Player 2 a b c a 1 0 0 1 2 0 player 1 b 0 1 1 0 1 0 c

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Player 2 A B C A (1 , 0) (0 , 1) (2 , 0) Player 1 B (0 , 1) (1 , 0) (1 , 0) C (1 / 2 , 0) (0 , 1) (2 , 2) a. Show that this game has a pure-strategy Nash equilibrium. Solution. The outcome (2,2) is a pure-strategy Nash equilibrium. 1
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b. What are the weakly dominated strategies? Solution. For player 1, strategy C is weakly dominated by strategy A. c. Iteratively remove the weakly dominated strategies. What is the resulting game? What are its pure-strategy Nash equilibria? Solution. After removing the last row, it easily follows that the reduced game has no pure-strategy Nash equilibria. Problem 4. (20 points) There are two political candidates competing in a selection. Each candidate i can spend s i 0 on advertisement which increases the probability of winning the election. Given a pair ( s 1 , s 2 ), the probability of candidate i winning is s i s 1 + s 2 . If no one spends, each candidate wins with probability 1 2 . The utility of winning is v for both candidates and the cost of spending s i is s i . a. Given the investment pair ( s 1 , s 2 ), what is the expected payoff of candidate 1? Solution. v i ( s 1 , s 2 ) = s i s 1 + s 2 v - s i given that s i 0 , s j > 0. The payoff is v 2 when s i = s j = 0. b. Find each player’s best response function. Solution. Consider player 1. Player 1 wants to maximize v i ( s 1 , s 2 ) and the first order optimality condition gives v ( s 1 + s 2 ) - s 1 v ( s 1 + s 2 ) 2 - 1 = 0 . Denote the best response correspondence by x . Then x satisfies x 2 + 2 xs 2 + s 2 2 - vs 2 = 0. c. Find the unique Nash equilibrium. Hint: Use the symmetry. Solution. Since the best response correspondences are symmetric for both players, s 1 = s 2 = s must be the unique Nash equilibrium for some s . Using the quadratic equation in part b, we have s 2 + 2 s 2 + s 2 - vs = 0 which gives s = v 4 . d. Now assume that the utility of winning is still v for candidate 1 and kv for candidate 2 where k > 1. What happens to the Nash equilibrium? Solution. Now there is no symmetry because players have different utilities. Denote the Nash equilibrium now by ( s 1 , s 2 ). Using part b, the best response correspondence for player 1 is s 2 1 + 2 s 1 s 2 + s 2 2 - vs 2 = 0 . Similarly, the best response correspondence for player 2 is s 2 2 + 2 s 1 s 2 + s 2 1 - kvs 1 = 0 .
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