Stable project allocation under distributional constraints Operations Research Perspectives

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Operations Research Perspectives

journalhomepage:www.elsevier.com/locate/orp

Stable project allocation under distributional constraints ^R

Kolos Csaba Ágoston

^a

, Péter Biró

^b,c,1,∗

, Richárd Szántó

^d

aDepartment of Operations Research and Actuarial Sciences, Corvinus University of Budapest, Fövám tér 13-15, Budapest H-1093, Hungary

bInstitute of Economics, Research Centre for Economic and Regional Studies, Hungarian Academy of Sciences, Budaörsi út 45, Budapest H-1112, Hungary

cDepartment of Operations Research and Actuarial Sciences, Corvinus University of Budapest, Hungary

dDepartment of Decision Sciences, Corvinus University of Budapest, Fövám tér 13-15, Budapest H-1093, Hungary

a rt i c l e i n f o

Article history:

Received 19 January 2018 Accepted 19 January 2018

Keywords:

Assignment Stable matching Two-sided markets Project allocation Integer linear programming

a b s t r a c t

Inatwo-sidedmatchingmarketwhenagentsonbothsideshavepreferencesthestabilityofthesolution istypicallythemostimportantrequirement.However,wemayalsofacesomedistributionalconstraints withregardtotheminimumnumberofassigneesorthedistributionoftheassigneesaccordingtotheir types.Thesetworequirementscanbechallengingtoreconcileinpractice.Inthispaperwedescribetwo realapplications,aprojectallocationproblemandaworkshopassignmentproblem,bothinvolvingsome distributional constraints.We used integer programmingtechniquesto findreasonablygood solutions withregardtothestabilityandthedistributionalconstraints.Ourapproachcanbeusefulinavarietyof differentapplications,suchasresidentallocationwithlowerquotas,controlledschoolchoiceorcollege admissionswithaffirmativeaction.

© 2018TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Acentralisedmatchingschemehasbeenusedsince1952inthe UStoallocatejuniordoctorstohospitals[40].Later,thesametech- nology has been used in school choice programs in large cities, such asNewYork[3]andBoston[4].Similar schemeshavebeen established inEuropeforuniversityadmissionsandschool choice as well. Forinstance, in Hungary both the secondary school and thehighereducationadmissionschemesareorganisednationwide, see [12] and[13], respectively. Furthermore, it can also be used to allocatecoursestostudents underpriorities[20].In theabove mentioned applications itis commonthat the preferences ofthe applicants andthe rankings of the parties on the other side are collectedbyacentralcoordinatorandaso-calledstableallocation iscomputedbasedonthematchingalgorithmofGaleandShapley [26].Two-sided matching markets, andthe above applicationsin particular, have beenextensively studied inthe last decades,see

R A preliminary version of this paper has appeared in the proceedings of the 10th Japanese-Hungarian Symposium on Discrete Mathematics and its Applications, 2017.

∗ Corresponding author.

E-mail addresses: kolos.agoston@uni-corvinus.hu (K.C. Ágoston), peter.biro@krtk.mta.hu (P. Biró), richard.szanto@uni-corvinus.hu (R. Szántó).

1Supported by the Hungarian Academy of Sciences under its Momentum Pro- gramme (LP2016-3/2017) and Cooperation of Excellences Grant (KEP-6/2017).

[43] and[35] for overviewsfrom gametheoretical andcomputa- tionalaspects,respectively.

InthispaperwedescribetworecentapplicationsattheCorvi- nusUniversityofBudapest,whereweusedasimilarmethodwith someinterestingcaveats.Inthefirstapplicationwehadtoallocate students to projects in such a way that the number ofstudents allocatedtoeach projectisbetweenalower andan upperquota, together with an additional requirementover the distribution of theforeignstudents.Thisisanaturalrequirementpresentinmany applications,suchastheJapaneseresidentallocationscheme[30]. Inthesecondapplicationwescheduledstudentstocompaniesfor solvingcasestudiesinaconference,andhereagainwefacedsome distributionalconstraints.

We decided to use integer programming techniques for solv- ing both applications. We had at least three reasons for choos- ing this technique. The first is that with IP formulations we can easilyencodethosedistributionalrequirementsthattheorganisers requested,sothissolutionmethodisrobusttoaccommodatespe- cialfeatures.Thesecondreasonisthatthecomputationalproblem becameNP-hardasthecompaniessubmittedlistswithties.Using tiesintherankingwasbyourrecommendationtothecompanies, becausetiesgiveusmoreflexibilitywhenfindingastablesolution underthedistributionalconstraints.Wedescribethisissuemorein detailshortly. Finally,ourthird reasonforchoosingIP techniques was that it facilitates multi-objective optimisation, e.g. finding a

https://doi.org/10.1016/j.orp.2018.01.003

2214-7160/© 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

most-stablesolution ifastablesolution doesnot existunderthe strictdistributionalconstraints.

The usageofintegerprogrammingtechniquesforsolvingtwo- sidedstable matching problems is very rare in the applications, and the theoretical studies on this topic have only started very recently. The reason is that the problems are relatively large in mostapplications,andtheGale-Shapleytypeheuristicsareusually abletofindstablesolutions,eveninpotentiallychallengingcases.

Aclassical example is the resident allocation problemwith cou- ples,whichhasbeenpresentintheUSapplicationfordecades,and itisstillsolved bytheRoth-Peransonheuristic[42].Theunderly- ingmatchingproblemisNP-hard [39],butheuristic solutionsare quitesuccessfulinpractice,see also[14]on theScottishapplica- tion.However, integer programmingandconstraint programming techniqueshavebeendevelopedveryrecentlyandtheyturnedout tobepowerfulenoughtosolvelarge randominstances[15,18,21]. Similarly encouraging results have been obtained for some spe- cialcollegeadmissionproblems,whicharepresentintheHungar- ianhighereducationsystem.Thesespecialfeatures alsomakethe problemNP-hard ingeneral,butatleastoneofthesechallenging features,turned out to be solvableeven withreal data involving morethan 150,000 applicants[6].Finally, the last paperthat we highlightwithregardtothistopicdealswiththeproblemoffind- ingstablesolutions inthepresenceofties[34].However, weare not aware of anypapers that would studyIP techniques for the problemofdistributionalconstraints.

Distributional constraints are present in many two-sided matching markets. In the Japanese resident allocation the gov- ernmentwants to ensure that the doctors are evenly distributed acrossthe country, andto achieve this they imposed lower quo- tas on the number of doctors allocated in each region [27,30–

32]. Distributional objectives can also appear in school choice programs, wherethe decision makers want to control the socio- ethnical distribution of the students [2,17,22,23,33]. Nguyen and Vohra[37] studied a special casewhere soft constraints are im- posed on the proportion of different types of students. Further- more,thesamekindofrequirements areimplemented incollege admissionschemeswithaffirmativeaction[1]suchastheBrazilian collegeadmissionsystem[8]andtheadmissionschemetoIndian engineeringschools[9].

