A simple rule for the evolution of cooperation on graphs and social networks

Abstract

A fundamental aspect of all biological systems is cooperation. Cooperative interactions are required for many levels of biological organization ranging from single cells to groups of animals. Human society is based to a large extent on mechanisms that promote cooperation. It is well known that in unstructured populations, natural selection favours defectors over cooperators. There is much current interest, however, in studying evolutionary games in structured populations and on graphs. These efforts recognize the fact that who-meets-whom is not random, but determined by spatial relationships or social networks. Here we describe a surprisingly simple rule that is a good approximation for all graphs that we have analysed, including cycles, spatial lattices, random regular graphs, random graphs and scale-free networks: natural selection favours cooperation, if the benefit of the altruistic act, b, divided by the cost, c, exceeds the average number of neighbours, k, which means b/c > k. In this case, cooperation can evolve as a consequence of 'social viscosity' even in the absence of reputation effects or strategic complexity.

Fig. 1

The rules of the game. Each individual occupies the vertex of a graph and derives a payoff, P, from interactions with adjacent individuals. A cooperator (blue) pays a cost, c, for each neighbor to receive a benefit, b. A defector (red) pays no cost and provides no benefit. The fitness of a player is given by 1 − w + wP, where w measures the intensity of selection. Strong selection means w = 1. Weak selection means w ≪ 1. For ‘death-birth’ updating, at each time step, a random individual is chosen to die; subsequently the neighbors compete for the empty site proportional to their fitness. In this example, the central vertex will change from a defector to a cooperator with a probability F_C/(F_C +F_D), where the total fitness of all adjacent cooperators and defectors is F_C = 4(1 − w) + (10b − 16c)w and F_D = 4(1 − w) + 3bw, respectively.

Fig. 2

The simple rule, b/c > k, is in good agreement with numerical simulations. The parameter k denotes the degree of the graph, which is given by the (average) number of neighbors per individual. The first row illustrates the type of graph for (a) k = 2 and (b–e) k = 4. The second and third rows show simulation data for population sizes N = 100 and N = 500. The fixation probability, ρ, of cooperators is determined by the fraction of runs where cooperators reached fixation out of 10⁶ runs under weak selection, w = 0.01. Each type of graph is simulated for different (average) degrees ranging from k = 2 to k = 10. The arrows mark b/c = k. The dotted horizontal line indicates the fixation probability 1/N of neutral evolution. The data suggest that b/c > k is necessary but not sufficient. The discrepancy is larger for non-regular graphs (d,e) with high average degree (k = 10). This is not surprising given that the derivation of the rule is for regular graphs and in the limit N ≫ k. Note that the larger population size, N = 500, gives better agreement.

Fig. 3

Some intuition for games on graphs. (a) For ‘death-birth’ (DB) updating, we must consider a cooperator and a defector competing for an empty site. The pair-approximation calculation shows that for weak selection the cooperator has one more cooperator among its k − 1 other neighbors than the defector. Hence, the cooperator has a higher chance to win the empty site if b/c > k. (b) For ‘birth-death’ (BD) updating, we must consider a cooperator-defector pair competing for the next reproduction event. Again the cooperator has one more cooperator among its k − 1 other neighbors than the defector, but the focal cooperator is also a neighbor of the defector. Hence, both competitors are linked to the same number of cooperators, and therefore the defector has a higher payoff. For BD updating, selection does not favor cooperation. (c) On a cycle (k = 2), the situation is simple. A direct calculation, for weak selection and large population size, leads to the following results. For BD updating, the boundary between a cluster of cooperators and defectors tends to move in favor of defectors. For DB updating, the cooperator cluster expands if b/c > 2. For IM updating, the cooperator cluster expands if b/c > 4.