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midterm2006solutions

midterm2006solutions - MS&E 246 Game Theory with...

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MS&E 246: Game Theory with Engineering Applications Midterm Exam Feryal Erhun & Ramesh Johari Winter, 2006 Solutions 1. (10 points; 2.5 points each) Select true or false; you do not need to justify your answer. (a) In an extensive form game, every SPNE is an NE and every NE is an SPNE. False – every SPNE is an NE, but not every NE is necessarily an SPNE. (b) Every dynamic game of perfect information has an SPNE in pure strategies. True – it can be found using backward induction. (c) If M is a weakly dominant strategy for player 1 and L is a weakly dominant strategy for player 2, then (M,L) is a NE. True – each is a best response to the other. (d) Every game has at least one efficient NE. False – see the Prisoner’s Dilemma. 2. (30 points) Consider the strategic form games and answer each question separately. (a) Consider the following game. Player 2 L R T (1,1) (0,0) Player 1 M (1,1) (2,1) B (0,0) (2,1) i. Does player 1 have a strictly/weakly dominant strategy? Answer: (3 points) Player 1 does not have a strictly dominant strategy. How- ever, he has a weakly dominant strategy, M. ii. Does player 2 have a strictly/weakly dominant strategy? Answer: (3 points) Player 2 does not have a strictly/weakly dominant strategy. iii. Is this game dominance solvable? Answer: (4 points) This game is not dominance solvable. (b) Consider the following game. 1
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Player 2 L M R T (3,2) (4,0) (1,1) Player 1 M (2,0) (3,3) (0,0) B (1,1) (0,2) (2,3) i. Iteratively eliminate all the strictly dominated strategies. Answer: (5 points) For player 1, T strictly dominates M, hence we eliminate M of player 1. Now, R strictly dominates M, hence we eliminate M of player 2. We obtain the following. Note that, for part (iii), we are going to use this game as a rational player will not play a dominated strategy as a part of a NE.
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