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03solution

# 03solution - Prof William H Sandholm Department of...

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–1– Prof. William H. Sandholm Department of Economics University of Wisconsin March 13, 2003 Midterm Solutions – Economics 713 1. Suppose to the contrary that τ 1 strictly dominates σ 1 . Then u 1 ( τ 1 . s 2 ) > u 1 ( σ 1 , s 2 ) for all s 2 S 2 . Therefore, since G is zero sum, u 2 ( τ 1 , s 2 ) < u 2 ( σ 1 , s 2 ) for all s 2 S 2 , and so min ρ 1 1 S ( max s S 2 2 u 2 ( ρ 1 , s 2 )) max s S 2 2 u 2 ( τ 1 , s 2 ) < max s S 2 2 u 2 ( σ 1 , s 2 ). That is, σ 1 does not minmax player 2. (Alternatively, one could prove this statement by appealing to the Minmax Theorem. The Minmax Theorem tells us that if σ 1 is a minmax strategy for player 1, then ( σ 1 , σ 2 ) is a Nash equilibrium, where σ 2 is some minmax strategy for player 2. Hence, σ 1 is a best response to σ 2 , and so cannot be a strictly dominated strategy.) 2. This is a bad idea. While a player will never choose a dominated strategy in a Nash equilibrium of a normal form game, it is quite possible for him to play a dominated stage game action in a SPE of a repeated game. The “grim trigger” equilibrium of the repeated prisoner’s dilemma provides one example. 3. (i) This condition says that the change in a player’s payoffs when he unilaterally deviates is always identical to the change in potential.

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