L(λ) � l, μ(λ) � si, we (a) assign a lotterynumber P(l) and (b) give top second-round priority pλi �ni at their first-round assignment si and unchangedpriority pλj � pλj at all other schools sj �� si. PLDA(P) isthe two-round mechanism obtained using the reassign-ment mechanism DAη((�λ, pλ)λ∈Λ,P ◦ L).

We use CPπ,i to denote the second-round cutoff for

priority class π in school i under PLDA(P).We highlight two particular PLDA mechanisms.

The RLDA (reverse lottery) mechanism uses the re-verse permutation R(x) � 1 − x, and the FLDA (for-ward lottery)mechanism,which preserves the originallottery order, uses the identity permutation F(x) � x.By default, school districts often use a decentralizedversion of the FLDA mechanism, implemented viawaitlists. In this paper, we provide evidence that sup-ports using the centralized RLDA mechanism in a schoolsystem such as that in NYC, where a large proportionof vacated seats is revealed close to or after the startof the school year and where reassignments are costlyfor both students and the school administration.

The PLDA mechanisms are an attractive class oftwo-round assignment mechanisms for a number ofreasons. They are intuitive to understand and simple toimplement in systems already using DA. (A decen-tralized implementation would be even simpler to in-tegrate with current practice; the currently usedwaitlistmechanism for reassignments can be retained with thesimple modification of permuting the lottery num-bers just before waitlists are constructed.) In addition,we will show that the PLDA mechanisms have desir-able incentive and efficiency properties, which we nowdescribe.

Any reassignment mechanism that takes away astudent’s initial assignment against his or her will is

impractical. Thus, we require our mechanism to re-spect first-round guarantees:

Definition 5. A two-round mechanism (or a second-round assignment μ) respects guarantees if every student(weakly) prefers his or her second-round assignment tohis or her first-round assignment—that is, μ(λ)�λμ(λ)for every λ ∈ Λ.

One of the reasons for the success of DA in practiceis that it respects priorities: if a student is not assignedto a school he or she wants, it is because that schoolis filled with students with higher priority at thatschool. This leads to the following natural require-ment in our two-round context.

Definition 6. A two-round mechanism (or a second-round assignment μ) respects priorities (subject to guar-antees) if ∀si ∈ S, every eligible student λ ∈ Λ such thatsi �λ μ(λ), and every student λ′ such that μ(λ′) � si ��μ(λ′) it holds that λ′ is eligible for si and pλ

′i ≥ pλi .

Thus, our definition of respecting priorities (subjectto guarantees) requires that every student who wasupgraded to a school s in the second-roundmust havea (weakly) higher priority at that school than everyeligible student λ who prefers s to his or her second-round assignment.We now turn to incentive properties. In the school

choice problem, it is reasonable to assume that stu-dents will be strategic in how they interact with themechanism at each stage. Hence, it is desirable thatwhenever a student (with consistent preferences) re-ports preferences, conditional on everything that hashappened up to that point, it is a dominant strategyfor him or her to report truthfully. To describe theproperties formally, we start by fixing an arbitraryprofile offirst- and second-round preferences (�−λ, �−λ)for all the students other than student λ. For anypreference report of student λ in the first round, heor she will receive an assignment that is probabilisticbecause of the tie-breaking lottery; then, after ob-serving his or her first-round assignment and lotterynumber and his or her updated outside option, heor she can submit a second-round preference report,based onwhich his or herfinal assignment is computed.This leads to two natural notions of strategy-proofness.

Definition 7. A two-round mechanism is strongly strat-egy-proof if for each student λ (with consistent prefer-ences) truthful reporting is a dominant strategy—that is,for each realization of lottery numbers (including his orher own lottery number) and profile of first- and second-round reported preferences of the students other than λ,reporting his or her preferences truthfully in each of thetwo rounds is a best response for student λ.

Our definition of strong strategy-proofness is ratherdemanding: it requires that no student be able to


manipulate the mechanism even if he or she has fullknowledge of the first- and second-round preferencesof all other students and the lottery numbers.We shallalso consider a weaker version of strategy-proofnessthat applies when a manipulating student does notknow the lottery number realizations when he or shesubmits his or her first-round preference report andlearns all lottery numbers only after the end of the firstround. In that case, each student views his or her first-round assignment as a probability vector; his or hersecond-round assignment is also random but is a de-terministic function of the first-round outcome, thesecond-round reports, and the first-round lotterynumbers. We make precise the notion of a successfulmanipulation in this setting as follows.

Definition 8. A two-roundmechanism isweakly strategy-proof if the following conditions hold:

• Knowing the specific realization of first-round as-signments (and lottery numbers) and the second-roundpreferences of the students other thanλ, it is optimal forstudent λ to submit his or her second-round prefer-ence truthfully, given what the other students do.

• For each student λ (with consistent preferences),and for each profile of first- and second-round pref-erences of the students other than λ, the probabilitythat student λ is assigned to one of his or her top kschools in the second round is maximized when he orshe reports truthfully in the first round (assumingtruthful reporting in the second round), for each k �1, 2, . . . ,N.

In other words, in each stage of the dynamic game,the outcome from truthful reporting stochasticallydominates the outcomes of all other strategies. Weemphasize that the uncertainty in the first-round as-signment is solely due to the lottery numbers, whichstudents initially do not know.

Note that a two-round mechanism that uses thefirst-round assignment as the initial endowment for amechanism such as top trading cycles in the secondround will not be two-round strategy-proof becausestudents can benefit from manipulating their first-round reports to obtain a more popular initial as-signment that they could use to their advantage in thesecond round.

Finally, we discuss some efficiency properties. Tobe efficient, clearly a mechanism should not leaveunused any seats that are desired by students.

Definition 9. A two-round mechanism is nonwasteful ifno student is assigned to a school for which he or she iseligible that he or she prefers less than a school not atcapacity; that is, for each student λ ∈ Λ and schoolssi, sj, if μ(λ) � si and sj�λsi and pλj ≥ 0, then η(μ(sj)) � qj.

It is also desirable for a two-roundmechanism to bePareto efficient. We do not want any students to be

able to improve their utility by swapping probabilityshares in second-round assignments. However, wealso require that our reassignmentmechanism respectguarantees and priorities (see Definitions 5 and 6),which is incompatible with Pareto efficiency even in astatic, one-round setting.14 This motivates the follow-ing definitions. Consider a second-round assignmentμ. A Pareto-improving cycle is an ordered set of types(θ1, θ2, . . . , θm) ∈ Θm, sets of students (Λ1,Λ2, . . . ,Λm) ∈ Λm, and schools (s1, s2, . . . , sm) ∈ Sm such thatη(Λi) > 0 and si+1�θi si (where we define sm+1 � s1), forall i, and such that for each i, θλ � θi, μ(λ) � si for allλ ∈ Λi.Let p be the second-round priorities obtained by

giving each student λ a top second-round prioritypλi � ni at their first-round assignment μ(λ) � si (ifsi ∈ S) and unchanged priority pλj � pλj at all otherschools sj �� si. We say that a Pareto-improving cycle(in a second-round assignment) respects (second-round)priorities if pθi

si+1 ≥ pθi+1si+1 for all i (where we define

θm+1 � θ1).

Definition 10. A two-round mechanism is constrainedPareto efficient if the second-round assignment has noPareto-improving cycles that respect second-roundpriorities.