Finally,thereisarecentlineofresearchbymathematicianson so-calledclassifiedstablematchings,wheretheproblemoffinding astablesolutionunderlowerandupperquotasovercertaintypes ofapplicants. Huang [28] gavean efficientalgorithm for laminar setsystems,whichwasgeneralisedbyFleinerandKamada[25] is amatroidframework,andfurtherextendedbyYokoi[44]forpoly- matroids. Finally, Yokoi provided an efficient method for finding an envy-free matching for so-called paramodular lower and up- perquotafunctions, ifsucha solutionexists,andshealsoproves thattheproblemisNP-hard inthegeneralsetting.Amodelwith one-sidedpreferencesandditributionalconstraintswasstudiedre- centlyin[7].

When stablesolutions donotexist forthestrict distributional constraintsthenwe eitherneedtorelax stabilityortoadjust the distributionalconstraints.Inthisstudywewillconsiderthetrade- off between these twogoals,anddevelopsomereasonablesolution concepts.

Here,webrieflydescribeourdefinitionsandsolutionconcepts, thepreciseformulationswillfollowaswe developourmodeland solutionconceptsunderextendingsetsofconstraints.Inourmodel the applicants submit their strict preferences on the companies andthecompaniesprovideweakrankingsovertheapplicants.The companies have lower and upper quotas respecting the number ofassignees. Amatching isfeasible if itrespects these quotas. A matching is stable if for any applicant-company pair not in the matching either the applicant prefers her matching or the com-

panyhas filledits upperquota withweaklyhigher rankedappli- cants.Amatchingisenvy-free ifnoapplicant hasa justifiedenvy towardsanotherapplicant,meaningthatsheprefersthecompany wheretheotherapplicantisadmittedtoherassignmentandsheis alsorankedstrictly higherbythat companythan theotherappli- cant.An envy-freematchingmaybewasteful,meaning thatthere canbeunfilledcompaniesthatarepreferredbysomeapplicantsto theirassignments.Amatchingisstableifandonlyifitisenvy-free andnon-wasteful. When the applicantshave typesthen we may alsohavelowerandupperquotaswithrespecttothetypes,which havetobe obeyedforthefeasibility ofthematching.Thesequo- tasmayapplyforindividualcompanies(asinourfirstapplication), forsetsofcompanies,orforallcompanies(asinoursecondappli- cation).Inourmodel(andmotivatingapplications)theapplicants arepartitionedaccordingtotheirtypes(suchasdomesticandfor- eignstudents). Amatchingiswithin-typeenvy-free ifthereisno justifiedenvybetweenanytwostudentsofthesametype.

Regarding the solution concepts, we are focusing on “almost stability”. A stablematchingmaynot exist when bothlower and upperquotasareimposed.Inthiscaseanaturalsolutionistolook foranenvy-freematching,whichisasnon-wastefulaspossible.If envy-freematchingdoesnotexisteither,thenwemaywanttofind afeasiblematchingwherethenumberofpairswithjustifiedenvy isminimised.Iftheapplicantshavetypesandanenvy-freematch- ing does not exist, then we can look for within-type envy-free matchings.Thissolutionisguaranteedtoexistundersomenatural assumptions, whichare satisfiedinourapplications (Theorem 1).

We can also characterise these matchings by the usage of type- specific scores, wherethe applicantsofcertain types can getex- tra scores (Theorem 2). Finally,among the within-type envy-free matchings wemaywant tominimiseenvy acrosstypes,i.e.min- imisethe pairsof applicantswithdifferenttypesthat havejusti- fiedenvies. Inthisminimisation we cansimply take thenumber ofsuchpairs,oralternativelywecanconsidertheintensityofthe envy (howmuch highertherejectedapplicant iscomparedto an unfairlyacceptedapplicant)andwemayaimtominimisethetotal intensityoftheenvies.

Wedevelopedintegerprogrammingformulationstosolvethese problemsarisingfromtworealapplications,andwereporttheso- lutionsthatweobtainedinourcasestudies.

2. Definitionsandpreliminaries

Many-to-one stable matching markets have been defined in manycontexts intheliterature.Intheclassicalcollegeadmissions problem by Gale and Shapley [26] the students are matched to colleges. In the computer science literature this problem setting is typically called Hospital / Residents problem (HR), due to the NationalResidentMatchingProgram(NRMP)andotherrelatedap- plications. In our paper we will refer the two sets as applicants A=

{

^a1,...,an

}

^{and companies C}=

{

^c1,...cm

}

^{. Let u}j denote the upperquotaofcompanyc_j.

Regardingthepreferences,we assumethat theapplicantspro- vide strict rankings over the companies, but the companies may have tiesin their rankings. The preferencelists of theapplicants maybeincompleteinourmodel(sonotalltheapplicant-company pair is possible), but in our applications the preference lists are complete,andthisconditionisalsousedinsomeofourtheoreti- calresults.ThismodelissometimesreferredtoasHospital/Resi- dentsproblemwithTies(HRT)inthecomputer scienceliterature, seee.g. [35].Inourcontext,letr_ij denotetherankofcompanyc_j ina_i’spreferencelist,meaningthatapplicant a_i prefersc_jto c_k if andonlyifr_ij<r_ik.Lets_ij beanintegerrepresentingthescoreofa_i bycompanyc_j,meaningthata_iispreferredovera_kbycompanyc_j ifs_ij>s_kj.Notethatheretwoapplicantsmayhavethesamescore ata company, so s_{i j}=s_{k j} is possible.Let s¯denote themaximum

possiblescoreatanycompanyandletEbethesetofapplications.

Amatchingisasubset ofapplications,whereeachapplicantisas- signed to at mostone company andthe number of assignees at eachcompanyislessthanorequaltotheupperquota.Amatching iscomplete ifevery studentisallocated.A matchingissaid tobe stableifforanyapplicant-companypairnotincludedinthematch- ing eithertheapplicant ismatchedto amorepreferredcompany orthecompanyfilleditsupperquotawithapplicantsofthesame orhigherscores.

Inthe classicalcollegeadmission problem,that we referto as HR, astablesolutionisguaranteed toexist,andthetwo-versions of the Gale-Shapley algorithm [26] findeither a student-optimal oracollegeoptimalsolution,respectively.Furthermore,thisalgo- rithm can be implemented to run in linear time in the number of applications.Moreover, thestudent-proposing variantwasalso provedtobestrategyproofforthestudents[40],whichmeansthat nostudentcanevergetabetterpartnerby submittingfalsepref- erences.Finally,the so-calledRuralHospitals Theorem[41]states that the samestudents are matchedin every stablesolution,the numberofassigneesdoesnotvaryacrossstablematchingsforany college,andforthelesspopularcollegeswheretheupperquotais notfilledthesetofassignedstudentsisfixed.

