The limits of weak selection and large population size in evolutionary game theory.

*(English)*Zbl 1378.91029Summary: Evolutionary game theory is a mathematical approach to studying how social behaviors evolve. In many recent works, evolutionary competition between strategies is modeled as a stochastic process in a finite population. In this context, two limits are both mathematically convenient and biologically relevant: weak selection and large population size. These limits can be combined in different ways, leading to potentially different results. We consider two orderings: the \(wN\) limit, in which weak selection is applied before the large population limit, and the \(Nw\) limit, in which the order is reversed. Formal mathematical definitions of the \(Nw\) and \(wN\) limits are provided. Applying these definitions to the Moran process of evolutionary game theory, we obtain asymptotic expressions for fixation probability and conditions for success in these limits. We find that the asymptotic expressions for fixation probability, and the conditions for a strategy to be favored over a neutral mutation, are different in the \(Nw\) and \(wN\) limits. However, the ordering of limits does not affect the conditions for one strategy to be favored over another.

##### MSC:

91A22 | Evolutionary games |

92D15 | Problems related to evolution |

60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |

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\textit{C. Sample} and \textit{B. Allen}, J. Math. Biol. 75, No. 5, 1285--1317 (2017; Zbl 1378.91029)

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##### References:

[1] | Abramowitz M, Stegun IA (1964) Handbook of mathematical functions: with formulas, graphs, and mathematical tables, vol 55. Courier Corporation, New York · Zbl 0171.38503 |

[2] | Allen, B; Nowak, MA, Games on graphs, EMS Surv Math Sci, 1, 113-151, (2014) · Zbl 1303.91040 |

[3] | Antal, T; Scheuring, I, Fixation of strategies for an evolutionary game in finite populations, Bull Math Biol, 68, 1923-1944, (2006) · Zbl 1296.92238 |

[4] | Bladon, AJ; Galla, T; McKane, AJ, Evolutionary dynamics, intrinsic noise, and cycles of cooperation, Phys Rev E, 81, 066122, (2010) |

[5] | Blume, LE, The statistical mechanics of strategic interaction, Games and economic behavior, 5, 387-424, (1993) · Zbl 0797.90123 |

[6] | Bomze, I; Pawlowitsch, C, One-third rules with equality: second-order evolutionary stability conditions in finite populations, J Theor Biol, 254, 616-620, (2008) · Zbl 1400.92359 |

[7] | Broom M, Rychtár J (2013) Game-theoretical models in biology. Chapman & Hall/CRC, Boca Raton · Zbl 1264.92002 |

[8] | Chen, YT, Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs, Ann Appl Probab, 23, 637-664, (2013) · Zbl 1267.91019 |

[9] | Fisher RA (1930) The genetical theory of natural selection. Oxford University Press, Oxford · JFM 56.1106.13 |

[10] | Haldane, JBS, A mathematical theory of natural and artificial selection, part V: selection and mutation, Math Proc Camb Philos Soc, 23, 838-844, (1927) · JFM 53.0516.05 |

[11] | Harsanyi JC, Selten R et al (1988) A general theory of equilibrium selection in games, vol 1. MIT Press Books, Cambridge · Zbl 0693.90098 |

[12] | Helbing, D, A stochastic behavioral model and a ‘microscopic’ foundation of evolutionary game theory, Theor Decis, 20, 149-179, (1996) · Zbl 0848.90147 |

[13] | Hofbauer J, Sigmund K (1998) Evolutionary games and replicator dynamics. Cambridge University Press, Cambridge · Zbl 0914.90287 |

[14] | Ibsen-Jensen, R; Chatterjee, K; Nowak, MA, Computational complexity of ecological and evolutionary spatial dynamics, Proc Nat Acad Sci, 112, 15,636-15,641, (2015) |

[15] | Imhof, LA; Nowak, MA, Evolutionary game dynamics in a wright-Fisher process, J Math Biol, 52, 667-681, (2006) · Zbl 1110.92028 |

[16] | Jeong, HC; Oh, SY; Allen, B; Nowak, MA, Optional games on cycles and complete graphs, J Theor Biol, 356, 98-112, (2014) · Zbl 1412.91007 |

[17] | Kimura, M, Diffusion models in population genetics, J Appl Probab, 1, 177-232, (1964) · Zbl 0134.38103 |

[18] | Ladret, V; Lessard, S, Evolutionary game dynamics in a finite asymmetric two-deme population and emergence of cooperation, J Theor Biol, 255, 137-151, (2008) · Zbl 1400.91064 |

[19] | Lessard, S; Ladret, V, The probability of fixation of a single mutant in an exchangeable selection model, J Math Biol, 54, 721-744, (2007) · Zbl 1115.92046 |

