Figure 2.

The relationship between the fixation probabilities $({\rho }_{{\text{TFT}}_{n-1}},{\rho }_{\text{ALLD}})$ and the frequency of TFT_n−1 at the unstable equilibrium (p*). If (3.5) holds, then $N{\rho }_{{\text{TFT}}_{n-1}}>1>N{\rho }_{\text{ALLD}}$, i.e. selection favours TFT_n−1 replacing ALLD and opposes ALLD replacing TFT_n−1. If (3.6) holds, then $N{\rho }_{\text{ALLD}}>1>N{\rho }_{{\text{TFT}}_{n-1}}$, i.e. selection favours ALLD and opposes TFT_n−1. If neither (3.5) nor (3.6) is satisfied, then both $N{\rho }_{{\text{TFT}}_{n-1}}$ and Nρ_ALLD are less than 1, i.e. selection opposes the fixation of either strategy. We also find that (3.7) is equivalent to ${\rho }_{{\text{TFT}}_{n-1}}>{\rho }_{\text{ALLD}}$. All these relationships hold for large population size. Although ALLD is evolutionarily stable against invasion by TFT_n−1 in the deterministic model, in our stochastic model, the probability that a single mutant of TFT_n−1 takes over an ALLD population can exceed 1/N.