School Choice: A Mechanism Design Approach

By ATILA ABDULKADIROGLU AND TAYFUN SONMEZ*

A central issue in school choice is the design of a student assignment mechanism.Education literature provides guidance for the design of such mechanisms but doesnot offer specific mechanisms. The flaws in the existing school choice plans resultin appeals by unsatisfied parents. We formulate the school choice problem as amechanism design problem and analyze some of the existing school choice plansincluding those in Boston, Columbus, Minneapolis, and Seattle. We show that theseexisting plans have serious shortcomings, and offer two alternative mechanismseach of which may provide a practical solution to some critical school choice issues.(JEL C78, D61, D78, I20)

School choice is one of the widely discussedtopics in education.1 It means giving parents theopportunity to choose the school their child willattend. Traditionally, children are assigned topublic schools according to where they live.Wealthy parents already have school choice,because they can afford to move to an area with

good schools, or they can enroll their child in aprivate school. Parents without such means, un-til recently, had no choice of school, and had tosend their children to schools assigned to themby the district, regardless of the school qual-ity or appropriateness for the children. As aresult of these concerns, intra-district andinter-district choice programs have become in-creasingly popular in the past ten years.2 Intra-district choice allows parents to select schoolsthroughout the district where they live, andinter-district choice allows them to send theirchildren to public schools in areas outside theirresident districts. In 1987, Minnesota becamethe first state to oblige all its districts to estab-lish an inter-district choice plan (Allyson M.Tucker and William F. Lauber, 1995). Today,several states offer inter-district and intra-district choice programs.

Since it is not possible to assign each studentto her top choice school, a central issue inschool choice is the design of a student assign-ment mechanism.3 While the education litera-ture stresses the need for rigorous studentassignment mechanisms and provides guidance

* Abdulkadiroglu: Department of Economics, ColumbiaUniversity,NewYork,NY10027(e-mail:aa2061@columbia.edu); Sonmez: Department of Economics, Koc University,Sarıyer, 80910, Istanbul, Turkey (e-mail: tsonmez@ku.edu.tr). The previous version of this paper was entitled“School Choice: A Solution to the Student AssignmentProblem.” We are grateful to Michael Johnson for a discus-sion that motivated this paper. We thank Elizabeth Caucutt,Steve Ching, Julie Cullen, Dennis Epple, Roger Gordon,Matthew Jackson, Tarık Kara, George Mailath, Paul Mil-grom, Thomas Nechyba, Ben Polak, Phil Reny, Al Roth,Ariel Rubinstein, Larry Samuelson, Mark Schneider, JaySethuraman, Lones Smith, Rohini Somanathan, Ennio Stac-chetti, Mark Stegeman, William Thomson, Utku Unver,Steve Williams, Susan Zola, seminar participants at BilkentUniversity, Columbia University, HEC School of Manage-ment, Harvard-MIT, Hong Kong University, Koc Univer-sity, the University of Michigan, NYU, Sabancı University,the conference “Axiomatic Resource Allocation Theory” atNamur, 1999 North American Meetings of the EconometricSociety at Madison, Summer in Tel Aviv 1999, SED 2002Conference on Economic Design in New York, and espe-cially to three anonymous referees whose suggestions sig-nificantly improved the paper. We thank Michael Furchtgottand Ting Wu for excellent research assistance. Abdulka-diroglu gratefully acknowledges the research support ofSloan Foundation and Sonmez gratefully acknowledges theresearch support of the National Science Foundation viaGrant Nos. SBR-9709138 and SES-9904214. Any errors areour own responsibility.

1 Milton Friedman (1955, 1962) initiates the schoolchoice literature.

2 For empirical investigation of various issues in schoolchoice see Mark Schneider et al. (2000). Another line ofdiscussion has been on private school vouchers. See Caro-line M. Hoxby (1994), Dennis Epple and Richard E. Ro-mano (1998), Cecilia E. Rouse (1998), Thomas J. Nechyba(2000), and Raquel Fernandez and Richard Rogerson(2003).

3 Indeed, one of the key obstacles identified by the criticsof school choice concerns student selection to overde-manded schools (see Donald Hirch, 1994, p. 14).

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for design (see, for example, Michael J. Alvesand Charles V. Willie, 1990; Office of Educa-tional Research and Improvement, 1992; andTimothy W. Young and Evans Clinchy, 1992,Ch. 6), it does not offer specific mechanisms.4

Many of the real-life school choice plans haveprotocols and guidelines for the student assign-ment without explicit procedures. Saul Yanof-sky (see Office of Educational Research andImprovement, 1992, p. 19), the former superin-tendent of the White Plains Public Schools,states: “You need to have a set of proceduresthat are very explicit, with rules.”

The lack of rigorous procedures invites selec-tive interpretation and it often results in evasiveaction by the students and their parents. Con-sider the following statement by the SupremeCourt of Mississippi which affirms the judge-ment of a circuit court against a school district(Supreme Court of Mississippi, 2001)5:

We agree that the denial of Gentry’stransfer cannot be based on the alleged“middle-school” transfer policy sincethere is no written record outlining itssubstance. Such a denial based on thisvague policy would clearly be arbitraryand capricious.

Along similar lines consider the followingsummary of a court case concerning a schoolchoice plan in Wisconsin (Court of Appeals ofWisconsin, 2000)6:

McMorrow v. State Superintendent ofPublic Instruction

Respondent applied under open enroll-

ment, Wis. Stat. 118.51 (1997–98), to at-tend high school in a district where he didnot live. His application was denied andappellant affirmed, concluding that thedenial was supported by substantial evi-dence based on lack of class space; thus, itwas not arbitrary or unreasonable. Thecircuit court reversed. When three othercontinuing students were accepted eventhough space was not available, relianceon class size guidelines to deny respon-dent enrollment was arbitrary. The courtaffirmed. There was no substantial evi-dence to support appellant’s findings offact, and appellant erroneously inter-preted statutory provisions which gavepreference to continuing students onlywhen spaces were available in the firstplace. Under the statute, when therewere more applicants than spaces avail-able, admission selection was to be on arandom basis. Thus, accepting three stu-dents in spite of class size guidelinesand denying a fourth that same excep-tion without any explanation was arbi-trary and unreasonable.

Outcome: Judgement affirmed. StateSuperintendent of Public Instructions(SSPI) erred when it found that the openenrollment statute supported preferentialtreatment of three continuing studentswhen no class space was available; andwhen based on that finding, SSPI errone-ously concluded that denying respon-dent’s application for class space reasonswas not arbitrary or unreasonable.

Other school choice programs, such as thosein Boston, Minneapolis, and Seattle, are accom-panied by explicit procedures. However, each ofthese procedures has serious shortcomings. Un-der these procedures students with high priori-ties at specific schools lose their prioritiesunless they list these schools as their topchoices. Consequently, students and their par-ents are forced to play very complicated admis-sions games, and often, misrepresenting theirtrue preferences is in their best interest. This isnot only confusing to students and their parents,but also results in inefficient allocation ofschool seats.

In this paper we propose two competing stu-dent assignment mechanisms, each of whichmay be helpful in dealing with these criticalschool choice issues. A natural starting point is

4 This is a major problem for school choice programs inother countries as well. For example, Gulam-Husien Mayet(1997), the former Chief Welfare Adviser for the InnerLondon Education Authority, indicates that the lack ofsynchronization and transparency in admissions to Londonpublic schools is a major problem for parents and localauthorities.