We remark that this is the same notion of effi-ciency that is satisfied by static, single-round DA-STB(Definition 3)—the resulting assignment has no Pareto-improving cycles that respect priorities. In other words,the constrained Pareto-efficiency requirement is in-formally to be “as efficient as static DA.”We also notehere that as a result of the requirement to respectsecond-round priorities, Pareto-improving cycles con-sidered must include only reassigned students.Finally, for equity purposes, it is desirable that a

mechanism be anonymous.

Definition 11. A two-round mechanism is anonymous ifstudents with the same first-round assignment and thesame first- and second-round preference reports havethe same distribution over second-round assignments.

We show that PLDA mechanisms satisfy all theaforementioned properties.

Proposition 1. PLDA mechanisms respect guarantees andpriorities and are nonwasteful, constrained Pareto efficient, andanonymous. Moreover, if student preferences are consistent,PLDA mechanisms are strongly two-round strategy-proof.

Wewill show in Section 3.1 that in a settingwithoutpriorities, the PLDA mechanisms are the only mech-anisms that satisfy all these properties (and some ad-ditional technical requirements), even ifwe only requireweak strategy-proofness (Theorem 3).Finally, it is simple to show that the natural coun-

terparts to PLDA mechanisms in a discrete setting


(with a finite number of students) respect guaranteesand priorities and are nonwasteful, constrained Paretoefficient, and anonymous. We make these claims for-mal in the online appendix and also provide an in-formal argument that the discrete PLDA mechanismsare also approximately strategy-proof when the num-ber of students is large.

3. Main ResultsIn this section, we show that the defining character-istic of a PLDA mechanism—the permutation of lot-teries between the two rounds—can be chosen toachieve desired operational goals.We first provide anintuitive order condition and show that, under thiscondition, all PLDA mechanisms give the same exante allocative efficiency. Thus, when the primitivesof the market satisfy the order condition, it is possibleto pursue secondary operational goals without sac-rificing allocative efficiency. Next, in the context ofreassigning school seats at the start of the school year,we consider the specific problem of minimizing reas-signment and show that when the order conditionis satisfied, reversing the lottery minimizes reas-signment among all centralized PLDA mechanisms.In Section 5, we empirically demonstrate using datafrom NYC public high schools that reversing thelottery minimizes reassignment (among a subclass ofcentralized PLDA mechanisms) and does not signif-icantly affect allocative efficiency, even when theorder condition does not hold exactly. Our resultssuggest that centralized RLDA is a good choice ofmechanism when the primary goal is to minimizereassignments while providing a second-round as-signment with high allocative efficiency. In Section 3.1,we provide an axiomatic justification for PLDA mech-anisms, and later in Section 6, we discuss how thechoice of lottery permutation can be used to achieveother operational goals, such as maximizing the num-ber of students with improved assignments.

We begin by defining the order condition, whichwe will need to state our main results.

Definition 12. The order condition holds on a set of prim-itives (S, q,Λ, η) if for every priority class π, the first- andsecond-round school cutoffs under RLDA within thatpriority class are in the same order—that is, for allsi, sj ∈ S,

Cπ,i > Cπ,j ⇒ CRπ,i ≥ CR

π,j.

Weemphasize that the order condition is a conditionon the market primitives—namely, school capacitiesand priorities and student preferences (though check-ing whether it holds involves investigating the outputof RLDA). We may interpret the order condition asan indication that the relative demand for the schoolsis consistent between the two rounds. Informally

speaking, this means that the revelation of the out-side options does not change the order in whichschools are overdemanded. One important settingwhere the order condition holds is the case of uniformdropouts and a single priority type. In this setting, eachstudent independently with probability ρ either re-mains in the system and retains his or her first-roundpreferences in the second round or drops out of thesystem entirely; student first-round preferences andschool capacities are arbitrary. In Section 4, we use theuniform dropouts setting to provide some intuitionfor our general results.To compare the allocative efficiency of different

mechanisms, we define type equivalence of assign-ments. In words, two second-round assignments aretype equivalent if the masses of different studenttypesθ assigned to each school are the same across thetwo assignments.

Definition 13. Two second-round assignments μ and μ′are said to be type equivalent if

η λ ∈ Λ : θλ � θ, μ(λ) � si{ }( )

� η λ ∈ Λ : θλ � θ, μ′(λ) � si{ }( )

∀θ ∈ Θ and si ∈ S.

In our continuum model, if two two-round mech-anisms produce type-equivalent second-round as-signments, we may equivalently interpret them asproviding each individual student of type θ with thesame ex ante distribution (before lottery numbers areassigned) over assignments.Our first main result is the surprising finding that

all PLDAs are allocatively equivalent.

Theorem 1 (Order Condition Implies Type Equivalence). Ifthe order condition (Definition 12) holds, all PLDA mech-anisms produce type-equivalent second-round assignments.

Thus, if the order condition holds, the measure ofstudents of type θ ∈ Θ assigned to each school in thesecond round is independent of the permutation P.We remark that type equivalence does not imply anequal (or similar) amount of reassignment (e.g., seeFigure 1) because type equivalence depends only onthe second-round assignment, whereas reassignment(Definition 2) measures the difference between thefirst- and second-round assignments. This brings usto our second result.

Theorem 2 (Reverse Lottery Minimizes Reassignment). Ifall PLDAmechanisms produce type-equivalent second-roundassignments, then RLDA minimizes the measure of reas-signed students among PLDA mechanisms.

The intuition for this result is that reversing thelottery shortcuts improvement chains and moves afew students many schools up their preference listinstead of a large number of students a few schoolsup their preference list. In particular, we show that


RLDA never reassigns both a student of type θ into aschool si and another student of type θ out of the sameschool si.

Proof of Theorem 2. Fix θ � (�θ, �θ, pθ) ∈ Θ and si ∈ S.We show that among all type-equivalent mechanisms,RLDA minimizes the measure of students with type θwho were reassigned to si because it never reassignsboth a student of type θ into a school si and anotherstudent of type θ out of si.

For every permutation P, let the measures of stu-dents with type θ leaving and entering school siin the second round under PLDA(P) be denoted by�P � η({λ ∈ Λ : θλ � θ, μ(λ) � si, μP(λ) �� si}) and eP �η({λ ∈ Λ : θλ � θ, μ(λ) �� si, μP(λ) � si}), respectively.As a result of type equivalence, there is a constant c,independent of P, such that �P � eP − c. We will showthat eR ≤ eP for all permutations P, specifically byshowing that either �R � 0 and eR � c or eR � 0.

If both eR > 0 and �R > 0, then students of type θwho entered si in the second round of RLDA hadworse first- and second-round lottery numbers thanstudents of type θ who left si in the second round ofRLDA, which contradicts the reversal of the lottery.Because eP � �P + c ≥ c and eP ≥ 0, this completes theproof. □

Our results present a compelling case for using thecentralized RLDA mechanism when the main goalsare to achieve allocative efficiency and minimize thenumber of reassigned students. Theorems 1 and 2show that when the order condition holds, central-ized RLDA is unequivocally optimal in the classof PLDA mechanisms because all PLDA mecha-nisms give type-equivalent assignments and centralizedRLDAminimizes the number of reassigned students. Inaddition, we remark that the order condition can bechecked easily by running RLDA (e.g., on historicaldata).15 Finally, even if the order condition does nothold, RLDAmoves as few students as possible to reachthe RLDA assignment.

