The replicator dynamics for multilevel selection in evolutionary games

  • Daniel B. CooneyEmail author


We consider a stochastic model for evolution of group-structured populations in which interactions between group members correspond to the Prisoner’s Dilemma or the Hawk–Dove game. Selection operates at two organization levels: individuals compete with peer group members based on individual payoff, while groups also compete with other groups based on average payoff of group members. In the Prisoner’s Dilemma, this creates a tension between the two levels of selection, as defectors are favored at the individual level, whereas groups with at least some cooperators outperform groups of defectors at the between-group level. In the limit of infinite group size and infinite number of groups, we derive a non-local PDE that describes the probability distribution of group compositions in the population. For special families of payoff matrices, we characterize the long-time behavior of solutions of our equation, finding a threshold intensity of between-group selection required to sustain density steady states and the survival of cooperation. When all-cooperator groups are most fit, the average and most abundant group compositions at steady state range from featuring all-defector groups when individual-level selection dominates to featuring all-cooperator groups when group-level selection dominates. When the most fit groups have a mix of cooperators and defectors, then the average and most abundant group compositions always feature a smaller fraction of cooperators than required for the optimal mix, even in the limit where group-level selection is infinitely stronger than individual-level selection. In such cases, the conflict between the two levels of selection cannot be decoupled, and cooperation cannot be sustained at all in the case where between-group competition favors an even mix of cooperators and defectors.


Multilevel selection Evolutionary game theory Replicator dynamics 



I would like to thank Carl Veller for initial discussions and advice on the problem of multilevel selection in evolutionary games. I am grateful to Carl Veller, Simon Levin, Joshua Plotkin, Chai Molina, and an anonymous referee for helpful comments on the manuscript and to Peter Constantin, Robin Pemantle, Fernando Rossine, Dylan Morris, George Constable, Chadi Saad-Roy and Gergely Boza for helpful discussions.