5 Pascagoula Municipal Separate School District v. W.Harvey Barton and Renee Barton, as Parents and NextFriends of William Gentry Barton, A Minor. Supreme Courtof Mississippi, No. 2000-CC-00035-SCT, decided on Feb-ruary 1, 2001.

6 Michael E. McMorrow, Petitioner-Respondent, v. StateSuperintendent of Public Instructions, John T. Benson, Re-spondent-Appellant. Court of Appeals of Wisconsin, No.99-1288, decided on July 25, 2000.

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studying how similar allocation problems arehandled in real life as well as in the mechanismdesign literature. A closely related problem isthe allocation of dormitory rooms (or on-campus housing facilities) to students (AanundHylland and Richard Zeckhauser, 1979).7 Thefollowing mechanism, known as the randomserial dictatorship, is almost exclusively used inreal-life applications of these problems (Ab-dulkadiroglu and Sonmez, 1998, 1999): Orderthe students with a lottery and assign the firststudent her top choice, the next student her topchoice among the remaining slots, and so on.This mechanism is not only Pareto efficient, butalso strategy-proof (i.e., it cannot be manipu-lated by misrepresenting preferences), and itcan accommodate any hierarchy of seniorities.So why not use the same mechanism to allocateschool seats to students? The key difficulty withthis approach is the following: Based on stateand local laws, the priority ordering of a studentcan be different at different schools. For exam-ple:

● students who live in the attendance area of aschool must be given priority for that schoolover students who do not live in the school’sattendance area;

● siblings of students already attending aschool must be given priority; and

● students requiring a bilingual program mustbe given priority in schools that offer suchprograms.

Therefore a single lottery cannot be used toallocate school seats to students. It is thisschool-specific priority feature of the problemthat complicates the student assignment pro-cess. A student assignment mechanism shouldbe flexible enough to give students differentpriorities at different schools. This point directsour attention to another closely related problem,namely the college admissions problem (DavidGale and Lloyd S. Shapley, 1962).

College admissions problems have been ex-

tensively studied (see Alvin E. Roth andMarilda A. O. Sotomayor, 1990, for a survey)and successfully applied in British and Ameri-can entry-level labor markets (see Roth, 1984,1991). The central difference between the col-lege admissions and school choice is that incollege admissions, schools themselves areagents which have preferences over students,whereas in school choice, schools are merely“objects” to be consumed by the students. Thisdistinction is important because the educationof students is not and probably should not beorganized in a market-like institution. A studentshould not be rejected by a school because ofher personality or ability level. Despite this im-portant difference between the two models,school preferences and school priorities are sim-ilar mathematical objects and the college admis-sions literature can still be very helpful indesigning an appealing student admissionsmechanism.

The central notion in the college admissionsliterature is stability: There should be no un-matched student-school pair (i, s) where stu-dent i prefers school s to her assignment andschool s prefers student i to one or more of itsadmitted students. This mathematical propertyis equivalent to the following appealing prop-erty in the context of school choice, whereschools do not have preferences but instead theyhave priorities: There should be no unmatchedstudent-school pair (i, s) where student i pre-fers school s to her assignment and she hashigher priority than some other student who isassigned a seat at school s. Therefore a stablematching in the context of college admissionseliminates justified envy in the context of schoolchoice. Moreover it is well-known that thereexists a stable matching which is preferred toany stable matching by every student in thecontext of college admissions (Gale and Shap-ley, 1962). Since only the welfare of studentsmatters in the context of school choice, thismatching Pareto-dominates any other matchingthat eliminates justified envy. This series ofobservations motivate the following student ad-missions mechanism to the school choice prob-lem: Interpret school priorities as preferencesand select the student optimal stable matchingof the induced college admissions problem. Werefer to this mechanism as the Gale-Shapleystudent optimal stable mechanism. Since 1998,

7 See also Lin Zhou (1990), Lars-Gunnar Svensson(1999), Haluk I. Ergin (2000), Szilvia Papai (2000), JamesSchummer (2000), Anna Bogomolnaia and Herve Moulin(2001), Eiichi Miyagawa (2001, 2002), Lars Ehlers (2002),Ehlers et al. (2002), Andrew McLennan (2002), and Ab-dulkadiroglu and Sonmez (forthcoming).

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a version of this mechanism has been used inthe United States hospital-intern market (Rothand Elliot Peranson, 1997, 1999).

Gale-Shapley student optimal stable mecha-nism has one additional very appealing feature:It is strategy-proof. That is, truthful preferencerevelation is a dominant strategy for the stu-dents. In particular, students and their parentsdo not need to worry about losing their prioritiesas a consequence of reporting their truthful pref-erences. Gale-Shapley student optimal stablemechanism relieves students and their parentsof devising complicated admissions strategies.

However, Gale-Shapley student optimal sta-ble mechanism is not “problem free” in thecontext of school choice. While it Pareto-dominates any other mechanism that eliminatesjustified envy, its outcome may still be Pareto-dominated. That is because there is a potentialconflict between complete elimination of justi-fied envy and Pareto efficiency (see Example 1in Section II, subsection A). This observationmotivates the following question: Could therebe a milder interpretation of the priorities whichin turn does not cause a conflict with Paretoefficiency? The answer to this question is affir-mative. Suppose that if student i1 has higherpriority than student i2 for school s, that doesnot necessarily mean that she is entitled a seat atschool s before student i2. It rather representsthe opportunity to get in school s. If i1 hashigher priority than i2, then she has a betteropportunity to get in school s, other thingsbeing equal. This milder requirement is compat-ible with Pareto efficiency: Find all students,each of whom has the highest priority at aschool. Among these individuals there is agroup of students, all of whom can be assignedtheir top choices by trading their priorities. As-sign all such students their top choices and oncethey are removed, proceed in a similar waystarting with the students who have the highestpriorities among the remaining students. Werefer to this Pareto-efficient mechanism as thetop trading cycles mechanism. When all schoolshave the same priority ordering (say an orderingobtained from a common lottery draw), thismechanism reduces to the random serial dicta-torship which is commonly used in the alloca-tion of on-campus housing facilities. The toptrading cycles mechanism is a natural extensionof this mechanism, an extension which allows

for different priorities at different schools: Thesimple insight of assigning objects to agents oneat a time based on their priority can be simplyextended by assigning objects to top tradingcycles one cycle at a time based on priorities. Asin the case of the Gale-Shapley student optimalstable mechanism, the top trading cycles mech-anism is also strategy-proof. Therefore, thechoice between these two competing mecha-nisms depends on the structure and interpreta-tion of the priorities. In some applicationspolicy makers may rank complete eliminationof justified envy before full efficiency, thenGale-Shapley student optimal stable mechanismcan be used in those cases. Efficiency may beranked higher by others, and the top tradingcycles mechanism can be used in suchapplications.

One of the major concerns about the imple-mentation of school choice plans is that theymay result in racial and ethnic segregation atschools. Because of these concerns, choiceplans in some districts are limited by court-ordered desegregation guidelines. This versionof school choice is known as controlled choice.In many school districts (such as in Boston priorto 1999, as well as in Columbus and Minneap-olis) controlled choice constraints are imple-mented by imposing racial quotas at publicschools. An important advantage of both mech-anisms is that they can be easily modified toaccommodate controlled choice constraints byimposing racial quotas. Moreover, the modifiedmechanisms are still strategy-proof and themodified top trading cycles mechanism is con-strained efficient.

