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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 4, - 5 1, - 3 - 1, - 6 M 2, - 1 - 2, 0 0, 1 B 7, 6 0, 3 - 4, 8 Answer : ( T,m ) and ( M,r ). Q2 : Consider the game of Battle of Sexes, B A Boxing Opera Boxing 5, 3 0, 0 Opera 0, 0 3, 5 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 ) = 5 p A p B + 3(1 - p A )(1 - p B ) = (8 p B - 3) p A + 3 - 3 p B Hence, A’s best response is BR A ( p B ) = { 0 } , when p B < 3 8 entire [0 , 1] , when p B = 3 8 { 1 } , when p B > 3 8 . B’s expected utility is given ( p A ,p B ) is EU B ( p A ,p B ) = 3 p A p B + 5(1 - p A )(1 - p B ) = (8 p A - 5) p B + 5 - 5 p A 1

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