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1、Dynamic Games of Complete InformationYongqin Wang,CCES, FudanFall, 20072Dynamic games of complete informationnPerfect informationA player knows Who has made What choices when she has an opportunity to make a choicenImperfect informationA player may not know exactly Who has made What choices when she
2、 has an opportunity to make a choice.3Subgame-perfect Nash equilibriumnA Nash equilibrium of a dynamic game is subgame-perfect if the strategies of the Nash equilibrium constitute or induce a Nash equilibrium in every subgame of the game.nA subgame of a game tree begins at a singleton information se
3、t (an information set containing a single node), and includes all the nodes and edges following the singleton information set, and does not cut any information set; that is, if a node of an information set belongs to this subgame then all the nodes of the information set also belong to the subgame.4
4、Find subgame perfect Nash equilibria: backward inductionnWhat is the subgame perfect Nash equilibrium?Player 1LRPlayer 2LR2, 2, 0Player 2LR3L”R”3L”R”3L”R”3L”R”1, 2, 33, 1, 22, 2, 12, 2, 10, 0, 11, 1, 21, 1, 15Find subgame perfect Nash equilibria: backward inductionnWhat is the subgame perfect Nash e
5、quilibrium?Player 1LRPlayer 2LR2, 2, 0Player 2LR3L”R”3L”R”3L”R”3L”R”1, 2, 33, 1, 22, 2, 12, 2, 10, 0, 11, 1, 21, 1, 16Bank runs (2.2.B of Gibbons)nTwo investors, 1 and 2, have each deposited D with a bank.nThe bank has invested these deposits in a long-term project. If the bank liquidates its invest
6、ment before the project matures, a total of 2r can be recovered, where D r D/2.nIf banks investment matures, the project will pay out a total of 2R, where RD.nTwo dates at which the investors can make withdrawals from the bank.7Bank runs: timing of the gamenThe timing of this game is as followsnDate
7、 1 (before the banks investment matures)Two investors play a simultaneous move gameIf both make withdrawals then each receives r and the game endsIf only one makes a withdrawal then she receives D, the other receives 2r-D, and the game endsIf neither makes a withdrawal then the project matures and t
8、he game continues to Date 2.nDate 2 (after the banks investment matures)Two investors play a simultaneous move gameIf both make withdrawals then each receives R and the game endsIf only one makes a withdrawal then she receives 2R-D, the other receives D, and the game endsIf neither makes a withdrawa
9、l then the bank returns R to each investor and the game ends.8Bank runs: game tree1WNW2WNW12WNWWNWWNW2a subgameOne subgame-perfect Nash equilibrium( NW W, NW W )Wr, rNWDate 1Date 2W: withdrawNW: not withdraw2D, 2rD2rD, DR, RD, 2RD2RD, DR, R9Bank runs: game tree1WNW2WNW12WNWWNWWNW2One subgame-perfect
10、 Nash equilibrium( W W, W W )Wr, rNWDate 1Date 2W: withdrawNW: not withdraw2D, 2rD2rD, DR, RD, 2RD2RD, DR, Ra subgame10Tariffs and imperfect international competition (2.2.C of Gibbons)(关税和不完美信息的国际竞争）nTwo identical countries, 1 and 2, simultaneously choose their tariff rates, denoted t1, t2, respect
11、ively.nFirm 1 from country 1 and firm 2 from country 2 produce a homogeneous product for both home consumption and export. nAfter observing the tariff rates chosen by the two countries, firm 1 and 2 simultaneously chooses quantities for home consumption and for export, denoted by (h1, e1) and (h2, e
12、2), respectively.nMarket price in two countries Pi(Qi)=aQi, for i=1, 2. nQ1=h1+e2, Q2=h2+e1.nBoth firms have a constant marginal cost c.nEach firm pays tariff on export to the other country.11Tariffs and imperfect international competition厂商1的利润厂商1的利润函数厂商2的利润函数12Tariffs and imperfect international c
13、ompetition国家1的福利国家2的福利消费者剩余企业利润税收消费者剩余企业利润税收13Backward induction: subgame between the two firms关税即定下的博弈14Backward induction: （逆向归纳法）subgame between the two firms库尔诺纳什均衡15Backward induction: whole game国家博弈：将厂商的纳什均衡战略代入到国家的目标函数子博弈精炼的要求。16Tariffs and imperfect international competition子博弈精炼纳什均衡的战略组合子博弈
14、精炼纳什均衡的结果17Repeated game（重复博弈）nA repeated game is a dynamic game of complete information in which a (simultaneous-move) game is played at least twice, and the previous plays are observed before the next play.（前一阶段的收益可被观察到） nWe will find out the behavior of the players in a repeated game.18Two-stage
