forthcoming in The Axiomatic Method Principles and

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ACrashCourseinImplementationTheoryMatthewO.JacksonDecember1997,thisrevisionAugust2000forthcominginTheAxiomaticMethod:PrinciplesandApplicationstoGameTheoryandResourceAllocation,editedbyWilliamThomson.AbstractTheselecturesaremeanttofamiliarizetheaudiencewithsomeofthefunda-mentalresultsinthetheoryofimplementationandprovideaquickprogressiontosomeopenquestionsintheliterature.HSS228-77,Caltech,PasadenaCA91125,USA.Homepage::jacksonm@hss.caltech.edu.ThispaperprovidesawrittenversionoflecturesgivenattheNATOAdvancedStudyInstituteonGameTheoryandResourceAllocation:TheAxiomaticApproach,whichtookplaceatSUNYStonyBrookinJulyof1997.IthanktheorganizersandespeciallyWilliamThomsonfororganizingtheinstituteandtheparticipantsforfeedbackonthelectures.IthankSalvadorBarberafordetailedcommentsonanearlierversionofthismanuscript.11IntroductionTherearemanyeconomic,social,andpoliticalsituationswhereindividualsinteracttomakedecisionsthataectthemcollectively.Examplesrangefromvotingtoelectrepresentativesorchooseapublicpolicy,totradinginamarket.Basedontheirpref-erencesoverthepossibleoutcomesoftheinteraction,individualsmayactstrategicallyinordertoinuencetheoutcometotheiradvantage.Forinstance,inanelectionavotermightvoteforhisorhersecondrankedcandidateifthevoter’sfavoritecandidatehasnochanceofbeingelected;orinanauctionabuyermayselectabidconsideringtrade-osbetweentheprobabilityofwinningtheauctionandthepricetobepaid.Thespecicdesignoftheinstitutionthroughwhichindividualsinteract,forinstancetherulesoftheelectionorauction,canhaveaprofoundimpactonthestrategicbehaviorofthemembersofthesocietyandontheoutcomesoftheprocess.Implementationtheoryisastudyoftherelationshipbetweenthestructureoftheinstitutionthroughwhichindividualsinteractandtheoutcomeofthatinteraction.Gametheoryplaysacentralroleinthemodelingofthestrategicinteractionstudiedinimplementationtheory.Inmanyapplicationsofgametheory,thegamemodelinginteractionistakenasgivenandanalyzedtopredicttheactionsofindividualsandtheresultingoutcome.Inimplementationtheory,insteadoftakingthegameasgiven,itissomethingtobedesigned.Often,onethinksofthedesiredoutcomesasthegivenandanalyseswhetherthereexistgameformsforwhichthestrategicpropertiesinduceindividualsto(always)chooseactionsthatleadtothedesiredoutcomes.Anexampleofanimplementationquestionis:howcanwedesignanauctiontobesurethattheindividualwhomosthighlyvaluesanobjectissuretobethewinneroftheauction?1Inthisview,implementationtheoryisanormativebranchandgametheoryisapositivebranchofthesametree,andimplementationtheoryisthedesignorreverseengineeringprocessassociatedwithgametheory.Ofcourse,thisviewisabitcaricatured,butindicatesthatthereisacloserelationshipbetweenthetoolsandunderstandingsdevelopedinimplementationtheoryandgametheory.Togetafeelingforthetypeofquestionsthatareanalyzedinimplementationtheoryletusstartbylookingataclassicexample.Considerasocietyorcommitteeholdinganelectiontoselectoneoutofasetofcandidates.Eachmemberofthesocietyhasapreferencerankingoverthecandidates.Thesocietymayhavecertainaspirations1Forananalysisofthisparticularproblemfromamechanismdesignpointofview,seeDasguptaandMaskin(1997).Iwilldiscusstherelationshipanddierencebetweenmechanismdesignandimplementation.2regardingwhichcandidateshouldbeselectedasafunctionofthepreferencesofthemembersofthesociety.Forinstance,itmaywishtoavoidselectingacandidatewhoisParetodominatedbyanothercandidate(i.e.,acandidaterankedlowerthananothercandidatebyallmembersofsociety).ItmayalsowishtoselectaCondorcetwinner(acandidatewhodefeatsanyothercandidateinapairwisecomparisonaccordingtoamajorityofvoters’preferences)ifsuchacandidateexists.Ifthesewerethegoalsofthesocietythentheimplementationquestionwouldbe,\Doesthereexistanelectionprocedureforwhichforeachpossibleproleofpreferencerankingsofthevoters,eachequilibriumoutcomeoftheelectionprocedurewouldbeParetooptimalandCondorcetconsistent?Inordertoanswerthisquestiononehastomakeprecisewhatanelectionprocedureisandwhatequilibriumoutcomesare.Thisisthepointatwhichgametheoretictoolsareused.Theelectionprocedureismodeledasagameformorwhatiscommonlyreferredtoasamechanismintheliterature.Itspeciesasetofpossibleactionsormessagesthateachmemberofsocietycanuse,andthentheoutcome(inthiscasethecandidateelected)asafunctionoftheactionsormessagessentbythemembersofsociety.Forinstanceonecouldhaveeachmemberofsocietysubmittheirrankingofthecandidates.Iftherearemcandidates,thenonecouldawardacandidatempointsforeachvoterwhosesubmittedrankingplacesthemhighest,m1pointsforeachvoter’ssubmittedrankingplacesthemsecondhighest,andsoon.Theelectedcandidateistheonewhohasthemostpoints,withtiesbrokenaccordingtosomepre-speciedrule.ThisisthemechanismcorrespondingtoBorda’s(1781)scoringmethod.Ofcourse,itmaynotbeinasocietymember’sbestinteresttoreporttheirtruepreferenceranking.Thisiswhereequilibriumconceptsfromgametheoryareusedtomakepredictionsconcerningstrategicbehavior.AsolutionconceptsuchasNashequilibriumcanbeusedtopredictthepreferencerankingsthatwillbereportedbyvotersas