Problem_Set_4__Take_home_Sol

# Problem_Set_4__Take_home_Sol - Kaushik Basu Spring 2008...

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Kaushik Basu Spring 2008 Econ 367: Game-Theoretic Methods Problem Set 4 1. (a) Describe the mixed strategy Nash equilibrium (that is, one that actually involves mixing two strategies) in the game shown below. L R U 4, 4 0, 0 D 0, 0 2, 2 Answer: Suppose P1 plays U with probability p and P2 plays L with probability q. P1 will be indifferent between the strategies U and D when 4q = 2(1-q), i.e., when q = 1/3. Similarly, P2 will be indifferent between the strategies L and R when 4p = 2(1-p), i.e., when p = 1/3. Therefore, the mixed strategy Nash equilibrium (that is, one that actually involves mixing two strategies) in the game above is (p*, q*) = (1/3, 1/3). (b) In the game described below (as always, player 1 chooses between rows and 2 between columns) locate all the pure strategy Nash equilibria. Now allow players to mix their strategies. Is there a Nash equilibrium in which player 1 mixes the two strategies? Is there one in which player 2 mixes strategies? L R U 4, 2 0, 2 D 6, 0 0, 2 Answer: There are two pore strategy Nash equilibria: (U,R) and (D,R). Yes. Suppose P1 plays U with probability p and P2 plays L with probability q. Take (p,q) = (1/2, 0). When P2 is playing the pure strategy R (i.e., playing L with probability 0), P1 is indifferent between his strategies U and D and thus does not deviate from playing p = ½. On the other hand, when P1 plays U with p = ½, P2’s expected payoff from playing L is 1 which is less than 2 - his expected payoff

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## This note was uploaded on 09/30/2008 for the course ECON 3670 taught by Professor Basu during the Spring '08 term at Cornell.

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Problem_Set_4__Take_home_Sol - Kaushik Basu Spring 2008...

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