Next, we give examples of when the order conditionholds and does not hold and illustrate the resultingimplications for type equivalence. We illustrate thesein Figure 3.

Example 1. There are N � 2 schools, each with a singlepriority group. School s1 has lower capacity and isinitially more overdemanded. Student preferences aresuch that when all students who want only s2 drop out,the order condition holds, and when all students whowant only s1 drop out, then s2 becomes more over-demanded under RLDA, and the order condition doesnot hold.

School capacities are given by q1 � 2, q2 � 5. There ismeasure 4 of each of the four types of first-roundstudent preferences. Let θi denote the student type

that finds only school si acceptable, and let θi,j denotethe type thatfinds both schools acceptable and preferssi to sj. (We will define the second-round preferencesof each student type below; each typewill either leavethe system completely or keep the same preferences.)If we run DA-STB, the first-round cutoffs are (C1,C2) � (34 , 12).Suppose that all type θ2 students leave the system

and all students of other types stay in the system andkeep the same preferences as in the first round. Thisfrees up 2 units at s2. Under RLDA, the second-roundcutoffs are (CR

1 , CR2 ) � (1, 34). In this case, the order con-

dition holds, and FLDA and RLDA are type equiva-lent. It is simple to verify that both FLDA and RLDAassign measure μ(s) of students of type (θ1, θ1,2, θ2,1)to school s, where

μF � μR � μ(s1), μ(s2)( ) � (1, 1, 0), (0, 2, 3)( ).Suppose that all typeθ1 students leave the system andall students of other types stay in the system and keepthe same preferences as in the first round. This freesup 1 unit at s1. Under RLDA, no new students areassigned to s2, and the previously bottom-ranked (butnow top-ranked) measure 1 of students who find s1acceptable are assigned to s1. Hence, the second-roundcutoffs are (CR

1 , CR2 ) � (78 , 1). In this case, the order con-

dition does not hold. Type equivalence also does nothold because the FLDA and RLDA assignments are

μF � (2, 0, 0), 1/3, 7/3, 7/3( )( ),μR � (1.5, 0.5, 0), (1, 2, 2)( ).

3.1. Axiomatic Justification of PLDA MechanismsWe have shown that PLDA mechanisms satisfy anumber of desirable properties. Namely, PLDA mech-anisms respect guarantees and priorities and aretwo-round strategy-proof (in a strong sense), non-wasteful, constrained Pareto efficient, and anony-mous. In this section, we show that in a setting with asingle priority class, the PLDA mechanisms are theonly mechanisms that satisfy all these properties aswell as two mild technical conditions on the sym-metry of the mechanism, even when we require onlythe weaker version of two-round strategy-proofness.

Definition 14. A two-round mechanism satisfies theaveraging axiom if for every type θ and pair of schools(s, s′), the randomization of the mechanism does notaffect the measure of students with type θ assigned to(s, s′) in the first and second rounds, respectively. Thatis, for all θ, s, s′, there exists a constant cθ,s,s′ such thatη({λ ∈ Λ : θλ � θ, μ(λ) � s, μ(λ) � s′}) � cθ,s,s′ w.p. 1.

In other words, the averaging axiom requires thatdespite potential randomness in how the mechanismproduces both first- and second-round assignments


for individual students, the distribution over (first-round assignment, second-round assignment) for stu-dents of a given type is deterministic. The averagingaxiom serves the purpose of excluding mechanismswith aggregate randomness, which adds unnecessarycomplexity to the assignment process. For example, amechanism that with probability 1/2 runs FLDA andwith probability 1/2 runs RLDA satisfies all our otheraxioms besides averaging and violates averaging byadding unnecessary uncertainty. (Instead of relyingon a coin-flip decision between FLDA and RLDA,the designer could choose the better option betweenthe two based on their goals.) The averaging axiomprecludes such mechanisms, and its particular tech-nical form facilitates a sharp characterization of PLDAmechanisms.

Definition 15. A two-round mechanism is nonatomicif any single student changing his or her preferences

has no effect on the assignment probabilities of otherstudents.Our characterization result is the following.

Theorem3. Suppose that student preferences are consistentand student types have full support (Assumptions 1 and 2).A nonatomic two-round assignment mechanism with firstround DA-STB respects guarantees and is nonwasteful,(weakly) two-round strategy-proof, constrained Pareto effi-cient, anonymous, and averaging if and only if the second-round assignment is given by PLDA.

We remark that we require two-round strategy-proofness only for students whose true preferencetype is consistent. This is because preference incon-sistencies across rounds can lead to conflicts betweenthe desired first-round assignment with respect tofirst-round preferences and the desired first-roundguarantee with respect to second-round preferences,making it unclear how to even define a best response.

Figure 3. (Color online) In Example 1, FLDA and RLDA Are Type Equivalent When the Order Condition Holds and GiveDifferent Assignments to Students of Every Type When the Order Condition Does Not Hold

Notes. The initial economy and first-round assignment are depicted on the top left. On the right, we show the second-round assignments underFLDA and RLDAwhen type θ2 students (who want only s2) drop out and when type θ1 students (who want only s1) drop out. Students towardthe left have larger first-round lottery numbers. The patterned boxes above each column of students indicate the affordable sets for students inthat column. When students who want only s2 drop out, the order condition holds, and FLDA and RLDA are type equivalent. When studentswho want only s1 drop out, s2 becomes more overdemanded in RLDA, and FLDA and RLDA give different ex ante assignments to students ofevery remaining type.


Moreover, it is reasonable to assume that studentswho are sophisticated enough to strategize aboutmisreporting in the first round in order to affect theguarantee structure in the second round will alsoknow their second-round preferences over schools inS (i.e., everything except where they rank their out-side option) at the beginning of the first round andhence will have consistent preferences.16 We remarkalso that the “only if” direction of this result is the onlyplace where we require the full support assumption(Assumption 2).

The main focus of our result is the effect of cross-round constraints. By assumption, the first-roundmechanism is DA-STB. It is relatively straightfor-ward to deduce that the second-round mechanismalso has to be DA-STB. Strategy-proofness in the sec-ond round, together with nonwastefulness, respectingpriorities and guarantees, and anonymity, constrainsthe second round to be DA, with each student givena guarantee at the school to which he or she wasassigned in the first round, and constrained Paretoefficiency forces the tie breaking to be in the sameorder at all schools. The cross-round constraints aremore complicated but can be understood using af-fordable sets. A student’s affordable set is the set ofschools that he or she can choose to attend—that is,the first-round affordable set is the set of schools forwhich he or she meets the first-round cutoff, and theaffordable set is the set of schools for which he or shemeets the first- or second-round cutoff. The set ofpossible affordable sets is uniquely determined bythe order of cutoffs. By carefully using two-roundstrategy-proofness and anonymity, we show that astudent’s preference type does not affect the jointdistribution over his or her first-round affordableset and affordable set, and hence his or her second-round lottery is a permutation of his or her first-roundlottery that does not depend on his or her prefer-ence type.

