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Dominant Strategy Double Auction Mechanisms: Design and Implementation

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Dominant Strategy Double Auction Mechanisms: Design and Implementation
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ZHU, LEON YANG ( Author, Primary )
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2008

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Allocative efficiency ( jstor )
Auctions ( jstor )
Bid prices ( jstor )
Game theory ( jstor )
Linear programming ( jstor )
Mechanism design ( jstor )
Optimal solutions ( jstor )
Prices ( jstor )
Social welfare ( jstor )
Transaction costs ( jstor )

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University of Florida
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University of Florida
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Copyright Leon Yang Zhu. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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8/31/2006
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DOMINANTSTRATEGYDOUBLEAUCTIONMECHANISMS:DESIGNANDIMPLEMENTATIONByLEONYANGZHUADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2005

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Copyright2005byLeonYangZhu

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IdedicatethisworktoHimwhoprovidesallthese.

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ACKNOWLEDGMENTSFirstofall,IwouldliketothankZuo-JunMaxShen.HehasbeenagreatsupervisorandmentorthroughoutmyfouryearsattheUniversityofFlorida,andIthankhimfortheexperienceswehaveshared.Heallowedmetheroomtothinkcreatively,buthewasalwaystherewhenIneededadviceorguidance.IamalsogratefultoRavindraAhuja,whohelpedmecometotheUnitedStatesandbuiltmyoptimizationfoundationviahisgreatbookandcourse|LinearProgrammingandNetworkFlows.Iwouldalsoliketoacknowledgemyothercommitteemembers,DonaldHearnandGaryKoehler,fortheirtimeandguidance.SpecialthanksgotoDavidSappingtonandStevenSlutskyfromtheEco-nomicsDepartment,aswellasotherfacultymembersintheCollegeofBusinessAdministration,whohelpedmedevelopabalancebetweenintuitionandtech-niques.IwouldliketothankmycollaboratorsGangDerekChen,CatherineShuZhang,LianQi,RogerLezhouZhan,andShanLi,whohavebeenapleasuretoworkwith.Iamalsogratefultothefaculty,sta,andstudentsoftheIndustrialandSystemsEngineeringDepartmentattheUniversityofFloridaforhelpingmakemyexperiencehereunforgettable.Finally,myutmostappreciationgoestomyfamilymembersandmydearestJeanYujiaoQiao,whoseloveandunderstandingwerekeyingredientsofthiswork. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. vii LISTOFFIGURES ................................ viii ABSTRACT .................................... ix CHAPTER 1INTRODUCTION .............................. 1 2LITERATUREREVIEW .......................... 6 2.1TheAuctionGame .......................... 6 2.1.1AnIntroductiontoGameTheory .............. 7 2.1.2TheMechanismDesignProblem ............... 10 2.1.3Strategy-proofAuctionMechanisms ............. 14 2.2TheAssignmentGame ........................ 20 3MULTI-STAGEDESIGNAPPROACH ................... 25 3.1ExistingApproach .......................... 25 3.2DrawbacksoftheExistingApproaches ............... 26 3.3Multi-StageDesignApproach .................... 27 4AGENTCOMPETITIONDOUBLEAUCTIONMECHANISM ..... 29 4.1ModelandNotation .......................... 29 4.2BuyerCompetitionMechanismandSellerCompetitionMechanism 32 4.3Theorems ............................... 33 4.4Proofs ................................. 34 5SIMPLEEXCHANGEENVIRONMENTWITHTRANSACTIONCOSTS ..................... 41 5.1ModelandMechanism ........................ 42 5.2ImpactofTransactionCosts ..................... 43 5.3AsymptoticEciencyProperty ................... 48 5.4Remarks ................................ 51 5.4.1False-nameBid ........................ 51 5.4.2IncentiveonRevealingtheTransactionCosts ........ 55 v

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5.4.3Non-quasi-linearSocialWelfare ................ 56 6BILATERALEXCHANGEENVIRONMENTWITHTHESINGLEOUTPUTRESTRICTION ............ 58 6.1ModelandSingleOutputRestriction ................ 59 6.2BuyerCompetitionLPMechanism ................ 61 6.3AnEnhancedBC-LPMechanism .................. 66 6.4ModiedBuyerCompetitionMechanism .............. 71 6.5AsymptoticEciencyTheorems ................... 73 6.6AnExample .............................. 75 6.7Proofs ................................. 77 6.7.1ProofsofPropositions6.2.1and6.2.2 ............ 77 6.7.2ProofsofTheorems6.2.4and6.2.6 .............. 79 6.7.3ProofofProposition6.2.3 ................... 89 6.7.4ProofofTheorem6.2.7 .................... 90 6.7.5ProofofTheorem6.4.1 .................... 94 6.7.6ProofsofTheorems6.5.1and6.5.2 .............. 96 7IMPLEMENTATIONANDCOMPARISON ................ 102 7.1ModelandExchangeEnvironment .................. 103 7.2MechanismsunderTradeReductionApproach ........... 108 7.2.1KSM-TRmechanism ..................... 108 7.2.2Tradereductionmechanism .................. 110 7.3ImplementationoftheMechanisms ................. 113 7.3.1ImplementationoftheAC-DAMechanism ......... 113 7.3.2ImplementationoftheBC-LPandMBCMechanisms ... 116 7.4ImplementationandApplicabilityComparison ........... 119 7.5EciencyandPayosComparison .................. 121 7.5.1ComparisonundertheSimpleExchangeEnvironment ... 122 7.5.2ComparisonundertheBilateralExchangeEnvironment .. 124 7.6Proofs ................................. 127 8CONCLUDINGREMARKS ......................... 133 REFERENCES ................................... 135 BIOGRAPHICALSKETCH ............................ 140 vi

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LISTOFTABLES Table page 5{1ExpectedRevenue/SocialWelfarewithoutFalse-nameBids ...... 54 5{2ExpectedRevenue/SocialWelfarewithSellerCoalition ......... 54 6{1NotationfortheProofs .......................... 86 7{1ImplementationNeedsoftheMechanisms ................ 121 7{2EciencyandPayoComparisonSummary ............... 125 7{3ParameterSettings ............................. 126 7{4ComparisonsoftheMechanisms ..................... 126 vii

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LISTOFFIGURES Figure page 1{1Buyers,Sellers,andPotentialTrades .................. 2 2{1PossibleTransactionsunderMcAfee'sMechanism ........... 19 3{1DiagramforMulti-StageApproach ................... 27 5{1PossibleTransactionStructures ..................... 44 5{2ExamplewithLowTransactionCosts .................. 45 5{3ExamplewithHighTransactionCosts .................. 46 6{1ExamplefortheBC-LPMechanism ................... 66 6{2AnExchangeEnvironment ........................ 75 6{3IllustrationoftheFlows ......................... 82 7{1BilateralExchangeEnvironmentsandtheCorrespondingMechanisms 104 viii

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDOMINANTSTRATEGYDOUBLEAUCTIONMECHANISMS:DESIGNANDIMPLEMENTATIONByLeonYangZhuAugust2005Chair:Zuo-JunMax"ShenMajorDepartment:IndustrialandSystemsEngineeringMillionsoftransactionsarearrangedoveronlinemarketplaceseveryday.Becausetransactionsofidenticalcommoditiesarecarriedoutinmanyisolatedauctions,apotentialcustomermayneedtodecidewhethertocontinuecompetinginthecurrentauctionortogiveupandmoveontothenextduringthebiddingprocess.Togetagooddeal,thebuyerhastospeculatetheotherbidders'actionandthepricemovementoftheitem.Sinceitisnotclearhowthebuyersshouldbidtomaximizetheirpayos,thispracticeplacesahugedecisionburdenonthebuyersanddetersbuyerparticipation.Consequently,theburdeninvolvedcanoutweighthesavingsforcustomersandresultinthelossofcustomersforonlinemarketplaces.Toregaincustomers,marketplaceshavetosimplifycustomers'decision-making.Thispaperproposesanewmulti-stagedoubleauctiondesignapproachtoaccomplishthistask,evenwhenshippingandhandlingcosts,andsalestaxesaredierentacrossvariouspossibletransactions.Thenovelmulti-stageapproachrenderstruthfuldoubleauctionmechanisms,whichsimpliescustomers'decision-making,asbiddingone'struevaluationprivateinformationisthebeststrategy ix

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foreachindividualbuyerandseller.Thisdesignapproachisthenappliedtoprocurementmarketplacesandnewtruthfuldoubleauctionmechanismsforprocurementauctionareproposed.Comparedtootherknowndoubleauctionmechanisms,weshowthattheresultingmechanismsalsoachievehighersocialwelfareandindividualpayos. x

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CHAPTER1INTRODUCTIONThisdissertationfocusesonauctionmechanismdesignforthebilateralexchangeenvironments.Suchmechanismsapplytocontextsinwhichamarketmaker,ortheauctioneer,providesmatchingservicetomultiplebuyersandsellerswhilehavinglittleornoinformationaboutthepreferencesorvaluationsofhiscustomers.Forexample,thesemechanismscanapplytothecurrentcustomer-to-customeronlineauctionmarket,inwhichabuyerbidstheitemofinterestsequentiallyindierentauctionstillhe/shewinstheitemorgivesup.Duringthebiddingprocess,abuyerneedstodecidewhethertocontinuecompetinginthecurrentauctionortogiveupandmoveontothenext.Togetagooddeal,thebuyerhastospeculatetheotherbidders'actionandthepricemovementoftheitem.Sinceitisnotclearhowthebuyersshouldbidtomaximizetheirpayos,thispracticeplacesahugedecisionburdenonthebuyersanddetersbuyerparticipation.Consequently,thecurrentapproachmayfailtocapturethesurplusfromthetradesandhurtthewelfareofbothbuyersandsellers.Ascustomersaredrivenbytheirpayos,thiscurrentpracticemayalsoendangerthemarketmaker'srevenueinthelongrun.Littlethoughthasbeengiventoaidcustomers'decisionprocessesandfacili-tatethetransactions.Here,wefocusonthedoubleauctiondesignapproachesanddesigndoubleauctionmechanismstocounterspeculateandimprovesocialwelfare.Sinceself-interestedcustomerstypicallyknowmoreabouttheirpreferencesthanthemarketmaker,totacklethedesignproblem,wehavetoconsiderthegametheoryaspectofthesituation.Itisnotatrivialproblemtopredictcustomers'behavior,letalongtodesigntherightmechanismtoimprovesocialwelfare.Even 1

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2 inaverysimpletwo-sideexchangeenvironment,thecustomer-to-customeronlinemarketinwhicheachcustomerbuys/sellsoneunitofanitem,thedesignchallengeisfurthercomplicatedbytheheterogeneousnessofcustomers,whonotonlyhavevariousvaluationsbutmayalsoincurdierenttransactioncostsduetoshipping,handling,salestax,etc. Figure1{1:Buyers,Sellers,andPotentialTrades Figure 1{1 showsapotentialcustomer-to-customermarket.Theemptycirclesrepresentbuyers,andthesolidcirclesrepresentsellers,whilethelinesbetweenthebuyersandthesellersrepresentthepotentialtradeswithvariousshipping,handling,andsalestaxcosts.Weproposeanoveldoubleauctiondesignapproach,andconstructmechanismsapplicabletothetwofollowingexchangeenvironments: Simpleexchangeenvironmentwithtransactioncosts,and Bilateralexchangeenvironmentwiththesingleoutputrestriction.Chapter 2 reviewstheauctionandgametheoryliterature.Startingwithabriefintroductiontogametheory,wepresentsomemajorresultsonthemechanismdesignproblem.Acentralconceptinmechanismdesignistruthfulmechanism,in

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3 whicheachagenthasincentivetobidtruthfully.Wereviewthetruthfulmecha-nismsforone-sidedauction,wherewehavemultiplebuyersandasinglesellerwhoalsoactsastheauctioneer,andfordoubleauction,wherewehavemanybuyersandmanysellersandathird-partyauctioneer.Afterthestudyofauctionasanon-cooperativegame,wetakeadetourandexaminetheresultsontheassignmentgameinwhichthecurrentexchangeenvironmentismodelledasacooperativegamewithcompleteinformation.Theresultsundercooperativegamerendermeasuresofthebargainingpoweroftheagentsandshedlightonhowtosharethesurplusfairly.InChapter 3 ,weexaminethedoubleauctionapproaches.Duringthemech-anismdesign,weneedtomaketwodecisions:itheallocationdecision:howtheresourcesareallocated;iiandthepricingdecision:howmucheachcustomershouldpay/receive.Sinceitisunpromisingtodesignatruthfulmechanismunderthecurrentapproachforthesimpleexchangeenvironmentwithtransactioncosts,weproposethemulti-stagedesignapproach,inwhichwebeginwithapartialpricingdecision,thentheallocationdecision,andnallythepricingdecision.Thisdesignapproachenablesaprocedurethatisapplicabletotheenvironmentinwhicheachpotentialtransactionmayincurdierenttransactioncosts.InChapter 4 ,weproposetheagentcompetitiondoubleauctionAC-DAmechanismunderthemulti-stagedesignapproach.Withoutimposingdetailedstructureassumptionsontheformulationofthesocialwelfare,weshowthatthisresultingmechanismpossessesthedesiredproperties.Whenthesocialwelfaresatisesthecomplementarity-substitutabilityconditions[ 57 ],truthfulrevelationofpersonalvaluationisadominantstrategyforeachagentunderthisdoubleauctionmechanism.Furthermore,theauctioneerwillnotfaceadecitifthesocialwelfareformulationisalsoquasi-linear.

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4 Chapter 5 appliestheagentcompetitiondoubleauctionmechanisminthecurrentcustomer-to-customeronlineauctionmarketinwhichthecomplementarity-substitutabilityconditionsindeedhold.Wendthattheexistenceofthetrans-actioncostsmaysegmentthemarket,prohibitthetrades,andincuralossonsocialwelfare.However,thislosscanbealleviatedandbecomesnegligibleinalargemarket.Thus,astheauctionbecomeslarge,almostallfeasiblesurpluseswillberealized.Aftertheanalysisontransactioncosts,weprovideacomprehensivediscussiononfalse-namebids,self-reportofthetransactioncosts,andthetwo-parttariinthisonlineauctionmarket.InChapter 6 ,westudytheproblemofdesigningtruthfuldoubleauctionmechanismsfore-marketplaceswithmanybuyersandsellers,especiallytheindustrialprocurementsettingwheresellersaresmallinsizeandhavelittleornomarketpower.Sincethecomplementarity-substitutabilityconditions[ 57 ]nolongerholdinthisgeneralbilateralexchangeenvironment,weproposenewmechanismsthatarestrategy-proof,weaklybudget-balanced,individual-rational,andasymptoticallyecientunderthemulti-stagedesignapproach.Inaddition,oneofthemechanisms,achievesasymptoticeciencybyjustsolvingtwolinearprograms.ThisissignicantbecausetheeciencymaximizationproblemwithcompleteinformationisNP-hard.Chapter 7 recapitulatesthevariousexchangeenvironmentsofinterestanddetailstheimplementationofvariousdoubleauctionmechanismsproposedinthispaper.Afterreviewingthedoubleauctionmechanismsunderthetradereductionapproach,wecomparetheimplementation,applicability,eciency,andpayosofthemechanismsunderthesetwoapproaches,thetradereductionapproachandthenewlyproposedmulti-stagedesignapproach.Asshowninthisdissertation,themulti-stagedesignapproachisapplicabletodierentexchangeenvironmentswithsimplerimplementationsaswellashigher

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5 ecienciesandpayos.Thesechaptersprovideasolidfoundationforfutureresearchindoubleauctionmechanismdesign,andwealsomakeseveralsuggestionsforresearchdirectionsinChapter 8 .

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CHAPTER2LITERATUREREVIEWThemainpurposeofgametheoryistoconsidersituationswhereagents'decisionsarestrategicreactionstootheragents'actions.Gametheorycanberoughlydividedintotwobroadareas:non-cooperativeorstrategicgamesandco-operativeorcoalitionalgames.Non-cooperativegametheoryisconcernedwiththeanalysisofstrategicchoices,whileco-operativegametheoryinvestigateshowasuccessfulcoalitionshoulddivideitsproceeds.Sincethecustomer-to-customeronlinemarketistypicallyanonymousandanonymousnesspreventscoalitions,wewilldesigndoubleauctionmechanismsundernon-cooperativegametheoryframework.Theremainderofthischapterisorganizedasfollows.Section 2.1 focusesonnon-cooperativegametheoryandhowtomodelanauctionasanon-cooperativegame.StartingwithSection 2.1.1 ,wepresentabriefintroductiontogametheory.ThenwegooversomemajorresultsonthemechanismdesignprobleminSection 2.1.2 .Acentralconceptinmechanismdesignistruthfulmechanism,inwhicheachagenthasincentivetobidtruthfully.InSection 2.1.3 ,wereviewthetruthfulmechanismsforboththeone-sidedauctionandthedoubleauction.Afterthestudyofauctionasanon-cooperativegame,wetakeadetourinSection 2.2 andexaminetheresultsontheassignmentgameinwhichthesituationismodelledasacooperativegamewithcompleteinformation. 2.1 TheAuctionGameSimplystated,anauctionisamethodofallocatinggoodswithanexplicitsetofrulesdeterminingresourceallocationandpricesonthebasisofbidsfromthemarketparticipants. 6

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7 TheanalysisofauctionsasgamesofincompleteinformationoriginatedinCounterspeculation,Auctions,andCompetitiveSealedTenders"[ 60 ],inwhichWilliamVickreysuggestedusingsecond-priceauctiontocounterspeculateandimprovesocialwelfare.Vickreyshowsthatundersecond-priceauction,truthfulrevelationofpersonalvaluationisadominantstrategyforeachbidder.Becauseofthisproperty,hewasabletopredictagents'behaviorandclaimthatthisauctionachievesthemostecientsocialoutcome.FollowingVickrey,theauctionresearchfocusesonthefollowingquestions:Whataregoodbiddingstrategiesforthebiddersunderacertainauctionmechanism?Whataretheformsofauctionslikelytobringtheauctioneergreaterrevenuesthanothers?Andhowdoestheprivateinformationconstrainthewaysinwhichsocialdecisionscanrespondtoindividualpreferences?Beforeweinvestigatethesequestions,weneednotationandconceptsofthegametheory,thetheoryofrationalbehaviorforinteractivedecisionproblems. 2.1.1 AnIntroductiontoGameTheoryTobegin,consideranenvironmentwithsetofagents,orplayers,N=f1;2;;ng.TheseagentsmustmakeacollectivechoicefromsomesetXofpossibleoutcomes.Priortothechoice,however,eachagentiprivatelyobserveshispreferenceoverthealternativeinX.Formally,wemodelthisbysupposingthatagentiprivatelyobservesaparameter,orsignal,ithatdetermineshispreference,orutilityuix;iforx2X.Becauseiisobservedonlybyagenti,weareinasettingcharacterizedbyincompleteinformation.Wewillrefertoiastheagent'stype.Letidenotethesetofpossibletypesforagenti.Furthermore,letidenotethesetofagenti'spossibleactionsinthegame. Denition2.1.1 [Strategy]Astrategyforagentiisafunctionsi:i!i.

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8 Thenotionofstrategyisacentralconceptofgametheory.Astrategyisacompletecontingentplan,ordecisionrule,thatdenotestheactionanagentwillselectineverystateoftheworld.Adeterministicstrategyforagenti,whichwenowcallapurestrategy,speciesadeterministicchoicesiiforeachtype.LetSibethesetofagenti'spurestrategies.Onewayfortheagenttorandomizeistochooserandomlyoneelementfromthisset.Thiskindofrandomizationgivesrisetowhatiscalledamixedstrategy.Wewilloftennditconvenienttorepresentaproleofagents'strategychoicesinann-playergamebyavectors=s1;;sn,wheresiisthestrategychosenbyagenti.Wewillalsosometimeswritethestrategyprolesassi;s)]TJ/F23 7.97 Tf 6.587 0 Td[(i,wheres)]TJ/F23 7.97 Tf 6.586 0 Td[(iisthen)]TJ/F15 11.955 Tf 12.721 0 Td[(1vectorofstrategiesforagentsotherthaniandletS)]TJ/F23 7.97 Tf 6.587 0 Td[(idenotek6=iSk.Similarly,let)]TJ/F23 7.97 Tf 6.587 0 Td[(idenotethetypeofeveryagentexceptiand)]TJ/F23 7.97 Tf 6.587 0 Td[(i=k6=ik.Everyproleofstrategiesfortheagentss=s1;;sninducesanoutcomeofthegame.Thus,foranyproleofstrategiess,wecandeducethepayosreceivedbyeachagent.Letuis1;;sn;iuix;idenotetheutilityofagentiattheoutcomexofthegame,giventypeiandstrategiesproles=s1;;snselectedbyeachagent.AgameisspeciedintermsofstrategiessetSiandtheirassociatedpayosui.Thebasicmodelofagentrationalityingametheoryisthatofautilitymaximizer.Anagentwillselectastrategythatmaximizeshisutility,givenhistypei,beliefsaboutthestrategiesofotheragents,andstructureofthegame.Themostwell-knownsolutionconceptisthatofaNashequilibrium[ 44 ],whichstatesthatinequilibriumeveryagentselectsautility-maximizingstrategygiventhestrategyofeveryotheragent.

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9 Denition2.1.2 [Nashequilibrium]Astrategyproles=s1;;snisinNashequilibriumifforeveryi=1;;n,uisii;s)]TJ/F23 7.97 Tf 6.587 0 Td[(i)]TJ/F23 7.97 Tf 6.586 0 Td[(i;iuis0ii;s)]TJ/F23 7.97 Tf 6.587 0 Td[(i)]TJ/F23 7.97 Tf 6.586 0 Td[(i;iforalls0i6=siand)]TJ/F23 7.97 Tf 6.587 0 Td[(i2)]TJ/F23 7.97 Tf 6.587 0 Td[(i.InaNashequilibrium,eachagentplay'sstrategychoiceisabest-responsetothestrategyactuallyplayedbyhisopponents.ToplayaNashequilibrium,everyagenttypicallyneedsperfectinformationaboutthepreferencesofeveryotheragent.However,wearestudyinganincompleteinformationgame.Therefore,letusassumethattheagentsshareacommonknowledgeaboutthejointprobabilitydistribution1;;n,andinequilibrium,everyagentselectsastrategytomaximizeexpectedutilitygiventhestrategyofeveryotheragent.ThissolutionconceptiscalledBayesian-Nashequilibrium. Denition2.1.3 [Bayesian-Nash]ABayesian-Nashequilibriumisastrategyproles=s1;;snifforeveryagenti=1;;n,E)]TJ/F24 5.978 Tf 5.756 0 Td[(iuisii;s)]TJ/F23 7.97 Tf 6.587 0 Td[(i)]TJ/F23 7.97 Tf 6.586 0 Td[(i;iE)]TJ/F24 5.978 Tf 5.756 0 Td[(iuis0ii;s)]TJ/F23 7.97 Tf 6.587 0 Td[(i)]TJ/F23 7.97 Tf 6.587 0 Td[(i;iforalls0i6=siand)]TJ/F23 7.97 Tf 6.587 0 Td[(i2)]TJ/F23 7.97 Tf 6.586 0 Td[(i.InaBayesian-Nashequilibrium,eachagentmustbeplayingabest-responsetothedistributionofhisopponents'strategiesanddoesnotnecessarilyplayabest-responsetotheactualstrategiesoftheotheragents.Astrongersolutionconceptisthedominantstrategyequilibrium.Inadominantstrategyequilibrium,everyagentchoosesadominantstrategythatmaximizesutilityforallpossiblestrategiesofotheragents. Denition2.1.4 [Dominant-strategy]Astrategysi2Siisaweaklydominantstrategyforplayeriifforalls0i6=si,wehave

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10 uisii;s)]TJ/F23 7.97 Tf 6.587 0 Td[(i)]TJ/F23 7.97 Tf 6.586 0 Td[(i;iuis0ii;s)]TJ/F23 7.97 Tf 6.587 0 Td[(i)]TJ/F23 7.97 Tf 6.587 0 Td[(i;iforalls)]TJ/F23 7.97 Tf 6.586 0 Td[(i2S)]TJ/F23 7.97 Tf 6.586 0 Td[(iand)]TJ/F23 7.97 Tf 6.587 0 Td[(i2)]TJ/F23 7.97 Tf 6.587 0 Td[(i.Dominant-strategyequilibriumisaveryrobustsolutionconcept,becauseitmakesnodistributionassumptionsabouttheinformationavailabletotheagents,anddoesnotrequireanagenttobelievethatotheragentswillbehaverationallytoselecthisownoptimalstrategy. 2.1.2 TheMechanismDesignProblemInthissection,weprovideanintroductiontothemechanismdesignproblemthatstudieshowtheindividualpreferencescanbeelicited,andtheextenttowhichtheinformationrevelationproblemconstrainsthewaysinwhichsocialdecisionscanrespondtoindividualpreferences. Denition2.1.5 [Mechanism]Amechanism)-332(=S1;;Sn;gisacollectionofnstrategysetsS1;;Snandanoutcomefunctiong:S1Sn!X.Amechanismdenesthestrategiesavailable,andgovernstheprocedureformakingthecollectivechoice.Themechanism)-326(combinedwiththepossibletypes,thejointprobabilityfunctionoverthetypes,andutilityfunctionsuiinducesagameofincompleteinformation.Becausetheoutcomeofthegamedependsontherealizationsofagents'types=1;;n,weintroducethenotionofasocialchoicefunction. Denition2.1.6 [Socialchoicefunction]Socialchoicefunctionf:!Xchoosesanoutcomef2X,giventypes1;;n.Inotherwords,givenagenttypes,wemayliketochooseoutcomef.Themechanismdesignproblemistoimplementrulesofagame"toimplementthesolutiontothesocialchoicefunctiondespitetheagent'sself-interest.However,foragivensocialchoicefunctionf,anagentmaynotndittobeinhisbestinteresttorevealthisinformationtruthfully.Looselyspeaking,amechanism

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11 implementssocialchoicefunctionfifthereisanequilibriumofthegameinducedbythemechanismthatyieldsthesameoutcomesasfforeachpossibleproleoftypes=1;;n. Denition2.1.7 [Mechanismimplementation]Mechanism)-346(=S1;;Sn;gimplementssocialchoicefunctionfifthereisanequilibriumstrategypro-les1;;snofthegameinducedby)]TJ/F32 11.955 Tf 11.499 0 Td[(suchthatgs11;;snn=f1;;nforall1;;n21;;n.Onedesirablefeatureforasocialchoicefunctiontosatisfyisthepropertyofexposteciency. Denition2.1.8 [Paretooptimal]Thesocialchoicefunctionf:1n!XisParetooptimaliffornoprole=1;;nisthereanx2Xsuchthatuix;iuif;iforeveryi,anduix;i>uif;iforsomei.Inauctionenvironment,typicallywehavesidepaymentsamongagents.Averycommonassumptionisthatagentshavequasi-linearutilityfunctions. Denition2.1.9 [Quasi-linearpreferences]Aquasi-linearutilityfunctionforagentiwithtypeiisoftheform:uix;i=vix;i+tiwherevix;iisagenti'sutilitygiventheallocationxandnosidepayments,andtiisthemonetarytransfertoorfrom,ifti<0agenti.Underquasi-linearpreferences,thesocialchoicefunctionfisParetooptimalifandonlyiftheaggregatesocialwelfareismaximized.Asocialchoicefunctionisecientif: Denition2.1.10 [Eciency]Socialchoicefunctionf:1n!Xisecientiffornoprole=1;;nisthereanx2XsuchthatPni=1vix;i>Pni=1vif;i.

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12 Correspondingly,wesaythatamechanismisecientif: Denition2.1.11 [Ecientmechanism]Mechanism)]TJ/F32 11.955 Tf 11.499 0 Td[(isecientifitimplementsanecientsocialchoicefunctionf.Toinsureagents'participation,wemustsatisfytheinterimindividual-rationalityIRconstraints. Denition2.1.12 [Individualrationality]Amechanism)]TJ/F32 11.955 Tf 11.498 0 Td[(isinterimindividual-rationalifforallpreferencesiitimplementsasocialchoicefunctionfwithE)]TJ/F24 5.978 Tf 5.756 0 Td[(iuif;iuiiwhereuiiistheexpectedutilityfromnon-participation.Therearealsotwoothertypesofindividual-rationality.Inamechanisminwhichanagentcanwithdrawoncehelearnstheoutcome,expostindividual-rationalityismoreappropriate,inwhichtheagent'sexpectedutilityfrompartici-pationmustbeatleasthispayofromnon-participationforallagents'types.Inamechanisminwhichanagentmustchoosetoparticipatebeforeheevenknowshisownpreferences,exanteindividual-rationalityisappropriate,inwhichtheagent'sexpectedutilityinthemechanism,averagedoverallpossiblepreferences,mustbeatleasthisexpectedpayofromnon-participation.Expostindividual-rationalityisstrongerthaninterimindividual-rationality;whileinterimindividual-rationalityisstrongerthanexanteindividual-rationality.Intheauctionliterature,uiistypicallyassumedtobenormalizedtozero;therefore,wesayanauctionmechanismisindividual-rationaliftheexpectedutilityinthemechanismisnon-negative.Individual-rationalitydrawsthepotentialagentsintotheauction.Meanwhile,theauctioneeralsofacesmonetarytransfersandexpectsanon-negativepayo.Thisleadstothebudgetbalanceconstraint.Similartotheindividual-rationalproperty,wehaveexanteandexpostbudgetbalance.

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13 Denition2.1.13 [exantebudgetbalance]Mechanism)]TJ/F32 11.955 Tf 11.499 0 Td[(isexantebudget-balancediftheexpectedequilibriumnettransferstotheauctioneerarenon-negativeoverthejointdistributionofagentpreferences. Denition2.1.14 [expostbudgetbalance]Mechanism)]TJ/F32 11.955 Tf 11.499 0 Td[(isexpostbudget-balancediftheequilibriumnettransferstotheauctioneerarenon-negativeforallagentpreferences.Onelastimportantmechanismproperty,denedfordirectrevelationmecha-nisms,isincentivecompatibility. Denition2.1.15 [Directrevelationmechanism]AdirectrevelationmechanismisamechanisminwhichSi=iforalliandg=fforall21n.Inotherwords,inadirectrevelationmechanismthestrategyofagentiistoreporttype^i=sii,basedonhisactualpreferencesi.Letsii=idenotethetruth-revealingstrategyfori=1;;n. Denition2.1.16 [Incentive-compatible]Adirect-revelationmechanism)]TJ/F32 11.955 Tf 11.499 0 Td[(isincentive-compatibleifs1;;snisaBayesian-Nashequilibriumofthegameinducedbythemechanism.Inanincentive-compatiblemechanism,everyagenthasincentivetoreporthistruepreferencesifhebelievesthatallotheragentswouldalsoreporttheirtruepreferences.Astrongernotationiscalledstrategy-proofordominant-strategyincentive-compatibleiftruth-revelationisadominant-strategyequilibrium: Denition2.1.17 [strategy-proof]Adirect-revelationmechanism)]TJ/F32 11.955 Tf 11.498 0 Td[(isstrategy-proofifs1;;snisadominant-strategyequilibriumofthegameinducedbythemechanism.Dominant-strategyimplementationisaveryrobustconceptasitplacesnoassumptionontheagents'beliefsorthejointdistribution.Onemajorresultinmechanismdesignistherevelationprinciple[ 41 ],whichstatesthatunderquiteweakconditionsanymechanismcanbetransformedintoan

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14 equivalentincentive-compatibledirect-revelationmechanismthatimplementsthesamesocial-choicefunction.Therevelationprincipleenablesustofocusononlyincentive-compatibledirect-revelationmechanismsduringthemechanismdesign.Thisisnottosaythattruth-revelationiseasytoachieve,butthatifanindirect-revelationand/ornon-truthfulmechanismimplementsasocial-choicefunction,thenwewouldalsoexpectadirect-revelationtruthfulmechanismtodothesamething.Therevelationprinciplefacilitatesthetheoreticalanalysisofwhatispossible,andofwhatisimpossible,inmechanismdesign. Theorem2.1.18 Supposethatthereexistsamechanism)]TJ/F32 11.955 Tf 11.499 0 Td[(thatimplementsthesocial-choicefunctionfindominantstrategies.Thenfistruthfullyimplementableindominantstrategies. Theorem2.1.19 Supposethatthereexistsamechanism)]TJ/F32 11.955 Tf 11.499 0 Td[(thatimplementsthesocial-choicefunctionfinBayesianNashequilibrium.ThenfistruthfullyimplementableinBayesian-Nashequilibrium. 2.1.3 Strategy-proofAuctionMechanismsSinceitishardtoevaluatethejointtypedistributionandassertthateachcustomerholdsthesamebeliefsaboutthejointtypedistribution,wearemoreinterestedinstrategy-proofauctionmechanisms,inwhichnoassumptionisplacedontheagents'beliefsorthejointdistribution.Inthereminderofthischapter,letispecifythevaluationofagenti.Tofacilitatetherepresentation,weseparatethesocialchoicefunctionfintoaresourceallocationchoicecomponentxandsidepaymentcomponentspimadefromorto,ifpi<0agenti:f=x;p1;;pnforpreferences=1;;n.Underquasi-linearvaluation,wehaveuix;i=vix;i)]TJ/F22 11.955 Tf 12.536 0 Td[(piwherevix;iequalsiifagentiholdstheresourceinoutcomex,andzerootherwise.Theauctioneer'spayoisPni=1pi.

