ChapterTwenty Eight GameTheory I APrelude GamesandChineseWisdoms 田忌赛马借刀杀人 Makeuseofotherpersontogetridofanenemy 暗渡陈仓 doonethingundercoverofanother 破釜沉舟 Burnone sboats bridge 不战而屈人之兵一报还一报 TitforTat 空城计 presentingaboldfronttoconcealaweakdefence TypesofGamesandConceptsofEquilibrium GameTheoryI Contents SimultaneousGames 28 1DominantStrategies 28 1BestResponseCurves 29 1NashEquilibrium 28 2MixedStrategies 28 3 29 2 GameTheoryI Contents Examples Gamesofcoordination 28 4 29 3BattleofthesexesAssuranceGamesChickenBoxedpigs Table29 9 Prisoner sdilemmaExamples GamesofCompetition 29 4 Examples GamesofCoexistence 29 5 GameTheoryII Contents SequentialGames 28 7 Strategies NormalFormandNashEquilibrium 28 7Extensiveform 28 7 SubgamesandSubgamePerfectNashEquilibrium SPNE GameTheoryII Contents Examples GamesofCommitment 28 8 29 6EntryDeterrenceKindlyKidnapperSavingandSocialSecurity SoftBudgetConstraints DewatripontandMaskin 1995 Examples Bargaining 29 7RepeatedGamesandEnforcingCartel 28 5 28 6 27 11 GameTheory Gametheorymodelsstrategicbehaviorbyagentswhounderstandthattheiractionsaffecttheactionsofotheragents SomeApplicationsofGameTheory Thestudyofoligopolies industriescontainingonlyafewfirms Thestudyofcartels e g OPECThestudyofexternalities e g usingacommonresourcesuchasafishery Thestudyofmilitarystrategies Simultaneous MoveGamesandSequentialGames Inthisclass westudysimultaneous movegames inwhichallplayersmoveonlyonceandatthesametime Innextclass westudysequentialgames whereoneplayergetstomovefirstandtheotherplayerresponds andtheycanmovemorethanonce WhatisaGame A Normal form gameconsistsofasetofplayers 1 2 I asetofstrategiesforeachplayer Si si foranyi thepayoffstoeachplayerforeverypossiblelistofstrategychoicesbytheplayers ui si s i si Si s i S i foranyi irepresentsallplayersexcepti Two PlayerGames Agamewithjusttwoplayersisatwo playergame Wewillstudyonlygamesinwhichtherearetwoplayers eachofwhomcanchoosebetweentwostrategies AnExampleofaTwo PlayerGame TheplayersarecalledAandB i A BPlayerAhastwostrategies called Up and Down SA Up down PlayerBhastwostrategies called Left and Right SB Left Right Thetableshowingthepayoffstobothplayersforeachofthefourpossiblestrategycombinationsisthegame spayoffmatrix Exampleno 1 Thisisthegame spayoffmatrix PlayerB PlayerA PlayerA spayoffisshownfirst PlayerB spayoffisshownsecond Exampleno 1 E g ifAplaysUpandBplaysRightthenA spayoffis0andB spayoffis1 Thisisthegame spayoffmatrix PlayerB PlayerA L R U D 1 2 2 1 0 1 1 0 Exampleno 1 AndifAplaysDownandBplaysRightthenA spayoffis1andB spayoffis0 Thisisthegame spayoffmatrix PlayerB PlayerA L R U D 1 2 2 1 0 1 1 0 Exampleno 1 PlayerB PlayerA Aplayofthegameisapairsuchas U R wherethe1stelementisthestrategychosenbyPlayerAandthe2ndisthestrategychosenbyPlayerB Exampleno 1 Whatplaysarewelikelytoseeforthisgame A It ssimpleinthisgame They re D L Why PlayerB PlayerA DominantStrategy DominantStrategy Astrategywhichisoptimalfortheplayernomatterwhattheotherplayerdoes DisplayerA sdominantstrategy LisplayerB sdominantstrategy inourexample DominantStrategy Ifthereisadominantstrategyforeachplayer thenwewouldpredictthatitwouldbetheequilibriumoutcomeofthegame Thus AplaysDandBplaysL or D L isacombinationofequilibriumstrategies TheequilibriumoutcomeisAreceiving2andBreceiving1 or 2 1 DominatedStrategies Q1 Whatisthecounterpartofadominantstrategyforaplayer A Adominatedstrategy Whenastrategyisdominatedbysomeotherstrategy itcouldnotbeoptimal thusneverchoose fortheplayernomatterwhattheotherplayerdoes Exampleno 2 Whatplaysarewelikelytoseeforthisgame PlayerB PlayerA BestResponse Awaytodealwiththisgameisfindingoutthebestresponseforeachplay Foranygivenchoicetheotherplayercanmake yourbestresponseisthechoicethatmaximizesyourpayoff Notnecessarilyunique Exampleno 2 IfplayerBplaysL PlayerAgets1ifplaysU 0ifplaysD PlayerB PlayerA Exampleno 2 IfplayerBplaysL PlayerAgets1ifplaysU 0ifplaysD SoplayerA sbestresponsegivenplayerBplayingLisU Writeas bA sB L U PlayerB PlayerA Exampleno 2 IfplayerBplaysR playerA sbestresponseisD bA sB R D PlayerB PlayerA Exampleno 2 IfplayerAplaysU PlayerB sbestresponseisL bB sA U L PlayerB PlayerA Exampleno 2 IfplayerAplaysD PlayerB sbestresponseisR bB sA D R PlayerB PlayerA Exampleno 2 Whatplaysarewelikelytoseeforthisgame A U L and D R PlayerB PlayerA NashEquilibrium ApairofstrategiesisaNashEquilibriumifA schoiceisoptimal givenB schoice andB schoiceisoptimalgivenA schoice I e apairofstrategiesisaNashEquilibriumifeachofthemisabestresponsetotheother NashEquilibrium OurexamplehastwoNashequilibria U L and D R since for U L bA L UandbB U Lfor D R bA R DandbB D R WhyNashEquilibriumReasonable InaNashequilibrium thebeliefsandtheactionsoftheplayersaremutuallyconsistent Nashequilibriumisaself enforcingagreement Nashequilibriumisaevolutionarystablesolution ESS orastablesocialconvention SeeGamesofcoexistencebelow Exampleno 3 PlayerA Whatplaysarewelikelytoseeforthisgame 0 0 0 1 1 0 1 3 U D L R PlayerB Exampleno 3 PlayerA ThisgamehasnopurestrategyNashequilibrium butitdoeshaveaNasheq