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, 109 (26), 10409-13

Iterated Prisoner's Dilemma Contains Strategies That Dominate Any Evolutionary Opponent

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Iterated Prisoner's Dilemma Contains Strategies That Dominate Any Evolutionary Opponent

William H Press et al. Proc Natl Acad Sci U S A.

Abstract

The two-player Iterated Prisoner's Dilemma game is a model for both sentient and evolutionary behaviors, especially including the emergence of cooperation. It is generally assumed that there exists no simple ultimatum strategy whereby one player can enforce a unilateral claim to an unfair share of rewards. Here, we show that such strategies unexpectedly do exist. In particular, a player X who is witting of these strategies can (i) deterministically set her opponent Y's score, independently of his strategy or response, or (ii) enforce an extortionate linear relation between her and his scores. Against such a player, an evolutionary player's best response is to accede to the extortion. Only a player with a theory of mind about his opponent can do better, in which case Iterated Prisoner's Dilemma is an Ultimatum Game.

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
(A) Single play of PD. Players X (blue) and Y (red) each choose to cooperate (c) or defect (d) with respective payoffs R, T, S, or P as shown (along with the most common numerical values). (B) IPD, where the same two players play arbitrarily many times; each has a strategy based on a finite memory of the previous plays. (C) Case of two memory-one players. Each player’s strategy is a vector of four probabilities (of cooperation), conditioned on the four outcomes of the previous move.
Fig. 2.
Fig. 2.
(A) Markov matrix for the memory-one game shown in Fig. 1C. (B) The dot product of any vector f with the Markov matrix stationary vector v can be calculated as a determinant in which, notably, a column depends only on one player’s strategy.
Fig. 3.
Fig. 3.
Evolution of X’s score (blue) and Y’s score (red) in 10 instances. X plays a fixed extortionate strategy with extortion factor formula image. Y evolves by making small steps in a gradient direction that increases his score. The 10 instances show different choices for the weights that Y assigns to different components of the gradient, i.e., how easily he can evolve along each. In all cases, X achieves her maximum possible (extortionate) score.

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