College admissions with ties and common quotas: Integer programming approach

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European Journal of Operational Research

journalhomepage:www.elsevier.com/locate/ejor

Innovative Applications of O.R.

College admissions with ties and common quotas: Integer programming approach ^R

Kolos Csaba Ágoston

^a

, Péter Biró

^a,b,1,∗

, Endre Kováts

^c

, Zsuzsanna Jankó

^a,b

aDepartment of Operations Research and Actuarial Sciences, Corvinus University of Budapest, H-1093, F ˝ovám tér 13-15., Budapest, Hungary

bInstitute of Economics, Research Centre for Economic and Regional Studies, Hungarian Academy of Sciences, H-1097, Tóth Kálmán u. 4., Budapest, Hungary

cBudapest University of Technology and Economics, H-1111, M ˝uegyetem rakpart 3., Budapest, Hungary

a rt i c l e i nf o

Article history:

Received 28 January 2020 Accepted 19 August 2021 Available online xxx Keywords:

Assignment Stable matching College admission Distributional constraints Integer programming

a b s t r a c t

Admissiontouniversities isorganised inacentralisedschemeinHungary.Inthispaperweinvestigate twomajorspecialitiesofthisapplication:tiesandcommonquotas.Atieoccurwhensomestudentshave thesamescoreataprogramme.Ifnotenoughseatsareavailableforthelasttied groupofapplicants ataprogrammethentherearethreereasonablepoliciesusedinpractice:1)allmustberejected,asin Hungary2)allcanbeaccepted, asinChile3)alotterydecideswhichstudentsareacceptedfromthis group,asinIreland.Eventhoughstudent-optimalstablematchingscanbecomputedefficientlyforeach oftheabovethreecases,wedeveloped(mixed)integerprogramming(IP)formulationsforsolvingthese problems,andcomparedthesolutionsobtainedbythethreepoliciesforarealinstanceoftheHungarian application from2008.Inthecase ofHungarycommonquotasarisefromthe facultyquotasimposed ontheirprogrammesandfromthenationalquotassetforstate-financedstudentsineachsubject.The overlappingstructureofcommonquotasmakesthecomputationalproblemoffindingastablesolution NP-hard, even forstrictrankings.In thecase oftiesand commonquotasweproposetwo reasonable stablesolutionconceptsfortheHungarianandChileanpolicies.Wedeveloped(mixed)IPformulations forsolvingthesestablematchingproblemsandtestedtheirperformanceonthelargescalerealinstance from2008andalsoforonefrom2009undertwodifferentassumptions.Wedemonstratethatthemost generalcaseisalsosolvableinpracticebyIPtechnique.

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1. Introduction

GaleandShapleygaveastandardmodelforcollegeadmissions (Gale & Shapley,1962), andsuggestedstablematching foritsso- lution. Intuitively,a matchingis stableif an applicationto acol- lege is rejected because the college is already full with higher ranked students. Gale and Shapley showed that a stable match- ing can always be found by the deferred-acceptance algorithm, which runsinlinear time inthe numberofapplications, seee.g.

Manlove (2013).Moreover,the student-orientedvariantresultsin

R Earlier results of this paper have been presented in two conference papers [5,6].

∗Corresponding author.

E-mail addresses: kolos.agoston@uni-corvinus.hu (K.C. Ágoston), peter.biro@krtk.mta.hu (P. Biró), endre.kovats.92@gmail.com (E. Kováts), zsuzsanna.janko@uni-corvinus.hu (Z. Jankó).

1 Supported by the Hungarian Academy of Sciences under its Momentum Pro- gramme ( Engineering Economics in Matching Markets , no. LP2021-2), and by the Hungarian Scientific Research Fund – OTKA (no. K129086 ).

astudent-optimalstablematching,meaningthatnostudentcould geta betterassignment inanyother stablematching.The theory of stable matchings has intensively been studied since 1962 by mathematicians/computer scientists (see e.g. Manlove, 2013) and economists/gametheorists(see e.g. Roth& Sotomayor,1990). The Gale-Shapleyalgorithm hasalsobeen usedinpractice all around theworld (Biró, 2017), firstin 1952inthe USresident allocation programme,calledNRMP(Roth, 1984),thenalsoinschoolchoice, e.g. in Boston (Abdulkadiro˘glu, Pathak, & Roth, 2005a) and New York(Abdulkadiro˘glu,Pathak,Roth,&Sönmez,2005b).InHungary, the nationaladmission scheme forsecondary schoolsfollows the original Gale-Shapleymodelandalgorithm (Biró,2014a), andthe highereducationadmission schemealsouses aheuristic solution basedontheGale-Shapleyalgorithm(Biró,2014b).

The Hungarian higher education admission scheme have at leastfour important specialfeatures: presenceof ties, lower and common quotas, and paired applications. The students submit preference list on the university programmes they apply to, and

https://doi.org/10.1016/j.ejor.2021.08.033

0377-2217/© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Please citethisarticleas:K.C.Ágoston,P.Biró,E.Kovátsetal.,Collegeadmissionswithtiesandcommonquotas:Integerprogramming

K.C. Ágoston, P. Biró, E. Kováts et al. European Journal of Operational Research xxx (xxxx) xxx

are ranked according to their scores, whichare of integer values currentlyintherangeof[0,500] higherscoremeaningbetterper- formance. The solution by the coordinating agency is announced in termsof cutoff scoresto be understood asfollows: every stu- dent isadmitted tothebest programmeofher preferencewhere she achievedthe cutoff score.Atie canoccur whentwo ormore studentshavetheverysamescoreataprogrammetheyapplyfor.

