高级微观经济学博弈论讲义(复旦大学CCES Yongqin Wang)

StaticGamesofCompleteInformation Lecture1 NashEquilibrium PureStrategyYongqinWang yongqinwang CCES FudanUniversity Dec 2006 FudanUniversity GameTheory Lecture1 2 OutlineofStaticGamesofCompleteInformation IntroductiontogamesNormal form orstrategic form representationIteratedeliminationofstrictlydominatedstrategiesNashequilibriumApplicationsofNashequilibriumMixedstrategyNashequilibrium Dec 2006 FudanUniversity GameTheory Lecture1 3 Agenda WhatisgametheoryExamplesPrisoner sdilemmaThebattleofthesexesMatchingpenniesStatic orsimultaneous move gamesofcompleteinformationNormal formorstrategic formrepresentation Dec 2006 FudanUniversity GameTheory Lecture1 4 Whatisgametheory Wefocusongameswhere ThereareatleasttworationalplayersEachplayerhasmorethanonechoicesTheoutcomedependsonthestrategieschosenbyallplayers thereisstrategicinteractionExample Sixpeoplegotoarestaurant Eachpersonpayshis herownmeal asimpledecisionproblemBeforethemeal everypersonagreestosplitthebillevenlyamongthem agame Dec 2006 FudanUniversity GameTheory Lecture1 5 Whatisgametheory Gametheoryisaformalwaytoanalyzestrategicinteractionamongagroupofrationalplayers oragents GametheoryhasapplicationsEconomicsPoliticsSociologyLawetc Dec 2006 FudanUniversity GameTheory Lecture1 6 ClassicExample Prisoners Dilemma Twosuspectsheldinseparatecellsarechargedwithamajorcrime However thereisnotenoughevidence Bothsuspectsaretoldthefollowingpolicy Ifneitherconfessesthenbothwillbeconvictedofaminoroffenseandsentencedtoonemonthinjail Ifbothconfessthenbothwillbesentencedtojailforsixmonths Ifoneconfessesbuttheotherdoesnot thentheconfessorwillbereleasedbuttheotherwillbesentencedtojailforninemonths Dec 2006 FudanUniversity GameTheory Lecture1 7 Example Thebattleofthesexes Attheseparateworkplaces ChrisandPatmustchoosetoattendeitheranoperaoraprizefightintheevening BothChrisandPatknowthefollowing Bothwouldliketospendtheeveningtogether ButChrispreferstheopera Patpreferstheprizefight Dec 2006 FudanUniversity GameTheory Lecture1 8 Example Matchingpennies Eachofthetwoplayershasapenny TwoplayersmustsimultaneouslychoosewhethertoshowtheHeadortheTail Bothplayersknowthefollowingrules Iftwopenniesmatch bothheadsorbothtails thenplayer2winsplayer1 spenny Otherwise player1winsplayer2 spenny Dec 2006 FudanUniversity GameTheory Lecture1 9 Static orsimultaneous move gamesofcompleteinformation Asetofplayers atleasttwoplayers Foreachplayer asetofstrategies actionsPayoffsreceivedbyeachplayerforthecombinationsofthestrategies orforeachplayer preferencesoverthecombinationsofthestrategies Player1 Player2 Playern S1S2 Snui s1 s2 sn foralls1 S1 s2 S2 sn Sn Astatic orsimultaneous move gameconsistsof Dec 2006 FudanUniversity GameTheory Lecture1 10 Static orsimultaneous move gamesofcompleteinformation Simultaneous moveEachplayerchooseshis herstrategywithoutknowledgeofothers choices Completeinformation ongame sstructure Eachplayer sstrategiesandpayofffunctionarecommonknowledgeamongalltheplayers