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1、高级微观-博弈论Econ711February32014Peter NormanAssignment41. Consider a prisoners dilemma with stage game payors c dC2;20;3D3;01;1played twice.1.Sketch the extensive form.2. How many nodes are there in the extensive form?How many information sets?How many purestrategies?3. Write down the payomatrix for the
2、 reduced normal form.Assume that there is no discounting.4. Find all pure strategy Nash equilibria of the reduced normal form game.5. What are all Nash equilibria for the regular normal form?For each equilibrium,specify the equilibrium outcome and determine which equilibria are subgame perfect.6. Wh
3、ich strategies survive iterated elimination of strictly dominated strategies?Which strategiessurvive iterated elimination of weakly dominated strategies?7. Now consider an arbitrary.nite horizon T:Prove that no player can play C on the equilibrium path.8. Given an arbitrary.nite horizon T,write down
4、 a strategy pro.le that is a Nash equilibrium,but notsubgame perfect.2. Consider the normal form gamec d A d BC3;3 2;4 2;4D A4; 22;2 1; 1D B4; 2 1; 10;0l. Find all Nash equilibria(pure and mixed).2.Suppose that the stage game is played twice and that payors are not discounted.Construct a subgameperf
5、ect equilibrium in which(C;c)is played in the.rst period?3.Suppose that the stage game is repeated4times.Construct a subgame perfect equilibrium where(D A;c)is played in the.rst period?4.Consider the following outcome path(D A;d B);(D B;d A);(D A;d A);(D A;d A);:;(D A;d A):Con-struct a strategy pro.
6、le that supports this as a subgame perfect equilibrium in the T repeated game.What is the smallest value of T for this to work3. Consider an in.nite repetition of c dC2;20;3D3;01;1and assume payors are discounted by 2(0;1):1. For large enough,construct a subgame perfect equilibrium where(C;c)is play
7、ed in every periodontheequilibriumhttp:/ Nash reversion in your construction and.nd a critical value for .2. Change the stage game top c dP 1; 1 1; 1 1;0C 1; 12;20;3D0; 13;01;1Assuming strategies relying on Nash reversion,under which condition on can a subgame perfect equilibrium where(C;c)is played
8、 in every period on the equilibrium path be supported.3. Given a normal form game(N;S;u),let u i = min si max siu i(s i;s i)be the(pure strategy)minmaxvalue for player i:What are the minmax values in the two games above.4. Construct a strategy with the following properties:1)(C;c)is played in every
9、period on the outcomepath;2)if(D;c)is played when(C;c)should have been played(P;d)is played in the next period;if (C;d)is played when(C;c)would have been played(D;p)is played in the next period.For any any history where(P;d)or(D;p)has been played at least1plays D and2plays d:Write down a fully speci
10、.ed strategy for each player so that these properties hold and.nd a critical value for so that the strategy pro.le is subgame perfect.4. Consider a Cournot game with inverse demand p(y) = 12 y and constant marginal cost c=3:1.Find the static Nash equilibrium in the one shot Cournot game and.nd the“c
11、artel outcome”thatmaximizes industry pro.ts.2.Suppose the game is repeated T1times.Find all subgame perfect equilibria.3.Suppose that payors are discounted by and that the game is repeated in.nitely.Derive a conditionunder which it is possible to sustain industry pro.ts at the level of the cartel ou
12、tcome in every period as a subgame perfect equilibrium using Nash reversion strategies.4.Suppose that your condition on fails,so that the cartel outcome cannot be supported by Nashreversion.Can you sustain any other outcome than the stage game Nash if you restrict attention to Nash reversion strateg
13、ies.5. Let G be an arbitrary stage game and let G T be the T-fold repetition.A strategy pro.le is history independent if for every t the continuation strategies satisfy s j h t=s j hOfor every h t;h0t2H t:1. True/False?Any history independent strategy that is subgame perfect must induce stage game N
14、ashequilibrium play in every period.(provide a proof or counter example)2. True/False?Any history independent strategy pro.le that induces Nash equilibrium play in everyperiod is subgame perfect.(proof or counter example)3. True/False?Every strategy pro.le that induces stage game Nash equilibrium play in every subgameis subgame perfect.(proof or counter example).
