<p>Suppose two people are playing a game (Bill and Amanda). Each player has two

Question

<p>Suppose two people are playing a game (Bill and Amanda). Each player has two possible actions: Left; and<br />Right. The payos are reported in the table below; the rst number is Amanda's payo and the second is<br />Bill's.<br />Bill<br />Left Right<br />Amanda Left 6,6 11,2<br />Right 2,11 9,9<br />a) What is the pure strategy Nash equilibrium of this game?<br /><br />Now suppose this game is played twice, once today and once tomorrow (repeated prisoners' dilemma).Further, suppose that because of discounting, a payoff tomorrow of X, is only worth (.9)X today.<br />b) What is Amanda's payoff when both players play the Nash equilibrium on both days?<br /><br />Suppose Bill makes the following credible claim "I will play "Right" on each day. However, if on the first day I play "Right" and Amanda plays "Left," I will play "Left" for the rest of time."<br />c) What is Amanda's payoff if she plays "Right" on both days.<br />d) What is Amanda's payoff if she plays "Left" on the first day and plays "Right" on the second day?<br />e) What is Amanda's optimal strategy?</p>

Explanation / Answer

Suppose two people are playing a game (Bill and Amanda). Each player has two possible actions: Left; and Right. What is the pure strategy Nash equilibrium of this game? RIght, Right Now suppose this game is played twice, once today and once tomorrow (repeated prisoners' dilemma). Further, suppose that because of discounting, a payoff tomorrow of X, is only worth (.9)X today. What is Amanda's payoff when both players play the Nash equilibrium on both days? Amanda's pay-off today is 9+8.1=17.1 Suppose Bill makes the following credible claim "I will play "Right" on each day. However, if on the first day I play "Right" and Amanda plays "Left," I will play "Left" for the rest of time." What is Amanda's payoff if she plays "Right" on both days. Same as above What is Amanda's payoff if she plays "Left" on the first day and plays "Right" on the second day? Today Amanda's payoff is 11+1.8=12.8 What is Amanda's optimal strategy? Play right the first day to get the pay-off of 9. Since Bill saw you play right the first day, he will play right the second day, but since this is the end of the game, you play left to get the pay-off of 11 (9.9 discounted).

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