EconS491_Recitation5_Spring2011

EconS491_Recitation5_Spring2011 - #5 , 1

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1 EconS 491 – Review Session #5 Backward induction and subgame perfection Exercise 1 ­ Chater 15 ­ Watson Solve the game by using backward induction. Starting from the terminal nodes, the smallest proper subgame we can identify is depicted below:
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2 In this subgame player 3 chooses his action without observing player 2’s choice. In order to find the NE of this subgame we must represent it in its normal (matrix) form: Hence, the NE of this subgame predicts that players 2 and 3 choose strategy profile (C, X). We can now plug the payoff triple resulting from the NE of this subgame, (4,3,1), at the end of the branch indicating that player 1 chose action I, as follows. O 1 I 2, 2, 2 4 , 3, 1 From the subgame (C, X) Then the SPNE is (I, C, X)
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3 2. Exercise 5 ­ Chapter 15 ­ Watson a. Draw this game’s extensive form tree for k=5 b. Use backward induction to find the subgame perfect equilibrium Working backward, it is easy to see that in round 5, player 1 will choose S ሺ3 ൐ 0ሻ. Thus, in round 4 player 2 will choose S ሺ4 ൐ 3 ܽ݊݀ 4 ൐ 0ሻ. Continuing in this fashion, we find that, in equlibrium, each player will choose S any time he is on the move. c. Describe the backward induction outcome of this game for any finite integer k For any finite k, the backward induction outcome is that player 1 chooses S in the first round and each player receives on dollar. Because if neither player choose to step by the end of the kth round, then both players obtain zero ሺ1 ൐ 0ሻ.
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4 Exercise 8 ­ Chapter 15 ­ Watson a. Describe the strategy spaces of the players ܵ ൌ ሼܣ, ܤሽ ൈ ሺ0, ∞ሻൈሺ0 ,∞ሻ Each player selects A or B, picks a positive number when (A, B) is chosen, and picks a positive number when (B, A) is chosen. 5 5 u 1 u 2 -1 -1 P 1 and P 2 Simultaneously choosing A or B AA AB BA BB + + + + + + 2 1 2 1 2 1 2 1 1 1 x x x x x x x x + + + + + + 2 1 2 1 2 1 2 1 1 1 x x x x x x x x b. Compute the NE of this game It is easy to see that when one of the players chooses A and the other selescts B, then 0൏ ݔ ൅ݔ 1൅ݔ ൏1 And that ݔ ՜ 1 ܽݏ ሺݔ ሻ ՜ ∞ Thus, each has a higher payoff when both choose A. Further, B ሺെ1, െ1ሻ will never be selected in
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EconS491_Recitation5_Spring2011 - #5 , 1

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