博弈論的歷史發(fā)展

模擬性思考題

Rousseau:DiscourseontheOriginandBasisofEqualityamongMen:Ifagroupofhunterssetouttotakeastag,theyarefullyawarethattheywouldallhavetoremainfaithfullyattheirpostsinordertosucceed;butifaharehappenstopassnearoneofthem,therecanbenodoubtthathepursueditwithoutqualm,andthatoncehehadcaughthisprey,hecaredverylittlewhetherornothehadmadehiscompanionmisstheirs.0、策略性思維的基本概念（一）策略性思維的意義策略性思維：人類的內(nèi)在優(yōu)勢策略性思維的兩面性策略性思維的不可回避性策略性思維：基本的生活技巧0、策略性思維的基本概念（二）策略性博弈實例例一：西班牙叛亂Roman:PompeyandPiusRebellion:SertoriusandHirtuleius0、策略性思維的基本概念實例二：囚徒困境(Prisoner’sDilemma)實例三：性別戰(zhàn)(theBattleoftheSexes)實例四：鷹鴿博弈(theHawkandDove)實例五：猜硬幣博弈（Matchingpennies）實例六：齊威王田忌賽馬實例七：小偷與守衛(wèi)博弈實例八：智豬博弈0、策略性思維的基本概念（三）策略性思維的系統(tǒng)性研究：博弈論的歷史發(fā)展（四）博弈論的性質(zhì)博弈論是對理性、智能決策者之間沖突與合作的模型研究，是尋求決策者最佳反應(yīng)策略的一種理論體系0、策略性思維的基本概念（五）博弈論的基本假設(shè)個人主義理性智能共同知識0、策略性思維的基本概念（六）策略性思維的基本框架

決策與博弈

博弈的基本類型1、同時行動或序慣行動(arethemovesinthegamesequentialorsimultaneous)2、沖突還是合作(aretheplayers’interestsintotalconflictoristheresomecommonality)3、一次博弈還重復(fù)博弈,參與人是否變化(isthegameplayedonceorrepeatedly,andwiththesameorchangingopponents)0、策略性思維的基本概念

博弈的基本類型4、信息是否完備和完全(dotheplayershavefullofequalinformation)5、規(guī)則是否可變(aretherulesofthegamefixedormanipulateable)6、協(xié)議是否可實施(Areagreementstocooperateenforceable)

博弈論的運用解釋（Explanation）預(yù)測(prediction)建議（Adviceorprescription）博弈分類（Categories）一、完備信息的同時行動博弈（StaticGamesofCompleteInformation

）二、完全信息的序慣行動博弈（DynamicGamesofCompleteInformation

）三、不完全信息的同時行動博弈（StaticGamesofIncompleteInformation

）四、不完全信息的序慣行動博弈（DynamicGamesofIncompleteInformation

）五、高級專題（AdvancedTopics）1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

1.1策略式博弈導(dǎo)論與嚴格占優(yōu)IntroductiontoGamesinStrategicFormandIteratedStrictDominance

1.1.1博弈的策略式(Strategic-FormGames

)三要素參與人（players)可選策略(thestrategiesavailabletoeachplayers)支付(thepay-offs):

1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

CommonknowledgeUsuallywealsoassumethatallplayersknowthestructureofthestrategicform,andknowthattheiropponentsknowit,andknowthattheiropponentsknowthattheyknow,andsoonadinfinitum.Thatis,thestructureofthegameiscommonknowledge,aconceptexaminedmoreformallyinchapter14.Thischapterusescommonknowledgeinformally,tomotivatethesolutionconceptofNashequilibriumanditeratedstrictdominance.Aswillbeseen,commonknowledgeofpayoffsonitsownisinfactneithernecessarynorsufficienttojustifyNashequilibrium.

