Strategic decision theory

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StrategicDecisionTheoryProblemSet2Due5.00PMonThursday,September21st.PleasesubmitaPDF,hand-writtenortypedonbrightspace.RemarkYoumaycollaboratewithasmallstudygroup.Ifso,pleaseindicatewhoyoucollaboratedwithandwriteupyouranswersindividuallysothatitisclearthatyouunderstandthesolutionanditisnotsimplyacopyofsomeoneelse’ssolution.Problem1-NashEquilibriaIFindalltheNashequilibria(bothpureandmixed)ofthefollowinggame.XYA3,01,1B4,20,3C2,24,0Problem2-NashEquilibriaIIFindalltheNashequilibria(bothpureandmixed)ofthefollowinggame.XYZA2,11,02,0B1,22,31,1C0,31,13,1Problem3-NashEquilibriaIIIArmyAhasasingleplanewithwhichitcanstrikeoneofthethreepossibletargets.ArmyBhasoneanti-aircraftgunthatcanbeassignedtooneofthetargets.Thevalueoftargetkisvkwithv1>v2>v3>0.ArmyAcandestroyatargetifandonlyifthetargetisundefendedandAattacksit.ArmyAwishestomaximizetheexpectedvalueofthedamageandArmyBwishestominimizeit.Notethatthisisazero-sumgame.1.Formulatethesituationasanormalformormatrixgame.2.Supposev1=5,v2=4,v3=3.FindanequilibriuminwhichArmyAattacksallthreetargetswithpositiveprobability.Forextracreditarguethatallequilibriahavethisproperty.3.Supposev1=3,v2=2,v3=1.FindanequilibriuminwhichArmyAattacksonlythefirsttwotargetswithpositiveprobability.Forextracreditshowthatthisgamehasauniqueequilibrium.1
IntroductiontoGameTheoryStrategicDecisionTheoryFall20231
NormalFormGamesIGibbons:Chapter1.IWhatisagame?Soccer?Boardgames?Poker,Bridge?Oligopolisticcompetition?IAllcanbemodeledasgames.IBoardgames,cardgames,etc.withtheirpreciselydenedrulesofplayandclearlyspeciedoutcomesareclosesttothemathematicalformulationofagame.ISinglepersondecisionproblem:maxs12S1u1(s1;a)whereaissomeexogenouslygivenlevelofaparameter.IMulti-persondecisionproblem:maxs12S1u1(s1;s2)wheres2isnotexogenouslygivenbutratherendogenouslydeterminedalso,asthechoicevariableoftheotherplayer(inatwo-persongame).Moregenerally,otherplayers.2
Prisoners’DilemmaDefinitionAgameGisspeciedbyasetofplayersi=1;:::;nandforeachplayeri,astrategyspace/setSiandapayofunctionuiwhichindicatesi’spayoforeachpossibleprole/combinationofplayers’strategies(S1;:::;Sn).Iui:S1


Sn!R.IG=f(Si;ui):i=1;2;:::;ng.ITwoplayernitestrategycase:Matrixdepiction(asabove)possibleandconvenient.Rowplayer(1)andColumnPlayer(2).Payosincorrespondingboxes.Ineachbox,therstelementisrow’s(1’s)payoandthesecondcolumn’s(2’s).3
Prisoners’DilemmaCont’dIAnalysis:DominantstrategytoconfessIItdoesstrictlybetterthananyalternativestrategyagainstanystrategyofopponent.IDoesplayer1’spayodependonwhatplayer2does?IBeautifulexamplealthoughstrategicreasoningistrivial.INoteinecientoutcome.IComparewithcompetitiveeconomicenvironment.4
DominanceDenitionsIDefinitionAstrategysi2Siisstrictlydominatedifthereexistsastrategys0i2Sisuchthatui(s0i;si)>ui(si;si)forallsi2Si.INotation:si(s1;:::;si1;si+1;:::;sn)andSiS1