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15 Inthefollowing,wereviewthestrategy-proofauctionmechanismsforone-sidedauctionandfordoubleauction. 2.1.3.1One-sidedauctionmechanismsInanone-sidedauction,wehavemultiplebuyersandasingleseller.Theselleractsastheauctioneer,whilethebuyersaretheagents,ortheplayers,oftheauctiongame.Sometimes,asinanindustrialprocurementauctionenvironment,wemayalsohaveaone-sidedauctionwithmanysellersandonebuyer,whoactsastheauctioneer.Fortheone-sidedauction,Vickrey[ 60 ],Clarke[ 11 ],andGroves[ 19 ],proposetheVickrey-Clarke-GrovesVCGfamilyofmechanismsspecifyalltheecientandstrategy-proofdirect-revelationmechanisms.Sinceallthesepapersconcerngovernmentregulationandorganizationbehavior,andarenotconstrainedforatransactionenvironment,theVCGmechanismsdonotmaximizetheauctioneer'srevenue,butthesocialwelfare.However,whentheauctioneerhasonlyoneitemtosellandthebuyersindependentlydrawtheirtypesfromthesamedistribution,Myerson[ 42 ]showsthattheauctioneercanmaximizehisrevenuebysimplyaddingareserveprice,whichcanbeachievedbysubmittingactitiousbid.Therefore,weonlyfocusontheVCGmechanismsinthissection.Vickrey[ 60 ]suggestsusingasecond-priceauctiontocounterspeculateandimprovesocialwelfare.Consideranenvironmentwithonesellerandnbuyerswhowanttheitemoeredbytheseller.Inasecond-priceauction,thebidderwhosubmittedthehighestbidisawardedtheobjectbeingsoldandpaysapriceequaltothesecondhighestamountbid.Thesecond-priceauctioncanbeimplementedinasealedbidfashion;thatis,eachagentsubmitshisbidinasealedenvelopeanddoesnotknowothers'bids;or,thesecond-priceauctioncanbeimplementedinaopenbidfashion,whereagentsoutbidseachotheruntilthebidpricestopsbecause

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16 thepriceshouldstopatthesecondhighestvaluationplussomesmallpositiveamount.Itisstraightforwardtoseethatindividual-rationalityholdsbecausethesecond-highestbidpriceisnogreaterthanthehighestbidprice,whichequalsthewinneragent'svalueinequilibrium.Also,truth-revelationisadominantstrategyintheVickreyauctionbecauseanagent'sbiddetermineswhetherhewinstheitem,butnottheactualpricehepays.Thepricethatanagentpaysiscompletelyindependentofhisbidprice,andevenifanagentknowsthesecond-highestbid,biddinghistruevalueisstillinhisbestinterestasheisfacingatake-it-or-leave-it"oer.Furthermore,notethatthestrategy-proofnessholdswithoutquasi-linearutilityassumption,andthequasi-linearutilityassumptionisonlyusedtoguaranteethattheallocationisecient.TheClarkemechanism[ 11 ]generalizestheideaofVickreyauctionandstudyageneralenvironmentwithquasi-linearutilityassumption.InaClarkemechanism,theallocationistheecientallocation,whileeachagent'spayoissettoequalhismarginalcontributiontothecoalitionofthewhole.Thatis,foranagentk,hispayoiswN)]TJ/F22 11.955 Tf 12.201 0 Td[(wNnfkg,whereNistheagentsetandwspeciestheworthofthecoalitions.Sincemoreagentsenlargetheregionofthefeasibleallocations,eachagent'spayoisnon-negative.Thatis,individual-rationalityholds.Underquasi-linearassumption,thepriceeachtradingagentpaysorreceivesisthecriticalpriceonwhichheisinvolvedintheoptimalallocationandisindependentofhisownbidprice.Therefore,truth-revelationisadominantstrategy.Thecriticalprice,whichequalsbk)]TJ/F15 11.955 Tf 12.583 0 Td[(wN)]TJ/F22 11.955 Tf 12.582 0 Td[(wNnfkgfortradingagentkwithbidpricebk,iscalledtheVCGprice.Sincetheallocationisecient,theClarkemechanismisecient.Inasense,theClarkemechanismisanexpostindividual-rational,strategy-proof,andecientmechanismthatalsomaximizesthepaymentsmadebytheagentstotheauctioneer.

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17 Groves[ 19 ]considerscoordinationofagentswithquasi-linearpreferencesinanorganization,whereui,theexpectedutilityfromnon-participationmaybedierentfromzero.Therefore,thoughtheGrovesmechanismalsoimplementstheecientallocation,agentk'spayomaydierfromhismarginalcontributiontothecoalitionofthewholebysomefunctionhk^)]TJ/F23 7.97 Tf 6.586 0 Td[(k,where^)]TJ/F23 7.97 Tf 6.587 0 Td[(karethereportedtypesbyallagentotherthank.Sincehk^)]TJ/F23 7.97 Tf 6.586 0 Td[(kisindependentfromagentk'sreport,truth-revelationisadominantstrategy.TheGrovesmechanismscapturethestructureofallecient,strategy-proofdirect-revelationmechanismsforagentswithquasi-linearpreferences. Theorem2.1.20 TheGrovesmechanismsaretheonlyecientandstrategy-proofmechanismsforagentswithquasi-linearpreferencesandgeneralvaluationfunctions,amongstalldirect-revelationmechanisms. 2.1.3.2DoubleauctionmechanismsUnliketheone-sidedauction,inadoubleauction,wehavemultiplebuyersandsellers,hereafterbothcalledagents,andathird-partyauctioneerwhoholdstheauction.Thesimplestexchangeenvironmentfordoubleauctionisthesimpleexchangeenvironment. Denition2.1.21 [simpleexchange]Asimpleexchangeenvironmentisoneinwhichtherearebuyersandsellers,sellingsingleunitsofthesamegood.Wehaveseenthatintheone-sidedauction,theseller,ortheauctioneercanachievesocialeciency.However,underadoubleauction,wehavetomakesurethatthebudgetbalanceconstraintisalsosatised.Becausetheagentsareself-interested,weshouldnotexpectanecientallocationingeneral.Hurwicz[ 24 ]showsthatitisimpossibletoimplementanecient,budget-balanced,andstrategy-proofmechanisminasimpleexchangeenvironment.ThisresultisfurtherstrengthenedbyMyersonandSatterthwaite[ 43 ].Theyshowedtheimpossibilityof

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18 havinganecient,individual-rational,incentive-compatible,andbudget-balancedmechanism.Atrstglimpse,onemaygiveupthedoubleauctionapproachandresorttoone-sidedauctionmechanismsbecauseoftheirtheoreticeciencyresult.However,thiseciencypropertyisachievedwithassumptionsandcompromises.First,inaVCGmechanism,thesellermayselltheitemunderhisvaluation,thatis,thesellermaygiveawaytoomuchofhisprottoinducethebuyerstoreporttheirtruevaluations.Thissituationalsohappensinaprocurementauctionwheretheauctioneermaypaytoomuch.Second,thesellermayhavelittleincentivetoimplementaneciency-maximizingmechanismbecausehemaytrytomaximizehisownrevenueinsteadofthesocialwelfare,thatis,theincentivesforeciencyandrevenuearenotaligned.Ontheotherhand,inadoubleauctionenvironment,itisrecognizedthateciency-maximizingmechanismsthatbenetthewholesystemaremorelikelytoattractbothbuyersandsellersandgeneratehigherrevenuesfortheauctioneersorthee-marketplacesinthelongrunMilgrom[ 39 ],WiseandMorrison[ 64 ];therefore,theincentiveforrevenueisalignedwithsocialwelfare.Moreover,thebudgetbalanceconstraintpreventstheauctioneerfromoverpayingtheagentstoachieveeciencyandforcesustodesignamechanismwithfairorreasonabletransactionprices.Thereareveryfewstrategy-proofdoubleauctionmechanismsintheliterature.McAfee[ 33 ]showsamechanismthatisstrategy-proof,budget-balanced,andasymptoticallyecientinthesimpleexchangeenvironment.Asymptoticeciencymeansthatthewelfarelossunderthemechanismcomparedtothemaximumfeasiblesocialwelfareconvergesto0asthenumberofbuyersand/orsellersapproachinnity;thus,astheauctionbecomeslarge,almostallfeasiblesurpluswillberealized.InMcAfee'smechanism,allthebuybidpricesarerankedfromhightolow,f1f2fm;andallthesellbidpricesarerankedfrom

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19 lowtohigh,g1g2gn.Letqbetheoptimaltradesize,themaximumnumbersatisfyingfqgq,whichisthenumberoftradesintheecientallocations.McAfee'smechanismmakeseitherqorq)]TJ/F15 11.955 Tf 11.529 0 Td[(1transactiondependingonthefollowingcondition:iIffqfq+1+gq+1 2gq,qtransactionsaremadebetweenthehighestqbuybidsandthelowestqsellbids,thetransactionpricesarefq+1+gq+1 2foralltradingagents.2Otherwise,q)]TJ/F15 11.955 Tf 12.181 0 Td[(1transactionsaremadebetweenthehighestq)]TJ/F15 11.955 Tf 12.181 0 Td[(1buybidsandthelowestq)]TJ/F15 11.955 Tf 12.448 0 Td[(1sellbids,thetransactionpricesarefqforalltradingbuyersandgqforalltradingsellers. Figure2{1:PossibleTransactionsunderMcAfee'sMechanism Notethatinfactwecanhaveastrategy-proof,budget-balanced,andasymp-toticallyecientmechanismwithsimplerstructurebyalwaysconductingq)]TJ/F15 11.955 Tf 12.615 0 Td[(1tradesatbuypricefqandsellpricegq.BabaioandWalsh[ 7 ]summarizethisapproachasthetradereductionapproach,inwhichtheallocationdecisioniscon-structedbyremovingtradesfromtheecientallocation,andthepricingdecisionismadeaccordingtothehighestandlowestlosingbids.Accordingtothisapproach,

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20 theyproposeaknownsingle-mindedtradereductionKSM-TRmechanism,whichisstrategy-proof,weaklybudget-balanced,andindividual-rationalforaclassofsupplychainformationproblemwherebuyersmayacquireabundleofdierentitemsinsteadofsingleunitofthesameitem.ThisapproachisalsousedinBabaioetal.[ 6 ],inwhichtheystudiedasimpleexchangeenvironmentwithmultiplelocationsandtransportationcosts.Whenabuyertradesaselleratadierentlocation,alocationdependenttransactioncostincurs.Forthisenvironment,theyproposeatradereductionmechanismTRM,whichisstrategy-proof,individual-rational,andbudget-balanced.TheTRMremovesfewertransactionsfromtheecientallocationcomparedtoanaiveapplicationofthetradereductionapproach.Inadierentresearchdirection,Huangetal.[ 23 ]examineanexchangeenvironmentinwhichtherearebuyersandsellers,exchangingmultipleunitsofthesamegood.Byrankingallthebidprices,theygeneralizetheresultofMcAfeeanddesignadoubleauctionmechanismthatisstrategy-proof,budget-balanced,andindividual-rationalforthemulti-unitexchangeenvironment. 2.2 TheAssignmentGameInthissection,wetakeadetourandvisitsomeclassicresultsoftheassign-mentgame.Themajordierencebetweenourcurrentsettingandanassignmentsgameisthattheassignmentgameisacooperativegamewithcompleteinforma-tion.Atthispointitwouldbemoreconvenienttogivetheformaldenitionofthecooperativegame. Denition2.2.1 [Cooperativegame]Agameincoalitionalformconsistsof1asetNtheagents,and2afunctionv:2N!Rsuchthatv;=0.AsubsetofNiscalledacoalition;vSiscalledtheworthofthecoalitionS.

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21 AssignmentgameswereintroducedbyShapleyandShubik[ 58 ]asamodelforabilateralmarketwithtransferableutility.TheagentsetconsistsoftheunionoftwonitedisjointsetsI[J,whereIisthesetofbuyersandJisthesetofsellers.Givenanon-negativematrixA=aiji;j2IJ,whereaijisthejointprotthatthepairi;jcanobtainiftheytrade,wecandeneacooperativegamebydeterminingtheworthofcoalitionS[T,whereSIandTJ.LetUS;TbethesetofmatchingsbetweenSandT,denewS;T=maxfPi;j2aijj2US;TgandwS;;=w;;T=0.ThematchingprobleminducesacooperativegameI[J;w.Sinceanauctionmechanismneedstospecifyanallocationandthepayos,weinvestigatethepointsolutionconceptsforthisgame.Astheassignmentgameisacooperativetransferableutilitygame,theagentssharetheworthofthewholeaggregationinallthesolutionconcepts,becauseotherwisethesolutionisnotParetooptimalandwecanimprovetheagents'payos.TheimpossibilityresultinSection 2.1.3 showsthatanindividual-rational,budget-balancedmechanismdoesnotalwaysachievetheecientallocations;thus,weshouldnotexpectamechanismtoimplementanysolutionconceptoftheassignmentgame.Nevertheless,thesesolutionconceptsrendermeasuresofthebargainingpoweroftheagentsandprovideinsightonhowtosharethesurplusfairly.ThewellknownsolutionconceptsforpointsolutionsaretheShapleyvalue[ 56 ]andthenucleolus[ 55 ].Wedeneandinvestigatetheseconceptsonebyone: Denition2.2.2 [Substitute]iandj,elementsofN,aresubstitutesinvifforallScontainingneitherinorj,vS[fig=vS[fjg. Denition2.2.3 [Nullagent]AnelementiofNiscalledanullagentifvS[fig=vSforallSN. Denition2.2.4 [EN]ENistheEuclideanspacewhosedimensionisthecardinal-ityofN,andwhosecoordinatesareindexedbythemembersofNthemselves.

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22 Denition2.2.5 [Shapleyvalue]LetN=f1;2;;ngandletGNbethesetofallgameswhoseagentsetisN.AShapleyvalueonNisafunction:GN!ENsatisfyingthefollowingconditions:1.Symmetrycondition:ifiandjaresubstitutesinvthenvi=vj.2.Nullagentcondition:ifiisanullagent,thenvi=0.3.Eciencycondition:PNi=1vi=vN.4.Additivitycondition:v+wi=vi+wi.Shapley[ 56 ]establishstheexistenceanduniquenessoftheShapleyvalue.LetuschecktheShapleyvalueforasimpleassignmentgame.Consideramarketwithonesellerandtwobuyersand3,wherewf1;2;3g=wf1;2g=wf1;3g=1andwS=0forallotherSf1;2;3g.Wehavew=2 3;1 6;1 6.Thisallocationofthepayounderthecooperativegameiscountertotheintuitionfromthenon-cooperativegame,whereweexpectthecompetitionofthebuyerstoreducetheirpayotozero,andthesellertocaptureallthesurplus.ThisisamajordrawbackoftheShapleyvalueasitmaylieoutsidethecore,ageneralsolutionconceptofthecooperativegames.Beforewegivethedenitionofthecore,letusrstexaminethenucleolus. Denition2.2.6 [Payovector]ApayovectorisamemberofENsuchthatPi2Nxi=vN.LetxSdenotesPi2Sxi. Denition2.2.7 [x]ForgameN;v,letxbeavectorinR2jNj,thecompo-nentsofwhicharethenumbersvS)]TJ/F22 11.955 Tf 11.972 0 Td[(xS,arrangedaccordingtotheirmagnitude,whereSrunsoverthesubsetsofN. Denition2.2.8 [Nucleolus]ThenucleolusofthegameN;visthesetofallxsuchthatxyaccordingthelexicographicalorderonR2jNjforally.WemaythinkvS)]TJ/F22 11.955 Tf 12.642 0 Td[(xSasameasureofdissatisfactionofacoalitionS,andthenucleolusisthesetofpointsthatminimizestheoveralldissatisfactionin

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23 alexicographicalfashion.Schmeidler[ 55 ]provesthatthenucleolusisnonempty,andthenucleoluscontainsonlyonepoint.Forthepreviousexample,thenucleolusoersthedesiredoutcome;0;0.However,thenucleolusingeneralisnoteasytocomputeandoersverylittleinsightoneachagent'spayo.Here,insteadofexaminingthepointsolutionconcepts,weanalyzethestructureofthecore,thegeneralsolutionconceptofthecooperativegames. Denition2.2.9 [Core]ThecoreofthegameN;visthesetofallthepayovectorsxsuchthatvSPi2SxiforallSN.Thecoreisthesetofallocationsthatcannotbeimproveduponbyanycoalition.ShapleyandShubik[ 58 ]provethatthecoreoftheassignmentgameI[J;wisnonempty,andSolymosiandRaghavan[ 59 ]showthatthenucleolusisalwaysacoreallocationintheassignmentgame.LetCwdenotethecoreoftheassignmentgameI[J;w.CwcanberepresentedintermsofanoptimalmatchinginI[J.Moreover,ifwedenote ui=maxfuiju;v2Cwg,andu i=minfuiju;v2Cwgforalli2I,anddenote vi=maxfviju;v2Cwg,andv i=minfviju;v2Cwgforallj2J.Ithappensthatallagentsonthesamesideofthemarketachievetheirmaximumcorepayosinthesamecoreallocation.Asaconsequence,therearetwospecialextremecoreallocations:inoneofthem u;v ,eachbuyerachieveshismaximumcorepayoandintheother,u ; v,eachsellerdoes.Fromamathematicalprogrammingviewpoint,themaximumandminimumcorepayosofanagentishismaximumandminimumshadowpricesinthelinearrelaxationformulation.Leonard[ 25 ]provesthatthismaximumpayoofanagentinthecoreoftheassignmentgameishismarginalcontribution, ui=wI[J)]TJ/F22 11.955 Tf 12.081 0 Td[(wInfig[Jforalli2Iand vj=wI[J)]TJ/F22 11.955 Tf 12.114 0 Td[(wI[Jnfjgforallj2J.Thatis, u; varetheVCGpayosoftheagents.

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24 Ontheotherhand,theminimumpayosalsofolloweasily.Ifweintroduceanadditionbuyerisuchthataij=aijforallj2J,andenlargewtowaccordingtoAI[figJ,u i=wI[fig[J)]TJ/F22 11.955 Tf 12.208 0 Td[(wI[J.SinceiandiaresubstitutesintheenlargedgameI[fig[J;w,theminimumpayosindicateswhetheriisabletoacquireonemorematching.Similarly,v j=wI[J[fjg)]TJ/F22 11.955 Tf 12.33 0 Td[(wI[Jforallj2J,wherejisanadditionsellersuchthataij=aijforalli2I.Theseminimumpayoscanbeusedinthepricingdecisioninthelatermulti-stagedesignapproach.

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CHAPTER3MULTI-STAGEDESIGNAPPROACHTherearetwodecisionsanauctionmechanismneedstomake:theallocationdecisionandthepricingdecision.Mechanismdesignistoconstructasolutionforthesedecisionstoachievedesiredproperties.Inthischapter,weexaminethepossibledoubleauctiondesignapproachesonthesetwodecisions.Theremainderofthischapterisorganizedasfollows.Section 3.1 summarizestheexistingdesignapproaches.Section 3.2 pointsoutwhythesedesignapproachfallsshortoftheneedunderthesimpleexchangeenvironmentwithtransactioncosts.Section 3.3 proposesthemulti-stagedesignapproachasadesignsolutiontoourproblem. 3.1 ExistingApproachTheexistingstrategy-proofdoubleauctionmechanismsrstmakethealloca-tiondecision,andthenthepricingdecisionfollows.Inordertobestrategy-proofandbudget-balanced,givenagents'bids,thesemechanismsselectasubsetoftradesfromtheecientallocation,whicharegenerallyachievedbyremovingtheleastprotabletradesfromtheecientallocation.Sinceeachagenthasotheragentswhoareperfectsubstitutesinthesemodels,thosebidsintheremovedtradesbecomereferenceprices,andthepricingdecisionismadebysettingthetrans-actionpricesequaltothereferenceprices.ThisisthetradereductionapproachsummarizedbyBabaioandWalsh[ 7 ].ThisapproachisextendedinBabaioetal.[ 6 ],wheretheystudiedasimpleexchangeenvironmentwithmultiplelocationsandtransportationcosts.Theyshowedthatinsomecasesthereferencepricescanbecalculatedfromthebidprices 25

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26 insteadofsettingequaltosomelosingbidprices.Therefore,fewertradescanberemovedandhighereciencycanbeachieved.Anotherintuitiveapproachistomakethepricingdecisionrst.Ifwecancalculateathresholdpriceforeachbuyersuchthatshewinsanitemifsheoutbidsthisthresholdpriceandlosesifsheunderbids,andviceversafortheseller,thentheresultingmechanismisstrategy-proof.AnaiveexampleistosetallthethresholdpricesequaltotheVCGprices,andthenthenalallocationistheecientallocation.However,underthiscase,theauctioneergenerallywillhaveabudgetdecit.Sinceweknowtheallocationisecient,wecanalsosaythatallocationdecisionismaderstinthisexample. 3.2 DrawbacksoftheExistingApproachesThetradereductionapproachworkswellwhenagentsareperfectsubstitutes.However,whenagentsareheterogeneous,itisnotclearhowtoremovetradesfromtheecientallocationorcalculatethereferencepricesunderthetradereductionapproach.Eventhoughthemechanismby[ 6 ]successfullyhandledthelocationdependentcosts,itmakesnotransactionifeachtrademayincurdierentcosts.Therefore,itisunpromisingtousethetradereductionapproach,orrstgureoutanallocationdecisiontodesignamechanismforanenvironment,inwhichwemayhavevariouscostduetoshipping,handling,salestax,etc.Ontheotherhand,noknownmechanismrstmakesthepricingdecision,thentheallocationdecision,sinceittypicallyleadstoasupply/demandvolumemismatcheveninthesimpleexchangeenvironment.Moreover,asagentsareheterogeneousandtransactioncostsvary,itisnotclearhowamechanismcandeterminethebuyingorsellingpricesforeachindividualagentandguaranteebudgetbalancewithhigheciencywithoutinformationontheallocationdecisionandhowmucheachtransactioncosts.Forthesereasons,ifwemakethepricing

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27 Figure3{1:DiagramforMulti-StageApproach decisionrst,theresultingstrategy-proofbudget-balancedmechanismisunlikelytohavesuperioreciency.Therefore,thesetwointuitiveapproachesfailtoprovideasatisfactorysolutionforthecurrentsetting.Analternativeisneeded. 3.3 Multi-StageDesignApproachHere,weproposeanovelmulti-stageapproachasshowninFigure 3{1 .Underthisapproach,amechanismbeginswithapricingdecisionforonesideandeliminatessomeagentsfromtheauction,thenmakestheallocationdecision,andnalizesthepricingdecisionfortheothersidesubsequently.Forexample,amechanismcanrstsetthethresholdprice/transactionpriceforeachbuyerandremovethebuyerswhobidlowerthantheirthresholdprices,thatis,rstmakethepricingdecisiononthebuyerside.Then,theallocationdecisioncanbemadebychoosinganecientallocationamongtheremainingbuyersandtheoriginalsellers.Finally,themechanismmakesthepricingdecisiononthesellerside,thatis,setsthetransactionpricesforeachtradingseller.Theideaistoselectasmallgroupofsellerswhoarehighlycompetitiveduringtherststage,suchthatwhen

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28 wetakeanecientallocationamongoriginalbuyersandthesesellers,allofthesellersgettransaction.Andhopefully,thetransactionpricesforthewinningbuyerscanbedeterminedbythecompetitionsamongbuyersinthenalallocation,sothatbuyershavenoincentivetospeculateeventhoughtheirbidscaninuencethesellers'thresholdpricesandconsequentlythenalallocationasshownintheFigure 3{1 .Duringtherststageofthedesign,wefaceatrade-o.Ontheonehand,thelowerthethresholdpricesforsellersintherststage,themorelikelythisapproachwillwork.Ontheotherhand,thelowerthethresholdpricesforsellersintherststage,thelesstheallocationeciencywillberealized.Aswewillshownext,bysettingthesethresholdpriceswisely,strategy-proof,budget-balanced,andindividual-rationalmechanismscanbeobtained.Thene-tunedpricingdecisiononthesellersidesecuresanon-negativepayofortheauctioneer,guaranteestransactionsforcompetitivesellers,anddetersspeculationsofbuyersalthoughtheirbidscaninuencethesellers'thresholdpricesandconsequentlythenalallocation.

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CHAPTER4AGENTCOMPETITIONDOUBLEAUCTIONMECHANISMInthischapter,weproposetheagentcompetitiondoubleauctionAC-DAmechanismunderthemulti-stagedesignapproach.WeshowthattheAC-DAmechanismisstrategy-proof,expostindividual-rational,andexpostbudget-balancedifthequasi-linearsocialwelfaresatisesthecomplementarity-substitutabilityconditions[ 57 ].Theremainderofthischapterisorganizedasfollows.Section 4.1 denesthemodelandrelatednotation.Section 4.2 proposesthetwoversionsoftheagentcompetitiondoubleauctionAC-DAmechanism:thebuyercompetitionmechanismandthesellercompetitionmechanism.Section 4.3 showsthattheAC-DAmechanismisstrategy-proof,expostindividual-rational,andexpostbudget-balancedundermildconditions.TheproofsforSection 4.3 arecompiledinSection 4.4 . 4.1 ModelandNotationLetIdenotethegroupofbuyers,andJthegroupofsellers,hereafterbothcalledagents.Wealsorefertoabuyerasshe"andasellerashe".Letusconsiderageneralbilateralexchangeenvironment,whereeachbuyertriestoacquireacertainbundleandeachsellertriestosellacertainbundle.Weassumeaprivatevaluemodel,thatis,eachagentknowshis/hervaluationofthebundleofinterest,butnotothers'.Ifweassumequasi-linearpreferences,whereanagent'sutilityisthedierencebetweenhis/hervaluationoftheitemhe/shesellsbuysandtheamountofmoneyhe/shereceivespays,thenwecandeneaquasi-linearsocialwelfare,whichisthesummationoftheauctioneer'spayoandeachindividualagent'sutility.Here,weallowthepossibilityofotherformsof 29

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30 socialwelfareandadoptaweakerassumption:thefunctionvalueofthesocialwelfareiszeroifthereisnotransaction,andforeachofotherpossibleallocations,thesocialwelfarefunctionisanadditivemeasureoverfunctionsofindividualagent'sbid.Notethatthequasi-linearsocialwelfaresatisesthiscondition.Considerthemaximumsocialwelfareamongallthepossibleallocations,weareinterestedintheexchangeenvironmentswhosemaximumsocialwelfaresatisesthecomplementarity-substitutabilityconditionsinShapley[ 57 ]: Property1 Ifkandl2Iaretwodierentbuyers,thenV+Vfk;lgV)]TJ/F23 7.97 Tf 6.587 0 Td[(k+V)]TJ/F23 7.97 Tf 6.586 0 Td[(l. Property2 Ifkandl2Jaretwodierentsellers,thenV+Vfk;lgV)]TJ/F23 7.97 Tf 6.586 0 Td[(k+V)]TJ/F23 7.97 Tf 6.587 0 Td[(l. Property3 Ifk2Iisabuyerandl2Jisaseller,thenV+Vfk;lgV)]TJ/F23 7.97 Tf 6.586 0 Td[(k+V)]TJ/F23 7.97 Tf 6.587 0 Td[(l.whereVisthemaximumfeasiblesocialwelfare,V)]TJ/F23 7.97 Tf 6.586 0 Td[(histhemaximumfeasiblesocialwelfarewithoutagenthforh2I[J,andVfk;lgisthemaximumfeasiblesocialwelfarewithoutagentkandl.Property 1 meansthatbuyersaresubstitutesforeachother.Similarly,Property 2 saysthatsellersaresubstitutesforeachother.Property 3 ,thecom-plementarycondition,impliesthatthecontributionfromasysteminwhichonlynewbuyersareallowedtoenterplusthecontributionfromasysteminwhichonlynewsellersareallowedtoenterislessthanthecontributionfromasysteminwhichbothbuyersandsellersareallowedtoenter.Inthispaper,weusethetermbid"todenotebothabuyerandaseller'sdeclaration.Beforeintroducingthemechanisms,weneedtodenethefollowingnotation. Notation fi Thebidpriceofbuyeri,i2I. gj Thebidpriceofsellerj,j2J. VI0;J0 ThemaximumfeasiblesocialwelfareregardingtothebidsofbuyersetI0andsellersetJ0.

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31 V)]TJ/F23 7.97 Tf 6.587 0 Td[(kI0;J0 ThemaximumfeasiblesocialwelfareregardingtothebidsofagentsetI0[J0nfkgk2I0[J0. VkI0;J0 ThemaximumfeasiblesocialwelfareregardingtothebidsofagentsetI0[J0andonemoreagentwhoisidenticaltoagentkk2I0[J0. p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(kI0;J0 TheinmumsupremumofbidpricesofbuyersellerksatisfyingVI0;J0>V)]TJ/F23 7.97 Tf 6.586 0 Td[(kI0;J0. p+kI0;J0 TheinmumsupremumofbidpricesofbuyersellerksatisfyingVkI0;J0>VI0;J0.Noteifagentkisabuyer,whenwesaythatthereisonemoreagentwhoisidenticaltoagentk,itmeansthatwehaveanadditionalbuyerwhoacquiresthesamebundleasagentk'sandsubmitsabidpricethesameasagentk's.Furthermore,whensellerstradewiththisadditionalbuyer,theyincurthesamecostsorbenetsaswithagentk.Similarly,ifagentkisaseller,whenwesaythatthereisonemoreagentwhoisidenticaltoagentk,itmeansthatwehaveanadditionalsellerwhodisposesthesamebundleasagentk'sandsubmitsabidpricethesameasagentk's.Furthermore,whenbuyerstradewiththisadditionalseller,theyincursamecostsorbenetsaswithagentk.Forsimplicityofrepresentation,wemaydroptheparametersI0;J0whenthereferencestothebuyersetandthesellersetareobvious.Also,inthedenitionofp+kandp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,theinmumofanemptysetispositiveinnity,andthesupremumofanemptysetisnegativeinnity.Morecommentsonp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kandp+kareinorder.Essentially,p)]TJ/F15 11.955 Tf 10.987 1.793 Td[(isthepriceatwhichanagentisabletotradeinsomeecientallocation.Inparticular,whenwehavethequasi-linearutilityassumption,p)]TJ/F15 11.955 Tf 10.987 1.794 Td[(istheagent'sVCGprice.Thereasonthatweadoptthecurrentdenitionsistobeabletospecifyp+andp)]TJ/F15 11.955 Tf 10.987 1.793 Td[(evenifwedonothavethequasi-linearutilityassumption.IfweemployanecientallocationandcompletethetransactionsattheVCGprices,theauctioneerwilllikelyfacea

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32 decit.Toavoidthis,theauctioneerneedstoretainsomeearningsfromtheagents.Becauseofthis,wedenep+,whichisthepriceatwhichanagentisabletosellbuyonemoreidenticalbundleinsomeecientallocation.Theauctioneermayretainsomeearningsbylettingallorpartoftheagentstradeatp+,insteadofp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(.Underthecomplementarity-substitutabilityconditions,p+isgreaterthanorequaltop)]TJ/F15 11.955 Tf 10.986 1.794 Td[(foreachbuyer,andp+islessthanorequaltop)]TJ/F15 11.955 Tf 10.987 1.794 Td[(foreachseller.Thisisintuitivesincep+isderivedunderanenvironmentwithmorecompetition. 4.2 BuyerCompetitionMechanismandSellerCompetitionMechanismNowwepresentourdoubleauctionmechanism.Weletthep+sbethethresh-oldpricesforthepricingdecisionofonesideofthemarket.Sincethesepricesareobtainedthroughctitiouscompetition,wecallthemechanismtheAgentCom-petitionDoubleAuctionAC-DAMechanism.Weprovidetwodierentversions,asellercompetitionmechanismandabuyercompetitionmechanism. BuyerCompetitionBCMechanism: { Eachagentsubmitsonesealedbid. { Forbuyeri2I,ifherbidfiislessthanp+iI;J,sheiseliminatedfromtheauction.Let~Idenotethesetofremainingbuyers,fijfip+iI;J;i2Ig. 1 { Theitemsareallocatedamongtheremainingagents~IandJinthemostecientway. { Thetradingbuyerkpaysp+kI;J,andthetradingsellerlreceivesp)]TJ/F15 11.955 Tf 7.084 1.794 Td[(l~I;J. SellerCompetitionSCMechanism: 1Wemayalsoeliminatebuyeriifherbidisnomorethanp+i,thatis,weset~I=fijfi>p+iI;J;i2Ig.Theresultingmechanismmaintainstheproperties,andshouldberecognizedasavariantofthebuyercompetitionmechanism.

