2008SpringHomework4Answers - SABANCI UNIVERSITY ECON 201 A...

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SABANCI UNIVERSITY ECON 201 – A GAMES AND STRATEGY ASSIGNMENT 4 ANSWERS 1. ( 25 points ). Bonnie and Clyde are playing a simultaneous-move stage game (a prisoner’s dilemma to which a third strategy is added) as shown below . They play this game two times; they learn the first-round strategies and then play the second- round . Is there a SPE in which the players cooperate in the first round? If yes show it, if not, explain why there is none. Clyde Bonnie Cooperate Don’t Cooperate Defect Cooperate 1 , 1 -2 , 2 -5, -5 Don’t Cooperate 2, -2 0 , 0 -5, -5 Defect -5, -5 -5 -5 -1, -1 Answer: The Stage game has two Nash equilibria: (Don’t, don’t) and (defect, defect). Note that the payoffs in the second Nash equilibrium are both -1, whereas those in the first equilibrium are 0. In the finitely repeated game, the first Nash equilibrium can be used as a “punishment path”. Let us see if it is possible to have as SPE the outcome written in the question. We know than in ANY SPE, the last round play must be one of the two Nash equilibria. So, the strategy could be: “Play Coop in the first round. If in round 1 the outcome is (coop, coop), play don’t coop in round two. If any outcome other than (coop, coop) is observed in round 1, then play defect in round 2.” If both players adopt this strategy, would any one of them deviate? If your rival is playing coop in round one, the best deviation is to don’t cooperate; this will increase your payoff from 1 to 2. But in the second round your rival will play defect, to which defect is the best response, and you will get -1, instead of 0. So the benefit from the best deviation is 2 – δ(-1), which is larger than the benefit from no deviation, 1 + δ(0) for all δ smaller than one (the two payoffs are equal only if δ = 1, when the payers do not discount future payoffs). We conclude that it is impossible to have an SPE in which
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This note was uploaded on 03/16/2012 for the course FENS 101 taught by Professor Selçukerdem during the Fall '12 term at Sabancı University.

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2008SpringHomework4Answers - SABANCI UNIVERSITY ECON 201 A...

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