When extending the classical collegeadmission problem with thepossibilityofhavingtiesinthecolleges’rankings,thatwere- ferred toasan HRTinstance,theexistence ofastablesolutionis still guaranteed,since wecanbreakthetiesarbitrarily,andasta- ble solutionforthe strict preferences isalso stableforthe origi- nalones. However,now thesetofmatchedstudentsandthesize of the stable matchings can vary. Take justthe following simple example:we havetwoapplicants, a₁ anda₂ firstapplyingtocol- legec₁withthesamescoreandapplicanta₂alsoappliestocollege c₂ asher second choice.Here,ifwe breakthe tieatc₁ infavour ofa₁ then wegetthematchinga₁c₁,a₂c₂,whilst ifwe breakthe tieinfavourofa₂ thentheresultingstablematchingisa₂c₁ (thus a₁ isunmatched).Theproblemoffinding amaximumsize stable matchingturnedouttobeNP-hard[36],andhasbeenstudiedex- tensivelyinthecomputerscienceliterature,seee.g.[35].Notethat when the objective of an application is to find a maximum size stable matching, such asthe Scottish resident allocation scheme [29],thenthemechanismisnotstrategyproof.Toseethis,wejust havetoreconsidertheabove example,andassumethat originally a₁alsofoundc₂ acceptableandwouldrankitsecond,justlikea₂. By removing c₂ fromher list,a₁ is now guaranteed to get c₁ in the maximum size stablesolution, however,forthe original true preferencesa₂wouldhaveanequalchance togetherfirstchoice c₁.

2.1. Introductionoflowerquotas

Inourfirstapplicationtheorganisersoftheprojectallocations wanted to ensurea minimumnumber ofstudents foreach com- pany. Similar requirements have been imposed for the Japanese regions withregardtothenumberofresidentsallocatedthere.In ourmodel,weintroducealowerquotal_jforeachcompanyc_jand werequirethatinafeasiblematchingthenumberofassigneesat anycompany isbetweenthe lower andupperquotas. Stabilityis defined asbefore.We refer to thesetting withstrict preferences asHospitals/ResidentsproblemwithLowerquotas(HRL)andthe casewithtiesisreferredtoasHospitals/Residentsproblemwith TiesandLowerQuotas(HRTL).

Regarding HRL, the Rural Hospitals Theorem impliesthat the existence of a stablematching that obeysboth the lower an up- per quotascanbe decidedefficiently.Thisisbecausewejustfind one stablematchingby consideringthe upperquotas only, andif thelower quotas areviolated thenthere existsnostablesolution under these distributional constraints. This problem can be still

solvedefficientlywhenthesetsofcompanieshavecommonlower andupperquotasinalaminarsystem,see[25].

However, the problem of deciding the existence of a stable matching for HRTL is NP-hard. To see this, we just have to re- markthat theproblemoffinding acomplete stablematchingfor HRTwith unit quotas isalso NP-hard [36], soifwe require both lowerandupperquotas tobe equaltooneforallcompanies then thetwoproblemsareequivalent.Furthermore,nomechanismthat findsa stablematching wheneverthereexists one can be strate- gyproof.²

2.2.Addingtypesanddistributionalconstraints

In our first application, the organisers want to distribute the foreignstudentsacross theprojectsalmost equally.Inoursecond application,therearetargetnumbersforthetotalnumberofHun- garian,Europeanandotherparticipantsandtherearealsospecific lower quotas for Hungarian students by some companies. These applicationsmotivate ourproblemswithapplicant typesanddis- tributionalconstraints.

LetT =

{

^T1,...,T^p

}

^{bethesetoftypes,wheret(a}i)denotesthe type of applicant a_i. For a company c_j, let l^k_j andu^k_j denote the lowerandupperquotaforthenumberofassigneesoftypeT^k.Fur- thermore,we may alsoset lower and upperquotas for anytype ofapplicantsfora setofcompanies. Inparticular, wedenote the lowerandupperquotasforthetotalnumberofapplicantsoftype T^kassignedinthematchingbyL^kandU^k,respectively.Thesetof feasibilityconstraintsforthematchingisnowextendedwiththese lowerandupperquotas.Yet,theoriginalstabilitycondition,which doesnotconsiderthetypesoftheapplicants,remainsthesame.

3. Solutionconceptsandintegerprogrammingformulations

Inallofourformulationsweusebinaryvariablesx_ij∈{0,1}for eachapplicationcomingfromapplicanta_itocompanyc_j.Thiscan beseenasacharacteristicfunctionofthematching,wherex_{i j}=1 correspondstothecasewhena_iisassignedtoc_j.

Whendescribingtheintegerformulations,firstwekeepthesta- bilityconditionfixedwhileweimplementthesetofdistributional constraints.Thenweinvestigatethewaysonecanrelaxstabilityor findmost-stablesolutionsunderthedistributionalconstraints.

3.1. Findingstablesolutionsunderdistributionalconstraints

In this subsection we gradually add constraints to the model whilekeepingtheclassicalstabilitycondition.

ClassicalHRinstance

Firstwe describe thebasicIP formulationforHR describedin [10].Thefeasibilityofamatchingcanbeensuredwiththefollow- ingtwosetsofconstraints.

j:(^ai,cj)∈E

x_{i j}≤1foreacha_i∈A (1)

2Strategyproofness is an important desiderata in matching markets. However, there are many applications where the mechanisms used are not (fully) strate- gyproof, see [11] . The solution concepts that we use in our approach, stability and envy-freeness seem to provide some guarantee against manipulation by naïve agents. This is because stability (and envy-freeness) can be validated by the cutoff scores, which are the scores of the weakest admitted applicants at the companies (or universities in college admissions). The naïve agents believe that they cannot affect the cutoff scores, which is actually a realistic assumption in many large ap- plications, and thus under that assumption submitting their true preferences is ob- viously the best strategy. We had no complain reported about this aspect of our mechanism from the students’ side.

i:(^ai,cj)∈E

xi j≤ujforeachcj∈C (2)

Notethat(1)impliesthatnoapplicantcanbeassignedtomore thanone company, and(2) impliesthat the upper quotas of the companiesarerespected.³

To enforcethestabilityof afeasible matchingwe canuse the followingconstraint.

k:rik≤ri j

xik

·uj+

h:(ah,cj)∈E,sh j>si j

xh j≥ujforeach

(

^ai,cj

)

∈E (3)

Notethatforeach(a_i,c_j)∈E,ifa_iismatchedtoc_jortoamore preferredcompanythenthefirsttermprovidesthesatisfactionof theinequality.Otherwise,whenthefirsttermiszero,thenthesec- ondtermisgreaterthanorequaltotherighthandsideifandonly iftheplacesatc_jarefilledwithapplicantswithhigherscores.

