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Economics 703 Advanced Microeconomics Prof. Peter Cramton Problem Set 1: Suggested Answers 1. All the pure strategy equilibria are symmetric (i.e., each player plays the same strategy), since otherwise there exists a player that can do better by deviating. Namely, any player that is not the closest to the mean can do better by moving closer. There are a continuum of such equilibria: any s [0,1] will do. Let x = Pr(1 plays 0), y = Pr(2 plays 0), and z = Pr(3 plays 0). Player 1 will only randomize if she is indifferent between 0 and 1; hence, we look for randomizations y and z that make player 1 indifferent: s 1 = 0 yields 1/3 with prob. yz 1/2 with prob. y(1-z) + z(1-y) 0 with prob. (1-y)(1-z). s 1 = 1 yields 0 with prob. yz 1/2 with prob. y(1-z) + z(1-y) 1/3 with prob. (1-y)(1-z). Therefore, 1's expected payoffs are equal iff yz = (1-y)(1-z) iff y + z = 1. Similarly, 2 will randomize iff x + z = 1, and 3 will randomize iff x + y = 1. Hence, x = y = z = 1/2 is the unique mixed strategy equilibrium.

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