Public goods games on any population structure

TL;DR

This work develops a comprehensive analytical framework for the evolution of cooperation in multiplayer public goods games (PGGs) on arbitrary population structures, accurate under weak selection. By mapping groups formed with neighbors and applying coalescent-time based metrics ($\uptau_{ij}$, $\Upsilon_{ij}$ and their variants) across update rules (PC, DB, BD) and payoff schemes (average vs accumulated), the authors derive explicit threshold conditions $r^\star$ for the emergence of cooperation. They demonstrate that PGGs robustly promote cooperation on a wide range of networks, including star graphs and various random and empirical networks, highlighting a fundamental advantage over pairwise donation games. The results are validated by extensive simulations and extended to extensions with accumulated payoff and to the donation game, offering a unified, network-structure-dependent theory for group interactions in realistic systems with potential applications to social, ecological, and biological networks.

Abstract

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.

Figures (11)

• Figure 1: The public goods game (PGG) on a general network. (A) Each agent organizes a PGG in the group of itself and its neighbors. (B) In each PGG, payoffs for cooperation and defection are calculated by Eq. (\ref{['eq_singlepgg']}). Obviously, $\uppi_D>\uppi_C$, defectors always have higher payoffs than cooperators, seemingly driving cooperation towards defection. (C) Since each agent organizes a PGG, an agent plays not only the PGG organized by itself, but also the PGGs organized by its neighbors. We average the payoffs from these PGGs as an agent's actual payoff. (D) The emergence of spatial reciprocity. For the presented two agents, their actual payoffs $f_C>f_D$ when $r>360/89$, driving defection towards cooperation.
• Figure 2: Cooperation conditions for PGGs on homogeneous and heterogeneous networks. (A) Star graphs support cooperation in PGGs as simple as when $r>4$, and the essence of which is $2\times 2$: there are 2 groups for a leaf, with 2 players in its own group. When using accumulated payoffs, connecting the hubs of two or more star graphs leads to $r^\star_\text{accu}=1$ under the DB rule, which maximally supports cooperation. (B) Comparison with other networks,including square lattices with von Neumann (left) and Moore (right) neighborhoods, joint stars with any number of hubs, and ceiling fans. The $r^\star$ values reported here are for infinite population size; see \ref{['sec_appl']} for finite size. (C) Agent-based simulations confirm the theoretical predictions on $L=5$ square lattice with von Neumann neighborhood, $n=9$ star, $m=3$ & $n=9$ joint stars, and $n=9$ ceiling fans. The dots represent the average cooperation fraction in the steady states (Methods), while the dashed lines are theoretical $r^\star$ over which $\uprho_C>1/N$.
• Figure 3: Effects of local structures on cooperation in PGGs. (A) Critical synergy factor $r^\star$ as a function of connecting probability $p$ on random graphs (ER) erdos1959random. The increasing average degree inhibits cooperation. (B) $r^\star$ as a function of rewiring probability $p$ on small-world networks (WS) watts1998collective with $d=2$. The increasing clustering coefficient promotes cooperation. (C) $r^\star$ as a function of generated degree distribution heterogeneity $\upgamma$ on scale-free networks (BA) barabasi1999emergencekrapivsky2000connectivity with $m=2$. The increasing degree heterogeneity initially promotes but ultimately inhibits cooperation. In A--C, each point on dashed lines is the average result of 1,000 randomly generated networks (totaling 58,000 in A, 101,000 in B, 101,000 in C). All networks are of size $N=100$ and are connected. (D) Agent-based simulations on selected sample networks confirm theoretical predictions. For ER, $p=4/99$; for WS, $d=2$, $p=0.5$; for BA, $m=2$, $\upgamma=2$. (E) Visualization of the sample networks in agent-based simulations.
• Figure 4: Cooperation emerges consistently in PGGs on almost all networks under different update rules, outperforming pairwise DGs. (A) Among all 12,111 networks of sizes $3\leq N\leq 8$, the fraction of networks classified by their critical synergy factors. Almost all networks have $0<r^\star \leq 30$ in supporting cooperation. The symbol * means that the only structure that does not support cooperation is the fully connected network. In contrast, for pairwise DGs, cooperation is only possible under DB update, with more than half of networks not supporting cooperation. (B) The ranks of all 11,117 networks of size $N=8$ in supporting cooperation for PGGs and DGs under PC and DB update rules. The star graph ranks the top 0.97% (PC) and 7.11% (DB). The results in PGGs are consistent under different update rules, while in pairwise DGs they are quite different. (C) The best and worst networks of size $N=8$ for cooperation in PGGs. The best networks differ with update rules. The worst two networks are consistent under all update rules, which are the fully connected (right) and a similar network (left).
• Figure 5: PGGs could be a promising interaction mode for cooperation on empirical networks. We analyze PGGs on four empirical networks, where the cooperation conditions are consistently relaxed across various model details in PGGs. In contrast, the conditions for cooperation are strict and inconsistent across different model details in pairwise DGs. The examined empirical systems include: (A) 16 tribes of the Gahuku--Gama alliance structure of the Eastern Central Highlands of New Guinea nrread1954cultures. (B) 29 seventh-grade students in Victoria, Australia, connected by whom they would prefer to work with vickers1981representing. (C) The networks of trophic interactions that occur among the 69 major taxonomic groups of everglades habitats in South Florida ecosystems nrulanowicz1998networkmelian2004food. (D) The network of body contact interactions among 17 North American barn swallows nrlevin2016stress.