Figure 3.

Comparison of the frequency of TFT_n−1 at the unstable polymorphic equilibrium in the deterministic model (p^*) with the conditions ρ_ALLD>1/N, ${\rho }_{{\text{TFT}}_{n-1}}<{\rho }_{\text{ALLD}}$ and ${\rho }_{{\text{TFT}}_{n-1}}<1/N$ when N is large. The solid line represents the frequency of TFT_n−1 at the unstable polymorphic equilibrium, p*, in the deterministic model. The other three lines represent the thresholds of p* above which ρ_ALLD>1/N (dotted line), ${\rho }_{{\text{TFT}}_{n-1}}<{\rho }_{\text{ALLD}}$ (dashed line) or ${\rho }_{{\text{TFT}}_{n-1}}<1/N$ (dot-dashed line). (a) The situation when b/c and m are constant (b/c=1.5 and m=30). The solid line is below the dot-dashed line, which means that p* is smaller than the threshold above which ${\rho }_{{\text{TFT}}_{n-1}}<1/N$, if and only if n≤5. Hence, ${\rho }_{{\text{TFT}}_{n-1}}>1/N$ is satisfied if and only if n≤5. Similarly, ${\rho }_{{\text{TFT}}_{n-1}}>{\rho }_{\text{ALLD}}$ holds if and only if n≤14, while ρ_ALLD<1/N holds if and only if n≤29. In this case, the evolution of cooperation becomes more difficult with increasing group size. (b) The situation when b/(nc) and m are constant (b/(nc)=0.51 and m=30). The condition ${\rho }_{{\text{TFT}}_{n-1}}>1/N$ is satisfied if and only if 3≤n≤57, while ${\rho }_{{\text{TFT}}_{n-1}}>{\rho }_{\text{ALLD}}$ and ${\rho }_{{\text{TFT}}_{n-1}}<1/N$ are always satisfied unless n=2. In this case, the situation under which TFT_n−1 more readily replaces ALLD than vice versa and the situation under which cooperation is maintained are both more likely to be realized when the group size is larger.