Anarchy Stability and Utopia Creating Better Matchings - PDF document

OurResults.Weinitiateaninvestigationoftheques-tionsdescribedaboveinthecontextoftwo-sidedmatchings,andgiveboththeoreticalandexperimentalresults.Specif-ically,westudytheeectsofdierentnetworkstructuresandutilitydistributionsonthepriceofanarchy:theratioofsocialutilitiesachievedbystableandoptimalmatchingsrespectively.Wendthatinmostcasesthestablematchingattainsclosetotheoptimalsocialwelfare(generallyabove90%).Wecharacterizesomesituationswherethepriceofanarchycanbemoresubstantial,andthenstudyapoten-tialmeansofincentivizinggoodstablematchingsinSection5.Weconsiderapproximatestability,whichcorrespondstotheadditionofaswitchingcosttothemechanism,sothatanagentwouldhavetopayinordertodeviatefromthecurrentmatching.Weshowboththeoreticallyandexper-imentallythattheadditionofasmallswitchingcostcangreatlyimprovethepriceofanarchy.Finally,inSection6weconsiderseveralgreedyalgorithmsforpartner-switching,andshowexperimentallythattheyconvergequicklytosta-bilityforsomesimpleyetnaturaldistributionsofutilities,aswellasproveconvergenceguarantees.2.MATCHING,STABILITY,ANDSOCIALWELFAREMatching,theprocessofagentsformingbenecialpart-nerships,isoneofthemostfundamentalsocialprocesses.Examplesofmatchingwithself-interestedagentsrangefrombasicsocialactivities(marriage,mateassignment[9]),tothecoreofeconomicactivity(matchingemployeesandem-ployers[17]),torecentinnovationsinhealthcare(matchingkidneydonorsandrecipients[4]).Theprocessofmatchingcanbeextremelycomplex,since(1)agentscanhavecompli-catedpreferences,and(2),inmostsocialapplicationsagentsareself-interested:theycaremostlyabouttheirownwelfare,andwouldnotobeyacentralizedmatchingalgorithmunlessitwastotheirbenet.Forthisreason,theoutcomesofmatchingprocessesareusuallyanalyzedintermsofstability,therequirementthatnocollectionofagentscouldformagrouptogether,andbe-comebetterothantheyarecurrently[22].Fortheclassic\stablemarriage"problem[13],thiscorrespondstothelackofdesireofanypairtodroptheircurrentpartnersandin-steadmatchwitheachother.Stablematchingalgorithmshavebeenusedinmanyapplicationsincludingmatchingmedicalresidentswithhospitals[21],studentswithsoror-itiesandschools[1,19],andonlineuserswithservers.Whilestablematchingsmaybenaturaloutcomes,desir-ableforvariousreasons,therearefewguaranteesonthequalityandsocialwelfareofstablematchings.Mostre-searchonmatchingsofself-interestedagentshasfocusedon(1)deningoutcomeswithstabilityasthegoal(mostoftheworkonthedesignoftwo-sidedmatchingmarketsattemptstodoexactlythisbydeningproblemsappropri-ately[22]),(2)computingstableoutcomesandunderstand-ingtheirproperties(rangingfromtheseminalworkofGaleandShapley[13]toalgorithmsthattryandcompute\opti-mal"matches,forexamplebyminimizingtheaveragepref-erencerankingofmatchedpartners[16]),and(3)design-ingtruthfulpreference-revealingmechanisms(suchasintheNewYorkCity[2]andBostonpublicschoolmatches[3]).Questionsaboutthesocialwelfareofstablematchingshavebeenlessstudied.1Therehasbeenalmostnoresearchonconstructingsociallydesirablestableoutcomes,partlybe-causeinmostsituationsonecannotinstructself-interestedagentsonwhattodoinordertoengineersuchoutcomes,sinceanagentwillonlyfollowinstructionsifitbenetsthempersonally.Anincreasingbodyofliteratureinbehavioraleconomicsandsocialscience(e.g.