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33 { Eachagentsubmitsonesealedbid. { Forsellerj2J,ifhisbidgjisgreaterthanp+jI;J,heiseliminatedfromtheauction.Let~Jdenotethesetofremainingsellers,fjjgjp+jI;J;j2Jg. 2 { TheitemsareallocatedamongtheremainingagentsIand~Jinthemostecientway. { Thetradingbuyerkpaysp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kI;~J,andthetradingsellerlreceivesp+lI;J. 4.3 TheoremsDuetothesymmetry,weuseAC-DAmechanismtorefertoboththesellercompetitionmechanismandthebuyercompetitionmechanism.Surprisingly,withoutanyrestrictionontheformulationofthesocialwelfare,wehavetherstsetofourmajortheorems: Theorem4.3.1 TheAC-DAmechanismisexpostindividual-rationalifthesocialwelfaresatisesthecomplementarity-substitutabilityconditions. Theorem4.3.2 TheAC-DAmechanismisstrategy-proofifthesocialwelfaresat-isesthecomplementarity-substitutabilityconditions. Theorem4.3.3 TheAC-DAmechanismisexpostbudget-balancedifthesocialwelfaresatisesthecomplementarity-substitutabilityconditionsandisnomorethanthecorrespondingquasi-linearsocialwelfare. 2Wemayalsoeliminatesellerjifhisbidisnolessthanp+j,thatis,weset~J=fjjgj
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34 Ifwehaveabilateralexchangemarketinwhichtheagentsonthesamesidearesubstitutes,andtheagentsonthedierentsidesarecomplements,thenTheorems 4.3.1 and 4.3.2 implythatwecanconstructastrategy-proofandexpostindividual-rationaldoubleauctionmechanism.Thestrategy-proofnessandindividual-rationalitypropertiesdonotrequirethateachagenthasaquasi-linearutility,aseachofthemisfacingatake-it-or-leave-it"oer.Furthermore,ifthesocialwelfarefunctionisquasi-linearorisnomorethanthecorrespondingquasi-linearsocialwelfare,Theorem 4.3.3 guaranteesthemechanismisexpostbudget-balanced.Duetothelengthesoftheproofs,theyaredeferredtothenextsection. 4.4 ProofsWeonlyprovetheresultsforthesellercompetitionmechanism,anditisunderstoodthatalltheoremsaboutthesellercompetitionmechanismalsohold,mutatismutandis,forthebuyercompetitionmechanism.Recall~Jisthesetofremainingsellers,let^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k=p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kI;~J.WerstproveTheorems 4.3.1 and 4.3.2 ,notethatquasi-linearassumptionisnotrequiredintheseproofs,weonlyrequirethatthesocialwelfarefunctionisanadditivemeasureoverfunctionsofindividual'sbid.Itisrelativelystraightforwardtoseethatthesellercompetitionmechanismisalwaysexpostindividual-rational.ProofofTheorem 4.3.1 :notethatifsellerktrades,hisbidgkmustbenomorethanp+kbecauseotherwisehewouldbeeliminated.Sincehistransactionpriceisp+k,hisutilityfromparticipation,p+k)]TJ/F22 11.955 Tf 12.128 0 Td[(gk,isnon-negative.Becauseallthenon-tradingsellershaveutilityzero,themechanismisexpostindividual-rationalforallthesellers.Ifbuyerltrades,sheisinvolvedintheecientallocationofagentsetI[~J.Thatis,herbidflmustbenolessthan^p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(l.Sincehertransactionpriceis^p)]TJ/F15 11.955 Tf 7.084 1.794 Td[(l,

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35 herutilityfromparticipation,fl)]TJ/F15 11.955 Tf 13.745 0 Td[(^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(l,isnon-negative.Becauseallthenon-tradingbuyershaveutilityzero,themechanismisexpostindividual-rationalforallthebuyers.Therefore,themechanismisexpostindividual-rational. Toshowthatthemechanismisstrategy-proof,werstshowthattruth-revelationisadominantstrategyforallthesellers.Thisisaccomplishedbythefollowinglemmas. Lemma4.4.1 p+kp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kforsellerk2J.Proof:Consideranexchangeenvironmentwithonemoresellerwhoisidenticaltok.ByProperty 2 ,wehaveVk+V)]TJ/F23 7.97 Tf 6.587 0 Td[(kV+V.Thus,Vk)]TJ/F22 11.955 Tf 11.212 0 Td[(VV)]TJ/F22 11.955 Tf 11.211 0 Td[(V)]TJ/F23 7.97 Tf 6.586 0 Td[(k.Sincep+kandp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(karethesupremaofthebidpricessuchthatVk)]TJ/F22 11.955 Tf 11.778 0 Td[(V>0andV)]TJ/F22 11.955 Tf 11.778 0 Td[(V)]TJ/F23 7.97 Tf 6.586 0 Td[(k>0,respectively,wehavep+kp)]TJ/F15 11.955 Tf 7.085 1.794 Td[(k. Thefollowinglemmaimpliesthatifsellerk2J0Jbidsgk,whichislowerthanp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kI;JheisinvolvedineveryecientallocationforagentsetI[J,sellerkisinvolvedineveryecientallocationforagentsetI[J0. Lemma4.4.2 Fork2J0J,V)]TJ/F22 11.955 Tf 11.955 0 Td[(V)]TJ/F23 7.97 Tf 6.587 0 Td[(kV)]TJ/F23 7.97 Tf 6.587 0 Td[(JnJ0)]TJ/F22 11.955 Tf 11.955 0 Td[(V)]TJ/F21 7.97 Tf 6.587 0 Td[(JnJ0[fkg.Proof:ItsucestoprovetheresultforthecasejJ0j=jJj)]TJ/F15 11.955 Tf 19.041 0 Td[(1.SupposelistheuniqueelementinJnJ0.FromProperty 2 ,wehaveV+Vfk;lgV)]TJ/F23 7.97 Tf 6.587 0 Td[(k+V)]TJ/F23 7.97 Tf 6.587 0 Td[(l,i.e.,V)]TJ/F22 11.955 Tf 12.011 0 Td[(V)]TJ/F23 7.97 Tf 6.587 0 Td[(kV)]TJ/F23 7.97 Tf 6.587 0 Td[(l)]TJ/F22 11.955 Tf 12.011 0 Td[(Vfk;lg.NoteifgkV)]TJ/F23 7.97 Tf 6.586 0 Td[(k.Thus,V)]TJ/F23 7.97 Tf 6.587 0 Td[(l>Vfk;lg,whichmeansthatsellerkisinvolvedineveryecientallocationforagentsetI[J0. Lemma4.4.3 Ifsellerkbidslowerthanp+k,hetradeshisbundle.Proof:FromLemma 4.4.1 ,weknowifsellerkbidslowerthanp+k,hisbidisalsolowerthanp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k.Notethataftertheeliminationphase,theagentsetbecomesI[~Jwhere~JJandk2~J.ByLemma 4.4.2 ,sellerkisinvolvedineveryecientallocationforagentsetI[~J.Therefore,sellerktrades.

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36 Theorem4.4.4 Eachsellerhasaweaklydominantstrategytobidtruthfully.Proof:Notethatp+kisdeterminedbyagentsetI[Jnfkg;thus,itisindependentofthebidpricegk.Also,ifsellerktrades,hisrevenueisp+k.Thus,ifsellerk'svaluationishigherthanp+k,heprefersnottotrade,whichcanbeachievedbybiddinghisvaluation.Ifsellerk'svaluationislowerthanp+k,hepreferstotradeatpricep+k,whichcanalsobeachievedbybiddinghisvaluation.Ifsellerk'svaluationisequaltop+k,heisindierentbetweentradingandnottrading.Thus,itisaweaklydominantstrategyforallthesellerstobidtruthfully. Nowweshowthattruth-revelationisadominantstrategyforallthebuyers.Theresultbuildsonthefollowinglemmas.Thenextlemmaimpliesthatifbuyerkbidsanamountfk,whichishigherthanp)]TJ/F15 11.955 Tf 7.085 1.794 Td[(kI;J0forJ0J,buyerkisinvolvedineveryecientallocationforagentsetI[J. Lemma4.4.5 GivenasubsetJ0Jandabuyerk2I,V)]TJ/F22 11.955 Tf 12.545 0 Td[(V)]TJ/F23 7.97 Tf 6.586 0 Td[(kV)]TJ/F23 7.97 Tf 6.587 0 Td[(JnJ0)]TJ/F22 11.955 Tf -401.267 -23.908 Td[(V)]TJ/F21 7.97 Tf 6.586 0 Td[(JnJ0[fkg.Proof:ItsucestoprovetheresultforthecasejJ0j=jJj)]TJ/F15 11.955 Tf 19.041 0 Td[(1.SupposelistheuniqueelementinJnJ0.FromProperty 3 ,wehaveV+Vfk;lgV)]TJ/F23 7.97 Tf 6.586 0 Td[(k+V)]TJ/F23 7.97 Tf 6.587 0 Td[(l,orV)]TJ/F22 11.955 Tf 12.298 0 Td[(V)]TJ/F23 7.97 Tf 6.586 0 Td[(kV)]TJ/F23 7.97 Tf 6.587 0 Td[(l)]TJ/F22 11.955 Tf 12.299 0 Td[(Vfk;lg.Notethatifbuyerkbidsanamountfk,whichishigherthanp)]TJ/F15 11.955 Tf 7.084 1.794 Td[(kI;J0,V)]TJ/F23 7.97 Tf 6.587 0 Td[(l>Vfk;lg.Thus,V>V)]TJ/F23 7.97 Tf 6.587 0 Td[(land,buyerkisinvolvedineveryecientallocationforagentsetI[J. Lemma4.4.6 Ifbuyerk2Ibidsanamountfk,whichislowerthanp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(k,thenshedoesnottradeinanyecientallocationofagentsetI[J0,whereJ0J.Proof:ItsucestoprovetheresultforthecasejJ0j=jJj)]TJ/F15 11.955 Tf 19.041 0 Td[(1.SupposelistheuniqueelementinJnJ0.Wehavefkfkinsteadoffkforsomesmall>0.ThenwestillhaveV=V)]TJ/F23 7.97 Tf 6.587 0 Td[(kandVfk;lg=V)]TJ/F23 7.97 Tf 6.586 0 Td[(lforthenewbid.ThismeansthatthevalueofV)]TJ/F23 7.97 Tf 6.586 0 Td[(lremainsthesame,sinceVfk;lgisindependentofk'sbid.Thus,buyerkdoesnottradeinanyecient

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37 allocationforagentsetI[J0,otherwiseV)]TJ/F23 7.97 Tf 6.587 0 Td[(lincreasesasbuyerkincreasesherbidtop)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k)]TJ/F22 11.955 Tf 11.956 0 Td[(. Lemma4.4.7 p+lforl2Jdoesnotincreaseifonebuyerlowersherbid.Proof:Sincewedenep+lasthesupremumofsellerl'spricetomakeVl>V,itsucestoshowthatVl)]TJ/F22 11.955 Tf 12.381 0 Td[(Vdoesnotincreasewhensomebuyerklowersherbid.Letbuyerklowerherbidcontinuously.BothVlandVeitherdecreaseorremainthesamewhenfkdecreases.IfVdecreasesasfkdecreases,buyerkisinvolvedineveryecientallocationforagentsetI[J.DuetoLemma 4.4.5 ,ifthereisonemoresellerwhobidsthesamepriceassellerl'sprice,buyerkstilltrades;thus,Vldecreasesatthesamerate,andVl)]TJ/F22 11.955 Tf 12.041 0 Td[(Vdoesnotincreasewhensomebuyerklowersherbid. Nowletusconsiderthesetofremainingsellersinthelimitasbuyerk'sbidapproachesinnity.ThislimitingsetiswelldenedbyLemma 4.4.7 .Sincethereareonlynitelymanypossibleremainingsellersets,ifbuyerkbidshighenough,thesetofremainingsellersisthelimitingset.Weuse~Jktodenotethislimitingremainingsellersetanduse~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(ktodenotep)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kI;~Jk.Wehavethefollowinglemma: Lemma4.4.8 Thebuyerkwhobidshigherthan~p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(ktradesthebundleatprice~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(k.Proof:Weknowifbuyerkbidshigherthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,sheisinvolvedineveryecientallocationforagentsetI[~Jk.FromLemma 4.4.5 ,weknowthatbuyerkisinvolvedineveryecientallocationforVI;JandVlI;Jforanysellerl2J.Thus,Vl)]TJ/F22 11.955 Tf 12.152 0 Td[(Vandp+lI;Jremainthesame,aslongasbuyerkbidshigherthan~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(k.Weclaimthatthesetofremainingsellersis~Jkbecausethesetofremainingsellersis~Jkifbuyerkbidshighenough.Sincebuyerkbidshigherthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,sheisinvolvedineveryecientallocationforagentsetI[~Jk.Therefore,shetradesatprice~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k.

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38 Corollary4.4.9 Theremainingsellersetis~Jkaslongasbuyerkbidshigherthan~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(k.Furthermore,p+jI;Jremainsthesameforallj2~Jk. Theorem4.4.10 Eachbuyerhasaweaklydominantstrategytobidtruthfully.Proof:Wehavealreadyshownthatifbuyerkbidshigherthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,shetradesatprice~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k.Supposeshebidslowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,thenthesetofremainingsellers~Jbecomesasubsetof~JkbyLemma 4.4.7 .Furthermore,shedoesnottradeinanyecientallocationforagentsetI[J0sinceshebidslowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kbyLemma 4.4.6 .Thus,buyerkdoesnottradeifshebidslowerthan~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(k.Nowifshebids~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(k,thesetofremainingsellers~Jisasubsetof~Jk,andshemayeithernottradeortradeatprice~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(k.Thus,ifhervaluationoftheitemislowerthan~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(k,sheprefersnottotrade,whichcanbeachievedbybiddinghervaluation.Ifhervaluationishigherthan~p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(k,shepreferstotradeatprice~p)]TJ/F15 11.955 Tf 7.084 1.794 Td[(k,whichcanalsobeachievedbybiddinghervaluation.Ifhervaluationisequalto~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,sheisindierentbetweentradingandnottrading.Thus,itisaweaklydominantstrategyforallthebuyerstobidtruthfully. ProofofTheorem 4.3.1 :byTheorems 4.4.4 and 4.4.10 ,themechanismisstrategy-proof. Nowweshowthatthesellercompetitionmechanismisweaklybudget-balancedifthesocialwelfarenotonlysatisesthecomplementarity-substitutabilityconditions,butalsoisnomorethanthecorrespondingquasi-linearsocialwelfare.Since~JJ,p+jI;J~p+jbyLemma 4.4.2 ,thatis,noselleriseliminatediftheauctionstartswithagentsetI[~J.Notethatthetradingpricesforthebuyersremainthesame,whilethetradingpricesofthesellersincreasefromp+jI;Jsto~p+js,andtheauctioneer'srevenuedoesnotincrease.Thus,itsucestoprovethebudgetbalanceresultonlyforthespecialcaseJ=~J.Notethatinsuchanenvironment,theallocationdeterminedbyourmechanismisecient.

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39 Thefollowinglemmaimpliesthatifbuyerk2I0Ibidsfk,whichishigherthanp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kI;JsheisinvolvedineveryecientallocationforagentsetI[J,buyerkisinvolvedineveryecientallocationforagentsetI0[J. Lemma4.4.11 Forbuyerk2I0I,V)]TJ/F22 11.955 Tf 11.955 0 Td[(V)]TJ/F23 7.97 Tf 6.587 0 Td[(kV)]TJ/F23 7.97 Tf 6.587 0 Td[(InI0)]TJ/F22 11.955 Tf 11.956 0 Td[(V)]TJ/F21 7.97 Tf 6.586 0 Td[(InI0[fkg.Proof:TheproofissimilartotheproofofLemma 4.4.2 ,exceptthatweuseProperty 1 insteadofProperty 2 . Lemma4.4.12 Supposebuyerkbidsfk>p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kI;J,andshelowersherbidtop)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kI;J+kk>0,thenithesetofremainingsellersandp+jI;Jforsellerj2Jdonotchange;iiforbuyeriwithfi>p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(iI;J,p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(iI;Jdoesnotchange;iiitheoriginalecientallocationisstillecient.Proof:iSinceJ=~J,thisisjustarestatementofCorollary 4.4.9 .Toproveii,notethatsincefk>~p)]TJ/F15 11.955 Tf 7.084 1.794 Td[(k,buyerkisinvolvedineveryecientallocationforagentsetI[J=I[~J.ByLemma 4.4.11 ,sheisalsoinvolvedineveryecientallocationforagentsetI[J=I[~Jnfigfori2Iandi6=k.Asbuyerklowersherbid,bothVandV)]TJ/F23 7.97 Tf 6.586 0 Td[(ichangebythesameamount,p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(iI;Jofbuyeri,whobidshigherthanoriginalp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(iI;J,remainsunchanged.Foriii,sincebiddinghigherthanp)]TJ/F15 11.955 Tf 7.085 1.794 Td[(kI;JguaranteesthatbuyerkisinvolvedineveryecientallocationforagentsetI[J,thedecreaseofsocialwelfareisdeterminedbytheamountofbiddecrease.Thus,iiifollows. Lemma4.4.13 Supposesellerkbidsgk,whichislessthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,andheraiseshisbidto~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k)]TJ/F22 11.955 Tf 12.242 0 Td[(kk>0.Thenifortradingsellerj6=k,p+jI;Jdoesnotchange;iitheoriginalecientallocationisstillecient.Proof:iIfsellerkchangeshisbidtop+kI;J)]TJ/F22 11.955 Tf 13.173 0 Td[(k,Vdecreases,andVjdecreasesbyatmostthesameamountforanyothersellerj2J.Thus,p+jI;Jforsellerj2Jdoesnotdecrease,andnoselleriseliminatedafterthischange.

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40 Forii,sincebiddinglowerthanp+kI;Jguaranteesthatsellerkisinvolvedineveryecientallocation,thesocialwelfaredecreasesaccordingtotheamountchangedinthebid.Thus,iifollows. Wedenotethetradingbuyersetas^Iandthetradingsellersetas^J.Wenowprovethemaintheoremforbudgetbalance.RecallthatwehaveJ=~J.ProofofTheorem 4.3.1 :Supposeweletalltradingbuyerswhobidhigherthanp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kI;Jlowertheirbidstop)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J+kforsomek>0onebyone.ThenbyLemma 4.4.12 ,theoriginalallocationremainsecientandp+jI;Jremainsthesameforsellerj2J.Afterthat,byLemma 4.4.13 ,wecanletallofthetradingsellerswhobidlowerthanp+kI;Jraisetheirbidstop+kI;J)]TJ/F22 11.955 Tf 10.428 0 Td[(kforsomek>0onebyoneandstillkeeptheoriginalallocationecient.Sincetheoriginalallocationisstillecient,itisbetterthanmakingnotransaction.SincethesocialwelfareVisnomorethanthecorrespondingquasi-linearsocialwelfare,underquasi-linearsocialwelfarefunction,wehavePi2^Ip)]TJ/F15 11.955 Tf 7.085 1.793 Td[(iI;J+i)]TJ/F28 11.955 Tf 12.673 8.967 Td[(Pj2^Jp+jI;J)]TJ/F22 11.955 Tf 12.673 0 Td[(j)]TJ/F22 11.955 Tf 12.673 0 Td[(CQ0,whereCQisthetransactioncoststoexecutetheallocationQ.SincetheksarearbitrarypositivenumbersandJ=~J,Pi2^I~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(iPj2^Jp+jI;J+CQ,i.e.,thetotalcashinowisnolessthanthetotalcashoutow.Thus,thesellercompetitionmechanismisweaklybudget-balanced.

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CHAPTER5SIMPLEEXCHANGEENVIRONMENTWITHTRANSACTIONCOSTSNow,letusrevisitthecurrentcustomer-to-customeronlineauctionmarketandapplytheAC-DAmechanismtothismarket.Forsimplicity,werstmodelthismarketasasimpleexchangeenvironmentinwhicheachagentwantstobuyselloneunitofthesameitem.Weassumethatwhenbuyeritradeswithsellerj,transactioncostdi;jisincurredduetoshipping,handling,salestax,etc.Thetransactioncostscanbecommonknowledgeoronlyknownbytheauctioneer,andthesecostsresultinproductdierentiation.Theremainderofthischapterisorganizedasfollows.Section 5.1 speciesthesimpleexchangemodelandshowsthattheAC-DAmechanismmaintainsallthedesiredpropertiesunderthisenvironment.Section 5.2 investigatestheimpactoftransactioncosts,whichmaysegmentthemarket,prohibitthetrades,andincuralossonsocialwelfare.However,Section 5.3 showsthatthislosscanbealleviatedandbecomesnegligibleinalargemarket.Thus,astheauctionbecomeslarge,almostallfeasiblesurpluseswillberealized.Section 5.4 providesacomprehensivediscussiononfalse-namebids,self-reportofthetransactioncosts,andthetwo-parttariinthisonlineauctionmarket. 41

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42 5.1 ModelandMechanismGivenagents'bidsfisi2Iandgjsj2J,wecanformulateaquasi-linearsocialwelfareVI;J:P:MaximizeV=Pifixi)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pjgjyj)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pi;jdi;jzi;jSubjecttoPjzi;j=xiforeachi2IPizi;j=yjforeachj2Jxi2f0;1gforeachi2Iyi2f0;1gforeachj2Jzi;j2f0;1gforeachi2Iandj2Jwherexiandyjdenotewhetherbuyeri,orsellerj,respectively,enteratransactionornot;zi;jdenoteswhetherbuyeritransactswithsellerjornot.Duetothenetworkpropertyoftheformulation,thismaximizationproblemcanbesolvedeciently.Furthermore,underthisquasi-linearformulation,p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kistheVCGprice,andwecancalculateitbythefollowingmethod:letthebuyersellerkbidapricephighlowenoughsothatV>V)]TJ/F23 7.97 Tf 6.586 0 Td[(k.Thenp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k=p)]TJ/F15 11.955 Tf 10.633 0 Td[(V)]TJ/F22 11.955 Tf 10.633 0 Td[(V)]TJ/F23 7.97 Tf 6.586 0 Td[(kifkisabuyer,andp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(k=p+V)]TJ/F22 11.955 Tf 10.747 0 Td[(V)]TJ/F23 7.97 Tf 6.587 0 Td[(kifkisaseller.Whenbuyerihasvaluationhigherthanp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(i,andshetradesatp)]TJ/F15 11.955 Tf 7.085 1.794 Td[(i,herpayoisV)]TJ/F22 11.955 Tf 12.692 0 Td[(V)]TJ/F23 7.97 Tf 6.586 0 Td[(i,hermaximumcorepayointheassignmentgame.Similarly,whensellerjhasvaluationlowerthanp)]TJ/F15 11.955 Tf 7.085 1.794 Td[(j,andhetradesatp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j,hispayoisV)]TJ/F22 11.955 Tf 12.261 0 Td[(V)]TJ/F23 7.97 Tf 6.587 0 Td[(j,hismaximumcorepayointheassignmentgame.Wecanalsocalculatep+kinasimilarapproach:letthebuyersellerkbidapricephighlowenoughsothatVk>V.Thenp+k=p)]TJ/F15 11.955 Tf 12.354 0 Td[(Vk)]TJ/F22 11.955 Tf 12.353 0 Td[(Vifkisabuyer,andp+k=p+Vk)]TJ/F22 11.955 Tf 12.361 0 Td[(Vifkisaseller.Whenbuyerihasvaluationhigherthanp+i,andshetradesatp+i,herpayoisVi)]TJ/F22 11.955 Tf 12.178 0 Td[(V,herminimumcorepayointheassignmentgame.Similarly,whensellerjhasvaluationlowerthan

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43 p+j,andhetradesatp+j,hispayoisVj)]TJ/F22 11.955 Tf 12.154 0 Td[(V,hisminimumcorepayointheassignmentgame.Shapley[ 57 ]showsthatthesocialwelfareformulationPofthesimpleex-changeenvironmentsatisesthecomplementarity-substitutabilityconditions.Therefore,theAC-DAmechanisminducedbyformulationPisapplicabletothisenvironment. Theorem5.1.1 TheAC-DAmechanismisstrategy-proof,expostindividual-rational,andexpostbudget-balancedinthesimpleexchangeenvironmentwithtransactioncosts. 5.2 ImpactofTransactionCostsIfthereisnotransactioncost,wecanusetherankingmethodtoillustratethebehavioroftheAC-DAmechanism.Withoutlossofgenerality,considerthesellercompetitionmechanismandsupposetherearembuyoersf1f2fmandnselloersg1g2gn.Letqbetheoptimaltradesize.Thesellercompetitionmechanismmakeseitherqorq)]TJ/F15 11.955 Tf 12.665 0 Td[(1transactions,dependingonthefollowingconditions:1Ifgqfq+1:Fortheqsellerswhobidnohigherthanfq+1,theirp+sequalfq+1.Thus,theysurvivetheeliminationphase.Forallothersellers,theirp+sequalmaxfgq;fq+2g.Thus,theyareeliminated.qtransactionsoccuramongthehighestqbuyoersandthelowestqselloers.Thepriceforalltradingbuyersisfq+1,andthepriceforalltradingsellersisalsofq+1.2Ifgq>fq+1andgq=gq)]TJ/F21 7.97 Tf 6.586 0 Td[(1:Forthosesellerswhobidnohigherthangq=gq)]TJ/F21 7.97 Tf 6.586 0 Td[(1,theirp+sequalgq,andtheysurvivetheeliminationphase.Forallothersellers,theirp+sequalgq,andtheyareeliminated.qtransactionsoccuramongthehighestqbuyoersandthelowestqselloers.Thepriceforalltradingbuyersisgq,andthepriceforalltradingsellersisalsogq.

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44 3Ifgq>gq)]TJ/F21 7.97 Tf 6.587 0 Td[(1andgq>fq+1:Fortheq)]TJ/F15 11.955 Tf 11.301 0 Td[(1sellerswhobidnohigherthangq)]TJ/F21 7.97 Tf 6.587 0 Td[(1,theirp+sequalgq.Thus,theysurvivetheeliminationphase.Forallothersellers,theirp+sequalmaxfgq)]TJ/F21 7.97 Tf 6.586 0 Td[(1;fq+1g,andtheyareeliminated.q)]TJ/F15 11.955 Tf 12.195 0 Td[(1transactionsoccuramongthehighestq)]TJ/F15 11.955 Tf 12.015 0 Td[(1buyoersandthelowestq)]TJ/F15 11.955 Tf 12.015 0 Td[(1selloers.Thepriceforalltradingbuyersisfq,andthepriceforalltradingsellersisgq. Figure5{1:PossibleTransactionStructures Thesellercompetitionmechanismeitherachievesanecientallocationormissesatmostoneleastprotabletradeincomparisontotheecientallocationiftherearenotransactioncosts.Similarly,thebuyercompetitionmechanismmissesatmostoneleastprotabletrade.Therefore,theAC-DAmechanismisasymptoticallyecientasthenumberoftradesapproachesinnity.NowweusetheexampleinFigure 5{2 toillustratehowtheAC-DAmecha-nismmaybehaveunderasimpleexchangeenvironmentwithtransactioncosts.Inthisexample,wehavetwosellersandthreebuyers,andapplythesellercompeti-

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45 tionmechanism.Thetransactioncostmatrixdi;jisgivenatthetopofthegure.Theagentshavetheincentivestotruthfullybidtheirvaluations,whichareshowninthegraphbelowthetable.Inthisexam-ple,thep+softhesellersare88and90,respectively,andbothsellerssurvivetheeliminationphase.Thenaccordingtotheecientallocation,themechanismexecutestwotransactionsbetweenbuyer1andseller1,andbuyer2andseller2.Thepaymentsfrombuyer1and2are92and94,respectively.Therevenuesforseller1and2are88and90,respectively. di;j 1 2 1 4 7 2 6 4 3 9 6 sssssXXXXHHHHg2=85g1=86f3=96f2=95f1=97 Figure5{2: ExamplewithLowTransactionCosts Notethatthesellercompetitionmechanismachievestheecientallocationinthisexample.Itisinterestingtoseehowtransactioncostsaectthemaximumeciencyofamechanism.WeillustratethiseectbytheexampleshowninFigure 5{3 .ThisexampleisthesameastheoneinFigure 5{2 exceptthatthetransactioncostsareabout70%higher.ThenewtransactioncostmatrixisgiveninFigure 5{3 .Inthisexample,p+softhesellersare85and86,respectively.Thesellercompetitionmechanismexecutesonlyonetransactionbetweenbuyer2atbuyingprice93andseller2atsellingprice86insteadoftwointhepreviousexample,whiletheecientallocationallowstwotransactions,i.e.,betweenbuyer1andseller1,andbetweenbuyer2andseller2.

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46 Notethatduetothehightransactioncosts,weessentiallyhavetwoseparatemarkets:therstmar-ketconsistsofbuyer1andseller1,whilethesecondonehasbuyers2,3andseller2.ByMyersonandSatterthwaite[ 43 ],noincentive-compatible,budget-balancedmechanismiscapableofalwaysinducingthetransactionfortherstmarket.Thus,itisunlikelythatthereisanystrategy-proof,budget-balancedmechanismthatleadstoatransactionbetweenbuyer1andseller1atafairprice.Inthissense,thesellercompetitionmechanismachievesthehighestpossibleeciencyinthisexample. di;j 1 2 1 7 12 2 10 7 3 15 10 sssssXXXXHHHHg2=85g1=86f3=96f2=95f1=97 Figure5{3: ExamplewithHighTransactionCosts Thisexampleillustratesthatthetransactioncostscanpreventsomepossibletransactionsbyseparatingthebuyersandsellersintosmallmarkets,andtheseparationresultsinalossofeciency.Thiseectcanbeaggravatedifthereareaninnitenumberoftransactioncosts.Todemonstratethispoint,letusassumethatFisafamilyofcontinuousdistributionswithsupportcontainedin[0;a]forsomeconstanta.Supposedis-tributionsFandGaredrawnfromFaccordingtosomestochasticprocess.Leteachbuyer'svaluationindependentlyfollowF,andeachseller'svaluationinde-pendentlyfollowG.Now,weassumethattheagentsareindependentlygeneratedfromsomecompactdomainHaccordingtosomecontinuousdistributionU,andthatthetransactioncost,di;j,isthedistancebetweenbuyeriandsellerj.Thatis,thetransactioncostddependsonxandy,thelocationsofiandj,respec-tively,x;y2H,anddx;yisametric:dissymmetric;dsatisesthetriangleinequality;anddx;y=0ifandonlyifx=y.Letusalsoassumethatdiscontinuous.

