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1、Chapter 11Game Theory(2)博弈论1经济学院Chapter 11 includes:2经济学院11.1roduction to Game Theory11.2 Nash Equilibrium(均衡)11.3 Subgame Perfect Nash Equilibrium（子博弈精炼均衡）11.4 Repeated Game（重复博弈）Overview of Last Class3经济学院Game Theory:roductionElements of a gameClassifications of GameDominant strategy（占优策略）Dominant

2、 strategy equilibrium（占优策略均衡）Nash equilibrium（均衡）Mixed StrategiesOutlines of Todays ClassDefinition of SubgamePerfect Equilibrium (SPE)(子博弈精炼均衡)Repeated GamesFini弈)Infiniy repeated games(有限重复博y repeated games（无限重复博弈）4经济学院Chapter 11 includes:5经济学院11.3 Subgame Perfect Nash Equilibrium（子博弈精炼均衡）Sequenti

3、al-move games:roductionThere is a strict order of playPlayers know what the onest moved before haveeA player must consider how others will reactIt will be convenient to describe sequential-move gamesusing the extensive-form represenion (tree)Important note: Actually, each type of game is capable ofe

4、ach type of represenion!6经济学院Exle: 仿冒与反仿冒博弈7经济学院A企业是仿冒企业，B企业是被仿冒企业。如果被仿冒企业采取措施制止，仿冒企业就会停止仿冒，如果被仿冒企业不采取措施制止，那么仿冒企业就会继续仿冒下去。假设仿冒最多进行两次The extensive form (game tree)1. Decinodes(决策结): corresponding to playersBranches（决策枝）: corresponding to actionsTerminal nodes（终点结）: corresponding to payoff2-1,-1AA3,00

5、,3NA1N2,2N28经济学院The Solution to Sequential-move GameBackward induction(逆向归纳法)2A-1,-1ANA3,00,31NN2,229经济学院How did we reason?: Player 2 chooses N after A and A after NSecond: Realising this, player 1 will choose ANotice: If no player is ever indifferent betn twoactions, backward induction produ ea uni

6、que10经济学院Definition of Subgame（子博弈）A subgame in an extensive-form game has the following properties:It begins at a node of the tree corresponding to an information set reduced to a singleton (the setcontains only ot)(单结)Itpasses all parts of the tree following thestarting node.It never divides an in

7、formation set.11经济学院Matching PenniesExtensive Form12 (1,-1) (-1,1)经济学院(-1,1)(1,-1)Matching PenniesExtensive Form(-1,1) (1,-1) (1,-1) 经济学院(-1,1)13Exle of a Game is extensive formPlayer 1Player 2has two strategies has 3 strategiesL and RL, M and RLRLRLLRRL MRRL MMM(6,1)(2,7) (7,8)(1,2) (3,1) (0,4)(6,1

8、)(2,7) (7,8)(1,2) (3,1) (0,4)The ordered pairs (1,2) gives the payoff of player 1 and the payoffof the second player as 2.he lefnd side diagram, player 2doesnt know what Player 1 does.he Righnd sides diagram,P2 has acs to P1s deci.14经济学院P2P2P1P1Exle: 仿冒与反仿冒博弈15经济学院Subgame Perfect Equilibrium (SPNE)

9、(子博弈精炼均衡)A NE is a SPNE if the strategies of the players yield a NE in every subgame, whether these subgamesare reached wiitiveprobability at the equilibrium or not.16经济学院Further Understanding SPNE如果在一个完美信息的动态博弈中，各博弈方的策略构成的一个策略组合满足：在整个动态博弈中及它的所有子博弈中都均衡，那么，这么策略组合称为该动态博弈的一个SPNE。子博弈是原博弈的一个分支，它本身可以作为一个独

10、立的博弈来进行分析。SPNE实现的条件：1.策略组合(s1*,s2*, sn*)是原博弈的均衡；2.策略组合(s1*,s2*, sn*)在每一个子博弈上给出均衡。17经济学院Sequential Game: Extensive FormAA playsB plays secondBLUDBRLR(0,0)(2,1)(1,8)(3,9)18经济学院Sequential Game: Extensive FormAUDBBLRLR(0,0)(2,1)(1,8)(3,9)(U,L) and (D,R) are both Nash equilibria Which is more likely to o