AssumptionsontheplayersRationalityPlayersaimtomaximizetheirpayoffsPlayersareperfectcalculatorsEachplayerknowsthatotherplayersarerational Dec 2006 FudanUniversity GameTheory Lecture1 11 Static orsimultaneous move gamesofcompleteinformation Theplayerscooperate No Onlynon cooperativegamesMethodologicalindividualismThetimingEachplayerichooseshis herstrategysiwithoutknowledgeofothers choices Theneachplayerireceiveshis herpayoffui s1 s2 sn Thegameends Dec 2006 FudanUniversity GameTheory Lecture1 12 Definition normal formorstrategic formrepresentation Thenormal form orstrategic form representationofagameGspecifies Afinitesetofplayers 1 2 n players strategyspacesS1S2 Snandtheirpayofffunctionsu1u2 unwhereui S1 S2 Sn R Dec 2006 FudanUniversity GameTheory Lecture1 13 Normal formrepresentation 2 playergame Bi matrixrepresentation2players Player1andPlayer2EachplayerhasafinitenumberofstrategiesExample S1 s11 s12 s13 S2 s21 s22 Dec 2006 FudanUniversity GameTheory Lecture1 14 Classicexample Prisoners Dilemma normal formrepresentation Setofplayers Prisoner1 Prisoner2 Setsofstrategies S1 S2 Mum Confess Payofffunctions u1 M M 1 u1 M C 9 u1 C M 0 u1 C C 6 u2 M M 1 u2 M C 0 u2 C M 9 u2 C C 6 Payoffs Dec 2006 FudanUniversity GameTheory Lecture1 15 Example Thebattleofthesexes Normal orstrategic formrepresentation Setofplayers Chris Pat Player1 Player2 Setsofstrategies S1 S2 Opera PrizeFight Payofffunctions u1 O O 2 u1 O F 0 u1 F O 0 u1 F O 1 u2 O O 1 u2 O F 0 u2 F O 0 u2 F F 2 Dec 2006 FudanUniversity GameTheory Lecture1 16 Example Matchingpennies Normal orstrategic formrepresentation Setofplayers Player1 Player2 Setsofstrategies S1 S2 Head Tail Payofffunctions u1 H H 1 u1 H T 1 u1 T H 1 u1 H T 1 u2 H H 1 u2 H T 1 u2 T H 1 u2 T T 1 Dec 2006 FudanUniversity GameTheory Lecture1 17 Example Tourists Natives Onlytwobars bar1 bar2 inacityCanchargepriceof 2 4 or 56000touristspickabarrandomly4000nativesselectthelowestpricebarExample1 Bothcharge 2eachgets5 000customersand 10 000Example2 Bar1charges 4 Bar2charges 5Bar1gets3000 4000 7 000customersand 28 000Bar2gets3000customersand 15 000 Dec 2006 FudanUniversity GameTheory Lecture1 18 Example Cournotmodelofduopoly Aproductisproducedbyonlytwofirms firm1andfirm2 Thequantitiesaredenotedbyq1andq2 respectively Eachfirmchoosesthequantitywithoutknowingtheotherfirmhaschosen ThemarketpriceisP Q a Q whereQ q1 q2 ThecosttofirmiofproducingquantityqiisCi qi cqi Thenormal formrepresentation Setofplayers Firm1 Firm2 Setsofstrategies S1 0 S2 0 Payofffunctions u1 q1 q2 q1 a q1 q2 c u2 q1 q2 q2 a q1 q2 c Dec 2006 FudanUniversity GameTheory Lecture1 19 OneMoreExample Eachofnplayersselectsanumberbetween0and100simultaneously Letxidenotethenumberselectedbyplayeri LetydenotetheaverageofthesenumbersPlayeri spayoff xi 3y 5Thenormal formrepresentation Dec 2006 FudanUniversity GameTheory Lecture1 20 SolvingPrisoners Dilemma ConfessalwaysdoesbetterwhatevertheotherplayerchoosesDominatedstrategyThereexistsanotherstrategywhichalwaysdoesbetterregardlessofotherplayers choices Payoffs Dec 2006 