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Budget Feasible Mechanisms Yaron Singer Computer Science Division University of California Budget Feasible Mechanisms Yaron Singer Computer Science Division University of California

Budget Feasible Mechanisms Yaron Singer Computer Science Division University of California - PDF document

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Budget Feasible Mechanisms Yaron Singer Computer Science Division University of California - PPT Presentation

berkeleyedu Abstract We study a novel class of mechanism design problems in which the outcomes are constrained by the payments This basic class of mechanism design problems captures many common economic situations and yet it has not been studied to ID: 25072

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parameterdomains,whereeachagent'sprivateinforma-tionisasinglenumber,designingtruthfulmechanismsoftenreducestodesigningmonotoneallocationrules,sincepaymentscanbecomputedviabinarysearch[21].Thisnolongerholdswhenthepaymentsarerestrictedbyabudget:Designingabudgetfeasibleallocationrulerequiresunderstandingitspayments,whichin-turndependontheallocationruleitself.Notsurprisingly,itseemsthatbudgetfeasiblemechanismsareverytrickytond.TheVCGmechanismdoesnotwork:ConsiderasimpleKnapsackinstancewhereallitemshaveidenticalvalues,andexceptforoneitemwhosecostequalsthebudget,allitemshavesmallcosts.TheVCGmechanismwillchoosethen�1small-costagents,payingthebudgettoeach.Thus,whilethismechanismreturnstheoptimalsolutionwithtotalcostwithinthebudget,thetotalpaymentwillbewayoverbudget(infact,(n�1)timesthebudget).Ingeneral,nothingcanwork:Consideraslightvariationoftheaboveproblem,inwhichallitemshavesmallcosts,andidenticalvaluesaslongasaparticularitemiisinthesolution,andotherwiseallhavevalue0(forexample,thinkofiasacorkscrewandtherestoftheitemsasbottlesofwine).Howwellcanabudgetfeasiblemechanismdohere?Ifthemechanismhasaboundedapproximationratioitmustalwaysguaranteetoincludeiinitssolution.Thishoweverimpliesthataslongasideclaresacostthatislessthanthemechanism'sbudget,themechanismincludesherinthesolution.Atruthfulmechanismmustthereforesurrenderitsentirebudgettoi.Thisofcourseresultsinanunboundedapproximationratio.OurResultsThequestion,then,is:Whichclassesoffunctionshavebudgetfeasiblemechanismswithgoodapproximationproperties?Ourmainresultisarandomizedconstantfactorbudgetfeasiblemechanismthatisuniversallytruthfulforthequitegeneral,andimportant,classofnondecreasingsubmodularfunctions(Theorem4.8).Foraslightlybroaderclass,thatoffractionallysubad-ditivefunctions,weshowthatcomputationalconstraintsdictatealowerbound.Asshowninthesimpleexam-pleabove,superadditivefunctionsbringouttheclashbetweentruthfulnessandthebudgetconstraint.Onapositivenote,thethreeproblemsinthebeginningofthesectioncorrespondtosubclasses(additive,OXS,andcoverage)ofsubmodularmaximizationproblems.Weshowimprovedapproximationsfortheseproblemsandotherspecialcases.Wefurtherexplorethespaceofbudgetfeasiblemechanisms,showingseveralimpossibil-itiesaswellasacharacterizationundermorerestrictedconditions.RelatedWorkBudgetsinauctions:Budgetscameunderscrutinyinauctiontheory[9],[7],[13],[6]afterobservingbehaviorofbiddersinonlineautomatedauctions[2],aswellasinspectrumauctionswherebiddingisperformedbygroupsofstrategicexperts[7].Whilethesepioneeringworkshighlightthesignicanceandchallengesthatbudgetsintroducetomechanismdesign,theyrelatetoanentirelydifferentconceptthantheonewestudyhere.Whiletheseworksstudytheimpactofbudgetsonstrategicbidders,ourinterestistoexplorethebudget'seffectonthemechanism.Thesepapers,however,dopointoutthecomplexityinducedbybudgetconstraintsinmechanismdesign,andtheneedforapproximations.Frugality:Inrecentyearsatheoryoffrugalityhasbeendevelopedwiththegoalofprovidingmech-anismsforprocurementauctionsthatadmitminimalpayments[8],[3],[15],[11],[26].Budgetfeasibilityandfrugalityarecomplementaryconcepts.Frugalityisaboutbuyingafeasiblesolutionatminimumcost—therearenopreferencesamongthesolutions,andthegoalistominimizepayments.Inoursettingwehavenopreferencesamongpayments—aslongastheyarebelowthebudget—butwedocareaboutthevalueofthesolutions.Thetwoapproachesarecomplementaryalsoinanotherimportantsense:inourlastsectionweshowthatforalltheproblemsstudiedinthefrugalityliteraturetherearenobudgetfeasiblemechanisms.CostSharing:Somewhatconceptuallyclosertoourworkisthesubjectofcostsharing,inwhichagentshaveprivatevaluesforaservice,thereisanondecreasingcostforallocatingtheservicetoagents,andthegoalistomaximizetheagents'valuationsunderthecost(see[14]forasurvey).Theproportionalsharemechanismwestudyinthispaperisinspiredby[20]and[25].Therelationshipbetweencostsharingandoursetting,how-ever,isquitelimited:wearenotaimingtooptimizeafunctionoftheagentsprivateinformationunderapubliccostfunction,butratheroptimizeapublicfunctionunderconstraintsdictatedbyagents'privateinformationandaxedbudget.Here,ourgoalisnon-utilitarian—weaimtomaximizethebuyer'sdemand,whichisindependentoftheagents'utilities.SubmodularMaximization:Fromapurealgorith-micperspective,evenunderacardinalityconstraint,maximizingasubmodularfunctioniswellknowntobeNP-hard,andan1�1=eapproximationratiocanbeachievedbygreedilytakingitemsbasedontheirmarginalcontribution[23].Whenitemshavecosts,vari-ationsofgreedyonmarginalcontributionnormalizedbycostcanachieveconstantfactorapproximations,andeventheoptimal1�1=eratio[16],[17].Forsubmodularmaximizationproblemsthatcanbeexpressedasintegerprograms,roundingsolutionsoflinearandnonlinearpro- Denition3.1:AfunctionV:2[n]!R+issym-metricsubmodularifthereexistr1:::rn0,suchthatV(S)=PjSji=1ri.ConsiderthefollowingallocationrulefM:Sortthenbidssothatc1c2:::cn,andconsiderthelargestksuchthatckB=k.Thatis,kistheplacewherethecurveoftheincreasingcostsintersectsthehyperbolaB=k.Thesetallocatedhereisf1;2;:::;kg.Thatis,fM=f1;2;:::;kg.Thisisobviouslyamono-toneallocationrule:anagentcannotbeexcludedwhendecreasingherbid.IntheAppendixweshowthatpayingeachagenti=minfB=k;ck+1gresultsinatruthfulmechanism.2Observethatthisallocationrulehasthepropertyweseek:summingoverthepaymentsthatsupporttruthfulnesssatisesthebudgetconstraint.Hencethisgivesusabudgetfeasiblemechanism.Importantly,thisisalsoagoodapproximationoftheoptimumsolution:Theorem3.2:Theabovemechanismhasapproxima-tionratiooftwo.Proof:Observethattheoptimalsolutionisobtainedbygreedilychoosingthelowest-priceditemsuntilthebudgetisexhausted.Bythedownwardslopingproperty,toprovetheresultitsufcestoshowthatthemechanismreturnsatleasthalfoftheitemsinthegreedysolution.Assumeforpurposeofcontradictionthattheoptimumsolutionhas`items,andthemechanismreturnslessthan`=2.Itfollowsthatcd`=2e�2B=`.Notehowever,thatthisisimpossiblesinceweassumethatcd`=2e:::c`,andP`i=d`=2eciBwhichimpliesthatcd`=2e2B=`,acontradiction. InSectionVweshowthatnobetterapproximationratioispossible.Thisisrathersurprising,giventhesimplicityofthefull-informationproblem,andillustratestheintricaciesofbudgetfeasibility.IV.GENERALSUBMODULARFUNCTIONSWenowturntothegeneralcaseofnondecreasingsubmodularfunctions.AdemandfunctionVisnondecreasingifSTimpliesV(S)V(T).Denition4.1:V:2[n]!R+issubmodularifV(S[fig)�V(S)V(T[fig)�V(T)8ST:Ingeneral,submodularfunctionsmayrequireexpo-nentialdatatoberepresented.WethereforeassumethebuyerhasaccesstoavalueoraclewhichgivenaqueryS[n]returnsV(S)(seerelatedworksectionformore2Itisratherinterestingthatthesecondtermisneeded;weshowintheAppendixthatthemechanismbreaksdowninitsabsence.discussiononsubmodularmaximization).Indesigningtruthfulmechanismsforsubmodularmaximizationprob-lems,thegreedyapproachisanaturalt,sinceitismonotonewhenagentsaresortedaccordingtotheirincreasingmarginalcontributionsrelativetocost:themarginalcontributionofanagentigivenasubsetSisVijS:=V(S[fig)�V(S).Inthemarginalcontribution-per-costsortingthei+1agentistheagentjforwhichVjjSi=cjismaximizedoverallagentsNwhereSi=f1;2:::;ig,andS0=;.TosimplifynotationwewillwriteViinsteadofVijSi�1.Thissorting,inthepresenceofsubmodularity,implies:V1=c1V2=c2:::Vn=cn:(1)NoticethatV(Sk)=PikViforallk.A.TheProportionalShareAllocationRuleThemechanismfromtheprevioussectionforthelimitedsymmetriccasecanbegeneralizedappropriatelytoworkforvariousclassesinthesubmodularfamilyoffunctions.Denition4.2:ForabudgetBandsetofagentsNwithcostvectorc,thegeneralizedproportionalshareal-locationrule,denotedfM(c;B;N)sortsagentsaccord-ingto(1)withcostsvectorcandbudgetBandallocatestoagentsf1;:::;kgthatrespectciBVi=V(Si).Observethatthisconditionismetforeveryf1;:::;igwhenik.Forconcretenessconsiderthecaseofadditivevalu-ations(KnapsackfromtheIntroduction):eachagentisassociatedwithaxedvalueviandV(S)=Pi2Svi.Herethemarginalcontributionofeachagentisindepen-dentoftheirplaceinthesorting,andwesimplyhavethatVi=viforallagentsi2[n].InthiscasefMproducesabudget-feasiblemechanism.Thereasonis,itassuresusthatforeachagenti,thethresholdpaymentsoffM,denotedidonotexceedtheagent'sproportionalshare:0i=minnViB Pi2SVi;Vick+1 Vk+1o:whichallowsbudgetfeasibility,aswellasindividuallyrationality:0ici.Thisseemstomakethepropor-tionalshareallocationruleanidealcandidatetoobtainbudgetfeasiblemechanisms.Indeed,withsomeminoradjustments,formanyproblemswithfunctionsinthesubmodularclass(e.g.symmetric,Knapsack,Match-ing,)thisgeneralapproachworkswellandproducesbudgetfeasiblemechanismswithgoodapproximationguarantees(seethefollowingsectionformoredetails).Furthermore,aswediscussedabove,theproportionalsharemechanismisoptimalissomecases,andinsomerestrictedenvironmentsourcharacterizationsshowthat