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M210_sol - 14.12 Game Theory Midterm II Solution Prof...

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14.12 Game Theory °Midterm II Solution 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 play the in°nitely repeated game with the following stage game: In every payo/ vector in the °gure, the °rst entry is for Alice, the second entry is for 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) , and all the previous moves are observable. For each strategy pro°le below °nd the range of ° under which the strategy pro°le is a subgame-perfect equilibrium. (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: We will show that the range for which the proposed strategy pro°le 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 ; which cannot be satis°ed for any ° 2 (0 ; 1) . Therefore, Alice has an incetive to deviate and take a day o/ if anyone shirked in the past.
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