TiesareneverbrokeninHungary,eitherallornoneofthestudents inthetieareadmitted,dependingonthecutoff score.Lowerquo- tas are minimum requirements forthe number of admitted stu- dents for each programme, which are set by the universities to maketheeducationeconomical.Applicationsbythestudentsalso includethecontracttermofthestudy,i.e.,whethertheirstudyis funded by thestate or privately.Forevery programmethere isa commonupperboundforthenumberofadmittedstudentsunder anycontract term,andtherearealso nationwidecommonquotas forthenumberofstudentsgettingstatefundsineachsubject(e.g., Chemistry).Finally,thestudentsapplyingforteachers’programmes shouldapply forpairsofprogrammes(suchasMath-Physics).The latterfeature wasre-introducedinthe applicationin2010,butit was not present in 2008 and2009, the years our analysis bears oninthispaper.Furtherdetailsoftheapplicationcanbefoundin Biró (2014b).

Each ofthe three specialfeatures (lowerandcommon quotas andpairedapplications)makestheproblemNP-hard(Biró,Fleiner, Irving, & Manlove, 2010), only the case of tiesis resolvable effi- ciently (Biró & Kiselgof, 2015). In a recent paper (Ágoston, Biró,

& McBride,2016) Ágostonetal.studiedtheusage ofintegerpro- gramming techniques for finding stable solutions withregard to each ofthesefourspecialfeatures separately,andtheysolvedthe case of lower quotas for the real instance of 2008. We refer to this instance as 2008-Educatio instance, which wasprovided for research purposeby thecoordinatinggovernmental agency called Educatiokht,andcontainedalltherelevantupperandlowerquo- tasinan anonymdataset.Inthisfollow-upworkwedevelop and test new IP formulations for the case of ties and common quo- tas separately andthen also forthecase whenboth features are present. So, the ultimategoal of this work was to suggest a so- lution concept forthe collegeadmission problemwhereties and commonquotasarealsopresent,andtoprovideintegerprogram- ming formulations that are suitable to compute this solution for large scale applications, such as theHungarian university admis- sionschemewithover100,000students.

Thepresenceoftiesandequaltreatmentpolicy(i.e.,notbreak- ing theties)isalsoa featureintheChilean universityadmissions (Rios, Larroucau, Parra, & Cominetti, 2014). However, the policy used there is more permissive than the Hungarian one, since if two studentswiththesamescorearecompetingforthelastseat ataprogrammethentheyarebothacceptedinChile,butbothre- jectedinHungary,whilstarandomtie-breaking isusedinIreland (Chen,2012) todecidewhich studentwillbeadmitted.Theseso- lution conceptshavebeenstudied theoreticallyinBiró & Kiselgof (2015) underthenameofH-stabilityandL-stability.Theintuitive resultprovedinthatpaperisthatwhenstudent-optimalstableso- lutionsare comparedforthesameinstancethenthecutoff scores are atleastashighinIreland asin Chile,andatleastashighin HungaryasinIreland.Sothestudentsarealwaysgettingtheworst assignments in Hungary, a better assignment in Ireland, andthe best one in Chile.In thispaperwe quantify thesedifferenceson theHungarianuniversityadmissioninstancesfrom2008and2009, presentedinSection5.

Commonquotas arealsopresentinmanyotherapplications.A recentpaper(Baswana,Chakrabarti,Chandran,Kanoria,&Patange, 2019) describes the admission to Engineering Colleges in India, wherecommonquotasareusedfordifferent,possiblyoverlapping types,justasintheHungariancase.Thismeansthatastablesolu-

tionmaynotexistandtheproblemisNP-hard(Biró etal.,2010), thustheauthorshaveproposedaheuristicalgorithm.Interestingly theauthorswereawareofthepossibilityofusingIPsolutionsfor this problem, asdescribed in Ágostonet al. (2016) for the Hun- gariancase²,but they decided not to use that approach because ofthe possibly long run time. In this paperwe demonstrate the caseofcommonquotasistractableforlargeinstances,evenifthe quotasareoverlapping andtheproblemisfurthercomplicatedby the presence ofties, asin theHungarian case. The Indian appli- cations have also been studied by Sönmez & Yenmez (2019a,b), wherethecaseofnestedsetsystemshavebeenprovedtobesolv- ablebyageneraliseddeferred-acceptancealgorithm,whichcorre- spondstothefindingofBiró etal.(2010)ontheHungariancollege admissions. Furthermore, thesame kind ofrequirements are im- plemented in college admission schemes with affirmative action, suchastheBraziliancollegeadmissionsystem(Aygün&Bo,2013).

Similar distributional requirements are present for the Israeli Mechinot gap-year programs (Gonczarowski, Kovalio, Nisan, &

Romm, 2019), where the authors developed and implemented a new Gale-Shapley type heuristic solution for the application.

Ágoston, Biró, & Szántó (2018b) used integer programmingtech- niques for allocating students to companies at CEMS universities undercomplexdistributionalconstraintswithrespecttothetypes ofstudents.Distributional constraintsarepresentinschool choice programmes as well, where the decision makers want to con- trolthesocio-ethnicaldistributionofthestudents(Abdulkadiro˘glu, 2005;Abdulkadiro˘glu&Ehlers,2007;Bo,2016;Echenique&Yen- mez, 2015; Ehlers, Hafalir, Yenmez, & Yildirim, 2014). Another well-documented case is the Japanese resident allocation, where thegovernment wantsto ensure that thedoctors areevenly dis- tributed across the country. They imposed lower quotas on the numberofdoctorsallocatedineachregion (Goto,Kojima, Kurata, Tamura,&Yokoo,2017;Kamada&Kojima,2014;2017a;2017b).

Assignments problemsare extensivelystudied in theOR liter- ature (see e.g. Pentico, 2007). There are many examples ofprac- tical matching problems, such as papers assignment to review- ers(Garg,Kavitha, Kumar,Mehlhorn, & Mestre, 2010), course al- location(Diebold&Bichler,2017),marriageassignment(Cao,Frag- niére,Gautier,Sapin,&Widmer,2010)andkidneyexchanges(Biró, vandeKlundert,& Manlove,2019). However,theusageofinteger programming techniques is relatively new for two-sided match- ing markets under preferences. This may well be caused by the good performance of the Gale-Shapley type heuristics in prac- tice(see e.g. Roth &Peranson, 1999). Withtheir shortruntimes, theyapparentlyhavebeenpreferredoverintegerprogrammingap- proaches tosolve thesometimes largeinstances. Besidesa previ- ouspaper(Ágostonetal., 2016)motivatedby theHungarianuni- versityadmissions, therewereonlyacoupleofstudiesinthisdi- rectionforfindingmaximumsizeweaklystablematchingsinresi- dentallocationproblemwithties(Delormeetal.,2019;Kwanashie

& Manlove, 2014), forfinding stable matchingin thepresence of couples(Biró,McBride,&Manlove,2014),andunderdistributional constraints(Ágostonetal.,2018b).