The mechanism design approach has recentlybeen very fruitful in many real-life resourceallocation problems. Important examples in-clude the design of FCC spectrum auctions [seeJohn McMillan (1994); Peter Cramton (1995);R. Preston McAfee and McMillan (1996); PaulMilgrom (2000)] and the redesign of the Amer-ican hospital-intern market [see Roth and Per-anson (1999); Roth (2002)]. This paper, to thebest of our knowledge, is the first paper toapproach the school choice problem from amechanism-design perspective. We believe thisapproach may be helpful in some critical schoolchoice issues.

The organization of the rest of the paper is asfollows: In Section I, we introduce the school

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choice model, give examples of real-life schooladmissions mechanisms, and illustrate theirshortcomings. In Section II, we introduce thetwo proposed mechanisms and analyze theirproperties. In Section III, we introduce con-trolled choice, modify the proposed mecha-nisms, and analyze them. In Section IV, weconclude. Finally, we present an example andinclude the omitted proofs in the Appendix.

I. School Choice

In a school choice problem there are a num-ber of students, each of whom should be as-signed a seat at one of a number of schools.Each school has a maximum capacity but thereis no shortage of the total seats. Each studenthas strict preferences over all schools, and eachschool has a strict priority ordering of all stu-dents. Here, priorities do not represent schoolpreferences but they are imposed by state orlocal laws. For example, in several states astudent who has a sibling already attending aschool is given priority for that school by theeducation codes. Similarly, students who livewithin walking proximity of a school are givenpriority. For each school, the priority betweentwo students who are identical in each relevantaspect is usually determined by a lottery.

The school choice problem is closely relatedto the well-known college admissions problemintroduced by Gale and Shapley (1962). Thecollege admissions problem has been exten-sively studied (see Roth and Sotomayor, 1990,for a survey) and successfully applied in theAmerican and British entry-level labor markets(see Roth, 1984, 1991). The key difference be-tween the two problems is that in school choice,schools are objects to be “consumed” by thestudents, whereas in college admissions,schools themselves are agents who have prefer-ences over students.

The outcome of a school choice problem is anassignment of schools to students such that eachstudent is assigned one school and no school isassigned to more students than its capacity. Werefer each such outcome as a matching. Amatching is Pareto efficient if there is no othermatching which assigns each student a weaklybetter school and at least one student a strictlybetter school. A student assignment mechanismis a systematic procedure that selects a matching

for each school choice problem. A student as-signment mechanism is a direct mechanism if itrequires students to reveal their preferencesover schools and selects a matching based onthese submitted preferences and student priori-ties. A student assignment mechanism is Paretoefficient if it always selects a Pareto-efficientmatching. A direct mechanism is strategy-proofif no student can ever benefit by unilaterallymisrepresenting her preferences.

Since it is not possible to assign each studenther top choice, a central issue in school choiceis the design of a student assignment mecha-nism (Hirch, 1994). In this paper we proposetwo direct student assignment mechanisms withdifferent strengths. Depending on the prioritiesof policy makers, either mechanism can bepractically implemented in real-life applicationsof school choice problems. Before we introduceand analyze these mechanisms, we describeand analyze some real-life student assignmentmechanisms.

A. Boston Student Assignment Mechanism

One of the common mechanisms is the directmechanism that is used by the city of Boston.The mechanism that we next describe has beenin use in Boston since July, 1999. Prior to thatanother version of the same mechanism thatimposed racial quotas was in use (United StatesDistrict Court for the District of Massachusetts,2002).8 Variants of the same mechanism arecurrently used in Lee County, Florida,9 Minne-apolis (Steven Glazerman and Robert H. Meyer,1994), and Seattle,10 among other school districts.

The Boston student assignment mechanismworks as follows:11

1. Each student submits a preference ranking ofthe schools.

8 Boston’s Children First et al. v. Boston School Com-mittee et al. United States District Court for the District ofMassachusetts, Civil Action Number: 99-11330-RGS, de-cided on January 25, 2002.

9 See http://www.lee.k12.fl.us/dept/plan/Choice/faqs.htm#13.

10 See page 12 of http://www.seattleschools.org/area/eso/elementaryenrollmentguide20022003.pdf.

11 See http://boston.k12.ma.us/teach/assign.asp. See alsothe case, Boston’s Children First et al. v. Boston SchoolCommittee cited in footnote 8.

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2. For each school a priority ordering is deter-mined according to the following hierarchy:

• First priority: sibling and walk zone.• Second priority: sibling.• Third priority: walk zone.• Fourth priority: other students.

Students in the same priority group are or-dered based on a previously announcedlottery.

3. The final phase is the student assignmentbased on preferences and priorities:

Round 1: In Round 1 only the first choices ofthe students are considered. For each school,consider the students who have listed it astheir first choice and assign seats of theschool to these students one at a time fol-lowing their priority order until either thereare no seats left or there is no student leftwho has listed it as her first choice.

Round 2: Consider the remaining students.In Round 2 only the second choices of thesestudents are considered. For each schoolwith still available seats, consider the stu-dents who have listed it as their secondchoice and assign the remaining seats tothese students one at a time following theirpriority order until either there are no seatsleft or there is no student left who has listedit as her second choice.

In general at

Round k: Consider the remaining students.In Round k only the kth choices of thesestudents are considered. For each schoolwith still available seats, consider the stu-dents who have listed it as their kth choiceand assign the remaining seats to these stu-dents one at a time following their priorityorder until either there are no seats left orthere is no student left who has listed it asher kth choice.

The major difficulty with the Boston studentassignment mechanism is that it is not strategy-proof. Even if a student has very high priority atschool s, unless she lists it as her top choice sheloses her priority to students who have listed s

as their top choices. Hence the Boston studentassignment mechanism gives very strong incen-tives to students and their parents to misrepre-sent their preferences by improving ranks ofthose schools for which they have high prior-ity.12 This point is also observed by Glazermanand Meyer (1994) for Minneapolis:

It may be optimal for some families to bestrategic in listing their school choices.For example, if a parent thinks that theirfavorite school is oversubscribed and theyhave a close second favorite, they may tryto avoid “wasting” their first choice on avery popular school and instead list theirnumber two school first.

Another difficulty with the Boston studentassignment mechanism concerns efficiency. Ifstudents submit their true preferences, then theoutcome of the Boston student assignmentmechanism is Pareto efficient. But since manyfamilies are likely to misrepresent their prefer-ences, its outcome is unlikely to be Paretoefficient.

B. Columbus Student Assignment Mechanism

The mechanism used by Columbus CitySchool District is not a direct mechanism and itworks as follows:13

1. Each student may apply to up to three dif-ferent schools.

2. For some schools, seats are guaranteed forstudents who live in the school’s regularassignment area and the priority among re-maining applicants is determined by a ran-dom lottery. For the remaining schools, thepriority among all applicants is determinedby a random lottery.

