Fig. 1: Strategy frequencies versus synergy factor r for a small network size N = 200².

In spatial optional public goods games, finite-size effects in small networks near phase transition boundaries are addressed by introducing mutation, producing results consistent with larger networks. Dots indicate outcomes without mutation, while solid lines represent outcomes with a mutation rate of μ = 10⁻⁶, with the colours blue, orange, and green represent cooperators (C), defectors (D), and loners (L), respectively. Black arrows at r = 2.0 and r = 4.6005 mark critical values from large networks (N = 2000²), as reported in ref. ³², separating phases of pure L, C + D + L, and C + D coexistence. Gray regions indicate discrepancies due to finite-size effects.