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Unformatted text preview: 6.972 Game Theory and Equilibrium Analysis Midterm Exam April 6, 2004; 1-2:30 pm Problem 1. (40 points) For each one of the statements below, state whether it is true or false. If the answer is true, explain why. If the answer is false, give a counterexample. Explanations and counterexamples are required for full credit. (2 and 3 are 6 points each, the rest are 7 points each.) 1. All strategic games with continuous payoff functions (i.e., u i ( a ) is continu- ous in a for all i ) and nonempty compact action spaces ( A i ’s are nonempty and compact) have a pure strategy Nash equilibrium. 2. Symmetric games have at least one symmetric Nash equilibrium. 3. A weakly dominated action cannot be used with positive probability in any mixed strategy Nash equilibrium. 4. Consider a variation of the ultimatum game, in which player 1 offers player 2 an amount x . If player 2 accepts, the payoffs are (1- x, x ); if she rejects, the payoffs are (1- x, 0). This game has a unique subgame perfect equilibrium in which player 1 offers x = 0 and player 2 accepts all offers. 5. Recall the infinite horizon bargaining game with alternating offers where player i ’s payoff to the terminal history ( x 1 , N, x 2 , N, . . . , x t , Y ) [ x i is a vector of two components, representing the division of one unit] is δ t- 1 i x t i with δ...
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- Spring '10
- Game Theory, Nash, Subgame perfect equilibrium, Equilibrium Analysis