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Unformatted text preview: Econ 171 Fall 2008 Final Exam - Solutions December 10 You have three hours to take this exam. Please answer all questions. Each question is worth 15 points, with the exception of number 3, which is worth 10 points. Point subtotals are in brackets. To obtain credit, you must provide arguments or work to support your answer. 1.  Find all Nash equilibria of the following game. L M R T 3 , 2 4 , 1 , 1 M 2 , 3 , 3 , B 1 , 1 , 2 2 , 3 Solution : For Player 1, T strictly dominates M , hence we can eliminate M of Player 1. Next, R strictly dominates M for Player 2, hence we eliminate M of Player 2. This leaves us with L R T 3 , 2 1 , 1 B 1 , 1 2 , 3 Of this game, there are two pure-strategy Nash equilibria, ( T,L ) and ( B,R , and one mixed-strategy Nash equilibrium, σ 1 = (2 / 3 , 1 / 3) and σ 2 = (1 / 3 , 2 / 3). 2.  Consider the extensive-form game represented below, and notice that two of the values for Player 3’s payoffs are left unspecified, a and b . For all of the questions below, you may restrict your attention to pure strategies. I O 2 , 2 , 2 1 C A 4 , 3 , 1 B 2 Y 7 , 5 ,a X 1 , 2 , 6 Y 5 , ,b X 3 , 2 , 1 3 (a)  One possibility for the missing payoffs is (5 , 0) (that is, a = 5 and b = 0). Another possibility is (0 , 5). Which of these possibilities would make it so that we can use backwards induction to find SPNE? (For your answer, write either “(5 , 0)” or “(0 , 5)”.) Very briefly state why we can use backwards induction with these payoffs, even though this game form has imperfect information. Solution : (5 , 0). With these payoffs, X dominates Y for Player 3, making it the only rational choice. 1 (b)  Find all SPNE of the game, using ( a,b ) = (5 , 0). If, in your answer above, you stated that these payoffs allow us to use backwards induction, then use backwards induction to find the answer. Solution : Backwards inducting, we have Player 3 choosing X at her information set, Player 2 choosing A , and Player 1 choosing I . This yields ( I,A,X ) as the unique SPNE. (c)  Find all SPNE of the game, using ( a,b ) = (0 , 5). If, in your answer to part (a), you stated that these payoffs allow us to use backwards induction, then use backwards induction to find the answer. Solution : For these payoffs, we cannot use backwards induction, so we identify the Nash equilibria in each subgame. In this case, it is easier to find the NE of the sole proper subgame, which begins with Player 2’s decision node, than to find the NE of the entire game. The subgame can be represented with the following matrix, which includes the payoffs only of Players 2 and 3, and has Player 2’s strategies in the rows and 3’s strategies in the columns....
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This note was uploaded on 12/26/2011 for the course ECON 171 taught by Professor Charness,g during the Fall '08 term at UCSB.
- Fall '08