  1. Aktipis CA, Boddy AM, Jansen G, Hibner U, Hochberg ME, Maley CC, Wilkinson GS (2015) Cancer across the tree of life: cooperation and cheating in multicellularity. Philos Trans R Soc B 370(1673):20140,219CrossRefGoogle Scholar
  2. Archetti M, Scheuring I (2011) Coexistence of cooperation and defection in public goods games. Evolution 65(4):1140–1148CrossRefGoogle Scholar
  3. Ball JM, Carr J, Penrose O (1986) The becker-döring cluster equations: basic properties and asymptotic behaviour of solutions. Commun Math Phys 104(4):657–692CrossRefzbMATHGoogle Scholar
  4. Bergstrom TC (2002) Evolution of social behavior: individual and group selection. J Econ Perspect 16(2):67–88CrossRefGoogle Scholar
  5. Blancas A, Wakolbinger A (2017) A representation for the semigroup of a two-level Fleming–Viot process in terms of the Kingman nested coalescent. Working paper.
  6. Bomze IM (1990) Dynamical aspects of evolutionary stability. Monatsh Math 110(3–4):189–206MathSciNetCrossRefzbMATHGoogle Scholar
  7. Böttcher MA, Nagler J (2016) Promotion of cooperation by selective group extinction. N J Phys 18: 063008CrossRefGoogle Scholar
  8. Boza G, Számadó S (2010) Beneficial laggards: multilevel selection, cooperative polymorphism and division of labour in threshold public good games. BMC Evolut Biol 10(1):336CrossRefGoogle Scholar
  9. Chalub FA, Souza MO (2014) The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations. J Math Biol 68(5):1089–1133MathSciNetCrossRefzbMATHGoogle Scholar
  10. Coombs D, Gilchrist MA, Ball CL (2007) Evaluating the importance of within-and between-host selection pressures on the evolution of chronic pathogens. Theor Popul Biol 72(4):576–591CrossRefzbMATHGoogle Scholar
  11. Dawidowicz AL, Łoskot K (1986) Existence and uniqueness of solution of some integro-differential equation. Ann Polon Math 1:79–87MathSciNetCrossRefzbMATHGoogle Scholar
  12. Dawkins R (1976) The selfish gene. Oxford University Press, OxfordGoogle Scholar
  13. Dawson DA (2018) Multilevel mutation-selection systems and set-valued duals. J Math Biol 76:295–378MathSciNetCrossRefzbMATHGoogle Scholar
  14. Dwyer G, Levin SA, Buttel L (1990) A simulation model of the population dynamics and evolution of myxomatosis. Ecol Monogr 60(4):423–447CrossRefGoogle Scholar
  15. Eigen M (1971) Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58(10):465–523CrossRefGoogle Scholar
  16. Evers JH (2016) Mild solutions are weak solutions in a class of (non) linear measure-valued evolution equations on a bounded domain. arXiv preprint arXiv:1606.01332
  17. Fontanari JF, Serva M (2013) Solvable model for template coexistence in protocells. EPL (Europhys Lett) 101(3):38,006CrossRefGoogle Scholar
  18. Fontanari JF, Serva M (2014a) Effect of migration in a diffusion model for template coexistence in protocells. Bull Math Biol 76(3):654–672MathSciNetCrossRefzbMATHGoogle Scholar
  19. Fontanari JF, Serva M (2014b) Nonlinear group survival in Kimura’s model for the evolution of altruism. Math Biosci 249:18–26MathSciNetCrossRefzbMATHGoogle Scholar
  20. Gilchrist MA, Coombs D (2006) Evolution of virulence: interdependence, constraints, and selection using nested models. Theor Popul Biol 69(2):145–153CrossRefzbMATHGoogle Scholar
  21. Gilchrist MA, Coombs D, Perelson AS (2004) Optimizing within-host viral fitness: infected cell lifespan and virion production rate. J Theor Biol 229(2):281–288MathSciNetCrossRefGoogle Scholar
  22. Haig D (2016) Intracellular evolution of mitochondrial DNA (mtDNA) and the tragedy of the cytoplasmic commons. BioEssays 38(6):549–555CrossRefGoogle Scholar
  23. Hogeweg P, Takeuchi N (2003) Multilevel selection in models of prebiotic evolution: compartments and spatial self-organization. Orig Life Evol Biosph 33(4):375–403CrossRefGoogle Scholar
  24. Jensen MK, Rigos A (2018) Evolutionary games and matching rules. Int J Game Theory 47:707–735MathSciNetCrossRefzbMATHGoogle Scholar
  25. Kimura M (1955) Solution of a process of random genetic drift with a continuous model. Proc Natl Acad Sci 41(3):144–150CrossRefzbMATHGoogle Scholar
  26. Levin S, Pimentel D (1981) Selection of intermediate rates of increase in parasite-host systems. Am Nat 117(3):308–315MathSciNetCrossRefGoogle Scholar
  27. Luo S (2014) A unifying framework reveals key properties of multilevel selection. J Theor Biol 341:41–52MathSciNetCrossRefGoogle Scholar
  28. Luo S, Mattingly JC (2017) Scaling limits of a model for selection at two scales. Nonlinearity 30(4):1682MathSciNetCrossRefzbMATHGoogle Scholar
  29. MacLean RC, Fuentes-Hernandez A, Greig D, Hurst LD, Gudelj I (2010) A mixture of “cheats” and “co-operators” can enable maximal group benefit. PLoS Biol 8(9):e1000,486CrossRefGoogle Scholar
  30. Markvoort AJ, Sinai S, Nowak MA (2014) Computer simulations of cellular group selection reveal mechanism for sustaining cooperation. J Theor Biol 357:123–133CrossRefGoogle Scholar