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

(^ai,cj)∈E

r_{i j}·x_{i j}

ModificationforHRT

When the companies can express ties the following modi- fied stabilityconstraints, together withthe feasibility constraints (1)and(2),leadtostablematchings.Notethatheretheonlydif- ferencebetweenthisandtheprevious constraintisthatthestrict inequalitys_hj>s_ijbecameweak.

k:rik≤ri j

x_ik

·uj+

h:(^ah,cj)∈E,sh j≥si j

x_{h j}≥ujforeach

(

^ai,cj

)

∈E (4)

Extensionwithlowerquotas

Here,weonlyaddthelowerquotasforeverycompany.

i:(ai,cj)∈E

x_{i j}≥l_j foreachc_j∈C (5)

Addingdistributionalconstraints

Asadditionalconstraintswerequirethenumberofassigneesof a particular type to be between the lower and upperquotas for thattypeatacompany.

i:t(ai)=T^k,(ai,cj)∈E

xi j≤u^k_j foreachcj∈CandT^k∈T (6)

i:t(^ai)=T^k,(^ai,cj)∈E

x_{i j}≥l^k_j foreachc_j∈C andT^k∈T (7)

Wecanalsoaddsimilarconstraintsforsetsofcompanies,orfor theoverallnumberofassigneesatcertain typesatallcompanies.

Wedescribethelatter,aswewilluseitwhensolvingoursecond application.

i,j:t(^ai)=T^k,(^ai,cj)∈E

x_{i j}≤U^k foreachT^k∈T (8)

i,j:t(ai)=T^k,(ai,cj)∈E

x_{i j}≥L^kforeachT^k∈T (9)

3These conditions are standard for the assignment problem as well, see a survey on this problem and its variants [38] and an interesting application on marriage markets [19] .

3.2. Relaxingstability

Addingadditionalconstraintstotheproblemcancausethelack of astable matching,even if we addedsome flexibility withthe ties.

Onewaytofindamost-stablesolutionistointroducenonneg- ativedeficiencyvariables,d_ij foreachapplicationandaddthemto theleftsideofthestabilityconstraint(4).Byminimisingthesum ofthesedeficiencies asafirst objectivewe can obtaina solution whichisclosetobestable.

k:rik≤ri j

x_ik

·uj+

h:(^ah,cj)∈E,sh j≥si j

x_{h j}+di j≥ujforeach

(

^ai,cj

)

∈E

(10) Notethat here, ifa pair(a_i,c_j) isblockingforthe assignment thenwe needtoadd morecompensationd_ij ifthenumberofas- signeesatc_jthatthecompanypreferstoa_iislarge.Thisapproach canbe reasonableifwe wanttoavoidtherefusal ofa verygood candidate at a company. We call this solution as matching with minimumdeficiency.

Alternatively,ifwejustwanttominimisethenumberofblock- ingpairsthenwecansetd_ijtobebinaryandminimisethesumof thesevariablesunderthefollowingmodifiedconstraints.

k:rik≤ri j

x_ik

·u_j+

h:(^ah,cj)∈E,sh j≥si j

x_{h j}+d_{i j}·u_j

≥u_jforeach

(

^ai,c_j

)

∈E (11) Here,every blockingpairshouldbe compensatedby thesame amount,sothenumberofblockingpairs inminimised.Notethat thisconcepthasalreadybeenstudiedintheliteratureforvarious modelsunderthenameofalmoststablematchings,seee.g.[18].

3.3. Adjustinguppercapacities,envy-freematchings

Adifferentwayofenforcingthelowerquotaistorelaxstability byartificiallydecreasingthecapacitiesofthecompanies.Thiswas alsothe solution inthe residentallocation schemein Japan[30], wherethegovernmentintroducedartificialupperquotas foreach ofthehospitals, sothat ineachregion thesumoftheseartificial upperboundssummeduptothetargetcapacityforthatregion.In thecaseofourmotivatingexampleofprojectallocation,onesim- plewayofachievingthelowerquotaswasbyreducingtheupper quotasateverycompany.

In this solution what we essentially get is a so-called envy- freematching, studiedin [5,45,46]. Foramatching Mapplicanta_i has justified envy towards a_j if a_i prefers M(a_j) to M(a_i) and a_i isranked strictly higherthan a_j atM(a_j). Ifa matchingis freeof justified envy thenwe call it envy-free.A matchingthat isstable withrespecttotheartificialupperquotas,isenvy-freefortheorig- inalquotas.Thismeansthat theonlyblockingpairsthat mayoc- curwithregardtotheoriginalupperquotasareduetotheempty slotscreatedbythedifferencebetweentheoriginalandtheartifi- cialquotas,thatwecallopen-slotblockings.

However,onemaynotwanttoreducetheupperquotasofthe companies inthesameway,perhaps some morepopularcompa- niesshouldbeallowedtohavemorestudentsthanthelesspopu- larones.Furthermore,maybethedecisiononwhichupperquotas should be reducedshould be madedepending on their effect of satisfyingthelowerquotas(orotherrequirements).Thus,wemay not want to set the artificial upper quotas in advance, but keep them as variables, by ensuring envy-freeness in a different way.

Onealternativewayofenforcingenvy-freenessisbythefollowing

setofconstraints.

k:rik≤ri j

xik≥xh j

∀ (

^ai,cj

)

,

(

^ah,cj

)

∈E,si j>sh j (12)

Constraints(12)willensureenvy-freeness,bymakingsurethat if applicant a_h is assigned to company c_j and applicant a_i has higher scorethan a_h at c_j then a_i mustbe assignedto c_jor toa morepreferredcompany.

3.4. Within-typepriorities

So far we haveonly considered differentapproachesof relax- ing stabilityor enlargingthe set offeasible solutions in orderto satisfy the distributionalconstraints. In thissubsection we study alternative solutionconcepts andmethods forthe casewhenthe distributionalconstraintsaretype-dependent.Thisisthecasealso inour motivatingapplication, wherespecialrequirements areset fortheforeignstudentsassignedtothecompanies.

When the numberofstudents ofa type doesnot achieve the minimumrequiredata placethenthereare twowell-known ap- proaches.Forinstanceinaschoolchoicescenario,wheretheratio ofansocio-ethnicgroupshouldbeimproved(seee.g.[2])thenone possible affirmativeaction isto increase thescores ofthat group ofstudentsasmuchasneeded. Theotherusual solutionistoset somereservedseatstothosestudents(seee.g.[8]).

