Cooperation in two-dimensional mixed-games

Evolutionary game theory is a common mathematical framework to study the evolution of cooperation in selfish systems, specially using the Prisoners Dilemma game, where it is usually assumed that the same game is played in all interactions. Here, we investigate a model where the game that is played by two individuals is uniformly drawn from a sample of two different games at each iteration. Using the master equation approach we show that the random mixture of two games is equivalent to play the average game when (i) the strategies are statistically independent of the game distribution and (ii) the transition rates are linear functions of the payoffs. This result still holds using Pair-Approximation for a small cluster of 8 sites arranged in a square lattice. We also use Monte-Carlo simulations in a two dimensional lattice to investigate the scenario when the two above conditions do not hold, i.e. we use the Fermi-Dirac distribution for the transition rates. We find that even outside of such conditions, several quantities characterizing the mixed-games are still the same as the ones obtained in the average game when the two games are not very different. Also we find interesting results regarding how the heterogeneity of the games played can increase the final fraction of cooperators above the usual mean game limit. We would like to thanks FAPEMIG and CNPq for the financial support given.