M210_sol

M210_sol - 14.12 Game Theory Midterm II Solution 11/18/2010...

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11/18/2010 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. Find all subgame-perfect equilibria of the following game in which Player 2 plays a pure strategy: Answer: There are two subgames in this game: a proper subgame starting from the node in which Player 3 makes a move, and the whole game itself. The normal form representation of the proper subgame is given by 1 n 3 L R a 1,0 0,1 b 0,1 1,0 : This game has the unique Nash equilibrium 1 2 a + 1 2 b; 1 2 L + 1 2 R ± . Therefore, we can replace this subgame with the payo± vector 1 4 0 @ 1 1 0 1 A + 1 4 0 @ 0 2 1 1 A + 1 4 0 @ 0 1 1 1 A + 1 4 0 @ 1 2 0 1 A = 0 @ 1 2 3 2 1 2 1 A : The normal form representation of the new reduced form game is given by 1 n 2 x y A 1 2 ; 3 2 1 2 ; 3 2 B 2 ,1 0,0 C 1,1 0,0 : Clearly this game has two Nash equilibria in which Player 2 plays a pure strategy, namely, ( B; x ) and ( A; y ) . In conlusion, there are two subgame-perfect equilibria 1
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of the game in which Player 2 plays a pure strategy: 1 2 Ba + 1 2 Bb; x; 1 2 L + 1 2 R ± and 1 2 Aa + 1 2 Ab; y; 1 2 L + 1 2 R ± . Note that we cannot eliminate strategy y of Player 2 be- cause it is not strictly dominated (strategies x and y give the same payo/ to Player 2 if Player 1 plays A ). (Those who found only one NE got 15 points out of 25 points.) 2. Alice is a small business owner. She has two workers, named Bob and Colin. The three Bob, and the third entry is for Caroline. Each player±s payo/ in the repeated game is the discounted some of his or her payo/s in the stage game with discount factor 2 (0 ; 1) (The range may be empty.) Show your work. (a) (10 Points) Alice Hires and Bob and Colin both Work until any of the workers Shirk; Alice Hires and Bob and Colin both Shirk thereafter. Answer: is a SPE is empty. There are two states which we call "Good" and "Bad". In "Good" state, Alice hires, Bob and Colin work. In "Bad" state, Alice hires, Bob and Colin shirk. We never transit away from "Bad" state. Using single-deviation principle, Alice does not have an incentive to deviate (Day o/ instead of Hire) in "Bad" state if 2 2 1 X t =1 t ± 1 2 1 X t =1 t ;
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This note was uploaded on 10/31/2011 for the course 18 18.445 taught by Professor Liewang during the Spring '11 term at MIT.

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M210_sol - 14.12 Game Theory Midterm II Solution 11/18/2010...

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