Our result mirrors similar large market cutoffcharacterizations for single-round mechanisms byLiu and Pycia (2016) and Ashlagi and Shi (2014),which show, in settings with single and multiple pri-ority types, respectively, that a mechanism is non-atomic, strategy-proof, symmetric, and efficient (in eachpriority class) if and only if it can be implemented bylottery-plus-cutoff mechanisms, which provide ran-dom lottery numbers to each student and admit them totheir favorite school for which they meet the admissioncutoff. We obtain such a characterization in a two-round setting using the fact that the mechanism re-spects guarantees and introducing an affordable setargument to isolate the second round from the first.This simplification allows us to employ argumentssimilar to those used in Liu and Pycia 2016 andAshlagi and Shi 2014 to show that the first- and

second-round mechanisms can be individually charac-terized using lottery-plus-cutoff mechanisms.

4. Intuition for Main ResultsIn this section,we provide some intuition for ourmainresults. This section may be skipped at a first readingwithout loss of continuity.A key insight is thatwe simplify analysis by shifting

away from assignments, which depend on prefer-ences, to considering the schools that a student canattend, which are independent of his or her prefer-ences. Specifically, if we define the affordable set foreach student as the set of schools for which he or shemeets either the first- or second-round cutoffs, theneach student is assigned to his or her favorite school inhis or her affordable set at the end of the secondround, and changing the student’s preferences doesnot change his or her affordable set in our contin-uum model. Moreover, affordable sets and prefer-ences uniquely determine demand.The main technical idea that we use in establishing

our main results is that the order condition is equiva-lent to the following seemingly much more powerful“global” order condition.

Definition 16. We say that PLDA(P) satisfies the localorder condition on a set of primitives (S, q,Λ, η) if, forevery priority classπ, the first- and second-round schoolcutoffs within that priority class are in the same orderunder PLDA(P). That is, for all si, sj ∈ S,

Cπ,i > Cπ,j ⇒ CPπ,i ≥ CP

π,j.

We say that the global order condition holds on a set ofprimitives (S, q,Λ, η) ifa. (Consistency across rounds) PLDA(P) satisfies

the local order condition on (S, q,Λ, η) ∀P.b. (Consistency across permutations) For every

priority class π, for all pairs of permutations P,P′ andschools si, sj ∈ S ∪ {sN+1}, it holds that CP

π,i > CPπ,j ⇒

CP′π,i ≥ CP′

π,j.

In other words, the global order condition requiresthat all PLDAmechanisms result in the same order ofschool cutoffs in both rounds. Surprisingly, if thecutoffs are in the same order in both rounds underRLDA, then they are in the same order in both roundsunder any PLDA.

Theorem 4. The order condition (Definition 12) holds for aset of primitives (S, q,Λ, η) if and only if the global ordercondition holds for (S, q,Λ, η).The affordable set framework can provide some

intuition as to why Theorem 4 holds. Under the re-verse permutation, the sets of schools that enter astudent’s affordable set in thefirst and second rounds,respectively, are maximally misaligned. Hence, ifas required by the local order condition the cutoff


order is consistent across both rounds under the re-verse permutation, then the cutoff order should alsobe consistent across both rounds under any otherpermutation.

4.1. Global Order Condition ImpliesType Equivalence

The affordable set framework can also be used toshow that when the global order condition holds, allPLDA mechanisms produce type-equivalent assign-ments. Fix a mechanism and suppose that the first-and second-round cutoffs are in the same order. Theneach student λ’s affordable set is of the form Xi �{si, si+1, . . . , sN} for some i � i(λ), where schools areindexed in decreasing order of their cutoffs for therelevant priority group pθ

λ, and the probability that a

student receives some affordable set is independent ofhis or her preferences. Moreover, because affordablesets are nested, X1 ⊇ X2 ⊇ · · · ⊇ XN , and because thelottery order is independent of student types, thedemand for schools is uniquely identified by the pro-portion of students whose affordable set contains sifor each i. When the global order condition holds, thisis true for every PLDAmechanism individually, whichprovides enough structure to induce type equiva-lence. We formalize this argument in the online ap-pendix (Lemmas 1 and 2).

4.2. Uniform DropoutsWenow introduce a special case of ourmodel. For thisspecial case, we prove that the order condition holds.It follows that all PLDA mechanisms give type-equivalent assignments.

Definition 17 (Informal). A market satisfies uniformdropouts if there is exactly one priority group at eachschool, students leave the system independentlywith some fixed probability ρ, and the students whoremain in the system retain their preferences.

Intuitively, in the uniform dropouts model, eachstudent drops out of the system with probabilityρ—for example, because he or she leaves the city afterthe first round for reasons that are independent of theschool choice system. The second-round problem canthus be viewed as a rescaled version of the first-roundproblem. This suggests that schools should fill in thesame order regardless of the choice of permutationbecause the measure of remaining students whowereassigned to each school si in the first round is (1 − ρ)qi,the measure of students of each type θ assigned toeach school is scaled down by 1 − ρ, the capacityof each school is still qi, and the measure of studentsof each type θ who are still in the system is scaleddown by 1 − ρ.

Formally, let student types bedefinedbyθ � (�θ, �θ, 1)(which we will write as θ � (�θ, �θ) because there

are no priorities). We define uniform dropouts withprobability ρ by

ζ θ � �θ, �θ( )

∈ Θ :�θ��, �θ � sN+1 � . . .{ }( )

� ρζ θ � �θ, �θ( )

∈ Θ :�θ��{ }( )

,

ζ θ � �θ, �θ( )

∈ Θ :�θ��, �θ ��{ }( )

� 1 − ρ( )

ζ θ � �θ, �θ( )

∈ Θ :�θ��{ }( )

. (3)That is, all students with probability ρ find the out-side option sN+1 the most attractive in the secondround and otherwise retain the same preferences inthe second round.17

Theorem 5. In any market with uniform dropouts(Definition 17), the global order condition (Definition 16)holds.

We prove Theorem 5 in the online appendix. Themain steps of the proof mirror those used to proveTheorem 4, albeit in a simpler and more transparentsetting, and can provide the interested reader with ataste of our more general proof techniques.

Remark. We can also use the uniform dropouts settingto explore what happens when students’ assignmentsaffect their preferences. We say that a market satisfiesuniform dropouts with inertia if there is exactly onepriority group at each school, students leave the systemindependently with some fixed probability ρ, studentsremain and wish to stay at their first round assignmentwith some fixed probability ρ′ (have “inertia”), andstudents otherwise remain and retain their first-roundpreferences.18 It can be shown that in such amarket, theglobal order condition always holds, and RLDA min-imizes reassignment among all type-equivalent alloca-tions. Moreover, if all students are assigned in the firstround, it can also be shown that PLDA mechanismsproduce type-equivalent allocations.

5. Empirical Analysis ofPLDA Mechanisms

In this section, we use data from the NYC high schoolchoice system to simulate and evaluate the perfor-mance of centralized PLDAmechanisms under differentpermutations P. The simulations indicate that althoughour theoretical results are for markets satisfying the ordercondition, they are real-world relevant. Different choicesof P yield similar allocative efficiency: the number ofstudents assigned to their kth choice for each rank kand the number of students remaining unassigned aresimilar for different permutations P. At the same time,the difference in the number of reassigned students issignificant and is minimized under RLDA.Motivated by current practice, we also simulate de-

centralized versions of FLDA and RLDA. In a versionwhere students take time to vacate previously assigned


seats, reversing the lottery increases allocative efficiencyduring the early stages of reassignment and decreasesthe number of reassignments at every stage. However,in a versionwhere students take time to decide on offersfrom the waitlist, the efficiency comparisons are re-versed.19 In both versions, both FLDA andRLDA tooktens of stages to converge. Our simulations suggestthat decentralized waitlist mechanisms can achievesome of the efficiency gains of a centralized mecha-nism but incur significant congestion costs, and the ef-fects of reversing the tie-breaking order before con-structingwaitlists will depend on the specific time andinformational constraints of the market.