FudanUniversity GameTheory Lecture1 21 Definition strictlydominatedstrategy Dec 2006 FudanUniversity GameTheory Lecture1 22 Example Twofirms ReynoldsandPhilip sharesomemarketEachfirmearns 60millionfromitscustomersifneitherdoadvertisingAdvertisingcostsafirm 20millionAdvertisingcaptures 30millionfromcompetitor Dec 2006 FudanUniversity GameTheory Lecture1 23 2 playergamewithfinitestrategies S1 s11 s12 s13 S2 s21 s22 s11isstrictlydominatedbys12ifu1 s11 s21 u1 s12 s21 andu1 s11 s22 u1 s12 s22 s21isstrictlydominatedbys22ifu2 s1i s21 u2 s1i s22 fori 1 2 3 Dec 2006 FudanUniversity GameTheory Lecture1 24 Definition weaklydominatedstrategy Dec 2006 FudanUniversity GameTheory Lecture1 25 Strictlyandweaklydominatedstrategy Arationalplayerneverchoosesastrictlydominatedstrategy Hence anystrictlydominatedstrategycanbeeliminated Arationalplayermaychooseaweaklydominatedstrategy Theorderofeliminationdoesnotmatterforstrictdominanceelimination pindownthesameequilibrium butdoesforweakone Dec 2006 FudanUniversity GameTheory Lecture1 26 Iteratedeliminationofstrictlydominatedstrategies Ifastrategyisstrictlydominated eliminateitThesizeandcomplexityofthegameisreducedEliminateanystrictlydominatedstrategiesfromthereducedgameContinuedoingsosuccessively Dec 2006 FudanUniversity GameTheory Lecture1 27 Iteratedeliminationofstrictlydominatedstrategies anexample Player1 Player2 Middle Up Down Left Right Dec 2006 FudanUniversity GameTheory Lecture1 28 Example Tourists Natives Onlytwobars bar1 bar2 inacityCanchargepriceof 2 4 or 56000touristspickabarrandomly4000nativesselectthelowestpricebarExample1 Bothcharge 2eachgets5 000customersand 10 000Example2 Bar1charges 4 Bar2charges 5Bar1gets3000 4000 7 000customersand 28 000Bar2gets3000customersand 15 000 Dec 2006 FudanUniversity GameTheory Lecture1 29 Example Tourists Natives Payoffsareinthousandsofdollars Dec 2006 FudanUniversity GameTheory Lecture1 30 OneMoreExample Eachofnplayersselectsanumberbetween0and100simultaneously Letxidenotethenumberselectedbyplayeri LetydenotetheaverageofthesenumbersPlayeri spayoff xi 3y 5 Dec 2006 FudanUniversity GameTheory Lecture1 31 OneMoreExample Thenormal formrepresentation Players player1 player2 playern Strategies Si 0 100 fori 1 2 n Payofffunctions ui x1 x2 xn xi 3y 5Isthereanydominatedstrategy Whatnumbersshouldbeselected Dec 2006 FudanUniversity GameTheory Lecture1 32 Newsolutionconcept Nashequilibrium Thecombinationofstrategies B R hasthefollowingproperty Player1CANNOTdobetterbychoosingastrategydifferentfromB giventhatplayer2choosesR Player2CANNOTdobetterbychoosingastrategydifferentfromR giventhatplayer1choosesB Dec 2006 FudanUniversity GameTheory Lecture1 33 Newsolutionconcept Nashequilibrium Thecombinationofstrategies B R hasthefollowingproperty Player1CANNOTdobetterbychoosingastrategydifferentfromB giventhatplayer2choosesR Player2CANNOTdobetterbychoosingastrategydifferentfromR giventhatplayer1choosesB Dec 2006 FudanUniversity GameTheory Lecture1 34 NashEquilibrium idea NashequilibriumAsetofstrategies