Social goods dilemmas in heterogeneous societies


Prosocial behaviours are encountered in the donation game, the prisoner’s dilemma, relaxed social dilemmas and public goods games. Many studies assume that the population structure is homogeneous, meaning that all individuals have the same number of interaction partners or that the social good is of one particular type. Here, we explore general evolutionary dynamics for arbitrary spatial structures and social goods. We find that heterogeneous networks, in which some individuals have many more interaction partners than others, can enhance the evolution of prosocial behaviours. However, they often accumulate most of the benefits in the hands of a few highly connected individuals, while many others receive low or negative payoff. Surprisingly, selection can favour producers of social goods even if the total costs exceed the total benefits. In summary, heterogeneous structures have the ability to strongly promote the emergence of prosocial behaviours, but they also create the possibility of generating large inequality.

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Fig. 1: Social goods and prosocial behaviours.
Fig. 2: Production and absorption of social goods.
Fig. 3: Four update rules driving evolutionary dynamics through imitation.
Fig. 4: Heterogeneous graphs allow for efficient evolution of prosocial behaviour.
Fig. 5: Evolution of prosocial behaviour can result in widespread inequality.
Fig. 6: Diffuse versus concentrated benefits.
Fig. 7: Effects of reciprocity on the selection of prosocial behaviours.
Fig. 8: Beyond weak selection on the star graph.

Data availability

This study has no associated data.

Code availability

Custom code that supports the findings of this study is available from the corresponding authors upon request.


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We thank J. Plotkin for constructive feedback and B. Fotouhi and C. Hilbe for helpful conversations. This work was supported by the Army Research Laboratory (grant W911NF-18-2-0265), the Bill & Melinda Gates Foundation (grant OPP1148627), the John Templeton Foundation (grant 61443), the National Science Foundation (grant DMS-1715315), and the Office of Naval Research (grant N00014-16-1-2914). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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A.M. designed the study and derived the initial results; A.M., B.A. and M.A.N. analysed the model; A.M., B.A. and M.A.N. wrote the main text; and A.M. and B.A. wrote the supplementary information.

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Correspondence to Alex McAvoy or Martin A. Nowak.

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Extended data

Extended Data Fig. 1 Evolution of producers of ff-goods with 0 < bc 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 < bc). 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.

Extended Data Fig. 2 Evolution of producers can be possible for ff-goods but not for pp-goods.

a, PC updating on Erdös-Rényi graphs of size N = 100 for various edge-inclusion probabilities, p. If p is sufficiently small, the critical benefit-to-cost ratio is positive for both pp- and ff-goods, but for slightly larger p values this ratio can be positive for ff-goods and negative for pp-goods. In the latter case, producers cannot evolve under any b/c ratio for pp-goods, but they can evolve for ff-goods as long as b/c is sufficiently large. b, PC updating on small-world networks with different rewiring probabilities, p. Again, there are many examples for which the critical benefit-to-cost ratio is positive for ff-goods but negative for pp-goods.

Extended Data Fig. 3 Distributions of critical ratios on small graphs.

There are 11,989,763 undirected, unweighted graphs of size at most N = 10. Of those that can support the evolution of prosocial behaviors, the critical benefit-to-cost ratios are given for PC updating, a, and DB updating, b.

Extended Data Fig. 4 Heterogeneous graphs allow efficient evolution of prosocial behaviour.

a, The distribution of the critical benefit-to-cost ratio under PC updating for 106 preferential-attachment graphs of size N = 100 (see Methods). For ff-goods, these structures often have critical benefit-to-cost ratios that are less than one. However, the critical ratio for pp-goods is always greater than one. b, When \(b/c={\left(b/c\right)}^{* }\), these graphs result in a majority (but not all) of the population being worse-off in the all-producer state than in the all-non-producer state.

Extended Data Fig. 5 Fixed costs on a dense cluster of stars.

Consider a population consisting of m stars, each of size n, connected by a complete graph at their hubs. Provided m > 1, this structure results in extremely low critical ratios under both PC and DB updating when n is large. Illustrated here is the case in which m = 5. This structure has the interesting property that ff-goods result in lower critical thresholds than pf-goods (both of which are lower than that of pp-goods, which is not depicted here). Qualitatively, the results are similar for both PC and DB updating, with the exception that producers can never be favored by selection on the star (m = 1) under DB updating but can be favored under PC updating. We derive explicit formulas for \({\left(b/c\right)}^{* }\) for any m and n in the SI.

Extended Data Fig. 6 Graphs of size N≤10 that most easily support prosocial behavior.

For PC and DB updating, we illustrate the 100 graphs with the lowest positive critical ratios for pp- and ff-goods. In each case, the graphs are colored according to their critical ratios. In these examples, ff-goods result in lower critical ratios than pp-goods, and DB updating tends to give lower ratios than PC updating.

Extended Data Fig. 7 Division of a society into two factions.

To illustrate the effects of the division of real-world interaction topologies on evolutionary dynamics, we consider Zachary’s karate club97, a, and the subsequent split of the karate club into two disjoint groups98, b and c. Under PC updating, producers can evolve on all three networks only in the case of ff-goods. Moreover, even for the two populations (a and b) in which both ff- and pf-goods can evolve, this split swaps the rankings of the two. In particular, the critical ratio for pf-goods is lower in a but that of ff-goods is lower in b. The threshold for all individuals to be better off in the all-A state than in the all-B state, \({\left(b/c\right)}_{* }\), is lowered by the split.

Extended Data Fig. 8 Summary of main examples.

A good is wealth-producing (w) if the total payoff (sum of all benefits minus sum of all costs) is positive when everyone in the population is a producer. It is harmful (h) if at least one individual has a negative payoff in the all-producer state. For three kinds of social goods (pp, ff, and pf) and update rules (PC, DB, and IM), this table summarizes when a good can be wealth-producing and/or harmful, as well as when such a good can evolve. Notably, these results are not influenced much by the choice of update rule.

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.

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McAvoy, A., Allen, B. & Nowak, M.A. Social goods dilemmas in heterogeneous societies. Nat Hum Behav (2020).

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