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The Iterated Prisoner's Dilemma on a Cycle

Pavlov, a well-known strategy in game theory, has been shown to have some advantages in the Iterated Prisoner's Dilemma (IPD) game. However, this strategy can be exploited by inveterate defectors. We modify this strategy to mitigate the exploitation. We call the resulting strategy Rational Pavlov. This has a parameter p which measures the "degree of forgiveness" of the players. We study the evolution of cooperation in the IPD game, when n players are arranged in a cycle, and all play this strategy. We examine the effect of varying p on the convergence rate and prove that the convergence rate is fast, O(n log n) time, for high values of p. We also prove that the convergence rate is exponentially slow in n for small enough p. Our analysis leaves a gap in the range of p, but simulations suggest that there is, in fact, a sharp phase transition.