GameTheory

12/23/20191GameTheoryGametheorywasdevelopedbyJohnVonNeumannandOscarMorgensternin1944-Economists!Oneofthefundamentalprinciplesofgametheory,theideaofequilibriumstrategieswasdevelopedbyJohnF.Nash,Jr.(ABeautifulMind),aBluefield,WVnative.Gametheoryisawayoflookingatawholerangeofhumanbehaviorsasagame.12/23/20192ComponentsofaGameGameshavethefollowingcharacteristics:PlayersRulesPayoffsBasedonInformationOutcomesStrategies12/23/20193TypesofGamesWeclassifygamesintoseveraltypes.Bythenumberofplayers:BytheRules:BythePayoffStructure:BytheAmountofInformationAvailabletotheplayers12/23/20194GamesasDefinedbytheNumberofPlayers:1-person(orgameagainstnature,gameofchance)2-personn-person(3-person&up)12/23/20195GamesasDefinedbytheRules:Thesedeterminethenumberofoptions/alternativesintheplayofthegame.Thepayoffmatrixhasastructure(independentofvalue)thatisafunctionoftherulesofthegame.Thusmanygameshavea2x2structuredueto2alternativesforeachplayer.12/23/20196GamesasDefinedbythePayoffStructure:Zero-sumNon-zerosum(andoccasionallyConstantsum)Examples:Zero-sumClassicgames:Chess,checkers,tennis,poker.PoliticalGames:Elections,WarNon-zerosumClassicgames:Football(?),D&D,VideogamesPoliticalGames:PolicyProcess12/23/20197GamesdefinedbyinformationIngamesofperfectinformation,eachplayermovessequentially,andknowsallpreviousmovesbytheopponent.Chess&checkersareperfectinformationgamesPokerisnotInagameofcompleteinformation,therulesareknownfromthebeginning,alongwithallpossiblepayoffs,butnotnecessarilychancemoves12/23/20198StrategiesWealsoclassifythestrategiesthatweemploy:Itisnaturaltosupposethatoneplayerwillattempttoanticipatewhattheotherplayerwilldo.HenceMinimax-tominimizethemaximumloss-adefensivestrategyMaximin-tomaximizetheminimumgain-anoffensivestrategy.12/23/20199IteratedPlayGamescanalsohavesequentialplaywhichlendstomorecomplexstrategies.(Tit-for-tat-alwaysrespondinkind.Tat-for-tit-alwaysrespondconflictuallytocooperationandcooperativelytowardsconflict.12/23/201910GameorNashEquilibriaGamesalsooftenhavesolutionsorequilibriumpoints.Theseareoutcomeswhich,owingtotheselectionofparticularreasonablestrategieswillresultinadeterminedoutcome.Anequilibriumisthatpointwhereitisnottoeitherplayersadvantagetounilaterallychangehisorhermind.12/23/201911SaddlepointsTheNashequilibriumisalsocalledasaddlepointbecauseofthetwocurvesusedtoconstructit:anupwardarchingMaximingaincurveandadownwardarcforminimumloss.Drawin3-d,thishasthegeneralshapeofawesternsaddle(ortheshapeoftheuniverse;andifyouprefer)..12/23/201912SomeSimpleExamplesBattleoftheBismarkSeaPrisoner’sDilemmaChicken12/23/201913TheBattleoftheBismarckSeaSimple2x2GameUSWWIIBattleJapaneseOptionsSailNorthSailSouthUSOptionsReconNorth2Days2DaysReconSouth1Day3Days12/23/201914TheBattleoftheBismarckSeaJapaneseOptionsSailNorthSailSouthMinimaofRowsUSOptionsReconNorth2Days2Days2ReconSouth1Day3Days1MaximaofColumns2312/23/201915TheBattleoftheBismarckSea-examinedThisisanexcellentexampleofatwo-personzero-sumgamewithaNashequilibriumpoint.EachsidehasreasontoemployaparticularstrategyMaximinforUSMinimaxforJapanese).Ifbothemploythesestrategies,thentheoutcomewillbeSailNorth/WatchNorth.12/23/201916DecisionTreeJapaneseSailNorthSailSouthSearchNorth2SearchSouth1SearchNorth2SearchSouth3DecisionTreeVersionofBattleofBismarkSea12/23/201917ThePrisonersDilemmaThePrisoner’sdilemmaisalso2-persongamebutnotazero-sumgame.Italsohasanequilibriumpoint,andthatiswhatmakesitinteresting.ThePrisoner'sdilemmaisbestinterpretedviaa“story.”12/23/201918ASimplePrisoner’sDilemmaPrisonerA~ConfessConfessPrisonerB~Confess-1-10-10Confess-100-5-512/23/201919AlternatePrisoner’sDilemmaLanguagePrisonerACooperateDefectPrisonerBCooperate-1-10-10Defect-100-5-5UsesCooperateinsteadofConfesstodenoteplayercooperationwitheachotherinsteadofwithprosecutor.12/23/201920WhatCharacterizesaPrisoner’sDilemmaPrisonerACooperateDefectPrisonerBCooperateRewardRewardTemptSuckerDefectSuckerTemptPunishPunishUsesCooperateinsteadofConfesstodenoteplayercooperationwitheachotherinsteadofwithprosecutor.12/23/201921WhatmakesaGameaPrisoner’sDilemma?WecancharacterizethesetofchoicesinaPDas:Temptation(desiretodouble-crossotherplayer)Reward(cooperatewithotherplayer)Punishment(playitsafe)Sucker(theplayerwhoisdouble-crossed)AgameisaPrisoner’sDilemmawhenever:TRPSOrTemptationRewardPunishmentSucker12/23/201922WhatistheOutcomeofaPD?ThesaddlepointiswherebothConfessThisistheresultofusingaMinimaxstrategy.Twoaspectsofthegamecanmakeadifference.ThegameassumesnocommunicationThestrategiescanbealteredifthereissufficienttrustbetweentheplayers.12/23/201923SolutionstoPD?TheRewardoptionisthejointoptimalpayoff.CanPrisoner’sreachthis?MinimaxstrategiesmakethisimpossibleArethereotherstrategies?12/23/201924IteratedPlayThePDisasingledecisiongameinwhichtheNashequilibriumresultsfromadominantstrategy.Initeratedplay(aseriesofPDs),conditionalstrategiescanbeselected12/23/201925TheTheoryofMetagamesMetagamesstepbackfromthegameandlookattheotherplayersstrategyStr