oneforeachplayer suchthateachplayer sstrategyisbestforher giventhatallotherplayersareplayingtheirequilibriumstrategies Dec 2006 FudanUniversity GameTheory Lecture1 35 Definition NashEquilibrium Dec 2006 FudanUniversity GameTheory Lecture1 36 2 playergamewithfinitestrategies S1 s11 s12 s13 S2 s21 s22 s11 s21 isaNashequilibriumifu1 s11 s21 u1 s12 s21 u1 s11 s21 u1 s13 s21 andu2 s11 s21 u2 s11 s22 Dec 2006 FudanUniversity GameTheory Lecture1 37 FindingaNashequilibrium cell by cellinspection Player1 Player2 Middle Up Down Left Right Dec 2006 FudanUniversity GameTheory Lecture1 38 Example Tourists Natives Payoffsareinthousandsofdollars Dec 2006 FudanUniversity GameTheory Lecture1 39 OneMoreExample Thenormal formrepresentation Players player1 player2 playern Strategies Si 0 100 fori 1 2 n Payofffunctions ui x1 x2 xn xi 3y 5WhatistheNashequilibrium Dec 2006 FudanUniversity GameTheory Lecture1 40 Bestresponsefunction example IfPlayer2choosesL thenPlayer1 sbeststrategyisM IfPlayer2choosesC thenPlayer1 sbeststrategyisT IfPlayer2choosesR thenPlayer1 sbeststrategyisB IfPlayer1choosesB thenPlayer2 sbeststrategyisR Bestresponse thebeststrategyoneplayercanplay giventhestrategieschosenbyallotherplayers Dec 2006 FudanUniversity GameTheory Lecture1 41 Example Tourists Natives whatisBar1 sbestresponsetoBar2 sstrategyof 2 4or 5 whatisBar2 sbestresponsetoBar1 sstrategyof 2 4or 5 Payoffsareinthousandsofdollars Dec 2006 FudanUniversity GameTheory Lecture1 42 2 playergamewithfinitestrategies S1 s11 s12 s13 S2 s21 s22 Player1 sstrategys11isherbestresponsetoPlayer2 sstrategys21ifu1 s11 s21 u1 s12 s21 andu1 s11 s21 u1 s13 s21 Dec 2006 FudanUniversity GameTheory Lecture1 43 UsingbestresponsefunctiontofindNashequilibrium Ina2 playergame s1 s2 isaNashequilibriumifandonlyifplayer1 sstrategys1isherbestresponsetoplayer2 sstrategys2 andplayer2 sstrategys2isherbestresponsetoplayer1 sstrategys1 Dec 2006 FudanUniversity GameTheory Lecture1 44 UsingbestresponsefunctiontofindNashequilibrium example M isPlayer1 sbestresponsetoPlayer2 sstrategyL T isPlayer1 sbestresponsetoPlayer2 sstrategyC B isPlayer1 sbestresponsetoPlayer2 sstrategyR L isPlayer2 sbestresponsetoPlayer1 sstrategyT C isPlayer2 sbestresponsetoPlayer1 sstrategyM R isPlayer2 sbestresponsetoPlayer1 sstrategyB Dec 2006 FudanUniversity GameTheory Lecture1 45 Example Tourists Natives Payoffsareinthousandsofdollars UsebestresponsefunctiontofindtheNashequilibrium Dec 2006 FudanUniversity GameTheory Lecture1 46 Example Thebattleofthesexes OperaisPlayer1 sbestresponsetoPlayer2 sstrategyOperaOperaisPlayer2 sbestresponsetoPlayer1 sstrategyOperaHence Opera Opera isaNashequilibriumFightisPlayer1 sbestresponsetoPlayer2 sstrategyFightFightisPlayer2 sbestresponsetoPlayer1 sstrategyFightHence Fight Fight isaNashequilibrium Dec 2006 FudanUniversity GameTheory Lecture1 47 Example Matchingpennies HeadisPlayer1 sbestresponsetoPlayer2 sstrategyTailTailisPlayer2 sbestresponsetoPlayer1 sstrategyTailTailisPlayer1 sbestresponsetoPlayer2 sstrategyHeadHeadisPlayer2 