×

zbMATH — the first resource for mathematics

Assortment and the evolution of cooperation in a Moran process with exponential fitness. (English) Zbl 1405.91035
Summary: We study the evolution of cooperation in a finite population interacting according to a simple model of like-with-like assortment. Evolution proceeds as a Moran process, and payoffs from the underlying cooperator-defector game are translated to positive fitnesses by an exponential transformation. These evolutionary dynamics can arise, for example, in a nest-structured population with rare migration. The use of the exponential transformation, rather than the usual linear one, is appropriate when interactions have multiplicative fitness effects, and allows for a tractable characterisation of the effect of assortment on the evolution of cooperation. We define two senses in which a greater degree of assortment can favour the evolution of cooperation, the first stronger than the second: (i) greater assortment increases, at all population states, the probability that the number of cooperators increases, relative to the probability that the number of defectors increases; and (ii) greater assortment increases the fixation probability of cooperation, relative to that of defection. We show that, by the stronger definition, greater assortment favours the evolution of cooperation for a subset of cooperative dilemmas: prisoners’ dilemmas, snowdrift games, stag-hunt games, and some prisoners’ delight games. For other cooperative dilemmas, greater assortment favours cooperation by the weak definition, but not by the strong definition. We also show that increasing assortment expands the set of games in which cooperation dominates the evolutionary dynamics. Our results hold for any strength of selection.

