S2010_final_sols

S2010_final_sols - Econ 171 Spring 2010 Final Exam June 8 You have three hours to take this exam Please answer all 5 questions for a maximum of 100

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Unformatted text preview: Econ 171 Spring 2010 Final Exam June 8 You have three hours to take this exam. Please answer all 5 questions, for a maximum of 100 points. Point totals and subtotals are indicated in brackets. To obtain credit, you must provide arguments or work to support your answer. 1. [10] Find all Nash equilibria of the following game. X Y Z A 2 ,- 1 4 , 2 2 , B 3 , 3 , 1 , 1 C 1 , 2 2 , 8 5 , 1 Solution : There are three equilibria. ( B,X ) and ( A,Y ) are PSNE and (( 4 5 , 1 5 , 0) , ( 1 2 , 1 2 , 0)) is a mixed-strategy Nash equilibrium. 1 2. [20] Consider the extensive-form game represented below. C A 5 , 2 B 1 Y 6 , 2 X 2 , 6 Y 2 , 6 X 6 , 2 2 (a) [5] Which solution concept is the appropriate one to apply to this game? Find the set of equilibria using that concept. Solution : Perfect-Bayesian Equilibrium is the appropriate solution concept (be- cause it discards equilibria that are not sequentially rational). The only kind of PBE is of the following form: ( A, ( p, 1- p )), where p = Pr( X ) ∈ [ 1 4 , 3 4 ], and player 2’s beliefs are given by μ 2 ( B ) = μ 2 ( C ) = 1 2 . (b) [5] Suppose that we delete the information set between player 2’s two decision nodes. (Plugging the values ( a,b,c ) = (5 , 6 , 2) into the figure below shows the extensive form of this game.) In other words, suppose that 2 can actually observe whether 1 chose B or C . What solution concept should we apply now? Find the unique equilibrium under that concept. Solution : Now we should apply SPNE, which yields ( A,XY ). (c) [5] Now let ( a,b,c ) = (3 , 1 , 3). Find all PSNE of this game. Solution : The PSNE are ( A,XX ), ( A,XY ), and ( C,XY ). (d) [5] Which of these are subgame-perfect? Solution : ( A,XY ) and ( C,XY ) are subgame-perfect. C A a, 2 B 1 Y 6 , 2 X 2 , 6 2 Y c, 6 X b, 2 2 2 3. [10] Suppose two players play one of the two normal-form games shown below. Player 1 knows which game is being played, but player 2 thinks that it is Game (a) with...
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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.

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S2010_final_sols - Econ 171 Spring 2010 Final Exam June 8 You have three hours to take this exam Please answer all 5 questions for a maximum of 100

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