A Primer in Game Theory.docx

A Primer in Game Theory.docx

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APrimerinGameTheory

APrimerinGameTheory

APrimerinGameTheoryRobertGibbons

ContentsPrefacexj1StaticGamesofCompleteInformation11.1BasicTheory:

Normal-FormGamesandNashEquilibrium21.1.ANormal-FormRepresentationofGames....21.1.8><#004699'>BIteratedEliminationofStrictlyDominatedStrategies41.1.CMotivationandDefinitionofNashEquilibrium81.2Applications141.2.ACournotModelofDuopoly141.2.<#004699'>BBertrandModelofDuopoly211.2.CFinal-OfferArbitration221.2.DTheProblemoftheCommons271.3AdvancedTheory:

MixedStrategiesandExistenceofEquilibrium291.3.AMixedStrategies291.3.<#004699'>BExistenceofNashEquilibrium331.4FurtherReading481.5Problems481.6References512DynamicGamesofCompleteInformation552.1DynamicGamesofCompleteandPerfectInformation572.1.ATheory:

BackwardsInduction572.1.<#004699'>BStackelbergModelofDuopoly612.1.CWagesandEmploymentinaUnionizedFirm642.1.DSequentialBargaining682.2Two-StageGamesofCompletebutImperfectInformation^vn

Contentsbe12.A;Theory:

SubgamePerfection714.2.<#004699'>Bjob-MarketSignaling190IB4.2.CCorporateInvestmentandCapitalStructure.205BankRuns73-CTariffsandImperfectInternationa4.2.DMonetaryPolicy208lCompetition754.3OtherApplicationsofPerfectBayesian2.2.DTournaments79Equilibrium.2102.3RepeatedGames824.3.ACheap-TalkGames210;ATheory:

Two-StageRepeatedGames824.3.<#004699'>BSequentialBargainingunderAsymmetric2.3.<#004699'>BTheory:

InfinitelyRepeatedGames88Information218:

.CCollusionbetweenCoumotDuopolists....1024.3.CReputationintheFinitelyRepeated23-DEfficiencyWages107Prisoners’Dilemma224ETime-ConsistentMonetaryPolicy1124.4RefinementsofPerfectBayesianEquilibrium2332.4DynamicGamesofCompletebut4.5FurtherReading244ImperfectInformation1154.6Problems245-;.AExtensive-FormRepresentationofGames..1154.7References2532.4.<#004699'>BSubgame-PerfectNashEquilibrium1222.5FurtherReading129Index7^716Problems13017References138StaticGamesofIncompleteInformation1433.1Theory:

StaticBayesianGamesandBayesianNashEquilibrium1443.1.AAnExample:

CournotCompetitionunderAsymmetricInformation1443.1.<#004699'>BNormal-FormRepresentationofStaticBayesianGames1463.1.CDefinitionofBayesianNashEquilibrium...14931Applications1523.2.AMixedStrategiesRevisited1523.2.<#004699'>BAnAuction1553.2.CADoubleAuction1583.3TheRevelationPrinciple1643.4FurtherReading1683.5Problems1693.6References1721DynamicGamesofIncompleteInformation173IntroductiontoPerfectBayesianEquilibrium175SignalingGames183PerfectBayesianEquilibriuminSignalingGames183

PrefaceGametheoryisthestudyofmultipersondecisionproblems.Suchproblemsarisefrequentlyineconomics.Asiswidelyappreciated,forexample,oligopoliespresentmultipersonproblemseachfirmmustconsiderwhattheotherswilldo.Butmanyotherap?

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plicationsofgametheoryariseinfieldsofeconomicsotherthanindustrialorganization.Atthemicrolevel,modelsoftradingprocesses（suchasbargainingandauctionmodels）involvegametheory.Atanintermediatelevelofaggregation,laborandfinan?

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cialeconomicsincludegame-theoreticmodelsofthebehaviorofafirminitsinputmarkets（ratherthanitsoutputmarket,asinanoligopoly）.Therealsoaremultipersonproblemswithinafirm:

manyworkersmayvieforonepromotion;severaldivisionsmaycompeteforthecorporation’sinvestmentcapital.Finally,atahighlevelofaggregation,internationaleconomicsincludesmodelsinwhichcountriescompete（orcollude）inchoosingtariffsandothertradepolicies,andmacroeconomicsincludesmodelsinwhichthemonetaryauthorityandwageorpricesettersinteractstrategicallytodeterminetheeffectsofmonetarypolicy.Thisbookisdesignedtointroducegametheorytothosewhowilllaterconstruct（oratleastconsume）game-theoreticmodelsinappliedfieldswithineconomics.Theexpositionemphasizestheeconomicapplicationsofthetheoryatleastasmuchasthepuretheoryitself,forthreereasons.First,theapplicationshelpteachthetheory;formalargumentsaboutabstractgamesalsoap?

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pearbutplayalesserrole.Second,theapplicationsillustratetheprocessofmodelbuildingtheprocessoftranslatinganinfor?

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maldescriptionofamultipersondecisionsituationintoaformal,game-theoreticproblemtobeanalyzed.Third,thevarietyofap?

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plicationsshowsthatsimilarissuesariseindifferentareasofeco?

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nomics,andthatthesamegame-theoretictoolscanbeappliedinXI

PrefacevixmiIlearnedgametheoryfromDavidKreps,JohnRoberts,andeachsetting.InordertoemphasizethebroadpotentialscopeofBobWilsoningraduateschool,andfromAdamBrandenburger,tiietheory,conventionalapplicationsfromindustrialorganizationDrewFudenberg,andJeanTiroleafterward.Iowethetheoreti?

