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Unformatted text preview: Economics 109 Midterm Examination Prof. Watson, Fall 2005 You have one hour and twenty minutes to complete this examination. You may not use your notes or any books during the examination. Write your answers in the spaces provided on the answer sheet that has been distributed separately. You may use the scratch paper that has been distributed, but show your work (and write the key analytical steps) on your answer sheet and, when you have finished the examination, submit only your answer sheet. 1. Write your name in the designated space on the answer sheet. 2. Consider the following strategic setting. There are three players, numbered 1, 2, and 3. Player 1 has two cards, labelled King and Ace. At the beginning of the game, player 1 deals one of the cards to player 2 and the other card to player 3. That is, player 1 either gives the Ace to player 3 and the King to player 2 (call this the action A) or the King to player 3 and the Ace to player 2 (action K). Player 2 observes the card dealt to him; player 3 does not get to see the card dealt to her. Player 2 then must decide between switching cards with player 3 (S) or not (N). Player 3 observes whether player 2 made the switch, but does not see her card. Finally, player 3 responds to the question “Is your card the Ace” by saying either “yes” (Y) or “no” (N). If player 3 correctly states whether her card is the Ace, then she obtains a payoff of 1 and the other players get 0; otherwise, players 1 and 2 each gets a payoff of 1 and player 3 obtains 0. Represent this game in the extensive form (draw the game tree). 3. Convert the following extensive form game into the normal form (draw and label the matrix): 4. For the two normal form games pictured on your answer sheet, find the sets of rationalizable strategies and the Nash equilibria. Designate the Nash equilibria by circling the cells corresponding to equilibrium strategies. Designate the rationalizable sets by striking out strategies that are iteratively dominated and by describing the sets with the proper notation. If you need a mixed strategy to dominate a pure strategy, specify which mixed strategy you use. 5. Consider a two player game and suppose that s * and t * are Nash equilibrium strategy profiles in the game. Must it be the case that { s * 1 ,t * 1 }×{ s * 2 ,t * 2 } is a weakly congruous strategy set? Briefly explain why or why not. 1 6. For the normalform game pictured below, answer the following questions. (You do not need to show any work; these questions will be graded only on the basis of whether your final answers are correct.) (a) Name the Nash equilibria of this game. (b) Which of the Nash equilibria are efficient? (c) Calculate the set BR 1 ( μ 2 ) for the belief μ 2 = (3 / 7 , , 2 / 7 , 2 / 7). (That is, μ 2 puts probability 3 / 7 on w, zero probability on x, 2 / 7 on y, and 2 / 7 on z.) 7. Consider a Cournot duopoly (two firms) game in which the demand curve is given by p = 24 2 q 1 2 q 2 and the firms produce at a cost of 4 per unit. Write firm 1’s payoff (profit) function,and the firms produce at a cost of 4 per unit....
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 Fall '08
 WATSON
 Economics, Game Theory, ........., player, Nash, Prof. Watson

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