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1 E CONOMICS S-1050 H ARVARD U NIVERSITY S UMMER 2009 S UMMER S CHOOL P ROBLEM S ET 3 A NSWERS due July 23 , in class Show all calculations, explain all answers (up to 5 lines), and fully label graphs. Also: (a) staple sheets, (b) acknowledge assistance, (c) highlight final answers, and (d) write neatly! 1. Consider the following game: - 1.5 - L C R U 2, 2 6, 1 1, 1 M 1, 6 5, 5 1, 1 D 1, 1 1, 1 4, 4 a) If the game is played once, what are the pure strategy Nash Equilibria? NE = { (U,L), (D,R) } - 0.5 - b) If the game is played twice, and the results of the first stage is known before the second stage begins, how many strategies does each player have? In Stage-I, the first player has 1 node, with 3 actions. Though the second player appears to have 3 nodes, one each for each of the 3 actions of the first player, because the game is simultaneous within each stage (as shown by the normal form above), all the 3 nodes are part of an information set. Hence, the second player has only 1 node, as well. In Stage-II, first player has 9 possible nodes, one for each of the 3 × 3 outcomes of Stage-I, giving her a total of 1 + 9 = 10 nodes in the game. But for each node of the first player, the second player still has the information set (as explained above). Hence, the second player in this stage has 9 possible nodes, as well, for a total of 10 nodes in the game. (Note that the either the row or the column player can be the “first”player.) Each player has 3 actions (at each node) so each player has 3 10 strategies - 0.5 - .
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2 c) Can the payoff (5,5) be achieved in the first stage of a subgame perfect Nash Equilibria? If yes, find strategies that do so. If not, prove why not. Following Gibbons, pp. 84-7: Assume that in the first stage, the players anticipate that second stage outcome will be a NE of the stage game. Suppose that the players anticipate that (D,R) will be played in the second stage, if (M,C) was the outcome of the first stage, and that (U,L) will be played in the second stage after any of the 8 other outcomes in the first stage. Then the players’ first stage interaction is depicted in the matrix below, where (4,4) has been added to the (M,C) cell, and (2,2) has been added to the other 8 cells: L C R U 4, 4 8, 3 3, 3 M 3, 8 9, 9 3, 3 D 3, 3 3, 3 6, 6 The resulting game has 3 SPNE (in pure strategies): (U,L) in which the player 1 has played U in the first stage and U in the second, and player 2 has played L followed by L; (D,R) in which player 1 has played D followed by U and 2 has played R followed by L; and (M,C) in which player 1 has played M then D, and 2 has played C then R. Thus, The payoff (5,5) can be achieved in the first stage of a SPNE of the repeated game if: Player 1: Play M in stage I; in stage II, play D if the stage I outcome was (M,C); otherwise play U. Player 2:
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