Si1Si+1


Sn.IDefinitionThestrategysi2Siisastrictlydominating/dominantstrategyforplayeri,ifforallsi2Sisuchthatsi6=siitisthecasethatui(si;si)>ui(si;si)forallsi2SiIForthecorrespondingweakdenitionsreplace\>”with\”and,inaddition,requirestrictinequalityforsomesi2Si.5
InteractiveReasoning6
InteractiveReasoningCont’d7
IterativeRemovalofStrictlyDominatedStrategiesIWedenethisprocedure(informally)asfollows:IStep1:RemoveallstrictlydominatedstrategiesofallplayersfromG.ThisyieldsthenewgameG1whichisidenticaltoGexceptthatthestrictlydominatedstrategiesofGhavebeenremoved.I…IStepn:RemoveallstrictlydominatedstrategiesofallplayersfromGk1.ThisyieldsthenewgameGkwhichisidenticaltoGk1exceptthatthestrictlydominatedstrategiesofGk1havebeenremoved.IProcedureterminateswhenGk+1=Gk,thatis,whennostrategyofGkisstrictlydominated.IFact:Ski6=forallk.IProcedureterminatesinatmostPni=1(jSij1)steps.8
KnowledgeAssumptionsforIterativeRemovalIPlayeriRationalmeansPlayerioptimizesgivenwhatiknowsIShorthand:R=\Rational”andS=\Statement”IKnowledgeAssumptionsIS1:1knows\ownpayos”andisRIS2:2knows\ownpayosand1’spayoandknowsS1″andisRIStep3:1knows\ownpayosand2’spayoandS1andS2″andisRIStep4:2knowseverythingthatheandotherplayersneedtoknowinallprecedingstepsandisR9
Cont’dIDefinition(informal):Someinformationiscommonknowledgeifforanysequenceofplayersi;j;:::;kthestatement\iknowsthatjknowsthat:::thatkknows”istrue.INotethattheelementsofthechaini;j;:::;kneednottobedistinctandthechainmaybeofanylength.IOurstandardassumptionisthat:IThestructureofthegame(setofplayers,strategiesandpayos)andtheplayers’rationalityiscommonknowledge.IIterativeremovalofdominatedstrategiesispersuasive,indeedunavoidableundertheaboveassumption.IStrongassumption.IIntuitivelyspeakinginformationiscommonknowledgeamongstagroupofplayersifitispubliclyannounced.10
Example:Guessthenumberofblackhats.ISeeGeanakoplos(1992){ratheradvancedandnotrequired|foranextensivediscussionofcommonknowledgeanditsimplications.STORY:Fivestudentsareseatedaroundatable.Ateacherwalksintotheroomwithaboxofhatsandplacesahatoneachoftheirheads.HatsareknowntobeeitherBlackorWhite.Eachpersoncanseeeveryoneelse’shatbutnottheirown.Nocommunicationisallowed.Theteacherannouncesthatatleastonestudentiswearingablackhat,setsoatimerwhichbeepsin5minutesandevery5minutesthereafter,asksthentoguessthecorrectnumberofblackhatsandleavestheroom.Guessescanonlybemadewhenthetimerbeeps.Theobjectiveistoguesscorrectlyasquicklyaspossible.11
NashEquilibriumIDiscreteQuantitySettingOligopoly.INodominatedstrategiesINASHequilibriumDefinition:Astrategyprole(s1;


;sn)2S1


SnisaNashequilibriumifforalliui(si;si)ui(si;si)forallsi2SiIJohnNash,NobelPrizeinEconomics,1996.12
BattleoftheSexes13
WhatisEquilibrium?IEquilibriumdenotesastateofrestorstabilityofaphysicalorothersystem.IANashequilibriumissuchastateinthesensethatatanequilibriumnosingleplayerhasareasontodeviatefromtheiractionchoiceintheequilibrium.IThisstatementistrueatanygivenstrategyproleifandonlyiftheproleisaNashequilibrium.14
EquilibriumCont’dIAnyuniquepredictionofbehaviorwillunravel/self-destruct,unlessthepredictedproleisanequilibriumprole(underthenaturalassumptionthattheplayersknowthepredictionalso)IIndynamicsituations,whenthesamegameisplayedovertimebydierentplayersoranonymously,ifthebehaviorsettlesdowntoaxedprolethatagentsexpecttobeplayedinthefuture,suchaprolemustbeaNashequilibriumprole.Examples:trac
ows,bargainingprotocols,socialconventionslikewaysofgreeting,LANGUAGE!Conceptisnow\universally”used!thisitselfisaNE!15
MixedStrategiesMatchingPenniesIEquilibrium?16
MixedStrategiesCont’dIMixedextensionofG=(Si;ui:i=1;:::;n).IAmixedstrategypiforplayeriisaprobabilitydistributionoveri’spurestrategiessi2Si.Ipi:Si![0;1]suchthatPsipi(si)=1.ILetPidenotethesetofmixedstrategiesforplayeriUi(pi;pi)=Xs1Xs2


Xsnui(s1;s2;


;sn)p1(s1)p2(s2)


pn(sn)IWhenn=2andi=1U1(p1;p2)=Xs1Xs2u1(s1;s2)p1(s1)p2(s2)IIndependentrandomization17
MixedStrategyEquilibriumIBestResponse:Thestrategyp0iisabestresponsetop(orpi)ifUi(p0i;pi)Ui(p00i;pi)forallp00i2PiBRi(p)fp0i:p0iisabestresponsetopigIDenition1:Themixedstrategyprole(p1;p2;:::;pn)isaNashequilibriumifforalliUi(pi;pi)Ui(pi;pi)forallpi2PiIEquivalently,foralli,pi2BRi(p)IDeneVi(si;pi)=Xsiui(si;si)p1(s1)


pi1(si1)pi+1(si+1)


