Iterated Prisoner's Dilemma: Pay-off variance

The iterated Prisoner's Dilemma (IPD) is usually analysed by evaluating arithmetic mean pay-offs in an ESS analysis. We consider several points that the standard argument does not address. Finite population size and finite numbers of matches in the IPD game lead us to consider both pay-off variance and the sampling process in the evolutionary game. We provide a general form for the pay-off variance of a Markov strategist in the IPD game, and present a general analysis of the initial invasion process of an 'all defection strategist' (ALLD) into a 'tit-for-tat' (TFT) strategist population by considering stochastic processes. Finite population size, strategic error and the variances of pay-offs alter the prediction concerning the initial invasion of ALLD compared with the standard Evolutionarily Stable Strategy (ESS) analysis. Even though TFT gets the larger arithmetic mean, the variance of its pay-off is also larger when the expected iterations of the game are sufficiently large. Therefore, the boundary of the parameter of the probability of game continuity, w, above which ALLD does not have advantage to invade into the TFT population, becomes a bit larger than predicted by the deterministic model.