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1、Chapter 11Game Theory(2)博弈論1經濟學院Chapter 11 includes:2經濟學院11.1roduction to Game Theory11.2 Nash Equilibrium(均衡)11.3 Subgame Perfect Nash Equilibrium（子博弈精煉均衡）11.4 Repeated Game（重復博弈）Overview of Last Class3經濟學院Game Theory:roductionElements of a gameClassifications of GameDominant strategy（占優(yōu)策略）Dominant

2、 strategy equilibrium（占優(yōu)策略均衡）Nash equilibrium（均衡）Mixed StrategiesOutlines of Todays ClassDefinition of SubgamePerfect Equilibrium (SPE)(子博弈精煉均衡)Repeated GamesFini弈)Infiniy repeated games(有限重復博y repeated games（無限重復博弈）4經濟學院Chapter 11 includes:5經濟學院11.3 Subgame Perfect Nash Equilibrium（子博弈精煉均衡）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經濟學院Exle: 仿冒與反仿冒博弈7經濟學院A企業(yè)是仿冒企業(yè)，B企業(yè)是被仿冒企業(yè)。如果被仿冒企業(yè)采取措施制止，仿冒企業(yè)就會停止仿冒，如果被仿冒企業(yè)不采取措施制止，那么仿冒企業(yè)就會繼續(xù)仿冒下去。假設仿冒最多進行兩次The extensive form (game tree)1. Decinodes(決策結): corresponding to playersBranches（決策枝）: corresponding to actionsTerminal nodes（終點結）: corresponding to payoff2-1,-1AA3,00

5、,3NA1N2,2N28經濟學院The Solution to Sequential-move GameBackward induction(逆向歸納法)2A-1,-1ANA3,00,31NN2,229經濟學院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經濟學院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)(單結)Itpasses all parts of the tree following thestarting node.It never divides an in

7、formation set.11經濟學院Matching PenniesExtensive Form12 (1,-1) (-1,1)經濟學院(-1,1)(1,-1)Matching PenniesExtensive Form(-1,1) (1,-1) (1,-1) 經濟學院(-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經濟學院P2P2P1P1Exle: 仿冒與反仿冒博弈15經濟學院Subgame Perfect Equilibrium (SPNE)

9、(子博弈精煉均衡)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經濟學院Further Understanding SPNE如果在一個完美信息的動態(tài)博弈中，各博弈方的策略構成的一個策略組合滿足：在整個動態(tài)博弈中及它的所有子博弈中都均衡，那么，這么策略組合稱為該動態(tài)博弈的一個SPNE。子博弈是原博弈的一個分支，它本身可以作為一個獨

10、立的博弈來進行分析。SPNE實現(xiàn)的條件：1.策略組合(s1*,s2*, sn*)是原博弈的均衡；2.策略組合(s1*,s2*, sn*)在每一個子博弈上給出均衡。17經濟學院Sequential Game: Extensive FormAA playsB plays secondBLUDBRLR(0,0)(2,1)(1,8)(3,9)18經濟學院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?經濟學院19Sequential Game: Extensive FormAUDBBLRLR(0,0)(2,1)(1,8)(3,9)If A plays U then B plays L; A gets 3經濟學院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經濟學院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經濟學院ScenarioTwSuccereal(st, crispy) cerealssful only if each firm produoneSt will sell betterBoth still profitable with only one producer23經濟學院Exle:Modified Produ

13、ct Choice ProblemFirm 2CrispyStCrispyFirm 1St24經濟學院-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經濟學院-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經濟學院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經濟學院Threats（）, Commitments（承諾）,and Credibility（度）Strategic MovesWhtions can a firm take to gainadvantagehe marketplace?Deter entry（進入）Induce（引誘） competitors to reduce output, leave, raise priceImplicit agreementst benefit other f