Inourprojectallocationapplicationourrequirementistohave atleastoneforeignstudentassignedtoeverycompany.Ifinasta- ble solutionthiscondition wouldbe violated fora popular com- pany thatranks the foreignstudents low then we can tryto en- force the admission ofa foreignstudent by increasingthe scores oftheforeignstudentsatthiscompany.Byadjustingthescoresof acertaintypeofstudentsatacompanywemeanthatweincrease (ordecrease) thescores ofthesestudentsatthatcompanybythe sameamountofpoints.Wecallamatchingstablewithtype-specific scores,ifthematchingisstableforsometype-specificallyadjusted scores. Thesecond approach isto devoteone placeateach com- panytoforeignstudents.Forthisoneseattheforeignstudentswill havehigherprioritythanthelocalsirrespectiveoftheirscores,but fortherestofthespacestheusualscore-basedrankingsapply.We call this concept as stable matching with reserved seats for types. Notethatneitherofthesetwoconceptscanalwaysensurethatwe getatleastone foreignstudentateach company, sincethey may allhavehighscoresandtheymayalldislikeaparticularcompany.

However, this situation changes ifwe alsoallow to decrease the scoresofagroupofstudents.Wewilldescribethiscaseafterdis- cussingthethirdapproach.

Finally, as a third approach, we can also extend the concept ofenvy-free matchings fortypes.We donot requireanystability withregardtostudentsofdifferenttypes,butwedorequireenvy- freenessforstudents ofthesametype.Thustheso-calledwithin- typeenvy-freematchingswillbethosewhosatisfythefollowingset ofconstraints.⁴

k:rik≤ri j

xik≥xh j

∀ (

^ai,cj

)

,

(

^ah,cj

)

∈E,si j>s_{h j},t

(

^ai

)

=t

(

^ah

)

=T^k,T^k∈T (13) Thatis,ifa_ianda_h havethesame typeanda_h isassignedto c_j then the higherranked a_i must alsobe assignedto c_j orto a morepreferredcompany.Notethat withthismodificationwe ex- tendthesetoffeasiblesolutionscomparedtothesetofenvy-free matchings.Anotherimportantobservationthatismotivatedbyour projectallocationproblemisthatundersomerealisticassumptions

4This solution concept was called within-type -compatibility by Echenique and Yenmez [22] .

awithin-typeenvy-freematchingalwaysexists,thatwewillshow inthefollowingtheorem.

Theorem1. Supposethatallthecompanies areacceptable toevery studentand that the sum of the lower quotas with regard to each typeislessthanorequaltothenumberofstudentsofthattype,and thesumofthelowerquotasacross typesforacompanyis lessthan orequaltotheupperquotaofthatcompany,thenacompletewithin- typeenvy-freematchingalwaysexistsandcanbefoundefficiently.

Proof. Weconstruct awithin-type envy-free matchingseparately foreachtype andthenwe mergethemattheendoftheprocess.

WhenconsideringaparticulartypeT^k,wesetartificialupperquo- tasatthecompaniestobeequaltothetype-specificlowerquotas (i.e.l^k_j forcompanyc_j) andwefindastablematchingM_kforthis type. Thisstablematching must exist,since we assumedthat all thecompanies areacceptabletoevery studentandthenumberof studentsineverytypeisatleastasmuchasthesumofthelower quotas forthat type. We create matchingM by mergingthe sta- blematchings forthetypes,i.e.M=M₁∪M₂∪· · · ∪Mp.Notethat noupperquotaisviolatedinM,sinceweassumedthatthesumof thelowerquotasacrosstypesforanycompanyc_jislessthanequal totheupperquotaofc_j.BythestabilityofM_kforeverytypeT^kit followsthat matching Mis within-typeenvy-free. Ifthere isstill a company c_j, where the overall lower quota (l_j) is not yet met, then weincrease an artificial upperquota forsome type T^k atc_j sothatthereisstillsome unmatchedapplicantsofthistype.This adjustment will affect the corresponding stable matching M_k for thistype,andthereforealsoMbyallocatingonemoreapplicantof type T^k toc_jin bothM_k andM. Sincethetotal numberofappli- cantsisgreaterorequaltothesumofthelowerquotas,wemust beable toachieve thelowerquotas atall companiesinthisway.

Finally,if thereare still some unmatchedapplicants then we in- creasesome artificial upperquotas for their types one-by-oneat anycompanyc_j,wheretheoriginalupperquotaisnotyetreached in M. At the end of this iterative process we obtain a complete within-typeenvy-freematching,M.

We note that there is a closely related solution concept in- troducedby Yokoi [45] which results in a within-type envy-free matching when restrictedto our model, that we describe inde- tailsbelow. Themodel studiedin that paperisthe moregeneral so-called classifiedstable matching problem where each student canhaveseveraltypes(e.g.gender,fieldofstudy,nationality)and thelower andupperquotas areset foreverytype. Whenputting theirmoregeneralmodelinourcontext astudenta_ihasjustified envy towards another student a_k at company c_j if a_k is assigned to c_j,a_i prefers c_j to her assignment, c_j ranks a_i higher than a_k, andnolower andupper quotais violated foranytype when re- placinga_k witha_i atc_j.It iseasy to seethat underthe assump- tionsofTheorem1anenvy-freematchingalwaysexistsasdefined by Yokoi and such a solution is a within-type envy-free match- ingaccordingtoourdefinitions.Finallyweremarkthatthismodel ofYokoiisoriginatedfromtheclassifiedstablematchingproblem introducedin [28],andfurther generalisedin[25,44].Acommon featureofthesepapersthatthelaminarnatureofthesetrequire- mentsmakes theproblempolynomial timesolvable.Acloselyre- latedmodelwasstudied in[24]without thelaminarassumption, wheretheproblemwasproved toby NP-hardandwassolvedby integerprogrammingtechniques.

Letusabbreviate a complete within-type envy-free matching as CWTEFM.Now,wewillcomparethisconceptofCWTEFMwithsta- blematchingswithtype-specificscoresandobservethattheyare essentiallythesame.

Theorem2. UndertheassumptionsofTheorem1acompletematch- ingiswithin-typeenvy-freeifandonlyifitisstablewithtype-specific scores.

Proof.Suppose first that M is a complete stable matching with type-specificscores, we willseethat Misalsowithin-typeenvy- freebydefinition.Supposeforacontradictionthatthereisa stu- dent a_i who has justified envy against student a_h of the same type at company c_j,i.e. a_h is assignedto c_j whilst a_i has higher scoreatc_jthana_handa_iisassignedtoalesspreferredcompany.

Thiswouldmeanthatthepair{a_i,c_j}isblockingfortheadjusted scores,sincebothstudentsgetthesameadjustmentatc_j,contra- dictingwiththestabilityofM.

