02 exams

# 02 exams - Economics 109 Midterm Examination Prof Watson...

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Economics 109 Midterm Examination Prof. Watson, Fall 2002 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. When you have f nished the examination, submit only your answer sheet. Basic Questions 1. Write your name in the designated space on the answer sheet. In the space marked “version,” write the following number: 3. 2. Consider the following strategic setting. There are three players, numbered 1, 2, and 3. At the beginning of the game, players 1 and 2 simultaneously make decisions, each choosing between “X” and “Y.” If they both choose “X” then the game ends and the payo f vector is (1 , 0 , 0); that is, player 1 gets 1, player 2 gets 0, and player 3 gets 0. If they both choose “Y” then the game ends and the payo f vector is (0 , 1 , 0); that is, player 2 gets 1 and the other players get 0. If one player chooses “X” while the other chooses “Y,” then player 3 must guess which of the players selected “X.” That is, player 3 chooses between “1” and “2.” Player 3 makes his selection knowing only that the game did not end following the choices of players 1 and 2. If player 3 guesses correctly, then both he and the player who selected “X” each obtains a payo f of 2, and the player who selected “Y” gets 0. If player 3 guesses incorrectly, then everyone gets a payo f of 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): 1 L M Q P 1, 2 2 4, 1 8, 0 0, 3 0, 2 X Y X Y 2 4. For each of the normal form games pictured on your answer sheet, f nd 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. 1

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More Challenging Questions 5. Consider an asymmetric Cournot duopoly game, where the two f rms have di f erent costs of production. Firm 1 selects quantity q 1 at a production cost of 2 q 1 . Firm 2 selects quantity q 2 and pays the production cost 4 q 2 . The market price is given by p =12 q 1 q 2 . (a) What are the payo f (pro f t) functions of the two f rms? (b) Calculate the f rms’ best-response functions BR 1 ( q 2 )and 2 ( q 1 ) and graph them. (c) Calculate the Nash equilibrium of this game and designate it on your graph. 6. Consider the normal form game pictured here: 2, 0 1, 3 5, 4 1, 3 A B X Y 2 1 Z 5, x 6, 2 All of the payo f numbers are speci f ed, with the exception of that denoted by x .F indanumb e r for x such that the following three statements are all true: (B, X) is a Nash equilibrium, (A, Z) is an e cient strategy pro f le ,and ,forthebe l ie f μ 1 =( 1 2 , 1 2
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02 exams - Economics 109 Midterm Examination Prof Watson...

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