Thepapermostcloselyrelatedtoourworkistherecentstudy of Delorme et al. (2019), where IP techniques have been devel- oped andtestedto solveatwo-sidedstablematchingproblemin a realapplication, pairing children withadoptive families. OurIP formulations forsolving the classical college admissionsproblem (in their terminology, the Hospitals/Residents problem) are very similar. Wealsofindthat introductionofadditionalvariablesand

2The authors wrote “In principle, one could appeal to the integer programming method devised by Biró and McBride (2014) for this problem, that finds a stable outcome when it exists. However, such an approach was untenable in practice due to complexity, relative opaqueness, and the likelihood of an unreasonably large run time on our large problem.”

binary cutoff scoresvariables drasticallyimproves theefficiencyof the IP solution.Thereafter they focuson theNP-hard problemof finding a maximum size weakly stable matching in case of ties, whilst weinvestigatetheHungarianequaltreatment policywhen considering thetiesandalsothefeatureofcommonquotas,both present in the Hungarian application. The presence of common quotasmakestheproblemNP-hard.WetestourIPformulationson the2008and2009instancesofHungarianuniversityadmission.

Realinstancesanalysed

Besides the 2008-Educatio instance, we have also had access to the Hungarian university admission datafrom anothersource, the KRTK Databank for the years 2001–2017. However the lat- ter instances donot contain capacity constraints,andthe identi- fiers for the programmes also differ from the 2008-Educatio in- stance. Nevertheless, using the 2008-Education instance, we ex- tendedourcomputationalanalysesforthe2009-KRTKinstanceun- der two reasonableassumptions afterlinking the2008-Education and2008-KRTKinstancesandthenthe2008-KRTKand2009-KRTK instances.³Thelinkageofthetwo2008instancesinvolvedmatch- ing of the students and programmes of the two instances. This wasnot straightforward dueto some limitationsof theinstances (e.g. the KRTK instance contained only the first six applications of each student) and the possibility that the two instances re- flectdifferentsnapshotsoftheapplications.Nevertheless,approx- imately 98,5% ofthe programmes have been identified. Then we alsoneededtomatchtheprogrammesinthe2008and2009KRTK instances, which was also non-trivial due to the changes in the listofprogrammesoffered (sometimesonlythe nameoftheuni- versity orthefacultyhaschanged,buttheprogrammesremained essentially the same, which required manual checks). When the linkagebetweentheprogrammesofthe2008-Educatioand2009- KRTKinstanceswere ready,weaddedthecapacityconstraintsfor the 2009-KRTKinstance intwo reasonableways: a)we usedthe same constraints as in the 2008-Educatio instance, b) we used thenumberofadmittedstudentsin2009forallprogrammesand common quotas identified from the 2008-Educatio instance. We refer to these two cases as2009-KRTK-previous and2009-KRTK- admitted,respectively.

Regarding the main statistics of the instances, in the 2008- Educatioinstancewehave81,427applicants,353,618applications, 3298programmes,2275facultyquotas,and206nationalcommon quotas. Whilstinthe2009-KRTKinstancewe have105,739 appli- cants,310,346applications,2992programmes,1828facultyquotas, and197nationalcommonquotas.

Ourcontribution

Ourresearchisafollow-upoftheworkofÁgostonetal.(2016), where the same application, the Hungarian university admission was studied with its four special features: ties, common quotas, lower quotas andpaired applications.InÁgostonetal.(2016)the specialfeatureswereconsideredone-by-oneandtheirmainresult wasapracticallytractableIPsolutionfortheNP-hardcaseoflower quotas, demonstratedonthe 2008-Educationinstance.Inthispa- per we continuethe investigation andfirst we lookmore deeply into the classical collegeadmission problem, where we compare several (mixed) IP formulations.The cutoff score formulation(al- readydescribedinÁgostonetal.(2016))turnsouttobe viableto solveforthe2008-Educatioinstanceevenwithoutanypreprocess- ing. For thestill efficientlysolvable caseofties we findthat the

3This data matching challenge was conducted as part of a student project, the details are available in a BSc thesis upon request.

newbinarycutoff formulation (thatis similartothe IPsuggested inDelorme etal.(2019)) performs thebest amongtheIP-s stud- ied. We then compare the solutions of the Hungarian, Irish and Chilean policies for the caseof ties. Confirming the theories de- scribedinBiró &Kiselgof(2015),wefindthatindeedthestudent- optimalcutoff scoresare alwaysthehighestinHungary,followed by theIrishcutoffs andthe lowestin Chile,ifconsidered forthe same instance. Finally, we define stability through cutoff scores forthecaseoftiesandcommonquotas withrespectto theHun- garianandChileanpolicies,andwe proposeIP formulationswith binary cutoff score variables. We find that these IP formulations work well forthe 2008-Educatioinstance.We comparethe solu- tionswithrespecttotheHungarian,IrishandChileanpolicies.We also extended the computational analyses for the 2009-KRTKin- stanceafterlinking the2008-Education and2008-KRTKinstances aswell asthe2008and2009KRTKinstancesundertwoassump- tions:a) by using the 2008 quotas for2009 everywhere, and b) setting the quotas equal to the number of students admitted. In orderto speedup thecomputationswe introduced apreprocess- ingphasethatfixessomevariablesintheIPmodelcorresponding tostudents’ applicationsthat are eithersurelyacceptedorsurely rejectedinthestablesolutions.