pn(sn)IExpectedpayotoiofpurestrategysigiventhatotherplayersplayaccordingtothemixedstrategyprolepi.Then:Ui(pi;pi)=XsiVi(si;pi)pi(si)18
MixedStrategyEquilibriumCont’dILet’sspellthisoutforn=2andi=1.ThenV1(s1;p2)=Xs2u1(s1;s2)p2(s2)IConsequently,U1(p1;p2)=Xs1Xs2u1(s1;s2)p1(s1)p2(s2)=Xs1p1(s1)Xs2u1(s1;s2)p2(s2)=Xs1p1(s1)V1(s1;p2)19
Cont’dIDenition2:Aproleofmixedstrategies(p1;p2;:::;pn)isaNashequilibrium,ifandonlyifforalliandallsi2Sisuchthatpi(si)>0,Vi(si;pi)=ViwhereVi=maxsiVi(si;pi)IThatis,anypurestrategywhichisusedwithstrictlypositiveprobabilitymusthaveatleastashighanexpectedpayoasanyotherpurestrategy,giventhemixedstrategieswhichotherplayersareusing.IIndierenceCondition:Allpurestrategiesusedwithstrictlypositiveprobabilitybyimusthaveequalpayos.IIdentifyapurestrategysiwiththedegeneratemixedstrategywhichchoosessiwithprobabilityone.IDoesanequilibriuminpurestrategiescontinuetobeanequilibriumwhenmixedstrategiesareallowed?20
ExampleExampleNotationIp!probabilityof1Iqk!probabilityofkBothplayersmustmixinequilibrium.21
EquivalenceofTwoDenitionsRecallthatforanyp(pi;pi),Vi(si;pi)=Xsiui(si;si)p1(s1)


pi1(si1)pi+1(si+1)


pn(sn)Ui(pi;pi)=XsiVi(si;pi)pi(si)LetSi(pi)=fsi:Vi(si;pi)Vi(s0i;pi)foralls0i2Sig.SinceSiisnite,Si(pi)isnonempty.Fixp.LetVi=maxsiVi(si;pi).Thenforanypi:Ui(pi;pi)=Xsipi(si)Vi(si;pi)Viwithequalityifandonlyifpi(si)>0=)si2Si(pi)9>>>=>>>;(1)Let~sisatisfyVi(~si;pi)=Viand~pi(
)bedenedby~pi(~si)=1,~pi(si)=08si6=~si.ThenUi(~pi;pi)=Vi.22
D1=)D2IfpisatisesD1thenUi(pi;pi)Ui(~pi;pi)=ViButby(1),Ui(pi;pi)Vi.Hence,Ui(pi;pi)=Vi.By(1)itthenfollowsthatVi(si;pi)=Viforanysi2Sisuchthatpi(si)>0.Thatis,pi(
)satisestheconditionsofD2.23
D2=)D1IfpisatisestheconditionofD2thenby(1)Ui(pi;pi)=ViButthenby(1)italsofollowsthatUi(pi;pi)Ui(pi;pi)forallpi.Thatis,pisatisesD1.24
Whymix?IInourexample1’smixingiscalibratedtomake2indierentbetweenthepurestrategiesinthesupportof2’sequilibriummixedstrategy,andvice-versa.Thisisageneralfeatureofmixedstrategyequilibria.IInparticularplayeri’spayoisthesameforanyprobabilitydistributionoverthepurestrategieswhichiuseswithpositiveprobabilityinherequilibriummixedstrategy.IWhymix?Longrundynamic`story’.IIngeniousjusticationbyHarsanyi.Willpresentlater.25
Example26
ExtensionofIterativeDominancetoMixedStrategiesIcf.tothedenitionprovidedforpurestrategiesINowGkisstillidentiedwiththesetofundominatedpurestrategiesSki;i=1;:::;natthekthstage.IWesaythatatstagek,si2Skiisstrictlydominatedifthereexistsamixedstrategyp0iwithsupportSkisuchthatUi(p0i;si)>Ui(si;si)8si2SkiIGk+1isidenticaltoGkexceptforremovalofthestrictlydominatedpurestrategiesofGk.IAgain,terminatesinatmostnPi=1(jSij1)steps.27
StrictDominanceandNashEquilibriumFact:Let(p1;p2;


;pn)beamixedstrategyNashequilibrium.Foranyilet~sibeapurestrategysuchthatpi(~si)>0.Then~siisnotaiterativelystrictlydominatedstrategy.Furthermorethesestatementsaretrueifindeningdominanceanditerativedominanceweallowpurestrategiestobestrictlydominatedbymixedstrategies.28
StrictDominanceandNashEquilibriumCont’dCorollary:Supposeiterativeeliminationofstrictlydominatedstrategiesleadstoauniqueiterativelyundominatedstrategy~siforplayeri.ThenforanymixedstrategyNashequilibrium(p1;p2;


;pn),pi(~si)=1;playericannotuseanyotherstrategywithpositiveprobability,inanyNashequilibrium.29
ExistenceofEquilibriumIInallourexamplesthusfarwehavebeenabletondatleastonepureormixedstrategyNashEquilibrium.IDidwegetlucky?No!IAdeepresultduetoJohnNashguaranteestheexistenceofanequilibriuminmixedstrategies(whenapurestrategyisseenasaspecialcaseofamixedstrategy,i.e.onewhichpicksthepurestrategywithprobabilityone).ITheorem:Everyniten-playernormalformgamehasanequilibriuminmixedstrategies.I(SeeGibbons,p.45)30

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