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midterm2 2008 solutions

midterm2 2008 solutions - 14.12 Game Theory Midterm II Prof...

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14.12 Game Theory Midterm II 11/15/2007 Prof. Muhamet Yildiz Instructions. This is an open book exam; you can use any written material. You have one hour and 20 minutes. Each question is 25 points. Good luck! 1. Compute all the subgame-perfect equilibria in pure strategies for the following game: 1 2 1 2 L M R l r a b a b 0 2 0 1 2 0 0 -1 x y x y 2 1 1 0 -1 10 1 0 ANSWER: First, notice that the game has two subgames: a proper subgame originated when player 1 plays L, and the whole game itself. Since we are looking for SPEs, we need to make sure that the equilibria we find are NE of every subgame. So then, lets restrict our search to NE of the proper subgame first. This game can be represented by l r x (2,-1) (1,1) y (1,10) (0,0) where the underlined values are best responses. Then, we can see that the only NE of this subgame is ( x , r ): 1
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Figure 1: Having solved for the NE of this subgame, the game reduces to the second figure above. This game is represented by a b L (1,1) (1,1) M (0,2) (0,1) R (2,0) (0,-1) where again the BR are underlined. We can see that this game has 2 NE, which are the SPE we are looking for. They can be written as ( Lx , br ),and ( Rx , ar ). 2 Consider the infinitely repeated game with the following stage game: Chicken Lion Chicken Lion 3,3 1,4 4,1 0,0 All the previous actions are observed, and each player maximizes the discounted sum of his stage payoff with discount factor δ =0 . 99 . For each strategy profile below check if it is a subgame-perfect equilibrium. (You need to state your arguments clearly; you will not get any points for Yes or No answers.) (a) (10 points) There are two modes: Cooperation and Fight. The game starts in the Cooperation mode. In Cooperation mode, each player plays Chicken. If both players play Chicken, then they remain in the Cooperation mode; otherwise they go to the Fight mode in the next period. In the Fight mode, both play Lion, and they go back to the Cooperation mode in the following period (regardless of the actions). 2
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ANSWER: It is not SPE because in the fight mode each player has an incentive to deviate. In the fight mode, according to the strategy profile both players play Lion and get 0 and then go back to the cooperation mode where they both get 3 forever. This yields present value of 0+3 δ/ (1 δ ) to each player. If a player deviates in the fight mode and plays chicken, his payoff is 1 in period t and then they will still go back to the cooperation in period t+1 and obtain 3 forever. This yields a higher present value of 1+3 / (1 δ ) . b) (15 points) There are three modes: Cooperation, P1 and P2. The game starts in the Cooperation mode. In the Cooperation mode, each player plays Chicken. If they play (Chicken, Chicken) or (Lion, Lion), then they remain in the Cooperation mode in the next period. If player i plays Lion while the other player plays Chicken, then in the next period they go to P i mode. In P i mode player i plays Chicken while the other player plays Lion; they then go back to Cooperation mode (regardless of the actions).
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