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Problem_Set_4__Solutions

# Problem_Set_4__Solutions - Kaushik Basu Spring 2008 Econ...

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Kaushik Basu Spring 2008 Econ 367: Game-Theoretic Methods Problem Set 4 [This is a take-home examination.—Total 10 points. Your answer must be turned in (hard copies, please) by 21 st February, 5 pm. You may hand it over to Vidya or Sarah after class on 21 st Feb. Do not discuss the questions with anybody till the answers are posted on the course website.] 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

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Problem_Set_4__Solutions - Kaushik Basu Spring 2008 Econ...

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