5.1. DataWe use data from the high school admissions processin NYC for the academic year 2004–2005 as follows.

5.1.1. First-Round Preferences. In our simulation, wetake the first-round preferences � of every student tobe the preferences they submitted in the main roundof admissions. The algorithm used in practice is es-sentially strategy-proof (see Abdulkadiroglu et al.2005a), justifying our assumption that reported prefer-ences are true preferences.20

5.1.2. Second-Round Preferences. In our simulation,students either drop out of the system entirely in thesecond round or maintain the same preferences. Stu-dents are considered to drop out if the data do notrecord them as attending any public high school inNYC the following year (this was the case for about 9%of the students each year).21

5.1.3. School Capacities and Priorities. Each school’scapacity is set to the number of students assigned toit in the first-round assignment in the data. This is alower bound on the true capacity but lets us computethe final assignment under PLDA with the true ca-pacities because the occupancy of each school withvacant seats decreases across rounds in our setting.School priorities over students are obtained directlyfrom the data. (We obtain similar results in simula-tions with no school priorities.)

5.2. SimulationsIn a setting with a finite number of students, DA-STBuses an iterative process of student application andschool tentative acceptance to assign students accord-ing to student preferences and school preferencerankings after tie breaking, as described in Section 2.2.PLDA mechanisms are reassignment mechanismsthat run DA-STB with modified school preferences pin the second round: for each school si, students λ ∈ Λfor whom μ(λ) � si are given additional priority ni atschool si to produce updated priorities p, and ties

within the updated priority groups p are brokenaccording to the permuted lottery P ◦ L (in favor of thestudent with the larger permuted lottery number).

5.2.1. Centralized PLDA. We first consider the fol-lowing family of centralized PLDA mechanisms, pa-rameterized by a single parameter α that smoothlyinterpolates between RLDA and FLDA. Each studentλ receives a uniform independent and identically dis-tributed (i.i.d.) first-round lottery number L(λ) (anormal variable with mean 0 and variance 1) thatgenerates a uniformly random lottery order.22 Thesecond-round “permuted lottery” of λ is given byαL(λ) + L(λ), where L(λ) is a new i.i.d. normal variablewith mean 0 and variance 1, and α is identical for allstudents. RLDA corresponds to α � −∞, and FLDAcorresponds to α � ∞. For a fixed real α, every re-alization of second-round scores corresponds to somepermutation of first-round lottery numbers, with αroughly capturing the correlation of the second-roundorder with that of the first round. We quote averagesacross simulations.

5.2.2. Decentralized PLDA. In order to evaluate theperformance of waitlist systems, we experiment withtwo different versions of managing the waitlists ina decentralized fashion. The starting point for bothversions is the first-round assignment of students toschools. Each school maintains an ordered waitlist ofstudents constructed as follows: any student who is notassigned to his or her most-preferred school is auto-matically included in the waitlist at every school he orshe strictly prefers to his or her first-round assignment,and each school orders its waitlist in priority order,breaking ties using the chosen permutation of the first-round lottery numbers. We emphasize that studentscould appear on multiple waitlists and that individualschools do not know the preferences of the students,the priority of a student at another school, or thenumber of available seats at other schools.

Version 1. The algorithm proceeds iteratively in stages.The residual capacity of a school at the beginning of astage is its original capacity minus the number of stu-dents in the system who are assigned to that school atthat time. (Thus, if a school was filled to capacity in thefirst round, its residual capacity at the beginning of thefirst stage of the second round would simply bethe number of its assigned students who dropped outof the system.) At the beginning of any stage �, eachschool si makes an offer of admission to the top q�istudents on itswaitlist (or to all students on itswaitlistif there are fewer students), where q�i is the residualcapacity of school i at the beginning of stage �. Notethat these include offers to students who have droppedout of the system because the school is not aware of this


fact. A student who drops out of the system rejects alloffers; students still in the system accept their most-preferred school among their first-round assignmentand all offers received at this stage of the secondround and reject every other offer. The schools updatetheir waitlists and residual capacity, and the algo-rithm proceeds to the next stage.

There can be two types of “wasted” offers at eachstage of version 1: a school may make an offer to astudent who has already dropped out of the systembecause the school does not know that the student hasdropped out (recall that when a student drops out,only his or her assigned school in the first round isnotified), and multiple schools may make offers to astudent at the same stage of the second round, and the“losing” schools come to know only at the end of thestage. Such wasted offers are avoided by version 2 ofthe algorithm described next.

Version 2. The algorithm proceeds iteratively in stages.The residual capacity of a school at the beginning of astage is its original capacity minus the number of stu-dents in the system who are assigned to that school atthat time. At each stage �, the school-proposing DAalgorithm (Definition 18) is run, with the school ca-pacities being the residual capacities at that stage,student preferences restricted to schools that the studentstrictly prefers to his or her current assignment, andties broken using the chosen permutation of first-roundlottery numbers.

In contrast to version 1, here the underlying prem-ise is that students who have dropped out or thosewho receive multiple offers immediately reject all buttheir most-preferred offer, thus allowing schools tomake more offers within the stage than their residual

capacity permits if some offers are rejected. However,as in version 1, schools adjust their residual capacityonly at the end of the stage. That is, a student whoaccepts an offer from a school at some stage � in thesecond round does not notify his or her previouslyassigned school until the end of the stage (equiva-lently, that school is allowed to fill this student’s spotonly in stage � + 1).Version 1 of the decentralized PLDA mechanisms

mirrors a decentralized process where students taketime tomake decisions. However, it does so in a rathernaive fashion by assuming that students take thesame unit of time (one stage) both to directly respondto offers (i.e., accept or reject an offer) and to indirectlyrespond (i.e., notify a school that they were previ-ously assigned to that they have been assigned to adifferent school). Version 2 captures a decentralizedprocess where direct responses to offers are instan-taneous, so rejected offersmay result inmany offers ina single stage, whereas updates to schools that areindirectly affected happen only at the end of eachstage. Accordingly, the efficiency outcomes at a givenstage of version 2 dominate those of version 1 at thesame stage because more information is communi-cated during each stage.In practice, we expect that the dynamics of waitlist

systems would lie somewhere on the spectrum betweenthese two extreme versions of decentralized PLDA.

