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Unformatted text preview: 6.254 Game Theory with Engineering Applications Midterm April 8, 2008 Problem 1 : (35 points) Consider a game with two players, where the pure strategy of each player is given by x i [0 , 1]. Assume that the payoff function u i of player i is given by u i ( x 1 ,x 2 ) = a i x i + I{ x i < x j } , where 0 < a i 1 and I is the indicator function given by I{ x i < x j } = ( 1 if x i < x j 0 otherwise . (a) Does this game have a pure strategy Nash equilibrium? Verify your answer. (b) Show that there cannot be an atom at any x (0 , 1] in either players mixed strategy at an equilibrium. (c) Find all mixed strategy Nash equilibria. Problem 2 : (30 points) Consider the following oligopoly competition model: There are two firms. Each firm i simultaneously chooses price p i [0 , 1]. Assume that the demand function for firm i , given by d i ( p 1 ,p 2 ), is twice differentiable. Each firm is interested in maximizing his profits (assume for simplicity that there is no cost)....
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 Spring '10
 AsuOzdaglar

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