Layoutofthepaper

In Section 2, we start by investigating the basicGale-Shapley caseandtestingdifferentIP formulationsforasimplifiedinstance ofthe 2008-Educationinstance. We findthat the cutoff formula- tions perform better than the standard ones regarding their run time. In Section 3,we consider the special feature of tiesunder theHungarianpolicy,wherethequotasarestrictly obeyed,sothe lastgroup ofstudentswiththesamescore(thatcannotfitinthe quota) is rejected. Here we observe that the cutoff formulation withbinaryvariablesoutperformsthecutoff formulationwithcon- tinuousvariables.Then, inSection 4,wedescribe IP formulations alsofortheChileanpolicy,wherethelastgroupofstudentsisstill accepted (without whom there remains an empty seat, butwith whom the quota may be violated). We compare the results ob- tainedfortheHungarian,IrishandChileanpolicies.Thenweturn our attentionto common quotas, which are presentin the Hun- garianapplicationin a structure that makethe problemNP-hard to solve (Biró et al., 2010). We test different IP-s for solving the problemunderstrictpreferencesinSection5.Finally,inSection6, we tackle the real case when both ties andcommon quotas are present.WedevelopIP-sagainforboththeHungarianandChilean policies and we compare the results forboth the 2008-Educatio andthe2009-KRTKinstances.WeconcludeinSection7.

2. TheGale-Shapleymodel

Inthissectionwe providevarious IPformulationsfortheclas- sicalGale-Shapleycollegeadmissionmodelandthenwetestthese formulationsonthe2008-Educatioinstance.

2.1. Definitionsandpreliminaries

Inthe classicalcollegeadmissionsproblemby Gale & Shapley (1962)thestudentsareassignedtocolleges.⁴ Inthefollowingwe willrefertothetwosetsasapplicantsA=

{

^a1,...,an

}

^andcolleges

C=

{

^c1,...c_m

}

^{.Throughoutthe}manuscript,weusetheconvention ofi=1,...,nand j=1,...,m. Letu_j denote theupper quota of

4In the computer science literature this problem setting is typically called Hospi- tal/Residents problem (HR), due to the National Resident Matching Program (NRMP) and other related applications.

K.C. Ágoston, P. Biró, E. Kováts et al. European Journal of Operational Research xxx (xxxx) xxx

college c_j. Regarding the preferences,we assume that the appli- cants provide strict rankings over thecolleges, wherer_{i j} denotes therankoftheapplication(^ai,c_j)^inapplicantai’spreferencelist.

Wesupposethatthestudentsarerankedaccordingtotheirscores at the colleges, so college c_j ranks applicant a_i according to her score s_{i j} atc_j,where thegreater thescore themore preferredis the studentby the college. LetE⊆A×C denote theset ofappli- cations. Amatching isasetofapplications,whereeach studentis admitted to atmostone collegeandeach collegehasatmostas many assignees asits quota. Fora matching M let M(^ai) ^denote the collegewherea_i isadmitted to (or∅if a_i isnot allocatedto anycollege)andletM(^cj)^{denotethesetofapplicantsadmittedto} c_jinM.ThusthefeasibilityofamatchingM⊂Emeansthatforev- eryapplicanta_i,

|

^M(^ai)

|

≤1andforeverycollegec_j,

|

^M(^cj)

|

≤u_j. AmatchingM⊂E isstableifforanyapplication(^ai,c_j)^outsideM eithera_iprefersM(^ai)^tocjorc_jfilleditsseatswithu_japplicants who all have higher scores than a_i has. The deferred-acceptance algorithm ofGale andShapley provides a student-optimal stable matchinginlineartime(Gale&Shapley,1962).

The notion of cutoff scores is important for both the classical Gale-Shapleymodelanditsgeneralisationswithtiesandcommon quotas.Lett_jdenotethecutoff scoreofcollegec_jandlettdenote asetofcutoff scoresforallcolleges.Astudenta_iisadmissibletoa collegec_jwithcutoff scoret_jifs_{i j}≥t_j.WesaythatmatchingMis impliedbycutoff scorestifeverystudentisadmittedtothemost preferredcollegeinherlist,wheresheisadmissible(i.e.,achieved the cutoff score).Wesay that aset ofcutoff scorestcorresponds to a matching M ift implies M. Fora matching M an applicant a_i has justified envytowards another applicant a_k atcollege c_j if M(^ak)=c_j,a_iprefersc_jtoM(^ai)^andaiisrankedhigherthana_kat c_j (i.e.s_{i j}>s_{k j}). Amatchingwithnojustifiedenvyiscalledenvy- free(seeWu&Roth,2018andYokoi,2020).

It is not hard to see that a matching is envy-free ifand only ifitisimpliedbysome cutoff scores(Ágoston&Biró,2017).Note that an envy-free matching can be wasteful in the sense that it can leave manydesiredseatsempty (in factthe emptymatching is alsoenvy-free).More precisely,whena studenta_iprefersc_jto M(^ai) ^andcj isnot saturated(i.e.

|

^M(^cj)

|

<u_j) then we saythat M iswasteful. Bydefinitionit followsthat amatching isstableif and onlyif itis envy-free andnon-wasteful (see also Azevedo&

Leshno,2016).Toachievenon-wastefulnesswecanrequirethecut- off of any unsaturated college to be minimum (zero in our case).

Alternatively, we may require that no cutoff score be decreased withoutviolatingthequotaofthatcollege,whilekeepingtheother cutoff scores.Furthermore,wemayalsosatisfythelattercondition by ensuring that we select thestudent-optimal envy-free match- ing,whichisthesameasthestudent-optimalstablematching(Wu

& Roth,2018).Toobtainthissolutionweonlyneedtouseanap- propriateobjectivefunction.Wewillusetheabovedescribedcon- nectionswhendevelopingourIPs.

2.2. IPformulations

Herewewilldescribethreedifferentformulations.

