Chromodynamics of cooperation in finite populations

Abstract

Background: The basic idea of tag-based models for cooperation is that individuals recognize each other via arbitrary signals, so-called tags. If there are tags of different colors, then cooperators can always establish new signals of recognition. The resulting "chromodynamics" is a mechanism for the evolution of cooperation. Cooperators use a secret tag until they are discovered by defectors who then destroy cooperation based on this tag. Subsequently, a fraction of the population manages to establish cooperation based on a new tag.

Methodology/principal findings: We derive a mathematical description of stochastic evolutionary dynamics of tag-based cooperation in populations of finite size. Benefit and cost of cooperation are given by b and c. We find that cooperators are more abundant than defectors if b/c > 1+2u/v, where u is the mutation rate changing only the strategy and v is the mutation rate changing strategy and tag. We study specific assumptions for u and v in two genetic models and one cultural model.

Conclusions/significance: In a genetic model, tag-based cooperation only evolves if a gene encodes both strategy and tag. In a cultural model with equal mutation rates between all possible phenotypes (tags and behaviors), the crucial condition is b/c > (K+1)/(K-1), where K is the number of tags. A larger number of tags requires a smaller benefit-to-cost ratio. In the limit of many different tags, the condition for cooperators to have a higher average abundance than defectors becomes b > c.

Figure 1

Payoff matrix for chromodynamics of cooperation. Interactions that lead to nonzero payoffs only occur between individuals using the same tag. For a given tag, defectors always dominate cooperators. By continuously changing the ’secret handshake’ ( = tag), cooperators can run away from defectors. For a cultural model, it turns out if b/c>(K+1)/(K−1), then cooperators can run faster than defectors.

Figure 2

Evolutionary chromodynamics in finite populations. The red cooperator population is invaded by red defectors at t≈1200. At t≈4500, cooperation is established based on blue tags. Blue defectors invade at t≈6000. The time unit is given by one individual learning event (pairwise comparison). For example, after t = 5000 each individual had 100 learning events on average. The following parameters are used: population size N = 50, intensity of selection β = 1.0, cost of cooperation c = 0.5, benefit from cooperation b = 1.0, mutation rate u = 0.01.

Figure 3

Consider a pleiotropic gene that encodes both strategy and tag. The first n bits encode the strategy according to a parity rule: if the sum of the first n bits is even, the strategy is cooperation, otherwise it is defection. The last m bits encode the tag. Each sequence encodes a different tag. Hence there are 2^m possible tags. There is an overlapping region of L bits which affect both the strategy and the tag. This setup allows evolution of tag based cooperation if b/c>(2n−L)/L is fulfilled.

Figure 4

In a model with 2K phenotypes consisting of a pair of strategy and tag, cooperation evolves depending on the benefit to cost ratio. For small mutation rates, the critical benefit to cost ratio for evolution of cooperation is given by b/c>(K+1)/(K−1) (red line). If the benefit to cost ratio exceeds this critical value, then cooperators are more abundant than defectors averaged over time. With increasing mutation rates, the populations become more mixed which favors defectors. Hence, the critical benefit to cost ratio increases with a higher mutation rate, as shown for u = 0.01 and u = 0.001. In all cases, the critical benefit to cost ratio decreases with the number of tags K and converges to 1 for β→∞. The following parameters are used: population size N = 100, intensity of selection β = 0.1, cost of cooperation c = 0.2, averages over 10⁸ time steps.