12 If a mechanism is not strategy-proof, that does notnecessarily mean that it can be easily manipulated. Forexample, Roth and Urial G. Rothblum (1999) show thatalthough the hospital optimal stable mechanism can bemanipulated by the interns, it is unlikely that such anattempt will be successful and hence truthful preferencerevelation is still in the best interest of the interns. In case ofthe Boston student assignment mechanism, the situation isquite different and the parents are warned to be careful howthey use their top choices.

13 See http://www.columbus.k12.oh.us/applications/FAQ.nsf/(deadline)?openview#19.

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3. For each school, available seats are offeredto students with the highest priority by alottery office and the remaining applicationsare put on a waiting list. After receiving anoffer a student has three days to accept ordecline an offer. If she accepts an offer, sheis assigned a seat; she then must declineoffers from other schools and she is removedfrom the waiting list of other schools towhich she has applied. As soon as seatsbecome available at schools because of de-clined offers, the lottery office makes offersto students on the waiting lists.

The Columbus student assignment mecha-nism is similar to the entry-level market forclinical psychologists in the United States (seeRoth and Xiaolin Xing, 1997). The market forclinical psychologists is more decentralized andeach employer makes its offers via telephone,whereas a centralized lottery office makes alloffers on behalf of each school for student as-signment at Columbus.

As in the case of the Boston student assign-ment mechanism, the optimal application strat-egy of students is unclear under the Columbusstudent assignment mechanism. When a familygets an offer from its second or third choice, itis unclear whether the optimal strategy isdeclining this offer or accepting it. Similarly,the optimal list of schools to apply for isunclear. Hence, in Columbus, families areforced to play a very difficult game on a verycrucial issue.

Another major difficulty with the Columbusstudent assignment mechanism concerns efficien-cy: Consider two students, each of whom hold anoffer from the other’s first choice. Since they donot know whether they will receive better offers,they may as well accept these offers, and this inturn yields an inefficient matching.

II. Two Competing Mechanisms

We are now ready to propose two alternativemechanisms to the school choice problem.

A. Gale-Shapley Student Optimal StableMechanism

As we have already emphasized, the schoolchoice problem is closely related to the college

admissions problem: In school choice, schoolsare not agents and they have priorities overstudents, whereas in college admissions,schools are agents and they have preferencesover students. One promising idea is interpret-ing school priorities as preferences and applyingthe following version of the Gale-Shapley de-ferred acceptance algorithm (Gale and Shapley,1962):

Step 1: Each student proposes to her firstchoice. Each school tentatively assigns its seatsto its proposers one at a time following theirpriority order. Any remaining proposers are re-jected.

In general, at

Step k: Each student who was rejected in theprevious step proposes to her next choice. Eachschool considers the students it has been hold-ing together with its new proposers and tenta-tively assigns its seats to these students one at atime following their priority order. Any remain-ing proposers are rejected.

The algorithm terminates when no studentproposal is rejected and each student is assignedher final tentative assignment. We refer to theinduced direct mechanism as Gale-Shapley stu-dent optimal stable mechanism. (See the Ap-pendix for a detailed example.)

Gale-Shapley student optimal stable mech-anism is used in Hong Kong to assign collegeseats to high school graduates, and since1998, a version of it is used in the Americanhospital-intern market (Roth and Peranson,1997, 1999).

The central notion in the college admissionsliterature is stability, which is equivalent to thefollowing natural requirement in the context ofschool choice: There should be no unmatchedstudent-school pair (i, s) where student i pre-fers school s to her assignment and she hashigher priority than some other student who isassigned a seat at school s. Therefore, a stablematching in the context of college admissionseliminates justified envy in the context of schoolchoice.

As it is suggested by its name, Gale-Shapleystudent optimal stable mechanism is stable inthe context of college admissions and therefore

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it eliminates justified envy in the context ofschool choice. It also has a number of additionalvery plausible properties.

PROPOSITION 1 (Gale and Shapley,1962): Gale-Shapley student optimal stablemechanism Pareto-dominates any other mech-anism that eliminates justified envy.

PROPOSITION 2 (Lester E. Dubins andDavid A. Freeman, 1981; Roth, 1982a): Gale-Shapley student optimal stable mechanism isstrategy-proof.

Nevertheless, Gale-Shapley student optimalstable mechanism is not “problem free.” Thefollowing example due to Roth (1982a) showsthat there is a potential trade-off between sta-bility and Pareto efficiency.

Example 1: There are three students i1, i2, i3,and three schools s1, s2, s3, each of which hasonly one seat. The priorities of schools and thepreferences of students are as follows:

s1 : i1 � i3 � i2 i1 : s2 s1 s3

s2 : i2 � i1 � i3 i2 : s1 s2 s3

s3 : i2 � i1 � i3 i3 : s1 s2 s3 .

Let us interpret the school priorities as schoolpreferences and consider the associated collegeadmissions problem. In this case there is onlyone stable matching:

� i1 i2 i3

s1 s2 s3� .

But this matching is Pareto-dominated by

� i1 i2 i3

s2 s1 s3� .

Here agents i1 and i2 have the highest prioritiesfor schools s1 and s2, respectively. So there isno way student i1 can be assigned a school thatis worse than school s1 and hence she shall beassigned either s2 or s1. Similarly there is noway student i2 can be assigned a school that isworse than school s2 and hence she shall beassigned either s1 or s2. Thus students i1 and i2should share schools s1 and s2 among them-selves. Stability forces them to share these

schools in a Pareto-inefficient way: This is be-cause if students i1 and i2 are assigned schoolss2 and s1 respectively, then we have a situationwhere student i3 prefers school s1 to her assign-ment s3 and she has a higher priority for schools2 than student i2 does.

As Example 1 shows, complete eliminationof justified envy may conflict with Pareto effi-ciency.14 If policy makers rank complete elim-ination of justified envy above Paretoefficiency, then Gale-Shapley student optimalstable mechanism is a very well-behavedmechanism.15

B. Top Trading Cycles Mechanism

The stability notion strictly eliminates all jus-tified envy. Next we consider a milder interpre-tation of the priorities, which in turn does notcause a conflict with Pareto efficiency. Supposethat if student i1 has higher priority than studenti2 for school s, that does not necessarily meanthat she is entitled a seat at school s beforestudent i2. It rather represents the opportunity toget into school s. If i1 has higher priority thani2, then she has a better opportunity to get intoschool s, other things being equal.

Next, we introduce a competing mechanismwhich is Pareto efficient but which does notcompletely eliminate justified envy. Looselyspeaking, the intuition for this mechanism isthat it starts with students who have the highestpriorities, and allows them to trade the schoolsfor which they have the highest priorities in casea Pareto improvement is possible. Once thesestudents are removed, it proceeds in a similar

14 In many school districts, strict priorities are obtainedwith the help of a single tie-breaking lottery in addition tofundamental policy considerations. In others, strict prioritiesare obtained with the help of several tie-breaking lotteries,typically one for each school. Using a single tie-breakinglottery might be a better idea in school districts that adoptGale-Shapley student optimal stable mechanism, since thispractice eliminates part of the inefficiency: In this case, anyinefficiency will be necessarily caused by a fundamentalpolicy consideration and not by an unlucky lottery draw. Inother words, the tie-breaking will not result in additionalefficiency loss if it is carried out through a single lottery(while that is likely to happen if the tie-breaking is inde-pendently carried out across different schools).