5.3. ResultsThe results of our centralized PLDA computationalexperiments based on 2004–2005 NYC high schooladmissions data appear in Table 1 and Figure 4.Figure 4 shows that the mean number of reassign-ments is minimized at α � −∞ (RLDA) and increases

Table 1. Centralized PLDA Simulation Results: 2004–2005 NYC High School Admissions

αReassignments

NUnassigned

%k � 1%

k ≤ 2%

k ≤ 3%

Round 1 (noreassignment)

0 9.31 50.14 64.14 72.44

Round 2FLDA: ∞ 7,797 5.89 55.41 69.85 78.038.00 7,606 5.90 55.40 69.85 78.026.00 7,512 5.90 55.40 69.85 78.034.00 7,325 5.89 55.38 69.84 78.022.00 6,863 5.89 55.33 69.81 78.020.00 5,220 5.87 54.96 69.65 77.97−2.00 3,686 5.81 54.52 69.37 77.82−4.00 3,480 5.79 54.47 69.33 77.78−6.00 3,433 5.79 54.46 69.32 77.77−8.00 3,416 5.79 54.45 69.31 77.77RLDA: −∞ 3,391 5.79 54.45 69.30 77.75

Notes. We show the mean percentage of students remaining unassigned or getting at least their kthchoice averaged across 100 realizations for each value of α. All percentages are out of the total number ofstudents remaining in the second round. The data contained 81,884 students, 74,366 students remainingin the second round, and 652 schools. The percentage of students who dropped out was 9.18%. Thevariation in the number of reassignments across realizations was ∼ 100 students.


with α, which is consistent with our theoretical resultin Theorem 2. The mean number of reassignments isas large as 7,800 under FLDA compared with just3,400 under RLDA.

Allocative efficiency appears not to vary muchacross values of α: the number of students receiving atleast their kth choice for each 1 ≤ k ≤ 12 and thenumber of unassigned students vary by less than 1%of the total number of students. There is a slight trade-off between allocative efficiency resulting from reas-signment and allocative efficiency from assigningpreviously unassigned students, with the percentageof unassigned students and percentage of studentsobtaining their top choice both decreasing inα by about0.1%and 1%of students, respectively.23We further findthat for most students, the likelihoods of getting oneof their top k choices under FLDA and under RLDAare very close to each other. (For instance, for 87% ofstudents, these likelihoods differ by less than 3% forall k.) This is consistent with what we would expectbased on our theoretical finding of type equivalence(Theorem 1) of the final assignment under differentPLDA mechanisms.

The results of our decentralized PLDA computa-tional experiments appear in Table 2. When imple-menting PLDAs in a decentralized fashion, our mea-sures of congestion can be more nuanced. We let areassignment be amovement of a student from a schoolin S to a different school in S, possibly during aninterim stage of the second round, and let a temporaryreassignment be amovement of a student from a schoolin S ∪ {sN+1} to a different school in S that is not his orher final assignment. We will also be interested in thenumber of stages it takes to clear the market.

In the first version of decentralized PLDAs, FLDAreassigns more students than RLDA but far out-performs RLDA in terms of minimizing congestionand maximizing efficiency. FLDA takes, on average,17 stages to converges, whereas RLDA requires 33stages. FLDA performs 780 temporary transfers,whereas RLDA performs 2,420, creating much moreunnecessary congestion. FLDA takes two and fivestages to achieve 50% and 90%, respectively, of the totalincrease in the number of students assigned to their topschool, whereas RLDA takes three and nine stages,respectively. FLDA also dominates RLDA in terms ofthe number of students assigned to one of their top kchoices in the first � stages, for all k and all �, and thepercentage of unassigned students in the first � stagesfor almost all small �.24

In the second version of decentralized PLDAs,FLDA still reassigns more students and now achievesless allocative efficiency than RLDA during the initialstages of reassignment. RLDA has fewer unassignedstudents by stage � than FLDA for all �. RLDA alsodominates FLDA in terms of the number of studentsassigned to one of their top k choices in the first twostages and achievesmost of its allocative efficiency bythe second stage, improving the allocative efficiencyby fewer than 100 students from that point onward. Inthe limit, FLDA is still slightly more efficient thanRLDA, so for large �, FLDA achieves higher allocativewelfare than RLDA after � stages. However, FLDAalso requires more stages to converge, taking, on av-erage, 12 stages compared with 9 stages for RLDA.Our empirical findings havemixed implications for

implementing decentralized waitlists. Our clearestfinding is the benefit of centralization in reducing

Figure 4. Number of Reassigned Students vs. α

Note. The number of reassigned students under the extreme values of α—namely, α � ∞ (FLDA) and α � −∞ (RLDA)—are shown by dotted lines.


congestion. In most school districts, students aregiven up to a week to make decisions. If students takethis long both to reject undesirable offers and to vacatepreviously assigned seats, our simulations on NYCdata suggest that in the best case the market couldtake at least 4 months to clear. Even if students makequick decisions, if it takes them a week to vacate theirpreviously assigned seats, our simulations suggestthat the market would take at least 2 months to clear.In both cases, the congestion costs are prohibitive. If,despite these congestion costs, a school districtwishesto implement decentralized waitlists, our results sug-gest that the optimal permutation for the second-roundlottery for constructing waitlists will depend on theinformational constraints in the market.

5.4. Strategy-Proofness of PLDAOne of the aspects of the DA mechanism that makesit successful in school choice in practice is that it isstrategy-proof. While we have shown that PLDAmechanisms are two-round strategy-proof in a con-tinuum setting, it is natural to ask to what extent

PLDA mechanisms are two-round strategy-proof inpractice. We provide a numerical upper bound on theincentives to deviate from truthful reporting usingcomputational experiments based on 2004–2005 NYChigh school data and find that, on average, a negli-gible proportion of students (< 0.01%) could benefitfrom misreporting within their consideration set ofprograms. Specifically, 0.8% of sampled students couldmisreport in a potentially beneficial manner in at least 1of 100 sampled lotteries, and no students could bene-fit in more than 3 of 100 sampled lotteries from mis-reporting. Moreover, for 99.8% of lotteries, the pro-portion of students who could successfully manipulatetheir report is at most 1%.25

6. Proposals and Discussion6.1. Summary of FindingsWe have proposed the PLDA mechanisms as a classof reassignment mechanismswith desirable incentiveand efficiency properties. These mechanisms can beimplemented with a centralized second round at thestart of the school year or with a decentralized second

Table 2. Decentralized PLDA Simulation Results: 2004–2005 NYC HighSchool Admissions

αReassignments Unassigned k � 1 k ≤ 2 k ≤ 3

Total (temporary) % % % %

Round 1 (no reassignments) 0 9.31 50.14 64.17 72.45Round 2 FLDA, version 1Stage 1 3,461 (447) 7.89 52.68 66.62 74.47Stage 2 2,126 (206) 7.04 53.93 68.03 76.14Stage 3 1,258 (80) 6.55 54.60 68.83 76.96Stage 4 727 (30) 6.27 54.97 69.28 77.42Stage 5 425 (11) 6.11 55.18 69.53 77.68Total (stage ≈ 17) 8,590 (780) 5.87 55.46 69.87 78.05Round 2 RLDA, version 1Stage 1 1,004 (835) 7.85 51.38 65.70 74.09Stage 2 1,077 (577) 7.18 52.24 66.72 75.15Stage 3 838 (369) 6.78 52.82 67.39 75.83Stage 4 640 (234) 6.52 53.23 67.86 76.30Stage 9 180 (24) 5.97 54.22 69.02 77.45Total (stage ≈ 33) 5,818(2419) 5.79 54.51 69.37 77.80Round 2 FLDA, version 2Stage 1 4,139 (457) 7.62 53.21 67.14 75.21Stage 2 2,333 (166) 6.69 54.50 68.66 76.75Stage 3 1137 (42) 6.24 55.06 69.35 77.48Stage 4 511 (9) 6.03 55.30 69.65 77.80Total (stage ≈ 12) 8,503 (677) 5.89 55.47 69.87 78.04Round 2 RLDA, version 2Stage 1 2,863 (199) 6.15 54.14 68.85 77.24Stage 2 489 (17) 5.88 54.38 69.16 77.58Stage 3 165 (2) 5.82 54.46 69.26 77.69Stage 4 63 (0) 5.79 54.49 69.30 77.73Total (stage ≈ 9) 3,624 (220) 5.79 54.51 69.33 77.76