11、ccur?经济学院19Sequential Game: Extensive FormAUDBBLRLR(0,0)(2,1)(1,8)(3,9)If A plays U then B plays L; A gets 3经济学院20Sequential Game: Extensive FormAUDBBLRLR(0,0)(2,1)(1,8)(3,9)If A plays U then B plays L; A gets 3 If A plays D then B plays R; A gets 2经济学院21Sequential Game: Extensive FormAUDBBLRLR(0,0)

12、(2,1)(1,8)(3,9)If A plays U then B plays L; A gets 3 If A plays D then B plays R; A gets 2ipates B, so (U,L) is likely Nash equilibriumA22经济学院ScenarioTwSuccereal(st, crispy) cerealssful only if each firm produoneSt will sell betterBoth still profitable with only one producer23经济学院Exle:Modified Produ

13、ct Choice ProblemFirm 2CrispyStCrispyFirm 1St24经济学院-5, -510, 2020, 10-5, -5Modified Product Choice ProblemQuestionWhat is the likelye if both make theirFirm 2CrispyStCrispydecisindependently, simultaneously, and without knowledge of theFirm 1Stothersentions?25经济学院-5, -510, 2020, 10-5, -5Modified Pro

14、duct Choice ProblemAmet Firm 1 willroduce its newcereal(a sequential game).QuestionWhat will be thee of this game?26经济学院The Extensive Form of a GameProduct Choice Game in Extensive FormThe Advantage of Movinghis product-choice game, there is a clearadvantage to moving Firm 1 decides produ.t cereals

15、at Crispy-5, -5CrispyFirm 2St10, 2020, 10Firm 1CrispyStFirm 2St-5, -527经济学院Threats（）, Commitments（承诺）,and Credibility（度）Strategic MovesWhtions can a firm take to gainadvantagehe marketplace?Deter entry（进入）Induce（引诱） competitors to reduce output, leave, raise priceImplicit agreementst benefit other f

16、irm28经济学院Threats, Commitments, and CredibilityHow To Make theMoveDemonstrate Commitment（宣布承诺）Firm 1 must constrain his behavior to theextent Firm 2 is convinced committedt he is29经济学院Threats, Commitments, and CredibilityEmpty（Incredibility） Threats（虚 (不可置信的承诺)If a firm will be worse off if it charge

17、s a low price, the threat of a low price is not crediblehe eyes of the competitors.）30经济学院Threats, Commitments, and CredibilityScenario（假定如下情况）Race Car Motors, Inc. (RCM) producarslty carFar Out Engines (FOE) produspeengines and sells most of them to RCMSequential game with RCM as the leaderFOE has

18、noer to threaten to build bigsince RCM controls output.31经济学院Production Choice ProblemRace Car MotorsSmall carsBig carsSmall enginesFar Out EnginesBig engines32经济学院3, 63, 01, 18, 3Threats, Commitments, and CredibilityQuestionHow could FOE force RCM to shift to big cars?33经济学院Modified Production Choi

19、ce ProblemRace Car MotorsSmall carsBig carsSmall enginesFar Out EnginesBig engines34经济学院0, 60, 01, 18, 3Modified Production Choice ProblemQuestions1)What is the risk of this strategy?2)How could irrational behaviiveFOE someer to control output?35经济学院Classification of Strategic MovesClassification of

20、 Strategic Moves:Unconditional Moves: MoveConditional Moves:(a) Threat（with you.）: punish others who fail to cooperate(1)Compellent Threat（强制性）:threat toinduomeone to action.(2)Deterrent Threat（威慑性someone from taking an action）: threat to prevent(b) Promises（承诺）: offer a reward who cooperativewith y

21、ou.Compellent vs. deterrent pre经.济学院36Application:金矿开采博弈策略：乙的最佳策略是不借， 甲的最佳策略是不分；其分的策略是不诺的的承37经济学院Application:金矿开采博弈甲乙双方策略：乙的完整策略是在第一阶段选择“借”，如第二阶段甲选择 “不分”，第三阶段选择“打”官司。甲的完整策略是第二阶段选择“分”38经济学院Application:金矿开采博弈乙在第三阶段选择“打”官司就不是一个“的的”，而是一个EmptyThreats。策略：乙的最优选择是第一阶段不借；甲的最优策略是第二阶段不分。39经济学院Chapter 11 includ

22、es:40经济学院11.4 Repeated Game（重复博弈）Repeated games（重复博弈）So far we have considered gamest are playely onceHowever, in real life the same games are played by the same players over and over againTwo kinds of repeated games:1.2.FiniInfiniy repeated (played a fixed number of times)y repeated (played an inde

