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Homework 6 Answer (Total 20 points) Q1 (3pt.): Find all the pure-strategy Nash equilibria of the game below. B A l m r T - 5, - 3 1, - 2 4, - 1 M - 2, 1 4, 0 2, 0 B - 4, 5 3, 6 3, 4 Answer : ( T, r ) and ( M, l ). Q2 : Consider the game of Battle of Sexes, B A Boxing Opera Boxing 7, 4 0, 0 Opera 0, 0 7, 4 Find all the Nash equilibria. There are two pure-strategy equilibria and one mixed- strategy equilibrium. (2pt. for each pure-strategy equilibrium, and 4pt. for the mixed-strategy equilibrium) Answer : Let p A be the probability that A goes to Baxing, let p B be the probability that B goes to Boxing. A’s expected utility is given ( p A , p B ) is EU A ( p A , p B ) = 7 p A p B + 4(1 - p A )(1 - p B ) = (11 p B - 4) p A + 4 - 4 p B Hence, A’s best response is BR A ( p B ) = { 0 } , when p B < 4 11 entire [0 , 1] , when p B = 4 11 { 1 } , when p B > 4 11 . B’s expected utility is given ( p A , p B ) is EU B ( p A , p B ) = 4 p A p B + 7(1 - p A )(1 - p B ) = (11 p A - 7) p B + 7 - 7 p A 1

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Hence, B’s best response is BR B ( p A ) = { 0 } , when p A < 7 11 entire [0 , 1] , when p A = 7 11 { 1 } , when p A > 7 11 .
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