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Unformatted text preview: Economics 101: practice problems 1 solutions 1. (NicholsonSnyder, 8.1) Consider the following game: D E F A 7 , 6 5 , 8 , B 5 , 8 7 , 6 1 , 1 C , 1 , 1 4 , 4 a. Find the purestrategy Nash equilibria (if any). Solution: we compute purestrategy Nash equilibria in the usual way, underlining conditional best responses. D E F A 7 , 6 5 , 8 , B 5 , 8 7 , 6 1 , 1 C , 1 , 1 4 , 4 The only sell with cooccurring underlines is ( C,F ); this is the unique purestrategy Nash equi librium. b. Find the mixedstrategy Nash equilibrium in which each player randomizes over just the first two actions. Solution: let σ j i be the probability that player i plays strategy j . We use indifference conditions to find E [ u 1 ( A,σ 2 )] = E [ u 1 ( B,σ 2 )] σ D 2 u 1 ( A,D ) + σ E 2 u 1 ( A,E ) = σ D 2 u 1 ( B,D ) + σ E 2 u 1 ( B,E ) 7 σ D 2 + 5 σ E 2 = 5 σ D 2 + 7 σ E 2 2 σ D 2 = 2 σ E 2 1 σ E 2 = σ E 2 = ⇒ σ E 2 = 1 2 E [ u 2 ( D,σ 1 )] = E [ u 2 ( E,σ 1 )] σ A 1 u 2 ( D,A ) + σ B 1 u 2 ( D,B ) = σ A 1 u 2 ( E,A ) + σ B 1 u 2 ( E,B ) 6 σ A 1 + 8 σ B 1 = 8 σ A 1 + 6 σ B 1 2 σ B 1 = 2 σ A 1 1 σ A 1 = σ A 1 = ⇒ σ A 1 = 1 2 Then the unique mixedstrategy Nash equilibrium is [( 1 2 , 1 2 ) , ( 1 2 , 1 2 )]. c. Compute players’ expected payoffs in the equilibria found in parts (a) and (b). Solution: the expected payoffs from the purestrategy Nash equilibrium ( C,F ) are (4 , 4), as presented in the payoff matrix. We compute the expected payoffs from the mixedstrategy Nash January 14, 2011 1 Economics 101: practice problems 1 solutions equilibrium as E [ u 1 ( σ * 1 ,σ * 2 )] = σ A 1 σ D 2 u 1 ( A,D ) + σ A 1 σ E 2 u 1 ( A,E ) + σ B 1 σ D 2 u 1 ( B,D ) + σ B 1 σ E 2 u 1 ( B,E ) = 1 4 (7) + 1 4 (5) + 1 4 (5) + 1 4 (7) = 6 E [ u 2 ( σ * 2 ,σ * 1 )] = σ D 2 σ A 1 u 2 ( D,A ) + σ D 2 σ B 1 u 2 ( D,B ) + σ E 2 σ A 1 u 2 ( E,A ) + σ E 2 σ B 1 u 2 ( E,B ) = 1 4 (6) + 1 4 (8) + 1 4 (8) + 1 4 (6) = 7 Then player 1’s expected payoff is 6 and player 2’s expected payoff is 7. Notably, they are both better off in this mixedstrategy Nash equilibrium than in the purestrategy Nash equilibrium. 2. (NicholsonSnyder, 8.2) The mixedstrategy Nash equilibrium in the Battle of the Sexes may depend on the numerical values for the payoffs. To generalize this solution, assume that the payoff matrix for the game is given by B allet B oxing B allet K, 1 , B oxing , 1 ,K where K ≥ 1. Show how the mixedstrategy Nash equilibrium depends on the value of K . Solution: we apply indifference conditions to solve for mixedstrategy Nash equilibrium. To simplify notation, let L represent Ba L let, and X represent Bo X ing. E [ u 1 ( L,σ 2 )] = E [ u 1 ( X,σ 2 )] σ L 2 u 1 ( L,L ) + σ X 2 u 1 ( L,X ) = σ L 2 u 1 ( X,L ) + σ X 2 u 1 ( X,X ) σ L 2 ( K ) = σ X 2 Kσ L 2 = 1 σ L 2 = ⇒ σ L 2 = 1 K + 1 Appealing to the nearly symmetric structure of player 2’s payoffs, we see that player 1’s mixture would have σ X 1 = 1 K +1 . Then the mixedstrategy Nash equilibrium of this game, as a function of....
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 Winter '08
 Buddin
 Economics, Game Theory, Nash

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