# Exam 2 - Nash Equilibrium 1 2 Opera 1 2 0 0 2 1Movie There...

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2, 1 0, 0 0, 0 1, 2 Opera Movie Opera Movie 2 1 (O,O) is the best response for both players, so it is a Nash Equ (O,M) is not the best response for both players, so it is not a Nash E (M, O) is not the best response for both players, so it is not a Nash E (M, M) is the best response for both players, so it is a Nash Equilib M L K J Z Y X 5, 2 0, 4 3, 3 5, 6 3, 1 3, 7 7, 5 4, 4 3, 5 7, 5 8, 3 5, 6 2 1 This is a strong equilibrium since players would not want to move anywhere else These are weak equilibriums since players would like to move to another equilibrium 2, 2 0, 3 3, 0 1, 1 Cooperate Defect Cooperate Defect 2 1 Strict Nash Equilibrium Nash Equilibriums 0, 0 3, 1 1, 3 2, 2 Hawk Dove Hawk Dove 2 1 Strict Nash Equilibriums 2, 2 0, 0 0, 0 1, 1 A B A B 2 1 There are no Nash Equilibriums! 1, -1 -1, 1 -1, 1 1, -1 Heads Tails Heads Tails 2 1 This is firm 1’s best response equation This is firm 2’s best response equation 1, -1 -1, 1 -1, 1 1, -1 Heads Tails Heads Tails 2 1 There is a mixed strategy Nash where each player plays H or T 50% of the time. 1 - q q 1, -1 -1, 1 -1, 1 1, -1 Heads Tails Heads Tails 2 1 p 1 - p B C F by a combination of C and B. Likewise, F is also dominated by a mixed strategy for player 2. So what is this mixed strateg B C F 2, 3 3, 2 3, 2 0, 5 5, 0 2, 3 2, 3 2, 3 0, 5 0, 5 3, 2 3, 2 2, 3 C B C B 2 1 Payoffs: Player 1: 0q + 3(1-q) 3q + 2(1-q) Player 2: 5p + 2(1-p) 2p + 3(1-p) Player 1 Plays: (0, ¼, ¾) Player 2 Plays: (0, ¼, ¾) S3 S2 S1 S1 6, - 2, - 1, - 2, - 5, - 4, - 7, - 2, - 3, - S3 S2 at the other player will also choose a strategy that limits his losses. Maxmin equilibrium (and also happens to be the Nash in this game) 3, 2 0, 4 6, 1 1, 3 X Y A B 2 1 Nash Equilibrium There are no strictly dominated strategies in this game. Iterated dominance doesn’t help; any belief that you may have about the other person is rational in this game. But just because it’s rational doesn’t mean the belief is correct! Nash Equilibrium: a strategy profile (strategy for each player) is a Nash equilibrium if and only if each player’s strategy is a best response to the strategies of the other players.

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• Spring '09
• KRISHNA,VIJAYXU,HAIQING

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