Extended Data Fig. 1: Evolution of producers of ff-goods with 0 < b≤c on the star (PC updating).
The star may be viewed as a special case of the rich club, in which there is just a single ‘rich’ individual (m = 1). a, invasion and fixation of a mutant producer arising in a leaf under PC updating. This producer has a payoff of − c, and the non-producer at the hub gets b. Through drift, this producer can take the hub and propagate a small portion of producers to the leaves. Once there are \(k>c/b+1/\left(N-1\right)\) producers at the periphery, a central producer’s payoff exceeds that of everyone else in the population and selection favors the further spread of producers. b, invasion and fixation of a mutant non-producer arising in a leaf. As soon as a non-producer captures the hub, selection favors the proliferation of non-producers. However, when there is just a single non-producer in the population, a producer at the hub has a much greater payoff than everyone else in the population (even when 0 < b≤c). Thus, relative to the initial invasion of a producer in a, selection acts much more strongly against the initial invasion of a non-producer in b. For any fixed b, c > 0, these effects become strong enough as N grows that we find ρA > ρB.