Suppose nowthatMisaCWTEFM.Letusadjustthe scoresof thestudents accordingto their types ateach company such that theweakeststudentsadmittedhavethesamescoresacrosstypes.

MatchingMisstablewithregardtotheadjustedscores,becauseif astudenta_iisnotadmittedtoacompanyc_jandanybetterplace ofherpreferencethenitmustbethecasethatherscoreatc_jwas lessthan orequal to the score of the weakest assigned student ofthesametype atc_j,whichmeansthat theadjustedscoreofa_i atc_jislessthanorequalto theadjustedscoreofevery assigned studentatc_j.

Insteadofusingtheabovedescribedprocessesofsettingtype- specific artificial upper quotas or making adjustments for the scores ofdifferent types,we can also get a CWTEFMdirectly by anIPformulation.Weshallsimplyusethefeasibilityanddistribu- tionalconstraintstogetherwith(13)andwithan objectivefunction maximisingthenumberofstudentsassigned.Thisapproachisnot justmore robust than the above described two heuristics, butit hasalsothe advantage that wecan enforce additionaloptimality orfairnesscriteria.Asanadditionalfairnesscriterionwemayaim tominimisetheenvyacrosstypes.Wecanachievethisbyadding deficiency variables to the left hand side of constraints (12) for students ofdifferent types, asdescribedin (14) below, andthen minimisingthesum ofthedeficiencies. We refer tothis solution asMin#E-CWTEFM,thatiscompletewithin-typeenvy-freematching withminimumnumberofenvyacrosstypes.

k:rik≤ri j

x_ik+d_ih^j ≥x_{h j}

∀ (

^ai,c_j

)

,

(

^ah,c_j

)

∈E,t

(

^ai

)

=t

(

^ah

)

⁽¹⁴⁾

However, wemayfindanenvy morejustified,ifthescoredif- ference between the two applicants involved is higher. Thus, by takingthescoredifferencesastheintensitiesoftheenvies,wecan alsoaimtofindarefinedsolutionwherethetotalintensitiesofthe enviesisminimised,byusingthefollowingobjectivefunction:

(

^si j−s_{h j}

)

·d_ih^j.

We call the corresponding solution complete within-type envy- freematching withminimum envyintensitiesacross types,abbrevi- atedasMinEI-CWTEFM.

If the solution is still not unique then we can further refine it, by considering two additional objectives. Regarding the wel- fareof thestudents, we maywant to minimisethe totalrank of thestudents,leadingtoaPareto-optimalassignmentforthemun- dertheconstraints.WedenotethesesolutionsasMinRank-Min#E- CWTEFM and MinRank-MinEI-CWTEFM, depending whether we minimised the number of envies or the envy intensities in the previous round. Finally, an alternative objective can be to min- imisethenumberofblockingpairsduetoopen slots.Thiscanbe achievedbyaddingbinarydeficiencyvariablestothefirsttermof theleftsideofthestabilityconstraints,asfollows.

k:rik≤ri j

xik+di j

·uj+

h:(ah,cj)∈E

xh j≥uj foreach

(

^ai,cj

)

∈E

(15) We can then minimise the sum of these deficiency variables andfind a matching within the restricted solution set that min- imises the number of open-slot blockings. We denote these so- lutions as MinOSB-Min#E-CWTEFM and MinOSB-MinEI-CWTEFM, depending whether we minimised the number of envies or the envyintensities.

4. Firstapplication:CEMSprojectallocation

CEMS Alliance is a global co-operation of leading business schools, multinational corporations and social partners in higher educationdomain.TheseentitiesruntogethertheCEMSMasterin InternationalManagement(MIM) one-yeargraduateprogramthat isaccessibleforgraduatestudentsofthepartnerinstitutionsin29 countriesinfivecontinents.Duringtheone-year-programstudents spend one semester at their home institution and one semester atanotherpartnerinstitutionsomewhereabroad,andtheyalways learninaninternationalenvironment.CEMSMIMhasbeenranked asaleadingmasterprogrambyFinancialTimesinrecentyears.

Withintheframework oftheMIMprogrameachstudentmust carry out a businessproject duringtheSpringsemester account- ing for15ECTS credits (thatis half ofthe workloadof theentire semester). The consultancy-like projects are designed as real life learningexperience.Businessprojectsaredone insmallgroupsof 3–6 students in which ideally at least one student comes from a foreign school, hence business project teams are culturally di- verse.Businessprojects are offeredandsupervisedby the corpo- rate partners throughout the semester and they usually last for threemonths.

Students learn about the business projects during a kickoff event atthe beginning of the semester from company represen- tatives andthey also receivewritten descriptions ofthe projects.

Afterthekickoff eventcorporatepartnersevaluateallstudentsac- cordingtotheirCV-s,andstudentsalsorankthebusinessprojects in the same time. The school assigns students to the individual projectsbasedontheseevaluationsandrankings.

At Corvinus University of Budapest the authors of this paper havebeengiventhetaskofredesigningtheallocationmechanism in2016.Inpreviousyearsthemechanismwasasimpleimmediate acceptancemechanism(alsoknownastheBostonmechanism[4]), wherethestudentssubmittedtheirCV-stotheirfirstchoicecom- panies,thecompaniesevaluated thecandidatesandthentheyac- ceptedthebestcandidatesuptotheirquotasandrejectedtherest.

TherejectedstudentsthensubmittedtheirCV-stofurthercompa- nies,butthosecompanieswhichhavealreadyfilledtheirpositions didnotacceptmoreapplications.Thismechanismwasheavilycrit- icized intheliteratureon schoolchoicedueto itsunfairnessand also because this mechanism is highly manipulable, therefore in many cities it hasbeen replaced by other algorithms, mainly by thedeferredacceptance(orGale-Shapley)algorithm,seee.g.[4].

4.1. Solutionplan

In 2016 therewere 25 students, including20 localand 5for- eignstudents,and5companies.Theinitialupperquotas wereset to6andthelowerquotasweresetto4atallcompanies.Thepro- grammecoordinatordecidedtosetanupperquotaof2forthefor- eignstudentsateachcompany toenforcediversity. In2017there was a slight change in the distributional criteria, the number of studentsallocatedto eachcompanywassettobe between3and

Table 1

The results of the 2016 matching run with the number of all and for- eign students assigned to the companies and the total rank of the stu- dents.

