Answered: Consider the following game in which…

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.6P

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Transcribed Image Text:1. Consider the following game in which each of the two players has two possible actions, cooperate with each other (C) and defect (D, i.e., Don't cooperate with each other). The monetary payoffs to each action profile are given in the following table. Player1 / Player 2 Defect (D) Cooperate (C) 3, 0 Defect (D) 1, 1 Cooperate (C) 0,3 2, 2 However, the above table does not represent the true preferences of the players, as they care about what the other player earns as well as their own earnings. In particular, if mi is the amount of money earned by Player 1 and m2 is the amount of money earned by Player 2, then the payoff to Player 1 is mi + am2 ; where a 2 0. (Of course, when a = 0; the payoffs coincide with the ones in the above table). For example, Player 1's payoff to action profile (C,C) is u1 (C;C) = 2 + 2a; and Player 1's payoff to action profile (C,D) is u1 (C;D ) = 0+ 3a = 3a a) Write down the strategic form of this game for a= 1. Is this game a Prisoners' Dilemma game? b) Find the range of values of a for which the resulting game is the Prisoners' Dilemma. For values of a for which the game is not the Prisoners' Dilemma, find its Nash equilibria. c) Now assume that the game presented in the above table (i.e, when a = 0) is played infinitely many times between the two players. For what values of discount factor o, if any, the following strategies constitute subgame perfect equilibria? (i) Players play C in period 1 and after any history in which nobody had ever played D (i.e., if there was no deviation). They play D after any other history. (ii) Choose C in period 1 and then do whatever your opponent did last period.

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