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Unformatted text preview: Economics 109 Midterm Examination Prof. Watson, Fall 2004 You have 50 minutes to complete this examination. You may not use your notes, calculators, or any books during the exami nation. Write your answers, including all necessary derivations, in the spaces provided on the answer sheet that has been dis tributed separately. You may use the scratch paper that has been distributed but submit only your answer sheet. You do not need to show any work in your answers to questions 15; these questions will be graded only on the basis of whether your final answers are correct. 1. Write your name in the designated space on the answer sheet. In the space marked “version,” write the following number: 3. 2. In the extensiveform game pictured on the right, how many (pure) strategies does player 2 have? 3. In the normalform game pictured on the right, is player 1’s strategy M dominated? If so, describe a strategy that dominates it. If not, describe a belief to which M is a best response. 4. The normalform game pictured on the right represents a sit uation in tennis, whereby the server (player 1) decides whether to serve to the opponent’s forehand (F), center (C), or backhand (B) side. Simultaneously, the receiver (player 2) decides whether to favor the forehand, center, or backhand side. Calculate the set of rationalizable strategies for this game. 5. Consider the normalform game pictured on the right. (a) What are the (pure strategy) Nash equilibria of this game? (b) Which of these equilibria are efficient? (c) Calculate BR 1 ( μ 2 ) for the belief μ 2 = (1 / 3 , 1 / 3 , 1 / 3). (d) Is the set X = { w , x }×{ b , c } weakly congruent? 1 6. Consider a game between two boxers. The players simulta neously choose quantities of steroids to take. Denote player 1’s quantity by s 1 and denote player 2’s quantity by s 2 , where we assume s 1 ,s 2 ≥ 0. The players’ payoffs are given by: u 1 ( s ) = 10 s 1 + s 1 s 2 s 2 1 20 s 2 u 2 ( s ) = 10 s 2 + s 1 s 2 s 2 2 20 s 1 . (a) Calculate the players’ bestresponse functions (the optimal s i as a function of s j ). (b) Calculate the Nash equilibrium of this game. (c) Is the Nash equilibrium efficient? Explain why or why not. 7. Consider a game in which, simultaneously, player 1 selects a number x ∈ [2 , 8] and player 2 selects a number y ∈ [2 , 8]. The payoffs are given by: u 1 ( x,y ) = 2 xy x 2 u 2 ( x,y ) = 4 xy y 2 . Calculate the rationalizable strategy profiles for this game. Show your logic. 2 Version: Economics 109 Midterm Examination Answer Sheet, Fall 2004, Prof. Watson 1. Your name....................................................................................._________________________________ 2. How many strategies does player 2 have (circle one): 1 2 3 4 5 6 7 8 10 16 64 256 3. Is M dominated? Circle one: YES NO If so, name a strategy that dominates it: If not, name a belief to which M is a best response: 4. The set of rationalizable strategies: R = 5. (a) The Nash equilibria are:...
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 Fall '08
 WATSON
 Economics, Game Theory, ........., player, Prof. Watson

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