05solution - Prof. William H. Sandholm Department of...

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–1– Prof. William H. Sandholm Department of Economics University of Wisconsin March 10, 2005 Midterm Solutions – Economics 713 1. Let v i be player i ’s minmax value: v i = min ! " i # $ A " i max a i # A i u i ( a i , a " i ) . Let F = convex hull({ u ( a ): a A }) and IR = { v : v i > v i for all i }. The Classic Folk Theorem says that if v F IR , then for all δ close enough to one, there is a Nash equilibrium of G ! ( " ) with payoffs v . ( ii ) Let u ( ˆ a ) = v . Suppose each player follows this grim trigger strategy: “Play ˆ a i so long as there has never been a period in which exactly one player has deviated; if player j was the first player to unilaterally deviate, then do your part in minmaxing j forever.” Then on the equilibrium path, player i cannot benefit from deviating if v i > (1 ! )max a # A u i ( a ) + v i . Since v i > v i , this inequality holds so long as is close enough to 1. 2. ( i ) v K if and only if v = ( u 1 ( T , σ 2 ), u 1 ( B , 2 )) for some 2 Δ S 2 . ( ii ) c * = 2. This is player 1’s minmax value: by playing 1 2 b + 1 2 c , the strategy that generates (2, 2), player 2 ensures that player 1 cannot obtain a payoff higher than 2. If you draw a picture of K , you will see that player 2 cannot restrict 1 to a lower payoff. ( iii ) p * = ( 2 3 , 1 3 ) and d * = 2. ( iv ) Let 1 * = p *. Then ( p * · v 2 for all v K ) is equivalent to ( u 1 *( 1 *, 2 ) 2 for all 2 S 2 ). Thus, player 1’s maxmin payoff is at least 2; by part ( ii ), it must be exactly 2. ( v ). Define n -strategy analogues of the sets J , K , and L ( c ). Let c * be the largest value of c such that int( L ( c )) and int( K ) do not intersect. (Notice that if, for example, player 1 has a dominated strategy, L ( c *) K may not include ( c *, … , c *); this is why we need the set L ( c *).) If player 2 chooses a 2 that generates a point in
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05solution - Prof. William H. Sandholm Department of...

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