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largelyhavebeenreplacedbyapplicationsfromlabor,macro,andotherappliedfieldsineconomics.1calperspectiveinthisbooktothem.ThefocusonapplicationsWandotheraspectsofthepedagogicalstyle,however,arelargelyewilldiscussfourclassesofgames:

staticgamesof5><#880000'>com?

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pleteinformation,dynamicgamesofcompleteinformationduetothestudentsintheMITEconomicsDepartmentfrom1985,staticgameto1990,whoinspiredandrewardedthecoursesthatledtothissofincompleteinformation,anddynamicgamesofincom?

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pleteinformation.（Agamehasincompletebook.Iamverygratefulfortheinsightsandencouragementallinformationifoneplayerdoesnotknowanotherplayer’sthesefriendshaveprovided,aswellasforthemanyhelpful<#880000'>com?

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payoff,suchasinanauc?

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tionwhenonebidderdoesnotknowmentsonthemanuscriptIreceivedfromJoeFarrell,MiltHarris,howmuchanotherbidderiswillingtopayforthegoodbeingsold.）CorrespondingtotheseGeorgeMailath,MatthewRabin,AndyWeiss,andseveralanony?

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fourclassesofgameswillbefournotionsofequilibriumingames:

mousreviewers.Finally,IamgladtoacknowledgetheadviceandNashequilibrium,subgame-perfectNashequilibrium,BayesianencouragementofJackRepcheckofPrincetonUniversityPressandNashequilibrium,andperfectBayesianequilibrium.financialsupportfromanOlinFellowshipinEconomicsattheNa?

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TwotionalBureauofEconomicResearch.（related）waystoorganizeone’sthinkingabouttheseequi?

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libriumconceptsareasfollows.First,onecouldconstructse?

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quencesofequilibriumconceptsofincreasingstrength,wherestronger（i.e.,morerestrictive）conceptsareattemptstoeliminateimplausibleequilibriaallowedbyweakernotionsofequilibrium.Wewillsee,forexample,thatsubgame-perfectNashequilibriumisstrongerthanNashequilibriumandthatperfectBayesianequi?

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libriuminturnisstrongerthansubgame-perfectNashequilib?

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rium.Second,onecouldsaythattheequilibriumconceptofin?

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terestisalwaysperfectBayesianequilibrium（orperhapsanevenstrongerequilibriumconcept）,butthatitisequivalenttoNashequilibriuminstaticgamesofcompleteinformation,equivalenttosubgame-perfectionindynamicgamesofcomplete（andper?

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fect）information,andequivalenttoBayesianNashequilibriuminstaticgamesofincompleteinformation.Thebookcanbeusedintwoways.Forfirst-yeargraduatestu?

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dentsineconomics,manyoftheapplicationswillalreadybefamil?

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iar,sothegametheorycanbecoveredinahalf-semestercourse,leavingmanyoftheapplicationstobestudiedoutsideofclass.Forundergraduates,afull-semestercoursecanpresentthetheoryabitmoreslowly,aswellascovervirtuallyalltheapplicationsinclass.Themainmathematicalprerequisiteissingle-variablecal?

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culus;therudimentsofprobabilityandanalysisareintroducedasneeded.’AgoodsourceforapplicationsoigametheoryinindustrialorganizationisMie’sTheTheon/offnctusfrialOrganizaiion（MITPress,1988）.

Chapter1StaticGamesofCompleteInformationInthischapterweconsidergamesofthefollowingsimpleform:

firsttheplayerssimultaneouslychooseactions;thentheplayersreceivepayoffsthatdependonthecombinationofactionsjustcho?

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sen.Withintheclassofsuchstatic（orsimultaneous-move）games,werestrictattentiontogamesofcompleteinformation.Thatis,eachplayer’spayofffunction（thefunctionthatdeterminestheplayer’spayofffromthecombinationofactionschosenbytheplayers）iscommonknowledgeamongalltheplayers.Weconsiderdynamic（orsequential-move）gamesinChapters2and4,andgamesofincompleteinformation（gamesinwhichsomeplayerisuncertainaboutanotherplayer’spayofffunctionasinanauctionwhereeachbidder’swillingnesstopayforthegoodbeingsoldisun?

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knowntotheotherbidders）inChapters3and4.InSection1.1wetakeafirstpassatthetwobasicissuesingametheory:

howtodescribeagameandhowtosolvethere?

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sultinggame-theoreticproblem.Wedevelopthetoolswewilluseinanalyzingstaticgamesofcompleteinformation,andalsothefoundationsofthetheorywewillusetoanalyzerichergamesinlaterchapters.Wedefinethenormal-formrepresentationofagameandthenotionofastrictlydominatedstrategy.Weshowthatsomegamescanbesolvedbyapplyingtheideathatrationalplayersdonotplaystrictlydominatedstrategies,butalsothatinothergamesthisapproachproducesaveryimprecisepredictionabouttheplayofthegame（sometimesasimpreciseas;anythingcould1

2STATICGAMESOFCOMPLETEINFORMATIOND/ip?

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rTlionni3happen;）.WethenmotivateanddefineNashequilibriumaso?

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lutionconceptthatproducesmuchtighterpredictionseparatecellsandexplaintheconsequencesthatwillfollowfromsinaverybroadtheactionstheycouldtake.Ifneit