2016 profiles all/foreign total rank

Solution 1: MinRank-Stable 6 1 6 6 6 34

u i= 6 1 0 0 2 2

Solution 2: MinRank-Stable 6 2 6 6 5 35 l i= 2 , u i= 6 1 0 1 2 1 Solution 3: MinRank-Stable 6 4 5 5 5 40 u 1= 6 , u i= 5(i = 2 . 5) 1 0 0 2 2 Solution 4: MinRank-Stable 5 5 5 5 5 41

u i= 5 0 2 0 2 1

Solution 5: MinRank-EF 5 4 6 6 4 38 l i= 4 , u i= 6 0 2 1 2 0 Solution 6: MinOSB-EF 6 4 6 5 4 39 l i = 4 , u i= 6 1 1 1 2 0

6andatleastoneforeignstudentwasrequiredtobeallocatedto everycompany.

Ourfirst solutionplanwasto askthestudents torank allthe companies inastrict orderandtoask thecompanies toevaluate all the CV-sandrank thestudents weakly bygivingthem scores between1and10.⁵Ourintentionwithallowingtieswastoenlarge the set of stablesolutions, even though we understandthat this fairnessconceptisabitweaker,sincewemayacceptastudentand reject anotheronewiththesamescore.Allowing tiesalsomakes the problemNP-hardalready withlower quotas, aswe described intheintroduction.Yet,ifthetieswerenotallowedthenthesetof stable(andenvy-free)solutions wouldbe muchsmallerandthus itwouldbehardertosatisfythedistributionalconstraints.

We remark that theconditions of Theorem1 are satisfied for both 2016 and2017, sinceall the students haveto rank(andac- cept)all thecompanies andin2017we wererequiredto haveat leastone foreign studentateach company, wherethenumberof foreignstudentswasmorethanthenumberofcompanies.There- fore a complete within-type envy-free matching always existed.

Withinthissetofsolutionswedecidedtominimisethenumberof envies acrosstypes andtheir intensities astheprimal objectives.

As secondary objectives we tried to minimisethe total rank and thenumberofopen-slotblockings.

Finally,sinceinbothyearsitwaspossibletodecreasetheupper quotas atall companies by one(and setthem to5 insteadof6), we alsoexamined thesesolutions.Thiswasreasonableasallocat- ing very differentnumbers ofstudents tothe companies seemed problematic,especiallyifsomeofthemostpopularcompanieswas forcednottofillitsquota,whilelesspopularcompaniesdid.

4.2. Resultsin2016

The mostimportantresultsofthe 2016 matchingrun arecol- lectedinTable1.

In2016 theupperboundoftwofortheforeignstudents were always satisfiedwithoutconsideringit,soweleaveoutthisques- tionfromthediscussionandwefocusonlyoncommonlowerquo- tas.Wewerenotabletofindastablesolutionfortheoriginalquo- tas of4–6,since oneof thecompanies (number 2)wasvery un- popularandthehighestnumberofstudents thatwecould match thereinastablesolutionwas2,thisisSolution2inTable1.(For the record, we also checkedwhich would be the minimumrank solution amongthe stableones, that isSolution 1.)Therefore we decreasedtheupperquotasofallcompaniesto5,exceptthemost popular company (number 1) and found a stable matching with

5Most companies gave only integer scores, but some submitted half-integer scores as well, so ties indeed occurred.

minimumtotalrank(Solution3).Notethatthismatchingisenvy- free forthe original quotas. Finally we considered the possibility ofdecreasingalltheupperquotasto5,asdescribedinSolution4.

Fromthelattertwosolutionsthedecisionmakerdecidedtochoose Solution 4, since it wasnot substantially different from Solution 3and forthe companies itseemed tobe easier to communicate thecommondecreaseofupperquotas,comparedtothecasewhen onlyonecompanyhasalargernumberofstudents.

Recently,aftercarefullyinvestigatingthesolutionconceptsde- scribedinthispaper,wedidanothercheckonthepossibleresults andcomputedSolutions 5and6.Solution 5isanenvy-free solu- tionwhere the totalrank is minimised.It wasinteresting to ob- servethatthemostpopularcompany(number1) doesnot fillits upperquota,leadingtomanyopen-slotblockingsatthatcompany.

Solution6 isalsoenvy-free, butherethe open-slot blockingsare minimised,butthisresultedinasmalldecreaseinthetotalrank.

4.3.Resultsin2017

The results of the 2017 matching run are summarised in Table2.In2017thenumberofstudentswas40amongwhich13 were from abroad and the number of companies was8. Due to thehigher proportionof foreignstudents, the organisersdecided to require the allocation of at least one foreign student to each company.Theinitialcallsuggestedgroupsofsizesbetween3and 6,butinthisyearalsowe investigatedthesolutions whenevery upperquotawasdecreasedto5.Inthelattercasethelowerquo- tasfortheforeignstudentswerenotautomaticallysatisfied,sowe foundwithin-typeenvy-freesolutionsandthenasafirstobjective weeitherminimisedthenumberofenviesacrosstypesorwemin- imisedthe intensities ofthe envies.As a secondary objectivewe triedtominimisethetotalrank(therewasnoopen-slotsblocking whentheupperquotaswerecommonlysetto5).

Solution1isenvy-free,andthetotalrankisminimised.Asintu- itivelyexpected,thetwoleastpopularcompanieshaveonlythree studentsallocatedeach anda mediumpopularcompany hasfour students,whilstthepopularcompaniesreceivesixstudents.Solu- tions 2and3are bothwithin-type envyfree forupperquotas 5.

Solution2 minimises the numberof envies asthe first objective andthenthetotalrank.Solution3minimisestheintensitiesofthe enviesandthen the totalrank. (Notethat we also computedthe minimal envy solutions without requiring within-typeenvy free- ness,andessentiallywereceivedthesametwosolutions.)Itisin- terestingtoknowthatonlyonejustifiedenvywaspresentinboth Solutions2and3,andtheintensityofthisenvywas1inSolution 2and ¹₂ in Solution3.However, thesetwo solutions were rather different, andSolution 2 had much smaller total rank. Thus So- lution2wasclearlyfoundbetter than Solution3by thedecision maker.Whencomparingthefirsttwosolutions,thedecisionmaker selectedSolution2,duetothemorebalancedsizesofgroups.

4.4.Discussion,furtherquestions

Herewediscussourfindingsandpossiblequestionsforthefu- ture.

Importanceofthedistributionalrequirements.Wehavecon- sideredourdistributionalconstraintsashardbounds,theonlyre- laxationwetestedwasthecommondecreaseoftheupperquotas.

However, inmanyapplications the distributionalgoals are softer, andthusmaybeviolated.Forinstance,inschoolchoicetheexact proportionalitywithregardtoethnicityorgendermaybe toode- mandingandunnecessary tosatisfy,theseare ratherjustgeneral aims.Insuchsituationsonemayinsistonthestabilityortheenvy freenessofthesolutionandwanttosatisfythedistributionalcon- straintsasmuchaspossible.Finally,thetrade-off betweenfairness

Table 2

The results of the 2017 matching run with the number of all and foreign students assigned to the companies and the total rank of the students.

