Evolution of cooperation under \(N\)-person snowdrift games. (English) Zbl 1402.91061

Summary: In the animal world, performing a given task which is beneficial to an entire group requires the cooperation of several individuals of that group who often share the workload required to perform the task. The mathematical framework to study the dynamics of collective action is game theory. Here we study the evolutionary dynamics of cooperators and defectors in a population in which groups of individuals engage in \(N\)-person, non-excludable public goods games. We explore an \(N\)-person generalization of the well-known two-person snowdrift game. We discuss both the case of infinite and finite populations, taking explicitly into consideration the possible existence of a threshold above which collective action is materialized. Whereas in infinite populations, an \(N\)-person snowdrift game (NSG) leads to a stable coexistence between cooperators and defectors, the introduction of a threshold leads to the appearance of a new interior fixed point associated with a coordination threshold. The fingerprints of the stable and unstable interior fixed points still affect the evolutionary dynamics in finite populations, despite evolution leading the population inexorably to a monomorphic end-state. However, when the group size and population size become comparable, we find that spite sets in, rendering cooperation unfeasible.


91A22 Evolutionary games
91A06 \(n\)-person games, \(n>2\)
91B18 Public goods
Full Text: DOI HAL


[1] Axelrod, R.; Hamilton, W.D., The evolution of cooperation, Science, 211, 1390-1396, (1981) · Zbl 1225.92037
[2] Bach, L.A.; Helvik, T.; Christiansen, F.B., The evolution of n-player cooperation—threshold games and ESS bifurcations, J. theor. biol., 238, 426-434, (2006) · Zbl 1446.91021
[3] Boehm, C., Hierarchy in the forest: the evolution of Egalitarian behavior, (1999), Harvard University Press Cambridge, MA
[4] Boyd, R.; Richerson, P.J., The evolution of reciprocity in sizable groups, J. theor. biol., 132, 2, 337-356, (1988)
[5] Boyd, R.; Richerson, P.J., Culture and the evolutionary process, (1985), University of Chicago Press Chicago, IL
[6] Dawes, R.M., Social dilemmas, Annual review of psychology, 31, 169-193, (1980)
[7] Hamilton, W., Selfish and spiteful behaviour in an evolutionary model, Nature, 228, 1218-1220, (1970)
[8] Hamilton, W., 1975. In: Biosocial anthropology, Malaby Press, London, pp. 133-155.; Hamilton, W., 1975. In: Biosocial anthropology, Malaby Press, London, pp. 133-155.
[9] ()
[10] Hauert, C.; Michor, F.; Nowak, M.; Doebeli, M., The tragedy of the commons, Science, 162, 5364, 1243-1248, (2006)
[11] Hauert, C.; Michor, F.; Nowak, M.; Doebeli, M., Synergy and discounting of cooperation in social dilemmas, J. theor. biol., 239, 195, (2006) · Zbl 1446.91017
[12] Hauert, C.; Traulsen, A.; Brandt, H.; Nowak, M.; Sigmund, K., Via freedom to coercion: the emergence of costly punishment, Science, 316, 1905-1907, (2007) · Zbl 1226.91010
[13] Hofbauer, J.; Sigmund, K., Evolutionary games and population dynamics, (1998), Cambridge University Press Cambridge, UK · Zbl 0914.90287
[14] Kandori, M.; Mailath, G.; Rob, R., Learning, mutation, and long-run equilibria in games, Econometrica, 61, 29-56, (1993) · Zbl 0776.90095
[15] Karlin, S.; Taylor, H., A first course in stochastic processes, (1975), Academic Press London · Zbl 0315.60016
[16] Kollock, P., Social dilemmas: the anatomy of cooperation, Annu. rev. sociol., 24, 183-214, (1998)
[17] Macy, M.; Flache, A., Learning dynamics in social dilemmas, Proc. nat. acad. sci. USA, 99, 7229-7236, (2002) · Zbl 1355.91014
[18] Maynard-Smith, J., Evolution and the theory of games, (1982), Cambridge University Press Cambridge · Zbl 0526.90102
[19] Nowak, M., Five rules for the evolution of cooperation, Science, 314, 1560-1563, (2006)
[20] Nowak, M.; Sigmund, K., Evolutionary dynamics of biological games, Science, 303, 793-799, (2004)
[21] Ohtsuki, H.; Hauert, C.; Lieberman, E.; Nowak, M., A simple rule for the evolution of cooperation on graphs and social networks, Nature, 441, 502-505, (2006)
[22] Pacheco, J.M.; Santos, F.C.; Souza, M.O.; Skyrms, B., Evolutionary dynamics of collective action in n-person stag-hunt dilemmas, Proc. R. soc. B, 276, 1655, 315-321, (2009)
[23] Santos, F.; Pacheco, J.M., Scale-free networks provide a unifying framework for the emergence of cooperation, Phys. rev. lett., 95, 098104, (2005)
[24] Santos, F.; Pacheco, J.M.; Lenaerts, T., Evolutionary dynamics of social dilemmas in structured heterogeneous populations, Proc. nat. acad. sci. USA, 103, 3490-3494, (2006)
[25] Santos, F.; Santos, M.; Pacheco, J.M., Social diversity promotes the emergence of cooperation in public goods games, Nature, 454, 213-216, (2008)
[26] Schelling, T.C., Hockey helmets, concealed weapons, and daylight saving: a study of binary choices with externalities, J. conflict resolution, 17, 381, (1973)
[27] Skyrms, B., 2001. The stag hunt. In: Proceedings and Addresses of the American Philosophical Association, vol. 75, pp. 31-41.; Skyrms, B., 2001. The stag hunt. In: Proceedings and Addresses of the American Philosophical Association, vol. 75, pp. 31-41.
[28] Skyrms, B., The stag hunt and the evolution of social structure, (2004), Cambridge University Press Cambridge
[29] Sugden, R., The economics of rights, co-operation and welfare, (1986), Basil Blackwell Oxford, UK
[30] Traulsen, A.; Nowak, M.; Pacheco, J.M., Stochastic dynamics of invasion and fixation, Phys. rev. E, 74, 011909, (2006)
[31] Traulsen, A.; Nowak, M.; Pacheco, J.M., Stochastic payoff evaluation increases the temperature of selection, J. theor. biol., 244, 349-356, (2007) · Zbl 1450.91006
[32] Traulsen, A.; Pacheco, J.M.; Nowak, M., Pairwise comparison and selection temperature in evolutionary game dynamics, J. theor. biol., 246, 522-529, (2007) · Zbl 1451.92268
[33] Weir, P., Witness, (1985), Paramount Pictures USA
[34] Young, H., The evolution of conventions, Econometrica, 61, 57-84, (1993) · Zbl 0773.90101
[35] Zheng, D.; Yin, H.; Chan, C.; Hui, P., Cooperation behavior in a model of evolutionary snowdrift games with n-person interactions, Europhys. lett., 80, 18002-18006, (2007)
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