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Prelim_2_solutions

# Prelim_2_solutions - Kaushik Basu Spring 2008 Econ 367...

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Spring 2008 Econ 367. Game-Theoretic Method Prelim Exam 2 - Solutions [1 hour. Total 15 points] 1. Two players are going to play the Prisoner’s Dilemma described below an infinitely many times. Each player has a discount factor of 1, such that 0 1 1 < 1. C D C 5, 5 0, 6 D 6, 0 2, 2 (a) If 1 = 0, describe the subgame perfect equilibrium of this infinitely repeated game. Briefly explain your answer. Answer: 1 = 0 means that players don’t value their future at all. So at every period of the infinitely repeated game, they play the game as if they are playing a one- shot game. (Note that this does NOT mean that it actually is a one-shot game). In an infinitely repeated game, each subgame is an infinitely repeated game as well. In this case, where players don’t care about the future at all, each subgame reduces to the one-shot game. So the only subgame perfect equilibrium of the game is the strategy “to play D at every period” for both the players. (b) Suppose the two players decide to use the ‘trigger strategy,’ that is, each will begin by playing C and then in any period if they find that nothing but C’s have been played by both players in the past, they will play C. Under all other circumstance they will play D. Calculate the critical value of 1 such that for all values above that the trigger strategies constitute a Nash equilibrium. Answer: Stay: 5 5 5 5 5 Deviate: 6 2 2 2 2 Therefore PV(stay) = 5/(1 - 1) and PV(deviate) = 6 + 21/(1 - 1). The trigger strategies constitute a Nash equilibrium iff – PV(stay) ≥ PV(deviate) i.e., 5/(1 - 1) ≥ 6 + 21/(1 - 1) i.e., 1 ≥ ¼. Therefore the critical value of 1 such that for all values above that the trigger

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Prelim_2_solutions - Kaushik Basu Spring 2008 Econ 367...

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