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47 Thenwehavethefollowingresult: Theorem5.2.1 Givenanitebuyerset,theeciencyachievedbythesellercompetitionmechanismapproacheszeroalmostsurelyasthenumberofsellersapproachesinnity.Proof:Withoutlossofgenerality,assumethateverybuyerinthegivenbuyersethasapositiveprobabilityofmakingaprotabletransactionwithaseller,otherwisewecouldremovethisbuyer,asshedoesnotcontributetothesystem.SinceUiscontinuous,theprobabilityoftwobuyersbeingatthesamelocationiszero.Letd0betheminimumofdx;y,wherexandyarelocationsofdierentbuyers.Thend0ispositivewithprobability1.Set=d0=3>0,anddeneAx;=fyjdx;y<;y2Hg.Sincediscontinuous,theprobabilitythatsomesellerisfromtheareaAx;isalwayspositive.Duetothetriangleinequality,wehaveadisjointneighborhoodAxi;foreachbuyeriatxi.Letusdener=inffyjGy>0gandB=[r;r+].WehaveProbY2B>0forYdrawnfromdistributionG.Asthenumberofsellersapproachesinnity,thenumberofsellerswithavaluationbelongingtoBexceedsthenumberofbuyers,m,ineveryneighborhoodAxi;withprobability1.Weshowunderthisconditionthatnosellersurvivestheeliminationphase.Withnosellerleftintheremainingsystem,thesocialwelfareachievedbythesellercompetitionmechanismequalszero.Itsucestoshowthatthereisnoecientallocationinwhichsellerj,whosevaluationisr,cansellonemoreitemwiththiscondition.Notethattheprobabilityofsomesellerhavingavaluationlowerthanriszero.Supposethereissuchanecientallocationwheresellerj,withtwoitems,makestransactionswithbothi1andi2.SincetheneighborhoodAxi1;andAxi2;aredisjoint,withoutlossofgenerality,weassumethatsellerjdoesnotlocateatAxi1;.Sincethereareonlymbuyers,theremustbesomesellerj0inAxi1;whoisnotinvolvedintheecienttransactionwithavaluationbelongingtoB.However,bythetriangle

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48 inequality,onecanimprovethemaximumsocialwelfarebymakingtransactionsbetweeni1andj0insteadofi1andj.Wehaveacontradiction;thus,nosellersurvivestheeliminationphase,andthesocialwelfareachievedbythesellercompetitionmechanismiszeroalmostsurelyasthenumberofbuyersapproachesinnity. Thiscounterintuitiveresultshowstheimportanceofchoosingtherightmechanismtoachievehighereciency.Infact,thebuyercompetitionmechanismisasymptoticallyecientinsuchabuyer'smarket.Wediscusstheasymptoticeciencyresultfurtherinthenextsubsection. 5.3 AsymptoticEciencyPropertyNow,wewillshowthateventhoughtransactioncostshurteciency,thelosscanbemitigatedandbecomesnegligibleinalargemarket.Weadoptthesamedistributionandtransactioncostsassumptionsasintheprevioussection.Wehavethefollowingresult: Theorem5.3.1 TheAC-DAmechanismisasymptoticallyecientifbothmandnapproachinnityandm=napproachessomenumberpsatisfying00,sinceHiscompact,thereexistsanite-partitionA1;A2;;AkofHi.e.,apartitionsuchthatAlhasradiuslessthan.Weprovetheresultbyrstcalculatingthelimitingmaximumfeasiblesocialwelfareperagentwithcompleteinformationandthenshowingthatthesocialwelfareperagentachievedbythesellercompetitionmechanismalsoapproachesthislimitalmostsurely.Now,letusdener=inffyjGyGzg.Werstconsiderthosesellerswhobidlessthanr)]TJ/F22 11.955 Tf 10.432 0 Td[(inregionAi.WehaveGr)]TJ/F22 11.955 Tf 10.432 0 Td[(
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49 thanthenumberofbuyerswhobidhigherthanz.Thus,justbyrankingthepricesineachregionandtransactingaccordingtotheranking,onecanachievesocialwelfareperagentgreaterthanGr)]TJ/F24 5.978 Tf 5.756 0 Td[( GzR1zxdFx)]TJ/F23 7.97 Tf 6.587 0 Td[(pRr)]TJ/F24 5.978 Tf 5.756 0 Td[(0y+dGy 1+pasthetransactioncostforeachtransactionislessthan.AlsonotethatGr=GzsinceGisleftcontinuous,as!0,itisalmostsurethatthesocialwelfareachievedperagentisnolessthan CR1zxdFx)]TJ/F23 7.97 Tf 6.586 0 Td[(pRz0ydGy 1+pasthenumberofagentsapproachesinnity.Fromtheabovearguments,withcompleteinformation,thereisanallocationthatachievessocialwelfareperagentnolessthan Casthenumberofagentsapproachesinnity.Wenextshowthat Cisinfactalsoanupperboundonthesocialwelfareperagentforanyallocation.Thus,themaximumfeasiblesocialwelfareperagentwithcompleteinformationapproaches Calmostsurely.Toseethis,weconsiderthecaseinwhichweignorethetransactioncosts.Thesocialwelfareperagentwithcompleteinformationisnogreaterthan Calmostsurely,whichgivesusanupperboundforthecasewithtransactioncosts.Notethatasn!1,itisalmostsurethatineachregionAithereexistssomesellerwhosevaluationisbetweenr)]TJ/F22 11.955 Tf 12.505 0 Td[(andr.Moreover,thepercentageofthesesellerswhosevaluationsarebetweenr)]TJ/F22 11.955 Tf 12.202 0 Td[(andrapproachessomepositivenumberineachregion.Duetothetriangleinequality,asellerwhosevaluationislessthanr)]TJ/F15 11.955 Tf 12.994 0 Td[(2isalwaysmorecompetitivethanthesellerwhosevaluationisbetweenr)]TJ/F22 11.955 Tf 12.269 0 Td[(andrinthesameregion.Thus,ifinanecientallocation,somesellerwithvaluationbetweenr)]TJ/F22 11.955 Tf 11.97 0 Td[(andrcompletesatransaction,allsellersinthesameregionwithvaluationslessthanr)]TJ/F15 11.955 Tf 12.386 0 Td[(2mustalsogettransactions.Moreover,eachsellerwhosevaluationislessthanr)]TJ/F15 11.955 Tf 12.285 0 Td[(2survivesinthiscase.Bythecalculationintheaboveparagraph,weknowinanecientallocation,thepercentageofthesellerswhobidbetweenr)]TJ/F22 11.955 Tf 12.368 0 Td[(andrandgetatransactioncannotgotozero,otherwise,thelimitingmaximumfeasiblesocialwelfareperagentwithcompleteinformationisstrictlylessthan C.Thatis,itisalmostsurethereissomesellerwithvaluation

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50 betweenr)]TJ/F22 11.955 Tf 12.319 0 Td[(andrwhocompletesatransactionineachallocation.Thus,sellerswhosevaluationislessthanr)]TJ/F15 11.955 Tf 12.105 0 Td[(2surviveintheeliminationphase.Sincecanbearbitrary,bythesameargumentsasweusedintheaboveparagraph,weknowasn!1,thesocialwelfareperagentachievedbythesellercompetitionmechanismisnolessthan C.Notethat Cisanupperboundonthesocialwelfareperagentforanyallocation,thesocialwelfareperagentachievedbythesellercompetitionmechanismapproaches Calmostsurely.Asboththemaximumfeasiblesocialwelfareperagentwithcompleteinforma-tionandthesocialwelfareperagentachievedbythesellercompetitionmechanismapproach Calmostsurely,theratiobetweenthewelfareachievedbythesellercompetitionmechanismandthemaximumfeasiblesocialwelfarewithcompleteinformationconvergesto1asthenumberofagentsapproachesinnity.Thus,thesellercompetitionmechanismachievesasymptoticeciency. Theorem5.3.2 TheAC-DAmechanismisasymptoticallyecientifm!1,n!1,and00,weuseEtodenotethefollowingevent:theeciencyachievedbytheAC-DAmechanismishigherthan)]TJ/F22 11.955 Tf 12.642 0 Td[(maximumfeasiblesocialwelfarewithcompleteinformation.WeknowfromtheproofofTheorem 5.3.1 thatforany>0and01)]TJ/F22 11.955 Tf 12.861 0 Td[(ifthenumberofagentsislargeenough.Also,since00,ProbE>1)]TJ/F22 11.955 Tf 12.559 0 Td[(whenthenumberofagentsapproachesinnity.Byletting;!0,theratiobetweenthewelfareachievedbyAC-DAmechanismandthemaximumfeasiblesocialwelfarewithcompleteinformationconvergesto1almostsurelyasthenumberofagentsapproachesinnity,thatis,theAC-DAmechanismisasymptoticallyecient.

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51 Theasymptoticeciencytheoremshowsthataslongasthemarketislargeenoughandbalancedbetweenbuyersandsellers,boththebuyercompetitionmechanismandthesellercompetitionmechanismachievesatisfactoryeciency.Ifthemarketisunbalanced,saym napproacheszero,thenweshouldchoosethebuyercompetitionmechanismtoachieveasymptoticeciency.Similarly,ifm napproachesinnity,weshouldpickthesellercompetitionmechanismtoachieveasymptoticeciency. 5.4 RemarksUptonowwehaveconcentratedonasimpleexchangeenvironmentwithonlyonetypeofitem,inpractice,therearemanyotherscenariosthatcanmaketheauctionmorecomplicated.Forinstance,onebuyermayndseveraldierentitemstobeperfectsubstitutes,whileanotherbuyerisonlywillingtoacceptoneofthemduetothebrandnameeect.CantheAC-DAmechanismapplytothissetting?Theanswerisarmative.Aslongasthequasi-linearsocialwelfaresatisesthefollowingcomplementarity-substitutabilityconditions,theAC-DAmechanismisstrategy-proof,expostindividual-rational,andexpostbudget-balanced.Therearealsoothercomplicationsofthesimpleexchangeenvironment.Inthenext,wediscusswhetheragentshaveincentivestosubmitfalse-namebidsandhowthisimpactthesocialeciencyandindividualrevenue,wealsostudytheagents'incentiveoftruthfullyreportingthetransactioncosts,andthescenariowhentheauctioneerwantsatwo-parttariandthesocialwelfareisnotquasi-linear. 5.4.1 False-nameBidEventhoughwehaveshownthatitiseachagent'sbestinteresttosubmithis/herownvaluation,agents'maytrytomaximizetheirpayosbyothermeans,oneofwhichisafalse-namebid.Toseethepoweroffalse-namebids,consider

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52 whenthereisoneuniqueseller.Thesellerprefersanoptimalauctionthatmax-imizeshisrevenue,andhecanturnthesellercompetitionmechanismintoasecond-pricesealed-bidauctionwithreservepricebysubmittingafalsebuyer'sbid.Someauctionliteraturedevisesprotocolstopreventfalse-namebids.Butforanonlinecustomer-to-customermarketplace,wheretypicallyagentscanbidanonymously,false-namebidsarepartoftheagents'strategies.Therefore,weneedtostudytheagents'behavioronfalse-namebidsandunderstandtheirsimpact.Moredetailedexaminationsonfalse-namebidsareinorder.Forthesecond-pricesealed-bidauctionwhereonlybuyer'sbidsareaccepted,buyershavenoincentivetosubmitfalse-namebids,whiletheuniquesellercanuseeitherafalsebuyer'sbidorareservepricetomaximizehisrevenue.Fordoubleauctionmechanisms,bothbuyersandsellersinthemechanismsofMcAfee[ 33 ],Huangetal.[ 23 ],Babaioetal.[ 6 ],andBabaioandWalsh[ 7 ]mayhavetheincentivetosubmitbothfalsebuyer'sandfalseseller'bids.However,underthesellercompetitionmechanism,becausethemulti-stageapproachtreatsbuyersandsellersasymmetrically,agentshavenoincentivetosubmitfalseseller'sbids,whiletheymaywanttosubmitfalsebuyer'sbids. Theorem5.4.1 Noagenthasincentivetosubmitfalseseller'sbidsunderthesellercompetitionmechanisminthesimpleexchangeenvironmentwithtransactioncosts.Proof:Thecomplementarity-substitutabilityconditionsholdinthesimpleexchangeenvironmentwithtransactioncosts.Becauseofthis,whenthereisonemoreseller'sbid,thep+sforalltheoriginalsellersdonotincrease.Therefore,asellercannotusefalseseller'sbidstoboostuphisthresholdtransactionprice,andhasnoincentivetosubmitfalseseller'sbids.Asthep+sforalltheoriginalsellersdonotincrease,abuyercannotenlargethesetofremainingsellersviafalseseller'sbids.Thus,the~p)]TJ/F15 11.955 Tf 10.987 1.793 Td[(decreasesonlywhen

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53 thesubmittedfalse-namesurvivestheeliminationphase.Whenthevaluationdistributioniscontinuous,itisalmostsurelythatallthesellerbidsthatpasstheeliminationphasegettransaction,thatis,~p)]TJ/F15 11.955 Tf 10.987 1.793 Td[(decreasesonlywhenthebuyerisobligedtosellanitem.Notethatwhenthetransactioncostdsatisesthetriangleinequality,thebuyingpricemustbenolessthanthesellingprice.Thus,abuyerhasnoincentivetosubmitfalseseller'sbids. Next,westudytheincentiveforfalse-namebidsinasimpleexchangeenviron-mentwithouttransactioncosts.Supposesellers'valuationsofgoodsarenormalizedtozero.Alsoforillustrationpurpose,consideraseller'smarketwithnsellersandm=2nbuyers,whilebuyers'valuationsareindependentidenticalrandomvariables,uniformlydistributedon[0,1].Alltheresultsremainvalidwhenbuyers'valuationsareindependentidenticalrandomvariablesfollowingdistributionFandnm)]TJ/F22 11.955 Tf 11.956 0 Td[(Fr,whererisoptimalreservepricewithregardtoF.Weapplythesellercompetitionmechanism.Itisineachagent'sbestinteresttoreporthis/hervaluation.TheexpectedrevenuepersellerandtheexpectedsocialwelfareperselleraresummarizedinTable 5{1 ifthereisnofalse-namebid.Underthecurrentsetting,buyershavenoincentivetosubmitfalse-namebids,whilesellersmayimprovetheirrevenuesbysubmittingonefalsebuyer'sbids.Insteadofpinpointingtheequilibriumbehaviorwithfalse-namebids,weconstructanupperandalowerboundonsocialwelfareandsellers'revenues,takingintoaccountfalse-namebids.Notethatinanequilibrium,abuyershouldnotsubmitafalsebuyer'sbidhigherthanr.Andwheneachsellersubmitsafalsebuyer'sbidr,thesellercompetitionmechanismisequivalenttoanoptimalauctionforthesellercoalition.Thus,weobtainanupperboundforexpectedrevenuepersellerandalowerboundforexpectedsocialwelfareperseller.ThesenumbersaresummarizedinTable 5{2 .Ontheotherhand,whentherearenofalse-namebids,themechanismisecientinthecurrentsetting.Thisprovidesalowerboundfor

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54 expectedrevenuepersellerandanupperboundforexpectedsocialwelfareperseller.ThesenumbersaresummarizedinTable 5{1 .Theexpectedrevenueandtheexpectedsocialwelfareofanyequilibriumliebetweenthenumbersinthesetwotables. Table5{1:ExpectedRevenue/SocialWelfarewithoutFalse-nameBids numberofsellers/buyers 1/2 2/4 3/6 n=2n innity expectedrevenueperseller 1 3 2 5 3 7 1 2)]TJ/F21 7.97 Tf 21.13 4.707 Td[(1 4n+2 1 2 expectedsocialwelfareperseller 2 3 7 10 5 7 3 4)]TJ/F21 7.97 Tf 21.13 4.707 Td[(1 8n+4 3 4 Table5{2:ExpectedRevenue/SocialWelfarewithSellerCoalition numberofsellers/buyers 1/2 2/4 n=2n innity expectedrevenueperseller 5 12 9 20 1 2)]TJ/F21 7.97 Tf 21.13 4.707 Td[(1 8n+4 1 2 expectedsocialwelfareperseller 7 12 101 160 3 4)]TJ/F21 7.97 Tf 23.247 4.708 Td[(1 16n+8)]TJ/F21 7.97 Tf 22.148 4.708 Td[(1 22n+2)]TJ/F21 7.97 Tf 5.48 -4.379 Td[(2nn 3 4 Notethattheexpectedsocialwelfareandrevenuepersellerconvergewhennapproachesinnity.Theperformanceswithorwithoutfalse-namebidsconvergeaslongasnm)]TJ/F22 11.955 Tf 11.668 0 Td[(Fr,i.e.,theexpectednumberofbuyerswithvaluationhigherthanrisnolessthanthenumberofsellers.Sincethesetwosettingsprovidetheupperandlowerboundsforexpectedrevenueandexpectedsocialwelfare,theconvergenceimpliesthatifkeepingm=nconstantandnapproachesinnity,anyequilibriumperformanceapproachesthemaximumfeasiblesocialwelfareandthemaximumrevenuessimultaneously.Wesummarizetheresultinthefollowingproposition: Proposition5.4.2 Ifm=nisaconstantand)]TJ/F22 11.955 Tf 12.836 0 Td[(Frn=m,thedoubleauctionmechanismapproachesboththemaximumfeasiblesocialwelfareandthemaximumrevenuesasnincreases.Asshownintheexample,itisworthwhiletopointoutthattheexpectedimprovementbyusingfalse-namebidsapproacheszeroatrateOn)]TJ/F21 7.97 Tf 6.586 0 Td[(1,which

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55 impliesthatifthemarketislargeenough,itisprobablynotworththeeortsforsellerstocollectdetailedinformationonbuyers'evaluationsinordertosubmitfalse-namebids.Wehaveshownthatwhenthereisnotransactioncost,theincentiveforfalse-namebidsvanishesasthemarketbecomelargerandlargerwhensomeregularityconditionsaresatised.Weexpectthatthisresultholdsforgeneralenvironment;however,amoredetailedanalysisoftheincentiveforfalse-namebidsandtheirimpactisstillanopenresearchquestion. 5.4.2 IncentiveonRevealingtheTransactionCostsInourcurrentinformationsetting,theauctioneerknowsthetransactioncostsofanypotentialtrade.Thisseemsveryrestrictive;however,theauctioneermayaskeachsellerthetransactioncostsforeachgeographiclocationbeforehand,becausethetransactioncostsaretypicallyshipping,handling,andsalestaxcosts,whichcanbedeterminedbythesellerifheknowsthelocationsofthebuyers.Inthiscase,wehavetoexaminetheincentiveformisreportingthetransactioncosts.Thesellercanmisreporthisprivateinformationaboutthetransactioncosts.Say,heoverpricesallofhishandlingfeesby$2.However,astheAC-DAmech-anismisstrategy-proofandthepriceisdeterminedbytheseller'smarginalcontribution,thesellershouldrealizethatheshouldbid$2lowerthanhistruevaluationoftheitemifhecaresaboutthetotalrevenuethesalepriceplusthechargedtransactioncostsminustherealtransactioncosts.Thisisequivalenttothathebidsboththehandlingfeesandthevaluationtruthfully.Asellermayprotbyoverpricingsomeofhispossibletransactioncostswhileunderpricingothersifheknowsthedetailedinformationofbuyers'valuation.Considerasimpleenvironmentwithonesellerandtwobuyers,whiletherealtransactioncostsiszeroforallpossibletrades.Assumetheseller'svaluationisnormalizedatzero,whilebuyer1and2'svaluationis10and5,respectively.Ifthe

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56 sellerknowsthebuyers'valuations,hecanextractallofthe10frombuyer1ifhereportstotheauctioneerthatitcostshim5moretotradewithbuyer1.However,whensellershavethesamepriorvaluationdistribution,itisnotclearwhyasellershoulddiscriminatebetweenthebuyersbymisreportingthetransactioncosts.Returntothecasethatwithoneseller,twobuyersandnotransactioncosts,ifthesellerbelievesthatbothbuyersdrawtheirvaluationfromthesamedistribution,hehasnoincentivetomanipulatethereportofthetransactioncosts.Thedetailedanalysisofhisincentivecanbeveryinterestingandthisisstillanopenresearchquestion. 5.4.3 Non-quasi-linearSocialWelfareTocompensateforthecostofrunningthemarketplace,theauctioneerneedstochargeacommission.Typically,acommissionhastwocomponents,axedpartandavariablepart,whichisgeneralapercentageofthetransactionprice.Inthissection,wediscusshowtoincorporatethistwo-parttariintotheAC-DAmechanism.Lettbethexedchargeforeachtransaction.Sincewehavebothabuyingpriceandasellingpriceforeachtransaction,letanddenotethepercentagesoftariweimposedonthebuyerandseller,respectively.Furthermore,dierentbuyersmayhavedierentpercentagechargesii2I,whiledierentsellersmayhavedierentpercentagechargesjj2J.Theauctioneermaywanttoimplementthesedierencesinthepercentagechargetodistinguishagentsaccordingtotransactionrecords,credit,location,etc.Takingintoaccountthesecharges,apotentialgainoftransactionbetweenbuyeriandsellerjbecomes)]TJ/F22 11.955 Tf 12.702 0 Td[(ifi)]TJ/F15 11.955 Tf 12.702 0 Td[(+jgj)]TJ/F22 11.955 Tf 12.702 0 Td[(di;j)]TJ/F22 11.955 Tf 12.702 0 Td[(t.Themaximumfeasiblesocialwelfarecanbe

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57 formulatedas:P0:MaximizeV=Pi)]TJ/F22 11.955 Tf 11.955 0 Td[(ifixi)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pj+jgjyj)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pi;jdi;jzi;j)]TJ/F22 11.955 Tf 11.955 0 Td[(tSubjecttoPjzi;j=xiforeachi2IPizi;j=yjforeachj2Jxi2f0;1gforeachi2Iyi2f0;1gforeachj2Jzi;j2f0;1gforeachi2Iandj2JThisformulationisamatchingformulationandsatisesthecomplementarity-substitutabilityconditionsasshownbyShapley[ 57 ].Thus,theAC-DAmechanisminducedbyformulationP0isstrategy-proofandexpostindividual-rational.Furthermore,thesocialwelfareaccordingtoP0isnomorethanthecorrespondingquasi-linearsocialwelfareP.Thus,theAC-DAmechanisminducedbyformulationP0isalsoexpostbudget-balanced.Therefore,wecanimplementthecommissionsintothemechanismviaanon-quasi-linearsocialwelfarefunction.Thistechniquecanalsobeusedtoformulatetherealpercentagetransactioncostlikesalestax.

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CHAPTER6BILATERALEXCHANGEENVIRONMENTWITHTHESINGLEOUTPUTRESTRICTIONInthischapter,westudytheproblemofdesigningtruthfuldoubleauctionmechanismsfore-marketplaceswithmanybuyersandsellers,especiallytheindustrialprocurementsettingwheresellersaresmallinsizeandhavelittleornomarketpower.Sincethecomplementarity-substitutabilityconditions[ 57 ]nolongerholdinthisgeneralbilateralexchangeenvironment,weproposenewmechanismsthatarestrategy-proof,weaklybudget-balanced,individual-rational,andasymptoticallyecientunderthemulti-stagedesignapproach.Thesemechanismsarealsocapableofhandlingtransaction-relatedcosts,includingcostsrelatedtoproductquality,deliveries,andproductcustomization.Furthermore,oneofthemechanismsachievesasymptoticeciencybyjustsolvingtwolinearprograms.ThisissignicantconsideringthefactthattheeciencymaximizationproblemwithcompleteinformationisNP-hard.Theremainderofthischapterisorganizedasfollows.Section 6.1 speciestheexchangemodelandintroducesthesingleoutputrestriction.Section 6.2 proposesthebuyercompetitionLPmechanism,whichonlyneedtosolvelinearrelaxationofthesocialwelfare.Wethenimprovethismechanisminbothsocialeciencyandindividualpayos,andproposeanenhancedbuyercompetitionLPmechanisminSection 6.3 .Section 6.4 proposesanotherstrategy-proof,weaklybudget-balanced,andindividual-rationaldoubleauctionmechanism,themodiedbuyercompetitionmechanism.Section 6.5 providesasymptoticeciencyresultsforbothmechanisms. 58

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59 Section 6.6 demonstratesthebehaviorsofdierentmechanismsinonebilateralexchangeenvironment.TheproofsofthischapterarecompiledinSection 6.7 . 6.1 ModelandSingleOutputRestrictionAsbefore,letIdenotethegroupofbuyers,andJthegroupofsellers,hereafterbothcalledagents".Wealsorefertoabuyerasshe"andasellerashe".LetGdenotethesetofindivisiblecommodities.Weconsiderthefollowingbilateralexchangeenvironmentinwhicheachbuyerii2Iwantstopurchaseabundleofgoodsqi=qgig2Gandeachsellerjj2Jsuppliesabundleofgoodsqj=qgjg2G.Allqgksk2ISJ;g2Garenonnegativeintegerssincethecommoditiesareindivisible.Wealsoassumethatwhenbuyeripurchasesaunitofcommoditygfromsellerj,atransactioncostdi;j;gisincurred.Thetransactioncostincludescostsassociatedwithtransportation,quality,leadtime,customization,andthebuyer-vendorrelationship.Thus,eveniftwosellersprovidethesamecommodities,theymaystillbeheterogeneousduetothetransactioncosts.Thesetransactioncostscanbecommonknowledgeoronlyknownbytheauctioneer.Theauctioneerandeachagentmayormaynotknowthenumberofagentsinvolved,thestatisticaljointdistributionofthevaluations,oranyotherrelevantinformation.Weassumeaprivatevaluemodel,whereeachagent'svaluationofhis/herbundleisprivateinformation.Supposeallagentshavequasi-linearutility,thatis,ifanagentmakesnotransaction,hisorherutilitypayoiszero;otherwise,thepayoisthedierencebetweentheagent'svaluationofthebundleandtheamountofmoneytransferred.Theauctioneer'smonetarypayoisthetotalpaymentsfromthebuyers,lesstherevenuesofthesellersandtheneededtransactioncosts.Thesocialwelfareisthesummationoftheauctioneer'spayoandeachagent'sutility.Withoutlossofgenerality,wefocusonone-shotsealed-bidauctionmecha-nisms.Letfibethebidpriceofbuyeriforherbundle,andgjbethebidpriceof

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60 sellerj.Ifalltheagentsbidtruthfully,themaximumfeasiblesocialwelfarecanbeformulatedasthefollowingmixedintegerprogram:Q:MaximizePi2Ifixi)]TJ/F28 11.955 Tf 11.955 8.967 Td[(Pj2Jgjyj)]TJ/F28 11.955 Tf 11.955 8.967 Td[(Pi2I;j2J;g2Gdi;j;gzi;j;gSubjecttoPj2Jzi;j;g=qgixiforeachi2I;g2GPi2Izi;j;g=qgjyjforeachj2J;g2G0zi;j;gforeachi2I;j2J;g2Gxi2f0;1gforeachi2Iyj2f0;1gforeachj2Jwherexiandyjdenotewhetheranagenttradesintheauction,andzi;j;gspeciesthequantityofgoodgbuyeribuysfromsellerj.Thevariablesxi2I;yj2Jspecifytheresourceallocation.TheoptimizationproblemQanditscorrespondingdecisionproblem{theproblemofdecidingwhetherthereissomeallocationwithpositivewelfare{areNP-completeinthestrongsensebytransformingfrom3-PARTITIONproblem.Thus,itisunlikelythatanypolynomiallyimplementablemechanismwouldgeneratepositivewelfareinthegeneralbilateralexchangeenvironment.Inthischapter,wefocusontheindustrialprocurementsettingwheresellersaresmallinsizeandhavelittleornomarketpower.Specically,weconsiderane-marketplaceinwhicheachbuyerwantstopurchaseabundleofdierentitemsandeachsuppliersellercanproduceonlyasingleunitofonecommodity;thatis,Pg2Gqgj=1forj2J.Theassumptionwemakeaboutthesuppliersiscalledthesingleoutputrestriction[ 7 ].Theassumptionisnotaslimitedasitappearstobe.Forexample,asuppliertypicallyhasalimitedcapacity,andwemaynormalizethecapacityto1ifallthesuppliershavesimilarcapacities.Also,theauctioneermay

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61 restricttheamounteachsuppliercantradeinoneauction.Withthesingleoutputrestriction,weshowthatbiddingtruthfullyisthedominantstrategyforeachagentunderthelaterproposedmechanism.Evenifdierentsellershavedierentcapacitiesandareabletodeliverdierentamountsofcommodities,theagentshavelittleincentivetodeviatefromreportingtheirtruevaluationsundertheproposedmechanismaslongasthesellersaresmallinsizecomparedtothebuyersandhavelittleornomarketpower.Withthesingleoutputrestriction,theoptimizationproblemQisNP-hardbytransformingfromKNAPSACKproblem.But,thecorrespondingdecisionproblemofdecidingwhetherthereexistssomeallocationwithpositivewelfareispolynomialsolvable.Thissuggeststhatitispossibletodevelopapolynomiallyimplementablemechanismwithhigheciency. 6.2 BuyerCompetitionLPMechanismThekeydierencebetweenthebuyercompetitionmechanismandthebuyercompetitionLPmechanismisthatfortheallocationandpricingdecisions,thebuyercompetitionLPmechanismfocusesonthelinearrelaxationofthesocialwelfare:^QMaximizePi2Ifixi)]TJ/F28 11.955 Tf 11.955 8.967 Td[(Pj2Jgjyj)]TJ/F28 11.955 Tf 11.955 8.967 Td[(Pi2I;j2J;g2Gdi;j;gzi;j;gSubjecttoPj2Jzi;j;g=qgixiforeachi2I;g2GPi2Izi;j;g=qgjyjforeachj2J;g2G0zi;j;gforeachi2I;j2J;g2G0xi1foreachi2I0yj1foreachj2J

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62 Oneimmediatebenetofthislinearrelaxationisthattheresultingmechanismcanbesolvedecientlyandimplementedinpolynomialtime.Aswewillsee,thisformulationwill,infact,produceintegersolutionsinthenalallocationandinducetruth-telling.Beforeintroducingthemechanism,weneedtodenethefollowingnotation. Notation fi Thebidpriceofbuyeri. gj Thebidpriceofsellerj. di;j;g Thetransactioncostwhenbuyeripurchasesoneunitofcommoditygfromsellerj. ^VI0;J0 ThelinearrelaxationofsocialwelfareregardingtothebidsofbuyersetI0andsellersetJ0. ^V)]TJ/F23 7.97 Tf 6.587 0 Td[(kI0;J0 ThelinearrelaxationofsocialwelfareregardingtothebidsofagentsetI0SJ0nfkgk2I0SJ0. ^VkI0;J0 ThelinearrelaxationofsocialwelfareregardingtothebidsofagentsetI0SJ0andonemoreagentwhoisidenticaltoagentkk2I0SJ0. ^p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(kI0;J0 Theinmumsupremumofbidpricesofbuyersellerksatisfying^VI0;J0>^V)]TJ/F23 7.97 Tf 6.586 0 Td[(kI0;J0. ^p+kI0;J0 Theinmumsupremumofbidpricesofbuyersellerksatisfying^VkI0;J0>^VI0;J0.Forsimplicityofrepresentation,wemaydroptheparametersI0;J0whenthereferencestothebuyersetandthesellersetareobvious.Noteinthedenitionsof^p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(kand^p+kthattheinmumofanemptysetispositiveinnity,andthesupremumofanemptysetisnegativeinnity.When^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kand^p+karenite,thesepricescanbecalculatedviatheshadowprices.^p+iand^Viarecloselyrelatedtotheminimumshadowpriceoftheconstraintassociatedwithbuyeri,xi1.Weuse^V0i+todenotetheminimumshadow

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63 priceand^V0i)]TJ/F15 11.955 Tf 10.987 2.956 Td[(todenotethemaximumshadowpriceoftheconstraintassociatedwithbuyeri.Similarly,weuse^V0j+todenotetheminimumshadowpriceand^V0j)]TJ/F15 11.955 Tf -410.793 -20.952 Td[(todenotethemaximumshadowpriceoftheconstraintassociatedwithsellerj,yj1.Whentheminimumshadowprice^V0k+I;Jispositive,ifagentktradesat^p+kI;J,hisorherutilityequalstheminimumshadowprice^V0k+I;J.Thefollowingpropositionsformalizethisrelationship. Proposition6.2.1 Foralli2I,fi>^p+iI;Jifandonlyif^V0i+I;J>0. Proposition6.2.2 Foralli2I,if^V0i+I;J>0,then^p+iI;J=fi)]TJ/F15 11.955 Tf 13.742 3.022 Td[(^V0i+I;J.NowwepresenttheBC-LPmechanismunderthemulti-stagedesignapproach,inwhichweuse^p+sasthethresholdpricesforthebuyerside. Eachagentsubmitsonesealedbid. Forbuyeri2I,ifherbidfiisnomorethan^p+iI;J,sheiseliminatedfromtheauction.Let~Idenotethesetofremainingbuyers,~I=fijfi>^p+iI;J;i2Ig. Theitemsareallocatedamongtheremainingagents~IandJaccordingtotheoptimalsolutionto^V~I;J. Thetradingbuyerkpays^p+kI;J,andthetradingsellerlreceives^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(l~I;J.Becauseofthestructureofthemechanism,^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kand^V0k)]TJ/F15 11.955 Tf 10.986 3.109 Td[(arealsocloselyrelatedforeachtradingseller. Proposition6.2.3 Intheremainingsystem~IandJ,^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j~I;J=gj+^V~I;J)]TJ/F15 11.955 Tf -417.358 -20.886 Td[(^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J=gj+^V0j)]TJ/F15 11.955 Tf 7.085 2.955 Td[(~I;Jfortradingsellerj.Propositions 6.2.1 , 6.2.2 ,and 6.2.3 whoseproofsareinSection 6.7 enableustocalculate^p+isand^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(jsusingshadowprices.Furthermore,weonlyneedtocheckwhetherfi>^p+iI;Jforallthebuyers,thencalculate^p+sfortradingbuyersand^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(sfortradingsellers;wedonotneedtocalculateallthe^p+sand^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(s.

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64 SincetheBC-LPmechanismisbasedon^Q,thelinearrelaxationoffor-mulationQ,itisimportanttocheckwhethertheBC-LPmechanismrendersalegitimateallocationthatnobuyerreceivesapartialbundleandnosellersendsapartialbundle.Thefollowingtheoremshowsthatthiscanbeachievedbypickinganoptimalextremepointsolution: Theorem6.2.4 Alltheoptimalextremepointsolutionstoformulation^V~I;Jareinteger-valued.Theorem 6.2.4 andPropositions 6.2.1 , 6.2.2 ,and 6.2.3 leadtoapolynomialimplementationofthemechanism: Collectonesealedbidfromeachagent. Solvelinearprogram^VI;J. Foreachbuyeri,calculate^V0i+I;J,theminimumshadowpriceoftheconstraintassociatedwithbuyeri,xi1,in^VI;J.If^V0i+I;J>0,^p+iI;J=fi)]TJ/F15 11.955 Tf 13.742 3.022 Td[(^V0i+I;J.Otherwise,buyeriiseliminated. Solvelinearprogram^V~I;J,where~Iisthesetofremainingbuyers,andpickanoptimalextremepointsolution. Foreachtradingsellerj,calculate^p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(j~I;Jwhichequalsgj+^V0j)]TJ/F15 11.955 Tf 7.084 2.956 Td[(~I;J,where^V0j)]TJ/F15 11.955 Tf 7.085 2.956 Td[(~I;Jisthemaximumshadowpriceoftheconstraintassociatedwithsellerj,yj1,in^V~I;J. Conducttransactionsaccordingtotheoptimalsolutionto^V~I;J.Thetransactionpricefortradingbuyerkis^p+kI;J,andthetransactionpricefortradingsellerlis^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(l~I;J.So,toimplementthemechanism,weonlyneedtosolvetwolinearprograms^VI;Jand^V~I;Jandcalculatetheassociatedshadowprices.NextwepresentthemainresultsoftheBC-LPmechanism. Theorem6.2.5 TheBC-LPmechanismisexpostindividual-rational.

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65 Proof:Tradingbuyeribidsfiandpays^p+kI;J,sinceshesurvivesthebuyereliminationphase,fi>^p+kI;J.Thus,thedierencebetweenthebidpriceandthetransactionpricefortradingbuyerkispositive,andtheBC-LPmechanismisthereforeexpostindividual-rationalforeachtradingbuyer.Ifsellerltradesintheremainingsystemconsistingof~IandJ,hereceives^p)]TJ/F15 11.955 Tf 7.084 1.794 Td[(l~I;J.Forany>0,ifsellerlbidsgl)]TJ/F22 11.955 Tf 12.209 0 Td[(insteadofgl,^V~I;Jwillincrease,andwewillhave^V)]TJ/F23 7.97 Tf 6.587 0 Td[(l~I;J<^V~I;J.Thatis,^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(l~I;Jgl)]TJ/F22 11.955 Tf 12.347 0 Td[(forany>0.Thus,^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(l~I;Jgl;thismeansthatthesellerreceivesnolessthanhisbidprice,andtheBC-LPmechanismisexpostindividual-rationalforeachtradingseller.Sincethepayoofnon-tradingbuyersandsellersiszero,allthebuyersandsellersgetnonnegativepayos.Thus,theBC-LPmechanismisexpostindividualrationalforallbuyersandsellers. Theorem6.2.6 TheBC-LPmechanismisstrategy-proofinthebilateralexchangeenvironmentwiththesingleoutputrestriction. Theorem6.2.7 TheBC-LPmechanismisexpostweaklybudget-balancedinthebilateralexchangeenvironmentwiththesingleoutputrestriction.TheproofsofTheorems 6.2.6 and 6.2.7 areavailableinSection 6.7 .IntheproofofTheorem 6.2.6 ,wenoticethatforeachbuyer,acriticalthresholdpriceexistssuchthatshetradesherbundleifshebidsabovethispriceanddoesnottradeifshebidsbelow.Theoppositeistrueforeachseller.ThisobservationtellsusthatTheorems 6.2.5 , 6.2.6 ,and 6.2.7 donotdependontheassumptionthateachagent'sutilityisquasi-linear{eachagentonlyfacesatake-it-or-leave-it"situation.