16、irm28經濟學院Threats, Commitments, and CredibilityHow To Make theMoveDemonstrate Commitment（宣布承諾）Firm 1 must constrain his behavior to theextent Firm 2 is convinced committedt he is29經濟學院Threats, Commitments, and CredibilityEmpty（Incredibility） Threats（虛 (不可置信的承諾)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經濟學院Threats, Commitments, and CredibilityScenario（假定如下情況）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經濟學院Production Choice ProblemRace Car MotorsSmall carsBig carsSmall enginesFar Out EnginesBig engines32經濟學院3, 63, 01, 18, 3Threats, Commitments, and CredibilityQuestionHow could FOE force RCM to shift to big cars?33經濟學院Modified Production Choi

19、ce ProblemRace Car MotorsSmall carsBig carsSmall enginesFar Out EnginesBig engines34經濟學院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經濟學院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（強制性）:threat toinduomeone to action.(2)Deterrent Threat（威懾性someone from taking an action）: threat to prevent(b) Promises（承諾）: offer a reward who cooperativewith y

21、ou.Compellent vs. deterrent pre經.濟學院36Application:金礦開采博弈策略：乙的最佳策略是不借， 甲的最佳策略是不分；其分的策略是不諾的的承37經濟學院Application:金礦開采博弈甲乙雙方策略：乙的完整策略是在第一階段選擇“借”，如第二階段甲選擇 “不分”，第三階段選擇“打”官司。甲的完整策略是第二階段選擇“分”38經濟學院Application:金礦開采博弈乙在第三階段選擇“打”官司就不是一個“的的”，而是一個EmptyThreats。策略：乙的最優(yōu)選擇是第一階段不借；甲的最優(yōu)策略是第二階段不分。39經濟學院Chapter 11 includ

22、es:40經濟學院11.4 Repeated Game（重復博弈）Repeated games（重復博弈）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經濟學院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經濟學院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經濟學院Prisoners DilemmaEach player has a dominant strategyEquilibriumt arises from using dominantstrateg

25、ies(占優(yōu)策略) is worse for every playern theet would arise if everyplayer used her dominated strategy（劣勢策略） insteadPrivate rationality collective irrationalityGoal:To sustaeutually benefil cooperative經c濟e學n院tives to cheat44Duopoly CompetitionTwo firms: Firm 1 and Firm 2Two pri: low ($4) or high ($5 )45經

26、濟學院Prisoners DilemmaEquilibrium:$24 KFirm 2LowHighLowHighFirm 1Cooperation:經濟學院$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經濟學院Finite Repetitione the market relationship lasts for only TSupperiodsUse backward induction (逆向歸納法)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經濟學院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經濟學院Infinite Rep

29、etition（無限重復博弈）No last period, so no rollbackUse history-dependent strategiesTrigger strategies（觸發(fā)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經濟學院Two Trigge

30、r Strategies（兩種觸發(fā)策略）Grim Trigger Strategy（冷酷觸發(fā)策略）Cooperate until a rival deviatesOnce a deviation occurs, play non-cooperatively for the rest of the gameTit-for-Strategy（針鋒相對策略）Cooperate if your rival cooperated recent periodhe mostCheat if your rival cheated period經濟學院he most recent51Grim Trigger S

31、trategyIn any period t, a firm fa play:Zero deviations up toone of two histories oft po（不偏離合作點）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經濟學院Equilibrium in G

32、TS: Discounting（貼現(xiàn)）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經濟學院Equilibrium in GTS: Discounting（貼現(xiàn)）Discounting:present value o

33、f future profits is less of current profitsDiscounting rate isn value1 1 rNote: r iserest rate54經濟學院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經濟

34、學院Payoff Streamprofit40collude30cheat24tt+1t+2t+3time56經濟學院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經濟學院Sustainability（持續(xù)性）The minimum discount rate required to sustainthe collusive structuree depends on the payoff58經濟學院Tit-for-（以牙還牙策略）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經濟學院Axelrods Simulation弈模