Fig. 3: Strategy frequencies as a function of mutation rate.

The figure shows the frequencies of cooperators (C, blue), defectors (D, orange), and loners (L, green) as functions of the mutation rate μ, for two network sizes: N = 50² in (a–c) and N = 200² in (d–f). Error bars represent standard deviations from 100 independent runs, and horizontal dashed lines represent reference outcomes from a large lattice (N = 2000²), as reported in ref. ³². The shaded region highlights mutation rates for which outcomes converge to the large-lattice benchmark. At synergy factors near phase transition points (r = 2.02 in (a) and (d) and r = 4.6 in (c) and (f)), intermediate mutation rates effectively mitigate finite-size effects. In contrast, under stable conditions (r = 3 in (b) and (e)), only high mutation rates alter evolutionary outcomes.