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Unformatted text preview: Solutions to Problem Set 3 Problem 1 First note that there is no Nash equilibrium where one of the players plays a pure strategy and the other plays a completely mixed strategy, because the best response to any pure strategy is a pure strategy. Now, for player two to be willing to play a mixed strategy, player one must be randomizing so that the second player is indifferent between playing action P and C . If player one plays the mixed action ( P 1 , 1- P 1 ), then player 2s payoffs from playing actions P and C are as follows: P 1 u 2 ( P,P ) + (1- P 1 ) u 2 ( C,P ) = 5 P 1 P 1 u 2 ( P,C ) + (1- P 1 ) u 2 ( C,C ) = 2(1- P 1 ) respectively. Being indifferent implies: 5 P 1 = 2(1- P 1 ) P 1 = 2 7 Similarly, for player 1 to be willing to play a mixed strategy, player 2 must play P 2 = 2 7 . So if players 1 and 2 play ( P 1 , C 1 ) = ( 2 7 , 5 7 ) and ( P 2 , C 2 ) = ( 2 7 , 5 7 ), they would both be indifferent between playing P and C. This means that each players mixed strategy is a best response to the other players mixed strategy, so this is a Nash equilibrium. Problem 2 (a) Neither player has a strictly dominant strategy. For player 1, U is a best response to L, while U and D are both best responses to R. Since there is no single strategy which is a strict best response to every strategy of player 2, player 1 does not have a strictly dominant strategy. For player 2, L is a best response to U, while L and R are both best responses to D. Since there is no single strategy which is a strict best response to every strategy of player 1, player 2 does not have a strictly dominant strategy....
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This note was uploaded on 01/03/2012 for the course ECON 310-2 taught by Professor Sarver during the Spring '08 term at Northwestern.
- Spring '08