nicholson微观经济理论（答案+ppt+习题库）-100-03-033593-0

GAMETHEORYANDSTRATEGICEQUILIBRIUMUNCERTAINTYALSOAFFECTSINDIVIDUALDECISIONMAKINGINSITUATIONSTHATREQUIREDEALINGWITHOTHERPEOPLEWHENTHEUTILITYANINDIVIDUALRECEIVESDEPENDSONTHEACTIONSTHATOTHERSTAKE,HEORSHEMUSTMAKESOMESORTOFCONJECTUREABOUTWHATTHOSEACTIONSWILLBESUCHCONJECTURES,BYTHEIRVERYNATURE,INVOLVEUNCERTAINTYBUT,BYSTUDYINGTHEOPTIONSOPENTOALLOFTHEINDIVIDUALSINAPARTICULARSITUATION,ITMAYBEPOSSIBLETOMAKESOMEPROGRESSINANALYZINGUTILITYMAXIMIZINGBEHAVIORTHEFORMALTOOLSFORDOINGTHISCOMEFROM“GAMETHEORY”ASTHENAMEIMPLIES,THESUBJECTOFGAMETHEORYINVOLVESTHESTUDYOFSTRATEGIC,GAMELIKESITUATIONS,RANGINGFROMTHETRIVIALTICTACTOE,BLACKJACKTOTHEAWESOMELYCOMPLEXANTIMISSILEDEFENSEINTHISCHAPTERWEPROVIDEANINTRODUCTIONTOTHISTOPICANDSOMEILLUSTRATIONSOFTHEKINDSOFSTRATEGICEQUILIBRIATHATCANARISEINRELATIVELYSIMPLEGAMESTHESETOOLSWILLBEUSEFULATSEVERALPLACESLATERINTHEBOOK,ESPECIALLYINCHAPTER20,WHICHUSESGAMETHEORYTOSTUDYSTRATEGICINTERACTIONSAMONGRMSINANINDUSTRY10CHAPTERBASICCONCEPTSGAMETHEORYMODELSSEEKTOPORTRAYCOMPLEXSTRATEGICSITUATIONSINAHIGHLYSIMPLIEDANDSTYLIZEDSETTINGMUCHLIKETHEOTHERMODELSINTHISBOOK,GAMETHEORYMODELSABSTRACTFROMMOSTOFTHEPERSONALANDINSTITUTIONALDETAILSOFAPROBLEMINORDERTOARRIVEATAREPRESENTATIONOFTHESITUATIONTHATISMATHEMATICALLYTRACTABLETHISABILITYTOGETTOTHE“HEART”OFTHEPROBLEMISTHEGREATESTSTRENGTHOFTHISTYPEOFMODELINGANYSITUATIONINWHICHINDIVIDUALSMUSTMAKESTRATEGICCHOICESANDINWHICHTHENALOUTCOMEWILLDEPENDONWHATEACHPERSONCHOOSESTODOCANBEVIEWEDASAGAMEALLGAMESHAVETHREEBASICELEMENTS1PLAYERS2STRATEGIESAND3PAYOFFSGAMESMAYBECOOPERATIVE,INWHICHPLAYERSCANMAKEBINDINGAGREEMENTS,ORNONCOOPERATIVE,WHERESUCHAGREEMENTSARENOTPOSSIBLEHEREWEWILLBECONCERNEDPRIMARILYWITHNONCOOPERATIVEGAMESTHEBASICELEMENTSLISTEDBELOWAREINCLUDEDINSUCHGAMESPLAYERSEACHDECISIONMAKERINAGAMEISCALLEDAPLAYERTHESEPLAYERSMAYBEINDIVIDUALSASINPOKERGAMES,RMSASINOLIGOPOLYMARKETS,ORENTIRENATIONSASINMILITARYCONICTSALLPLAYERSARECHARACTERIZEDASHAVINGTHEABILITYTOCHOOSEFROMAMONGASETOFPOSSIBLEACTIONSTHEYMIGHTTAKE1USUALLY,THENUMBEROFPLAYERSISXEDTHROUGHOUTTHE“PLAY”OFAGAME,ANDGAMESAREOFTENCHARACTERIZEDBYTHENUMBEROFPLAYERSTHATIS,TWOPLAYER,THREEPLAYER,ORNPLAYERGAMESINTHISCHAPTERWEWILLPRIMARILYSTUDYTWOPLAYERGAMESANDWILLDENOTETHESEPLAYERSBYAORBONEOFTHEIMPORTANTASSUMPTIONSUSUALLYMADEINGAMETHEORYASINMOSTOFECONOMICSISTHATTHESPECICIDENTITYOFTHEPLAYERSISIRRELEVANTTHEREARENO“GOODGUYS”OR“BADGUYS”INAGAME,ANDPLAYERSARENOTASSUMEDTOHAVEANYSPECIALABILITIESORSHORTCOMINGSEACHPLAYERISSIMPLYASSUMEDTOCHOOSETHECOURSEOFACTIONTHATYIELDSTHEMOSTFAVORABLEOUTCOME,AFTERTAKINGTHEACTIONSOFHISORHEROPPONENTINTOACCOUNTSTRATEGIESEACHCOURSEOFACTIONOPENTOAPLAYERDURINGAGAMEISCALLEDASTRATEGYDEPENDINGONTHEGAMEBEINGEXAMINED,ASTRATEGYMAYBEAVERYSIMPLEACTIONTAKEANOTHERCARDINBLACKJACKORAVERYCOMPLEXONEBUILDALASERBASEDANTIMISSILEDEFENSE,BUTEACHSTRATEGYISASSUMEDTOBEAWELLDENED,SPECICCOURSEOFACTION2USUALLY,THENUMBEROFSTRATEGIESAVAILABLETOEACHPLAYERWILLBESMALLMANY246PARTIIICHOICEUNDERUNCERTAINTY1SOMETIMESONEOFTHEPLAYERSINAGAMEISTAKENTOBE“NATURE”FORTHISPLAYER,ACTIONSARENOT“CHOSEN”BUTRATHEROCCURWITHCERTAINPROBABILITIESFOREXAMPLE,THEWEATHERMAYAFFECTTHEOUTCOMESOFAGAME,BUTITISNOT“CHOSEN”BYNATURERATHER,PARTICULARWEATHEROUTCOMESAREASSUMEDTOOCCURWITHVARIOUSPROBABILITIESGAMESAGAINSTNATURECANBEANALYZEDUSINGTHEMETHODSDEVELOPEDINCHAPTER92INGAMESINVOLVINGASEQUENCEOFACTIONSFOREXAMPLE,MOSTBOARDGAMES,SUCHASCHESS,ASPECICATIONOFSTRATEGIESMAYINVOLVESEVERALDECISIONPOINTSEACHMOVEINCHESSASSUMINGPERFECTKNOWLEDGEOFHOWTHEGAMEISPLAYED,SUCHCOMPLEXPATTERNSCANOFTENBEEXPRESSEDBYCHOICESAMONGALARGEBUTNITESETOFPURESTRATEGIES,EACHOFWHICHSPECIESACOMPLETECOURSEOFACTIONUNTILTHEGAMEISCOMPLETEDSEEOURDISCUSSIONOF“EXTENSIVE”AND“NORMAL”FORMSANDDMKREPS,GAMETHEORYANDECONOMICMODELING,OXFORDUNIVERSITYPRESS1990,CHAPTER3ASPECTSOFGAMETHEORYCANBEILLUSTRATEDFORSITUATIONSINWHICHEACHPLAYERHASONLYTWOSTRATEGIESAVAILABLEINNONCOOPERATIVEGAMES,PLAYERSCANNOTREACHBINDINGAGREEMENTSWITHEACHOTHERABOUTWHATSTRATEGIESTHEYWILLPLAYEACHPLAYERISUNCERTAINABOUTWHATTHEOTHERWILLDOPAYOFFSTHENALRETURNSTOTHEPLAYERSOFAGAMEATITSCONCLUSIONARECALLED“PAYOFFS”PAYOFFSAREUSUALLYMEASUREDINLEVELSOFUTILITYOBTAINEDBYTHEPLAYERS,ALTHOUGHMONETARYPAYOFFSSAY,PROTSFORRMSAREOFTENUSEDINSTEADINGENERAL,ITISASSUMEDTHATPLAYERSCANRANKTHEPAYOFFSOFAGAMEORDINALLYFROMMOSTPREFERREDTOLEASTPREFERREDANDWILLSEEKTHEHIGHESTRANKEDPAYOFFATTAINABLEPAYOFFSINCORPORATEALLASPECTSASSOCIATEDWITHOUTCOMESOFAGAMETHESEINCLUDEEXPLICITMONETARYPAYOFFSANDIMPLICITFEELINGSBYTHEPLAYERSABOUTTHEOUTCOMES,SUCHASWHETHERTHEYAREEMBARRASSEDORGAINSELFESTEEMPLAYERSPREFERPAYOFFSTHATOFFERMOREUTILITYTOTHOSETHATOFFERLESSNOTATIONUSUALLYITISNOTNECESSARYTOWRITEDOWNAGAMEINFORMALNOTATIONALITERARYDESCRIPTIONOFTHESITUATIONWILLDOBUT,FORSTATINGRESULTSINACOMPACTWAY,SOMENOTATIONCANHELPCLARIFYMATTERSFOLLOWINGSTANDARDCUSTOM,WEWILLDENOTEAPARTICULARGAMEGBETWEENTWOPLAYERSAANDBBYGSA,SB,UAA,B,UBA,B,101WHERESAANDSBREPRESENTTHESETOFSTRATEGIESAVAILABLEFORPLAYERSAANDB,RESPECTIVELY,ANDUAANDUBREPRESENTTHEUTILITYOBTAINEDBYTHEPLAYERSWHENAANDBCHOOSEPARTICULARSTRATEGIESAH20666SA,BH20666SBNASHEQUILIBRIUMINGAMESINTHEECONOMICTHEORYOFMARKETS,THECONCEPTOFEQUILIBRIUMISDEVELOPEDTOINDICATEASITUATIONINWHICHBOTHSUPPLIERSANDDEMANDERSARECONTENTWITHTHEMARKETOUTCOMEGIVENTHEEQUILIBRIUMPRICEANDQUANTITY,NOMARKETPARTICIPANTHASANINCENTIVETOCHANGEHISORHERBEHAVIORTHEQUESTIONTHEREFOREARISESWHETHERTHEREARESIMILAREQUILIBRIUMCONCEPTSINGAMETHEORYMODELSARETHERESTRATEGICCHOICESTHAT,ONCEMADE,PROVIDENOINCENTIVESFORTHEPLAYERSTOALTERTHEIRBEHAVIORFURTHERDOTHESEEQUILIBRIATHENOFFERBELIEVABLEEXPLANATIONSOFTHEOUTCOMEOFGAMESALTHOUGHTHEREARESEVERALWAYSTOFORMALIZEEQUILIBRIUMCONCEPTSINGAMETHEORY,THEMOSTCOMMONLYUSEDAPPROACHWASORIGINALLYPROPOSEDBYCOURNOTSEECHAPTER19INTHENINETEENTHCENTURYANDGENERALIZEDINTHEEARLY1950SBYJNASH3UNDERNASHSPROCEDURE,APAIROFSTRATEGIES,SAY,A,B,ISDENEDTOBEANEQUILIBRIUMIFAREPRESENTSPLAYERASBESTSTRATEGYWHENBPLAYSB,ANDBCHAPTER10GAMETHEORYANDSTRATEGICEQUILIBRIUM2473JOHNNASH,“EQUILIBRIUMPOINTSINNPERSONGAMES,”PROCEEDINGSOFTHENATIONALACADEMYOFSCIENCES3619504849REPRESENTSBSBESTSTRATEGYWHENAPLAYSAFORMALLY,APAIROFSTRATEGIESISANASHEQUILIBRIUMIFUAA,BH11350UAAH11032,BFORALLAH11032H20690SA102ANDUBA,BH11350UBA,BH11032FORALLBH11032H20690SBEVENIFONEOFTHEPLAYERSREVEALSTHEEQUILIBRIUMSTRATEGYHEORSHEWILLUSE,THEOTHERPLAYERCANNOTBENETFROMKNOWINGTHISFORNONEQUILIBRIUMSTRATEGIES,THISISNOTTHECASEASWESHALLSEE,IFONEPLAYERKNOWSWHATTHEOTHERSSTRATEGYWILLBE,HEORSHECANOFTENBENETFROMTHATKNOWLEDGEANDCHOOSEANOTHERSTRATEGYTHISMAY,INTURN,REDUCETHEPAYOFFRECEIVEDBYTHEPLAYERWHOHASREVEALEDHISORHERSTRATEGY,PROVIDINGANINCENTIVETODOSOMETHINGELSENOTEVERYGAMEHASANASHEQUILIBRIUMPAIROFSTRATEGIESAND,INSOMECASES,AGAMEMAYHAVEMULTIPLEEQUILIBRIA,SOMEOFWHICHAREMOREPLAUSIBLETHANOTHERSSOMENASHEQUILIBRIAMAYNOTBEESPECIALLYDESIRABLEFORTHEPLAYERSINAGAMEAND