TheBaïou-Balinskiformulation

Firstwe describethebasicIP formulationfortheGale-Shapley model, proposed in Baïou & Balinski (2000). All of our formula- tions are basedon thebinaryvariables correspondingto applica- tions,wherex_{i j}=1denotesthattheapplication(^ai,c_j)^isaccepted in thesolution (andx_{i j}=0 denotesthat it isnot).The feasibility ofamatchingcanbeensuredwiththefollowingtwosetsofcon- straints,whicharepartofallourIPs.

j:(^ai,cj)∈E

x_{i j}≤1foreacha_i∈A (1)

i:(ai,cj)∈E

x_{i j}≤u_jforeachc_j∈C (2)

Notethat(1)impliesthatnoapplicantcanbeassignedtomore thanonecollege,whereas(2)impliesthatthe upperquotas ofthe collegesarerespected.

Toenforcethe stabilityofafeasible matchingwe can usethe followingconstraint.

k:r_ik≤ri j

x_ik

·u_j+

h:(^ah,cj)∈E,s_{h j}>si j

x_{h j}≥u_jforeach

(

^ai,c_j

)

∈E (3)

Notethatforeach(^ai,c_j)∈E,ifa_iismatchedtoc_jortoamore preferredcollegethenthefirsttermensuresthesatisfactionofthe inequality.Otherwise,whenthefirsttermiszero,thenthesecond termisgreater thanorequalto therighthandsideifandonlyif theplacesatc_jarefilledwithapplicantswithhigherscores.

Among the stable solutions we can choose the applicant- optimalonebyminimisingthefollowingobjectivefunction.

min

(^ai,cj)∈E

r_{i j}·x_{i j} (4)

Weabbreviatethisformulationbasedonconstraints(1),(2)and (3),andobjectivefunction (4)asSO-BB (referringtostudentopti- mal Baïou-Balinski model). This IP results in the student-optimal stablematching.

Thecutoff scoreformulation

Foreachcollegec_jwedefineanonnegativerealvariablet_j de- notingitscutoff score.

tj≤

1−xi j

·

(

^s¯+1

)

+si j foreach

(

^ai,cj

)

∈E (5) and

si j+

≤tj+

k:rik≤ri j

xik

·

(

^s¯+1

)

^foreach

(

^ai,cj

)

∈E (6)

wheres¯isan upperbound forthescores (currently500in Hun- gary)and

^{isasmallpositivenumber.5Here(5)impliesthatifa}

studenta_iisadmittedtocollegec_jthenherscore(s_{i j})hasreached the cutoff score. The second Eq. (6) ensures the envy-freeness, namelythatifa_iisnotadmittedtoc_jortoanybetteraccordingto herpreferencethen itmustbe thecasethat shehasnotreached thecutoff atc_j.Thusthesetwosetsofconditionscreatethecon- nectionbetweenthecutoff scoresandthematching,ensuringthat thematchingimpliedbythecutoff scoresisenvy-free.

To require stabilityof the matching we need to rule out the possibilityofblockingwithanempty seat(i.e.wastefulness).This canbeachievedbyforcingthecutoff scoreofunsaturatedcolleges tobeminimum(i.e.zeroinourcase)bythefollowingconstraints, where f_jisabinaryvariableindicatingwhetherc_jrejectsanystu- dentinthesolution.

f_j·u_j≤ (ai,cj)∈E:cj∈C

x_{i j}

∀

^cj∈C (7)

and

t_j≤ f_j

(

^s¯+1

) ∀

^cj∈C (8)

Our second IP is then constructed from feasibility constraints (1), (2), cutoff score constraints (5), (6), non-wastefulness con- straints (7),(8), andtheobjective function(4)enforcing student- optimality. We abbreviate this IP as SO-NW-CUT, referring to student-optimalnon-wastefulcutoff scores.

5Note in Ágoston et al. (2016) 1 was used instead of , but we found that the latter choice makes the constraints tighter and the solution more efficient.

Asanalternative,wecandropthenon-wastefulnessconstraints and enforce stability directly by obtaining the student-optimal envy-freematchingbyusingeitherofthefollowingobjectivefunc- tions.

min

cj∈C

t_j (9)

or

max

(ai,cj)∈E

(

^K−ri j

)

·xi j (10)

with alarge enough constant K. When combinedwiththe feasi- bility constraints(1),(2),andcutoff score constraints(5),(6),we abbreviate the IP using objectivefunction (9)as MIN-CUT, refer- ring tominimumcutoff scores.Likewise, whencombined withthe feasibilityconstraints(1),(2),andcutoff scoreconstraints(5),(6), we abbreviate theIP using objectivefunction (10)asMSMR-CUT, referring to maximum size minimum rank cutoff scores. Note that asexplainedearlierbothMIN-CUTandMSMR-CUTwilloutputthe student-optimalstablematching.

Thebinarycutoff scoreformulation

We canmakethecutoff formulationsdiscretebyreplacingthe continuous cutoff variables by binary variables, as follows.For a collegec_j,let S_jdenotethesetofscoresthe studentshavethere, i.e.S_j=

{

^si j:(^ai,c_j)∈E

}

^{.SupposealsothattheelementsofS}j are sorted in an increasing order, so S_j=

{

^s1j,s²_j,...,s^m_j

}

^{, where sk}j<

s^k_j⁺¹. Foreach collegec_j, let us now introduce

|

^Sj

|

^{binary cutoff}

variables:t¹_j,t²_j,...,t^m_j withthefollowingconstraints.

x_{i j}≤t^k_j foreach

(

^ai,c_j

)

∈E,s_{i j}=s^k_j (11) and

t^k_j≤t^k_j⁺¹ foreachk=1.

( |

^Sj

|

⁻¹

)

⁽¹²⁾

Here, t^k_j=0means that the cutoff scoreat c_j is greater than s^k_j.Furthermore,(12)ensuresthemonotonicityofthebinarycutoff variablesand(11)requiresthatanapplicationcanonlybeaccepted ifthecutoff scoreisreached,correspondingtothecontinuouscon- straint(5).Regardingthesecondcontinuousconstraint(6)weadd thefollowingsimplerequations.

1≤

h:rih≤ri j

xih+

(

¹−t^k_j

)

^foreach

(

^ai,cj

)

∈E,si j=s^k_j (13)

Therefore,constraints(11),(12)and(13)replace(5)and(6),and togetherwiththefeasibilityconstraints(1),(2)theymakethelink betweenthebinarycutoff scoresandtheenvy-freematchings.

Toachieve stability,wecan usethesametechniquesasinthe continuous case, with slightly modified conditions and objective functions.

AsthefirstIP,insteadofusingEqs.(7)and(8),wecanenforce the cutoff score being zero foreach unfilled collegec_j withthe followingconstraint.

(

¹−t¹_j

)

·uj≤ (ai,cj)∈E:cj∈C

xi j

∀

^cj∈C (14)

ThecorrespondingbinaryIPisthenconstructedfromfeasibility constraints(1),(2),cutoff scoreconstraints(11),(12)and(13),non- wastefulness constraints (14), andthe objective function (4) that enforces student-optimality.WeabbreviatethisIPasSO-NW-BIN- CUT,referringtostudent-optimalnon-wastefulbinarycutoff scores.

Alternatively, we can drop again the non-wastefulness con- straints and enforce stability directly by obtaining the student- optimalenvy-freematchingbyusingeitherthefollowingobjective function

max

cj∈C,k=1.|^Sj|

t^k_j (15)

or objective function (10). Combined with feasibility constraints (1),(2),andbinarycutoff scoreconstraints(11), (12)and(13) we obtain two IPs, the MIN-BIN-CUT and MSMR-BIN-CUT, referring tominimumbinary cutoff scores andmaximumsize minimumrank binary cutoff scores, both resulting in the student-optimal stable matching.

Envy-freeformulation

Itisalsopossibletoenforceenvy-freenesswithoutusingcutoff scores,asexplainedinÁgoston&Biró (2017),byusingthefollow- ingconstraints.

k:rik≤ri j

x_ik≥x_{h j}

∀ (

^ai,c_j

)

^,

(

^ah,c_j

)

^∈E,si j≥s_{h j} (16)

Theaboveconstraintmeansthatifastudenta_h isallocatedto collegec_j thenevery studenta_i,who hasascoreatc_j atleastas highasa_hhas,mustalsobeallocatedtoc_jortoabettercollegeof her preference. Combined withthe feasibilityconstraints (1),(2), andobjective function (10) the solution obtainedis the student- optimalstablematching.WeabbreviatethisformulationasMSMR- EF,referringtomaximumsizeminimumrankenvy-free.

Summaryofformulations

Wesummarisetheconstraintsneededforall ofthe(mixed)IP formulationsthatwetestedforthebasicGale-Shapleycollegead- missionmodelinTable1.

2.3. Computationalresults

Wetookthe2008-Educatioinstanceafterbreakingthetiesran- domly,byconsideringonlythefacultyquotasandkeepingonlythe highestranked applicationof each student for every programme (i.e. the application for either the state funded or the privately fundedseat).FortheimplementationweusedAMPLwithGurobi.

Aswecan seeinTable2,themostefficientformulationsused cutoff scores.EventhoughSO-NW-CUTneededtwiceasmanyvari- ablesasSO-BB,itsruntimewassmallerbyamagnitude.Notethat verysimilarfindingswerereportedinDelormeetal.(2019).Com- paring the continuous and binary cutoff score formulations, SO- NW-CUTandSO-NW-BIN-CUT,wecanobservethatthecontinuous versionperformedslightlybetterforthisbasicmodel.Thesimple MSMR-EFformulationdidnot terminate,sowe excludedthisfor- mulationfromfurtherconsiderationforthemoregeneralmodels.

3. Modelswithties

In many nationwide college admission programmes the stu- dents are ranked based on their scores, andties may appear. In Hungary,for instance,the students can obtain integerpoints be- tween 0and500 (the maximumwas144 until 2007),so tiesdo occur.Whentiesarepresentthenonewaytoresolvethisissueis to break tiesby lotteries, asdone in Ireland (soa lucky student with480pointmaybeadmittedtolawschool,whilst anunlucky studentwiththesame scoremaybe rejected).However,lotteries areoftenseenunfair,soinsomecountries,suchasHungary(Biró

&Kiselgof,2015)andChile(Riosetal.,2014)equaltreatmentpoli- ciesare used,meaningthat students withthesamescore areei- therallacceptedorallrejected. Thispolicy giveswaytotworea- sonable variants when deciding aboutthe last group of students withoutwhom thequotaisunfilledandwithwhomthe quotais violated.Inthe restrictivepolicy,usedinHungary,thequotas are never violated, so this last group of students is always rejected, whilstinChiletheyuseapermissivepolicyandtheyalwaysadmit thislastgroupofstudents.Forinstance,iftherearethreestudents, a₁,a₂anda₃,applyingtoaprogrammeofquota2withscores450,

K.C. Ágoston, P. Biró, E. Kováts et al. European Journal of Operational Research xxx (xxxx) xxx Table 1

The summary of (mixed) integer programming formulations for the classical Gale-Shapley model.

IP formulations (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

SO-BB √ √ √ √

SO-NW-CUT √ √ √ √ √ √ √

MIN-CUT √ √ √ √ √

MSMR-CUT √ √ √ √ √

SO-NW-BIN-CUT √ √ √ √ √ √ √

MIN-BIN-CUT √ √ √ √ √ √

MSMR-BIN-CUT √ √ √ √ √ √

MSMR-EF √ √ √ √

Table 2

The performance of (mixed) integer programming formulations for the classical Gale-Shapley model.

IP formulations #variables #constraints #non-0 elem. size(Kb) run time(s)

SO-BB 287,035 381,115 73,989,595 1,319,663 1139

SO-NW-CUT 291,935 673,050 2,463,808 69,464 81

MIN-CUT 289,485 668,150 2,169,423 64,254 5062

MSMR-CUT 289,485 668,150 2,169,423 69,846 2318

SO-NW-BIN-CUT 574,070 955,185 3,028,078 75,810 107 MIN-BIN-CUT 574,070 952,735 2,738,593 65,657 871 MSMR-BIN-CUT 574,070 952,735 2,738,593 66,467 4325

MSMR-EF n.a. n.a. n.a. 8,667,403 n.a.