agentinthesortingifshemeetstheconditioncjV0jB=V(Tj�1[fjg).Foragivenstagejinthissequentialallocation,wecanndthemaximalcostagentiwouldhavebeenablede-clareandbeallocated,ifshehadbeenconsideredbythemechanismatthisstage:Thevalueci(j)=Vi(j)cj=V0jisthemaximalcosticandeclarewhichwouldplaceheraheadofjinthesorting,andifthiscostdoesnotexceedi(j)=BV0i(j)=V(Tj�1[fig),themechanismwouldhaveallocatedtoagenti.Therefore,hadiappearedatstagej,theminimumbetweenthesevaluesisthemaximalcostshecandeclareandbeallocatedatthisstage.SinceVi(j)monotonicallydecreaseswithjwhilecj=V0jincreases,ci(j)mayhavearbitrarybehaviorasafunctionofj.However,aswenowshow,takingthemaximumofthesevaluesresultinthresholdpayments.Letrbetheindexin[k0+1]forwhichminfci(j);i(j)gismaximal.Declaringacostbelowici(r)guaranteesitobewithintherstrk0+1elementsinthesortingstageofthemechanism,withr�1itemsallocated.Sinceii(r),iwillbeallocated.Toseethatdeclaringahighercostpreventsifrombeingallocated,considerrstthecasewhereci(r)i(r).Ahighercostplacesiafterrinthesortingstageofthemechanism.Ifthemaximumofci(j)overallj2[k0+1]isci(r),reportingahighercostplacesiafteranelementwhichisnotallocatedandthereforeitwillnotbeallocated.Otherwise,ifci(r)ci(j),forsomejk0+1,bythemaximalityofritmustbethecasethat:BVi(j) V(Tj�1[fig)=i(j)ci(r)ci(j)andiwillnotbeallocatedasacostabovei(j)willnotmeettheallocationcondition.Inthesecondcasewhenci(r)&#x-278;i(r),ifristheindexwhichmaximizesi(j)overallindicesin[k0+1],reportingahighercostwillnotmeetthemechanism'sallocationconditionateachindexin[k0+1].Otherwise,ifthereissomeotherindexj2[k0+1]forwhichthismaximumisachieved,then:Vi(j)cj V0j=ci(j)i(r)i(j)andthusdeclaringahighercostinthiscaseplacesiafterjinthesorting,andthemechanismwillnotconsideri. Lemma4.5(IndividualRationality):ThemechanismfMisindividuallyrational,i.e.,cii.Proof:Observethat:(a)Vi(j)Vi(j+1)8j2N;(b)Tj=Sj8ji;(c)VijTi�1=Vi.Sincethethresholdpaymentisthemaximumoverallminfci(j);i(j)gin[k0+1],itisenoughtoshowthatciminfci(j);i(j)gforacertainjk0+1.Since(b)impliesthatik0+1wecanconsideri'sreplacementjwhichappearsintheithplaceinthemarginal-contribution-per-costsortingoverNnfig.Sincei2[k],anddueto(b)and(c)above,wehavethatciViB V((Si�1[fig)=VijTi�1B V(Ti�1[fig)=ci(j):Intheoriginalsorting,iappearsaheadofj(asimpliedfrom(b)),andthereforeitsrelativemarginalcontributionisgreater.Thus:ciVijSi�1cj VjjSi�1=VijTi�1cj V0jjTj�1=i(j):Itthereforefollowsthatciminfci(j);i(j)gi: E.PaymentBoundsThecharacterizationaboveallowsustoincludeaslightlymodiedversionoftheproportionalshareallo-cationruleinourmechanism,withthresholdpaymentsthatareguaranteedtobenomorethanaconstantfactorawayfromagents'proportionalcontribution.Thisisakeypropertywhichguidesthedesignofourmechanism.WewillrunthemodiedproportionalshareallocationruleoverasubsetoftheagentsNs,withaconstantfractionofthebudget,denotedB0.WedescribeNsandB0explicitlyinthefollowingsection,butforthepurposeofshowingthepaymentbounds,wecanthinkoftheseasanysubsetofagentsandanybudget.Inthismodiedversionoftheproportionalshareallocationrule,foris:=argmaxj2NsV(fjg),wesorttheagentsofNsnfisgaccordingtothemarginal-contribution-per-costorder,andallocatetoW=Sk[fisg,whereSkareallkagentsinNsthatrespecttheconditionciViB=V(Si[fisg).Thecharacterizationfromaboveeasilyextendstothiscaseusingi(j)=Vi(j)B=V(Tj�1[fi;isg).Underthismodicationwecanshowthefollowingdesirableboundonthethresholdpayments:Lemma4.6(PaymentBounds):Fori2Wnfisg:i6e�2 e�1ViB0 V(W):Proof:ForTk0asabove,letW0=Tk0[fisgandletrbetheindexforwhichi=minfci(r);i(r)g.Ifrk0,observethatthesortingimpliescr=V0rck0=V0k0andtherefore: Thepayment^hereis^is=B=2and^i=minfi;B=2gfori6=is,whereiarethepaymentsasdescribedforthemodiedproportionalshareallocationrule,usingthebudgetB0=B= ,for =(12e�4)=(e�1).Observethat isexactlytwicetheconstantfromtheboundinLemma4.6.Incaseiisallocated,Bisclearlyherthresholdpayment.IfWisallocated,fromthecharacterizationlemmaandthefactthatNsconsistsonlyofagentswithcostlessthanB=2,^asdescribedaboveareclearlythethresholdpayments.OnecanverifythepartitionofagentsismonotoneandsincepaymentsareboundedbyB=2agentsinN`cannotbenetbymisreportingtheircost.SincethemodiedproportionalshareallocationruleusesB0=B= asitsbudget,fromlemma4.6,wecanconcludethat:Xi^iB=2+ Xi6=isViB0 V(W)Bandthemechanismisthereforetruthfulandbudgetfeasible.Individualrationalityandmonotonicitywerediscussedabove,andthelowerboundasdescribedinthefollowingsectionapplieshereaswell.Finally,let =((3+2 )=(e�1))=(e�1).FromLemma4.7andthepartitionofagentswehave:OPT(c;B;N)OPT(c;B;NnN`)+V(fig)OPT(c;B;Ns)+2V(fig) V(W)+2V(fig)(2+ )maxnV(W);V(fig)owhichgivesusourdesiredapproximationratio. V.THESPACEOFBUDGETFEASIBLEMECHANISMSInthissectionweaddresssomenaturalquestionsthatarisewhenexploringbudgetfeasiblemechanisms.Welimitourdiscussiontodeterministicmechanisms.A.LowerBoundsandImprovedApproximationsForspecialcasesofsubmodularfunctions,specializedtechniquesbasedonthemainresultyieldbetterapproximationratios:Theorem5.1:ForKnapsackthereisabudgetfeasible5-approximationmechanism.ForMatchingthereisabudgetfeasible(5e�1 e�1)-approximationmechanism.Foreitherproblem,nobudgetfeasiblemechanismcanap-proximatewithinafactorbetterthan2�,foranyxed�0.Wedeferthedetailsofupperboundproofstothefullversionofthepaper.Wewillnowshowalowerboundthatisindependentofcomputationalassumptionsandshowsthatnoapproximationratiobetterthantwoispossible.Forourlowerboundwe'llconsideraverysimplefunction,V(S)=jSj.Observethatallthefunctionsinthesubmodularclasswehavediscussed,includingsymmetricsubmodularincludethisdemandvaluation.Proposition5.2:ForV(S)=jSj,nobudgetfeasiblemechanismcanguaranteeanapproximationof2�,forany�0.Proof:Supposewehavenitemswithcostsc1=c2==cn=B=2+,forsomepositiveB=2.Assume,forpurposeofcontradictionthatfisabudgetfeasiblemechanismwithapproximationratiobetterthan2.Inparticular,fhasaniteapproximationratioandmustthereforeallocatetoatleastoneagentinthiscase.W.l.o.g.,assumefallocatestoagent1.Bymonotonicity,agent1canreducehercosttoc01=0B=2�andremainallocated.Forthiscostvector,(c1;c�1),Myerson'scharacterizationimpliesthatthethresholdpaymentforagent1shouldbeatleastB=2+,byindividualrationalityandbudgetfeasibility,fcannotallocatetoanyotheragent.Observehoweverthattheoptimalfullinformationsolutioninthiscaseallocatestotwoagentswhichcontradictsf'sapproximationratioguarantee. B.LowerBoundonFractionallySubadditiveFunctionsInlightofthepositiveresultsforsubmodularfunctions,ournaturaldesirewouldbetoextendthemainresulttomoregeneralclassesofproblems.Wenowshowthereislittlehopeinthat,atleastinthevaluequerymodel.Letusconsiderfractionallysubadditivefunctions:Denition5.3:AfunctionV:2[n]!Riscalledfractionallysubadditiveifthereexistsanitesetofaddi-tivevaluationsfa1;:::;atgs.t.V(S)=maxi2[t]ai(S).Itisknownthateverysubmodularfunctioncanberepresentedasafractionallysubadditivefunction,andthatallfractionallysubadditivefunctionsaresubadditive[18].Usingasimplereductionfrom[19]weshowthatformechanismswhichusevaluequeryoracles,obtainingreasonableapproximationsinthecaseoffractionallysubadditivedemandsishard,regardlessofincentiveconsiderations.Theorem5.4:Inthecaseoffractionallysubadditivedemands,anyalgorithmwhichapproximateswithina undertheprolec00,thethresholdpriceforeachagentinSisatleastci�B=jSj.Toseethis,observethati'sthresholdpricemustbeatleastci,sincei2f(c0).Sincefisanonymous,andallagentsinSdeclarethesameprice,thethresholdpriceforeachagentinSmustalsobeatleastci.Thus,paymentstoagentsinSexceedthebudget,contradictingbudgetfeasibility. VI.DISCUSSIONThespaceofbudgetfeasiblemechanismsseemsquitebroadandinvitesforfurtherinvestigation.Therichnessofthesubmodularclassimpliestherearemanyproblemsforwhichbetterapproximationratiosareachievable.WehavebrieydiscussedsomespecicexampleswhichincludeKnapsack,Matching,Coverageandthesymmet-ricsubmodularcase,andbelievetherearemanymoreinterestingproblemstostudy.Furthermore,webelieveabetterapproximationratioisachievableforthegeneralcase,anditmaybepossibletoimprovetheanalysisoftheupperboundbyshowingtighterpaymentbounds.Whilewehavemadearststeptowardscharacteri-zationbyconsideringmorerestrictedmechanisms,webelieveamuchmoregeneralcharacterizationforbudgetfeasiblemechanismsawaitstoberevealed.Finally,itwouldbeinterestingtofurtherexplorethelowerboundsdictatedbybudgetfeasibility.Herewe'veshownseverallowerboundwhichareindependentofcomputationalassumptions,anditwouldbeinterestingtoextendthesetechniques.ACKNOWLEDGEMENTSTheauthorwishestothankChristosPapadimitriouforendlessdiscussionsandhelp.ToDaveBuchfuhrer,IftahGamzu,ArpitaGhosh,MohammadMahdian,GeorgePierrakos,AminSaberi,MichaelSchapira,MeromitSingerandMukundSundararajantheauthorwishestoexpressgratitudeformeaningfuldiscussionsandvalu-ableadvice.REFERENCES[1]AlexanderA.AgeevandMaximSviridenko.Pipagerounding:Anewmethodofconstructingalgorithmswithprovenperformanceguarantee.J.Comb.Optim.,8(3):307–328,2004.[2]GaganAggarwal,NirAilon,FlorinConstantin,EyalEven-Dar,JonFeldman,GereonFrahling,MonikaRauchHenzinger,S.Muthukrishnan,NoamNisan,MartinP´al,MarkSandler,andAnastasiosSidiropoulos.Theoryresearchatgoogle.SIGACTNews,39(2):10–28,2008.[3]AaronArcherand´EvaTardos.Frugalpathmechanisms.ACMTransactionsonAlgorithms,3(1),2007.