sbestresponsetoPlayer1 sstrategyHeadHence NONashequilibrium Dec 2006 FudanUniversity GameTheory Lecture1 48 Definition bestresponsefunction Dec 2006 FudanUniversity GameTheory Lecture1 49 Definition bestresponsefunction Playeri sbestresponsetootherplayers strategiesisanoptimalsolutionto Dec 2006 FudanUniversity GameTheory Lecture1 50 UsingbestresponsefunctiontodefineNashequilibrium Asetofstrategies oneforeachplayer suchthateachplayer sstrategyisbestforher giventhatallotherplayersareplayingtheirstrategies orAstablesituationthatnoplayerwouldliketodeviateifotherssticktoit Dec 2006 FudanUniversity GameTheory Lecture1 51 Cournotmodelofduopoly Aproductisproducedbyonlytwofirms firm1andfirm2 Thequantitiesaredenotedbyq1andq2 respectively Eachfirmchoosesthequantitywithoutknowingtheotherfirmhaschosen ThemarketpricedisP Q a Q whereaisaconstantnumberandQ q1 q2 ThecosttofirmiofproducingquantityqiisCi qi cqi Dec 2006 FudanUniversity GameTheory Lecture1 52 Cournotmodelofduopoly Thenormal formrepresentation Setofplayers Firm1 Firm2 Setsofstrategies S1 0 S2 0 Payofffunctions u1 q1 q2 q1 a q1 q2 c u2 q1 q2 q2 a q1 q2 c Dec 2006 FudanUniversity GameTheory Lecture1 53 Cournotmodelofduopoly HowtofindaNashequilibriumFindthequantitypair q1 q2 suchthatq1 isfirm1 sbestresponsetoFirm2 squantityq2 andq2 isfirm2 sbestresponsetoFirm1 squantityq1 Thatis q1 solvesMaxu1 q1 q2 q1 a q1 q2 c subjectto0 q1 andq2 solvesMaxu2 q1 q2 q2 a q1 q2 c subjectto0 q2 Dec 2006 FudanUniversity GameTheory Lecture1 54 Cournotmodelofduopoly HowtofindaNashequilibriumSolveMaxu1 q1 q2 q1 a q1 q2 c subjectto0 q1 FOC a 2q1 q2 c 0q1 a q2 c 2 Dec 2006 FudanUniversity GameTheory Lecture1 55 Cournotmodelofduopoly HowtofindaNashequilibriumSolveMaxu2 q1 q2 q2 a q1 q2 c subjectto0 q2 FOC a 2q2 q1 c 0q2 a q1 c 2 Dec 2006 FudanUniversity GameTheory Lecture1 56 Cournotmodelofduopoly HowtofindaNashequilibriumThequantitypair q1 q2 isaNashequilibriumifq1 a q2 c 2q2 a q1 c 2Solvingthesetwoequationsgivesusq1 q2 a c 3 Dec 2006 FudanUniversity GameTheory Lecture1 57 Cournotmodelofduopoly BestresponsefunctionFirm1 sbestfunctiontofirm2 squantityq2 R1 q2 a q2 c 2ifq2 a c 0 othwiseFirm2 sbestfunctiontofirm1 squantityq1 R2 q1 a q1 c 2ifq1 a c 0 othwise q1 q2 a c 2 a c 2 a c a c Nashequilibrium Dec 2006 FudanUniversity GameTheory Lecture1 58 Cournotmodelofoligopoly Aproductisproducedbyonlynfirms firm1tofirmn Firmi squantityisdenotedbyqi Eachfirmchoosesthequantitywithoutknowingtheotherfirms choices ThemarketpricedisP Q a Q whereaisaconstantnumberandQ q1 q2 qn ThecosttofirmiofproducingquantityqiisCi qi cqi Dec 2006 FudanUniversity GameTheory Lecture1 59 Cournotmodelofoligopoly Thenormal formrepresentation Setofplayers Firm1 Firmn Setsofstrategies Si 0 fori 1 2 nPayofffunctions ui q1 qn qi a q1 q2 qn c fori 1 2 n Dec 2006 FudanUniversity GameTheory Lecture1 60 Cournotmodelofoligopoly HowtofindaNashequilibriumFindthequantities q1 qn suchthatqi isfirmi sbestresponsetootherfirms quantitiesThatis q1 solvesMaxu1 q1 q2 qn q1 a