Evolutionary games on the lattice: Payoffs affecting birth and death rates

Abstract

This article investigates an evolutionary game based on the framework of interacting particle systems. Each point of the square lattice is occupied by a player who is characterized by one of two possible strategies and is attributed a payoff based on her strategy, the strategy of her neighbors and a payoff matrix. Following the traditional approach of evolutionary game theory, this payoff is interpreted as a fitness: the dynamics of the system is derived by thinking of positive payoffs as birth rates and the absolute value of negative payoffs as death rates. The nonspatial mean-field approximation obtained under the assumption that the population is well mixing is the popular replicator equation. The main objective is to understand the consequences of the inclusion of local interactions by investigating and comparing the phase diagrams of the spatial and nonspatial models in the four dimensional space of the payoff matrices. Our results indicate that the inclusion of local interactions induces a reduction of the coexistence region of the replicator equation and the presence of a dominant strategy that wins even when starting at arbitrarily low density in the region where the replicator equation displays bistability. We also discuss the implications of these results in the parameter regions that correspond to the most popular games: the prisoner's dilemma, the stag hunt game, the hawk-dove game and the battle of the sexes.