15 Recently, Ergin (2002) characterizes the conditionsunder which there is no conflict between complete elimina-tion of justified envy and Pareto efficiency.

736 THE AMERICAN ECONOMIC REVIEW JUNE 2003

way starting with the students who have thehighest priorities among those who remain. Thetop trading cycles mechanism is a direct mech-anism and for any priorities and reported pref-erences it finds a matching via the following thetop trading cycles algorithm.16

Step 1: Assign a counter for each school whichkeeps track of how many seats are still availableat the school. Initially set the counters equal tothe capacities of the schools. Each studentpoints to her favorite school under her an-nounced preferences. Each school points to thestudent who has the highest priority for theschool. Since the number of students andschools are finite, there is at least one cycle. (Acycle is an ordered list of distinct schools anddistinct students (s1, i1, s2, ... , sk, ik) where s1points to i1, i1 points to s2, ... , sk points to ik,ik points to s1.) Moreover, each school can bepart of at most one cycle. Similarly, each stu-dent can be part of at most one cycle. Everystudent in a cycle is assigned a seat at the schoolshe points to and is removed. The counter ofeach school in a cycle is reduced by one and ifit reduces to zero, the school is also removed.Counters of all other schools stay put.

In general, at

Step k: Each remaining student points to herfavorite school among the remaining schoolsand each remaining school points to the studentwith highest priority among the remaining stu-dents. There is at least one cycle. Every studentin a cycle is assigned a seat at the school thatshe points to and is removed. The counter ofeach school in a cycle is reduced by one and ifit reduces to zero the school is also removed.Counters of all other schools stay put.

The algorithm terminates when all studentsare assigned a seat. Note that there can be nomore steps than the cardinality of the set ofstudents. (See the Appendix for a detailedexample.)

The top trading cycles algorithm simply

trades priorities of students among themselvesstarting with the students with highest priorities.In the very special case where all schools havethe same priority ordering (for example, whenall schools use the same ordering from a singlelottery draw) this mechanism reduces to theserial dictatorship induced by this priority or-dering. That is, the first student is assigned hertop choice, the next student is assigned her topchoice among the remaining seats, and so on.Therefore, the top trading cycles mechanism isa generalization of this mechanism, extended ina way that allows for different priorities at dif-ferent schools.

A variant of the top trading cycles mecha-nism is proposed by Abdulkadiroglu and Son-mez (1999) in a model of house allocation withexisting tenants. In that model, there are exist-ing tenants who have squatting rights over theircurrent houses, and there are newcomers andvacant houses. The version proposed by Ab-dulkadiroglu and Sonmez (1999) is a specialcase of the mechanism presented here: In thatversion, there is a fixed priority ordering for allhouses but this ordering is slightly modified foreach occupied house by inserting its currentoccupant at the top.

In a closely related paper, Papai (2000) inde-pendently introduces the hierarchical exchangerules, which is a wider class of mechanisms.She characterizes the members of this class tobe the only mechanisms that are Pareto effi-cient, group strategy-proof, (i.e., immune topreference manipulation by a group of agents)and reallocation proof (i.e., immune to manip-ulation by misrepresenting the preferences andswapping the objects by a pair of agents).

The top trading cycles mechanism has a num-ber of very plausible properties. First of all,unlike the Gale-Shapley student optimal stablemechanism, it is Pareto efficient.

PROPOSITION 3: The top trading cyclesmechanism is Pareto efficient.

Another key desirable feature of the top trad-ing cycles mechanism is that, as in the case ofGale-Shapley student optimal mechanism, it isstrategy-proof. Therefore truthful preferencerevelation is a dominant strategy for all stu-dents. In particular, unlike the Boston studentassignment mechanism, students do not need to

16 This algorithm is inspired by Gale’s top trading cyclesalgorithm which is used to find the unique core allocation(Roth and Andrew Postlewaite, 1977) in the context ofhousing markets (Shapley and Herbert Scarf, 1974).

737VOL. 93 NO. 3 ABDULKADIROGLU AND SONMEZ: SCHOOL CHOICE

hesitate on reporting their truthful preferencesin fear of losing their priorities. Therefore bothour proposed mechanisms release an importantburden of finding the optimal application strat-egy over the shoulders of students and theirparents.

PROPOSITION 4: The top trading cyclesmechanism is strategy-proof.

The intuition for the strategy-proofness of thetop trading cycles mechanism is very simple.Suppose that a student leaves the algorithm atStep k when she reports her true preferences.Since she points to the best available seat ateach step of the algorithm, all the seats that sheprefers leave the algorithm before Step k and bymisrepresenting her preferences she cannot alterthe cycles that have formed at any step beforeStep k. So these better seats will leave thealgorithm before she does whether she reportsher true preferences or fake preferences. Thusshe can only hurt herself by a manipulation.Strategy-proofness of the core mechanism forhousing markets (Roth, 1982b), the top tradingcycles mechanism for house allocation with ex-isting tenants (Abdulkadiroglu and Sonmez,1999), and the hierarchical exchange functions(Papai, 2000) are all based on the same criticalobservation.

C. Which Mechanism Shall Be Chosen?

As we have already indicated, both mecha-nisms are strategy-proof, so the choice betweenthem depends on the structure and interpretationof the priorities. In some applications, policymakers may rank complete elimination of jus-tified envy before full efficiency, and Gale-Shapley student optimal stable mechanism canbe used in those cases. University admissions inTurkey is one such application (Michel Balinskiand Sonmez, 1999). In Turkey, priorities foruniversity departments are obtained via a cen-tralized exam and complete elimination of jus-tified envy is imposed by law. Depending on theapplication, Gale-Shapley student optimal sta-ble mechanism may have additional advantages.For example, consider a city which implementsseparate intra-district choice programs with aneventual target of eliminating the borders and

switching to an inter-district choice program.Furthermore, suppose that cross-district priori-ties will be lower than within-district prioritiesin the eventual program. In such applications,transition to an inter-district program is likely tomove smoother under the Gale-Shapley studentoptimal stable mechanism: The outcome pro-duced by the Gale-Shapley student optimal sta-ble mechanism under the inter-district programPareto-dominates the outcome produced by sep-arate intra-district choice programs (each ofwhich uses the Gale-Shapley student optimalstable mechanism). That is because the outcomeproduced by separate intra-district choice pro-grams is still stable under the inter-districtchoice program, provided that cross-district pri-orities are lower than within-district priorities.Hence, no student can possibly suffer from atransition to the inter-district choice programunder the Gale-Shapley student optimal stablemechanism. It is easy to construct an examplewhere the transition to an inter-district choiceprogram hurts some students under the top trad-ing cycles mechanism.17

In other applications, the top trading cyclesmechanism may be more appealing. Schoolchoice in Columbus is one such application.Recall the school priorities in Columbus: Forsome schools, students in the school’s regularassignment area have high priority and they areall guaranteed a seat and the priority among theremaining low-priority students is determinedby a random lottery. For the remaining schools,all students are in the same priority group andthe priority between them is determined by arandom lottery. Under the top trading cyclesmechanism, students who have high prioritiesfor their local schools are all guaranteed seatsthat are at least as good, provided that theytruthfully report their preferences. Any instabil-ity produced by the top trading cycles mecha-nism is necessarily due to the randomlyobtained priorities and in that case a milderinterpretation of the priorities may be more ap-pealing. In other cases the choice between thetwo mechanisms may be less clear and it de-pends on the policy priorities of the policymakers.