[25]),however,suggeststhatde-sirableoutcomescanbeachievedbygivingpeoplealittle\nudge"incertaindirections,perhapsbyalteringtheirin-centivesslightly,whilestillleavingthemwithfreedomtochoosetheirownactions.Smallchangesthatgreatlyim-proveasocialsystemareeasytoidentifyinsomesitua-tions:forexample,making401(K)plansopt-outratherthanopt-inincreasesparticipationdramatically.Findingsimilarchangesinmatchingscenariosismoredicultbecauseofthecomplexityofasystemwhereanyagent'sactionscantheoreticallyaectalargenumberofotheragents.Beforeaddressingthemechanismdesignquestionofhowtoachievebettersocialoutcomes,werstneedtoaddressthequestionofwhetherornotstablematchingcanleadtosubstantialsociallosses.Forthisquestiontomakesense,werstneedanobjectivefunctionthatmeasuresthequal-ityofamatching.Asmentionedintheintroduction,oneofthereasonswhythesocialqualityofstablematchingsisusuallynotaddressedisbecausetheagentsinquestionareassumedtohaveapreferenceorderingontheirpossiblepart-ners,withoutaspecicutilityfunctionthatstateshowgoodamatchwouldbe.Whiletherehasbeensomeworkonmea-suringthequalityofamatchingby,forexample,theaveragepreferencerankingofmatchedpartners[16],suchmeasurescansometimesbehardtojustify.Forexample,foranagentA,thesecondchoiceinitspreferenceordermightbealotworsethanitsrstchoice,whileforagentB,thesecondchoicemightbeonlyalittlebitworse.Measuressuchastheoneabovewouldmakenosuchdistinction.Inthispa-per,wearespecicallyconcernedwithcontextswhereeveryagenthasautilityfunction,notjustapreferenceordering:thatis,foreverypossiblepartnerv,anagenthasavalueU(v)specifyinghowhappyitwouldbetobematchedwithv.Weareespeciallyconcernedwithmeasuringthequalityofamatchingintermsofsocialwelfare:thetotalsumofutilitiesforalltheagents.Wewouldliketounderstandthesocialwelfareofstablematching.Thetradeobetweenstablematchingsandso-ciallyoptimalmatchingsisquantiedbythepriceofanar-chy:theratiobetweenthemaximumpossiblesocialutilityandtheutilitiesofequilibriumoutcomes(stablematchings).Understandingthepriceofanarchyisimportant,sinceitactsasaboundontheamountofimprovementinstablematchingsthatbettermechanismscouldyield.PriceofAnarchyBounds.Thepriceofanarchycanvarywidelydependingontheprobleminstanceandtheprefer-encestructure.Asanexample,Figure1illustratessomecaseswherethestablematchingishighlysociallysubopti-mal(discussedinmoredetailinthenextsection).Intwooftheunderlyingtypesofgraphstructures,thepriceofan-archyisatmosttwo(andtheboundcanbetight),while 1Asmentionedintheintroduction,oneofthedesiderataformatchingstudentswithschoolsormedicalstudentswithresidenciescanbetocomputethestablematchingthatisbest(typically)forthestudents,butthisisadierentnotionofwelfare. Symmetricedge-labeledpreferences Vertex-labeledpreferences Asymmetricedge-labeledpreferencesFigure1:Worst-caserealizationsofthepriceofanarchyindierentmodels.Ineachcasethesociallyoptimalmatchingisf(A;C);(B;D)gbuttheonlystablematchingpairsAandD.socialutilityofanystablematchingisatleastone-halfofthesocialutilityoftheoptimummatching.Inotherwords,thisobservationsaysexactlythatthepriceofanarchyisatmost2.Noticethatthesociallyoptimalmatchingissimplythemaximum-weightmatchinginthismodel.TheaboveobservationisaspecialcaseofTheorem1(provedinSection5),butitcanalsobeseentofollowfromtwofacts:(1)Anystablematchingcanbereturnedbyanalgorithmthatexaminesedgesgreedilybymagnitude,addingthemtothematchingiftheverticesinvolvedhavenotyetbeenmatched(theparticularstablematchingpro-duceddependsontheprocedureforbreakingtiesbetweenequal-weightededges),and(2)Anygreedysolutiontothemaximumweightedmatchingproblemiswithinafactoroftwooftheoptimalsolution.Notethatthisargumentholdsgenerally,evenfornon-bipartitegraphs.Figure1(a)pro-videsanexampleofagraphwherethisboundisachieved,showingthattheboundof2onthepriceofanarchyistight.Observation2.Invertexlabeledgraphsthesocialutilityofanystablematchingisatleastone-halfofthesocialutilityoftheoptimummatching.ThisisaconsequenceofTheorem2(seeSection5forfurtherdiscussion).Again,Figure1(b)providesanexampleofagraphwherethisboundisachieved.Observation3.Inasymmetricedge-labeledgraphs,thesocialutilityofthestablematchingcanbearbitrarilybadcomparedwiththesociallyoptimalmatching.ConsiderthecaseinFigure1(c){theutilityreceivedbyagentBfrombeingmatchedwithAgentDisarbitrarilyhigh,butthepairisnotpartofthestablematching,sothelossinutilitycanbeunbounded.Againthisargumentholdsfornon-bipartitegraphsaswell.Theseareworst-caseconstructions.Anaturalquestioniswhatthepriceofanarchyislikeinrealisticgraphswithdierentdistributionsoverutilities.Weexaminedseveraldierentdistributionsofutilitieswithinthethreemodelsde-scribedabove,andalsoconsidereddierentgraphstructuresinordertogetasenseofthepotentialpracticalimplicationsofthesepriceofanarchyresults.Weusedrandomdistribu-tionsoftheutilityvaluesonrandombipartite(andlaternon-bipartite)graphsofthedierenttypesdescribedabove,andcomputedboththemaximum-weightedstablematch-ing(thesociallyoptimalmatching)andastablematching Figure2:Averageratiooftherealizedstablematch-ingtothemaximumweightedmatchinginthreedif-ferentpreferencemodelswhenutilitiesaresampledatrandomfromexponentialanduniformdistribu-tionswiththesamemean(0.5:therateparameteris2fortheexponentialandthesupportoftheuniformis[0;1]).Reportedvaluesareaveragedover200runs.Thereare100agentsoneachsideofthematchingmarketinallcases.TheXaxisshowsthedegreeofeachnode.Notethattheratioisveryhigh,almostneverdroppingbelow85%,eveninindividualruns.usingtheGale-Shapleyalgorithm(inallcasesconsideredhere,exceptonedescribedinmoredetailbelow,thepropos-ingsidedoesnotaecttheoutcomeinexpectationbecausepreferencedistributionsaresymmetric).Figure2showsthatwhenutilitiesarerandomlydistributedaccordingtotwocommondistributions(exponentialanduniform,althoughthisresultseemstoberobustacrossmanydierentdistributions),thesociallossduetostabilityisnotparticularlyhighinanyofthethreemodelswedescribeabove.Thisisnotsurprisingforvertexlabeledgraphs{sinceanypersoninthematchingwillcontributethesametothetotalutilityregardlessofwhomtheyarematchedwith(forexample,everyperfectmatchingissociallyoptimal).Astheaveragedegreeofeachvertexincreases,thenumberofagentsgettingmatchedincreases,andtheratioquicklyreaches1,becauseallstablematchingsbecomeperfectatsomepoint.However,theresultisconsiderablymoresur-prisingfortheothertwocases,particularlyforasymmetricedge-labeledpreferences.Theonlycaseinwhichtheratiogoesbelow0:9isforexponentiallydistributedutilitieswithasymmetricedge-labeledpreferences(theratiostopsdeclin- Figure3:Averageratiooftherealizedstablematch-ingtothemaximumweightedmatchingwithtwodierentnon-bipartitegraphstructures:(1)smallworldnetworksand(2)preferentialattachmentnet-worksofdierentaveragedegree,bothwith100nodes.Utilitiesaresampledindependentlyfromanexponentialdistributionwithmean0:5.Resultsareaveragedover200runs.ingsignicantlybeyonddegree10).Forasymmetricedgelabeledgraphs,itmakessensethattheratiodeclinesasthedegreeofthegraphgetslarger,becauseitbecomespossibletoconstructmatchingsthataresociallymuchbetter.Ourexperimentsshowthatthevalueoftheoptimalmatchinggrowsquickly(sinceithasmoreoptionsavailable),whilethevalueofstablematchinggrowsslowly(sinceitishamperedbythestabilityconstraint).Theactualhighpercentageisquitesurprisinggiventhatintheory,theratiocouldbear-bitrarilybad.Theuniformdistributionratiosaregenerallyhigherthanthosefortheexponentialdistributionbecausetheuniformdistributionenforcesacompressionintherangeofhighutilitiesbycappingutilitiesat1.Figure3showsthatthehighratioisnotanaccidentofusingrandombipartitegraphs.Innon-bipartitegraphsthatareknownfortheirpowerinmodelingsocialandengineeringsystems,namelypreferentialattachmentnetworks[8]andsmall-worldnetworksonalattice[26],theresultsaresimilar,withthecomputedstablematchingachieving,onaverage,above95%ofthevalueofthesociallyoptimalmatching.ThisresultalsoholdsinlatticenetworksandinnetworksdenedinEuclideanspacewheretheutilityofamatchingforanypairistheinverseofthedistancebetweenthem.Thusitappearsthatinrandomgraphs,stablematchingsattainaveryhighproportionofthemaximumsocialutility.Therearehoweversomepreferencestructuresforwhichthisdoesnothold.Consideracasewheretheutilitiesreceivedbyonesideofthemarketaremuchhigherthanutilitiesre-ceivedbytheotherside.Inaddition,supposethatthesidewithlowerutilitiesismorepowerful,andisthereforeabletochoosethestablematchingoptimalforthoseonthatsideofthemarket(thesesituationscouldcorrespondtomanyinreallife{forexample,employersaremorepowerfulthanem-ployees).ThispowerstructureisimplementedbyrunningtheGale-Shapleyalgorithmwiththemorepowerfulsidebe-ingthesidethatproposes,whichresultsinthebeststablematchingfortheproposingside.Inthiscasetheratioofutilitiescanbesubstantiallylower,asseeninFigure4.Inotherwords,ifweonlycareaboutthewelfareofonesideofthemarket,therecanexiststablematchingsmuchworse Figure4:Averageratiooftherealizedstablematch-ingtothemaximumweightedmatchingwhentheutilitiesreceivedbythoseontheless\powerful"sideofthemarketare10000timesashighasthosere-ceivedbythoseonthemorepowerfulside,butthestablematchingistheoneoptimalforthemorepow-erfulside.Resultsareaveragedover200runs.Util-itiesareexponentiallydistributed.thantheoptimalones(althoughstillmuchbetterthanthetheoreticalboundofone-half).Whenanarchyisgood.Thepriceofanarchyisnottheonlyimportantmeasure.Ourexperimentssofarrevealthatthepriceofanarchyislowerforvertexlabeledgraphs,especiallyasthedegreegrows.Thisismostlybecauseanyperfectmatchingisso-ciallyoptimal.Asmoreandmoreverticesgetincludedinthematching,wegetcloserandclosertothesociallyop-timalmatching.Butthisisessentiallyacaseofscarcere-sources,andnosynergies{theaverageutilityreceivedbyeveryoneinaperfectmatchingisthevalueoftheaveragevertex{thereisnochancetomakeeveryonebetterobe-causesomepairsworkbettertogetherorlikeeachothermore.Ifpreferencesweremoreheterogeneous,therewouldbemoresuchsynergiesthatcouldbeexploited.Inordertoexplorethisfurther,weexperimentwithvaryingthelevelofhomogeneityinpreferencesbymakingpreferencesaconvexcombinationofvertex-labeledandasymmetricedge-labeledpreferences,whileholdingtheaveragevalueconstant.Inthiscasethevaluereceivedbyufrommatchingwithvisgivenbyw(v)+(1�)zwherebothw(v)andzaresam-pledfromexponentialdistributionswithmean0:5,butw(v)isanintrinsicfeatureofthenodevwhichisthesameforanyuthatisconnectedtov,whilezisidiosyncratic(indepen-dentlysampledforeachuthatisconnectedtov).Thenrepresentsthedegreeofhomogeneityofpreferences.Figure5showsthat,whiletheratioofstable-to-optimalutilitiesgoesupdramaticallyaspreferencesapproachpurehomo-geneity,thisisaccompaniedbyadeclineinaverageutilityreceivedbyeachindividual.Thisindicatesthathavingsomeheterogeneityinpreferencesisagoodthingforsociety:evenifitleadstoahigherpriceofanarchy,everyoneisbetterothantheywouldbeinalowerprice-of-anarchysociety.5.IMPROVINGSOCIALOUTCOMESInthissection,weconsiderhowtoimprovethequalityofstablematchings.Weconsider,boththeoreticallyandinsimulation,theadditionofaswitchingcosttothemecha-nismsothatanagentwouldhavetopayinordertodeviatefromthecurrentmatching.Wendthatitispossibletoim-provethequalityofsocialoutcomessubstantiallybymak- Figure5:Theratiooftherealizedstablematchingtothemaximumweightedmatching(goingupfromlefttoright,leftYaxis)andtheaverageutilityre-ceivedbyeachagent(goingdownfromlefttoright,rightYaxis)asafunctionofthedegreeofhomo-geneityofpreferences(0beingcompletelyhetero-geneous,i.e.asymmetricedge-labeled,and1beingcompletelyhomogeneous,i.e.vertex-labeled).Thegraphsarebipartite,containing100nodesoneachside,andthedegreeofeachvertexis10.Theaver-ageutilityofanyedgeremains0.5foreachsetting.Resultsareaveragedover200runs.ingonlysmallchangestotheincentivesoftheagents,andthuswithoutdrasticallychangingthenatureofthematch-ingmarket.Notethatinthecasesconsideredinthissection,thereisnochangeinpreferencesofthesortdiscussedim-mediatelyabove,sothepriceofanarchyisactuallyagoodproxyforsocial(dis)utility.5.1ApproximateStabilityandSwitchingCostsAnapproximateequilibriumisasolutionwherenoagentgainsmorethanasmallfactorinutilitybydeviating.Inthecaseofmatching,weconsiderthefollowingnotionofapproximately-stablematching.Denition1.Amatchingiscalled-stableiftheredoesnotexistapairofagentsnotmatchedwitheachotherwhowouldbothincreasetheirutilitybyafactorofmorethanbyswitchingtoeachother.If=1,thenthisisexactlyastablematching.An-stablematchingalsocorrespondstoastablesolutionifweassumethatswitchinghasacost.Inotherwords,inthepresenceofswitchingcosts,thesetofstablematchingsissimplythesetof-stablematchingswithoutswitchingcosts.Inthissectionweareconcernedwithunderstandinghowincreasingimprovesthequalityofstablematchings.Wearespecicallyconcernedwiththepriceofstability[6],whichistheratiooftheutilityofthebeststablematchingrelativetotheoptimummatching.Muchrecentworkinnetworkdesign[7]androuting[10,24]hasconsideredthepriceofstabilityinvariouscontexts.Thepriceofstabilityises-peciallyimportantfromthepointofviewofamechanismdesignerwithlimitedpower,sinceitcancomputethebeststablesolutionandsuggestittotheagents,whowouldim-plementthissolutionsinceitisstable.Therefore,thepriceofstabilitycapturestheproblemofoptimizationsubjecttothestabilityconstraint. Figure6:Ratioofthesocialutilitiesofbest-stableandsociallyoptimalmatchingsasafunctionofwhenthematchingsareconstructedaccordingtoouralgorithminsymmetricedge-labeledgraphs.Thedramaticincreasebetween=1and=1:1showsthatintroducingevensmallswitchingcostshasthepotentialtoproducesignicantsocialbene-ts.Preferencesweresampleduniformlyatrandomon[0;1].Belowwepresentvarioustheoreticalbounds,showingthatforsymmetricedge-labeledgraphs,therealwaysexistsan-stablematchingwithutilityofatleast 2OPT(whereOPTisthevalueoftheoptimummatching),andthatinvertex-labeledgraphs,therealwaysexistsan-stablematchingwithutilityatleast 