[4]ItaiAshlagi,ShaharDobzinski,andRonLavi.Anoptimallowerboundforanonymousschedulingmechanisms.InACMConferenceonElectronicCommerce,pages169–176,2009.[5]LiadBlumrosenandNoamNisan.Onthecomputationalpowerofiterativeauctions.InACMConferenceonElectronicCommerce,pages29–43,2005.[6]ChristianBorgs,JenniferT.Chayes,NicoleImmorlica,Mo-hammadMahdian,andAminSaberi.Multi-unitauctionswithbudget-constrainedbidders.InACMConferenceonElectronicCommerce,pages44–51,2005.[7]JeremyBulow,JonathanLevin,andPaulMilgrom.Winningplayinspectrumauctions.WorkingPaper.[8]MatthewCary,AbrahamD.Flaxman,JasonD.Hartline,andAnnaR.Karlin.Auctionsforstructuredprocurement.InSODA,pages304–313,2008.[9]ShaharDobzinski,RonLavi,andNoamNisan.Multi-unitauctionswithbudgetlimits.InFOCS,pages260–269,2008.[10]ShaharDobzinskiandMukundSundararajan.Oncharacteri-zationsoftruthfulmechanismsforcombinatorialauctionsandscheduling.InACMConferenceonElectronicCommerce,pages38–47,2008.[11]EdithElkind,AmitSahai,andKennethSteiglitz.Frugalityinpathauctions.InSODA,pages701–709,2004.[12]JoanFeigenbaum,ChristosH.Papadimitriou,RahulSami,andScottShenker.Abgp-basedmechanismforlowest-costrouting.InPODC,pages173–182,2002.[13]JonFeldman,S.Muthukrishnan,MartinP´al,andCliffordStein.Budgetoptimizationinsearch-basedadvertisingauctions.InACMConferenceonElectronicCommerce,pages40–49,2007.[14]KamalJainandMohammadMahdian.Costsharing.InNoamNisan,TimRoughgarden,EvaTardos,andVijayV.Vazirani,editors,AlgorithmicGameTheory.CambridgeUniversityPress,2007.[15]AnnaR.Karlin,DavidKempe,andTamiTamir.BeyondVCG:Frugalityoftruthfulmechanisms.InFOCS,pages615–626,2005.[16]SamirKhuller,AnnaMoss,andJoseph(Sef)Naor.Thebud-getedmaximumcoverageproblem.Inf.Process.Lett.,70(1):39–45,1999.[17]AndreasKrauseandCarlosGuestrin.Anoteonthebudgetedmaximizationofsubmodularfunctions.InCMUTechnicalReport,pagesCMU–CALD–05–103,2005.[18]BennyLehmann,DanielLehmann,andNoamNisan.Combi-natorialauctionswithdecreasingmarginalutilities.InACMconferenceonelectroniccommerce,2001.[19]VahabS.Mirrokni,MichaelSchapira,andJanVondr´ak.Tightinformation-theoreticlowerboundsforwelfaremaximizationincombinatorialauctions.InACMConferenceonElectronicCommerce,pages70–77,2008.[20]Herv´eMoulinandScottShenker.Strategyproofsharingofsubmodularcosts:budgetbalanceversusefciency.EconomicTheory,18(3),pages511–533,2001.[21]AhuvaMu'alemandNoamNisan.Truthfulapproximationmechanismsforrestrictedcombinatorialauctions.GamesandEconomicBehavior,64(2):612–631,2008.[22]R.Myerson.Optimalauctiondesign.MathematicsofOperationsResearch,6(1),1981.[23]G.L.Nemhauser,L.A.Wolsey,andM.L.Fisher.Ananalysisofapproximationsformaximizingsubmodularsetfunctionsii.Math.ProgrammingStudy8,pages73–87,1978.[24]NoamNisanandAmirRonen.Algorithmicmechanismdesign.GamesandEconomicBehaviour,35:166–196,2001.ApreliminaryversionappearedinSTOC1999.[25]TimRoughgardenandMukundSundararajan.Newtrade-offsincost-sharingmechanisms.InSTOC,pages79–88,2006.[26]KunalTalwar.Thepriceoftruth:Frugalityintruthfulmecha-nisms.InSTACS,pages608–619,2003.[27]WilliamVickrey.Counterspeculation,auctions,andcompetitivesealedtenders.TheJournalofFinance,16(1):8–37,1961.