Extended Data Fig. 9: Contributions by producers to a public pool can ameliorate payoff inequality.

For ff-goods, suppose that each producer (blue) donates θb to a pool (green) and \(\left(1-\theta \right)b\) to neighbors, a. If the total value of the public pool is divided among all members of the population (green arrows, b), then the situation can improve for those who are worst-off in the all-producer state. In particular, such a pool can result in a positive payoff to everyone in the population provided the contribution, quantified by θ, is sufficiently large. The trade-off is that this pool also increases the critical benefit-to-cost ratio required for producers to evolve by a multiplicative factor of \(1/\left(1-\theta \right)\) (see SI), illustrated in c on a star of size N = 100 under PC updating. For this population structure, d depicts the payoff of the poorest individual (‘leaf’ player, at the periphery of the star) in the prosocial (all-A) state as a function of the fraction contributed to the pool, θ, when b = 2 and c = 1. This payoff is negative when θ ≾ 1/2, which means that 99% of the population is better off in the asocial (all-B) state. However, when θ ≿ 1/2, all individuals are better off when producers proliferate.