1+OPT.Weprovideaconstructivealgorithmforndingsuchan-stablematching.Thisshowsthatbyincreasing,wecanimplementmuchbetterstablesolutionsthanfor=1,anddecreasethepriceofstabil-ity.Ourempiricalresultsusingthisalgorithmshowanevenmoredramaticimprovementthanpredictedbythetheoret-icalbounds.Forexample,Figure6showsthatfor=1:1wealreadyobtainatremendousimprovementinthequalityofstablematching,essentiallyobtainingstablematchingsthatareasgoodasamatchingwithmaximumutility.Thismeansthataddingaswitchingcostassmallasveortenpercentcanmakeanenormousdierenceinthequalityofstablematchings.Inmanysituations,itisreasonabletobe-lievethatacentralcontrollercancomputeagood-stablematching,assignagentstothatmatching,andonlyallowthemtodeviateonpaymentoftheswitchingcost.5.2Edge-labeledGraphsForedge-labeledgraphs,weprovebelowthatinthepres-enceofswitchingcostsofafactor,thepriceofanarchyisatmost2,butthepriceofstabilityisatmost2=.Thismeansthatasweincrease,therebegintobestablematchingsthatareworse,buttherealwaysexistsastablematchingthatisclosetooptimal.For=1,theseboundscoincide,givingustheresultthatallstablematchingsarewithinafactorof2fromthemaximumweightmatching.For=2,thisgivesustheeasilyveriablefactthattheoptimummatchingis2-stable.Theorem1.LetOPTbethevalueofthesociallyoptimalmatching.Inanyundirectededge-labeledgraph,thereexistsan-stablematchingwhosesocialutilityisatleast 2OPT.Furthermore,thesocialutilityofany-stablematchingis atleast1 2OPT.Proof.Denotebyw(M)theweightofamatchingM.First,noticethatthesociallyoptimalmatchingissimplythemaximumweightmatchinginthismodel,sincethesocialwelfareofamatchingisexactlytwiceitsweight.LetOPTdenotetheweightofthemaximumweightmatching,andprovethattheweightof-stablematchingsobeysthelowerboundsmentionedinthetheoremstatement.Werstprovethatforevery1,every-StableMatchinginGisofweightatleastOPT 2.LetMbean-stablematchinginG,andMbeamaximum-weightmatchinginG.Lete1=(u;v)beanarbitraryedgeinMnM.SinceMisan-stablematching,theremustbeeitheranedgee2=(u;w1)2Moranedgee3=(v;w2)2Msuchthatw(e1)w(e2)orw(e1)w(e3)(ifneitherweretrue,thenuandvcouldmatchtoeachotherandgainmorethanafactorofinutility).ThereforeforeveryedgeeinM,eithere2M,orthereisanedgee0ofMsharinganodewithesuchthatw(e)w(e0).SinceatmosttwoedgesofMcanshareanodewiththesameedgee0ofM(becauseMisamatching),thismeansthatifwesumtheaboveinequalities,weobtainw(M)2
w(M),asdesired.Wenowprovethattherealwaysexistsan-stablematch-ingMsuchthatw(M) 2w(M)bygivinganalgorithmforndingsuchamatching:SetM=MSorttheedgesofGinorderofdecreasingweight.Foreachedgee=(v1;v2)2Ginthisorder:Lete1;e2beedgestowhichv1;v2areincidentinM,re-spectively(iftheyexist)Ifw(e) isgreaterthanbothw(e1)andw(e2):Removee1ande2fromM.AddetoM.EndIfLoopThisalgorithmconsidersalledgesinthegraphinorderofdecreasingweight,andifthetwonodesintheedgecangainafactorofutilitybydeviatingtothisedge,thenweletthem.Ifanedgee1doesnotexist,thenforthenewedgeetobeaddedtothematching,allweneedisthatw(e) �w(e2).Calltheedgee=(v1;v2)inthealgorithmastheedgebeingcurrentlyexamined.Toprovecorrectness,wemustshowtwofacts:(i)Thealgorithmresultsinan-StableMatching.(ii)Theresultingmatchingisofweightatleastw(M) 2.Tobegintheproofof(i),noticethatMisamatching.Thisissimplybecausewheneverweaddanedge(u;v)toM,wealsoremovetheedgesincidenttothenodesuandv.SincewestartwithamatchingM,weknowthatMisamatchingateverypointinthealgorithm.ByLemma1,weknowthatifanedgee=(u;v)isinthematchingMimmediatelyafteritisexamined,thenitwillnotberemovedfromMlater.Noticealsothatifedgee=(u;v)isnotinthematchingMafteritisexamined,thenitwillneverbeaddedtoMlaterinthecourseofthealgorithm,becausethealgorithmonlyaddsedgestothematchingatthetimethatitisexaminingthem.Therefore,thenalmatchingMconsistsexactlyofedgesthatarekeptinMatthetimethealgorithmexaminesthem.Toshowthatthereturnedmatchingis-stable,supposetothecontrarythatthereisaninstabilityinthenalmatch-ingM,i.e.,anedgee1=(u;v)62Msuchthatw(e1)�w(e2)andw(e1)�w(e3),wheree2ande3aretheedgesofMincidenttouandv(whichmaynotexist).Sincee1isnotinthenalmatchingM,itcouldnothavebeenincludedinthematchingwhenitwasexamined.Thisimpliesthatatthistimetherewasanedgee02Mincidentto(withoutlossofgenerality)u,withw(e1)w(e0).Thisedgee0can-notstillbeinthematchingMattheendofthealgorithm'sexecution,sinceotherwisee1wouldnotformaninstabil-ity.Therefore,thealgorithmmusthaveremovededgee0atalaterpoint.Theonlyreasonwhyedgee0wouldbere-movedisifanedgee00wereaddedtothematching,withw(e00)�w(e0)w(e1).Sincethealgorithmconsiderstheedgesinorderofdecreasingweight,however,thisedgee00couldonlyhavebeenaddedbeforethealgorithmexaminededgee1,andsowehaveacontradiction.Wenowprove(ii).Ateachexaminationinthealgorithm,oneoftwothingscanoccur.ThetrivialcaseisthatnoedgeisformedsonochangeoccursinM.Theothercase,inwhichanewedgeeisaddedtothematching,addsanedgeofweightw(e)toMwhileremovingatmost2
w(e) .Theratioofthenewedgeweighttotheoldedgesweightisthereforew(e) 2
w(e) = 2.ByLemma1,onceanedgeisaddedtothematchingMbythealgorithm,itisneverremovedagain,sothetotalweightofthenalmatchingMisatleast 2w(M),asdesired,completingtheproofofTheorem1. Lemma1.Ifanedgee=(u;v)isinthematchingMimmediatelyafteritisexamined,thenitwillnotberemovedfromMlater.Proof.Supposetothecontrarythate=(u;v)2Mdirectlyafteritisexamined,butisnolongerinMatalaterpoint.Withoutlossofgenerality,assumethatewasremovedfromMbecausesomeedgee0=(u;w)wasadded.Forthistooccur,itmustbethatw(e0)�w(e).Butsince1,andthealgorithmexaminestheedgesinorderofdecreasingweight,thenthisadditionofedgee0couldonlyhaveoccurredbeforethealgorithmexaminede,acontradiction. 5.3VertexLabeledGraphsForvertexlabeledgraphs,resultssimilartoTheorem1hold:thepriceofanarchyisatmost1+andthepriceofstabilityisatmost(1+)=.For=1thisgivesustheobservationinSection4(noticethatwhileitiseasytoshowacorrespondencebetweenstablematchingsforedge-labeledandvertex-labeledgraphs,thesamedoesnotholdfor-stablematchings).Theorem2.LetOPTbethevalueofthemaximum-weightperfectmatching.Inanyvertex-labeledgraph,thereexistsan-stablematchingwhosesocialutilityisatleast 1+OPT.Furthermore,thesocialutilityofany-stablematchingisatleast1 1+OPT.Proof.Foranedgee=(u;v),denew(e)=w(u)+w(v),anddenotebyw(M)theweightofamatchingM.First,noticethatthesociallyoptimalmatchingissimplythemaximumweightmatchinginthismodel,sincethesocialwelfareofamatchingisexactlyequaltoitsweight.There-fore,weletOPTdenotetheweightofthemaximumweightmatching,andprovethattheweightof-stablematchings Figure7:Averagenumberofswitchesthegreedyal-gorithmmakesbeforetheresultingmatchingissta-bleforvertex-labeledandsymmetricedge-labeledgraphs.Notethequadraticgrowthforvertex-labeledandlineargrowthforedge-labeledgraphs.Utilitiesaresampledindependentlyfromanexpo-nentialdistributionwithmean0:5.Resultsareav-eragedover200runs.phase,uwillbematchedwithv(becauseuprefersvtoallitsotherneighborsandvprefersutoitsotherneighbors),andwecanremovevandufromthegraph.Therestoftheargumentisthesameasabove. Theabovetheoremsaysthatthesimpledecentralizedal-gorithmdescribedaboveconvergestoastablematchingintimeO(n2),sinceeachphasetakeslineartime.Notice,however,thatifinsteadofswitchingtoitsbestpartner,theagentssimplyswitchedtoarandomimprovingpartner,thesameargumentwouldguaranteeconvergencetoasta-blematchinginanexpectedtimeofO(n2d),wheredisthemaximumdegreeofthegraph.Inpractice(seeFigure7),onrandomutilitydistributionssimilartothosedescribedinprevioussection,theconver-gencetimeforvertex-labeledgraphsdoesindeedappeartobequadratic,butitisinterestingtoseethattheconvergencetimeforsymmetricedge-labeledgraphsseemstobelinear.Weconjecturethatthealgorithmconvergesinexpectedlin-eartimeforthesegraphs,perhapsbecausegoodedgesforonenodeareinexpectationalsogoodfortheothernodeintheedge,becauseofthesymmetricpreferences.Asymmetricedge-labeled.WhileTheorem3guaranteesconvergenceforthevertex-labeledandsymmetricedge-labeledutilities,thisisnotthecaseforasymmetricedge-labeledgraphs.Unfortunately,inthiscasethereareeasyexampleswherethisalgorithmcancycle.Inourexperiments,however,forsmalln(thenumberofnodesoneachside)thisalgorithmconvergedtoastablematchingonallbutasmallpercentageofcases,showingthatthebadscenariosarenot\typical."Asngetslarger,thisal-gorithmconvergesmoreandmorerarely(approximately2%lessforeveryadditionalnode),withconvergenceessentiallynon-existentforn=70.7.DISCUSSIONThispaperexploresthepricesofanarchyandofstabilityinmatchingmarkets.Wedemonstratethateventhoughthepriceofanarchycantheoreticallybehigh,whenutilitiesarerandomlysampled,thelossinsocialwelfarefromstrategicbehaviorislimited.Thisresultencompassesmanydier-entgraphandpreferencestructures,andisexperimentallyrobust.Whilethedownsideislimited,eventhisdownsidecanbealleviated:asignicantimprovementinsocialwel-farecanbeobtainedbysuggestingagoodmatchingandrequiringagentstopaysmallswitchingcoststodeviate.Weshowthistheoreticallyusinganalgorithmforconstruct-ingapproximatelystablematchings,andthendemonstratethatthealgorithmiseectiveinexperiments.Wealsoshowthatsimplegreedypartnerswitchingalgorithmscancon-vergequicklytostablematchingsinsomegraphstructures.Fromapracticalperspective,futureworkshouldincludeun-derstandingreal-worldutilitydistributionsandhowtheyaf-fectthesocialoutcomesofmatchingascomparedtorandomdistributionsofutilities.Fromamechanismdesignperspec-tive,itwouldbeinterestingtoexplorewhetheragentswouldchoosetoparticipateinaswitching-costbased,designer-suggestedmatchingmechanism.AcknowledgementsTheauthorswouldliketothankananonymousreviewerforhiscommentsonhowtoimprovethepaper.8.REFERENCES[1]A.Abdulkadiroglu,P.Pathak,andA.Roth.TheNewYorkCityHighSchoolMatch.AmericanEconomicReview,95(2):364{367,2005.[2]A.Abdulkadiroglu,P.Pathak,andA.Roth.Strategy-proofnessversusEciencyinMatchingwithIndierences:RedesigningtheNYCHighSchoolMatch.AmericanEconomicReview,2009.Toappear.[3]A.Abdulkadiroglu,P.Pathak,A.Roth,andT.Sonmez.TheBostonPublicSchoolMatch.AmericanEconomicReviewPapersandProceedings,95(2):368{371,2005.[4]D.Abraham,A.Blum,andT.Sandholm.Clearingalgorithmsforbarterexchangemarkets:enablingnationwidekidneyexchanges.InProceedingsofthe8thACMconferenceonElectroniccommerce,pages295{304.ACMPressNewYork,NY,USA,2007.[5]H.Ackermann,P.Goldberg,V.Mirrokni,H.Roglin,andB.Vocking.Uncoordinatedtwo-sidedmarkets.InProceedingsofthe9thACMConferenceonElectronicCommerce(EC),2008.[6]E.Anshelevich,A.Dasgupta,J.Kleinberg,E.Tardos,T.Wexler,andT.Roughgarden.Thepriceofstabilityfornetworkdesignwithfaircostallocation.InProc.FOCS,pages295{304,2004.[7]E.Anshelevich,A.Dasgupta,E.Tardos,andT.Wexler.Near-optimalnetworkdesignwithselshagents.InProceedingsSTOC,pages511{520.ACMPressNewYork,NY,USA,2003.[8]A.L.BarabasiandR.Albert.Emergenceofscalinginrandomnetworks.Science,286(5439):509{512,October1999.[9]G.Becker.ATreatiseOnTheFamily.FamilyProcess,22(1):127{127,1983.[10]G.ChristodoulouandE.Koutsoupias.OnthePriceofAnarchyandStabilityofCorrelatedEquilibriaofLinearCongestionGames.LectureNotesInComputerScience,3669:59,2005. 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