2017 profiles all/foreign total rank

Solution 1: MinRank-EF 6 3 6 6 6 3 6 4 66

l i= 3 , u i= 6 1 1 1 4 1 1 3 1

Solution 2: MinRank-Min#E-CWTEFM 5 5 5 5 5 5 5 5 85

u i= 5 , wEF, min 1 3 2 3 1 1 1 1

Solution 3: MinRank-MinEI-CWTEFM 5 5 5 5 5 5 5 5 105

u i = 5 1 4 1 2 1 1 1 2

anddistributionalgoalsmaybebalancedbyrelaxingbothrequire- mentsatthesametime.

Stability versus envy-freeness. Leaving some slots empty to satisfythedistributionalconstraintsisa naturalwaytorelax sta- bility. This is also used in the Japanese resident allocation pro- gramme,whereartificialupperquotashavebeensettothehospi- talsinordertosatisfytheregionallowerquotas[30].However,the open-slot blockingcan alsobe seen asunfair fromboth the stu- dents’andthecompanies’pointsofviews,especiallywhenapop- ularcompanyhastogiveupanintern.Notealsothattheopen-slot blockingsarerelativetotheoriginalquotas.Inourapplicationthe decisionmakerended upchoosing solutionsin bothyears where the upper quotas of the companies were commonly reduced by one.Thesesolutions admit a highnumberofopen-slot blockings regardingtheoriginal quotas, whilstifthey are envy freefor the originalquotasthentheyarealsostable(withnoopen-slotblock- ings)forthedecreasedquotas.Thusthesechosensolutionscanbe seen more fairfrom the students’ point ofview, as they do not regrettheirrejectionsbyacompanywithanopenslot.

Importance ofwithin-typeenvy-freeness.Inouranalyses we assumedthat within-type envy-freenessis an important require- mentthat we obeyedin all solutions. Note that inthe 2017 run wealsotestedthesolutionwhenthisrequirementwasrelaxedand wedidnotfindasignificantdifferenceinthesolutions.Itisanin- terestingquestionhowimportantthisrequirementis,andthean- swercandependontheactualapplication.Iftheseparationofthe typesissignificantandthereisabigdifferencebetweentheirper- formance (e.g.regarding the ethnicity incollege admission)then within-typeenvy-freenesscanbecrucial.

Minimising the number of justified envies or their intensi- ties.Inour2017run wehadasignificant differencebetweenour tworecommended solutionsbased on minimisingthenumber of justifiedenvies andtheir intensities, respectively.In ourcasethe former solution had much better total ranking for the students, butone can easily create an example where the opposite would happen.Iftheintensitiesofthe blockingareminimisedthen this meansthat theaveragedifference betweenthescoresofthe stu- dents who have envy towards one another is small.This can be moreacceptablethanhavinglargescoredifferences.Infact,ifthe maximumscoredifferenceisnothigherthantheoneinourappli- cation,thenwecouldsaythatthissolutioncouldbeseenastobe weaklystableifthescoringbythecompanies werelessfiner,say usedscorerange1–5insteadofthecurrentrangeof1–10.

Strict versusweakrankings.Using tiesintherankings ofthe companies was by our recommendation in order to enlarge the setof stable(or envy-free) matchings.However, in thiscase sta- bility(and envy-freeness)isweaker,the rejectionofastudentby acompanycanbeexplainedbytheadmittanceofanotherstudent withequal score orhigher. Thus, this can be seen unfair by the rejectedstudent,thereforeinmanyapplications(e.g.schoolchoice inNewYork,BostonandcollegeadmissionsinIrelandandTurkey) theties are broken by lotteries or by other random factors. Ties maketheproblemofsatisfyinglower quotas NP-hard,whilst this isapolynomial-timesolvableproblemforstrictrankings, seee.g.

[25].Furthermore,themechanismcanbecomehighlymanipulable bythestudentsfortiesdependingonthegoalsoftheoptimisation.

Incentive issues. A mechanism is strategy-proof for the stu- dents if neither of them can get a better match by submitting false preferences. This property holds for the student-proposing deferred-acceptancemechanismintheclassical collegeadmission modelof Gale andShapley (see e.g. [43]). Strategy-proofness can also be satisfied by modified variants ofthe deferred-acceptance mechanismforthecaseoflowerquotas,assuggestedalsoforthe Japanese residentallocations [27,30,31].However,ifwe allowties andweconsidergoalssuch asrank-minimisationthen ourmech- anismbecomesmanipulable.Asimplemanipulationstrategyfora medium-strongstudentcanbetoputhertopchoiceasfirstchoice, butinstead of puttingher true second choice in the second slot shecan putsome companieswhich arenot achievableforher in anystablesolution.Ifthereisanotherstudentwiththeverysame scoreand verysame preferencessubmitting her true preferences andthere isonlyone place left attheir mostpreferredcompany then therank-maximising algorithmwill assignthe manipulating student there, and the truth-telling student to the second com- pany,sincethealternativesolutionbyexchangingthetwostudents wouldresultinhighertotalrank.Despiteofthisissueofmanipu- lability, we believe that the expected gainsof manipulations are negligibleandtheirriskscanbe high,soinaBayesian senseitis unlikely thatastudent couldgeta positiveexpectedgainby ma- nipulating.However,weadmit thatthishypothesiswouldbevery hardtoproveformally.

Bounding the length of preferencelists. In2016 there were 25studentsand5companies,in2017therewere40studentsand 8 companies, so the screening costs of the companies have in- creased a lot. If this tendency will continuethen the organisers oftheprogrammemayneedtoreconsidertherequirementofpro- vidingfullrankings.Areasonablesolutioninsuch situationsisto have two rounds. In the first the students are required to rank a fixed number ofcompanies, say five), andit is not guaranteed thatallthestudentscanbeallocatedtoacceptablecompaniesthat they ranked.Inthesecond roundeithernopreferencesareasked from thestudents orthe organisers can elicit the preferences of the unmatchedstudents over the companies with remaining po- sitions. Thisisa standard techniquealsoinschool choice (e.g.in NewYork[3]),althoughherewewouldfacenewchallengestoen- surethesatisfactionofthedistributionalrequirements.

5. Secondapplication:workshopassignment

Afterrunningthe2016projectallocation,wereceivedverypos- itive feedbacks from the students, and in fact two students ap- proached us asking for a help in selecting andassigning confer- enceparticipantstocompaniesinvolvedinacasestudyworkshop withintheconference.

Thenumberofparticipantstobeselectedwas60,andtheyhad tobe assignedto threecompaniesina givenproportion, thefirst companyhadtoreceive16studentsandatleast8Hungarians,the second andthethird companieshadto receive22students each.

Therewere13pre-selectedstudents(thecountryleadersoftheor-