Extended Data Fig. 5: An unconditional strategy probC_p can invade classical DISC and pure DISC_q,M, but not mixed equilibria.

a-c, Competition between probabilistic unconditional cooperator (probC_p), classic discriminators (DISC), and aggregating discriminators (DISC_q,M) under scoring. The horizontal axis shows multiple values of probC_p’s probability of cooperation (p = 0 being ALLD and p = 1 being ALLC). The vertical axes represent the frequency of discriminators. Arrows show the flow of the evolutionary dynamics along the vertical axis. Filled-in circles represent stable equilibria; open circles represent unstable equilibria. a, probC_p can invade and coexist with the classic single-observation discriminator (DISC) when the probability of cooperating is large enough (p > 0.3). At low values of p ≤ 0.3, coexistence becomes bistability. b, When competing against ‘look twice, forgive once’, only a narrow range of high probabilities (0.8 < p < 1.0) allows probC_p to invade. At lower p ≤ 0.8, we find bistability. c, probC_p is also able to invade strict discriminators and coexist with them for p > 0.1, with a larger fraction of probC_p at this mixed equilibrium as p increases. d-e, Effects of probC_p when more strategies are present. The bars show the results of the competition between probC_p, ALLD, and DISC (for M = 1); and the competition between probC_p, ALLD, tolerant aggregating discriminator (q = 1/2), and strict aggregating discriminator (q = 2/2) (for M = 2). d, When discriminators use a single observation, the outcomes are very similar to the classic scenario with ALLC: if p is large enough (p > 0.3), there is coexistence between probC_p and DISC with the same weakness of the ‘scoring dilemma’ (i.e. ALLD can invade and take over). e, With multiple observations, for all values of p we tested, a mixture of tolerant and strict discriminators can coexist, have a non-trivial basin of attraction, and resist invasion by probC_p and ALLD. This is because having a few strict discriminators in the mix effectively increases the overall strictness of the population compared to a scenario where only DISC_1/2,2 is present. As a result, the overall ‘effective population tolerance’ becomes q_e > 1/2. This higher level of strictness enables the population to more successfully identify and punish probC_p when rare. As p increases, the basin of attraction towards such mixed discriminating equilibrium decreases. For all panels, the benefit-to-cost ratio is b/c = 5, with error rates of assessment and execution α = 0.02 and ε = 0.02, respectively.