The players play this game infinitely many times, and each discounts the future using a discount...

  

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The players play this game infinitely many times, and each discounts the future using a discount factor d < 1. Consider first a pair of Grim-Trigger (GT) strategies, as in class: players play C as long as each player has played C in the past; if someone ever played D, then each player plays D forever after. (a) What is the payoff from sticking to GT, if the opponent plays GT? (3 points) (b) For what discount factors is the pair of GT strategies a Subgame Perfect Equilib- rium? (3 points) Now, suppose that the players use the following "Eye for Eye" strategies. Start playing C, and play C "on-path", if both players played C in the last period. If player i ever deviates, do the following in the next round player i is supposed to play C while his opponent plays D. If both do that, they revert back to the "on-path" stage of playing C. If in this one round of punishment some player does not do the right thing, this is considered a deviation (and so he is punished in the next period, after which, if all goes well, players return to "on-path"). (e) For what discount factors, a player does not want to do the one-shot deviation, if the players are "on-path"? (3 points) (d) For what discount factors, the player that plays D ("punisher") in the punishment round does not want to do the one shot deviation? (2 points) (Hint: Focus on the first two terms in the discounted sequence of payoffs!) (e) For what discount factors, the player that plays C ("punished") in the punishment round does not want to do the one shot deviation? (3 points) (Hint: Focus on the first two terms in the discounted sequence of payoffs!) (f) Suppose now that you replace the "4" with a "6" in the payoffs. For what dis- count factors a pair of "Eye for Eye" strategies constitutes a Subgame Perfect Equilibrium? (4 points) 2 C D 1 C3, 3-1, 4 D4, -1 0,0 The players play this game infinitely many times, and each discounts the future using a discount factor d < 1. Consider first a pair of Grim-Trigger (GT) strategies, as in class: players play C as long as each player has played C in the past; if someone ever played D, then each player plays D forever after. (a) What is the payoff from sticking to GT, if the opponent plays GT? (3 points) (b) For what discount factors is the pair of GT strategies a Subgame Perfect Equilib- rium? (3 points) Now, suppose that the players use the following "Eye for Eye" strategies. Start playing C, and play C "on-path", if both players played C in the last period. If player i ever deviates, do the following in the next round player i is supposed to play C while his opponent plays D. If both do that, they revert back to the "on-path" stage of playing C. If in this one round of punishment some player does not do the right thing, this is considered a deviation (and so he is punished in the next period, after which, if all goes well, players return to "on-path"). (e) For what discount factors, a player does not want to do the one-shot deviation, if the players are "on-path"? (3 points) (d) For what discount factors, the player that plays D ("punisher") in the punishment round does not want to do the one shot deviation? (2 points) (Hint: Focus on the first two terms in the discounted sequence of payoffs!) (e) For what discount factors, the player that plays C ("punished") in the punishment round does not want to do the one shot deviation? (3 points) (Hint: Focus on the first two terms in the discounted sequence of payoffs!) (f) Suppose now that you replace the "4" with a "6" in the payoffs. For what dis- count factors a pair of "Eye for Eye" strategies constitutes a Subgame Perfect Equilibrium? (4 points) 2 C D 1 C3, 3-1, 4 D4, -1 0,0