443,and 443, respectively then in Hungary only a₁ is admitted, whilst in Chile all three of them. In Ireland, a₁ is admitted and theyusealotterytodecidewhethera₂ora₃willgetthelastseat.

3.1. Definitionsandpreliminaries

Stable matchingsforthecaseoftieswere definedthroughthe cutoff scores inBiró &Kiselgof(2015).Using cutoff scoresincase of ties makes the solution envy-free, meaning that no student a_i may be rejected from collegec_j if this collegeadmitted another studentwithascorelessthanorequaltothescoreofstudenta_i. Thisallocationconceptisalsocalledequaltreatment policy,since the admission of a student to a programme implies the admis- sion offertoall other studentswiththesame score.⁶ Hereagain, we have thesame equivalence betweenenvy-free matchings and matchinginducedbycutoff scores(Ágoston&Biró,2017),thatwe provehereforbeingself-contained.

Proposition1. Amatchingisenvy-freeforacollegeadmissionprob- lemwithtiesifandonlyifitisinducedbycutoff scores.

Proof. Givenan envy-free matchingM letusset thecutoff score of each collegeto be thescore ofthe weakest admitted student.

These cutoff scores will induce M. In the other direction, any matchinginducedbycutoff scoresisobviouslyenvy-free.

In this paperwe focus on the restrictive policy usedin Hun- gary,wherethestabilityofthematchingcanbedefinedbyadding anon-wastefulnessconditiontoenvy-freeness.Namely,amatching inducedbycutoff scoresisstableifnocollegecandecreaseitscut- off scorewithout violating itsquota,assumingthattheothercutoff scores remainthesame.We note thatthestabilityofa matching can equivalentlybedefinedbythelackofa setofblockingappli- cationsinvolvingonecollegeandasetofapplicantssuchthatthis set of applications would be accepted by all parties when com-

6Note that it is also possible to define envy-freeness and stability in a weaker form, where the rejection of a student is allowed when another student with the same score is accepted. These so-called weakly stable or weakly envy-free matchings are used in the Scottish resident scheme ( Irving & Manlove, 2008 ), and in a project allocation application at CEMS universities ( Ágoston et al., 2018b ), respectively.

pared to the applications of the matching considered. See more aboutthisconnectioninFleiner&Jankó (2014).⁷

Moreformally,fora collegec_j anda setofapplicationsX⊂E tothiscollegewedefine byCh_j(^X)⊆X thesetofapplicationsse- lected byc_j. Forthe caseof strictrankings thechoice functionis simple,if

|

^X

|

≤u_jthenCh_j(^X)=X,andif

|

^X

|

>u_jwhenc_jselects thetopu_japplicantsaccordingtotheirscores.Fortiesweconsider twochoicefunctions,Ch^H_j andCh^C_j correspondingtotheHungarian restrictive andtheChilean permissivepolicies. First wenote that for

|

^X

|

≤u_j we haveCh^H_j(^X)=Ch^C_j(^X)=X, the question is what happensfor

|

^X

|

>u_j. Forcutoff scoret_j let X^≥tj denote the sub- setofapplicationsinX wherethestudentshavescoret_jorhigher atc_j.Inthe HungarianpolicyCh^H_j(^X)=X^≥tj such that

|

^X≥tj

|

≤u_j and

|

^X≥tj−1

|

>u_j, thus the numberof students selected is never more than the quota, but the cutoff is minimal, i.e., decreasing thecutoff would implythe violationof thequota. Inthe Chilean policyCh^C_j(^X)=X^≥tj such that

|

^X≥tj

|

≥u_j and

|

^X≥tj+1

|

<u_j, thus thenumberofstudentsselectedisatleastasmuchasthequota, butthecutoff ismaximal,i.e.,increasingthecutoff wouldimply to haveemptyseats.

Biró &Kiselgof (2015)provedtwomaintheorems aboutstable matchings forcollegeadmissionswithties. Intheir first theorem they showed that a student-optimal anda student-pessimal sta- ble matching exist for both the restrictive policy (Hungary) and the permissive policy (Chile), where the cutoff scores are mini- mal / maximal, thus the matchings are the best / worst for all students,respectively. Furthermore,theyalsoproved theintuitive results that ifwe compare the student-optimal cutoff scores (or the student-pessimal ones) with respect to the three reasonable policies,namelytheHungarian(restrictive),theIrish(lottery),and theChilean (permissive),then theHungariancutoff scores areal- waysashighforeachcollegeastheChilean cutoff scoresandthe Irishcutoff scoresareinbetween.When consideringthestudent-

7There are many interesting properties that apply differently for the three poli- cies, as demonstrated in Fleiner & Jankó (2014) and Biró & Kiselgof (2015) . For in- stance, the corresponding choice functions are substitutable for all the three policies, but the irrelevance of rejected contracts property is violated for the Hungarian pol- icy, and the law of aggregate demand property is violated for both the Hungarian and Chilean policies. That is why neither of the latter two policies is strategyproof for the students, even though student-optimal solutions do exist.

optimal stable matching, it turns out to be also the student- optimalenvy-freematching,asthefollowingpropositiondescribes inÁgoston&Biró (2017).

Proposition 2. For the college admission problem with ties the student-optimal stable matching is also student-optimal among the envy-freematchingswithrespecttotheHungarian(orChilean)equal treatmentpolicy.

Proof. Assume bywayofcontradictionthatthereisan envy-free matching M,whereonestudent getsa betterassignment than in thestudent-optimalstablematchingM^s.Withoutlossofgenerality we may also assume that Mis not Pareto-dominatedby another envy-free matching M withthesame property(i.e., whereevery student wouldgetatleast asgoodan assignment andsomebody wouldgetastrictlybetterassignment).ByProposition1weknow that Mis inducedby some cutoff scoret.Notethat Mcannot be stable,sincethatwouldcontradicttothestudent-optimalityofM^s. Therefore thereisatleastone collegewhere thecutoff scorecan be decreasedsothatnewstudentswillbeadmitted there,butits quota isnot violated. Let the newcutoff score be t andlet the new matching implied be M. But then M is also envy-free and Pareto-dominatesM,acontradiction.

