Mutation enhances cooperation in direct reciprocity

Abstract

Direct reciprocity is a powerful mechanism for the evolution of cooperation based on repeated interactions between the same individuals. But high levels of cooperation evolve only if the benefit-to-cost ratio exceeds a certain threshold that depends on memory length. For the best-explored case of one-round memory, that threshold is two. Here, we report that intermediate mutation rates lead to high levels of cooperation, even if the benefit-to-cost ratio is only marginally above one, and even if individuals only use a minimum of past information. This surprising observation is caused by two effects. First, mutation generates diversity which undermines the evolutionary stability of defectors. Second, mutation leads to diverse communities of cooperators that are more resilient than homogeneous ones. This finding is relevant because many real-world opportunities for cooperation have small benefit-to-cost ratios, which are between one and two, and we describe how direct reciprocity can attain cooperation in such settings. Our result can be interpreted as showing that diversity, rather than uniformity, promotes evolution of cooperation.

Keywords: direct reciprocity; donation game; evolution of cooperation; mutation rate.

Fig. 1.

Evolutionary dynamics of cooperation. (A) In the donation game, players choose between cooperation and defection. Cooperation incurs a cost c and provides a benefit b to the coplayer. Defection incurs no cost and provides no benefit. (B) We consider the repeated donation game, in which players can use conditional strategies that depend on the outcome of previous rounds. (C) In each evolutionary update step, a focal individual, F, either explores a new random strategy (with probability u) or compares his own payoff to that of a random role model, R. He is then more likely to switch to the role model’s strategy if she performs better than him.

Fig. 2.

The effect of diversity on cooperation. In the limit of rare mutations (u → 0, black bars), the average cooperation rate increases very slowly with the benefit-to-cost ratio. Adding mutation (red bars) substantially enhances cooperation for small benefit-to-cost ratios. Parameters: β = 10; simulations are run for at least 10⁹ updates to get reliable averages.

Fig. 3.

Characteristic curve and the optimum diversity for reactive strategies. (A) The characteristic curve of an evolutionary process of cooperation and defection is the graph of average cooperation rate versus cost-to-benefit ratio. In the limit of rare mutations (u → 0, left), no substantial cooperation occurs for c/b > 0.5. For mutation rate u = 10⁻² (right), we observe substantial levels of cooperation even if c/b > 0.5, especially when N ≥ 50. (B) The average cooperation rate C as a function of the mutation rate u, for N = 100 (Left) and N = 200 (Right). When there are only mutations, u​ = ​1, the cooperation rate approaches C​ = ​0.5, regardless of the cost-to-benefit ratio c/b (Lemma 2 in SI Appendix, Note 2). As u goes down, the cooperation rate first drops toward zero, then jumps up toward one (here at around u ≈ 10⁻¹), and finally, it drops toward zero again (here at around u ≈ 10⁻⁴). We call these phenomena the “valley” and the “hump”. The optimum mutation rate is around u ≈ 10⁻². We use β = 10.

Fig. 4.

Frequencies of reactive strategies. (A–D) For 4 distinct values of the mutation rate, u = 10⁻⁴, 10⁻³, 10⁻², 10⁻¹, we simulate evolutionary dynamics for at least 10⁹ steps. We collect the appearing strategies into a 25 × 25 grid and plot their relative abundance as a heat map (Top row, log scale) and as a bar chart (Bottom row, linear scale). In all 4 cases, the strategies with nonnegligible frequency have either q ≈ 0 (orange peak, BottomLeft) or p ≈ 1 (blue peak, Right). For intermediate mutation rate, u = 10⁻² (third column), the blue peak has more mass than the orange one, and we observe high overall cooperation rates. Parameters: N = 100, c/b = 0.5, β = 10.

Fig. 5.

Evolution of the reduced three strategy system 𝒮₃ = {ALLD, GTFT, ALLC}. (A) Each point in the simplex represents a composition of the population. (B–I) Relative frequencies of the population composition (red is high) as u decreases from B, u = 10⁰ = 1 to i, u = 10^−2.8. (B) For u = 1, the population typically consists of roughly 1/3 of each of ALLD, GTFT, and ALLC and the average cooperation rate is 5.1/9 ≐ 0.57 (Theorem 1 in SI Appendix, Note 2). (E–F) For u ∈ [10^−1.2, 10^−1.6], the population typically consists mostly of GTFT players and the cooperation rate is ≈0.9. (H and I) For u ≤ 10^−2.4, virtually all players play ALLD virtually all the time. (J) The effects of mutation and selection on the total mass. Parameters: N = 100, c = 0.7, β = 10. For GTFT we use (1, 0.1).