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66 6.3 AnEnhancedBC-LPMechanism LetusrstlookattheexampleinFigure 6{1 ,inwhichtherearevesellers,twobuyers,andonecom-modity.Eachbuyerwantstwounitsofthecommodityandeachsellersuppliesoneunit.Transactioncostbetweenabuyerandaselleriszeroiftheyarelinkedinthegraphandpositiveinnityotherwise,andthebiddingpricesareshowninthegraph.Itturnsoutthat^p+forbothbuyersis6.Thus,bothofthemareeliminated,andnotransactiontakesplaceundertheBC-LPmechanism.sssssssHHHHHHHH f2=6f1=6g5=1g3=3g1=1g4=2g2=2 Figure6{1: ExamplefortheBC-LPMechanism Weproposethefollowingperturbationtechniquetoimprovetheeciencyforsituationslikethis.Fornotationalsimplicity,letusindexthebuyersbyiks,k=1;2;;jIj;andindexthesellersbyjks,k=1;2;;jJj.Weaddaperturbationfactorintoeachagent'sbidprice,thatis,wetreattheirbidsasfik+ikandgjk)]TJ/F22 11.955 Tf 12.892 0 Td[(jk,insteadoffikandgjk,foreachbuyerandseller,where1i1i2ijIjj1j2jjJj>0.WecallthecorrespondingBC-LPmechanismtheEnhancedBuyerCompetitionLPMechanism.Intheaboveexample,^p+forbothbuyersbecomes)]TJ/F15 11.955 Tf 13.266 0 Td[(2j3aftertheperturbation.Notethattheperturbedbidpricesforthebuyersare6+i1and+i2,respectively,bothofwhicharegreaterthan6)]TJ/F15 11.955 Tf 12.097 0 Td[(2j3.Therefore,bothofthemsurvive,andtwotransactionstakeplace.Buyer1receivesitemsfromsellers1and2,whilebuyer2receivesitemsfromsellers4and5.Eachbuyerpays6andeachsellerreceives3since1.Thisistheecientallocationforthesystem.WhyistheenhancedBC-LPmechanismanimprovementovertheBC-LPmechanism?Thisisbecauseafterperturbation,^p+smaychangebyadditionsand

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67 subtractionsoftheperturbationfactor,sosomeofthebuyerswhobidtheircriticalthresholdpricesmaysurvivetheeliminationphase.Theperturbationfactoressentiallyspeciesalexicographicorderamongthemultiplesolutionsto^VI;Jand^ViI;Ji2I.Thisorderisusedtodecidewhichoptimalextremepointsolutionthemechanismshouldpick.Tobetterun-derstandtherelationshipbetweentheperturbationtechniqueandthelexicographicorder,letusreexaminetheaboveexample.Afterweindexallthebuyersandsellers,theperturbationfactorsinducealexicographicorderamongalltheoptimalsolutionsto^VI;Jand^ViI;J.Whenthereismorethanoneoptimalsolution,wealwayspickonethatmaximizesx1.Ifx1ismaximizedinseveraloptimalsolu-tions,weselectonethatmaximizesx2.Ifbothx1andx2aremaximizedinseveraloptimalsolutions,weselectonethatmaximizesy1,andsoon,uptoy5.Therefore,theperturbationfactorsinducealexicographicorder.Ontheotherhand,eachlex-icographicorderspeciestheorderofmagnitudesoftheperturbationfactorsandhowtoindextheagents.Becauseofthesecorrespondences,wecaninterprettheenhancedBC-LPmechanismunderboththeperturbationfactorsrepresentationandthelexicographicorderrepresentation.Nowweapplybothrepresentationstotheaboveexample.Accordingtothebidpricesfisandgjs,^VI;Jhasuniqueoptimalsolution:x1=x2=y1=y2=y4=y5=1andy3=0.Neitherthelexicographicordernortheperturbationfactorsalterthisuniqueoptimalsolutionto^VI;J.Ontheotherhand,^Vi1I;Jhasmultipleoptimalsolutions:x2=y1=y2=y4=y5=1,x1=1+t=2,andy3=tfort2[0;1=2].Afterperturbation,^Vi1I;Jhasuniqueoptimalsolution:x2=y1=y2=y3=y4=y5=1,andx1=3=2.Since^VI;Jand^Vi1I;Jhavedierentuniqueoptimalsolutionswhile^Vi1I;Jhasalargerfeasibleregion,^Vi1I;J>^VI;J,andbuyer1survivestheeliminationphaseaftertheperturbation.Notethatwecanobtainthesameuniqueoptimalsolution

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68 to^Vi1I;Jusingtheinducedlexicographicorderanddetectthedierenceoftheoptimalsolutionswithoutexplicitlyapplyingtheperturbation.Moreover,theonlydierencebetween^VI;Jand^Vi1I;Jisthattheconstraintx11in^VI;Jbecomesx12in^Vi1I;J,thatis,theconstraintx11isrelaxedin^Vi1I;J.Therefore,ifwehavetheoptimalsolutionto^VI;J,weonlyneedtocheckwhetherthisoptimalsolutionchangesifwerelaxtheconstraintxi1insteadofsolving^Vi1I;Jfromscratch.Ingeneral,buyerisurvivesintheenhancedBC-LPmechanismifandonlyiftheoptimalsolutionbasedonthegivenlexicographicorderchangeswhenwerelaxtheconstraintxi1from^VI;J,becausethelatterisequivalentto^ViI;J>^VI;J.Sincethelexicographicorderinterpretationandtheperturbationfactorinterpretationareequivalent,wewillusewhateverishandyinthereminderofthispaper.TheseobservationsleadtoapolynomialimplementationoftheenhancedBC-LPmechanism: Collectonesealedbidfromeachagent. Generateanarbitrarylexicographicorder. Solvelinearprogram^VI;J,andpicktheoptimalsolutionbasedonthelexicographicorder. Foreachbuyeri,checkwhetherthisoptimalsolutionchangesifthecon-straintassociatedwithbuyeri,xi1,isrelaxed.Ifso,^p+iI;J=fi)]TJ/F15 11.955 Tf 14.462 3.022 Td[(^V0i+I;J,where^V0i+I;Jistheminimumshadowpriceoftheaboveconstraintin^VI;J;ifnot,buyeriiseliminated. Solvelinearprogram^V~I;J,where~Iisthesetofremainingbuyers,andpicktheoptimalsolutionbasedonthelexicographicorder.

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69 Foreachtradingsellerj,calculate^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j~I;J,whichequalsgj+^V0j)]TJ/F15 11.955 Tf 7.084 2.956 Td[(~I;J,where^V0j)]TJ/F15 11.955 Tf 7.085 2.955 Td[(~I;Jisthemaximumshadowpriceoftheconstraintassociatedwithsellerj,yj1,in^V~I;J; Conducttransactionsaccordingtotheoptimalsolutionto^V~I;J.Thetransactionpricefortradingbuyerkis^p+kI;J,andthetransactionpricefortradingsellerlis^p)]TJ/F15 11.955 Tf 7.084 1.794 Td[(l~I;J.ToimplementtheenhancedBC-LPmechanism,weonlyneedtosolvetwolinearprograms^VI;Jand^V~I;Jandcalculatetheassociatedshadowprices.TheenhancedBC-LPmechanisminheritsthedesiredpropertiesoftheBC-LPmechanism.Formally,wehave: Theorem6.3.1 TheenhancedBC-LPmechanismisstrategy-proof,expostindividual-rational,andexpostweaklybudget-balancedinthebilateralexchangeenvironmentwiththesingleoutputrestriction.Proof:Theproofoftheexpostweaklybudgetbalancepropertyremainsthesame.Sodotheproofsofthestrategy-proofnessandindividualrationalityproper-tiesforthesellers.Wenowprovethestrategy-proofnessandindividualrationalitypropertiesforthebuyers.Considertwoscenariosofbuyeri'sbidprice: Buyeribidshigherthan^p+iI;J:Weshowthatbuyeritradesherbundleat^p+iI;Junderthisscenario.Notethatifwerelaxtheconstraintassociatedwithbuyeri,theoptimalsolutionchangesastheoptimalobjectivefunctionvalueincreases.Thus,buyerisurvivesaccordingtotheprocedureoftheenhancedBC-LPmechanism.SinceallthesurvivingbuyerstradeintheBC-LPmechanism,buyeriacquiresherbundleat^p+iI;J. Buyeribidslowerthan^p+iI;J:Weshowthatbuyeridoesnottradeunderthisscenario.Supposeconstraintxi1,theconstraintassociatedwithbuyeri,isrelaxed,andbuyeribidsfi=^p+iI;J.Anyfeasiblesolution

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70 withxi>1isnobetterthananyoptimalsolutionto^VI;J.Asthebidpriceofbuyeridecreases,anyfeasiblesolutionwithxi>1isinferiortoanyoptimalsolutionto^VI;J.Thus,thereexistsnooptimalsolutionwithxi>1ifwerelaxtheconstraintassociatedwithbuyeri.Consequently,theoptimalsolutiondoesnotchange,andbuyeriiseliminated.Tosummarize,ifbuyeribidslowerthan^p+iI;J;shedoesnottrade;ifshebidshigherthan^p+iI;J,sheacquiresthebundleat^p+iI;J;andifshebids^p+iI;J,shedoeseither.Thetransactionpriceisnevermorethanthebidprice,thus,theenhancedBC-LPmechanismisexpostindividual-rationalforeachbuyer.Furthermore,biddingtruthfullyisaweaklydominatingstrategyforeachbuyer,thus,theenhancedBC-LPmechanismisalsostrategy-proofforeachbuyer. ThemostimportantadvantageoftheenhancedBC-LPmechanismisthatitcanachievehighereciencyandpayoscomparedtotheoriginalBC-LPmechanism.Formally,wehave: Theorem6.3.2 TheeciencyachievedbytheenhancedBC-LPmechanismisnolessthantheeciencyachievedbytheoriginalBC-LPmechanism.Proof:Wehaveseenthatthesetofremainingbuyers~Iislargerundertheen-hancedBC-LPmechanismenlargesthanundertheoriginalBC-LPmechanism.Bothmechanismsmaximize^V~I;Jinthenalallocation,buttheenhancedBC-LPmechanismachieveshighereciencysinceithasalargerfeasiblesolutionset. Theorem6.3.3 Eachagent'spayoundertheenhancedBC-LPmechanismisatleastashighasitundertheoriginalBC-LPmechanism.Proof:Foreachbuyer,herpayoiszeroiffi^p+iI;Jand^p+iI;J)]TJ/F22 11.955 Tf 12.575 0 Td[(fiiffi>^p+iI;Junderbothmechanisms.Therefore,eachbuyer'spayounder

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71 theenhancedBC-LPmechanismisatleastashighasitundertheoriginalBC-LPmechanism.Nowweconsiderthepayosofthesellers.Wehaveseenthatthesetofremainingbuyers~IislargerundertheenhancedBC-LPmechanismenlargesthanundertheoriginalBC-LPmechanism.Asalltheremainingbuyerstradeunderbothmechanisms,thetotaltransactionquantityundertheoriginalBC-LPmechanismisnomorethetotaltransactionquantityundertheenhancedBC-LPmechanismforeachcommodity.WhenasellerhaspositivepayoundertheoriginalBC-LPmechanism,hemustalsotradeundertheenhancedBC-LPmechanism.Notebothmechanismsaredeterministicmechanisms,wherethereissomecriticalpricesuchthatsellerjtradesatthiscriticalpriceifhisbidpriceislower,andhelosesthetransactionifhisbidpriceishigher.Thus,thecriticalpriceforeachsellerundertheenhancedBC-LPmechanismisnolessthanthecriticalpriceundertheoriginalBC-LPmechanism.Thepayoofatradingselleristhedierencebetweenthecriticalpriceandhisbidprice.Thus,eachseller'spayoundertheenhancedBC-LPmechanismisnolessthanhispayoundertheoriginalBC-LPmechanism. 6.4 ModiedBuyerCompetitionMechanismTheBC-LPmechanismreliesonthelinearrelaxation^Q,whoseoptimalsolutionmaybequitedierentfromtheoptimalsolutiontoQ.AninterestingquestionisthatwhetherwecandesignamechanismbytakingadvantageoftheinformationaboutthetrueproblemQ,namelytheecientallocationandassociatedp+sandp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(s.Becauseofthis,weproposethemodiedbuyercompetitionMBCmechanismunderthemulti-stagedesignapproach.SimilartotheenhancedBC-LPmechanism,weuseanarbitrarylexicographicordertouniquelydeterminetheoptimalsolution.WestatetheMBCmechanismasfollows:

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72 Collectonesealedbidfromeachagent. Generateanarbitrarylexicographicorder. CalculatetheVCGpaymentp)]TJ/F15 11.955 Tf 10.987 1.793 Td[(foreachagent. CalculatetheoptimalsolutiontoQbasedonthelexicographicorder. Removeallthebuyerswhoarenotinvolvedintheoptimalsolution.LetIdenotethesetoftradingbuyersintheoptimalsolution. Solvelinearprogram^VI;Jandpicktheoptimalsolutionbasedonthelexicographicorder. Foreachbuyeri,checkwhetherthisoptimalsolutionchangesifthecon-straintassociatedwithbuyeri,xi1,isrelaxed.Ifso,^p+iI;J=fi)]TJ/F15 11.955 Tf 14.462 3.022 Td[(^V0i+I;J,where^V0i+I;Jistheminimumshadowpriceoftheaboveconstraintin^VI;J;ifnot,buyeriiseliminated. Solvelinearprogram^V~I;J,where~Iisthesetofremainingbuyers,andpicktheoptimalsolutionbasedonthelexicographicorder. Foreachtradingsellerj,calculate^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j~I;Jbysolving^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(j=gj+^V0j)]TJ/F15 11.955 Tf 7.084 2.955 Td[(~I;J,where^V0j)]TJ/F15 11.955 Tf 7.085 2.955 Td[(~I;Jisthemaximumshadowpriceoftheconstraintassociatedwithsellerj,yj1,in^V~I;J. Conducttransactionsaccordingtotheoptimalsolutionto^V~I;J.ThetransactionpricefortradingbuyerkisthehigherofherVCGpriceand^p+kI;J,i.e.maxfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J;^p+kI;Jg;andthetransactionpricefortradingsellerlisthelowerofhisVCGpriceand^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j~I;J,i.e.minfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(lI;J;^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(l~I;Jg.TheMBCmechanismcanbeviewedasaBC-LPmechanismfollowingapreliminaryeliminationphasebasedontheoptimalsolutiontoformulationQandtheVCGprices.SimilartotheBC-LPmechanism,theMBCmechanismhascertaindesiredproperties.Formally,wehave:

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73 Theorem6.4.1 ThemodiedbuyercompetitionMBCmechanismisstrategy-proof,expostindividual-rational,andexpostweaklybudget-balancedinthebilateralexchangeenvironmentwiththesingleoutputrestriction. 6.5 AsymptoticEciencyTheoremsToevaluatetheeciencyofthemechanisms,weneedtospecifytheattributesoftheunderlyingbilateralexchangeenvironment.Weassumethatthereisanitenumberofcommodities,andthereexistsanumberMsuchthatPjGjg=1qgi
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74 bundletypetodenotewhetheranagentisasellerprovidingbundleq1;;qjGjPjGjg=1qg
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75 6.6 AnExampleInthissection,weuseanexampletoshowhowtheenhancedBC-LPmechanism,theMBCmechanism,andthebuyercompetitionmechanisminChapter 4 behaveunderthebilateralexchangeenvironmentwiththesingleoutputrestriction.TheexampleisillustratedinFigure 6{2 ,wheretherearetwocommodities,C1andC2.ForcommodityC1,thereare4sellers,allofwhichhavevaluationsnormalizedto0;forcommodityC2,thereare5sellers,wherethevaluationsare0,0,1,4,and4,respectively.Therearetwobuyers:buyer1wantsoneunitofC1andtwounitsofC2withvaluation9;buyer2wantstwounitsofC1andoneunitofC2withvaluation7.Theecientallocationforthisinstanceistoletthetwobuyerstransactwiththreelowestsellersofeachcommodity.Thesocialwelfareforthisallocationis15. Figure6{2:AnExchangeEnvironment Letusrstanalyzetheperformanceofthebuyercompetitionmechanismforthisexample.Underthismechanism,thethresholdpricesandthenaltransactionpricesforthemulti-stageapproacharecalculatedaccordingtotheintegerprogramQ.Ifallthebuyersandsellersbidtruthfully,itturnsoutthatthethresholdpriceforbothbuyers1and2is8.Sincebuyer2'sbidpriceislowerthanherthreshold

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76 price,sheiseliminated.Thecommoditiesareallocatedaccordingtotheecientsolutionamongbuyer1andthesellers,thatis,buyer1transactswithoneC1sellerandtwoC2sellers,whereeachofthesesellershasvaluation0.ThetransactionpriceforthisC1selleris0;thetransactionpricesfortheseC2sellersare1.Thetransactionpriceforbuyer1isherthresholdprice8.NotethattheC2sellerwithvaluation1hasincentivetomanipulatehisbid.Ifhebids3insteadof1,thesocialwelfarefortheecientallocationis13.Thethresholdpriceforbuyer1isstill8,whilethethresholdpriceforbuyer2becomes6.Then,bothbuyerssurvivetheeliminationphase,andthecommoditiesareallocatedaccordingtotheecientallocation,wheretwobuyerstransactwiththreelowestsellersofeachcommodity.ThetransactionpricefortheseC2sellersis4.TheC2sellerwithvaluation1isabletosellhisitematprice4ifhebids3insteadof1.Thus,thebuyercompetitionfailstobestrategy-proofinthebilateralexchangeenvironment.NowletusapplytheBC-LPmechanismtotheaboveexample,thatis,weuse^Q,thelinearrelaxationofthesocialwelfare,tocalculatethethresholdprices.Ifallthebuyersandsellersbidtruthfully,the^p+sforthebuyersare8and4,respectively.Bothbuyerssurvivetheeliminationphase,andthecommoditiesareallocatedaccordingtotheecientallocation.The^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(sfortradingsellersofC1andC2are0and4,respectively.TheBC-LPmechanismisstrategy-proof,thatis,biddingtruthfullyisaweaklydominantstrategyforeachbuyerandseller.NotethattheBC-LPmechanismalsoachievesstrongbudgetbalanceandtheecientallocationinthisexample.Sincep)]TJ/F15 11.955 Tf 10.987 1.793 Td[(iszeroforbothbuyers,theMBCmechanismrendersthesameallocationandthetransactionpricesastheBC-LPmechanism.

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77 6.7 Proofs 6.7.1 ProofsofPropositions 6.2.1 and 6.2.2 Inthissection,weexploretherelationshipbetween^V0i+and^p+i.Thefollowingnotationwillbeusedfrequently:xisacolumnvectorwithjIjentries,andtheithentryisxi;yisacolumnvectorwithjJjentries,andthejthentryisyj;0isacolumnvectorwhereallentriesare0;andekdenotesacolumnvectorwherethekthentryis1andallotherentriesare0.Letusstartbyreviewingsomeresultsforthefollowinglinearprogram:^VI;J;=MaximizePi2Ifixi)]TJ/F28 11.955 Tf 11.955 8.967 Td[(Pj2Jgjyj)]TJ/F28 11.955 Tf 11.955 8.967 Td[(Pi2I;j2J;g2Gdi;j;gzi;j;gSubjecttoPj2Jzi;j;g=qgixiforeachi2I;g2GPi2Izi;j;g=qgjyjforeachj2J;g2G0zi;j;gforeachi2I;j2J;g2G0x1+0y1+whereisacolumnvectorwithjIjentries,andisacolumnvectorwithjJjentries.Fortheremainderofthissection,wewillnotwriteI;Jifitisclearwhatwearereferringto.Notethataslongas+10and+10,thelinearprogramisfeasible.Furthermore,theobjectivefunction^V;isconcave,aswellaspiecewise-linearnondecreasingforeachcomponentofand. Lemma6.7.1 Foralli2I,fi>^p+iI;Jifandonlyif^Vei;0>^V0;0.Proof:^p+iistheinmumofbidpricesforbuyerithatmakes^Vi>^V.^Visthesameas^V0;0,while^Viissimply^Vei;0.If^Vei;0>^V0;0,fi^p+iaccordingtothedenitionof^p+i.Sinceboth^Vei;0and^V0;0arecontinuousfunctionsoffi,thereexistssome>0suchthat^Vei;0>^V0;0ifthecoecientintheobjectivefunctionisfi)]TJ/F22 11.955 Tf 12.611 0 Td[(

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78 insteadoffi.Then,fi)]TJ/F22 11.955 Tf 12.803 0 Td[(^p+iaccordingtothedenitionof^p+i.Thus,fi>^p+iif^Vei;0>^V0;0.Iffi>^p+i,accordingtothedenitionof^p+i,thereexistssomebidpricef0isuchthat^p+if0ifi.Also,ifwereplacethecoecientfiwithf0i,^Vei;0>^V0;0.Therefore,theoptimalsolutionfor^Vei;0musthavexi>1undercoecientf0i.Ifweincreasethecoecientfromf0itofi,^V0;0increasesbyatmostfi)]TJ/F22 11.955 Tf 12.012 0 Td[(f0i1=fi)]TJ/F22 11.955 Tf 12.012 0 Td[(f0i,and^Vei;0increasesbyatleastfi)]TJ/F22 11.955 Tf 12.012 0 Td[(f0ixi,whichisgreaterthanfi)]TJ/F22 11.955 Tf 11.955 0 Td[(f0i.Thus,^Vei;0>^V0;0iffi>^p+i. Since^V;ispiecewise-linearnondecreasingconcaveforeachcomponentofand,^Vrei;0isapiecewise-linearnondecreasingconcavefunctionofr,andtherightandleftderivativesof^Vrei;0mustexist.Denotetherightandleftderivativesof^Vrei;0atr=0as^V0i+and^V0i)]TJ/F15 11.955 Tf 7.084 2.956 Td[(,respectively.^V0i+and^V0i)]TJ/F15 11.955 Tf 10.986 2.956 Td[(aretheminimumandmaximumshadowpricesoftheconstraintassociatedwithbuyeri.Similarly,denotetherightandtheleftderivativesof^V0;rejatr=0by^V0j+and^V0j)]TJ/F15 11.955 Tf 7.085 2.955 Td[(,respectively.^V0j+and^V0j)]TJ/F15 11.955 Tf 10.986 2.955 Td[(aretheminimumandmaximumshadowpricesoftheconstraintassociatedwithsellerj.ProofofProposition 6.2.1 :ByLemma 6.7.1 ,fi>^p+iifandonlyif^Vei;0>^V0;0.Since^Vrei;0isanondecreasingconcavefunctionofr,^Vei;0>^V0;0ifandonlyiftherightderivativeof^Vrei;0withrespecttorat0ispositive;thatis,ifandonlyif^V0i+>0.Thus,foralli2I,fi>^p+iifandonlyif^V0i+>0. ProofofProposition 6.2.2 :ByProposition 6.2.1 ,if^V0i+>0,fi>^p+i,and^p+iisanitevaluethatisindependentofbidpricefi.AlsobyProposition 6.2.1 ,iffi=^p+i,thecorresponding^V0i+0.Since^Vrei;0isanincreasingfunctionofr,^V0i+0.Thus,^V0i+=0iffi=^p+i.Toprove^p+i=fi)]TJ/F15 11.955 Tf 14.677 3.022 Td[(^V0i+whenfi>^p+i,itsucestoshowthatiffiincreasesfrom^p+ito^p+i+forsome>0,thecorresponding^V0i+increasesfrom0to.

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79 Iffiisgreaterthan^p+i,byLemma 6.7.1 ,wehave^Vei;0>^V0;0;thus,xi1,theconstraintassociatedwithbuyeriinformulation^V0;0mustbetight.Asfiincreasesfrom^p+ito^p+i+,^V0;0increasesby.Foranyr>0,asfiincreasesfrom^p+ito^p+i+,^Vrei;0increasesbyatmost1+r;thus,^V0i+increasesbyatmost+r)]TJ/F15 11.955 Tf 12.24 0 Td[(=r=.Toshowthat^V0i+increasesfrom0to,itsucestoshowthatforanysmall>0,wecanndar>0suchthat^Vrei;0)]TJ/F15 11.955 Tf 14.593 3.022 Td[(^V0;0>)]TJ/F22 11.955 Tf 12.806 0 Td[(r.Nowsupposefi=^p+i+,then^Vei;0>^V0;0,andthereexistssomer>0suchthat^Vrei;0>^V0;0wheretheconstraintassociatedwithbuyeriinformulation^Vrei;0istight.Asfiincreasesfrom^p+i+to^p+i+,^Vrei;0increasesby+r)]TJ/F22 11.955 Tf 12.483 0 Td[(,while^V0;0increasesby)]TJ/F22 11.955 Tf 12.079 0 Td[(.Forr=r,^Vrei;0)]TJ/F15 11.955 Tf 13.865 3.022 Td[(^V0;0>)]TJ/F22 11.955 Tf 12.079 0 Td[(r,and^V0i+increasestoatleast)]TJ/F22 11.955 Tf 12.175 0 Td[(.Sincecanbearbitrarilysmall,^V0i+increasesfrom0toasfiincreasesfrom^p+ito^p+i+.Wecanconcludethatforalli2I,if^V0i+>0,^p+i=fi)]TJ/F15 11.955 Tf 13.741 3.022 Td[(^V0i+. 6.7.2 ProofsofTheorems 6.2.4 and 6.2.6 WeshowinthissectionthattheBC-LPmechanismrendersalegitimateallocationthatnobuyerreceivesapartialbundleandnosellersendsapartialbundle,andthismechanismisstrategy-proofinthebilateralexchangeenvironmentwiththesingleoutputrestriction.Letx=xi2Ibeanonnegativevector.Werstconsiderthefollowinglinearprogram^Mx:

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80 ^M:MaximizePi2Ifixi)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pj2Jgjyj)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pi2I;j2J;g2Gdi;j;gzi;j;gSubjecttoPj2Jzi;j;g=qgixiforeachi2I;g2GPi2Izi;j;g=qgjyjforeachj2J;g2G0zi;j;gforeachi2I;j2J;g2G0xixiforeachi2I0yj1foreachj2Jwherexis,yjs,andzi;j;gsaredecisionvariables.Sincethezerovectorisafeasiblesolution,weknowthat^Misbothfeasibleandbounded.^Misequivalentto^VI;Jifxi=1foralli2I.xtakesthisdefaultvalueintheremainderofthischapterunlesswespecicallyreassignanothervaluetoit.Letf^x;^y;^zg=f^xi2I;^yj2J;^zi2I;j2J;g2Ggbeanoptimalsolutionto^M. Lemma6.7.2 Letxi=1foralli2I.Iffi>^p+iI;J,then^xi=1.Proof:ByProposition 6.2.1 ,ifbuyeribidshigherthan^p+iI;J,thecorrespond-ingshadowpriceispositive,andximustequal1ineveryoptimalsolutionto^M. Let~x=~xi2Ibeanonnegativevectorsatisfying~x^xforanyoptimalsolutionf^x;^y;^zgto^M.WeconsiderthefollowinglinearprogramM~x:

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81 M:MaximizePi2Ifixi)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pj2Jgjyj)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pi2I;j2J;g2Gdi;j;gzi;j;gSubjecttoPj2Jzi;j;g=qgixiforeachi2I;g2GPi2Izi;j;g=qgjyjforeachj2J;g2G0zi;j;gforeachi2I;j2J;g2G0xi~xiforeachi2I0yj1foreachj2JMisalsofeasibleandbounded;thus,anoptimalsolutionexists.Misequivalentto^V~I;Jif~xiequalto1foralliwherefi>^p+iI;Jand0forallothervaluesofi.Lemma 6.7.2 guaranteesthat~x^xforanyoptimalsolutionf^x;^y;^zgto^M.Letfx;y;zg=fxi2I;yj2J;zi2I;j2J;g2GgbeanoptimalsolutiontoM.BytheFlowDecompositionTheoremAhujaetal.,1993,wecanviewthesolutionsasowssentfromsellerstobuyers.Letusconsiderthedierencebetweenanoptimalsolutionf^x;^y;^zgto^Mandanoptimalsolutionfx;y;zgtoM.Weknowthedierencecanberepresentedasnitepathowsandcycleows,thatis,ifwesendacertainamountofowsalongthesepathsandcycles,wecangetfx;y;zgfromf^x;^y;^zg.Moreover,ifweobtainthesepathsandcyclesaccordingtotheFlowDecompositionTheorem,wecansendpartialamountsoftheseowsalongthepathsandcycleswithoutviolatingthenon-negativeconstraintsofthearcs,astheowofeacharcwillbebetween^zandz.Eachpathconnectsasourcetoasink.Notethatnobuyercanbeasink,since^x~xx;thus,therearetwopossibilitiesforapathow:1asellertoaseller;2abuyertoaseller.Figure 6{3 illustrateswhattheowrepresentationsof^M,M,andthedierenceM)]TJ/F15 11.955 Tf 34.969 3.022 Td[(^Mmaylooklike.Inthisexample,theonlydierencebetween^M

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82 Figure6{3:IllustrationoftheFlows andMistheconstraintx~xinM,whichrequirestheinowofthemiddlebuyertobezero.Thedierenceintheowsof^MandMisrepresentedbytwoows:onegoesfromasellertoaseller,andtheothergoesfromabuyertoaseller. Lemma6.7.3 Foranyoptimalsolutionfx;y;zgtoM,x=~x.Proof:Considertheowdecompositionofthedierencebetweenfx;y;zgandf^x;^y;^zg.Consideralltheowsstartingfromaparticularbuyeriwhere~xi>xi.Whenwesendaproportionofalltheseowsfromf^x;^y;^zg,wegetanotherfeasiblesolutiontotheoriginalproblem^M,sinceeachselleronlyprovidesonetypeofcommodity.Sincexi<~xi^xiforeveryoptimalsolutionf^x;^y;^zgto^M,thegeneratedsolutioncannotbeoptimalfor^M,andtheobjectivefunctionvaluemustdecrease.Thus,ifxi<~xi,bysendingalltheseowsproportionallyinthereversedirections,wecangetafeasiblesolutiontoM,whichhashigherobjectivefunctionvaluethantheobjectivefunctionvalueoffx;y;zg.Thisisacontradiction!Thus,wemusthavex=~x. ProofofTheorem 6.2.4 :Setxiequalto1foralli2I.Set~xiequalto1foralliwherefi>^p+iI;Jandequalto0forallothervaluesofi.^MandMareequivalentto^VI;Jand^V~I;J,respectively.ByLemma 6.7.3 ,x=~xinevery

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83 optimalsolutiontoM.Giventhattheoptimalxisareinteger-valuedandthatthesingleoutputrestrictionappliestothesellers,thelinearprogramMisanetworkowproblem.Thus,alltheextremepointsoftheoptimalsolutionsetof^V~I;Jareinteger-valued. Next,weshowthattruthfulrevelationisadominantstrategyforeachbuyer. Lemma6.7.4 Ifbuyeri2Ibidshigherthan^p+iI;J,thenshetradesherbundle.Proof:Setxiequalto1foralli2I.Set~xiequalto1foralliwherefi>^p+iI;Jandequalto0forallothervaluesofi.ByLemma 6.7.3 ,weknowthatintheoptimalsolutiontoM,xi=~xi=1;thatis,buyeritradesherbundleforsure. Theorem6.7.5 Biddingtruthfullyisaweaklydominantstrategyforeachbuyer.Proof:Notethat^p+iI;JisdeterminedbytheagentsetISJnfkg,andthatitisindependentofthebidpricefi.Also,ifbuyeritrades,herpaymentis^p+iI;J.Thus,ifthebuyer'svaluationishigherthan^p+iI;J,shepreferstotrade,whichwillhappenifshebidshervaluationasshowninLemma 6.7.4 .Ifthebuyer'svaluationislowerthan^p+iI;J,sheprefersnottotrade,whichcanalsobeachievedbybiddinghervaluation.Ifthebuyer'svaluationisequalto^p+iI;J,sheisindierentbetweentradingandnottrading.Thus,biddingtruthfullyisaweaklydominantstrategyforeachbuyer. Thefollowinglemmasshowthattruthfulrevelationisadominantstrategyforeachseller. Lemma6.7.6 Thereisanoptimalsolutionfx0;y0;z0gtoMsuchthaty0^y.Proof:Considertheowdecompositionofthedierencebetweentheoptimalsolutionsfx;y;zgandf^x;^y;^zg.Especiallyconsideralltheowsfromasellertoanotherseller.Startingfromf^x;^y;^zg,wecansendtheseowsalongthepathtogetanotherfeasiblesolutionto^Msinceeachselleronlyprovidesonetypeof