Notes. We show themean number of reassignments (number ofmovements of a student from a school inS to a different school in S) and themean number of temporary reassignments (number of movements ofa student from a school in S ∪ {sN+1} to a different school in S that is not their final assignment) inparentheses. We also show mean percentage of students remaining unassigned or getting at least theirkth choice. All figures are averaged across 100 realizations for each value of α, and all percentages are outof the total number of students remaining in the second round. The data contained 81,884 students,74,366 students remaining in the second round, and 652 schools.


round via waitlists, and a suitable implementationcan be chosen depending on the timing of informa-tion arrival and subsequent congestion in the mar-ket. Moreover, the key defining characteristic of themechanisms in this class, the permutation used tocorrelate the tie-breaking lotteries between rounds,can be used to optimize various objectives. We pro-pose implementing centralized RLDA at the start ofthe school year because both in our theory and insimulations on data this allows us to maintain effi-ciency while eliminating the congestion caused bysequentially reassigning students and minimizes thenumber of reassignments required to reach an effi-cient assignment.

6.2. RLDA Is PracticalReversing the lottery between rounds is simple tounderstand and implement. It also has the nice propertyof being equitable in an intuitive manner because stu-dents who receive a poor draw of the lottery in the firstround are prioritized in the second round. This maymake RLDA more palatable to students than otherPLDA mechanisms. Indeed, Random Hall, a Massa-chusetts Institute of Technology (MIT) dorm, uses amechanism for assigning rooms that resembles thereverse-lottery mechanism we have proposed. Fresh-men rooms are assignedusing serial dictatorship.At theend of the year (after seniors leave), students can claimthe rooms vacated by the seniors using serial dicta-torship, where the initial lottery numbers (from theirfirst match) are reversed.26

6.3. Optimizing Other ObjectivesOur results suggest that PLDA mechanisms are anattractive class of mechanisms in more general set-tings, and the choice of PLDA mechanism will varywith the policy goal. If, for instance, it were viewedas more equitable to allow more students to receive(possibly small) improvements to their first-roundassignment, implementing FLDA optimizes this. Ourtype-equivalence result (Theorem 1) shows that whenstudents have consistent preferences and the ordercondition holds, this choice can be made withoutsacrificing allocative efficiency.

6.4. Discussion of Axiomatic CharacterizationOur characterization for PLDA mechanisms (Theo-rem 3) does not incorporate priorities. In amodel withpriorities, we find that natural extensions of our ax-ioms continue to describe PLDAmechanisms but alsoinclude undesirable generalizations of PLDA mech-anisms. Specifically, suppose that we add an axiomrequiring that for each school s, the probability that astudentwho reports a top choice of s then receives it inthe first or second round is independent of that stu-dent’s priority at other schools. This new set of axioms

describes a class of mechanisms that strictly includesthe PLDAmechanisms, as well as mechanisms wherethe permutation of the lottery can depend on students’priorities. Characterizing the class of mechanismssatisfying these axioms in the richer setting withschool priorities remains an open question. It may alsobe possible to characterize PLDA mechanisms in asettingwithpriorities using adifferent set of axioms.Weleave both questions for future research.

6.5. Finite MarketsIt is natural to ask what implications our results havefor finite markets. Azevedo and Leshno (2016) haveshown that if a sequence of (large) discrete economiesconverges to some limiting continuum economy witha unique stable matching (defined via cutoffs), thenthe stable matchings of the discrete economies con-verge to the stable matching of the continuum. Thissuggests that our theoretical results should approx-imately hold for large, discrete economies. As anexample, we provide a heuristic argument for whyPLDA mechanisms satisfy the strategy-proofness inthe large condition defined by Azevedo and Budish(2013). By definition, PLDA mechanisms satisfythe efficiency and anonymity requirements in finitemarkets as well. In the second round, it is clearly adominant strategy to be truthful, and intuitively, fora student to benefit from a first-round manipulation,his or her report should affect the second-roundcutoffs in a manner that gives him or her a second-round assignment that he or she would not have re-ceived otherwise. If the market is large enough, thecutoffs will converge to their limiting values, and theprobability that the student could benefit from such amanipulation would be negligible. (Indeed, in simu-lations on NYC high school data, we find that the av-erage proportion of students who can successfullymanipulate their report is < 0.01%; see Section 5.4.)A similar argument suggests that an approximateversion of our characterization result (Theorem 3)should hold for finite markets with no priorities.Our type-equivalence result (Theorem 1) and resultshowing that RLDA minimizes transfers (Theorem 2)should also be approximately valid in the largemarket limit.27

6.6. Inconsistent PreferencesAnother natural question is how to deal with incon-sistent student preferences. Narita (2016) observedthat in the current reapplication process in the NYCpublic school system, about 5% of students reappliedwith second-round preferences thatwere inconsistentwith their first-round reported preferences.28 PLDAsretain most of their desirable properties even whenstudents report inconsistent preferences in the sec-ond round. We believe that some of our insights on


optimizing allocative efficiency and reassignment re-main valid if a small fraction of students has an idio-syncratic change in preferences or if a small number ofnew students enter in the second round. We also be-lieve that our qualitative insights that reversing thelottery reduces reassignment and that the choice ofPLDA has smaller effects on allocative efficiency insettings where aggregate demand in the two roundsare well aligned and will continue to hold quite gen-erally. However, new effects may emerge if stu-dents have arbitrarily different preferences in the tworounds. In such settings, strategy-proofness is no lon-ger well defined. It can also be shown that the ordercondition is no longer sufficient to guarantee typeequivalence and optimality of RLDA, and the relativeefficiency of the PLDAmechanisms will depend on thedetails of school supply and student demand. Oursimulations further suggest that in such settings theremay be a more significant trade-off between decreas-ing the number of unassigned and reassigned studentsand increasing the efficiency of the final assignment.Mechanisms such asRLDA that prioritize studentswithpoor first-round assignments are likely to perform betteron the former, whereas mechanisms such as FLDA thatminimize the constraints imposed by first-round guar-antees will likely perform better on the latter. We leave atheoretical study of this trade-off for future research.

6.7. More Than Two RoundsFinally, what insights do our results provide for whenassignment is done in three or more rounds? For in-stance, one could consider mechanisms under whichthe lottery is reversed (or permuted) after a certainnumber of rounds and thereafter remains fixed. Atwhat stage should the lottery be reversed? Clearly,there are many other mechanisms that are reasonablefor this problem, and we leave a more comprehensivestudy of this question for future work.

AcknowledgmentsThe authors are grateful to Atila Abdulkadiroglu for in-troducing them to this problem, and Yusuke Narita andseminar participants at Caltech, Columbia, Cornell, Duke,Facebook Research, Georgia Tech, Google Research, JohnsHopkins, Hong Kong University of Science and Technology,Microsoft Research, Massachusetts Institute of Technology,Northwestern, National University of Singapore, Princeton,Purdue, Stanford, and Yale for their helpful comments. Theywould like to thank research assistants Paul Laliberte, AlexejProskynitopoulos, Pavithra Srinath, Avani Tanna, and CyndiaYu for their help with data analysis and simulations.