15、repeated gamenTwo-stage prisoners dilemmaTwo players play the following simultaneous move game twiceThe outcome of the first play is observed before the second play beginsThe payoff for the entire game is simply the sum of the payoffs from the two stages. That is, the discount factor is 1.（折现因子为1）Pl
16、ayer 2L2R2Player 1L11 , 15 , 0R10 , 54 , 419Game tree of the two-stage prisoners dilemma1L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R21+11+11+51+01+01+51+41+411115+10+15+50+05+00+55+40+40+15+10+55+00+05+50+45+44+14+14+54+04+04+54+44+4两阶段的总收益20Informal game tree of the two-st
17、age prisoners dilemma1L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2115005441111(1, 1)(5, 0)(0, 5)(4, 4)115005441150054411500544第一阶段收益第二阶段收益21Informal game tree of the two-stage prisoners dilemma1L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R2L1R12L2R22L2R21150054411
18、11(2, 2)(6, 1)(1, 6)(5, 5)115005441150054411500544子博弈22two-stage prisoners dilemmanThe subgame-perfect Nash equilibrium(L1 L1L1L1L1, L2 L2L2L2L2) Player 1 plays L1 at stage 1, and plays L1 at stage 2 for any outcome of stage 1.Player 2 plays L2 at stage 1, and plays L2 at stage 2 for any outcome of
19、stage 1.Player 2L2R2Player 1L11 , 15 , 0R10 , 54 , 423Finitely repeated gamenA finitely repeated game is a dynamic game of complete information in which a (simultaneous-move) game is played a finite number of times, and the previous plays are observed before the next play. n（有限次重复博弈的定义）nThe finitely
20、 repeated game has a unique subgame perfect Nash equilibrium if the stage game (the simultaneous-move game) has a unique Nash equilibrium. The Nash equilibrium of the stage game is played in every stage. n（有限次重复博弈的纳什均衡）24What happens if the stage game has more than one Nash equilibrium?Two players p
21、lay the following simultaneous move game twiceThe outcome of the first play is observed before the second play beginsThe payoff for the entire game is simply the sum of the payoffs from the two stages. That is, the discount factor is 1.Question: can we find a subgame perfect Nash equilibrium in whic
22、h M1, M2 are played? Or, can the two players cooperate in a subgame perfect Nash equilibrium?Player 2L2M2R2Player 1L11 , 15 , 00 , 0M10 , 54 , 40 , 0R10 , 00 , 03 , 3阶段存在多个纳什均衡？25Informal game tree1L1R122L2R2M2L2R2M2L2R2M22L1R122L2R2M2L2R2M2L2R2M22M1(1, 1)(5, 0)(0, 5)(4, 4)(0, 0)M1(0, 0)(0, 0)(0, 0)
23、(3, 3)1(1, 1)(5, 0)(0, 5)(0, 0)(0, 0)(0, 0)(0, 0)(3, 3)(4, 4)26Informal game tree and backward induction1L1R122L2R2M2L2R2M2L2R2M22L1R122L2R2M2L2R2M2L2R2M22M1(1, 1)(5, 0)(0, 5)(4, 4)(0, 0)M1(0, 0)(0, 0)(0, 0)(3, 3)1(1, 1)(5, 0)(0, 5)(0, 0)(0, 0)(0, 0)(0, 0)(3, 3)(4, 4)(1, 1)(1, 1)(1, 1)(3, 3)(1, 1)(1
24、, 1)(1, 1)(1, 1)(1, 1)+27Two-stage repeated gamePlayer 2L2M2R2Player 1L11 , 15 , 00 , 0M10 , 54 , 40 , 0R10 , 00 , 03 , 3nSubgame perfect Nash equilibrium:player 1 plays M1 at stage 1, and at stage 2, plays R1 if the first stage outcome is (M1,M2 ), or plays L1 if the first stage outcome is not ( M1
25、,M2 ) player 2 plays M2 at stage 1, and at stage 2, plays R2 if the first stage outcome is ( M1, M2 ), or plays L2 if the first stage outcome is not ( M1,M2 ) 是否合作？28Two-stage repeated gamePlayer 2L2M2R2Player 1L12 , 26 , 11 , 1M11 , 67 , 71 , 1R11 , 11 , 14 , 4nSubgame perfect Nash equilibrium:At s
26、tage 1, player 1 plays M1, and player 2 plays M2.At stage 2, player 1 plays R1 if the first stage outcome is ( M1, M2 ), or plays L1 if the first stage outcome is not ( M1, M2 ) player 2 plays R2 if the first stage outcome is ( M1, M2 ), or plays L2 if the first stage outcome is not ( M1, M2 ) The p
27、ayoffs of the 2nd stage has been added to the first stage game.29An abstract game: generalization of the tariff game (general form)nFour players: 1, 2, 3, 4. The timing of the game is as follows. nStage1: Player 1 and 2 simultaneously choose actions a1 and a2 from feasible action sets A1 and A2, res
28、pectively.nStage 2:After observing the outcome (a1, a2) of the first stage, Player 3 and 4 simultaneously choose actions a3 and a4 from feasible action sets A3 and A4, respectively.nThe game ends. nThe payoffs are ui(a1, a2, a3, a4), for i=1, 2, 3, 430An abstract game: informal game treeplayer 1Play
29、er 1 action set A1Stage 1Stage 2player 2player 3Player 2 action set A2Player 3 action set A3Player 4 action set A4player 4a1a2a3a4A smallest subgame following (a1, a2)31Backward induction: solve the smallest subgame32Backward induction: back to the rootplayer 1Player 1 action set A1Stage 1Stage 2player 2Player 2 action set A2a1a233Backward induction: back to the root