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84 commodity.Sincef^x;^y;^zgisoptimalfor^M,sendingtheseowsdoesnotincreasetheobjectivefunctionvalueof^M.Wewillshowbycontradictionthatsendingtheseowsdoesnotdecreasetheobjectivefunctionvalueeither.Supposesendingtheseowsdecreasestheobjectivefunctionvalue.Thensendingtheseowsinthereversedirectionshouldincreasetheobjectivefunctionvalue.Notestartingfromfx;y;zg,ifwesendtheseowsinthereversedirection,wecangetanotherfeasiblesolutionforM.Thus,wecanimprovetheoptimalsolutionfx;y;zgtoMbysendingalltheseowsinthereversedirections.Thatisacontradiction!Thus,sendingtheseowsineitherdirectiondoesnotchangetheobjectivefunctionvalue.Startingfromfx;y;zg,wecangetanoptimalsolutionfx0;y0;z0gtoMbysendingtheseowsinthereversedirections.Itisstraightforwardtocheckthatthisoptimalsolutionfx0;y0;z0gtoMsatisfyingy0^y. Lemma6.7.7 Thereisanoptimalsolutionf^x0;^y0;^z0gto^Msuchthaty^y0.Proof:TheproofissimilartoLemma 6.7.6 .Considertheowdecompositionofthedierencebetweentheoptimalsolutionsfx;y;zgandf^x;^y;^zg.Especiallyconsideralltheowsfromasellertoanotherseller.Startingfromf^x;^y;^zg,wecangetanoptimalsolutionf^x0;^y0;^z0gto^Msatisfyingy^y0bysendingalltheseowsalongthepaths. Let^pjdenotethemaximumbidpriceforsellerjsuchthatthereexistsanoptimalsolutionto^VI;Jwithyj=1.Werstshowthatifsellerjbidshigherthan^pj,hedoesnottrade. Lemma6.7.8 Ifsellerj2Jbidshigherthan^pj,thenhedoesnottrade.Proof:Setxiequalto1foralli2I.Set~xiequalto1foralliwherefi>^p+iI;Jandequalto0forallothervaluesofi.^MandMareequivalentto^VI;Jand^V~I;J,respectively.Sincesellerj2Jbidshigherthan^pj,nooptimalsolutionto^Mhasyj=1.ByLemma 6.7.7 ,Mhasnooptimalsolutionwithyj=1.Sincealltheextremepointsoftheoptimalsolutionsetof^V~I;Jare

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85 integer-valuedbyTheorem 6.2.4 ,yj=0ineveryoptimalsolution.Thatis,sellerjdoesnottradeintheremainingsystemconsistingof~IandJ. Lemma6.7.9 Noremainingbuyeriseliminatedifsellerjincreaseshisbid,aslongashisbidisnogreaterthan^pj.Proof:fi>^p+iI;Jisequivalentto^Vei;0)]TJ/F15 11.955 Tf 14.328 3.022 Td[(^V0;0>0byLemma 6.7.1 .Ifsellerjincreaseshisbidpriceby,aslongashebidsnomorethan^pj,^V0;0decreasesby,while^Vei;0decreasesbyatmost.Thus,thedierence^Vei;0)]TJ/F15 11.955 Tf 13.742 3.022 Td[(^V0;0isstillpositive,andbuyeriisnoteliminated. Nowletusconsiderthesetofremainingbuyersasthebidpricegjofsellerjgoesfrom^pjtonegativeinnity.ByLemma 6.7.9 ,thesetofremainingbuyersbecomessmallerandsmallerasthebidpricedecreases.Weuse~Ijtodenotethelimitofthesetofremainingbuyers.Notethatsincethereisonlyanitenumberofpossiblesetsofremainingbuyers,ifsellerjbidslowenough,thecorrespondingsetofremainingbuyersbecomesthelimitset,~Ij.Weuse~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(jtodenote^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(jIj;J,thesupremumofbidpricesofsellerjsatisfying^VIj;J>^V)]TJ/F23 7.97 Tf 6.586 0 Td[(jIj;J.Wehavethefollowinglemmas: Lemma6.7.10 Forallj2J,~p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(j^pj.Proof:Weprovethisbycontradiction.Assumethatforsomesellerj2J,^pj<~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j.ByLemma 6.7.9 ,ifsellerjbids^pj,allbuyersin~Ijsurvivetheeliminationphase;thatis,^Vei;0)]TJ/F15 11.955 Tf 14.07 3.022 Td[(^V0;0>0foralli2~Ij.Since^V0;0and^Vei;0i2~Ijarecontinuousfunctionsofbidpricegj,thereexistssome>0thatsatises^pj+<~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(jand^Vei;0)]TJ/F15 11.955 Tf 13.464 3.022 Td[(^V0;0>0foralli2~Ijifsellerjbids^pj+.Now,assumethatsellerjbids^pj+.Setxiequalto1foralli2I.Set~xiequalto1foralli2~Ijandequalto0forallothervaluesofi.Atthesevalues,^MandMareequivalentto^VI;Jand^V~Ij;J,respectively.ToapplyLemma 6.7.7 ,weneedtoshow~x^x,whichholdsbecause^Vei;0)]TJ/F15 11.955 Tf 14.175 3.022 Td[(^V0;0>0.Thatis,^xi=1foralli2~Ij,and~x^x.Sellerj'sbidpriceishigherthan^pj;thus,

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86 Table6{1:NotationfortheProofs NameDenition/Restriction x=xi2Isomenonnegativevector ^MxMaximizePi2Ifixi)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pj2Jgjyj)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pi2I;j2J;g2Gdi;j;gzi;j;gSubjecttoPj2Jzi;j;g=qgixiforeachi2I;g2GPi2Izi;j;g=qgjyjforeachj2J;g2G0zi;j;gforeachi2I;j2J;g2G0xixiforeachi2I0yj1foreachj2J f^xi2I;^yj2J;^zi2I;j2J;g2Gganoptimalsolutiontoformulation^M ~x=~xi2Ianonnegativevectorsatisfying~x^xforanyoptimalsolutionf^x;^y;^zgto^M M~xMaximizePi2Ifixi)]TJ/F28 11.955 Tf 11.955 8.967 Td[(Pj2Jgjyj)]TJ/F28 11.955 Tf 11.955 8.967 Td[(Pi2I;j2J;g2Gdi;j;gzi;j;gSubjecttoPj2Jzi;j;g=qgixiforeachi2I;g2GPi2Izi;j;g=qgjyjforeachj2J;g2G0zi;j;gforeachi2I;j2J;g2G0xi~xiforeachi2I0yj1foreachj2J fxi2I;yj2J;zi2I;j2J;g2GganoptimalsolutiontoformulationM ^pjthemaximumbidpriceforsellerjsuchthatthereexistsanoptimalsolutionto^VI;Jwithyj=1 ~Ijthelimitremainingbuyersetasthebidpricegjofsellerjgoestonegativeinnity ~p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(j^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(j~Ij;J,thesupremumofbidpricesofsellerjsatisfying^VIj;J>^V)]TJ/F23 7.97 Tf 6.586 0 Td[(jIj;J

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87 ^yj<1ineveryoptimalsolutionto^M.ByLemma 6.7.7 ,yj<1ineveryoptimalsolutiontoM.Also,fromLemma 6.7.3 ,x=~xineveryoptimalsolutiontoM.Giventhattheoptimalxisareinteger-valuedandthatthesingleoutputrestrictionappliestothesellers,Misanetworkowproblem;thus,alltheextremepointsfortheoptimalsolutionsetofMareinteger-valued.Sowemusthaveyj=0ineveryoptimalsolutiontoM.Thiscontradictstotheassumptionthatthebidprice^pj+islowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j.Thus,forallj2Jwithanite^pj,~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j^pj. Lemma6.7.11 Ifsellerjbidslowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j,hetrades.Proof:Setxiequalto1foralli2~Iandequalto0forallothervaluesofi.Set~xiequalto1foralli2~Ijandequalto0forallothervaluesofi.Atthesevalues,^MandMareequivalentto^V~I;Jand^V~Ij;J,respectively.ByLemma 6.7.4 ,eachremainingbuyerhasxi=1ineveryoptimalsolutionto^V~I;J,andbyLemma 6.7.10 ,ifsellerjbidslowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j,healsobidslowerthan^pj.FinallyfromLemma 6.7.9 ,weknowthatthesetofremainingbuyers~Icontains~Ij,thus,~x^x.Sincesellerjbidslowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j,yj>0ineveryoptimalsolutionfx;y;zgtoM.ByLemma 6.7.3 ,x=~xineveryoptimalsolutiontoM.Giventhattheoptimalxisareinteger-valuedandthatthesingleoutputrestrictionappliestothesellers,Misanetworkowproblem;thus,alltheextremepointsfortheoptimalsolutionsetofMareinteger-valued,andyj=1ineveryoptimalsolutiontoM.ByLemma 6.7.6 ,theoptimalsolutionto^Mmustalsohave^yj=1.Therefore,sellerjtradesifhebidslowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j. Lemma6.7.12 If~I6=~Ij,sellerjdoesnottrade.Proof:ByLemma 6.7.8 ,ifsellerjbidshigherthan^pj,hedoesnottrade.Now,considerthecaseinwhichthebidpricegjisnomorethan^pj.ByLemma 6.7.9 ,thesetofremainingbuyers~Ialwayscontains~Ij.Since~I6=~Ij,theremustbesomebuyerl2In~Ijinthesetofremainingbuyers~I.Foreachremainingbuyeri2~I,byLemma 6.7.1 ,^Vei;0>^V0;0.^V;isa

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88 continuousfunction;thusthereexistssomei>0suchthat^Vei+iel;0>^Vel;0.Becausethereisanitenumberofbuyers,thereexistsan>0suchthat^Vei+el;0>^Vel;0foralltheremainingbuyers~I.Thismeansthatintheoptimalsolutionsto^Vel;0,xi=1foralli2~Inflg.Sincebuyerlsurvivestheeliminationphase,Proposition 6.2.1 impliesthat^Vel;0>^V0;0,andtheoptimalxlineveryoptimalsolutionto^Vel;0isgreaterthan1.Nowsetxiequalto1foralli2Inflgandxl=1+.Set~xiequalto1foralli2~Iandequalto0forallothervaluesofi.Atthesevalues,^MandMareequivalentto^Vei;0and^V~I;J,respectively.Recallwehaveshownthatineveryoptimalsolution^x;^y;^zto^Vel;0,^xi=1foralli2~Inflgand^xl>1.Thus,~x^x.Notethatifsellerjbidslowenough,buyerliseliminatedand^Vel;0=^V0;0.Ifthebidpricegjincreases,aslongasgj^pj,^V0;0decreasesatthesamerate,and^Vei;0mustdecreaseataslowerratetohave^Vel;0>^V0;0.Thus,yj<1ineveryoptimalsolutionto^Vel;0.Since^yj<1ineveryoptimalsolutionf^x;^y;^zgto^M,byLemma 6.7.7 ,theoptimalsolutiontoMmusthaveyj<1.Furthermore,byLemma 6.7.3 ,x=~xineveryoptimalsolutiontoM.Giventhattheoptimalxisareinteger-valuedandthatthesingleoutputrestrictionappliestothesellers,Misanetworkowproblem;thus,alltheextremepointsfortheoptimalsolutionsetofMareinteger-valued.Sowemusthaveyj=0ineveryoptimalsolutiontoM.Thatis,sellerjdoesnottradewhenthesetofremainingbuyersisnot~Ij. Corollary6.7.13 Ifsellerjbidslowerthan~p)]TJ/F15 11.955 Tf 7.084 1.794 Td[(j,~I=~Ij. Theorem6.7.14 Biddingtruthfullyisaweaklydominantstrategyforeachseller.Proof:ByLemma 6.7.8 ,ifsellerjbidshigherthan^pj,hedoesnottrade.ByLemma 6.7.11 ,ifsellerjbidslowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j,hetrades.

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89 Now,considerthecaseinwhichthebidpricegjliesbetweenthesetwonumbers;i.e.~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(j
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90 Assellerjlowershisbidpricefrom~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(jto~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(j)]TJ/F15 11.955 Tf 12.914 0 Td[(,~I=~Ij,^V~I;Jincreasesby,and^V)]TJ/F23 7.97 Tf 6.587 0 Td[(j~I;Jremainsthesame.Theconstraintassociatedwithsellerjin^V~I;J0;rej)]TJ/F15 11.955 Tf 9.299 0 Td[(1r0istightaslongasthebidpriceislowerthan~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(j,so^V~I;J0;rejincreasesby+rfor)]TJ/F15 11.955 Tf 9.298 0 Td[(1r0assellerjlowershisbidprice.Thus,^V0j)]TJ/F15 11.955 Tf 7.085 2.955 Td[(~I;J=,and^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j=gj+^V~I;J)]TJ/F15 11.955 Tf 12.461 3.022 Td[(^V)]TJ/F23 7.97 Tf 6.587 0 Td[(j~I;J=gj+^V0j)]TJ/F15 11.955 Tf 7.085 2.955 Td[(~I;Jfortradingsellerj. 6.7.4 ProofofTheorem 6.2.7 Inthissection,weshowthattheBC-LPmechanismisexpostweaklybudget-balancedinthebilateralexchangeenvironmentwiththesingleoutputrestriction.Weaccomplishthisbyusingthefollowinglemmas. Lemma6.7.15 Forallj2J,xi=1foralli2~Iineveryoptimalsolutionto^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J.Proof:Ifsellerjdoesnottrade,yj=0intheoptimalsolutionto^V~I;J,and^V)]TJ/F23 7.97 Tf 6.587 0 Td[(j~I;J=^V~I;J.Sincexi=1foralli2~Iineveryoptimalsolutionto^V~I;J,thesameistrueforeveryoptimalsolutionto^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J.Nowconsiderthecasewheresellerjtrades.ByTheorem 6.7.5 ,sellerjbidsnomorethan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j,andthesetofremainingbuyers~I=~Ij.ByLemma 6.7.9 ,noremainingbuyeriseliminatedifsellerjincreaseshisbid,aslongashisbidisnogreaterthan^pj.ByLemma 6.7.10 ,~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j^pj,soifsellerjbids~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j,noremainingbuyeriseliminated.Thatis,^Vei;0>^V0;0foreachbuyeri2~IbyLemma 6.7.1 whengj=~p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(j.Nowassumethatsellerjraiseshisbidtoabove~p)]TJ/F15 11.955 Tf 7.084 1.794 Td[(j.Because^V;iscontinuous,thereexistssome>0suchthat^Vei;0>^V0;0aslongasgj<~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j+.Thus,ifsellerjbidsbetween~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(jand~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(j+,xi=1foralli2~Iineveryoptimalsolutionto^V0;0.Setxiequalto1foralli2I.Set~xiequalto1foralli2~Iandequalto0forallothervaluesofi.Atthesevalues,^MandMareequivalentto^VI;Jand^V~I;J,respectively.Sincexi=1for

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91 alli2~Iineveryoptimalsolutionto^V0;0,~x^x.ByLemma 6.7.3 ,x=~x;thatis,xi=xi=1fori2~Iineveryoptimalsolutionto^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J.Sincesellerjbidsabove~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j,yj=0ineveryoptimalsolutionto^V~I;J;thatis,^V~I;J=^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J.Notethattheoptimalsolutionsetfor^V)]TJ/F23 7.97 Tf 6.587 0 Td[(j~I;Jisindependentfrombidpricegj.Thereforewehavexi=1foralli2~Iineveryoptimalsolutionto^V)]TJ/F23 7.97 Tf 6.587 0 Td[(j~I;J. Lemma6.7.16 Ifsellerkbidslowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,thenyk=1ineveryoptimalsolutionto^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J.Proof:Weprovethislemmabycontradiction.Considertheowdecompositionofthedierencebetweenanoptimalsolutionto^V~I;Jandanoptimalsolutionto^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J,whereyk<1.ByLemmas 6.7.4 and 6.7.15 ,xi=1foralli2~Iineveryoptimalsolutionto^V~I;Jandto^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J.Thus,alltheowsmustgofromasellertoaseller.Sincekbidslowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,yk=1intheoptimalsolutionto^V~I;J,andyk<1intheoptimalsolutionto^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J.Therefore,thereexistsaowstartingfromsellerkintheowdecomposition.Startingfromtheoptimalsolutionto^V~I;J,sendingthisowinreversedirectiongivesusanotherfeasiblesolutionto^V~I;Jbecauseeachselleronlyprovidesonetypeofcommodity.Furthermore,sinceyk<1inthisfeasiblesolution,theobjectivefunctionvaluemustdecrease.Thus,startingfromtheoptimalsolutionto^V)]TJ/F23 7.97 Tf 6.587 0 Td[(j~I;J,thisowgivesusananotherfeasiblesolutionto^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;Jwithahigherobjectivefunctionvaluethanitsoptimalsolution.Thisisacontradiction!Thus,yk=1ineveryoptimalsolutionto^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J. Lemma6.7.17 Supposesellerkbidsgk,whichislessthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,andheraiseshisbidto~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k)]TJ/F22 11.955 Tf 12.179 0 Td[(kk>0.Thenithesetofremainingbuyersand^p+iI;Jforbuyeriinthesetdonotchange;ii~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(jfortradingsellerj6=kdoesnotchange;iiitheoriginalecientallocationof^V~I;Jisstillecient.

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92 Proof:iByTheorem 6.7.14 ,thesetofremainingbuyers~Iis~Ikaslongassellerkbidslowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j.Considerthecasewheresellerkraiseshisbidpriceto~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(k)]TJ/F22 11.955 Tf 11.955 0 Td[(k.ByLemma 6.7.1 ,^Vel;0>^V0;0foreachremainingbuyerl2~I.Because^V;iscontinuous,thereexistssome>0suchthat^Vel+ei;0>^Vei;0if0<1,sincei2~I.Nowsetxlequalto1foralll2Infigandxiequalto1+.Set~xiequalto1foralli2~Iandequalto0forallothervaluesofi.Atthesevalues,^MandMareequivalentto^Vei;0and^V~I;J,respectively.Sinceineveryoptimalsolutionf^x;^y;^zgto^Vel;0,^xl=1foralll2~Infig,and^xi>1,~x^x.Everyoptimalsolutionto^V~I;Jhasyk=1,aslongassellerkbidslowerthan~p)]TJ/F15 11.955 Tf 7.084 1.794 Td[(j.ByLemma 6.7.6 ,everyoptimalsolutionto^Vei;0<
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93 ^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(j~I;J=gj+^V~I;J)]TJ/F15 11.955 Tf 14.153 3.022 Td[(^V)]TJ/F23 7.97 Tf 6.586 0 Td[(j~I;J,and~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(jfortradingsellerjremainsthesame.iiiSincesellerktradesineveryecientallocationfor^V~I;Jifhebidslowerthan~p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kbyLemma 6.7.11 ,theamountofthesocialwelfaredecreasedisthesameastheamountequaltothechangeofhisbid.Thus,iiifollows. Lemma6.7.18 Supposebuyerkbidsfk>^p+kI;J,thenlowersherbidto^p+kI;J+kk>0.Weshowithesetofremainingbuyersand^p+iI;Jforbuyeriinthesetdonotchange;iitheoriginalecientallocation^V~I;Jisstillecient.Proof:iIfbuyerkbids^p+k+k,byLemma 6.7.1 ,^Vek;0>^V0;0.Because^V;iscontinuous,thereexistssome>0suchthatforalli2I,^Vek+ei;0>^Vei;0aslongas0<<.Thus,theconstraintassociatedwithbuyerkmustbetightin^Vei;0,andxk=1ineveryoptimalsolutionto^Vei;0.Nowconsiderthecasewherebuyerkbidshigherthan^p+kI;J+k.xkmuststillequal1ineveryoptimalsolutionto^Vei;0.Thus,ifbuyerklowersherbidpricefkto^p+kI;J+k,^Vei;0and^V0;0decreasethesameamount,^Vei;0)]TJ/F15 11.955 Tf 14.631 3.022 Td[(^V0;0remainsthesame,and^V0i+I;Jremainsthesame.ByProposition 6.2.1 ,thesetofremainingbuyersdoesnotchange.ByProposition 6.2.2 ,foralli2~I,^p+iI;J=fi)]TJ/F15 11.955 Tf 14.124 3.022 Td[(^V0i+I;J;thus,^p+iI;Jforbuyeriremainsthesame.iiByLemma 6.7.4 ,ifbuyerkbidshigherthan^p+kI;J,shetradesineveryecientallocationfor^V~I;J,andthesocialwelfaredecreasesbyanamountequaltothechangeofherbid.Thus,iifollows. ProofofTheorem 6.2.7 :ByLemma 6.7.17 ,ifweletallthetradingsellerswhobidlowerthan~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kraisetheirbidsto~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k)]TJ/F22 11.955 Tf 12.413 0 Td[(kforsomek>0onebyone,thesetofremainingbuyersdoesnotchangeandtheoriginalallocationintheremainingsystemisstillecient.Then,byLemma 6.7.18 ,wecanletallthe

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94 tradingbuyerswhobidhigherthan^p+kI;Jlowertheirbidsto^p+kI;J+kforsomek>0onebyonewhilekeepingtheoriginalallocationintheremainingsystemecient.Becausetheoriginalallocationintheremainingsystemisstillecient,theobjectivefunctionvaluemustbenon-negative,sincezeroisafeasiblesolution.WehaveP^p+iI;J+ixi)]TJ/F28 11.955 Tf 12.313 8.966 Td[(P~p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(j)]TJ/F22 11.955 Tf 12.313 0 Td[(jyj)]TJ/F28 11.955 Tf 12.312 8.966 Td[(Pdi;j;gzi;j;g0fortheoptimalallocationx;y;zintheremainingsystem.Sincetheksarearbitrarypositivenumbers,P^p+iI;Jxi)]TJ/F28 11.955 Tf 12.28 8.966 Td[(P~p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(jyj)]TJ/F28 11.955 Tf 12.281 8.966 Td[(Pdi;j;gzi;j;g0,i.e.theBC-LPmechanismisexpostweaklybudget-balanced. 6.7.5 ProofofTheorem 6.4.1 Inthissection,weprovetheexpostweaklybudgetbalance,expostindividualrationality,andstrategy-proofnesspropertiesoftheMBCmechanism.Inordertoprovetheseproperties,weneedthefollowinglemma: Lemma6.7.19 TradingsellerlundertheMBCmechanismalsotradesintheoptimalsolutiontoQ.Proof:ConsidertheowrepresentationofthedierenceoftheoptimalsolutiontoQandthenalallocation.Weprovebycontradiction.AssumethatsellerltradesinthenalallocationanddoesnottradeintheoptimalsolutiontoQ.Thentheremustbeaowsendingfromsellerltosomesellerjinthedierence.Sincebothsolutionsareinteger-valued,theowsendsexactlyoneunitofthecommoditysellerlsupplies.NotethatstartingfromtheoptimalsolutiontoQ,sendingthisunitowgivesusafeasiblesolutiontoQ.Duetotheperturbation/lexicographicorder,Qhasauniqueoptimalsolution,sothisfeasiblesolutionmustbeinferior.Now,startingfromthenalallocation,sendingthisowinreversedirectionimprovesthenalallocation.Thisisacontradiction.Thus,undertheMBCmechanism,eachtradingsellerisinvolvedinthetransactionundertheoptimalsolutiontoQ. ProofofTheorem 6.4.1 :

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95 expostweaklybudgetbalance :wecomparethefollowingtwoscenarios:1wehavebuyersetIandsellersetJ,andweapplytheMBCmechanism;2wehavebuyersetIandsellersetJ,andweapplytheBC-LPmechanism.Notethatbothscenariosproducesameallocations.Tradingbuyerkinscenario1paysatleastthepriceinscenario2,sincemaxfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J;^p+kI;Jg^p+kI;J,andtradingsellerlinscenario1receivesatmostthepriceinscenario2,sinceminfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(lI;J;^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(l~I;Jg^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(l~I;J.SincetheBC-LPmechanismisexpostweaklybudget-balanced,theMBCmechanismisalsoweaklybudget-balanced.expostindividualrationality :ConsiderthescenarioinwhichwehavebuyersetIandsellersetJ,andapplytheBC-LPmechanism.SincetheBC-LPmech-anismisexpostindividual-rational,tradingbuyerkmustbidnolessthanher^p+kI;J,whiletradingsellerlmustbidnomorethanhis^p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(l~I;J.Now,considerthescenarioinwhichwehavebuyersetIandsellersetJ,andweap-plytheMBCmechanism.Tradingbuyerkmustbidnolessthanp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;JbecauseshetradesintheoptimalsolutiontoQ.Thus,tradingbuyerk'sbidpricefkmaxfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J;^p+kI;Jg,andeachbuyer'spayoisnonneg-ative.ByLemma 6.7.19 ,tradingsellerlalsotradesintheoptimalsolutiontoQ,andmustbidnomorethanp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(lI;J.Thus,tradingsellerl'sbidpriceglminfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(lI;J;^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(l~I;Jg,andeachseller'spayoisnonnegative.Strategy-proofnessforthebuyerside :foreachbuyerk,p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kI;Jisdeter-minedbytheagentsetISJnfkg.IfbuyerktradesintheoptimalsolutiontoQ,Iisindependentofthebidpricefk.Thus,ifbuyerktrades,bothp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(kI;Jand^p+kI;Jareindependentofthebidpricefk.Bythestrategy-proofnessoftheBC-LPmechanism,buyerktradesatmaxfp)]TJ/F15 11.955 Tf 7.085 1.794 Td[(kI;J;^p+kI;Jgifshebidshigherthanmaxfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J;^p+kI;Jg,andshedoesnottradeifshebidslowerthanmaxfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J;^p+kI;Jg.Buyerkdoesnottradeortradesatmaxfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J;^p+kI;Jgifshebidsatmaxfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J;^p+kI;Jg.Thus,

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96 ifbuyerk'svaluationishigherthanmaxfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J;^p+kI;Jg,shepreferstotrade,whichcanbeachievedbybiddinghervaluation.Ifbuyerk'svalua-tionislowerthanmaxfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J;^p+kI;Jg,sheprefersnottotrade,whichcanalsobeachievedbybiddinghervaluation.Ifbuyerk'svaluationisequaltomaxfp)]TJ/F15 11.955 Tf 7.085 1.794 Td[(kI;J;^p+kI;Jg,sheisindierentbetweentradingandnottrading.Thus,biddingtruthfullyisaweaklydominantstrategyforeachbuyer.Strategy-proofnessforthesellerside :foreverysellerl,p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(lI;Jisdeter-minedbytheagentsetISJnflg.IfsellerltradesintheoptimalsolutiontoQ,Iisindependentofthebidpricegl.ByLemma 6.7.19 ,tradingsellerlmusttradeintheoptimalsolutiontoQ.Furthermore,ifsellerltrades,^p+l~I;JisindependentofthebidpriceglbecausetheBC-LPmechanismisastrategy-proofdeterministicmechanism.Thus,ifsellerltrades,bothp)]TJ/F15 11.955 Tf 7.085 1.794 Td[(lI;Jand^p+l~I;Jareindependentofthebidpricegl.Bythestrategy-proofnessoftheBC-LPmechanism,sellerltradesatminfp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(lI;J;^p+l~I;Jgifhebidslowerthanminfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(lI;J;^p+l~I;Jg.ByLemma 6.7.19 andthestrategy-proofnessoftheBC-LPmechanism,sellerldoesnottradeifhebidshigherthanminfp)]TJ/F15 11.955 Tf 7.085 1.794 Td[(lI;J;^p+l~I;Jg.Sellerldoesnottradeortradesatminfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(lI;J;^p+l~I;Jgifhebidsminfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(lI;J;^p+l~I;Jg.Therefore,ifsellerl'svaluationislowerthanminfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(lI;J;^p+l~I;Jg,hepreferstotrade,whichcanbeachievedbybiddinghisvaluation.Ifsellerl'svaluationislowerthanminfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(lI;J;^p+l~I;Jg,heprefersnottotrade,whichcanalsobeachievedbybiddinghisvaluation.Ifsellerl'svaluationisequaltominfp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(lI;J;^p+l~I;Jg,heisindierentbetweentradingandnottrading.Thus,biddingtruthfullyisaweaklydominantstrategyforeachseller. 6.7.6 ProofsofTheorems 6.5.1 and 6.5.2 Inthissection,weprovetheasymptoticeciencyoftheenhancedBC-LPmechanismandtheMBCmechanism.

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97 Assumethatthereareanitenumberofpossiblecommodities,andthatthereexistsanumberMsuchthatPjGjg=1qgi0.Ifbuyeriacquiresthesamebundlethatkdoesandbidshigherthanbuyerk,sincethereisnotransactioncost,wegetafeasiblesolutionto^ViI;Jwithahigherobjectivefunctionvaluecomparedtotheobjectivefunctionvalueof^VI;J.Thus,^ViI;J>^VI;J;buyerisurvivesandtradesthebundleinthenalallocationaccordingtoLemma 6.7.4 .SinceFisafamilyofcontinuousdistributions,theprobabilityoftwobuyerswiththesamevaluationacquiringthesamebundleiszero.Thus,withprobabilityone,atmostonetradetheleastvaluabletradeforeachpossiblebundleislostundertheBC-LPmechanismcomparedtotheoptimalsolutiontothelinearrelaxationformulation^VI;J.Thelinearrelaxationgivesanupperboundonthesocialwelfare,andthelossfromthefailedtradesisboundedalmostsurelyasbothFandGhaveboundedsupports.

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98 ProofofTheorem 6.5.1 :ByLemma 6.7.20 ,thesocialwelfarelossisboundediftherearenotransactioncosts.Asthesocialwelfareapproachesinnity,theratiobetweenthesocialwelfareachievedbytheBC-LPmechanismandthemaximumfeasiblesocialwelfareapproaches1.Thus,withouttransactioncosts,theBC-LPmechanismisasymptoticallyecientasthesocialwelfareapproachesinnity. Nowweanalyzethecasewithtransactioncosts.Weassumethatthetransac-tioncostdependsonlyonlocation,wherethelocationsoftheagentsareindepen-dentlydistributedaccordingtosomecontinuousdistributionUonsomecompactdomainH.Weassumethatthetransactioncostdi;j;gisthedistancebetweenbuyeriandsellerj.Thatis,thetransactioncostddependsoni'sandj'slocations,xandy,respectivelyx;y2H.Weassumethatthedistancefunctiondx;yisametric:dissymmetric,dsatisesthetriangleinequality,anddx;y=0ifandonlyifx=y.Letusalsoassumediscontinuous.Weuseanagentbundletypetodenotewhetheranagentisasellerprovidingbundleq1;;qjGjPjGjg=1qg0.WeassumethatBisnonempty;otherwise,theprobabilityofmakingatransactioniszero.Werstcalculateanupperboundonthemaximumfeasiblesocialwelfareperagent.Letbandgbethedecisionvariables,denotingthetransactionpercentageforagentsacquiringbandsupplyingg,respectively.Togetthisupperbound,we

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99 ignorethetransactioncostsandsolvethefollowingproblem:^E:MaximizePb2BdVFbb)]TJ/F28 11.955 Tf 11.955 8.967 Td[(Pg2GdVGggSubjecttog^pg=Pb2Bbbg^pbforallg2G0g1forallg2G0b1forallb2Bwhere^pband^pgaretherealizedproportionsoftheagentsacquiringbandsupply-ingg,respectively.dVFbbR11)]TJ/F23 7.97 Tf 6.586 0 Td[(b^F)]TJ/F21 7.97 Tf 6.586 0 Td[(1bxdx,anddVGggRg0^G)]TJ/F21 7.97 Tf 6.586 0 Td[(1gxdx,where^Fband^Ggaretherealizedvaluationdistributionsofbundlebandbundleg,respectively.dVFbanddVGgarethewelfarecontributionsperagentthatcomefromacquiringbundlebandsupplyingbundleg,respectively.LetVFbR11)]TJ/F23 7.97 Tf 6.586 0 Td[(F)]TJ/F21 7.97 Tf 6.586 0 Td[(1bxdxandVGgR0G)]TJ/F21 7.97 Tf 6.586 0 Td[(1gxdx.Asthenumberofagentsapproachesinnity,^pb!pb,^pg!pg,^Fb!Fb,and^Gg!Gg.dVFb!VFbanddVGg!VGg.Sinceinproblem^E,VFbs,andVGgsareallcontinuous,wecanobtainanupperlimitofthemaximumfeasiblesocialwelfareperagentbysolvingthefollowingproblem:E:MaximizePb2BVFbb)]TJ/F28 11.955 Tf 11.955 8.967 Td[(Pg2GVGggSubjecttogpg=Pb2Bbbgpbforallg2G0g1forallg2G0b1forallb2BNotethatthefeasibleregionisacompactconvexset,sinceifwehavetwofeasiblesolutions0and00,then0+00=2isalsoafeasiblesolution.Moreover,theobjectivefunctionisstrictlyconcave,sinceVFbsarestrictlyconcavewhileVGgsarestrictlyconvex.Thus,problemEhasauniqueoptimalsolution,andoptimalobjectivefunctionvaluegivestheupperlimit Cofthemaximumfeasible

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100 socialwelfareperagent.Also,asthenumberofagentsapproachesinnity,b!bandg!g.Therefore,F)]TJ/F21 7.97 Tf 6.587 0 Td[(1bbandG)]TJ/F21 7.97 Tf 6.586 0 Td[(1ggarethelimitequilibriumpricesforbundlebandbundleg,respectively.Nowweshowthatwithtransactioncosts,themaximumfeasiblesocialwelfareperagentconvergestothislimit C.SinceHiscompact,thereexistsanite-partitionA1;A2;;AkofHi.e.apartitionsuchthatAlhasaradiuslessthanforany>0.Ifwerestricttransactionstobewithineachpartition,thesocialwelfareperagentisnolessthan C)]TJ/F22 11.955 Tf 12.84 0 Td[(,sincethetransactioncostperagentislessthan.Ifthenumberofagentsislargeenough,wecangetafeasiblesolutionwherethewelfareperagentisnolessthan C)]TJ/F22 11.955 Tf 12.518 0 Td[(.Thus, C)]TJ/F22 11.955 Tf 12.517 0 Td[(isalowerboundforthelimitmaximumfeasiblesocialwelfareperagent.Sincecanbearbitrarilysmall,themaximumfeasiblesocialwelfareperagentconvergestothelimit C.Then,sinceallthevaluationdistributionsarecontinuous,wecanprovethatasthenumberofagentsapproachesinnityandthemaximumfeasiblesocialwelfareperagentapproaches C,thepercentageofthebuyerswhobidhigherthanthelimitequilibriumpriceforeachbundle,butfailtogetatransaction,mustapproachzerointheecientallocationbycontradiction,becauseotherwisethesocialwelfarewouldconvergetoanumberstrictlysmallerthan C.Therefore,foreachbundleineachpartitionAl,somebuyerwithvaluationnomorethanabovetheequilibriumpricetradesintheecientallocationalmostsurely.AlsonotethatthetransactioncostsofanagentwithineachpartitionAlareboundedbyM,sinceMisthelimitonhowmuchabuyercanacquire.NowconsiderbuyeriinpartitionAlwhobidsabovethelimitequilibriumpricebyM+1.ThereexistsanotherbuyerkinpartitionAlwhoacquiresthesamebundleasbuyeriandtradesintheecientallocationwithvaluationnomorethanabovetheequilibriumprice.Thistellsusthat^ViI;J>^VI;J,becausewecanimprovesocialwelfarebyreplacingbuyerkintheecientallocationwithanadditionalbuyerwhois

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101 identicaltoi.Thatis,whenweapplytheBC-LPmechanism,itisalmostsurelythateachbuyerwhobidsabovethelimitequilibriumpricebyM+1survivestheeliminationphase.ByLemma 6.7.4 ,allthesurvivingbuyerstradesinthenalallocation.Thus,wecanobtainalowerboundonsocialwelfareperagentundertheBC-LPmechanismbycalculatingthesocialwelfareofthesystemconsistingofthesurvivingbuyersandtheoriginalsellers.Also,asgoestozero,thelowerboundonthesocialwelfareperagentundertheBC-LPmechanismconvergesto C,becauseallthevaluationdistributionsarecontinuous.Since Cisalsoanupperboundforthesocialwelfareperagent,thesocialwelfareperagentachievedbytheBC-LPmechanismconvergesto C.SinceboththemaximumfeasiblesocialwelfareperagentandthesocialwelfareperagentundertheBC-LPmechanismconvergeto C,theratiobetweenthewelfareundertheBC-LPmechanismandthemaximumfeasiblesocialwelfareconvergesto1asthenumberofagentsapproachesinnity.Thus,theBC-LPmechanismisasymptoticallyecientifeveryagent'sbundletypeisindependentlydrawnfromthesamedistribution. Theproofsofotherasymptoticeciencypropertiesareomittedaswecanapplytheaboveargumentsinasimilarfashion.

