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 1-4 . Human society is based to a large extent on mechanisms that promote cooperation 5-7 . It is well known that in unstructured populations, natural selection favors defectors over cooperators. There is much current interest, however, for studying evolutionary games in structured populations and on graphs 8-17 . These efforts recognize the fact that who-meets-whom is not random, but determined by spatial relationships or social networks 18-24 . Here we describe a surprisingly simple rule, which is a good approximation for all graphs that we have analyzed, including cycles, spatial lattices, random regular graphs, random graphs and scale-free networks 25,26 : natural selection favors cooperation, if the benefit of the altruistic act, b, divided by the cost, c, exceeds the average number of neighbors, k. Therefore, cooperation can evolve as a consequence of 'social viscosity' even in the absence of reputation effects or strategic complexity.A cooperator is someone who pays a cost, c, for another individual to receive a benefit, b. A defector pays no cost and does not distribute any benefits. In evolutionary biology, cost and benefit are measured in terms of fitness. Reproduction can be genetic or cultural. In the latter case, the strategy of someone who does well is imitated by others. In an unstructured population, where all individuals interact equally likely with each other, defectors have a higher average payoff than unconditional cooperators. Therefore, natural selection increases the relative abundance of defectors and drives cooperators to extinction. These evolutionary dynamics hold for the deterministic setting of the replicator equation 27,28 and for stochastic game dynamics of finite populations 29 .In our model, the players of an evolutionary game occupy the vertices of a graph. The edges denote links between individuals in terms of game dynamical interaction and biological reproduction. We assume that the graph is fixed for the duration of the evolutionary dynamics. Consider a population of N individuals consisting of cooperators and defectors. A cooperator helps all individuals to whom it is connected. If a cooperator is connected to k other individuals and i of those are cooperators, then its payoff is bi -ck. A defector does not provide any help, and therefore has no costs, but it can receive the benefit from neighboring cooperators. If a defector is connected to j cooperators, then its payoff is bj.The fitness of an individual is given by a constant term, denoting the baseline fitness, plus the payoff that arises from the game. Strong selection means that the payoff is large compared to the baseline fitness; weak selection means the payoff is small compared to the baseline fitness. The idea behind weak selection is that many different factors contribute to the overall fitness of an individual, and th...

A symmetry of fixation times in evoultionary dynamics

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Network structure and the biology of populations

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Emergence and resilience of cooperation in the spatial prisoner's dilemma via a reward mechanism

“…This result, in turn, leaves open the question of the emergence of cooperation in a population with a majority of defectors. Recently, it has been shown (Ohtsuki et al, 2006) that, if the average number of connections in the interaction network is k, the condition b/c > k implies that selection favors cooperators invading defectors in the weak selection limit, i.e., when the contribution of the game to the fitness of the individual is very small. However, a general result valid for any intensity of the selection is still lacking.…”