  31. Mathis C, Ramprasad SN, Walker SI, Lehman N (2017) Prebiotic RNA network formation: a taxonomy of molecular cooperation. Life 7(4):38CrossRefGoogle Scholar
  32. Maynard Smith J (1982) Evolution and the theory of games. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  33. Maynard Smith J, Price GR (1973) The logic of animal conflict. Nature 246(5427):15CrossRefzbMATHGoogle Scholar
  34. McLoone B, Fan WTL, Pham A, Smead R, Loewe L (2018) Stochasticity, selection, and the evolution of cooperation in a two-level Moran model of the snowdrift game. Complexity.
  35. Nowak MA (2006) Five rules for the evolution of cooperation. Science 314(5805):1560–1563CrossRefGoogle Scholar
  36. Oechssler J, Riedel F (2001) Evolutionary dynamics on infinite strategy spaces. Econ Theory 17(1):141–162MathSciNetCrossRefzbMATHGoogle Scholar
  37. Oechssler J, Riedel F (2002) On the dynamic foundation of evolutionary stability in continuous models. J Econ Theory 107(2):223–252MathSciNetCrossRefzbMATHGoogle Scholar
  38. Ogura Y, Shimakura N (1987a) Stationary solutions and their stability for Kimura’s diffusion model with intergroup selection. J Math Kyoto Univ 27(2):305–347MathSciNetCrossRefzbMATHGoogle Scholar
  39. Ogura Y, Shimakura N (1987b) Stationary solutions and their stability for Kimura’s diffusion model with intergroup selection II. J Math Kyoto Univ 27(4):635–655MathSciNetCrossRefzbMATHGoogle Scholar
  40. Pacheco JM, Santos FC, Souza MO, Skyrms B (2009) Evolutionary dynamics of collective action in n-person stag hunt dilemmas. Proc R Soc Lond B Biol Sci 276(1655):315–321CrossRefGoogle Scholar
  41. Pruitt JN, Goodnight CJ (2014) Site-specific group selection drives locally adapted group compositions. Nature 514(7522):359CrossRefGoogle Scholar
  42. Pruitt JN, Goodnight CJ, Riechert SE (2017) Intense group selection selects for ideal group compositions, but selection within groups maintains them. Anim Behav 124:15–24CrossRefGoogle Scholar
  43. Puhalskii A, Reiman M, Simon B (2017) A large-population limit for a Markovian model of group-structured populations. arXiv preprint arXiv:1712.09119
  44. Shaffer Z, Sasaki T, Haney B, Janssen M, Pratt SC, Fewell JH (2016) The foundress’s dilemma: group selection for cooperation among queens of the harvester ant, Pogonomyrmex californicus. Sci Rep 6(29):828Google Scholar
  45. Simon B (2010) A dynamical model of two-level selection. Evolut Ecol Res 12(5):555–588Google Scholar
  46. Simon B, Nielsen A (2012) Numerical solutions and animations of group selection dynamics. Evolut Ecol Res 14(6):757–768Google Scholar
  47. Simon B, Pilosov M (2016) Group-level events are catalysts in the evolution of cooperation. J Theor Biol 410:125–136CrossRefzbMATHGoogle Scholar
  48. Simon B, Fletcher JA, Doebeli M (2013) Towards a general theory of group selection. Evolution 67(6):1561–1572CrossRefGoogle Scholar
  49. Souza MO, Pacheco JM, Santos FC (2009) Evolution of cooperation under n-person snowdrift games. J Theor Biol 260(4):581–588MathSciNetCrossRefzbMATHGoogle Scholar
  50. Szathmáry E, Demeter L (1987) Group selection of early replicators and the origin of life. J Theor Biol 128(4):463–486CrossRefGoogle Scholar
  51. Szathmáry E, Smith JM (1995) The major evolutionary transitions. Nature 374(6519):227–232CrossRefGoogle Scholar
  52. Takeuchi N, Hogeweg P (2009) Multilevel selection in models of prebiotic evolution II: a direct comparison of compartmentalization and spatial self-organization. PLoS Comput Biol 5(10):e1000,542MathSciNetCrossRefGoogle Scholar
  53. Takeuchi N, Hogeweg P (2012) Evolutionary dynamics of rna-like replicator systems: a bioinformatic approach to the origin of life. Phys Life Rev 9(3):219–263CrossRefGoogle Scholar
  54. Tarnita CE, Taubes CH, Nowak MA (2013) Evolutionary construction by staying together and coming together. J Theor Biol 320:10–22MathSciNetCrossRefzbMATHGoogle Scholar
  55. Traulsen A, Nowak MA (2006) Evolution of cooperation by multilevel selection. Proc Natl Acad Sci 103(29):10,952–10,955CrossRefGoogle Scholar
  56. Traulsen A, Sengupta AM, Nowak MA (2005) Stochastic evolutionary dynamics on two levels. J Theor Biol 235(3):393–401MathSciNetCrossRefGoogle Scholar
  57. Traulsen A, Shoresh N, Nowak MA (2008) Analytical results for individual and group selection of any intensity. Bull Math Biol 70(5):1410MathSciNetCrossRefzbMATHGoogle Scholar
  58. van Veelen M, Luo S, Simon B (2014) A simple model of group selection that cannot be analyzed with inclusive fitness. J Theor Biol 360:279–289CrossRefzbMATHGoogle Scholar
  59. Williams GC (1966) Adaptation and natural selection: a critique of some current evolutionary thought. Princeton University Press, PrincetonGoogle Scholar
  60. Wilson DS (1975) A theory of group selection. Proc Natl Acad Sci 72(1):143–146CrossRefzbMATHGoogle Scholar
  61. Young G, Belmonte A (2018) Fast cheater migration stabilizes coexistence in a public goods dilemma on networks. Theor Popul Biol 121:12–25. CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations