Social goods dilemmas in heterogeneous societies

Abstract

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.

Access options

from\$8.99

All prices are NET prices.

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.

References

1. 1.

Sigmund, K. The Calculus of Selfishness (Princeton Univ. Press, 2010).

2. 2.

Radzvilavicius, A. L., Stewart, A. J. & Plotkin, J. B. Evolution of empathetic moral evaluation. eLife 8, e44269 (2019).

3. 3.

Axelrod, R. The Evolution of Cooperation (Basic Books, 1984).

4. 4.

Szabó, G. & Töke, C. Evolutionary prisoner’s dilemma game on a square lattice. Phys. Rev. E 58, 69–73 (1998).

5. 5.

Abramson, G. & Kuperman, M. Social games in a social network. Phys. Rev. E 63, 030901 (2001).

6. 6.

Broom, M. & Rychtár, J. Game-Theoretical Models in Biology (Taylor & Francis, 2013).

7. 7.

Maynard Smith, J. Evolution and the Theory of Games (Cambridge Univ. Press, 1982).

8. 8.

Hauert, C. & Doebeli, M. Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428, 643–646 (2004).

9. 9.

Doebeli, M., Hauert, C. & Killingback, T. The evolutionary origin of cooperators and defectors. Science 306, 859–862 (2004).

10. 10.

Lloyd, W. F. Two Lectures on the Checks to Population (Oxford Univ. Press, 1833).

11. 11.

Hardin, G. The tragedy of the commons. Science 162, 1243–1248 (1968).

12. 12.

Szabó, G. & Hauert, C. Phase transitions and volunteering in spatial public goods games. Phys. Rev. Lett. 89, 118101 (2002).

13. 13.

Pinheiro, F. L., Vasconcelos, V. V., Santos, F. C. & Pacheco, J. M. Evolution of all-or-none strategies in repeated public goods dilemmas. PLoS Comput. Biol. 10, e1003945 (2014).

14. 14.

Peña, J., Wu, B., Arranz, J. & Traulsen, A. Evolutionary games of multiplayer cooperation on graphs. PLoS Comput. Biol. 12, e1005059 (2016).

15. 15.

Zhong, L.-X. et al. A generalized public goods game with coupling of individual ability and project benefit. Chaos Solitons Fract. 101, 73–80 (2017).

16. 16.

Nowak, M. A. & May, R. M. Evolutionary games and spatial chaos. Nature 359, 826–829 (1992).

17. 17.

Nakamaru, M., Matsuda, H. & Iwasa, Y. The evolution of cooperation in a lattice-structured population. J. Theor. Biol. 184, 65–81 (1997).

18. 18.

Lieberman, E., Hauert, C. & Nowak, M. A. Evolutionary dynamics on graphs. Nature 433, 312–316 (2005).

19. 19.

Ohtsuki, H., Hauert, C., Lieberman, E. & Nowak, M. A. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505 (2006).

20. 20.

Taylor, P. D., Day, T. & Wild, G. Evolution of cooperation in a finite homogeneous graph. Nature 447, 469–472 (2007).

21. 21.

Chen, Y.-T. Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs. Ann. Appl. Probab. 23, 637–664 (2013).

22. 22.

Débarre, F., Hauert, C. & Doebeli, M. Social evolution in structured populations. Nat. Commun. 5, 3409 (2014).

23. 23.

Santos, F. C. & Pacheco, J. M. Scale-free networks provide a unifying framework for the emergence of cooperation. Phys. Rev. Lett. 95, 098104 (2005).

24. 24.

Antal, T., Redner, S. & Sood, V. Evolutionary dynamics on degree-heterogeneous graphs. Phys. Rev. Lett. 96, 188104 (2006).

25. 25.

Gómez-Gardenes, J., Campillo, M., Floría, L. M. & Moreno, Y. Dynamical organization of cooperation in complex topologies. Phys. Rev. Lett. 98, 108103 (2007).

26. 26.

Sood, V., Antal, T. & Redner, S. Voter models on heterogeneous networks. Phys. Rev. E 77, 041121 (2008).

27. 27.

Cao, X.-B., Du, W.-B. & Rong, Z.-H. The evolutionary public goods game on scale-free networks with heterogeneous investment. Physica A 389, 1273–1280 (2010).