3.2. IPformulations

Firstwe describewhichofthe(M)IPformulationsfortheclas- sicalmodelworkunchangedforthecaseofties,andthenwegive somealternativeformulations.

PreviousIP-sthatworkforties

Fortherestrictive (Hungarian)equaltreatment policywe have to keep the original feasibility constraints (1), (2). Constraints (16) ensure envy-freeness immediately, and we can also achieve envy-freenessbyusingthesameconstraintswithcutoff scores((5), (6)),orwithbinarycutoff scores((11),(12)and(13)).

To secure stability, we can enforce the selection of the stu- dentoptimalenvy-freematchingbyusinganappropriateobjective function, asimplied by Proposition 2.Essentially all thestudent- optimal IP formulationsfor envy-freematchings that were previ- ously describedfortheGale-Shapleymodelwillleadtothissolu- tion, namelyMIN-CUT, MSMR-CUT,MIN-BIN-CUT, MSMR-BIN-CUT andMSMR-EF (however,weleave out thelatterfromthe simula- tionsduetoitsbadperformanceforthebasicmodel).

Alternativeformulations

Whenwewanttoavoidtheinclusionofobjectivefunctions,we mayalsoenforcestabilitydirectlybyaddingnewvariablesd_{i j} and constraints,asdescribedinÁgostonetal.(2016).Here d_{i j} isabi- nary variableshowingwhethera_iwould beadmitted to c_j ifthe cutoff scoredecreasedatc_jbyone.

d_ik≤

(

¹−x_{i j}

)

^foreach

(

^ai,c_j

)

∈E,

(

^ai,c_k

)

∈E,r_ik≥r_{i j} (17) Condition (17) implies that d_{i j} can only be one if student a_i prefersc_jtohercurrentassignment.

tj−1≤

1−di j

·

(

^s¯+1

)

+si j foreach

(

^ai,cj

)

∈E (18)

where (18)isamodification of(5),implyingthat d_{i j} can onlybe oneifa_ireachesthecutoff score,whendecreasedbyone.

Now,withthesenewvariableswecanalsoformulatethenon- wastefulnesscondition, where f_jwillagainindicatewhetherc_jis essentially full,meaning thatits cutoff scorecannot bedecreased withoutviolatingitsquota.Besideskeeping(8),wemodify(7)into thefollowingcondition.

f_j·

(

^uj+1

)

≤ (^ai,cj)∈E:cj∈C

(

^xi j+d_{i j}

) ∀

^cj∈C (19)

Tosummarise,togetherwiththebasicfeasibilityconditions(1), (2),and cutoff score constraints(5), (6),satisfaction of Eqs. (17), (18),(8),(19)resultinastablematchingwithrespecttotheHun- garianequal treatment policy. Tofind the student-optimalstable matchinginthiscontext,wemayagainuseobjectivefunction(10). WedenotethisformulationbySO-H-NW-CUT.

Binarycutoffs

Finally,wecanagainusebinaryvariablesforthecutoffs.Keep- ingthefeasibilityconstraints(1),(2),cutoff scoreconstraints(11), (12)and(13),we modifythenon-wastefulnessconstraints(14)as follows.

(

¹−t¹_j

)

·

(

^uj+1

)

≤ (^ai,cj)∈E:cj∈C

x_{i j}+d_{i j}

∀

^cj∈C (20)

Wekeep(17)andmodify(18)tothefollowingconstraints.

di j≤t_i^k+1−t_i^kforeach

(

^ai,cj

)

∈E,si j=s^k_j (21) Theseconstraintsmeanthatifd_{i j}=1thenthescoreoftheap- plicantis justbelowof thecutoff score. Using objective function (10), with feasibility constraints (1), (2), cutoff score constraints (11),(12)and(13),non-wastefulnessconstraints(17),(20)and(21), wegettheformulationdenotedbySO-H-NW-BIN-CUT.

3.3. Simulations

Weconsideredthe2008-Educationinstancewithtiesbytaking intoaccountonlythefacultyquotas andkeepingonlythehighest ranked applicationofeach student forevery programme(i.e. the applicationforeitherastatefundedorprivatelyfundedseat).The resultsare summarised inTable 3.We can observethat the best binaryIP formulationhas outperformed the best continuousfor- mulation.

4. Policycomparisonforties

First we set up an IP formulation for the Chilean policy,and then we compare the solutions obtainedby the Hungarian, Irish andChileanpoliciesinthe2008-Educatioinstance.

4.1. IPfortheChileanpolicy

Recall that here we admit the last group of students, with whom the quota is violated, but without whom some seats re- mainunfilled. Alternatively, we mayrequirethat afterdecreasing the cutoff score atany collegethe number ofadmitted students wouldbe strictly lessthan its quota.Toachieve thisin themost effectiveway, we usea similar formulationas SO-H-NW-CUTfor theHungarianpolicy.

Thecutoff scoreformulation

Hered_{i j} isabinaryvariableshowingthata_iisadmitted toc_j, butwouldberejectedifthecutoff scoreincreasedatc_jbyone.

di j≤xi j foreach

(

^ai,cj

)

∈E (22) Conditions(22)implythat d_{i j} canonly beone ifstudenta_i is admittedtoc_jintheactualmatching.

d_{i j}−1

·

(

^s¯+1

)

^+si j≤t_jforeach

(

^ai,c_j

)

^{∈E (23)}

where(23)impliesthat d_{i j} willonlybeone ifa_i isrejectedifthe cutoff increasesbyone.

Now, with these variables we can also formulate the non- wastefulness condition, where f_j will againindicates whether c_j is essentially full, meaning that there would be empty seats if