23、finite number of times)41经济学院Exle 1: prisoners dilemma repeated twiceAt theperiod, players choose simultaneously betnconfess (defect) or not confess (cooperate)After observing whappenedheperiod, theyagain choose simultaneously betn cooperate and defect42经济学院One could expect can achieve bettert if th

24、e game is repeated players esPlayers might be able to build trust and punish others for defectionWill this be the case?To find the subgame perfect Nash equilibrium, we solve the game starting from43经济学院Prisoners DilemmaEach player has a dominant strategyEquilibriumt arises from using dominantstrateg

25、ies(占优策略) is worse for every playern theet would arise if everyplayer used her dominated strategy（劣势策略） insteadPrivate rationality collective irrationalityGoal:To sustaeutually benefil cooperative经c济e学n院tives to cheat44Duopoly CompetitionTwo firms: Firm 1 and Firm 2Two pri: low ($4) or high ($5 )45经

26、济学院Prisoners DilemmaEquilibrium:$24 KFirm 2LowHighLowHighFirm 1Cooperation:经济学院$30 K4624 , 2440 , 1010 , 4030 , 30RepeatederactionRepeated eractionOngoing relationship betCurrent action affects futuren playerseractionsHistory-Dependent StrategiesChoose an action today dependent on thehistory oferact

27、ionCan history-dependent strategies help enforce mutual cooperation?47经济学院Finite Repetitione the market relationship lasts for only TSupperiodsUse backward induction (逆向归纳法)Tthperiod: no incentive to cooperateNo future loss to worry about in last periodT-1th period: no incentive to cooperateNo coope

28、rationperiod in any casethNo opportunity cost to cheating in period T-1Unraveling: logic goes back to period 1经济学院48Finite RepetitionCooperation is imsible if the relationshipn players is for a fixed and knownbetlength of time.Why do people cooperate even though theyt live forever?49经济学院Infinite Rep

29、etition（无限重复博弈）No last period, so no rollbackUse history-dependent strategiesTrigger strategies（触发策略）Begin by cooperatingCooperate as long as the rivals doUpon observing a defection（背叛）:immediay revert toriod of punishment ofspecified length in which everyone plays non- cooperatively50经济学院Two Trigge

30、r Strategies（两种触发策略）Grim Trigger Strategy（冷酷触发策略）Cooperate until a rival deviatesOnce a deviation occurs, play non-cooperatively for the rest of the gameTit-for-Strategy（针锋相对策略）Cooperate if your rival cooperated recent periodhe mostCheat if your rival cheated period经济学院he most recent51Grim Trigger S

31、trategyIn any period t, a firm fa play:Zero deviations up toone of two histories oft po（不偏离合作点）he next periodt poCharge the high priceOne or more deviations up toCharge the low price from periodt poon in everySince low, low is the Nash equilibrium, eachfirm isng the best it can52经济学院Equilibrium in G

32、TS: Discounting（贴现）ionDefinitactiven a discount f: value of an ent , the presence of payoffs inite se,quinf1,2 , ,.i.s.34 t1t 1 23.1234tExle 1ence of pafinite sequn: The present value of an iyoffs 11, 1, . ( t 1, for all t) is 1 .1 53经济学院Equilibrium in GTS: Discounting（贴现）Discounting:present value o

33、f future profits is less of current profitsDiscounting rate isn value1 1 rNote: r iserest rate54经济学院Equilibrium in GTSFTS to be an equilibrium, the present value ofcolluding must be greater cheatingn the present value ofPV(collude)= 30 + 30 + 30 + =PV(cheat)230/1- = 40 + 24 + 24 2+ =40 + 24 /1- 55经济

34、学院Payoff Streamprofit40collude30cheat24tt+1t+2t+3time56经济学院Equilibrium in GTSEquilibrium if: PV(collude) PV(cheat)30/1- 40 + 24 /1- 5/8 or r60%Cooperation istainable using grim triggerstrategies as long as r 5/8Or as long as $1 invested today returns less$1.60 next periodAs long as firms value the f

35、uture enoughn57经济学院Sustainability（持续性）The minimum discount rate required to sustainthe collusive structuree depends on the payoff58经济学院Tit-for-（以牙还牙策略）is nicerTit-for-n GTSa) If rival uses GTS, cooperate if:Colluding is better 30、30、30 b) If rival uses tit-for-Colluding is better 30、30、30 n cheating、, cooperate if:n cheating once、59经济学院Axelrods Simulation弈模