28. 28.

Maciejewski, W., Fu, F. & Hauert, C. Evolutionary game dynamics in populations with heterogenous structures. PLoS Comput. Biol. 10, e1003567 (2014).

29. 29.

Fan, R., Zhang, Y., Luo, M. & Zhang, H. Promotion of cooperation induced by heterogeneity of both investment and payoff allocation in spatial public goods game. Physica A 465, 454–463 (2017).

30. 30.

Allen, B. et al. Evolutionary dynamics on any population structure. Nature 544, 227–230 (2017).

31. 31.

Goldfarb, A. & Tucker, C. Digital economics. J. Econ. Lit. 57, 3–43 (2019).

32. 32.

Santos, F. C., Santos, M. D. & Pacheco, J. M. Social diversity promotes the emergence of cooperation in public goods games. Nature 454, 213–216 (2008).

33. 33.

Li, C., Zhang, B., Cressman, R. & Tao, Y. Evolution of cooperation in a heterogeneous graph: fixation probabilities under weak selection. PLoS ONE 8, e66560 (2013).

34. 34.

Stutzer, A., Goette, L. & Zehnder, M. Active decisions and prosocial behaviour: a field experiment on blood donation. Econ. J. 121, F476–F493 (2011).

35. 35.

Dunbar, R. I. M. Functional significance of social grooming in primates. Folia Primatol. 57, 121–131 (1991).

36. 36.

Horn, L., Scheer, C., Bugnyar, T. & Massen, J. J. M. Proactive prosociality in a cooperatively breeding corvid, the azure-winged magpie (Cyanopica cyana). Biol. Lett. 12, 20160649 (2016).

37. 37.

Wilkinson, G. S. Reciprocal food sharing in the vampire bat. Nature 308, 181–184 (1984).

38. 38.

Fudenberg, D. & Imhof, L. A. Imitation processes with small mutations. J. Econ. Theory 131, 251–262 (2006).

39. 39.

Fisher, R. A. The Genetical Theory of Natural Selection (Clarendon Press, 1930).

40. 40.

Maciejewski, W. Reproductive value in graph-structured populations. J. Theor. Biol. 340, 285–293 (2014).

41. 41.

Wild, G. & Traulsen, A. The different limits of weak selection and the evolutionary dynamics of finite populations. J. Theor. Biol. 247, 382–390 (2007).

42. 42.

Fu, F., Wang, L., Nowak, M. A. & Hauert, C. Evolutionary dynamics on graphs: efficient method for weak selection. Phys. Rev. E 79, 046707 (2009).

43. 43.

Wu, B., Altrock, P. M., Wang, L. & Traulsen, A. Universality of weak selection. Phys. Rev. E 82, 046106 (2010).

44. 44.

Wu, B., García, J., Hauert, C. & Traulsen, A. Extrapolating weak selection in evolutionary games. PLoS Comput. Biol. 9, e1003381 (2013).

45. 45.

Mullon, C. & Lehmann, L. The robustness of the weak selection approximation for the evolution of altruism against strong selection. J. Evol. Biol. 27, 2272–2282 (2014).

46. 46.

Iwasa, Y., Nakamaru, M. & Levin, S. A. Allelopathy of bacteria in a lattice population: competition between colicin-sensitive and colicin-producing strains. Evol. Ecol. 12, 785–802 (1998).

47. 47.

Forber, P. & Smead, R. The evolution of fairness through spite. Proc. R. Soc. B 281, 20132439 (2014).

48. 48.

Fudenberg, D. & Imhof, L. A. Monotone imitation dynamics in large populations. J. Econ. Theory 140, 229–245 (2008).

49. 49.

Roca, C. P., Cuesta, J. A. & Sánchez, A. Evolutionary game theory: temporal and spatial effects beyond replicator dynamics. Phys. Life Rev. 6, 208–249 (2009).

50. 50.

Traulsen, T., Semmann, D., Sommerfeld, R. D., Krambeck, H.-J. & Milinski, M. Human strategy updating in evolutionary games. Proc. Natl Acad. Sci. USA 107, 2962–2966 (2010).

51. 51.

Zhou, S. & Mondragon, R. J. The rich-club phenomenon in the internet topology. IEEE Commun. Lett. 8, 180–182 (2004).

52. 52.

Colizza, V., Flammini, A., Serrano, M. A. & Vespignani, A. Detecting rich-club ordering in complex networks. Nat. Phys. 2, 110–115 (2006).

53. 53.

McAuley, J. J., da Fontoura Costa, L. & Caetano, T. S. Rich-club phenomenon across complex network hierarchies. Appl. Phys. Lett. 91, 084103 (2007).

54. 54.

Fotouhi, B., Momeni, N., Allen, B. & Nowak, M. A. Conjoining uncooperative societies facilitates evolution of cooperation. Nat. Hum. Behav. 2, 492–499 (2018).

55. 55.

Ansell, C., Bichir, R. & Zhou, S. Who says networks, says oligarchy? Oligarchies as "rich club” networks. Connections 35, 20–32 (2016).

56. 56.

Dong, Y. et al. Inferring social status and rich club effects in enterprise communication networks. PLoS ONE 10, e0119446 (2015).

57. 57.

Vaquero, L. M. & Cebrian, M. The rich club phenomenon in the classroom. Sci. Rep. 3, 1174 (2013).

58. 58.

Ma, A., Mondragón, R. J. & Latora, V. Anatomy of funded research in science. Proc. Natl Acad. Sci. USA 112, 14760–14765 (2015).

59. 59.

Szell, M. & Sinatra, R. Research funding goes to rich clubs. Proc. Natl Acad. Sci. USA 112, 14749–14750 (2015).

60. 60.

Jiang, Z.-Q. & Zhou, W.-X. Statistical significance of the rich-club phenomenon in complex networks. New J. Phys. 10, 043002 (2008).

61. 61.

Newman, M. E. J. Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 74, 036104 (2006).

62. 62.

Laurance, W. F. Second thoughts on who goes where in author lists. Nature 442, 26 (2006).

63. 63.

Venkatraman, V. Conventions of scientific authorship. Science https://doi.org/10.1126/science.caredit.a1000039 (16 April 2010).

64. 64.

Bošnjak, L. & Marušić, A. Prescribed practices of authorship: review of codes of ethics from professional bodies and journal guidelines across disciplines. Scientometrics 93, 751–763 (2012).

65. 65.

McNutt, M. K. et al. Transparency in authors’ contributions and responsibilities to promote integrity in scientific publication. Proc. Natl Acad. Sci. USA 115, 2557–2560 (2018).

66. 66.

Trivers, R. L. The evolution of reciprocal altruism. Q. Rev. Biol. 46, 35–57 (1971).

67. 67.

Nowak, M. A. Five rules for the evolution of cooperation. Science 314, 1560–1563 (2006).

68. 68.

Press, W. H. & Dyson, F. J. Iterated prisoner’s dilemma contains strategies that dominate any evolutionary opponent. Proc. Natl Acad. Sci. USA 109, 10409–10413 (2012).

69. 69.

Stewart, A. J. & Plotkin, J. B. From extortion to generosity, evolution in the Iterated Prisoner’s Dilemma. Proc. Natl Acad. Sci. USA 110, 15348–15353 (2013).

70. 70.

Stewart, A. J. & Plotkin, J. B. Small groups and long memories promote cooperation. Sci. Rep. 6, 26889 (2016).

71. 71.

Hilbe, C., Chatterjee, K. & Nowak, M. A. Partners and rivals in direct reciprocity. Nat. Hum. Behav. 2, 469–477 (2018).

72. 72.

Ibsen-Jensen, R., Chatterjee, K. & Nowak, M. A. Computational complexity of ecological and evolutionary spatial dynamics. Proc. Natl Acad. Sci. USA 112, 201511366 (2015).

73. 73.

Hadjichrysathou, C., Broom, M. & Rychtár, J. Evolutionary games on star graphs under various updating rules. Dyn. Games Appl. 1, 386–407 (2011).

74. 74.

Taylor, P. D. Altruism in viscous populations—an inclusive fitness model. Evol. Ecol. 6, 352–356 (1992).

75. 75.

Wilson, D. S., Pollock, G. B. & Dugatkin, L. A. Can altruism evolve in purely viscous populations? Evol. Ecol. 6, 331–341 (1992).

76. 76.

Mitteldorf, J. & Wilson, D. S. Population viscosity and the evolution of altruism. J. Theor. Biol. 204, 481–496 (2000).

77. 77.

Irwin, A. J. & Taylor, P. D. Evolution of altruism in stepping-stone populations with overlapping generations. Theor. Popul. Biol. 60, 315–325 (2001).

78. 78.

Grafen, A. An inclusive fitness analysis of altruism on a cyclical network. J. Evol. Biol. 20, 2278–2283 (2007).

79. 79.

Lion, S. & van Baalen, M. Self-structuring in spatial evolutionary ecology. Ecol. Lett. 11, 277–295 (2008).

80. 80.

Tarnita, C. E., Antal, T., Ohtsuki, H. & Nowak, M. A. Evolutionary dynamics in set structured populations. Proc Natl Acad. Sci. USA 106, 8601–8604 (2009).

81. 81.

Nowak, M. A., Tarnita, C. E. & Antal, T. Evolutionary dynamics in structured populations. Phil. Trans. R. Soc. B 365, 19–30 (2009).

82. 82.

Rand, D. G., Nowak, M. A., Fowler, J. H. & Christakis, N. A. Static network structure can stabilize human cooperation. Proc. Natl Acad. Sci. USA 111, 17093–17098 (2014).

83. 83.

Débarre, F. Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations. J. Theor. Biol. 420, 26–35 (2017).

84. 84.

Su, Q., Zhou, L. & Wang, L. Evolutionary multiplayer games on graphs with edge diversity. PLoS Comput. Biol. 15, e1006947 (2019).

85. 85.

Su, Q., Li, A., Wang, L. & Stanley, H. E. Spatial reciprocity in the evolution of cooperation. Proc. R. Soc. B 286, 20190041 (2019).

86. 86.

Allen, B., Lippner, G. & Nowak, M. A. Evolutionary games on isothermal graphs. Nat. Commun. 10, 5107 (2019).

87. 87.

Freeman, J. The tyranny of structurelessness. Women’s Stud. Q. 41, 231–246 (2013).

88. 88.

Zhang, B., Li, C., De Silva, H., Bednarik, P. & Sigmund, K. The evolution of sanctioning institutions: an experimental approach to the social contract. Exp. Econ. 17, 285–303 (2014).

89. 89.

Allen, B. & Tarnita, C. E. Measures of success in a class of evolutionary models with fixed population size and structure. J. Math. Biol. 68, 109–143 (2014).

90. 90.

Allen, B. & McAvoy, A. A mathematical formalism for natural selection with arbitrary spatial and genetic structure. J. Math. Biol. 78, 1147–1210 (2019).

91. 91.

Su, Q., McAvoy, A., Wang, L. & Nowak, M. A. Evolutionary dynamics with game transitions. Proc. Natl Acad. Sci. USA 116, 25398–25404 (2019).

92. 92.

Barabási, A.-L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999).

93. 93.

Bollobás, B. Random Graphs (Cambridge Univ. Press, 2001).

94. 94.

Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

95. 95.

Leskovec, J. & Krevl, A. Stanford Large Network Dataset Collection (Stanford Network Analysis Project, 2014); http://snap.stanford.edu/data

96. 96.

Rossi, R. A. & Ahmed, N. K. The network data repository with interactive graph analytics and visualization. in Proc. 29th AAAI Conference on Artificial Intelligence http://networkrepository.com (Association for the Advancement of Artificial Intelligence, 2015).

97. 97.

Zachary, W. W. An information flow model for conflict and fission in small groups. J. Anthropol. Res. 33, 452–473 (1977).

98. 98.

Girvan, M. & Newman, M. E. J. Community structure in social and biological networks. Proc. Natl Acad. Sci. USA 99, 7821–7826 (2002).

Acknowledgements

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.

Author information

Authors

Contributions

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.

Corresponding authors

Correspondence to Alex McAvoy or Martin A. Nowak.

Ethics declarations

Competing interests

The authors declare no competing interests.

Editor recognition statement Primary Handling Editor: Charlotte Payne.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

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 < 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.

Supplementary information

Supplementary Information

Mathematical supplement to the main text.

Rights and permissions

Reprints and Permissions