MSC:
91A22 Evolutionary games
91A12 Cooperative games
92D15 Problems related to evolution
92D25 Population dynamics (general)
91A05 2-person games
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alger, I.; Weibull, J. W., Homo moralis-preference evolution under incomplete information and assortative matching, Econometrica, 81, 6, 2269-2302, (2013) · Zbl 1287.91041
[2] Allen, B.; Nowak, M. A., Games on graphs, EMS Surv. Math. Sci., 1, 1, 113-151, (2014) · Zbl 1303.91040
[3] Allen, B.; Nowak, M. A., Games among relatives revisited, J. Theor. Biol., 378, 103-116, (2015) · Zbl 1343.91004
[4] Allen, B.; Tarnita, C. E., Measures of success in a class of evolutionary models with fixed population size and structure, J. Math. Biol., 68, 1-2, 109-143, (2014) · Zbl 1280.92045
[5] Allen, B.; Nowak, M. A.; Dieckmann, U., Adaptive dynamics with interaction structure, Am. Nat., 181, 6, E139-E163, (2013)
[6] Archetti, M.; Scheuring, I., Reviewgame theory of public goods in one-shot social dilemmas without assortment, J. Theor. Biol., 299, 9-20, (2012) · Zbl 1337.91041
[7] Axelrod, R., The evolution of cooperation, (1984), Basic Books York
[8] Bergstrom, T. C., The algebra of assortative encounters and the evolution of cooperation, Int. Game Theory Rev., 5, 03, 211-228, (2003) · Zbl 1051.91076
[9] Bijma, P.; Aanen, D., Assortment, Hamilton’s rule and multilevel selection, Proc. R. Soc. Lond. B: Biol. Sci., 277, 1682, 673-675, (2010)
[10] Blume, L. E., The statistical mechanics of strategic interaction, Games Econ. Behav., 5, 3, 387-424, (1993) · Zbl 0797.90123
[11] Eshel, I.; Cavalli-Sforza, L. L., Assortment of encounters and evolution of cooperativeness, Proc. Natl. Acad. Sci. U.S.A., 79, 4, 1331-1335, (1982) · Zbl 0491.92024
[12] Ewens, W. J., Mathematical population genetics 1: theoretical introduction, vol. 27, (2004), Springer, New York · Zbl 1060.92046
[13] Fletcher, J. A.; Doebeli, M., A simple and general explanation for the evolution of altruism, Proc. R. Soc. Lond. B: Biol. Sci., 276, 1654, 13-19, (2009)
[14] Frank, S. A., Natural selection. vii. history and interpretation of kin selection theory, J. Evolut. Biol., 26, 6, 1151-1184, (2013)
[15] Fu, F.; Nowak, M. A.; Christakis, N. A.; Fowler, J. H., The evolution of homophily, Sci. Rep., 2, (2012)
[16] Fudenberg, D.; Imhof, L. A., Imitation processes with small mutations, J. Econ. Theory, 131, 1, 251-262, (2006) · Zbl 1142.91342
[17] Gokhale, C. S.; Traulsen, A., Evolutionary multiplayer games, Dyn. Games Appl., 4, 4, 468-488, (2014) · Zbl 1314.91033
[18] Gore, J.; Youk, H.; Van Oudenaarden, A., Snowdrift game dynamics and facultative cheating in yeast, Nature, 459, 7244, 253-256, (2009), ISSN 0028-0836
[19] Grafen, A., The hawk-dove game played between relatives, Anim. Behav., 27, 3, 905-907, (1979)
[20] Hamilton, W. D., The genetical evolution of social behaviour. I, J. Theor. Biol., 7, 1, 1-16, (1964)
[21] Hamilton, W. D., Selection of selfish and altruistic behaviour in some extreme models, (Eisenberg, J. F.; Dillon, W. S., Man and Beast: Comparative Social Behavior, (1971), Smithsonian Press Washington, DC), 57-91
[22] Hauert, C.; Michor, F.; Nowak, M. A.; Doebeli, M., Synergy and discounting of cooperation in social dilemmas, J. Theor. Biol., 239, 2, 195-202, (2006)
[23] Jansen, V. A.; Van Baalen, M., Altruism through beard chromodynamics, Nature, 440, 7084, 663-666, (2006)
[24] Jensen, M.K., Rigos, A., 2014. Evolutionary Games with Group Selection. Technical Report 14/9, University of Leicester.
[25] Kerr, B.; Godfrey-Smith, P.; Feldman, M. W., What is altruism?, Trends Ecol. Evol., 19, 3, 135-140, (2004)
[26] Lehmann, L.; Rousset, F., How life history and demography promote or inhibit the evolution of helping behaviours, Philos. Trans. R. Soc. Lond. B: Biol. Sci., 365, 1553, 2599-2617, (2010)
[27] Lion, S.; van Baalen, M., Self-structuring in spatial evolutionary ecology, Ecol. Lett., 11, 3, 277-295, (2008)
[28] McPherson, M.; Smith-Lovin, L.; Cook, J. M., Birds of a featherhomophily in social networks, Annu. Rev. Sociol., 415-444, (2001)
[29] Moran, P. A.P., Random processes in genetics, Math. Proc. Camb. Philos. Soc., 54, 1, 60-71, (1958) · Zbl 0091.15701
[30] Nathanson, C. G.; Tarnita, C. E.; Nowak, M. A., Calculating evolutionary dynamics in structured populations, PLoS Comput. Biol., 5, 12, e1000615, (2009), ISSN 1553-7358
[31] Nowak, M. A., Evolutionary dynamics, (2006), Harvard University Press, Cambridge · Zbl 1098.92051
[32] Nowak, M. A., Five rules for the evolution of cooperation, Science, 314, 5805, 1560-1563, (2006)
[33] Nowak, M. A., Evolving cooperation, J. Theor. Biol., 299, 1-8, (2012) · Zbl 1337.92152
[34] Nowak, M. A.; May, R. M., Evolutionary games and spatial chaos, Nature, 359, 6398, 826-829, (1992)
[35] Nowak, M. A.; Sasaki, A.; Taylor, C.; Fudenberg, D., Emergence of cooperation and evolutionary stability in finite populations, Nature, 428, 6983, 646-650, (2004)
[36] Ohtsuki, H., Evolutionary dynamics of n-player games played by relatives, Philos. Trans. R. Soc. Lond. B: Biol. Sci., 369, 1642, 20130359, (2014)
[37] 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)
[38] Okasha S., Martens, J., 2015. Hamilton’s rule, inclusive fitness maximization, and the goal of individual behaviour in symmetric two-player games. J. Evol. Biol.
[39] Peña, J.; Lehmann, L.; Nöldeke, G., Gains from switching and evolutionary stability in multi-player matrix games, J. Theor. Biol., 346, 23-33, (2014)
[40] Peña, J.; Wu, B.; Traulsen, A., Ordering structured populations in multiplayer cooperation games, J. R. Soc. Interface, 13, 20150881, (2016)
[41] Rapoport, A.; Chammah, A. M., Prisoner’s dilemma: A study in conflict and cooperation, (1965), University of Michigan Press Ann Arbor
[42] Rigos, A.; Nax, H. H., Assortativity evolving from social dilemmas, J. Theor. Biol., 395, 194-203, (2016) · Zbl 1343.91033
[43] Skyrms, B., Social dynamics, (2014), Oxford University Press Oxford
[44] Tarnita, C. E.; Ohtsuki, H.; Antal, T.; Fu, F.; Nowak, M. A., Strategy selection in structured populations, J. Theor. Biol., 259, 3, 570-581, (2009)
[45] Taylor, P. D., Inclusive fitness in a homogeneous environment, Proc. R. Soc. Lond. Ser. B: Biol. Sci., 249, 1326, 299-302, (1992)
[46] Taylor, P. D.; Day, T.; Wild, G., Evolution of cooperation in a finite homogeneous graph, Nature, 447, 7143, 469-472, (2007)
[47] Traulsen, A.; Shoresh, N.; Nowak, M. A., Analytical results for individual and group selection of any intensity, Bull. Math. Biol., 70, 5, 1410-1424, (2008) · Zbl 1144.92035
[48] Van Cleve, J., Social evolution and genetic interactions in the short and long term, Theor. Popul. Biol., 103, 2-26, (2015) · Zbl 1342.92147
[49] Van Cleve, J.; Akçay, E., Pathways to social evolutionreciprocity, relatedness, and synergy, Evolution, 68, 8, 2245-2258, (2014)
[50] Van Cleve, J.; Lehmann, L., Stochastic stability and the evolution of coordination in spatially structured populations, Theor. Popul. Biol., 89, 75-87, (2013) · Zbl 1302.92118
[51] Van Veelen, M., Group selection, kin selection, altruism and cooperationwhen inclusive fitness is right and when it can be wrong, J. Theor. Biol., 259, 3, 589-600, (2009)
[52] Van Veelen, M., The replicator dynamics with n players and population structure, J. Theor. Biol., 276, 1, 78-85, (2011) · Zbl 1405.91046
[53] van Veelen, M.; García, J.; Rand, D. G.; Nowak, M. A., Direct reciprocity in structured populations, Proc. Natl. Acad. Sci., U. S. A., 109, 25, 9929-9934, (2012) · Zbl 1355.91062
[54] Veller, C.; Hayward, L. K., Finite-population evolution with rare mutations in asymmetric games, J. Econ. Theory, 162, 93-113, (2016) · Zbl 1369.91023
[55] Wu, B.; Altrock, P. M.; Wang, L.; Traulsen, A., Universality of weak selection, Phys. Rev. E, 82, October (4), 046106, (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.