1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Wefocusourattentiononfinitegames,thatis,gameswhereisfinite;finitenessshouldbeassumedwhereverwedonotexplicitlynoteotherwise.Strategicformsforfinitetwo-playergamesareoftendepictedasmatrices,asinfigure1.1.Inthismatrix,players1and2havethreepurestrategieseach:U,M,D(up,middle,anddown)andL,M,R(left,middle,andright),respectively.Thefirstentryineachboxisplayer1’spayoffforthecorrespondingstrategyprofile;thesecondisplayer2’s.1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

MixedStrategy1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Notethatplayeri’spayofftoamixed-strategyprofileisalinearfunctionofplayer’smixingprobability

i,afactwhichhasmanyimportantimplications.Notealsothatplayeri’spayoffisapolynomialfunctionofthestrategyprofile,andsoinparticulariscontinuous.Last,notethatthesetofmixedstrategiescontainsthepurestrategies,asdegenerateprobabilitydistributionsareincluded.1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Forinstance,infigure1.1amixedstrategyforplayer1isavector(

1(U),

1(M),

1(D))suchthat

1(U),

1(M),

1(D)arenonnegativeand

1(U)+

1(M)+

1(D)=1.Thepayoffstoprofiles

1(1/3,1/3,1/3)and

2(0,1/2,1/2)are1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

1.1.2劣策略(DominatedStrategies)Isthereanobviouspredictionofhowthegamedescribedinfigure1.1shouldbeplayed?

Notethat,nomatterhowplayer1plays,Rgivesplayer2astrictlyhigherpayoffthanMdoes.Informallanguage,strategyMisstrictlydominated.Thus,a“rational”player2shouldnotplayM.Furthermore,ifplayer1knowsthatplayer2willnotplayM,thenUisabetterchoicethanMorD.Finally,ifplayer2knowsthatplayer1knowsthatplayer2willnotplayM,thenplayer2knowsthatplayer1willplayU,andsoplayer2shouldplayL.1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

一個純策略，即使其不嚴格劣于任何一個純策略，卻可能嚴格劣于一個混合策略ApurestrategymaybestrictlydominatedbyamixedstrategyevenifitisnotstrictlydominatedbyanypurestrategyHereplayer1’sstrategyMisnotdominatedbyU,becauseMisbetterthanUifplayer2movesR;andMisnotdominatedbyD,becauseMisbetterthanDwhen2movesL.However,ifPlayer1playsUwithprobability1/2andDwithprobability1/2,heisguaranteedanexpectedpayoffof1/2regardlessofhowplayer2plays,whichexceedsthepayoffof0hereceivesfromM.Hence,apurestrategymaybestrictlydominatedbyamixedstrategyevenifitisnotstrictlydominatedbyanypurestrategy.

1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

嚴格劣策略

(StrictlyDominatedPureStrategy)1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Sofarwehaveconsidereddominatedpurestrategies.Itiseasytoseethatamixedstrategythatassignspositiveprobabilitytoadominatedpurestrategyisdominated.

However,amixedstrategymaybestrictlydominatedeventhoughitassignspositiveprobabilityonlytopurestrategiesthatarenotevenweaklydominated.Figure1.3givesanexample.

1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Whenagameissolvablebyiteratedstrictdominanceinthesensethateachplayerisleftwithasinglestrategy,asinfigure1.1,theuniquestrategyprofileobtainedisanobviouscandidateforthepredictionofhowthegamewillbeplayed.Althoughthiscandidateisoftenagoodprediction,thisneednotbethecase,especiallywhenthepayoffscantakeonextremevalues.1.GamesinStrategicFormandNashEquilibriumIntheoriginalstory,twosuspectsarebeingseparatelyinterrogatedandinvitedtoconfess.Oneofthem,sayA,istold,“Iftheothersuspect,B,doesnotconfess,thenyoucancutaverygooddealforyourselfbyconfessing.ButifBdoesconfess,thenyouwoulddowelltoconfess,too;otherwisethecourtwillbeespeciallytoughonyou.Soyoushouldconfessnomatterwhattheotherdoes.”Bistoldtoconfess,withtheuseofsimilarreasoning.

AlconfessDon’tBobconfess0,02,-1Don’t-1,21,11.1.3ApplicationsoftheEliminationofDominatedStrategies

Example1.1ThePrisoner’sDilemma如果Bob的策略是Al的最優(yōu)反應(yīng)策略是坦白坦白不坦白坦白1.GamesinStrategicFormandNashEquilibrium

1.1.3ApplicationsoftheEliminationofDominatedStrategies

Manyversionsoftheprisoner’sdilemmahaveappearedinthesocialsciences.

