Anti-ALLC monoclonal antibody, clone 4E4

We explore a replicator-mutator model of the repeated Prisoner's Dilemma involving three strategies: always cooperate (ALLC), always defect (ALLD), and tit-for-tat (TFT). The dynamics resulting from single unidirectional mutations are considered, with detailed results presented for the mutations TFT -> ALLC and ALLD -> ALLC. For certain combinations of parameters, given by the mutation rate mu and the complexity cost c of playing tit-for-tat, we find that the population settles into limit cycle oscillations, with the relative abundance of ALLC, ALLD, and TFT cycling periodically. Surprisingly, these oscillations can occur for unidirectional mutations between any two strategies. In each case, the limit cycles are created and destroyed by supercritical Hopf and homoclinic bifurcations, organized by a Bogdanov-Takens bifurcation. Our results suggest that stable oscillations are a robust aspect of a world of ALLC, ALLD, and costly TFT; the existence of cycles does not depend on the details of assumptions of how mutation is implemented.

The repeated prisoner's dilemma (RPD) game has revealed how cooperation and competition arise among competitive players in long-run relationships. In the RPD game with no errors, zero-determinant (ZD) strategies allow a player to unilaterally set a linear relationship between the player's own payoff and the opponent's payoff regardless of the strategy of the opponent. On the other hand, unconditional strategies such as ALLD and ALLC also unilaterally set a linear relationship. However, little is known about the existence of such strategies in the RPD game with errors. Here, we analytically search for the strategies that enforce a linear payoff relationship under observation error in the RPD game. As a result, we found that, even in the case with observation errors, the only strategy sets that enforce a linear payoff relationship are either ZD strategies or unconditional strategies.