Consider again the repeated game where the stage game is the prisoner's dilemma game shown below...

  

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Consider again the repeated game where the stage game is the prisoner's dilemma game shown below and the repeated strategies a, T. and are as defined in Exercise 1. Player 1 C D Player 2 C 5,5 8,-8 D -8,8 0,0 (a) Explain how the payoffs for the strategy profile (o.o) compare with the payoffs for the strategy profile (T.T). (b) Find do> 0 such that TI(VT) TI(TT) whenever the discount factor do. That is, we want to show that for a large enough discount factor, player 1 cannot improve their own payoff by switching from 7 to when player 2 plays 7. (c) Let 3 denote the strategy "Always play D". Find do> 0 such that *I(B, T) ST1(T,T) whenever the discount factor d2 do. That is, we want to show that for a large enough discount factor, player I cannot improve their own payoff by switching from 7 to 8 when player 2 plays 7. (d) Note that in parts (b) and (c), we showed two possible strategy changes for player 1, neither of which could improve player 1's payoff over playing 7. It turns out that, for sufficiently large &, the strategy profile (7,7) is a Nash equilibrium of the repeated game. Describe, briefly, what we would need to show to prove this result. (e) Let denote the following generous tit-for-tat strategy: Begin by playing C in stage zero. Also play C in stage one. After this, play whatever the other played in the previous stage. This strategy is called generous or forgiving because it does not give up on cooperation if the other player begins with D. Find do such that whenever 8 > do. TI(C.V) > TI(T.V) This result can be used to show that, while (7,7) is a Nash equilibrium of the repeated game, it is not subgame perfect. Consider again the repeated game where the stage game is the prisoner's dilemma game shown below and the repeated strategies a, T. and are as defined in Exercise 1. Player 1 C D Player 2 C 5,5 8,-8 D -8,8 0,0 (a) Explain how the payoffs for the strategy profile (o.o) compare with the payoffs for the strategy profile (T.T). (b) Find do> 0 such that TI(VT) TI(TT) whenever the discount factor do. That is, we want to show that for a large enough discount factor, player 1 cannot improve their own payoff by switching from 7 to when player 2 plays 7. (c) Let 3 denote the strategy "Always play D". Find do> 0 such that *I(B, T) ST1(T,T) whenever the discount factor d2 do. That is, we want to show that for a large enough discount factor, player I cannot improve their own payoff by switching from 7 to 8 when player 2 plays 7. (d) Note that in parts (b) and (c), we showed two possible strategy changes for player 1, neither of which could improve player 1's payoff over playing 7. It turns out that, for sufficiently large &, the strategy profile (7,7) is a Nash equilibrium of the repeated game. Describe, briefly, what we would need to show to prove this result. (e) Let denote the following generous tit-for-tat strategy: Begin by playing C in stage zero. Also play C in stage one. After this, play whatever the other played in the previous stage. This strategy is called generous or forgiving because it does not give up on cooperation if the other player begins with D. Find do such that whenever 8 > do. TI(C.V) > TI(T.V) This result can be used to show that, while (7,7) is a Nash equilibrium of the repeated game, it is not subgame perfect.

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a Comparing Payoffs for Strategies o o and T T In the prisoners dilemma the strategy profile o o refers to both players always choosing the action coo... View the full answer