Evolutionary game theory studies frequency dependent selection. The fitness of a strategy is not constant, but depends on the relative frequencies of strategies in the population. This type of evolutionary dynamics occurs in many settings of ecology, infectious disease dynamics, animal behavior and social interactions of humans. Traditionally evolutionary game dynamics are studied in well-mixed populations, where the interaction between any two individuals is equally likely. There have also been several approaches to study evolutionary games in structured populations. In this paper we present a simple result that holds for a large variety of population structures. We consider the game between two strategies, A and B, described by the payoff matrix . We study a mutation and selection process. If the payoffs are linear in a, b, c, d, then for weak selection strategy A is favored over B if and only if σa + b > c + σd. This means the effect of population structure on strategy selection can be described by a single parameter, σ. We present the values of σ for various examples including the well-mixed population,games on graphs and games in phenotype space. We give a proof for the existenceof such a σ, which holds for all population structures and update rulesthat have certain (natural) properties. We assume weak selection, but allow anymutation rate. We discuss the relationship between σ and the criticalbenefit to cost ratio for the evolution of cooperation. The single parameter,σ, allows us to quantify the ability of a population structure to promotethe evolution of cooperation or to choose efficient equilibria in coordinationgames.
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