17 We are grateful to an anonymous referee who broughtthis observation to our attention.

738 THE AMERICAN ECONOMIC REVIEW JUNE 2003

III. Controlled Choice

Controlled choice in the United States at-tempts to provide choice to parents while main-taining the racial and ethnic balance at schools.In some states, choice is limited by court-ordered desegregation guidelines. In Missouri,for example, St. Louis and Kansas City mustobserve strict racial guidelines for the place-ment of students in city schools. There are sim-ilar constraints in other countries as well. Forexample, in England, City Technology Collegesare required to admit a group of students fromacross the ability range and their student bodyshould be representative of the community inthe catchment area (Hirch, 1994, p. 120).

In many school districts, controlled choiceconstraints are implemented by imposing racialquotas at public schools. For example, prior toJuly 1999, a version of the Boston student as-signment mechanism which uses racial quotaswas in use in the city of Boston. Similarly, inColumbus and Minneapolis, controlled choiceconstraints are implemented by imposing racialquotas. These quotas may be perfectly rigid orthey may be flexible. For example, in Minneap-olis, the district is allowed to go above or belowthe district-wide average enrollment rates by upto 15 percent points in determining the racialquotas. So consider a school district in Minne-apolis, where the average enrollment rates ofmajority students versus minority students are60 percent and 40 percent, respectively, andconsider a school with 100 seats. Racial quotasfor this school are 75 for majority students, and55 for minority students.

Both Gale-Shapley student optimal stablemechanism and the top trading cycles mecha-nism can be easily modified to accommodatecontrolled choice constraints by imposing type-specific quotas.

A. Gale-Shapley Student Optimal StableMechanism with Type-Specific Quotas

Suppose that there are different types of stu-dents and each student belongs to one type. Ifthe controlled choice constraints are perfectlyrigid then there is no need to modify the Gale-Shapley student optimal stable mechanism. Foreach type of students, one can separately imple-ment the mechanism in order to allocate the

seats that are reserved exclusively for that type.When the controlled choice constraints are flex-ible, consider the following modification of theGale-Shapley student optimal stable mechanismthat is studied by Abdulkadiroglu (2002) in thecontext of college admissions with affirmativeaction:18

Step 1: Each student proposes to her firstchoice. Each school tentatively assigns its seatsto its proposers one at a time following theirpriority order. If the quota of a type fills, theremaining proposers of that type are rejectedand the tentative assignment proceeds with thestudents of the other types. Any remaining pro-posers are rejected.

In general, at

Step k: Each student who was rejected in theprevious step proposes to her next choice. Eachschool considers the students it has been hold-ing together with its new proposers and tenta-tively assigns its seats to these students one at atime following their priority order. If the quotaof a type fills, the remaining proposers of thattype are rejected and the tentative assignmentproceeds with the students of the other types.Any remaining proposers are rejected.

This modified mechanism satisfies the fol-lowing version of the fairness requirement: Ifthere is an unmatched student-school pair (i, s)where student i prefers school s to her assign-ment and she has higher priority than someother student j who is assigned a seat at schools, then

1. students i and j are of different types, and2. the quota for the type of student i is full at

school s.

18 Abdulkadiroglu (2002) shows that flexible con-trolled choice constraints induce substitutable prefer-ences (Alexander S. Kelso, Jr. and Vincent P. Crawford,1982) in the context of college admissions. That is be-cause the role played by the flexible controlled choiceconstraints in the present context is analogous to the roleof discriminatory quotas (see Roth and Sotomayor, 1990,Proposition 5.22) in the context of college admissionsproblems. Gale-Shapley student optimal stable mecha-nism for the general case of substitutable preferences isdue to Roth (1991).

739VOL. 93 NO. 3 ABDULKADIROGLU AND SONMEZ: SCHOOL CHOICE

As an implication, the modified mechanismeliminates all justified envy between studentsof the same type. The above fairness require-ment is equivalent to stability in the contextof college admissions with affirmative action.Moreover, truthful preference revelation is adominant strategy for the students under themodified Gale-Shapley student optimal stablemechanism as well (Abdulkadiroglu, 2002).

PROPOSITION 5: Gale-Shapley student opti-mal mechanism with type-specific quotas isstrategy-proof.

In many real-life applications of controlledchoice, there are only two types of students (forexample, majority students and minority stu-dents). In such applications, the modified mech-anism is essentially a direct application of theoriginal mechanism with the following twist:Consider a school s with q seats and which hasquotas of q1, q2 for type 1, type 2 students,respectively. Clearly q � q1, q � q2, and q1 �q2 � q. In school s,

● q � q2 seats are reserved exclusively for type1 students,

● q � q1 seats are reserved exclusively for type2 students,

● and the remaining q1 � q2 � q seats arereserved for either type of students.

So it is as if there are three different schoolss1, s2, s3 where

● school s1 has q � q2 seats and student pri-orities are obtained from the original priori-ties by removing type 2 students and makingthem unacceptable at school s1,

● school s2 has q � q1 seats and student pri-orities are obtained from the original priori-ties by removing type 1 students and makingthem unacceptable at school s2, and

● school s3 has q1 � q2 � q seats and studentpriorities are the same as the original priori-ties.

Whenever there are two types of students, ourmodified mechanism

● divides each school into three schools as ex-plained above,

● extends each student’s preferences as follows:— for any school s, s1 is preferred to s2,

which is preferred to s3,— for any pair of schools s, t, if s is

preferred to t then each of s1, s2, s3 ispreferred to each of t1, t2, t3,

and

● selects the student optimal stable matching ofthe induced college admissions problem.

B. Top Trading Cycles Mechanismwith Type-Specific Quotas

As in the case of Gale-Shapley student opti-mal stable mechanism, there is no need to mod-ify the top trading cycles mechanism whencontrolled choice constraints are perfectly rigid.One can implement the top trading cycles mech-anism separately for each type of students.When the controlled choice constraints are flex-ible, the top trading cycles mechanism can bemodified as follows: For each school, in addi-tion to the original counter which keeps track ofhow many seats are available, include a type-specific counter for each type of students.

Step 1: For each school, set the counter equal tothe capacity of the school and set each type-specific counter equal to the quota of the asso-ciated type of students. Each student points toher favorite school among those which haveroom for her type (i.e., with a positive counterreading for her type). Each school points to thestudent with highest priority for that school.There is at least one cycle. Every student in acycle is assigned a seat at the school she pointsto and is removed. The counter of each schoolin a cycle is reduced by one. Depending on thestudent it is assigned to, the associated type-specific counter is reduced by one as well. Allother counters stay put. In case the counter of aschool (not the type-specific ones) reduces tozero, the school is removed as well. If there is atleast one remaining student, then we proceedwith the next step.

In general, at

Step k: Each remaining student points to herfavorite remaining school among those which

740 THE AMERICAN ECONOMIC REVIEW JUNE 2003

have room for her type, and each remainingschool points to the student with the highestpriority among remaining students. There is atleast one cycle. Every student in a cycle isassigned a seat at the school that she points toand is removed. The counter of each school in acycle is reduced by one and depending on thestudent it is assigned to, the associated type-specific counter is reduced by one as well. Allother counters stay put. In case the counter of aschool reduces to zero, the school is removed. Ifthere is at least one remaining student, then weproceed with the next step.

