Incentive compatible multi unit combinatorial auct

InentiveCompatibleMultiUnitCombinatorialAutionsYairBartalShoolofComputerSieneandEngineering,HebrewUniversity,Jerusalem91904,Israelyairs.huji.a.ilRiaGonenShoolofComputerSieneandEngineeringHebrewUniversity,Jerusalem91904,Israelrgonens.huji.a.ilNoamNisanShoolofComputerSieneandEngineering,HebrewUniversity,Jerusalem91904,Israelnoams.huji.a.ilJune8,2002AbstratTheproblemwedealwithinthispaperisamulti-unitombinato-rialaution:therearentypesofgoodsforsale,andforeahitherearekiunitsofgoodi.Weareinterestedintheasewhereeahbidderdesiresarelativelysmallnumberofunitsofeahgood.Thesimplestaseiswhereeahtypeofgoodhasexatlykunits(i.e.ki=kforalli),andeahbidderdesiresatmostasingleunitofeahgood.Weallthisaombinatorialautionwithk-dupliates.Inthemoregeneralase,weassumethatthereexistsalowerboundandanupperboundsuhthateahbidderdesiresatmostkiunitsofgoodiandeithernounitofgoodioratleastkiunitsofgoodi.Inpartiularthesimpleaseisaspeialasewith==1=k.Allourresultsarederivedforthegeneralase.Werstharaterizeinentiveompatiblemehanismsforombi-natorialautions(bothmulti-unit,andregular),forthegeneralasethatbiddersarenotlimitedtobenon-singleminded.Thenweshowthatasapurelyomputationalproblem,aombina-torialautionwithkdupliatesisNP-hardtoapproximatetowithin1afatorofO(n1=(k+1)).Thisgeneralizestheknowninapproxima-bilityresultforombinatorialautions(k=1).Wethenproeedtoderiveaninentiveompatiblegeneralalgorithmthatahievesanearlymathingapproximationfator.Fromthatgeneralalgorithmtwoasesarederived:AnOn-LinealgorithmandanO-Linealgorithm,bothin-entiveompatiblefornon-singlemindedbidderswithnearlymathingapproximationfator.1Introdution1.1CombinatorialAutionsInthispaperwestudymulti-unitombinatorialautions.Inaombinatorialautionnnon-identialitemsareautionedsimultaneouslytombidders;inamulti-unitombinatorialautionforeahgoodi2f1;:::;ng,thereisanumberkiofidentialunits.Theombinatorialnatureoftheautionsomesfromthefatthatbiddersvaluebundlesofitems.Abundleofitemsisavetor(d1:::dn),where0dikiisthenumberofitemsofgoodiinthebundle.Speially,eahbidderjhasavaluationfuntionvjthatassignsanon-negativevalueforeahbundleofitems,vj:f0:::k1g:::f0:::kng!R+.Thegoaloftheautionistomaximizethetotalsoialwelfare(surplus):ndanalloationthatpartitionstheavailableitemsintobundlesS1:::SmthatmaximizesPjvj(Sj).Mostoftheworkonombinatorialautionsdealswiththeaseofsingle-unitgoods,i.e.,whenforalli,ki=1,butthemoregeneralmulti-unitaseismostlysimilar.Theombinatorialautionproblemhasreentlyreeivedmuhattentionduebothtoitsmanyappliationsandtoitswideexpressibilitypowersthatallowittorepresentawiderangeofsituationsthatombinealgorithmiissueswithinentiveissues{aombinationthatseemstobeentraltomanyInternet-Computingtasks.Thereaderisreferrede.g.to[8,9,21℄forageneraloverview,andto,e.g.,[10,13,25,24,14℄andthemanyreferenesthereinforanintrodutionCombinatorialAutions.Thekeyhallengeinombinatorialautionsomesfromaombinationoftheomputationalintratabilityofthegeneralproblemtogetherwithinentiveompatibilityrequirements.Findingtheoptimalalloationinaombinatorialaution(withreasonableenodingofthevaluations)isom-putationallyintratable(eventoapproximatetowithinO(n12)[12,13℄).However,manyalgorithmitehniquesandheuristismaybeappliedtotheproblemyieldingalgorithmsthateitherworkforinterestingspeialasesorthatworkreasonablywell(fromanexperimentalpointofview)forhundreds2andeventhousandsofitems(e.g.[15,16,14,17,25℄).Thenotionofinentiveompatibilitytakesintoaountthefattheautioneerdoesnotreallyhaveaesstotheinput,i.e.tothevaluationsofthebidders.Rather,theautioneeronlyhasaesstothebids,whihmaybemanipulatedstrategiallybythebidders.Anautionisalledinentiveompatibleifbiddershavenoinentivetolieabouttheirvaluation,i.e.ifrationalbiddersalwaysbidtheirvaluation{inwhihasetheaution-eeratleasthastheproperinputaordingtowhihhemayalloatetheitems.(Formaldenitionsarepostponedtosetion2.)Thekeytehniqueusedfordesigninginentiveompatiblemehanismsisthe,soalled,VCG-mehanism[18,19,20℄thatrstdeterminestheoptimalalloationandthenhargesarefullyhosenpaymentsforthealloatedbundlesofitems.Hereomesthekeylashbetweeninentiveompatibilityandompu-tationalomplexity:ndingtheoptimalalloationisomputationallyin-tratableandhenetheVCG-mehanismisintratable.Furthermore,theVCGtehniquereallyreliesonthefatthatthealloationisoptimalandusinganon-optimalalloationtogetherwiththeVCGpaymentshemedoesnotyieldinentiveompatiblemehanisms.Theproblemwasrstnotiedin[2,9℄wasshowntobeessentiallyuniversalin[22℄,andwasfurtherstudiedin[26℄.Theonlypositiveresultsknownareforveryrestritedlassesofbid-ders,single-mindedbidders[2℄,thosethatdesireonlyasinglesubsetofitems.Forthespeialaseofsingle-mindedsingle-unitombinatorialau-tionsseveralpolynomial-timeinentiveompatiblemehanismsareknown[2,1℄,inludingonethatahievesanO(pn)approximation(thebestpos-sible).Nothingisknownforthegeneralase,andtodate,nonon-optimalinentiveompatibleombinatorialautionmehanismisknownforgen-eralbidders(whereoptimalitymaybewithinanyxedfamilyofalloations[22,26℄).Thislakofknowledgeispartofageneralprobleminmehanismdesign.Theonlynon-VCGmehanismsknownt