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CHAPTER7IMPLEMENTATIONANDCOMPARISONBesidesthispaper,considerableresearchhasaddressedissuesonthedesignofdoubleauctionmechanismsthatinducetruthfulrevelationofinformationfromself-interestedagentsandapplytovarioustradesettingsinvolvingmanybuyersandsellers.Inthispaper,weinvestigatetwotruthfuldoubleauctiondesignapproaches,oneinBabaioandWalsh[ 7 ]andthemulti-stagedesignapproach.Wecomparetheeectivenessoftheresultingmechanismsinvariousexchangeenvironments.Allthemechanismsofinterestassumetheprivatevaluemodel,inwhicheachagenthashis/herownprivateinformationaboutthevalueofthegoods.Fromagametheoryviewpoint,allthesemechanismshavethefollowingproperties:1strategy-proofness:truthfulrevelationofprivateinformationisadominantstrategyforeachagent;2expostindividual-rationality:eachagent'spayofromparticipationisnolessthanhisorherpayofromnon-participation;3expostweaklybudgetbalance:theauctioneer'spayoisnon-negative.Wecomparetheapplicabilityofthetwoapproaches;thatis,weexaminewhichdesignapproachcanoermechanismsapplicabletomoregeneralexchangeenvironments.Foreachexchangeenvironmentofinterest,wecomparealltheapplicablemechanismsbasedonthefollowingthreecriteria: Individualpayo:Thehighertheexpectedpayosare,themorelikelyamechanismistoattractindividualbuyersandsellers. Socialeciency:Thehighertheeciencyis,themorelikelyamechanismistogeneratehigherrevenuesfortheauctioneerortheauctionmarketplaceinthelongrunaspointedbyMilgrom[ 39 ]andWiseandMorrison[ 64 ]. 102

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103 ImplementationComplexity:Thelevelofcomplexitytoimplementthesemechanismsdetermineswhethertheycanbeappliedinsomepracticalsituations.Theremainderofthechapterisorganizedasfollows.InSection 7.1 ,wedescribethevariousexchangeenvironmentsofinterest.InSection 7.2 ,wepresentthetradereductionapproachandtheresultingmechanisms.InSection 7.3 ,wediscusstheimplementationoftheAC-DAmechanism,theBC-LPmechanism,andtheMBCmechanism.WethenevaluatetheimplementationandapplicabilityofthemechanismsinSection 7.4 .InSection 7.5 ,wefocusontotheeciencyandpayocomparisonoftheapplicablemechanismsunderboththesimpleexchangeenvironmentandthebilateralexchangeenvironment.TheproofsofthischapterarecompiledinSection 7.6 . 7.1 ModelandExchangeEnvironmentRecallthatIdenotesthegroupofbuyers,andJdenotesthegroupofsellers,andwerefertoabuyerasshe"andasellerashe".Figure 7{1 illustratesseveralbilateralexchangeenvironmentsandsomeknownmechanismsthatworkineachenvironment.Intheseexchangeenvironments,eachbuyerii2Iwantstopurchaseabundleofgoodsorasinglecommodity,andeachsellerjj2Jsuppliesabundleofgoodsorasinglecommodity.Weassumeaprivatevaluemodel,whereeachagenthashisorherownvaluationofthebundleheorsheisinterestedin.Weassumethatalltheagentshavequasi-linearutility,thatis,ifanagentmakesnotransaction,hisorherutilitypayoiszero;otherwise,thepayoisthedierencebetweentheagent'svaluationandtheamountofmoneytransferred.Theauctioneer'smonetarypayoisthetotalpaymentsfromthebuyerslesstherevenuesofthesellersandtherealizedtransactioncosts.Thesocialwelfareisthesummationoftheauctioneer'spayoandeachagent'sutility.

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104 Figure7{1:BilateralExchangeEnvironmentsandtheCorrespondingMechanisms Inallthemechanismsmentionedinthispaper,eachagentsubmitsonesealedbid.Letfibethebidpriceofbuyeri,i2I,andgjbethebidpriceofsellerj,j2J.Sincebiddingtruthfullyisadominantstrategyforeachagentinallthemechanismsofinterest,thebidpriceofeachagentisalsothetruevaluationofthatagent. EnvironmentA:Simpleexchangeenvironment Eachagentwantstobuy/selloneunitofthesameitemandthereisnotransactioncostinvolved. EnvironmentB:Simpleexchangeenvironmentwithtransactioncosts Eachagentwantstobuy/selloneunitofthesameitem.Whenbuyeritradeswithsellerj,transactioncostdi;jisincurred.Weassumethatthetransactioncostsarecommonknowledge. EnvironmentC:Bilateralexchangeenvironmentwiththesingleoutputrestriction Therearemultipleindivisiblecommoditiesinthisenvironment.Eachbuyerwantstopurchaseabundleofgoodsandeachsellercanproduceonlyasingleunitofonecommodity.

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105 EnvironmentD:Bilateralexchangeenvironmentwiththesingleoutputrestrictionandtransactioncosts. ThisisthesamesettingasinEnvironmentC,exceptwhenbuyeribuyscommoditygfromsellerj,transactioncostdi;j;gisincurred.AsChapter 5 ,EnvironmentB,thesimpleexchangeenvironmentwithtrans-actioncostscanbeusedtomodelthecurrentcustomer-to-customeronlinemarket,whileEnvironmentsCandD,thebilateralexchangeenvironmentswiththesingleoutputrestriction,canbeusedtomodeltheprocurementauctionenvironmentinwhichsellersaresmallinsizeandhavelittleornomarketpowerBabaioandWalsh[ 7 ]andChapter 6 .WeuseXYtodenoteEnvironmentYcontainsEnvironmentXasaspecialcase.ItiseasytoseethatAB,CD,AC,andBD.Furthermore,AD.ForEnvironmentsBandD,weassumethetransactioncostsarecommonknowledgethatcanincludetransportationcosts,vendorswitchcosts,orcostsduetoquality,timingofdeliveries,andcustomization.ForEnvironmentsAandB,ifallagentsbidtruthfully,themaximumfeasiblesocialwelfareforEnvironmentsAandBcanbeformulatedasbyformulationPdi;j=0;8i;jinEnvironmentA:P:MaximizePifixi)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pjgjyj)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pi;jdi;jzi;jSubjecttoPjzi;j=xiforeachi2IPizi;j=yjforeachj2Jxi2f0;1gforeachi2Iyi2f0;1gforeachj2Jzi;j2f0;1gforeachi2Iandj2J

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106 wherexiandyjdenotewhetheranagenttradesintheauction,andzi;jspecieswhethersellerjtradeswithbuyeri.Recallthat^PdenotesthelinearrelaxationofP.NotethatPisanetworkformulation,thus^Phasaninteger-valuedoptimalsolution,andPand^Phavethesameoptimalobjectivevalue.ForEnvironmentsCandD,weassumetherearemultipleindivisiblecom-moditiesinthisenvironment.LetGdenotethesetofindivisiblecommodities.Eachbuyerii2Iwantstopurchaseabundleofgoodsqi=qgig2G,whereqgisarenonnegativeintegerssincethecommoditiesareindivisible.Eachsellerjj2Jcanproduceonlyasingleunitofonecommodity;thus,sellerjsuppliesthebundleqj=qgjg2GsatisfyingtheconditionsthatqgjsarenonnegativeintegersandPg2Gqgj=1.TherestrictionPg2Gqgj=1iscalledthesingleoutputrestriction.Theproblemofndingthemaximumfeasiblesocialwelfarecanbeformulatedasthefollowingmixedintegerprogramming,ifallagentsbidtruthfully:Q:MaximizePi2Ifixi)]TJ/F28 11.955 Tf 11.956 8.966 Td[(Pj2Jgjyj)]TJ/F28 11.955 Tf 11.955 8.966 Td[(Pi2I;j2J;g2Gdi;j;gzi;j;gSubjecttoPj2Jzi;j;g=qgixiforeachi2I;g2GPi2Izi;j;g=qgjyjforeachj2J;g2G0zi;j;gforeachi2I;j2J;g2Gxi2f0;1gforeachi2Iyj2f0;1gforeachj2Jwherexiandyjdenotewhetheranagenttradesintheauction,andzi;j;gspeciestheamountofgoodgbuyeribuysfromsellerj.Recallthat^QdenotesthelinearrelaxationofQ.Beforewepresentandcomparethemechanismsunderdierentauctiondesignapproaches,itwillbehelpfultoreviewallthenotation.

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107 Notation fi Thebidpriceofbuyeri. gj Thebidpriceofsellerj. di;j;g Thetransactioncostwhenbuyeripurchasesoneunitofcommoditygfromsellerj. VI0;J0 ThemaximumfeasiblesocialwelfareregardingtothebidsofbuyersetI0andsellersetJ0. V)]TJ/F23 7.97 Tf 6.587 0 Td[(kI0;J0 ThemaximumfeasiblesocialwelfareregardingtothebidsofagentsetI0SJ0nfkgk2I0SJ0. VkI0;J0 ThemaximumfeasiblesocialwelfareregardingtothebidsofagentsetI0SJ0andonemoreagentwhoisidenticaltoagentkk2I0SJ0. p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(kI0;J0 TheinmumsupremumofbidpricesofbuyersellerksatisfyingVI0;J0>V)]TJ/F23 7.97 Tf 6.586 0 Td[(kI0;J0. p+kI0;J0 TheinmumsupremumofbidpricesofbuyersellerksatisfyingVkI0;J0>VI0;J0. ^VI0;J0 ThelinearrelaxationofsocialwelfareregardingtothebidsofbuyersetI0andsellersetJ0. ^V)]TJ/F23 7.97 Tf 6.587 0 Td[(kI0;J0 ThelinearrelaxationofsocialwelfareregardingtothebidsofagentsetI0SJ0nfkgk2I0SJ0. ^VkI0;J0 ThelinearrelaxationofsocialwelfareregardingtothebidsofagentsetI0SJ0andonemoreagentwhoisidenticaltoagentkk2I0SJ0. ^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI0;J0 Theinmumsupremumofbidpricesofbuyersellerksatisfying^VI0;J0>^V)]TJ/F23 7.97 Tf 6.586 0 Td[(kI0;J0. ^p+kI0;J0 Theinmumsupremumofbidpricesofbuyersellerksatisfying^VkI0;J0>^VI0;J0.Forsimplicityofrepresentation,wemaydroptheparametersI0;J0whenthereferencestothebuyersetandthesellersetareobvious.

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108 ItisstraightforwardtocheckthatV)]TJ/F23 7.97 Tf 6.586 0 Td[(kcorrespondstotheformulationPQwiththeright-hand-sideoftheagentk'sconstraintequalto0,andVkcorrespondstotheformulationPQwiththeright-hand-sideoftheagentk'sconstraintequalto2.p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kisthepriceatwhichagentkmaytradeinsomeecientallocation,thatis,p)]TJ/F15 11.955 Tf 7.085 1.794 Td[(ksaretheVCGpaymentsVickrey[ 60 ],Clarke[ 11 ],Groves[ 19 ]oftheagents. 7.2 MechanismsunderTradeReductionApproachInspiredbyMcAfee[ 33 ],oneapproachtodoubleauctiondesignistorstmaketheallocationdecision,andthenthepricingdecision.Underthisapproach,amechanismtypicallyselectsasubsetoftradesfromtheecientallocation.Inordertobestrategy-proofandbudget-balanced,thesubsetisgenerallydeterminedbyremovingtheleastprotabletradesfromtheecientallocation.Sinceeachagenthasotheragentswhoareperfectsubstitutesinthesemodels,thosebidsintheremovedtradesbecomereferenceprices,andthepricingdecisioncanbemadebysettingthetransactionpricesequaltothereferenceprices.ThisisthetradereductionapproachsummarizedbyBabaioandWalsh[ 7 ]. 7.2.1 KSM-TRmechanismBabaioandWalsh[ 7 ]proposeaknownsingle-mindedtradereductionKSM-TRmechanism.Thismechanismisstrategy-proof,expostindividual-rational,andexpostweaklybudget-balancedwhenappliedinEnvironmentC:thebilateralexchangeenvironmentwiththesingleoutputrestriction. 1 1InBabaioandWalsh[ 7 ],thetargetexchangeenvironmentisaclassofsupplychainformationproblemwithUniqueManufacturingTechnologiesUMTandsin-gleoutputrestriction,whichisequivalenttoabilateralexchangeenvironmentwiththesingleoutputrestriction.ThefollowinginterpretationofKSM-TRmechanismisalsoinitsequivalentforminsteadofitsoriginalnarration.

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109 TherststepoftheKSM-TRmechanismistopickanoptimalsolutiontoformulationQ.Incaseofmultipleoptimalsolutions,aperturbationtechniqueisadoptedtobreakthetieandselectauniqueoptimalsolution.Topresenttheperturbationtechnique,letusindexthebuyersbyiks,k=1;2;;jIj;andindexthesellersbyjks,k=1;2;;jJj.Aperturbationfactorisaddedintoeachagent'bidprice,thatis,wetreattheirbidsasfik+ikandgjk)]TJ/F22 11.955 Tf 12.67 0 Td[(jkinsteadoffikandgjkforeachbuyerandseller,where1i1i2ijIjj1j2jjJj>0.Afteraddingtheperturbationfactors,formulationQhasoneuniquesolution.Since1,thissolutionindeedisanecientallocationundertheoriginalbidprices.Theperturbationtechniquespeciestheruleofhowtopickanoptimalsolu-tiontoformulationQ:whenthereismorethanoneoptimalsolution,wealwayspickonethatmaximizesxi1.Ifxi1ismaximizedinseveraloptimalsolutions,weselectonethatmaximizesxi2.Ifbothxi1andxi2aremaximizedinseveraloptimalsolutions,weselectonethatmaximizesxi3,,xijIj,yj1,andsoon,uptoyjjJj.Therefore,theperturbationfactorsinducealexicographicorderamongtheoptimalsolutions,andtoindextheagentsandapplytheperturbationtechniqueisequiv-alenttosetalexicographicorder.Hence,wecanspecifyanoptimalsolutionbyalexicographicorderwithoutexplicitlyusingtheperturbationtechnique.Sincethesetwomethodsareequivalent,wewillusewhateverishandyinthereminderofthispaper.IntheKSM-TRmechanism,wealsoneedanewterm:market.Eachmarketrepresentsasetofagentswhoacquireorsupplythesamebundle.Whenwesaybuyeri'smarket,wemeanthesetofallbuyerswhoacquirethesamebundleasbuyeri;whenwesaysellerj'smarket,wemeanthesetofallsellerwhosupplythecommodityassellerj.

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110 Now,letusbrieyexplainhowtheKSM-TRauctionmechanismworks: Collectonesealedbidfromeachagent. Generateanarbitrarylexicographicorder. CalculatetheVCGpaymentp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(sforalltheagents. CalculatetheoptimalsolutiontoQbasedonthelexicographicorder. Removeallthebuyerswhoarenotinvolvedinthetransactionsspeciedbytheoptimalsolution. Foreachbuyer'market,ranktheremainingbuyersbytheirbidpricesfromhightolowandbreakthetiebasedonthelexicographicorder;setthismarket'sreferencepriceprequaltothebidpriceofthebuyerwhorankslast,andremovethisbuyerfromtheexchangesystem. CalculatethetotaldemandoftheremainingbuyersandthedemandDgforeachcommodity. Foreachseller'market,rankthesellersbytheirbidpricesfromlowtohighandbreakthetiebasedonthelexicographicorder;setthismarket'sreferencepriceprequaltothebidpriceoftheDg+1thbuyerinthismarket. ConducttransactionsbetweenalltheremainingbuyersandtherstDgsellersofcommodityg.ThetransactionpriceforabuyeristhehigherofherVCGpaymentandhermarket'sreferenceprice,i.e.maxfp)]TJ/F22 11.955 Tf 7.085 1.793 Td[(;prg.ThetransactionpriceforatradingselleristhelowerofthehisVCGpaymentandhismarket'sreferenceprice,i.e.minfp)]TJ/F22 11.955 Tf 7.085 1.793 Td[(;prg. 7.2.2 TradereductionmechanismBabaioetal.[ 6 ]proposeatradereductionmechanismTRM,whichisstrategy-proof,expostindividual-rational,andexpostweaklybudget-balancedwhenappliedinEnvironmentB:thesimpleexchangeenvironmentwithtransactioncosts.

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111 ThetradereductionmechanismneedstospecifyanoptimalsolutiontoformulationP,anditisimpliedthattheperturbationtechniqueisapplied.Sincethereisonlyonecommodityintheexchangeenvironment,weneedadierentmeaningforthetermmarketwhenwedealwithTRM.IntheTRM,amarketisasetofagentswhohavethesametransactioncostswhentradingwithanyotherbuyer/seller.Babaioetal.[ 6 ]implementadirectedgraphformulation.Inthegraph,eachmarketisformulatedasanodeandeachpossibletransactionisformulatedasanuncapacitatededgewithatransactioncost.Asinknodeisalsointroducedwhichconnectsallothernodeswithedges.Thecapacityofanedgefromthesinktoamarketisthenumberofthesellersinthismarket.Thecostofsendingoneunitowonthisedgeisthelowestseller'sbidinthismarket,andthecostofsendingonemoreunitowonthisedgeisthenextlowestseller'sbidinthismarket,therefore,thecostofthisedgeisaconvexfunction.Thecapacityofanedgefromamarkettothesinkeristhenumberofthebuyersinthismarket.Thecostofsendingoneunitowonthisedgeisthenegativeofthehighestbuyer'sbidinthismarket,andthecostofsendingonemoreunitowonthisedgeisthenegativeofthenexthighestbuyer'sbidinthismarket.Therefore,thecostofthisedgeisalsoaconvexfunction.Atthispoint,maximizationproblemPisrepresentedasaconvexminimalcostowproblem,andanallocationisrepresentedbyanintegerowinthegraph.Afterperturbation,theformulationPhasauniqueoptimalsolution.WeneedthefollowingdenitionsfromBabaioetal.[ 6 ]inordertorepresentthemechanism: Denition7.2.1 MarketsMiandMjareinadirectCommercialRelationshipCRifthereistradebetweenMiandMjintheecientallocation.

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112 Denition7.2.2 MarketsMiandMjareinanindirectCommercialRelationshipiftherearemarketsMk1;Mk2;;MklandtradesbetweenMiandMk1,betweenMk1andMk2,,andbetweenMklandMjintheecientallocation. Denition7.2.3 TheCommercialRelationshipComponentCRCofamarketMireferstoasubsetofmarkets,eachofwhichandMiareinadirectorindirectCommercialRelationship. Denition7.2.4 TheReducedResidualGraphRRGisagraphconsistingofallnodesoftheresidualgraphandthefollowingsubsetoftheedges,eachwithitscostintheresidualgraphservingastheedgelength. ForeachedgeMi;Mjsuchthatthereisowontheedge,weaddtheedgewithitscostanditsreversedresidualedgewiththenegatedcost. ForeachmarketMiweaddtheresidualedgescorrespondingtothetradingbuyerwiththeminimalvaluationifsuchbuyerexistsandthetradingsellerwiththemaximalcostifsuchsellerexists.NotethatnoneoftheedgesbetweenCRCs,aswellasedgescorrespondingtonon-tradingagentsareintheRRG.Also,theonlyedgesthatareretainedintheRRGbelongtothelowestvaluetradingagentsforeachclassbuyersorsellersineachmarket.Thetradereductionmechanismworksasfollows: Collectonesealedbidfromeachagent. Generateanarbitrarylexicographicorder. CalculatetheoptimalsolutiontoPbasedonthelexicographicorderandconstructtheresidualgraphandtheRRG. ForeachCRC,calculatetheminimalpositivecycleintheRRGandremoveitfromtheallocationbysendingaunitowalongthecycle. Conducttransactionsaccordingtothemodiedallocation.ThetransactionpriceforatradingbuyeristhedistancefromthesinktohermarketinRRG,

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113 andthetransactionpriceforatradingselleristhenegativedistancefromhismarkettothesinkinRRG.AspointedoutbyBabaioetal.[ 6 ],theminimalpositivecyclescanbefoundusingashortestpathalgorithm.Thecomplexityofthemechanismisthesumofthecomplexitiesoftheconvexminimumcostowalgorithmandthesortingalgorithmforeachmarket. 7.3 ImplementationoftheMechanismsInChapter 4 ,weproposetheAC-DAmechanismwithoutspecifyingsocialwelfareformulation.Here,weshowhowtoimplementtheAC-DAmechanismun-derEnvironmentBbytakingadvantageofthenetworkstructureoftheexchangeenvironment.WealsorecapitulatetheimplementationoftheenhancedBC-LPandMBCmechanismaswellastheirspecialsimpliedimplementationunderEnvironmentB. 7.3.1 ImplementationoftheAC-DAMechanismRecallthattheAC-DAmechanismhastwodierentversions:thebuyercompetitionmechanismandthesellercompetitionmechanism. BuyerCompetitionBCMechanism: { Eachagentsubmitsonesealedbid. { Forbuyeri2I,ifherbidfiislessthanp+iI;J,sheiseliminatedfromtheauction.Let~Idenotethesetofremainingbuyers,fijfip+iI;J;i2Ig. { Theitemsareallocatedamongtheremainingagents~IandJinthemostecientway. { Thetradingbuyerkpaysp+kI;J,andthetradingsellerlreceivesp)]TJ/F15 11.955 Tf 7.084 1.793 Td[(l~I;J. SellerCompetitionSCMechanism: { Eachagentsubmitsonesealedbid.

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114 { Forsellerj2J,ifhisbidgjisgreaterthanp+jI;J,heiseliminatedfromtheauction.Let~Jdenotethesetofremainingsellersfjjgjp+jI;J;j2Jg. { TheitemsareallocatedamongtheremainingagentsIand~Jinthemostecientway. { Thetradingbuyerkpaysp)]TJ/F15 11.955 Tf 7.084 1.794 Td[(kI;~J,andthetradingsellerlreceivesp+lI;J.Letusrstfocusonthebuyercompetitionmechanism.Alltheresultsapplytothesellercompetitionmechanism.Notethatinthebuyercompetitionmechanism,weonlyneedtocheckwhetherfip+iI;Jforallthebuyers,thencalculatep+sfortradingbuyersandp)]TJ/F15 11.955 Tf 7.084 1.794 Td[(l~I;Jsfortradingsellers;wedonotneedtocalculateallthep+sandp)]TJ/F15 11.955 Tf 7.085 1.794 Td[(s.Itturnsoutthatallthesepricecalculationscanbeincorporatedwithallocationdecisionifweuseanetworkrepresentation.Inthisnetwork,wehaveonenodeforeachagentandalsoasinknode.Forsellerjj2J,thereisanarcfromsinktosellerj'snodewithcapacity1andcostequaltogj.Forbuyerii2I,thereisanarcfrombuyeri'snodewithcapacity1andcostequalto)]TJ/F22 11.955 Tf 9.298 0 Td[(fi.Foreachbuyeriandeachsellerj,thereisanarcfromj'snodetoi'snodewithcapacity1andcostequaltodi;j.Notethatifsellerktradesinsomeecientallocation,p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kisequaltothedistancefromsinktok'snodeintheresiduegraph.Furthermore,ifbuyerkdoesnottradeinthisecientallocation,thenthedistancefromsinktok'snodeintheresiduegraphisp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(k,whichsatisesfkp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kp+k;andifbuyerktradesinsomeecientallocation,thenthedistancefromsinktok'snodeintheresiduegraphisthesmalleroffkandp+k.Hence,byexaminingthesedistances,wecanndallthevaluesofp+sforthebuyerswhobidshigherthanp+.Then,forotherbuyers,wecancheckwhetherp+k=fkornotbytestingthescenarioinwhichbuyerkbidsfk+>0

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115 insteadoffk.Thisobservationleadstothefollowingimplementationofthebuyercompetitionmechanism: Collectonesealedbidfromeachagent. CalculateanoptimalsolutiontoP. Foreachtradingbuyeri2I,calculatethedistancefromthesinktohernodeintheresidualgraph.Setp+iI;Jequaltothedistanceifthedistanceissmallerthanfi.Otherwise,calculateanoptimalsolutiontoPwithi'sbidpriceequaltofi+1,setp+iI;Jequaltothedistancefromthesinktohernodeintheresidualgraph.Removebuyeriifherbidfiislessthanp+iI;J. Calculateanecientallocationfortheremainingsystemconsistingoftheremainingbuyersandoriginalsellers,andconducttransactionsaccordingtothisallocation.Thetransactionpricefortradingbuyeriisp+iI;J,andthetransactionpricefortradingsellerjisthedistancefromthesinktohisnodeintheresiduegraphintheremainingsystem.StartingfromanoptimalsolutiontoP,whetherabuyertradesinsomeoptimalsolutioncouldbedeterminedbyashortestpathalgorithm.Thus,ifweformulatePasamaximumweightedbipartitematchingproblem,thecomplexityofthemechanismisdeterminedbythecomplexityofthematchingalgorithm.Iftherearemanyagentswiththesametransactioncost,theyformamarketastheydointheTRM.Inthiscase,wecanalsoformulatePasaconvexminimalcostowproblem,andthecomplexityofthemechanismisthesumofthecomplexitiesoftheconvexminimumcostowalgorithmandsortingalgorithmforeachmarket.Toimplementthealgorithmmorequickly,insteadofsolvingforthemostecientallocationintheremainingsystem,wecanobtainitbyremovingtheeliminatedbuyersandthecorrespondingtransactionsfromtheoptimalsolutiontoPviaashortestpathalgorithm.Then,weonlyneedtosolvethemaximumweighted

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116 bipartitematchingproblemortheconvexminimumcostowproblemonceforthebuyercompetitionmechanism.Thesellercompetitionmechanismisimplementedsimilarlyandhassimilarcomplexity. 7.3.2 ImplementationoftheBC-LPandMBCMechanismsSincetheenhancedBC-LPmechanismalwaysoershighereciencyandpayoscomparedtotheoriginalBC-LPmechanism,wewilluseBC-LPtodenoteitsenhancedversionintheremainderofthischapter.RecalltheimplementationoftheBC-LPmechanismunderEnvironmentD,thebilateralexchangeenvironmentwiththesingleoutputrestrictionandtransactioncosts: Collectonesealedbidfromeachagent. Generateanarbitrarylexicographicorder. Solvelinearprogram^VI;J,andpicktheoptimalsolutionbasedonthelexicographicorder. Foreachbuyeri,checkwhetherthisoptimalsolutionchangesifthecon-straintassociatedwithbuyeri,xi1,isrelaxed.Ifso,^p+iI;J=fi)]TJ/F15 11.955 Tf 14.462 3.022 Td[(^V0i+I;J,where^V0i+I;Jistheminimumshadowpriceoftheaboveconstraintin^VI;J;ifnot,buyeriiseliminated. Solvelinearprogram^V~I;J,where~Iisthesetofremainingbuyers,andpicktheoptimalsolutionbasedonthelexicographicorder. Foreachtradingsellerj,calculate^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j~I;J,whichequalsgj+^V0j)]TJ/F15 11.955 Tf 7.084 2.955 Td[(~I;J,where^V0j)]TJ/F15 11.955 Tf 7.085 2.955 Td[(~I;Jisthemaximumshadowpriceoftheconstraintassociatedwithsellerj,yj1,in^V~I;J. Conducttransactionsaccordingtotheoptimalsolutionto^V~I;J.Thetransactionpricefortradingbuyerkis^p+kI;J,andthetransactionpricefortradingsellerlis^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(l~I;J.ToimplementtheBC-LPmechanism,weonlyneedtosolvetwolinearprograms^VI;Jand^V~I;Jandcalculatetheassociatedshadowprices.

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117 WhenweimplementtheBC-LPmechanismunderEnvironmentB,thesimpleexchangeenvironmentwiththetransactioncosts,wecanhaveafasteralgorithmbytakingadvantageofthestructureoftheexchangeenvironment.Notethatthelinearrelaxationformulation^P^Qhasaninteger-valuedoptimalsolutionandthesameoptimalobjectivevalueasP.Therefore,^p+and^p)]TJ/F15 11.955 Tf 10.986 1.793 Td[(intheBC-LPmechanismareessentiallyequaltop+andp)]TJ/F15 11.955 Tf 10.986 1.794 Td[(intheAC-DAmechanism.SotoimplementtheBC-LPmechanisminthesimpleexchangeenvironmentwithtransactioncosts,thefollowingprocedurecanbeused: Collectonesealedbidfromeachagent. Applytheperturbationtechnique. CalculatetheoptimalsolutiontoPbasedontheperturbationandconstructtheresidualgraph. Removeeachbuyeri2IifshedoesnottradeintheoptimalsolutiontoP.Foreachremainingbuyeri,^p+iI;Jisequaltothedistancefromthesinktohernodeintheresidualgraph.Removebuyeriifherbidfiislessthan^p+iI;J. Calculatethemostecientallocationbasedontheperturbationfortheremainingsystemconsistingoftheremainingbuyersandoriginalsellers. Conductthetransactionaccordingtothemostecientallocationfortheremainingsystem.Thetransactionpricefortradingbuyeriis^p+iI;J,andthetransactionpriceforatradingselleristhedistancefromthesinktohisnodeintheresiduegraphintheremainingsystem.AsintheBCmechanism,insteadofsolvingforthemostecientallocationintheremainingsystem,wecanobtainitbyremovingtheeliminatedbuyersandthecorrespondingtransactionsfromtheoptimalsolutiontoP.Therefore,weonlyneedtosolvethemaximumweightedbipartitematchingproblemortheconvexminimumcostowproblemonce.

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118 TheMBCmechanismisequivalenttotheBC-LPmechanismunderEnvi-ronmentB,thesimpleexchangeenvironmentwiththetransactioncosts.Notethatthelinearrelaxationformulation^P^Qhasaninteger-valuedoptimalsolutionandthesameoptimalobjectivevalueasP.Therefore,^p+and^p)]TJ/F15 11.955 Tf 10.986 1.793 Td[(intheBC-LPmechanismareessentiallyequaltop+andp)]TJ/F15 11.955 Tf 10.987 1.793 Td[(intheAC-DAmechanism.Sincep)]TJ/F15 11.955 Tf -414.603 -22.114 Td[(isnomorethanp+forabuyerundertheAC-DAmechanism,theMBCmechanismisequivalenttotheBC-LPmechanismunderEnvironmentB.RecalltheimplementationoftheMBCmechanismunderEnvironmentD,whichcanbeviewedasaBC-LPmechanismfollowingapreliminaryeliminationphasebasedontheoptimalsolutiontoformulationQandtheVCGprices: Collectonesealedbidfromeachagent. Generateanarbitrarylexicographicorder. CalculatetheVCGpaymentp)]TJ/F15 11.955 Tf 10.987 1.793 Td[(foreachagent. CalculatetheoptimalsolutiontoQbasedonthelexicographicorder. Removeallthebuyerswhoarenotinvolvedintheoptimalsolution.LetIdenotethesetoftradingbuyersintheoptimalsolution. Solvelinearprogram^VI;Jandpicktheoptimalsolutionbasedonthelexicographicorder. Foreachbuyeri,checkwhetherthisoptimalsolutionchangesifthecon-straintassociatedwithbuyeri,xi1,isrelaxed.Ifso,^p+iI;J=fi)]TJ/F15 11.955 Tf 14.462 3.022 Td[(^V0i+I;J,where^V0i+I;Jistheminimumshadowpriceoftheaboveconstraintin^VI;J;ifnot,buyeriiseliminated. Solvelinearprogram^V~I;J,where~Iisthesetofremainingbuyers,andpicktheoptimalsolutionbasedonthelexicographicorder. Foreachtradingsellerj,calculate^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j~I;Jbysolving^p)]TJ/F15 11.955 Tf 7.084 1.793 Td[(j=gj+^V0j)]TJ/F15 11.955 Tf 7.084 2.955 Td[(~I;J,where^V0j)]TJ/F15 11.955 Tf 7.085 2.955 Td[(~I;Jisthemaximumshadowpriceoftheconstraintassociatedwithsellerj,yj1,in^V~I;J.