q1 q2 qn c subjectto0 q1 andq2 solvesMaxu2 q1 q2 q3 qn q2 a q1 q2 q3 qn c subjectto0 q2 Dec 2006 FudanUniversity GameTheory Lecture1 61 Bertrandmodelofduopoly differentiatedproducts Twofirms firm1andfirm2 Eachfirmchoosesthepriceforitsproductwithoutknowingtheotherfirmhaschosen Thepricesaredenotedbyp1andp2 respectively Thequantitythatconsumersdemandfromfirm1 q1 p1 p2 a p1 bp2 Thequantitythatconsumersdemandfromfirm2 q2 p1 p2 a p2 bp1 ThecosttofirmiofproducingquantityqiisCi qi cqi Dec 2006 FudanUniversity GameTheory Lecture1 62 Bertrandmodelofduopoly differentiatedproducts Thenormal formrepresentation Setofplayers Firm1 Firm2 Setsofstrategies S1 0 S2 0 Payofffunctions u1 p1 p2 a p1 bp2 p1 c u2 p1 p2 a p2 bp1 p2 c Dec 2006 FudanUniversity GameTheory Lecture1 63 Bertrandmodelofduopoly differentiatedproducts HowtofindaNashequilibriumFindthepricepair p1 p2 suchthatp1 isfirm1 sbestresponsetoFirm2 spricep2 andp2 isfirm2 sbestresponsetoFirm1 spricep1 Thatis p1 solvesMaxu1 p1 p2 a p1 bp2 p1 c subjectto0 p1 andp2 solvesMaxu2 p1 p2 a p2 bp1 p2 c subjectto0 p2 Dec 2006 FudanUniversity GameTheory Lecture1 64 Bertrandmodelofduopoly differentiatedproducts HowtofindaNashequilibriumSolvefirm1 smaximizationproblemMaxu1 p1 p2 a p1 bp2 p1 c subjectto0 p1 FOC a c 2p1 bp2 0p1 a c bp2 2 Dec 2006 FudanUniversity GameTheory Lecture1 65 Bertrandmodelofduopoly differentiatedproducts HowtofindaNashequilibriumSolvefirm2 smaximizationproblemMaxu2 p1 p2 a p2 bp1 p2 c subjectto0 p2 FOC a c 2p2 bp1 0p2 a c bp1 2 Dec 2006 FudanUniversity GameTheory Lecture1 66 Bertrandmodelofduopoly differentiatedproducts HowtofindaNashequilibriumThepricepair p1 p2 isaNashequilibriumifp1 a c bp2 2p2 a c bp1 2Solvingthesetwoequationsgivesusp1 p2 a c 2 b Dec 2006 FudanUniversity GameTheory Lecture1 67 Bertrandmodelofduopoly homogeneousproducts Twofirms firm1andfirm2 Eachfirmchoosesthepriceforitsproductwithoutknowingtheotherfirmhaschosen Thepricesaredenotedbyp1andp2 respectively Thequantitythatconsumersdemandfromfirm1 q1 p1 p2 a p1ifp1 p2 a p1 2ifp1 p2 0 ow Thequantitythatconsumersdemandfromfirm2 q2 p1 p2 a p2ifp2 p1 a p2 2ifp1 p2 0 ow ThecosttofirmiofproducingquantityqiisCi qi cqi Dec 2006 FudanUniversity GameTheory Lecture1 68 Bertrandmodelofduopoly homogeneousproducts Thenormal formrepresentation Setofplayers Firm1 Firm2 Setsofstrategies S1 0 S2 0 Payofffunctions Dec 2006 FudanUniversity GameTheory Lecture1 69 Bertrandmodelofduopoly homogeneousproducts Bestresponsefunctions pm a c 2 Dec 2006 FudanUniversity GameTheory Lecture1 70 Bertrandmodelofduopoly homogeneousproducts Bestresponsefunctions Firm1 sbestresponsetoFirm2 sp2 Firm2 sbestresponsetoFirm1 sp1 Dec 2006 FudanUniversity GameTheory Lecture1 71 Bertrandmodelofduopoly homogeneousproducts Bestresponsefunctions NashEquilibrium c c Dec 2006 FudanUniversity GameTheory Lecture1 72 Contributingtoapublicgood Twopersons person1andperson2 person1haswealthw1andperson2haswealthw2 Eachpersonchooseshowmuchtocontributewithoutknowingtheotherpersonhaschosen Theamou