Multiple strategies in structured populations

Many specific models have been proposed to study evolutionary game dynamics in structured populations, but most analytical results so far describe the competition of only two strategies. Here we derive a general result that holds for any number of strategies, for a large class of population structures under weak selection. We show that for the purpose of strategy selection any evolutionary process can be characterized by two key parameters that are coefficients in a linear inequality containing the payoff values. These structural coefficients, sigma(1) and sigma(2), depend on the particular process that is being studied, but not on the number of strategies, n, or the payoff matrix. For calculating these structural coefficients one has to investigate games with three strategies, but more are not needed. Therefore, n = 3 is the general case. Our main result has a geometric interpretation: Strategy selection is determined by the sum of two terms, the first one describing competition on the edges of the simplex and the second one in the center. Our formula includes all known weak selection criteria of evolutionary games as special cases. As a specific example we calculate games on sets and explore the synergistic interaction between direct reciprocity and spatial selection. We show that for certain parameter values both repetition and space are needed to promote evolution of cooperation.