There can be efficiency losses in both mech-anisms due to the controlled choice constraints.A matching is constrained efficient if there is noother matching that satisfies the controlledchoice constraints, and which assigns all stu-dents a weakly better school and at least onestudent a strictly better school. The outcome ofthe modified top trading cycles mechanism isconstrained efficient.

PROPOSITION 6: The top trading cyclesmechanism with type-specific quotas is con-strained efficient.

Moreover, truthful preference revelation isstill a dominant strategy under the modifiedmechanism.

PROPOSITION 7: The top trading cyclesmechanism with type-specific quotas isstrategy-proof.

IV. Conclusion

The Office of Educational Research and Im-provement (1992, pp. 19–20) emphasizes thefollowing seven factors on which student as-signment decisions should be based:

1. Racial Balance: Student assignment policiesshould respect the racial and ethnic propor-tions of the district.

2. Instructional Capacity: Student assignmentpolicies must take into consideration thedanger of creating an imbalance in the in-structional capacity of a school.

3. Replication Efforts: Popular programsshould be replicated and undersubscribed

schools be closed and then reopened as dis-tinctive schools created by collective efforts.

4. Space Availability: Schools must outlineclassroom use needs long before the schoolyear begins.

5. Neighborhood School Priority: A percentageof slots in a school should be reserved forneighborhood families, as long as racial bal-ance is maintained, to allow continuity forthe students and a connection for the schoolto the neighborhood.

6. Preference for Siblings: As a conveniencefor parents and to promote the sharing ofschool experience between brothers and sis-ters, preference for sibling requests shouldbe given some priority.

7. Gender Balance Considerations for CertainSchools.

Among these, items 1, 2, 5, 6, and 7 concernthe design of a student assignment mechanism.Both mechanisms that we propose respect eachof these factors: Racial balance and gender bal-ance can be achieved through type-specific quo-tas, instructional capacity overload is achievedthrough regular capacities, neighborhood schoolpriority and preference for siblings can beachieved through school-specific priorities.

In addition to these factors, Alves and Willie(1990), engineers of the Boston ControlledChoice Plan, emphasize the following objec-tives as essential elements of an effective con-trolled choice plan:

1. Eliminating, to the extent practical, all indi-vidual school attendance boundaries and/orgeocodes.

2. Allowing parents and students to make mul-tiple school selections but with no guaranteethat they will obtain their first-choice schoolsor programs of choice.

3. Ensuring complete honesty and integrity inthe disposition of all final assignment deci-sions.

Both mechanisms conform with these objec-tives as well: Students rank all schools (of coursewithout any guarantee of getting their top choices)and once the policies concerning school prioritiesare announced and these priorities determined, thefinal outcome is deterministic and does notleave any room for manipulation.

741VOL. 93 NO. 3 ABDULKADIROGLU AND SONMEZ: SCHOOL CHOICE

School choice is becoming increasingly com-mon in the United States. Cities having adoptedschool choice plans include Boston, Cambridge,Champaign, Columbus, Hartford, Little Rock,Minneapolis, Rockville, Seattle, White Plains,and parts of New York. Recent laws in severalstates require each school district to establish anintra-district school choice plan. Similarly, re-cent laws in Florida require each school districtto design a school choice plan, even if they donot implement it. An important difficulty indesigning such plans is the choice of an appeal-ing student assignment mechanism. Many of theschool choice plans that we find have protocolsand guidelines for the assignment of studentswithout explicit procedures. This gap offers op-portunities to manipulate these controlledchoice programs and results in appeals by un-satisfied parents. Jeffrey R. Henig (1994, p.212) states

The first step that districts must take toensure fair implementation of choicewithin and among schools is to make thecriteria for accepting and rejecting trans-fer requests clear and public. Vaguelyworded references to “maintaining racialbalance,” “avoiding overcrowding,” andmeeting children’s “individualized needs”invite selective interpretation unless theyare accompanied by practical definitions.

Other school choice programs, such as thosein Boston, Columbus, Minneapolis, and Seattleare accompanied by deterministic student as-signment mechanisms but these mechanismsare all vulnerable to preference manipulation.As a result, students and their families face adifficult task of finding optimal admissionsstrategies. Adopting either the Gale-Shapleystudent optimal stable mechanism or the toptrading cycles mechanism may provide a prac-tical solution to some of these critical schoolchoice issues.

APPENDIX

Example 2: This example illustrates the dy-namics of the Gale-Shapley student optimal sta-ble mechanism and the top trading cyclesmechanism. There are eight students i1, ... , i8and four schools s1, ... , s4. Schools s1, s2have two seats each and schools s3, s4 have

three seats each. The priorities of the schoolsand the preferences of the students are asfollows:

s1 : i1 � i2 � i3 � i4 � i5 � i6 � i7 � i8

s2 : i3 � i5 � i4 � i8 � i7 � i2 � i1 � i6

s3 : i5 � i3 � i1 � i7 � i2 � i8 � i6 � i4

s4 : i6 � i8 � i7 � i4 � i2 � i3 � i5 � i1

i1 i2 i3 i4 i5 i6 i7 i8

s2 s1 s3 s3 s1 s4 s1 s1

s1 s2 s2 s4 s3 s1 s2 s2

s3 s3 s1 s1 s4 s2 s3 s4

s4 s4 s4 s2 s2 s3 s4 s3

GALE-SHAPLEY STUDENT OPTIMALSTABLE MECHANISM:

Step 1: Students i2, i5, i7, i8 propose to schools1, student i1 proposes to school s2, students i3,i4 propose to school s3 and student i6 proposesto school s4.

School s1 tentatively assigns its seats to stu-dents i2, i5, and rejects students i7, i8. Sinceschool s1 is the only school with excess propos-als, all other students are tentatively assignedseats at schools that they propose.

Step 2: Having been rejected at Step 1, each ofstudents i7, i8 proposes to school s2 which istheir next choice. School s2 considers student i1whom it has been holding together with its newproposers i7, i8. School s2 tentatively assignsits seats to students i8, i7, and rejects student i1.

Step 3: Having been rejected at Step 2, studenti1 proposes to school s1 which is her nextchoice. School s1 considers students i2, i5whom it has been holding together with its newproposer i1. School s1 tentatively assigns itsseats to students i1, i2, and rejects student i5.

Step 4: Having been rejected at Step 3, stu-dent i5 proposes to school s3 which is her nextchoice. School s3 considers students i3, i4whom it has been holding together with itsnew proposer i5. Since school s3 has threeseats, it tentatively assigns its seats to thesestudents.

742 THE AMERICAN ECONOMIC REVIEW JUNE 2003

Since no student proposal is rejected at Step4, the algorithm terminates. Each student isassigned her final tentative assignment:

� i1 i2 i3 i4 i5 i6 i7 i8

s1 s1 s3 s3 s3 s4 s2 s2� .

TOP TRADING CYCLES MECHANISM:Let cs1

, cs2, cs3

, and cs4indicate the counters

of the schools.

Step 1: [see Figure A1].

