Figure 4: Evolutionary dynamics in the space of stochastic reactive strategies,.

(a,b) illustrate our approximation for the invariant distribution for two different cost values, c = 0.2 and c = 0.6. For the upper graphs, we have calculated the invariant distribution for the discretised state space, where the conditional cooperation probabilities of the reactive strategy are taken from the (finite) set {0, δ, 2δ, …, 1 − δ, 1}, using a grid size δ = 0.02. Areas in dark blue colour correspond to strategy regions that have a relatively high frequency in the invariant distribution. The lower graphs show the results of simulations for the Imhof-Nowak process⁵¹; each blue dot represents a strategy adopted by the resident population. Both methods confirm that when the cost of cooperation is small, e.g. c = 0.2, the resident strategies are either clustered around the lower left corner or around the right edge of the state space. As the cost increases, more weight is given to the lower edge. In (c) we show the strategy that is most favoured in the limit of weak selection, i.e., the strategy with the highest linear coefficient L(p) according to Eq. (13). The graph indicates that there are three parameter regions: for low cost values, a generous strategy is most favoured; for intermediate cost values, the most favoured strategy has only a positive cooperation probability if the co-player defected previously; and for high cooperation costs AllD is most favoured. Parameters: Population size N = 100, ε = 0.01, and w = 10; the Imhof-Nowak process was simulated over 5 · 10⁶ mutant strategies.