Self-organization in a simple model of adaptive agents playing 2X2 games with arbitrary payoff matrices

We analyze, both analytically and numerically, the self-organization of a system of "selfish" adaptive agents playing an arbitrary iterated pairwise game (defined by a 2X2 payoff matrix). Examples of possible games to play are: the Prisoner's Dilemma (PD) game, the Chicken game, the Hero game, etc. The agents have no memory, use strategies not based on direct reciprocity nor 'tags' and are chosen at random, i.e. geographical vicinity is neglected. They can play two possible strategies: cooperate (C) or defect (D). The players measure their success by comparing their utilities with an estimate for the expected benefits and update their strategy following a simple rule. Two versions of the model are studied: 1) the deterministic version (the agents are either in definite states C or D) and 2) the stochastic version (the agents have a probability c of playing C). Using a general Master Equation we compute the equilibrium states into which the system self-organizes, characterized by their average probability of cooperation c_{eq}. Depending on the payoff matrix, we show that c_{eq} can take five different values. We also consider the mixing of agents using two different payoff matrices an show that any value of c_{eq} can be reached by tunning the proportions of agents using each payoff matrix. In particular, this can be used as a way to simulate the effect a fraction d of "antisocial" individuals -incapable of realizing any value to cooperation- on the cooperative regime hold by a population of neutral or "normal" agents.