There are two cycles in Step 1: (s1, i1, s2, i3,s3, i5) and (s4, i6). Therefore students i1, i3, i5,i6 are assigned one slot at schools s2, s3, s1, s4,respectively, and removed. Since every schoolparticipates in a cycle, all counters are reducedby one for the next step.

Step 2: [see Figure A2].

There is only one cycle in Step 2: (s1, i2).Therefore student i2 is assigned one slot atschool s1 and removed. The counter of schools1 is reduced by one to zero and it is removed.All other counters stay put.

Step 3: [see Figure A3].

There is only one cycle in Step 3: (s3, i7, s2,i4). Therefore students i7, i4 are assigned oneslot at schools s2, s3, respectively, and removed.

FIGURE A3. TOP TRADING CYCLES ALGORITHM: STEP 3

FIGURE A1. TOP TRADING CYCLES ALGORITHM: STEP 1 FIGURE A2. TOP TRADING CYCLES ALGORITHM: STEP 2

743VOL. 93 NO. 3 ABDULKADIROGLU AND SONMEZ: SCHOOL CHOICE

The counters of schools s2 and s3 are reduced byone. Since there are no slots left at school s2 it isremoved. Counters of schools s3 and s4 stay put.

Step 4: [see Figure A4].

There is only one cycle in Step 4: (s4, i8).Therefore student i8 is assigned one slot atschool s4 and removed. There are no remainingstudents so the algorithm terminates. Altogetherthe matching it induces is

� i1 i2 i3 i4 i5 i6 i7 i8

s2 s1 s3 s3 s1 s4 s2 s4� .

PROOF OF PROPOSITION 3:Consider the top trading cycles algorithm.

Any student who leaves at Step 1 is assigned hertop choice and cannot be made better off. Anystudent who leaves at Step 2 is assigned her topchoice among those seats remaining at Step 2and since preferences are strict she cannot bemade better off without hurting someone wholeft at Step 1. Proceeding in a similar way, nostudent can be made better off without hurtingsomeone who left at an earlier step. Thereforethe top trading cycles mechanism is Paretoefficient.

The proof of Proposition 4 is similar to theproof of a parallel result in Abdulkadiroglu andSonmez (1999).19 The following lemma is thekey to the proof.

LEMMA: Fix the announced preferences of allstudents except i at Q�i � (Qj)j�I �{i}. Supposethat in the algorithm student i is removed atStep T under Qi and at Step T* under Q*i.Suppose T � T*. Then the remaining studentsand schools at the beginning of Step T are thesame whether student i announces Qi or Q*i.

PROOF OF LEMMA:Since student i fails to participate in a cycle

prior to Step T in either case, the same cyclesform and therefore the same students andschools are removed before Step T.

PROOF OF PROPOSITION 4:Consider a student i with true preferences Pi.

Fix an announced preference profile Q�i �(Qj)j�I �{i} for every student except i. We wantto show that revealing her true preferences Pi isat least as good as announcing any other pref-erences Qi. Let T be the step at which student ileaves under Qi , (s, i1, s1, ... , sk, i) be thecycle she joins, and thus school s be her assign-ment. Let T* be the step at which she leavesunder her true preferences Pi. We want to showthat her assignment under Pi is at least as goodas school s. We have two cases to consider:

Case 1: T* � T.

Suppose student i announces her true prefer-ences Pi. Consider Step T. By the Lemma, thesame students and schools remain in the marketat the beginning of this step whether student iannounces Qi or Pi. Therefore at Step T, schools points to student i1, student i1 points to schools1, ... , school sk points to student i. Moreover,they keep doing so as long as student i remains.Since student i truthfully points to her bestremaining choice at each step, she either re-ceives an assignment that is at least as good asschool s or eventually joins the cycle (s, i1,s1, ... , sk, i) and is assigned a slot at school s.

Step 2: T* � T.

By the Lemma the same schools remain inthe algorithm at the beginning of Step T*whether student i announces Qi or Pi. More-over, student i is assigned a seat at her bestchoice school remaining at Step T* under Pi.Therefore, in this case too her assignment under

19 Papai (2000) independently proves a similar result fora wider class of mechanisms.

FIGURE A4. TOP TRADING CYCLES ALGORITHM: STEP 4

744 THE AMERICAN ECONOMIC REVIEW JUNE 2003

the true preferences Pi is at least as good asschool s.

PROOF OF PROPOSITION 5:Immediately follows from Abdulkadiroglu

(2002) and a self-contained proof is availableupon request. The original strategy-proofnessresult by Dubins and Freeman (1981) and Roth(1982a) directly carries over to the case withtwo types of students since the modified mech-anism is a direct application of the originalmechanism as explained in Section III, subsec-tion A.

PROOF OF PROPOSITION 6:Consider the modified top trading cycles al-

gorithm. Any student who leaves at Step 1 isassigned her top choice and cannot be madebetter off. Any student who leaves at Step 2 isassigned her top choice among those schoolswhich has room for her type at Step 2 and sincepreferences are strict she cannot be made betteroff without hurting someone who left at Step 1.Proceeding in a similar way, no student can bemade better off without hurting someone wholeft at an earlier step. Therefore the top tradingcycles mechanism with type-specific quotas isconstrained efficient.

PROOF OF PROPOSITION 7:The Lemma preceding the proof of Proposi-

tion 4 as well as its proof are valid for themodified mechanism. Moreover the basic ele-ments of the proof of Proposition 4 carry over aswell.

Consider a student i with true preferences Pi.Fix an announced preference profile Q�i �(Qj)j�I�{i} for every student except i. We wantto show that revealing her true preferences Pi isat least as good as announcing any other pref-erences Qi. Let T be the step at which student ileaves under Qi , (s, i1, s1, ... , sk, i) be thecycle she joins, and thus school s be her assign-ment. Let T* be the step at which she leavesunder her true preferences Pi. We want to showthat her assignment under Pi is at least as goodas school s. We have two cases to consider:

Case 1: T* � T.

Suppose student i announces her true prefer-

ences Pi. Consider Step T. By the Lemma, thesame students and schools remain in the marketat the beginning of this step whether student iannounces Qi or Pi. Therefore at Step T, schools points to student i1, student i1 points to schools1, ... , school sk points to student i. Moreover,they keep doing so as long as student i remains.But at each step student i truthfully points to herbest choice among schools with an availableseat for her type. Therefore she either receivesan assignment that is at least as good as schools or eventually joins the cycle (s, i1, s1, ... , sk,i) and is assigned a slot at school s.

Case 2: T* � T.

By the Lemma the same schools remain inthe algorithm at the beginning of Step T*whether student i announces Qi or Pi. More-over, student i is assigned a seat at her bestchoice among schools with an available seat forher type remaining at Step T* under Pi. There-fore, in this case too her assignment under thetrue preferences Pi is at least as good as school s.

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Abdulkadiroglu, Atila and Sonmez, Tayfun.“Random Serial Dictatorship and the Corefrom Random Endowments in House Alloca-tion Problems.” Econometrica, May 1998,66(3), pp. 689–701.

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Alves, Michael J. and Willie, Charles V. “Choice,Decentralization and Desegregation: TheBoston ‘Controlled Choice’ Plan,” in Wil-liam Clune and John White, eds., Choice andcontrol in American education, volume 2:The practice of choice decentralization andschool restructuring. New York: FalmerPress, 1990, pp. 17–75.

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