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Repeated Sequential Prisoner's Dilemma

In this paper, we investigate a variant of the repeated Prisoners' Dilemma game: In each round of our model, the two players play a Sequential Prisoner's Dilemma. Specifically, player 1 acts firstly, and then, after seeing player 1's action, player 2 chooses her action. Player 1 has only one step memory and we assume that she has the priority to choose her strategy. According to the standard theory of the Markov decision process, we can calculate player 2's best strategy, and then player 1's. The conclusion shows that neither the Nash equilibrium strategy both defection in static game nor the Tit-For-Tat strategy, which is put forward by Axelrod, is the best strategy for player 1, and the player 2's strategy is to always cooperate. The cooperation rate of the game is determined by the ratio of the payoff when both player defects each other and the payoff when both player cooperate. And with the priority to choose strategy, player 1 can approximate her maximum equilibrium payoff despite the restriction on her strategy space.