Public goods games on any population structure

Understanding the emergence of cooperation in social networks has advanced through pairwise interactions, but the corresponding theory for group-based public goods games (PGGs) remains less explored. Here, we provide theoretical conditions under which cooperation thrives in PGGs on arbitrary population structures, which are accurate under weak selection. We find that a class of networks that would otherwise fail to produce cooperation, such as star graphs, are particularly conducive to cooperation in PGGs. More generally, PGGs can support cooperation on almost all networks, which is robust across all kinds of model details. This fundamental advantage of PGGs derives from self-reciprocity realized by group separations and from clustering through second-order interactions. We also apply PGGs to empirical networks, which shows that PGGs could be a promising interaction mode for the emergence of cooperation in real-world systems.

💡 Research Summary

This paper develops a general analytical framework for the evolution of cooperation in public‑goods games (PGGs) on arbitrary population structures, extending the well‑established theory of pairwise games on graphs to multiplayer interactions. The authors assume weak selection (δ ≪ 1), meaning that game payoffs only slightly perturb baseline fitness, and they model each individual i as organizing a group of size G_i = k_i + 1 that consists of itself and all its neighbors on an unweighted network. Within each group a standard PGG is played: cooperators contribute a cost c, the total contribution is multiplied by a synergy factor r > 1, and the resulting public good is divided equally among all G_i participants. The immediate payoffs are π_C = r·g_C·c/G_i − c for cooperators and π_D = r·g_C·c/G_i for defectors, where g_C is the number of cooperators in the group.

Each individual participates in G_i games (its own and those organized by its neighbors) and averages the resulting payoffs to obtain an actual payoff f_i. Fitness is then defined as F_i = exp(δ f_i). Strategy updating follows standard evolutionary rules; the main text focuses on the pairwise comparison (PC) rule, while death‑birth (DB) and birth‑death (BD) are treated in the supplementary material. In the PC rule a focal individual i randomly selects a neighbor j and adopts j’s strategy with probability W_{i←j}=1/(1+exp