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119 Conducttransactionsaccordingtotheoptimalsolutionto^V~I;J.ThetransactionpricefortradingbuyerkisthehigherofherVCGpriceand^p+kI;J,i.e.maxfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(kI;J;^p+kI;Jg;andthetransactionpricefortradingsellerlisthelowerofhisVCGpriceand^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j~I;J,i.e.minfp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(lI;J;^p)]TJ/F15 11.955 Tf 7.085 1.793 Td[(l~I;Jg. 7.4 ImplementationandApplicabilityComparisonWehaveintroducedthefollowingmechanisms:theTRM,BCandSCbothvariantsofAC-DA,KSM-TR,MBCandBC-LPmechanisms.SimilarlytotheBC-LPmechanism,wecandeneasellercompetitionLPmechanism,orSC-LPmechanisminshort,inwhichwerstremovesellerswhobidlowerthantheir^p+,thenconducttheecientallocationintheremainingsystem.NotethattheSC-LPmechanismmaintainsallthepropertiesunderthesimpleexchangeenvironmentwithtransactioncosts,butmayreachanallocationwithagentstradingpartialbundlesunderthebilateralexchangeenvironmentwiththesingleoutputrestriction.Therefore,weshouldonlyapplytheSC-LPmechanismunderthesimpleexchangeenvironment,orthesimpleexchangeenvironmentwithtransactioncosts.WerstcomparetheimplementationcomplexityunderEnvironmentA,thesimpleexchangeenvironment.Alltheabovemechanismsareapplicabletothisenvironment.Wecanndtheecientallocationbyrankingallthebidprices,andthesemechanismseitherallocateresourceecientlyorremoveoneleastprotabletransaction.Therefore,theimplementationcomplexityoftheimplementationisdeterminedbythesortingalgorithmadopted.Now,wecomparetheimplementationcomplexityunderEnvironmentB,thesimpleexchangeenvironmentwithtransactioncosts.AllthemechanismsexceptKSM-TRmechanismareapplicabletothisenvironment.Sinceformulation^PisequivalenttoformulationPunderthisenvironment,theMBCmechanism

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120 isequivalenttotheBC-LPmechanism.InSections 7.2 and 7.3 ,wehaveseenthatallthemechanismscantakeadvantageoftheproblemstructureandobtaintheoptimalsolutionofformulationPbysolvingamaximumweightedbipartitematchingalgorithmoraconvexminimumcostowalgorithm.Allthetransactionpricesarethendeterminedbythedistancelabelsviaashortestpathalgorithm.EachmechanismcanbeimplementedbyiterativelyremovingbuyersfromtheoptimalsolutiontoPviaashortestpathalgorithm.TheBC-LP/SC-LPmechanismhasaslightadvantageintermsofimplementa-tion.UnliketheAC-DAmechanism,theimplementationoftheBC-LPmechanismneedsnottocheckwhetherbuyeritradesinanyoptimalsolutiontoPsincethelexicographicorderdeterminesauniqueoptimalsolution.UnliketheTRM,theimplementationoftheBC-LPmechanismdoesnotrequireconstructingthereducedresidualgraph.ItalsorequireslesseorttoobtainthenalallocationbecauseitremoveslesstradesthantheTRMdoes,aswewillseeinTheorem 7.5.2 .Now,wecomparetheimplementationcomplexityunderEnvironmentC,thebilateralexchangeenvironmentwiththesingleoutputrestriction.TheKSM-TR,MBC,andBC-LPmechanismsareapplicabletothisenvironment.BoththeKSM-TRandtheMBCmechanismsrequirethesolutionofintegerformulationQ,whiletheBC-LPmechanismonlyneedtosolveitslinearrelaxation^Q.Therefore,theBC-LPmechanismhasagreatadvantageintermsofimplementation.Finally,wecomparetheimplementationcomplexityunderEnvironmentD,thebilateralexchangeenvironmentwiththesingleoutputrestrictionandtransactioncosts.OnlytheMBCandBC-LPmechanismsareapplicabletothisenvironmentwhilenoknownmechanismunderthetradereductionapproachcan.TheKSM-TRmechanismrequiresthesolutionofintegerformulationQ,whiletheBC-LPmechanismonlyneedtosolveitslinearrelaxation^Q.Onceagain,theBC-LPmechanismhasagreatadvantageintermsofimplementation.

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121 Table7{1:ImplementationNeedsoftheMechanisms EnvironmentAEnvironmentBEnvironmentCEnvironmentD TRMSortingAlgorithmNetworkProgram--BCandSCSortingAlgorithmNetworkProgram--KSM-TRSortingAlgorithm-IntegerProgram-MBCSortingAlgorithmNetworkProgramIntegerProgramIntegerProgramSC-LPSortingAlgorithmNetworkProgram--BC-LPSortingAlgorithmNetworkProgramLinearProgramLinearProgram -"denotesthatamechanismisnotapplicabletothespeciedenvironment. Table 7{1 summarizesthendingofthissection.WeuseNetworkProgram"todenotethatamechanismcanbeimplementedunderthespeciedenvironmentviaamaximumweightedbipartitematchingalgorithmoraconvexminimumcostowalgorithm.Similarly,Sortingalgorithm",LinearProgram",andIntegerProgram"denotesthatamechanismcanbeimplementedunderthespeciedenvironmentviaasortingalgorithm,alinearprogram,andanintegerprogram,respectively.WeseethattheBC-LPmechanismhasthelowestimplementationcomplexitywhileitisapplicabletoalltheenvironmentsofinterest. 7.5 EciencyandPayosComparisonNow,weinvestigatetheecienciesandthepayosofthemechanismsthatareapplicabletoEnvironmentBandEnvironmentC,thesimpleexchangeenvironmentwithtransactioncosts,andthebilateralexchangeenvironmentwithsingleoutputrestriction,respectively.Wendthatintermsofbothsocialeciencyandindividualpayos,themechanismsunderthemulti-stagedesignapproachdominatethecorrespondingmechanismsunderthetradereductionapproach.

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122 7.5.1 ComparisonundertheSimpleExchangeEnvironmentWehavethefollowingmechanismsthatareapplicableinthesimpleexchangeenvironmentwithtransactioncosts:theTRM,BCandSCbothvariantsofAC-DA,MBC,SC-LPandBC-LPmechanisms.Sinceformulation^PisequivalenttoformulationP,theMBCmechanismisequivalenttotheBC-LPmechanism,andwewillonlycomparethesocialeciencyandagentpayosoftheTRM,BCandSC,BC-LPandSC-LPmechanisms.WesaythatmechanismYdominatesmechanismXif,foranyinstance,mechanismYisatleastasecientasmechanismXandeachagent'spayoundermechanismYisatleastashighasitundermechanismX.WehavethefollowingtheoremwhoseproofisinAppendixalongwithotherproofs. Theorem7.5.1 TheBCmechanismdominatestheBC-LPmechanism.ThereareinstanceswheretheBCmechanismoutperformstheBC-LPmech-anism.Considerthefollowingexamplewithtwobuyers,twosellers,andnotransactioncosts.Thebidpriceforbothbuyersis1,andthebidpriceforbothsellersis0.IntheBCmechanism,p+forbothbuyersis1,bothbuyerssurvivetheeliminationstage,andthenalallocationmakestwotransactionatprice1.Thisallocationistheecientallocationforthesystem.IntheBC-LPmechanism,accordingtotheperturbationtechnique,wetakethebuyers'bidpricesas1+i1and1+i2,respectively,andthebuyers'p+sare1+i2and1+i1,respectively.Oneofthebuyersiseliminated,andthenalallocationmakesonetransactionwherethetradingbuyerpays1andthetradingsellerreceives0since1.Weseethatinthisexample,theBCmechanismachievesahighereciencythantheBC-LPmechanism. Theorem7.5.2 TheBC-LPmechanismdominatestheTRM.

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123 ThereareinstanceswheretheBC-LPmechanismachievestheecientallocationwhileTRMcannot.Considerthefollowingexamplewithonebuyer,twosellers,andnotransactioncosts.Thebidpriceforthebuyeris1,andthebidpriceforbothsellersis0.IntheBC-LPmechanism,^p+forthebuyeris0.Thebuyersurvivestheeliminationstage,andthenalallocationmakesonetransactionatprice0.Thisallocationistheecientallocationforthesystem.IntheTRM,thistransactionisremovedandnotransactiontakesplace.Weseethatinthisexample,theBC-LPmechanismachievesahighereciencythantheTRM.SincetheTRMissymmetricbetweenbuyersandsellers,bysymmetrybetweentheBC-LPandSC-LPmechanisms,wehave: Theorem7.5.3 TheSC-LPmechanismdominatestheTRM.Furthermore,duetothesymmetrybetweentheBCandSCmechanisms,wehave: Theorem7.5.4 TheSCmechanismdominatestheSC-LPmechanism.Sincethedominancerelationshipistransitive,byTheorems 7.5.1 and 7.5.2 ,andTheorems 7.5.4 and 7.5.3 ,wehave: Corollary7.5.5 TheBCmechanismdominatestheTRM. Corollary7.5.6 TheSCmechanismdominatestheTRM.Nowweshowthatthereisnobudget-balancedmechanismcandominateboththeBCmechanismandtheSCmechanism.Letusconsideranexamplewithtwobuyers,twosellers,andnotransactioncosts.Bothofsellers'bidsarezero,whilebothofbuyers'bidsare1.UndertheBCmechanism,eachsellerreceivespayo1.UndertheSCmechanism,eachbuyerreceivespayo1.Sincethemaximumsocialwelfareis2,thereisnobudget-balancedmechanismcanhavepayosequaltoorgreaterthan1forallfouragents.

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124 7.5.2 ComparisonundertheBilateralExchangeEnvironmentInthissection,wecomparethemechanismsthataredesignedforthebi-lateralexchangeenvironmentwiththesingleoutputrestriction.Wehavethreemechanismsthatareapplicabletothisenvironment:theKSM-TR,BC-LP,andMBCmechanisms.EventhoughtheBC-LPandMBCmechanismscanalsohan-dletransactioncosts,weassumenotransactioncostsinthissectionbecausetheKSM-TRmechanismunderthetradereductionapproachcannotbeappliedtoanenvironmentwithtransactioncosts.TheBC-LPmechanismcanbeimplementedpolynomially,whileboththeKSM-TRandMBCmechanismsneedtobeimplementedbycomputingtheVCGpricesandtheoptimalsolutiontoformulationQ,whichisNP-hard.WeareespeciallyinterestedincomparingtheeciencyandthepayosoftheKSM-TRandMBCmechanisms.Wehavethefollowingtheorem. Theorem7.5.7 TheMBCmechanismdominatestheKSM-TRmechanism.ThereareinstanceswheretheMBCmechanismachievestheecientallocationwhileKSM-TRmechanismcannot.Considerthefollowingexamplewithonebuyer,twosellers,andnotransactioncosts.Thebidpriceforthebuyeris1,andthebidpriceforbothsellersis0.TheVCGpriceforthebuyeris0,whiletheVCGpriceforbothsellersis0.IntheMBCmechanism,^p+forthebuyeris0,andthebuyertradesinthenalallocationforprice0.Thenalallocationistheecientallocationforthesystem.IntheKSM-TRmechanism,thistradeisremoved,andnotransactiontakesplace.Weseethatinthisexample,theMBCmechanismachievesahighereciencythantheKSM-TRmechanism.Nowweshowthatthereisnobudget-balancedmechanismcandominateboththeBC-LPmechanismandtheMBCmechanism.Letusconsideranexamplewithtwobuyers,sixsellers,onecommodity,andnotransactioncosts.Therstbuyerwantsveunitsandshebids5.Thesecondbuyerwantsthreeunitsandshebids

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125 Table7{2:EciencyandPayoComparisonSummary SimpleExchangeEnvironmentwithTransactionCostsTRMBC-LPBCTRMSC-LPSC BilateralExchangeEnvironmentwithSingleOutputRestrictionKSM-TRMBC XYmeansthatmechanismYdominatesmechanismXandmayoutperformmechanismXinsomeinstances. 4.Fiveofthesixsellers'bidsarezero,whilethelastseller'sbidis1.UndertheMBCmechanism,therstbuyergetstransaction,andveofthesixsellersreceivepayo1.UndertheBC-LPmechanism,thesecondbuyergetstransactionatprice1,andshereceivespayo1.Sincethemaximumsocialwelfareis5,thereisnobudget-balancedmechanismcandominateboththeBC-LPmechanismandtheMBCmechanism.Table 7{2 summariesoureciencyandpayocomparisonfromSections 7.5.1 and 7.5.2 .ThereareinstanceswheretheBC-LPmechanismandtheMBCmechanismoutperformeachother.TofullystudytheeciencyoftheBC-LPmechanism,weconductcomputationaltestsontheperformanceoftheKSM-TR,BC-LP,andMBCmechanismswiththeassumptionthattherearethreecommodities.Wetesttheperformanceofthesemechanismswhentherearedierentnumbersofmarkets,dierentnumbersofbuyerspermarket,anddierentvariancesinthevaluationdistribution.ParametersandnotationusedinthecomputationaltestsaresummarizedinTable 7{3 .Inthecomputationaltest,thenumberofmarketsissettobeeither5or10,andthebundleacquiredineachmarketisanintegertriplei;j;k,wherei;j,andkeachcorrespondstothedemandforonecommodity,andeachisindependentlydrawnfrom0to10.Ineachmarket,thenumberofbuyersiseither5or10,whilethenumberofsellersforeachcommodityequalstheexpectedtotaldemandofeachcommodity.Thevaluationsoftheagentsareindependentrandomvariables.The

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126 Table7{3:ParameterSettings Variablename Value NumberofMarkets 5MS,10ML NumberofBuyersinEachMarket 5NS,10NL StandardDeviationofSeller'sValuation 10S,20L BundleofEachMarketi;j;k i.i.d.integerU[0;10] NumberofSellers Thesizeoftheexpectedtotaldemand Valuationofsellers i.i.dN00;2 Valuationofbuyers i.i.d.Ni+j+k100;i+j+k2 integerU[a;b]"denotestheuniformdistributionoftheintegersin[a;b];N[;2]"denotesthenormaldistributionwithmeanandvariance2. Table7{4:ComparisonsoftheMechanisms KSM-TR BC-LP MBC A% E% B% A% E% B% A% E% B% MS;NS;S 81.9 0 0 94.9 23 33 99.5 62 83 MS;NS;L 81.8 0 0 95.1 27 40 99.6 63 83 MS;NL;S 94.8 0 0 95.9 20 30 99.8 75 83 MS;NL;L 94.8 0 0 96.2 28 45 99.8 72 82 ML;NS;S 83.0 0 0 96.6 20 27 99.8 73 87 ML;NS;L 83.0 0 0 96.6 17 27 99.9 77 90 ML;NL;S 95.0 0 0 97.2 13 20 99.8 67 87 ML;NL;L 95.3 0 0 97.5 7 14 99.9 57 86 valuationofasellerisnormallydistributedwithmean100andvariance2,whereiseither10or20.Thevaluationofabuyerisnormallydistributedwithmeani+j+k00andvariancei+j+k2,wherei;j;kisthebundleshewants.InTable 7{4 ,weuseA%todenotethepercentageofaverageeciencyachievedbythemechanism,E%todenotethepercentageofinstancesinwhichthemechanismyieldsecientallocation,andB%todenotethepercentageofinstancesinwhichthemechanismgivesthebesteciencyresultamongthethreemechanisms.Itisobservedthat,forallscenarios,MBCachievesthehighesteciency.Inallscenarios,MBCprovidesecientallocationsforoverhalfoftheinstances,aswellasthebestresultsamongthethreemechanismsinover80%oftheinstances.BC-LPisalsohighlydesirablebecauseitinvolvesonlysolvinglinearprogrammingproblemsandstillachievesover95%eciencyinallscenarios.The

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127 computationalresultsshowthattheeciencyachievedbyKSM-TRissubstantiallylowerthanboththeMBCandBC-LPmechanisms.Asthenumberofagentincreases,allthemechanismsbecomemoreecientastheyareallasymptoticallyecientinthecurrentcomputationaltestscheme.Forallthreemechanisms,theincreaseofvarianceseemstohaveonlytrivialimpact. 7.6 ProofsProofofTheorem 7.5.1 .WerstcomparetheeciencyoftheBCandBC-LPmechanisms.Let~IBCdenotethesetofremainingbuyersundertheBCmechanism,and~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LPdenotethesetofremainingbuyersundertheBC-LPmechanism.Wewillrstshowthat~IBC~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LPandthattheBCmechanismisatleastasecientastheBC-LPmechanism.Underthesimpleexchangeenvironment,p+kI;J=^p+kI;Jforeachagentk2ISJ.TheBCmechanismonlyeliminateseachbuyerwhobidslowerthanherp+,thatis,~IBC=fijfip+iI;J;i2Ig.BecausetheBC-LPmechanismappliestheperturbationtechnique,thesethresholdpricesmaydieradditionsandsubtractionsoftheperturbationfactors.Therefore,theBC-LPmechanismalsoeliminateseachbuyerwhobidslowerthanherp+,andmayeliminatebuyerswhosebidpricesequaltheirp+s.Thus,thesetofremainingbuyers~IBCundertheBCmechanismisatleastaslargeas~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LPundertheBC-LPmechanism.Sincebothmechanismsimplementthemostecientallocationoftheremainingsystem,theBCmechanismisatleastasecientastheBC-LPmechanism.Now,wecomparethebuyers'payos.Foreachnon-tradingbuyer,thepayoiszero.Fortradingbuyeri,thepayoisfi)]TJ/F22 11.955 Tf 12.037 0 Td[(p+iI;JundertheBCmechanism.Duetothestrategy-proofnessoftheBCmechanism,eachbuyerwhobidshigherthanherp+isinvolvedinatransaction.Thus,undertheBCmechanism,buyeri'spayoiszeroiffip+iI;Jandfi)]TJ/F22 11.955 Tf 12.235 0 Td[(p+iI;Jiffi>p+iI;J.Dueto

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128 thestrategy-proofnessoftheBC-LPmechanism,eachremainingbuyerisinvolvedinatransaction.Thus,undertheBC-LPmechanism,buyeri'spayoiszeroiffi^p+iI;Jandfi)]TJ/F15 11.955 Tf 13.098 0 Td[(^p+iI;Jiffi>p+iI;J.Underthesimpleexchangeenvironment,p+kI;J=^p+kI;Jforeachagentk2ISJ,andeachbuyerreceivesthesamepayounderboththeBCandBC-LPmechanisms.Now,weconsiderthesellers'payos.Wewillshowthateachseller'spayoundertheBCmechanismisatleastashighasitundertheBC-LPmechanism.SincetheBCmechanismisexpostindividual-rational,itsucestoshowthatforeachsellerwithapositivepayoundertheBC-LPmechanism,hispayoundertheBCmechanismisatleastashighashispayoundertheBC-LPmechanism.UndertheBC-LPmechanism,thepayoforanon-tradingselleriszero.Con-sidertradingsellerjwithapositivepayo.Thispayois^p)]TJ/F15 11.955 Tf 7.084 1.794 Td[(j~IBC)]TJ/F23 7.97 Tf 6.587 0 Td[(LP;J)]TJ/F22 11.955 Tf 12.126 0 Td[(gj=V~IBC)]TJ/F23 7.97 Tf 6.587 0 Td[(LP;J)]TJ/F22 11.955 Tf 12.068 0 Td[(V~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LP;Jnfjg.Shapley962showsthatthesimpleexchangeenvironmentwithtransactioncostssatisesthecomplementarity-substitutabilityconditions,whichguaranteethatV~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LP;J)]TJ/F22 11.955 Tf 12.064 0 Td[(V~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LP;JnfjgV~IBC;J)]TJ/F22 11.955 Tf -418.238 -23.908 Td[(V~IBC;Jnfjgif~IBC~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LP.Sincesellerjhasapositivepayoun-dertheBC-LPmechanism,V~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LP;J)]TJ/F22 11.955 Tf 13.197 0 Td[(V~IBC)]TJ/F23 7.97 Tf 6.587 0 Td[(LP;Jnfjg>0;there-fore,V~IBC;J)]TJ/F22 11.955 Tf 13.584 0 Td[(V~IBC;Jnfjg>0,thatis,sellerjmusttradeundertheBCmechanism,andsellerj'spayoundertheBCmechanismequalsp)]TJ/F15 11.955 Tf 7.085 1.793 Td[(j~IBC;J)]TJ/F22 11.955 Tf 12.659 0 Td[(gj=V~IBC;J)]TJ/F22 11.955 Tf 12.659 0 Td[(V~IBC;Jnfjg.Thisisnolessthanhispay-oundertheBC-LPmechanism. ProofofTheorem 7.5.2 .WerstcomparetheeciencyoftheBC-LPandTRMmechanisms.Let~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LPdenotethesetofremainingbuyersundertheBC-LPmechanism,and~ITRMdenotethetradingbuyersetundertheTRM.Wewillrstshowthat~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LP~ITRMandthattheBC-LPmechanismisatleastasecientastheTRM.

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129 Toprove~IBC)]TJ/F23 7.97 Tf 6.587 0 Td[(LP~ITRM,itsucestoshowthatallthetradingbuyersundertheTRMsurvivestheeliminationundertheBC-LPmechanism.Notethatbothmechanismsapplytheperturbationtechnique,soweknowthatanycyclelengthinthereducedresidualgraphisnon-zero.UndertheTRM,abuyeriisremovedfromthetransactionsifsheistherstbuyerstartingfromthesinkintheminimalpositivecycleintheReducedResidualGraphRRGforherCommercialRelationshipComponentCRC.LetsdenotethelastsellerstartingfromthesinkintheminimalpositivecycleintheRRGforhisCRC.Now,consideranybuyerb6=iintheCRC,bythedenitionofRRGandCRC,thereexistsapathfrombtos.ConsiderthecycleintheRRGconsistingoftheedgefromthesinktob,thepathfrombtos,andtheedgefromstothesink.Sinceperturbationtechniqueisapplied,thecyclehasanonnegativelength.Furthermore,sinceitbelongstotheresidualgraphrelatedtotheecientallocationintheoriginalsystem,thecyclelengthmustbepositive.Becausenotwocycleshavethesamelengthundertheperturbationtechnique,thecyclesink-b-s-sink"hasalengthgreaterthanthelengthoftheminimalpositivecyclesink-i-s-sink".Assumenowwehaveonemorebuyerwhoisidenticaltobuyerb.Theresidualgraphoftheoriginalallocationwouldhaveonemoreedgefrombtosinkwithlength/costequaltothenegativeofthebidpriceofb.Considerthecyclesink-i-s-b-sink",whichisthedierenceofthecyclessink-i-s-sink"andsink-b-s-sink".Thiscyclehasanegativelength,andthemaximumsocialwelfarewouldimproveifwehadonemorebuyerwhowasidenticaltobuyerb.ThisshowsthatVbI;J>VI;J.SincebothVbI;JandVI;Jarecontinuousfunctionsofbuyerb'sbidpricefb,fb>p+bI;Jbythedenitionofp+b,andbuyerbsurvivesundertheBC-LPmechanism.Thus,alltradingbuyersundertheTRMsurvivestheeliminationstageundertheBC-LPmechanism,thatis,~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LP~ITRM.Since

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130 theBC-LPmechanismimplementsthemostecientallocationoftheremainingsystem,itisatleastasecientastheTRM.Now,wecomparethebuyers'payos.Thestrategy-proofnessoftheBC-LPmechanismguaranteesthatalltheremainingbuyersin~IBC)]TJ/F23 7.97 Tf 6.587 0 Td[(LPareinvolvedintransactionsunderthismechanism.Thus,~IBC)]TJ/F23 7.97 Tf 6.587 0 Td[(LP~ITRMshowsthatifabuyertradesundertheTRM,shetradesundertheBC-LPmechanism.Notethatbothmechanismsaredeterministicmechanisms,thatis,thereissomecriticalpricesuchthatbuyeritradesatthiscriticalpriceifherbidpriceishigher,andshelosesthetransactionifherbidpriceislower.Thus,thecriticalpriceforeachbuyerundertheBC-LPmechanismisnomorethanthecriticalpriceundertheTRM.Thepayoofatradingbuyeristhedierencebetweenherbidpriceandthecriticalprice.Thus,eachbuyer'spayoundertheBC-LPmechanismisnolessthanherpayoundertheTRM.Now,weconsiderthesellers'payos.UndertheTRM,ifweremovesallnon-tradingbuyersintheexchangesystem,thecorrespondingresidualgraphhasnonegativecycle.Thus,thenalallocationundertheTRMisanecientallocationamongthetradingbuyerset~ITRMandoriginalsellersetJ.WewillshowthatifasellertradesundertheTRM,hetradesundertheBC-LPmechanism.Shapley962showsthatthesimpleexchangeenvironmentwithtransactioncostssatisesthecomplementarity-substitutabilityconditions,whichguaranteethatV~IBC)]TJ/F23 7.97 Tf 6.587 0 Td[(LP;J)]TJ/F22 11.955 Tf 13.031 0 Td[(V~IBC)]TJ/F23 7.97 Tf 6.586 0 Td[(LP;JnfjgV~ITRM;J)]TJ/F22 11.955 Tf 13.031 0 Td[(V~ITRM;Jnfjgif~IBC)]TJ/F23 7.97 Tf 6.587 0 Td[(LP~ITRM.Sinceperturbationtechniqueisapplied,eachtradingsellerjundertheTRMmusthaveV~ITRM;J)]TJ/F22 11.955 Tf 10.706 0 Td[(V~ITRM;Jnfjg>0;thus,V~IBC)]TJ/F23 7.97 Tf 6.587 0 Td[(LP;J)]TJ/F22 11.955 Tf -422.084 -23.907 Td[(V~IBC)]TJ/F23 7.97 Tf 6.587 0 Td[(LP;Jnfjg>0andsellerjtradesundertheBC-LPmechanism.Therefore,ifasellertradesundertheTRM,hetradesundertheBC-LPmechanism.Notebothmechanismsaredeterministicmechanisms,thatis,thereissomecriticalpricesuchthatsellerjtradesatthiscriticalpriceifhisbidpriceislower,andheloses

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131 thetransactionifhisbidpriceishigher.Thus,foreachseller,thecriticalpriceundertheBC-LPmechanismisnolessthanthecriticalpriceundertheTRM.Thepayoofatradingbuyeristhedierencebetweenthecriticalpriceandthebidprice.Thus,eachseller'spayoundertheBC-LPmechanismisnolessthanhispayoundertheTRM. ProofofTheorem 7.5.7 .WerstcomparethesocialeciencyoftheMBCandKSM-TRmechanism.Bothmechanismsapplythesameperturbationtech-nique/lexicographicorderandcalculatetheoptimalsolutiontoQ.LetIdenotethesetoftradingbuyersintheoptimalsolutiontoQ.EachtradingbuyeriintheKSM-TRmechanism,thatis,eachbuyerinIwhodoesnotranklastinhermarket,survivestheeliminationstageoftheMBCmechanism.Toseethis,iftheright-hand-sideoftheconstraintassociatedwithbuyeri,xi1,changesfrom1to2,i.e.,ifwehaveonemorebuyerwhoisidenticaltobuyeri,wecanimprovetheoptimalsolutionto^VI;Jbyusingthisadditionalbuyertoreplacethebuyerwhorankslastinthemarket.Thus,^ViI;J>^VI;J,andeachtradingbuyerintheKSM-TRmechanismsurvivestheeliminationstageoftheMBCmechanism.Thatis,thetradingbuyersetoftheKSM-TRmechanismisasubsetofthesetofremainingbuyers~IintheMBCmechanism.SincetheMBCmechanismimplementstheecientallocationoftheremainingsystemconsistingof~IandJ,theMBCmechanismisatleastasecientastheKSM-TRmechanism.Now,wecomparethebuyers'payos.Thestrategy-proofnessoftheMBCmechanismguaranteesthateachbuyerin~IisinvolvedinthetransactionsundertheMBCmechanism.Thus,ifabuyertradesundertheKSM-TRmechanism,shetradesundertheMBCmechanism.Notebothmechanismsaredeterministicmechanisms,wherethereissomecriticalpricesuchthatbuyeritradesatthiscriticalpriceifherbidpriceishigher,andshelosesthetransactionifherbid

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132 priceislower.Thus,thecriticalpriceforeachbuyerundertheMBCmechanismisnomorethanthecriticalpriceundertheKSM-TRmechanism.Thepayoofatradingbuyeristhedierencebetweenherbidpriceandthecriticalprice.Thus,eachbuyer'spayoundertheMBCmechanismisnolessthanherpayoundertheKSM-TRmechanism.Now,weconsiderthesellers'payos.SincethetradingbuyersetundertheKSM-TRmechanismisasubsetofthetradingbuyersetundertheMBCmechanism,thetotaltransactionquantityundertheKSM-TRmechanismisnomorethanthetotaltransactionquantityundertheMBCmechanismforeachcommodity.Foreachcommodity,wecandecidethetradingsellersbyrankingtheirbidpricesandbreakingthetiebasedontheperturbationtechnique/lexicographicorder;thus,thetradingsellersetundertheKSM-TRmechanismisasubsetofthetradingsellersetundertheMBCmechanism.Notebothmechanismsaredeterministicmechanisms,wherethereissomecriticalpricesuchthatsellerjtradesatthiscriticalpriceifhisbidpriceislower,andhelosesthetransactionifhisbidpriceishigher.Thus,thecriticalpriceforeachsellerundertheMBCmechanismisnolessthanthecriticalpriceundertheKSM-TRmechanism.Thepayoofatradingselleristhedierencebetweenthecriticalpriceandhisbidprice.Thus,eachseller'spayoundertheMBCmechanismisnolessthanhispayoundertheKSM-TRmechanism.

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CHAPTER8CONCLUDINGREMARKSInthispaper,weestablishamulti-stagedoubleauctionmechanismdesignapproach,underwhichweproposevarioustruthfulmechanismsfordierentexchangeenvironments.Wediscussboththesimpleexchangeenvironment,whichmodelsthecustomer-to-customeronlinemarkets,andthebilateralexchangeexchangeenvironment,whichmodelstheonlineprocurementmarketplaceswheresellersaresmallinsizeandhavelittleornomarketpower.Allthemechanismsfortheseenvironmentsachievestrategy-proofness,expostindividualrationality,expostweaklybudgetbalance,andasymptoticeciency.Wethendetailtheimplementationofvariousdoubleauctionmechanismsproposedinthispaperandcomparethesemechanismswithothertruthfulmech-anismsunderadierentdoubleauctiondesignapproach,thetradereductionapproach.Wefocusontheimplementation,applicability,eciency,andpayosofthemechanisms.Themulti-stagedesignapproachoersmechanismsapplicabletomorecomplicatedexchangeenvironments.Forexample,boththeBC-LPandMBCmechanismsareapplicabletotheprocurementmarketplaceswiththesingleoutputrestrictionandtransactioncosts,whilenoknownmechanismunderthetradere-ductionapproachiscapableofdoing.Themechanismsunderbothapproacheshavesimilarimplementationcomplexity,whilethemechanismsunderthemulti-stagedesignapproachdominatethemechanismsunderthetradereductionapproachinbothsocialeciencyandindividualpayosineachtypeofexchangeenvironmentofinterest.WealsowanttoemphasizetheBC-LPmechanism'scontributionintermsofimplementationasthismechanismcanbeimplementedbysolvingtwolinear 133

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134 programsandcalculatingtheassociatedshadowprices.Fromthecomputationalresults,weknowthattheBC-LPmechanismsuccessfullycapturesmostofthepossibleeciency.ThisissignicantconsideringthefactthattheeciencymaximizationproblemwithcompleteinformationisNP-hard.Thedoubleauctiondesignapproachwepresentservesasausefulfounda-tionforfurtherdevelopmentofgeneraldoubleauctions.Inthenearfuture,weplantodevelophighlyecientdoubleauctionmechanismsforamoregeneralexchangeenvironment.Possibletopicsincludesituationswithoutneitherthecomplementarity-substitutabilityconditionsnorthesingleoutputrestrictionandwithacomplexcoststructureoracombinatorialauctionstructure.

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BIOGRAPHICALSKETCHIwasbornonAugust2,1979,inShanghai,China.Aftermaintainingasteadyinterestinmathematicsandsciencethroughoutpreliminaryandmiddleschool,IattendedHighSchoolAliatedtoFudanUniversity,andthenShanghaiJiaotongUniversity,whereIreceivedaB.S.inelectricalengineeringin2001.Insearchofdeepercareerreward,Itookstepsinanewdirectionin2001andchosetopursueaPh.D.inindustrialandsystemsengineeringattheUniversityofFlorida.IwasawardedaAlumniFellowshipforthedurationofthedoctoralprogram.Duringmygraduatestudies,IacquiredaM.A.ineconomicsin2004,whichcomplementsmyexperienceandinterestinoptimizationandsupplychaindesign. 140