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Unformatted text preview: 6.254 Game Theory with Engineering Applications Midterm April 11, 2006 Problem 1 : (35 points) Consider a Bertrand competition between two firms, where each firm chooses a price p i [0 , 1]. Assume that one unit of demand is to be split between the two firms. The firm that charges the lower price gets all the demand. If the two firms charge the same price, the demand is split equally between the firms. (a) Find all pure strategy Nash equilibria. (b) Assume that firm 2 has a capacity constraint of 3 / 4. Does there exist a pure strategy Nash equilibrium? Verify your answer. (c) Assume that firm 2 has a capacity constraint of 3 / 4. Find a mixed strategy Nash equilibrium. Hint: First assume that player 2 has no atom in its mixed strategy. Write the profit of player 1 as a function of the cumulative distribution of player 2. Argue that the upper support of the distribution of player 2 is 1, and determine the lower support and the full distribution....
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 Spring '10
 AsuOzdaglar

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