Endnotes1 In the 2004–2005 school year, 9.22% of a total of 81,884 studentsdropped out of the public school system after the first round. Numbersfor 2005–2006 and 2006–2007 are similar.2A decentralized version of FLDA is used in most cities and in NYCkindergarten admissions.

3Capacity constraints are binding inmost schools.Most states imposemaximum class sizes and fund schools based on enrollment after thefirst 2–3weeks of classes, which incentivizes schools to enroll asmanystudents as permissible.4We describe the decentralized reassignment processes currentlyused in NYC kindergarten; Boston; Washington, DC; Denver;Seattle; New Orleans; and Chicago. A similar process was also usedin NYC high school admissions until a few years ago, when thesystem abandoned reassignments entirely, anecdotally because ofthe excessive logistical difficulties created by market congestion.5 Studentswho have accepted an offer off thewaitlist of one school areallowed to accept offers off the waitlists of other schools. Becauseregistration for one school automatically cancels the student’s previousregistrations, this would automatically release the seat the studentaccepted from the first school to other students on the waitlist.6 See, for example, Abdulkadiroglu et al. (2005a, b) for an overviewof the redesigns in New York City (2003) and Boston (2005), respec-tively. These were followed by NewOrleans (2012), Denver (2012), andWashington, DC (2013)—among others. See Abdulkadiroglu et al.(2017) for welfare analysis of the changes in NYC.7Our continuum model can be viewed as a two-round version ofthe model introduced by Azevedo and Leshno (2016). Continuummodels have been used in a number of papers on school choice; seeAgarwal and Somaini (2018), Ashlagi and Shi (2014), and Azevedoand Leshno (2016). Intuitively, one can think of the continuummodelas a reasonable approximation of the discrete model in Online Ap-pendix B when the number of students is large, although establishinga formal relationship between the discrete and continuum models isbeyond the scope of our paper.8Consistency is required tomeaningfully define strategy-proofness inour two-round setting because we require truthful reporting in thefirst round to be optimal for both the student’s first- and second-round assignments.9This can be justified via an axiomatization of the kind obtained byAl-Najjar (2004).10This ensures that students will report their full first-round pref-erences in the first round instead of truncating in the first round basedon their beliefs about their second-round preferences.11 Several alternative definitions of reassigned students—such ascounting students who are initially unassigned and end up at a schoolin S and/or counting initially assigned students who end upunassigned—could also be considered. We note that our resultscontinue to hold for all these alternative definitions.12Here we make the restriction that the second-round assignmentdepends on the first-round reports only indirectly, through the first-round assignment μ. We believe that this is a reasonable restriction,given that the second round occurs a significant period of time afterthe first round, and the mechanism should come across as fair to thestudents.13 If the demand function is continuously differentiable in the cutoffs,the assignment is unique. For an arbitrary demand function, theresulting assignment is unique for all but a measure zero set of ca-pacity vectors.14When schools have strict preferences, an assignment respects pri-orities if and only if it is stable, and it is well known that in two-sidedmatching markets with strict preferences, there exist preferencestructures for which every stable assignment can be Pareto improved(Erdil and Ergin 2008).15We are not suggesting that the mechanism should involve checkingthe order condition and then using centralized RLDA only if thiscondition is satisfied (based on the guarantee in Theorems 1 and 2).However, one could check whether the order condition holds onhistorical data and accordingly decide whether to use the centralizedRLDA mechanism or not.


16One obvious objection is that students may also obtain extra utilityfrom staying at a school between rounds, or equivalently, they mayhave a disutility for moving, creating inconsistent preferences wherethe school they are assigned to in the first round becomes preferred topreviously more desirable schools. We remark that Theorem 3 ex-tends to the case of students whose preferences incorporate addi-tional utility if they stay put, provided that the utility is the same atevery school for a given student or satisfies a similar noncrossingproperty.17We remark that there is a well-known technical measurability issuewith regard to a continuum of random variables and that this issuecan be handled; see, for example, Al-Najjar (2004).18This market is slightly beyond the scope of our general modelbecause the type of the student now also has to encode second-roundpreferences that depend on the first-round assignment—namely,whether they have inertia.19This is due to a phenomenon that occurs when the second round isdecentralized (not captured by our theoretical model), where underthe reverse lottery the students with the worst lottery in the firstround increase the waiting time for other students in the secondround by increase the waiting time for other students in the secondround by considering multiple offers off the waitlist that theyeventually decline.20The algorithm is not completely strategy-proof because studentsmay rank no more than 12 schools. However, only a very smallpercentage of students rank 12 schools. Another issue is that there issome empirical evidence that students do not report their true pref-erences even in school choice systemswith strategy-proofmechanisms;see, for example, Hassidim et al. (2015) and Narita (2016).21 For a minority of the students (9.2%−10.45%), attendance in thefollowing year could not be determined by our data, and hence weassume that they drop out randomly at a rate equal to the dropoutrate for the rest of the students (9.2%).22 School preferences are then generated by considering students inthe lexicographic ordering first in terms of priority and then by lotterynumber. We may equivalently renormalize the set of realized lotterynumbers to lie in the interval [0, 1] before computing scores.23 Intuitively, prioritizing students with lower lotteries has the de-sirable effect of decreasing the number of unassigned students be-cause it prioritizes more students who were not assigned in the firstround. However, it also decreases allocative efficiency by artificiallyincreasing the constraints imposed by first-round guarantees.24The mechanism driving the inefficiency of decentralized RLDA isthat all offers in the first few stages of round 2 are made to a smallnumber of students: thosewith theworst first-round lottery numbers.This inefficiency should be mitigated when students respond quicklytomost of their offers, aswe see in the second version of decentralizedPLDA.25These upper bounds were computed as follows: approximately2,700 students were sampled, and RLDA was run for each of thesestudents using 100 different sampled lotteries. For a given student, letS be the set of schools that were a part of the student’s first-roundpreferences in the data. We allowed the student to unilaterallymisreport in the first round, reporting atmost one school from S in thefirst round instead of his or her true preferences. We then counted thenumber of such students who by doing so could either (a) changetheir first-round assignment (for the worse) but second-round as-signment for the better or (b) create a rejection cycle. This provides aprovable upper bound on the number of students who can benefitfrom misreporting (and possibly reordering) a subset of S in the firstround. We omit the formal details in the interest of space.26The MIT Random Hall matching is more complicated becausesophomores and juniors can also claim the vacated rooms, but thelottery only gets reversed at the end of freshman year. Afterward, if a

sophomore switches rooms, his or her priority drops to the last placeof the queue.27 Specifically, consider a sequence of markets of increasing size. If theglobal order condition holds in the continuum limit, this should leadto approximate type-equivalence under all PLDAs and to RLDAapproximately minimizing transfers among PLDAs in the finitemarkets as market size grows.Moreover, if the order condition holds,then in large, finite economies and for every permutation P, the set ofstudents who violate a local order condition on PLDA(P) will be smallrelative to the size of the market.28About 7% of students reapplied, and about 70% of these reap-plicants reported second-round preferences that were inconsistentwith their first-round reported preferences.

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