Oneexampleismoralhazardinteams.Supposethattherearetwoworkers,i=1,2,andthateachcan“work”(si=1)or“shirk”(si=0).Thetotaloutputoftheteamis4(s1+s2)andissharedequallybetweenthetwoworkers.Eachworkerincursprivatecost3whenworkingand0whenshirking.With“work”identifiedwithCand“shirk”withD,thepayoffmatrixforthismoral-hazard-in-teamsgameisthatoffigure1.7,and“work”isastrictlydominatedstrategyforeachworker.1.GamesinStrategicFormandNashEquilibrium1.1.3ApplicationsoftheEliminationofDominatedStrategiesExamplePlayer2LeftMiddleRightPlayer1Up1,01,20,1Down0,30,12,01.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

ExampleSecond-PriceAuction1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

SolutiontotheSecond-PriceAuction1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

SolutiontotheSecond-PriceAuctionThus,itisreasonabletopredictthatbiddersbidtheirvaluationinthesecond-priceauction.Therefore,bidderIwinsandhasutilityvI-vI-1.Notealsothatbecausebiddingone’svaluationisadominantstrategy,itdoesnotmatterwhetherthebiddershaveinformationaboutoneanother’svaluations.Hence,ifbiddersknowtheirownvaluationbutdonotknowtheotherbidders’valuations,itisstilladominantstrategyforeachbiddertobidhisvaluation.1.GamesinStrategicFormandNashEquilibrium

1.2NashEquilibrium

1.2.1DefinitionsofNashEquilibriumANashequilibriumisaprofileofstrategiessuchthateachplayer’sstrategyisanoptimalresponsetotheotherplayers’strategies.1.GamesinStrategicFormandNashEquilibrium

Pure-StrategyNashEquilibriumApure-strategyNashequilibriumisapure-strategyprofilethatsatisfiesthesameconditions.Thatis,forallSinceexpectedutilitiesare“l(fā)inearintheprobabilities,”ifaplayerusesanondegeneratemixedstrategyinaNashequilibrium,hemustbeindifferentbetweenallpurestrategiestowhichheassignspositiveprobability.1.GamesinStrategicFormandNashEquilibrium

StrictNashEquilibrium1.

GamesinStrategicFormandNashEquilibrium

ConsistentPredictionsNashequilibriaare“consistent”predictionsofhowthegamewillbeplayed,inthesensethatifallplayerspredictthataparticularNashequilibriumwilloccurthennoplayerhasincentivetoplaydifferently.Thus,aNashequilibrium,andonlyaNashequilibrium,canhavethepropertythattheplayerscanpredictit,predictthattheiropponentspredictit,andsoon.Incontrast,apredictionthatanyfixednon-Nashprofilewilloccurimpliesthatatleastoneplayerwillmakea“mistake,”eitherinhispredictionofhisopponents’playor(giventhatprediction)inhisoptimizationofhispayoff.1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Wedonotmaintainthatsuchmistakesneveroccur.

Infact,theymaybelikelyinsomespecialsituations.Butpredictingthemrequiresthatthegametheoristsknowmoreabouttheoutcomeofthegamethantheparticipantsknow.ThisiswhymosteconomicapplicationsofgametheoryrestrictattentiontoNashequilibria.

1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

ThefactthatNashequilibriapassthetestofbeingconsistentpredictionsdoesnotmakethemgoodpredictions,andinsituationsitseemsrashtothinkthataprecisepredictionisavailable.

By“situations”wemeantodrawattentiontothefactthatthelikelyoutcomeofagamedependsonmoreinformationthanisprovidedbythestrategicform.Forexample,onewouldliketoknowhowmuchexperiencetheplayershavewithgamesofthissort,whethertheycomefromacommoncultureandthusmightsharecertainexpectationsabouthowthegamewillbeplayed,andsoon.1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

DominanceandNashEquilibriumThesamepropertyholdsforiterateddominance.Thatis,ifasinglestrategyprofilesurvivesiterateddeletionofstrictlydominatedstrategies,thenitistheuniqueNashequilibriumofthegame.1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

DominanceandNashEquilibriumConversely,anyNash-equilibriumstrategyprofilemustputweightonlyonstrategiesthatarenotstrictlydominated(or,moregenerally,dosurviveiterateddeletionofstrictlydominatedstrategies),becauseaplayercouldincreasehispayoffbyreplacingadominatedstrategywithonethatdominatesit.However,Nashequilibriamayassign

positiveprobabilitytoweaklydominatedstrategies.1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

