Question

If a prisoner's dilemma game is repeated 1000 times, is there a SPE where some player gets an average payoff strictly higher than his equilibrium payoff in the stage game? (Average payoff is calculated as the total payoff divided by the number of periods, 1000.)

Yes or no and say why or why not

Answer 1

No, it is not possible if both of the players are rational.

Reason

We can prove it by backward induction. In the last period, 1000 here, everyone will play their dominant strategy as there is no incentive to collude as they can not be punished in future. Given this, in the penultimate period, they should play dominant strategy since they expect the other players to play dominant strategy in the next turn and so on. By backward induction, for any last n periods, if they play dominant strategy, at the (n+1)th period from the last, they will play dominant strategy.

Since they always play dominant strategy in a finite prisoner's dilemma, they can not achieve a higher than average payoff.