,INSOMECASES,OTHEREQUILIBRIUMCONCEPTSMAYBEMOREREASONABLETHANTHOSEPROPOSEDBYNASHSTILL,WENOWHAVEANINITIALWORKINGDENITIONOFEQUILIBRIUMWITHWHICHTOSTARTOURSTUDYOFGAMETHEORYNASHEQUILIBRIUMSTRATEGIESAPAIROFSTRATEGIESA,BREPRESENTSANEQUILIBRIUMSOLUTIONTOATWOPLAYERGAMEIFAISANOPTIMALSTRATEGYINTHESENSEOFEQUATION102FORAAGAINSTB,ANDBISANOPTIMALSTRATEGYFORBAGAINSTA4ANILLUSTRATIVEDORMITORYGAMEASAWAYOFILLUSTRATINGTHEGAMETHEORETICAPPROACHTOSTRATEGICMODELING,LETSEXAMINEASIMPLEEXAMPLEINWHICHTWOSTUDENTSAANDBMUSTDECIDEHOWLOUDLYTOPLAYTHEIRSTEREOSINADORMEACHPERSONMAYCHOOSETOPLAYHISORHEREQUIPMENTEITHERLOUDLYLORSOFTLYSWEWISHTOEXAMINEPOSSIBLEEQUILIBRIUMCHOICESINTHISSITUATIONITSHOULDBESTRESSEDATTHEOUTSETTHATTHISGAMEISNOTESPECIALLYREALISTICITISINTENDEDFORPEDAGOGICPURPOSESONLYTHEGAMEINEXTENSIVEFORMFIGURE101ILLUSTRATESTHESPECICDETAILSOFTHEDORMGAMEINTHISGAME“TREE,”THEACTIONPROCEEDSFROMLEFTTORIGHT,ANDEACH“NODE”REPRESENTSADECISIONPOINTFORTHEPERSONINDICATEDTHERETHERSTMOVEINTHISGAMEBELONGSTOAHEORSHEMUSTCHOOSEADECIBELLEVEL,LORSBECAUSEBSDECISIONSOCCURTOTHERIGHTOFAS,THETREEINDICATESTHATBMAKESTHEDECISIONAFTERAATTHISSTAGE,TWOVERSIONSOFTHEDEFINITION248PARTIIICHOICEUNDERUNCERTAINTY4ALTHOUGHTHISDENITIONISSTATEDONLYFORTWOPLAYERGAMES,THEGENERALIZATIONTONPERSONSISSTRAIGHTFORWARDBUTNOTATIONALLYCUMBERSOMEGAMEAREPOSSIBLEDEPENDINGONWHETHERBKNOWSWHATCHOICEAMADEFIRSTWEWILLLOOKATTHECASEWHEREBDOESNOTHAVETHISINFORMATIONTHELARGEROVALSURROUNDINGBSTWODECISIONNODESINDICATESTHATBOTHNODESSHARETHESAMELACKOFINFORMATIONBMUSTCHOOSELORSWITHOUTKNOWINGWHATAHASDONELATERWEWILLEXAMINETHECASEWHEREBDOESHAVETHISINFORMATIONTHENUMBERSATTHEENDOFEACHTREEBRANCHINDICATEPAYOFFS,HEREMEASUREDINUTILITYTOTHESETWODORMMATESEACHPAIROFPAYOFFSLISTSASUTILITYRSTFOREXAMPLE,THEPAYOFFSINFIGURE101SHOWTHATIFACHOOSESSANDBCHOOSESL,UTILITYWILLBE6FORAAND4FORBOTHERPAYOFFSAREINTERPRETEDSIMILARLYTHEGAMEINNORMALFORMALTHOUGHTHEGAMETREEINFIGURE101OFFERSAUSEFULVISUALPRESENTATIONOFTHECOMPLETESTRUCTUREOFAGAME,SOMETIMESITISMORECONVENIENTTODESCRIBEGAMESINTABULARSOMETIMESCALLEDNORMALORSTRATEGICFORMTABLE101PROVIDESSUCHAPRESENTATIONFORTHEDORMITORYGAMEINTHETABLE,ASSTRATEGIESSORLARESHOWNATTHELEFT,ANDBSSTRATEGIESARESHOWNACROSSTHETOPPAYOFFSAGAINWITHASCOMINGRSTCORRESPONDINGTOTHEVARIOUSSTRATEGICCHOICESARESHOWNINTHEBODYOFTHETABLEFIGURE101ANDTABLE101CONVEYEXACTLYTHESAMEINFORMATIONABOUTTHISGAME,THOUGHUSUALLYITISMORECONVENIENTTOWORKWITHTHENORMALFORMDOMINANTSTRATEGIESANDNASHEQUILIBRIATABLE101MAKESCLEARTHATALOUDPLAYSTRATEGYISADOMINANTSTRATEGYFORPERSONBNOMATTERWHATSTRATEGYACHOOSES,THELSTRATEGYPROVIDESGREATERUTILITYTOBTHANCHAPTER10GAMETHEORYANDSTRATEGICEQUILIBRIUM249THEDORMITORYGAMEINEXTENSIVEFORMINTHISGAME,ACHOOSESALOUDLORASOFTSSTEREOVOLUME,THENBMAKESASIMILARCHOICETHEOVALSURROUNDINGBSNODESINDICATESTHATTHEYSHARETHESAMELACKOFINFORMATIONBDOESNOTKNOWWHATSTRATEGYAHASCHOSENPAYOFFSWITHASRSTARELISTEDATTHERIGHTASSSBBLLL7,55,46,46,3FIGURE101DOESTHESSTRATEGYOFCOURSE,BECAUSETHESTRUCTUREOFTHEGAMEISKNOWNTOBOTHPLAYERS,AWILLRECOGNIZETHATBHASSUCHADOMINANTSTRATEGYHENCE,AWILLOPTFORTHESTRATEGYTHATDOESTHEBESTAGAINSTBSCHOICEOFLASCANBESEENFROMTABLE101,AWILLCONSEQUENTLYALSOCHOOSETOPLAYHISORHERMUSICLOUDLYLCONSIDERATIONSOFSTRATEGYDOMINANCE,THEREFORE,SUGGESTTHATTHEAL,BLSTRATEGYCHOICEWILLBEMADEANDTHATTHERESULTINGUTILITYPAYOFFSWILLBE7TOAAND5TOBTHEAL,BLSTRATEGYCHOICEALSOOBEYSTHENASHCRITERIONFOREQUILIBRIUMIFAKNOWSTHATBWILLPLAYL,HISORHERBESTCHOICEISLSIMILARLY,IFBKNOWSAWILLPLAYLHISORHERBESTCHOICEISALSOLINDEED,BECAUSELISADOMINANTSTRATEGYFORB,THISISTHEBESTCHOICENOMATTERWHATADOESTHEAL,BLCHOICE,THEREFORE,MEETSTHESYMMETRYREQUIREDBYTHENASHCRITERIONTOSEEWHYTHEOTHERSTRATEGYPAIRSINTABLE101DONOTMEETTHENASHCRITERION,LETUSCONSIDERTHEMONEATATIMEIFTHEPLAYERSANNOUNCEAS,BL,THISPROVIDESAWITHACHANCETOBETTERHISORHERPOSITIONIFAKNOWSBWILLOPTFORL,HEORSHECANOBTAINGREATERUTILITYBYCHOOSINGLTHECHOICEAS,BLISTHEREFORENOTANASHEQUILIBRIUMNEITHEROFTHETWOOUTCOMESINWHICHBCHOOSESSMEETSTHENASHCRITERIONEITHERASWEHAVEALREADYPOINTEDOUT,NOMATTERWHATADOES,BCANIMPROVEITSUTILITYBYCHOOSINGLINSTEADBECAUSELSTRICTLYDOMINATESSORB,NOOUTCOMEINWHICHBPLAYSSCANBEANASHEQUILIBRIUMEXISTENCEOFNASHEQUILIBRIAALTHOUGHTHEDORMGAMEILLUSTRATEDINFIGURE101CONTAINSAUNIQUENASHEQUILIBRIUM,THATISNOTAGENERALPROPERTYOFALLTWOPERSONGAMESEXAMPLE101ILLUSTRATESASIMPLEGAMEROCK,SCISSORS,PAPERINWHICHNONASHEQUILIBRIUMEXISTS,ANDANOTHERGAMEBATTLEOFTHESEXESTHATCONTAINSTWONASHEQUILIBRIATHESEEXAMPLESMAKECLEAR,THEREFORE,THATTHENASHAPPROACHMAYNOTALWAYSIDENTIFYAUNIQUEEQUILIBRIUMSOLUTIONTOATWOPERSONGAMERATHER,ONEMUSTEXPLORETHEDETAILSOFEACHGAMESITUATIONTODETERMINEWHETHERTHEREEXISTBELIEVABLENASHEQUILIBRIAEXAMPLE101SAMPLENASHEQUILIBRIATABLE102ILLUSTRATESTWOFAMILIARGAMESTHATREECTDIFFERINGPOSSIBILITIESFORNASHEQUILIBRIAPARTAOFTHETABLEDEPICTSTHECHILDRENSNGERGAMEROCK,SCISSORS,PAPERTHEZEROPAYOFFSALONGTHEDIAGONALSHOWTHATIFPLAYERSADOPTTHESAMESTRATEGY,NOPAYMENTSAREMADEINOTHERCASESTHEPAYOFFSINDICATEA1PAYMENTFROM250PARTIIICHOICEUNDERUNCERTAINTYTHEDORMITORYGAMEINNORMALFORMBSSTRATEGIESLSASSTRATEGIESL7,55,4S6,46,3TABLE101LOSERTOWINNERUNDERTHEUSUALHIERARCHYROCKBREAKSSCISSORS,SCISSORSCUTPAPER,PAPERCOVERSROCKASANYONEWHOHASPLAYEDTHISGAMEKNOWS,THEREISNOEQUILIBRIUMANYSTRATEGYPAIRISUNSTABLEBECAUSEITOFFERSATLEASTONEOFTHEPLAYERSANINCENTIVETOADOPTANOTHERSTRATEGYFOREXAMPLE,ASCISSORS,BSCISSORSPROVIDESANINCENTIVEFOREITHERAORBTOCHOOSEROCKSIMILARLYAPAPER,BROCKOBVIOUSLYENCOURAGESBTOCHOOSESCISSORSTHEIRREGULARCYCLINGBEHAVIOREXHIBITEDINTHEPLAYOFTHISGAMECLEARLYINDICATESTHEABSENCEOFANASHEQUILIBRIUMBATTLEOFTHESEXESINTHEBATTLEOFTHESEXESGAME,AHUSBANDAANDWIFEBAREPLANNINGAVACATIONAPREFERSMOUNTAINLOCATIONSBPREFERSTHESEASIDEBOTHPLAYERSPREFERAVACATIONSPENTTOGETHERTOONESPENTAPARTTHEPAYOFFSINPARTBOFTABLE102REECTTHESEPREFERENCESHEREBOTHOFTHEJOINTVACATIONSREPRESENTNASHEQUILIBRIAWITHAMOUNTAIN,BMOUNTAINNEITHERPLAYERCANGAINBYTAKINGADVANTAGEOFKNOWINGTHEOTHERSSTRATEGYSIMILARCOMMENTSAPPLYTOASEASIDE,BSEASIDEHENCETHISISAGAMEWITHTWONASHEQUILIBRIAQUERYAREANYOFTHESTRATEGIESINEITHEROFTHESEGAMESDOMINANTWHYARENTSEPARATEVACATIONSNASHEQUILIBRIAINTHEBATTLEOFTHESEXESTABLE102THEREARE,HOWEVER,CERTAINTYPESOFTWOPERSONGAMESINWHICHANASHEQUILIBRIUMMUSTEXISTINTUITIVELY,GAMESINWHICHTHEPARTICIPANTSHAVEALARGENUMBEROFSTRATEGIESWILLOFTENOFFERSUFCIENTEXIBILITYTOENSURETHATATLEASTONENASHEQUILIBRIUMMUSTEXISTSUCHGAMESARISEINTWOCONTEXTSFIRST,GAMESINWHICHTHESTRATEGIESCHOSENBYAANDBAREALTERNATIVELEVELSOFASINGLECONTINUOUSVARIABLEINCLUDEAN“INNITE”NUMBEROFPOTENTIALSTRATEGIESSUCHGAMESAREGUARANTEEDTOHAVEANASHEQUILIBRIUMTHEMOSTIMPORTANTCLASSOFSUCHGAMESINVOLVESGAMESWHEREPLAYERSARETWORMSTHATMUSTCHOOSETHEPRICETHEYWILLCHARGEFORASINGLEPRODUCTSOMEGAMESOFTHISTYPE,TOGETHERWITHILLUSTRATIONSOFTHETYPESOFNASHEQUILIBRIATHEYEXHIBIT,AREDISCUSSEDINCHAPTER20CHAPTER10GAMETHEORYANDSTRATEGICEQUILIBRIUM251TWOSIMPLEGAMESAROCK,SCISSORS,PAPERNONASHEQUILIBRIABSSTRATEGIESROCKSCISSORSPAPERROCK0,01,H110021H110021,1ASSTRATEGIESSCISSORSH110021,10,01,H110021PAPER1,H110021H110021,10,0BBATTLEOFTHESEXESTWONASHEQUILIBRIABSSTRATEGIESMOUNTAINSEASIDEASSTRATEGIESMOUNTAIN2,10,0SEASIDE0,01,2TABLE102ANOTHERWAYINWHICHGAMESMAYCONTAINASUFCIENTLY“LARGE”NUMBEROFSTRATEGIESISTOPERMITPLAYERSTOUSE“MIXED”STRATEGIESINSUCHGAMES,THEREMAYBERELATIVELYFEW“PURE”STRATEGIESLIKETHEONESWEHAVEBEENEXAMININGPERHAPSONLYTWOBUTEACHPLAYERISPERMITTEDTOPLAYTHESEPURESTRATEGIESWITHCERTAIN,PRESELECTEDPROBABILITIESINTHEDORMGAME,FOREXAMPLE,AMIGHTIPACOINTODETERMINEWHETHERTOPLAYMUSICLOUDLYORSOFTLYTHATIS,HEORSHEWOULDPLAYEACHSTRATEGYWITHPROBABILITY12IFEACHPLAYERCANCHOOSETOPLAYTHEAVAILABLEPURESTRATEGIESWITHANYPROBABILITIESHEORSHEMIGHTCHOOSE,THEGAMEWILLBECONVERTEDINTOONEWITHANINNITENUMBEROFMIXEDSTRATEGIESAND,AGAIN,THEEXISTENCEOFANASHEQUILIBRIUMISENSUREDEXAMPLE102PROVIDESANILLUSTRATIONOFHOWTHECONSIDERATIONOFMIXEDSTRATEGIESCANADDTOTHENASHEQUILIBRIUMOUTCOMESINTHEBATTLEOFTHESEXESGAMEWEHAVEALREADYEXAMINEDEXAMPLE102BATTLEOFTHESEXESWITHMIXEDSTRATEGIESTOSHOWHOWTHEINTRODUCTIONOFMIXEDSTRATEGIESMAYADDNASHEQUILIBRIATOAGIVENGAME,LETSRETURNTOTHEBATTLEOFTHESEXESGAMEINEXAMPLE101SUPPOSETHATTHESPOUSESINTHEPROBLEMTIREOFCONSTANTBICKERINGABOUTVACATIONSANDDECIDETOLET“CHANCE”DECIDESPECICALLY,SUPPOSEADECIDESTOCHOOSEHISMOUNTAINSTRATEGYWITHPROBABILITYRANDSEASIDEWITHPROBABILITY1H11002RSIMILARLY,SUPPOSEBCHOOSESHERMOUNTAINSTRATEGYWITHPROBABILITYSANDSEASIDEPROBABILITYWITH1H11002SGIVENTHESEPROBABILITIES,THEOUTCOMESOFTHEGAMEOCCURWITHTHEFOLLOWINGPROBABILITIESMOUNTAINMOUNTAIN,RSMOUNTAINSEASIDE,R1H11002SSEASIDEMOUNTAIN,1H11002RSANDSEASIDESEASIDE1H11002R1H11002SASEXPECTEDUTILITYISTHENGIVENBYEUAH11005RS2H11001R1H11002S0H110011H11002RS0H110011H11002R1H11002S1103H110051H11002RH11002SH110013RSH110051H11002SH11001R3SH110021OBVIOUSLY,ASOPTIMALCHOICEOFRDEPENDSONBSPROBABILITY,SIFSH1102113,UTILITYISMAXIMIZEDBYCHOOSINGRH110050IFSH1102213,ASHOULDOPTFORRH110051ANDWHENSH1100513,ASEXPECTEDUTILITYIS23NOMATTERWHATVALUEOFRISCHOSENFIGURE102ILLUSTRATESASOPTIMALCHOICESOFRGIVENTHESEVARIOUSVALUESOFSFORSPOUSEB,EXPECTEDUTILITYISGIVENBYEUBH11005RS1H11001R1H11002S0H110011H11002SR0H110011H11002R1H11002S2104H110052H110022RH110022SH110013RSH110052H110022RH11001S3RH110022NOW,WHENRH1102123,BSEXPECTEDUTILITYISMAXIMIZEDBYCHOOSINGSH110050WHENRH1102223,UTILITYISMAXIMIZEDBYCHOOSINGSH110051ANDWHENRH1100523,BSEXPECTEDUTILITYISINDEPENDENTOFWHATSSHECHOOSESTHESEOPTIMALCHOICESAREALSOSHOWNINFIGURE102NASHEQUILIBRIAARESHOWNINFIGURE102BYTHEINTERSECTIONSOFTHEOPTIMALRESPONSECURVESFORAANDBTHATIS,THEINTERSECTIONSOBEYTHECONDITIONSSUMMARIZEDINEQUATIONS102NOTICETHATTHEREARETHREESUCHINTERSECTIONSTWOOFTHESEWEHAVESEENPREVIOUSLYRH110050,SH110050ANDRH110051,SH110051REPRESENTTHEJOINTVACATIONSTRATEGIESWEDISCUSSEDINEXAMPLE101BUTRH1100523,SH1100513ISANEWNASHEQUILIBRIUMTHATWASUNAVAILABLEBEFORETHEINTRODUCTIONOFMIXEDSTRATEGIESMOREGENERALLY,FIGURE102PROVIDESAHINTOFWHYGAMESWITHACONTINUUMOFSTRATEGIESMUSTHAVENASHEQUILIBRIAINGENERAL,CONTINUOUSOPTIMALRESPONSEFUNCTIONSWILLINTERSECTSOMEWHEREANDTHOSEINTERSECTIONSWILLBENASHEQUILIBRIA252PARTIIICHOICEUNDERUNCERTAINTYQUERYISTHEMIXEDSTRATEGYEQUILIBRIUMILLUSTRATEDINTHISPROBLEMPARTICULARLYDESIRABLETOTHEPLAYERSIFTHESPOUSESCOULDCOOPERATETOREACHADECISION,WOULDTHEYOPTFORSUCHAMIXEDSTRATEGYSOLUTIONUNFORTUNATELY,THEPROOFOFTHEEXISTENCEOFANASHEQUILIBRIUMINTWOPERSONGAMESWITHACONTINUUMOFSTRATEGIESISDIFCULTANDREQUIRESANUMBEROFTECHNICALASSUMPTIONSHENCE,WEWILLNOTATTEMPTTOPRESENTITHERERATHER,MOSTOFOURANALYSISOFTWOPERSONGAMESWILLINVOLVEEXPLICITLYSOLVINGFORANYNASHEQUILIBRIATHATMAYEXISTINTERESTEDREADERSMAYWISHTOEXPLOREMATHEMATICALLYSOPHISTICATEDEXISTENCEPROOFSONTHEIROWN5CHAPTER10GAMETHEORYANDSTRATEGICEQUILIBRIUM2535SEE,FOREXAMPLE,DFUDENBERGANDJTIROLE,GAMETHEORYCAMBRIDGE,MAMITPRESS,1992,SECTION13NASHEQUILIBRIAINMIXEDSTRATEGIESINTHEBATTLEOFTHESEXESGAMEWITHMIXEDSTRATEGIES,APLAYS“MOUNTAINS”WITHPROBABILITYR,ANDBPLAYSMOUNTAINSWITHPROBABILITYSTHEGURESHOWSEACHPLAYERSOPTIMALCHOICEGIVENTHEOTHERPLAYERSCHOICETHISGAMEHASTHREENASHEQUILIBRIADENOTEDBYE1RH110050,SH1100502RH110051,SH110051AND3RH1100523,SH1100513S1H11408301H1140832H1140831RASPROBABILITYASOPTIMALCHOICEBSPROBABILITYBSOPTIMALCHOICEE1E2E312H114083FIGURE102THEPRISONERSDILEMMANASHEQUILIBRIAARISEBECAUSEOFTHESTRATEGICUNCERTAINTIESINHERENTINASITUATIONNOTHINGGUARANTEESTHATTHESEEQUILIBRIAWILLBEESPECIALLYDESIRABLEFROMTHEPLAYERSPERSPECTIVESPROBABLYTHEMOSTFAMOUSEXAMPLEOFATWOPERSONGAMEWITHANUNDESIRABLENASHEQUILIBRIUMOUTCOMEISTHEPRISONERSDILEMMAGAME,RSTDISCUSSEDBYAWTUCKERINTHE19