For the prisoner’s dilemma in which: both cooperating gives each a payoff of 3, both defecting...

For the prisoner’s dilemma in which: both cooperating gives each a payoff of 3, both defecting gives each a payoff of 2, while when one defects and the other cooperates, the defector get 5 and the cooperator gets -1:

a) What condition on d can make the Grim-Trigger equilibrium sub-game perfect?

b) What condition on d is required to support an equilibrium in which: each is to cooperate in period 1 and in any other period as long as both have always cooperated. If either or both defect in any period, then each is to defect for the next 2 periods and then return to cooperating. Any deviation from cooperating when the strategy prescribes cooperation is to be followed by two periods of mutual defection and then a return to cooperation.

c) What about an equilibrium just like in part b) except that punishment only lasts one period?

d) How about an equilibrium in which cooperation is prescribed in every period except following a period in which one person has defected? If one person defected in the previous period when they were supposed to cooperate, then the defector is to cooperate, while the other player is to defect this period. In the next period, each is to return to cooperate. What requirement on d is necessary for this to be a sub-game perfect equilibrium?

Show that tit-for-tat is not sub-game perfect by showing that there is no value of d, the discount fact that will allow tit-for-tat to satisfy the one-period deviation principle for every history.

Expert Solution

Rahul Sunny answered 2 years ago

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