1.2.2ExamplesofPure-StrategyEquilibriaExample:CournotCompetition

1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Example:CournotCompetition(continued)1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Example:CournotCompetition(continued)1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Example:HotellingCompetition

1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Example:HotellingCompetition

1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Example:HotellingCompetition

1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Example:MajorityVotingTherearethreeplayers,1,2,and3,andthreealternatives,A,B,andC.Playersvotesimultaneouslyforanalternative;abstainingisnotallowed.Thus,thestrategyspacesareSi={A,B,C}.Thealternativewiththemostvoteswins;ifnoalternativereceivesamajority,thenalternativeAisselected.ThepayofffunctionsareThisgamehasthreepure-strategyequilibriumoutcomes:A,B,andC.Therearemoreequilibriathanthis:Ifplayers1and3voteforoutcomeA,thenplayer2’svotedoesnotchangetheoutcome,andplayer3isindifferentabouthowhevotes.Hence,theprofiles(A,A,A)and(A,B,A)arebothNashequilibriumwhoseoutcomeisA.(Theprofile(A,A,B)isnotaNashequilibrium,sinceifplayer3votesforBthenplayer2wouldprefertovoteforBaswell.)1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibriumRogerHeadsTailsBarryHeads-1,11,-1Tails1,-1-1,11.2.3NonexistenceofaPure-StrategyEquilibrium

Notallgameshavepure-strategyNashequilibria.TwoexamplesofgameswhoseonlyNashequilibriumisin(nondegenerate)mixedstrategiesfollow.MatchingPennisGame1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibriumPrincipalINIAgentS0,-hw,-wWw-g,v-w-hw-g,v-wInspectionGameApopularvariantofthe“matchingpennies”gameisthe“inspectiongame,”whichhasbeenappliedtoarmscontrol,crimedeterrence,andworkerincentives.1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

Battleofthesexes

WshowgameHshow1,20,0game0,02,11.2.4MultipleNashEquilibria,FocalPoints,andParetoOptimalityManygameshaveseveralNashequilibria.Whenthisisthecase,theassumptionthataNashequilibriumisplayedreliesontherebeingsomemechanismorprocessthatleadsalltheplayerstoexpectthesameequilibrium.2/31/31/32/31.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibriumChickenGameorhawk-doveGame

MikeGostraightTTurnawayWNeilGostraightT-1,-12,1TurnawayW1,20,01/21/21.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

FocalpointsSchelling’s(1960)theoryof“focalpoints”suggeststhatinsome“real-life”situationsplayersmaybeabletocoordinateonaparticularequilibriumbyusinginformationthatisabstractedawaybythestrategicform.Onereasonthatgametheoryabstractsawayfromsuchconsiderationsisthatthe“focalness”ofvariousstrategiesdependsontheplayers’cultureandpastexperiences.

Forexample,supposetwoplayersareaskedtonameanexacttime,withthepromiseofarewardiftheirchoicesmatch.1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

FocalPoints:Stag-huntGameAnotherexampleofmultipleequilibriaisthestag-huntgameweusedtobeginwithchapter,whereeachplayerhastochoosewhethertohunthare

byhimselfortojoinagroupthathuntsstag.SupposenowthatthereareIplayers,thatchoosingharegivespayoff1regardlessoftheotherplayers’actions,andthatchoosingstaggivespayoff2ifallplayerschoosestagandgivespayoff0otherwise.Thisgamehastwopure-strategyequilibria:“allstag”and“allhare.”

Nevertheless,itisnotclearwhichequilibriumshouldbeexpected.Inparticular,whichequilibriumismoreplausiblemaydependonthenumberofplayers.p

≥1/2;p8≥1/2,orp≥0.93

“allhare”risk-dominates“allstag.”

1.完全信息靜態(tài)博弈

GamesinStrategicFormandNashEquilibrium

ParetoOptimalityPlayer2LRPlayer1U9,90,8D8,07,7AlthoughriskdominancethensuggeststhataPareto-dominant

equilibriumneednotalwaysbeplayed,itissometimesarguedthatplayerswilli