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06solution

# 06solution - Prof William H Sandholm Department of...

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Prof. William H. Sandholm Department of Economics University of Wisconsin March 9, 2006 Midterm Exam Solutions – Economics 713 1. Statement (A) is always true: u 1 ( s 1 , ˆ s 2 ) ± min s 2 ² S 2 u 1 ( s 1 , s 2 ) for all s 1 ± S 1 and ˆ s 2 ± S 2 . ² max s 1 ± S 1 u 1 ( s 1 , ˆ s 2 ) ² max s 1 ± S 1 min s 2 ± S 2 u 1 ( s 1 , s 2 ) for all ˆ s 2 ± S 2 . ² min ˆ s 2 ± S 2 max s 1 ± S 1 u 1 ( s 1 , ˆ s 2 ) ± min ˆ s 2 ± S 2 max s 1 ± S 1 min s 2 ± S 2 u 1 ( s 1 , s 2 ) = max s 1 ± S 1 min s 2 ± S 2 u 1 ( s 1 , s 2 ) . Statement (B) is false in Matching Pennies: there, the minmax payoff is 1, which strictly exceeds the maxmin payoff of –1. 2. We first describe each player’s incentives: Player 1: B is strictly dominated. T is weakly preferred to M iff ± 2 ( R ) ² 1 2 . Player 2: L is weakly preferred to R iff 1 ± 2(1 – μ ( w ) ± 3 ( l )) iff μ ( w ) ± 3 ( l ) ± 1 2 . Player 3: l is weakly preferred to R iff μ ( y ) ² 1 3 . We now find all perfect Bayesian equilibria.

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