[20] | Lessard, S, On the robustness of the extension of the one-third law of evolution to the multi-player game, Dyn Games Appl, 1, 408-418, (2011) · Zbl 1252.91017 |

[21] | Moran, PAP, Random processes in genetics, Math Proc Camb Philos Soc, 54, 60-71, (1958) · Zbl 0091.15701 |

[22] | Nowak, MA; May, RM, Evolutionary games and spatial chaos, Nature, 359, 826-829, (1992) |

[23] | Nowak, MA; Sasaki, A; Taylor, C; Fudenberg, D, Emergence of cooperation and evolutionary stability in finite populations, Nature, 428, 646-650, (2004) |

[24] | Nowak, MA; Tarnita, CE; Antal, T, Evolutionary dynamics in structured populations, Philos Trans R Soc B Biol Sci, 365, 19-30, (2010) |

[25] | Ohtsuki, H; Nowak, MA, Evolutionary games on cycles, Proc R Soc B Biol Sci, 273, 2249-2256, (2006) · Zbl 1049.93517 |

[26] | Ohtsuki, H; Hauert, C; Lieberman, E; Nowak, MA, A simple rule for the evolution of cooperation on graphs and social networks, Nature, 441, 502-505, (2006) |

[27] | Ohtsuki, H; Bordalo, P; Nowak, MA, The one-third law of evolutionary dynamics, J Theor Biol, 249, 289-295, (2007) |

[28] | Saakian, DB; Hu, C-K, Solution of classical evolutionary models in the limit when the diffusion approximation breaks down, Phys Rev E, 94, 042422, (2016) |

[29] | Smith JM (1982) Evolution and the theory of games. Cambridge University Press, Cambridge · Zbl 0526.90102 |

[30] | Smith, JM; Price, GR, The logic of animal conflict, Nature, 246, 15-18, (1973) · Zbl 1369.92134 |

[31] | Szabó, G; Fáth, G, Evolutionary games on graphs, Phys Rep, 446, 97-216, (2007) |

[32] | Tarnita, CE; Ohtsuki, H; Antal, T; Fu, F; Nowak, MA, Strategy selection in structured populations, J Theor Biol, 259, 570-581, (2009) · Zbl 1402.91064 |

[33] | Taylor, C; Fudenberg, D; Sasaki, A; Nowak, M, Evolutionary game dynamics in finite populations, Bull Math Biol, 66, 1621-1644, (2004) · Zbl 1334.92372 |

[34] | Taylor, PD; Day, T; Wild, G, Evolution of cooperation in a finite homogeneous graph, Nature, 447, 469-472, (2007) |

[35] | Thomson BS, Bruckner JB, Bruckner AM (2001) Elementary real analysis. Prentice Hall Inc, Upper Saddle River · Zbl 1018.01004 |

[36] | Traulsen, A; Claussen, JC; Hauert, C, Coevolutionary dynamics: from finite to infinite populations, Phys Rev Lett, 95, 238701, (2005) |

[37] | Traulsen, A; Nowak, MA; Pacheco, JM, Stochastic dynamics of invasion and fixation, Phys Rev E, 74, 011909, (2006) |

[38] | Traulsen, A; Pacheco, JM; Imhof, LA, Stochasticity and evolutionary stability, Phys Rev E, 74, 021905, (2006) |

[39] | Traulsen, A; Pacheco, JM; Nowak, MA, Pairwise comparison and selection temperature in evolutionary game dynamics, J Theor Biol, 246, 522-529, (2007) |

[40] | Weibull JW (1997) Evolutionary game theory. MIT press, Cambridge · Zbl 0879.90206 |

[41] | Wright, S, Evolution in Mendelian populations, Genetics, 16, 97-159, (1931) |

[42] | Wu, B; Altrock, PM; Wang, L; Traulsen, A, Universality of weak selection, Phys Rev E, 82, 046106, (2010) |

[43] | Wu, B; García, J; Hauert, C; Traulsen, A, Extrapolating weak selection in evolutionary games, PLoS Comput Biol, 9, e1003381, (2013) |

[44] | Wu, B; Bauer, B; Galla, T; Traulsen, A, Fitness-based models and pairwise comparison models of evolutionary games are typically different-even in unstructured populations, New J Phys, 17, 023043, (2015) |

[45] | Zheng, X; Cressman, R; Tao, Y, The diffusion approximation of stochastic evolutionary game dynamics: Mean effective fixation time and the significance of the one-third law, Dyn Games Appl, 1, 462-477, (2011) · Zbl 1252.91022 |

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