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Bulletin of mathematical biology
Mar 12, 2025;87 (4): 51.

Exploring the Evolution of Altruistic Punishment with a PDE Model of Cultural Multilevel Selection.

Daniel B Cooney
Author Information
  1. Daniel B Cooney: Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL, USA. dbcoone2@illinois.edu. ORCID
PMID: 40072783 DOI: 10.1007/s11538-025-01422-4

Abstract

Two mechanisms that have been used to study the evolution of cooperative behavior are altruistic punishment, in which cooperative individuals pay additional costs to punish defection, and multilevel selection, in which competition between groups can help to counteract individual-level incentives to cheat. Boyd, Gintis, Bowles, and Richerson have used simulation models of cultural evolution to suggest that altruistic punishment and pairwise group-level competition can work in concert to promote cooperation, even when neither mechanism can do so on its own. In this paper, we formulate a PDE model for multilevel selection motivated by the approach of Boyd and coauthors, modeling individual-level birth-death competition with a replicator equation based on individual payoffs and describing group-level competition with pairwise conflicts based on differences in the average payoffs of the competing groups. Building off of existing PDE models for multilevel selection with frequency-independent group-level competition, we use analytical and numerical techniques to understand how the forms of individual and average payoffs can impact the long-time ability to sustain altruistic punishment in group-structured populations. We find several interesting differences between the behavior of our new PDE model with pairwise group-level competition and existing multilevel PDE models, including the observation that our new model can feature a non-monotonic dependence of the long-time collective payoff on the strength of altruistic punishment. Going forward, our PDE framework can serve as a way to connect and compare disparate approaches for understanding multilevel selection across the literature in evolutionary biology and anthropology.

Keywords

Altruistic punishment Evolutionary game theory Multilevel selection

References

  1. Alexiou K, Cooney DB (2024) Steady-state and dynamical behavior of a PDE model of multilevel selection with pairwise group-level competition. arXiv preprint arXiv:2411.08387
  2. Axelrod R, Hamilton WD (1981) The evolution of cooperation. Science 211(4489):1390���1396
  3. Baik KH (2008) Contests with group-specific public-good prizes. Soc Choice Welfare 30(1):103���117
  4. Bergstrom TC (2003) The algebra of assortative encounters and the evolution of cooperation. Int Game Theory Rev 5(03):211���228
  5. B��ttcher MA, Nagler J (2016) Promotion of cooperation by selective group extinction. New J Phys 18(6):063008
  6. Bowles S, Gintis H (2004) The evolution of strong reciprocity: cooperation in heterogeneous populations. Theor Popul Biol 65(1):17���28
  7. Boyd R, Richerson PJ (1989) The evolution of indirect reciprocity. Soc Netw 11(3):213���236
  8. Boyd R, Richerson PJ (1990) Group selection among alternative evolutionarily stable strategies. J Theor Biol 145(3):331���342
  9. Boyd R, Gintis H, Bowles S, Richerson PJ (2003) The evolution of altruistic punishment. Proc Natl Acad Sci 100(6):3531���3535
  10. Boyd R, Gintis H, Bowles S (2010) Coordinated punishment of defectors sustains cooperation and can proliferate when rare. Science 328(5978):617���620
  11. Carrillo JA, Chen L, Wang Q (2022) An optimal mass transport method for random genetic drift. SIAM J Numer Anal 60(3):940���969
  12. Chalub FA, Souza MO (2014) The frequency-dependent wright-fisher model: diffusive and non-diffusive approximations. J Math Biol 68:1089���1133
  13. Chalub FA, Santos FC, Pacheco JM (2006) The evolution of norms. J Theor Biol 241(2):233���240
  14. Choi JK, Bowles S (2007) The coevolution of parochial altruism and war. Science 318(5850):636���640
  15. Clutton-Brock TH, Parker GA (1995) Punishment in animal societies. Nature 373(6511):209���216
  16. Cooney DB (2019) The replicator dynamics for multilevel selection in evolutionary games. J Math Biol 79(1):101���154
  17. Cooney DB (2020) Analysis of multilevel replicator dynamics for general two-strategy social dilemma. Bull Math Biol 82(6):1���72
  18. Cooney DB (2022) Assortment and reciprocity mechanisms for promotion of cooperation in a model of multilevel selection. Bull Math Biol 84(11):126
  19. Cooney DB, Mori Y (2022) Long-time behavior of a PDE replicator equation for multilevel selection in group-structured populations. J Math Biol 85(2):12
  20. Cooney DB, Rossine FW, Morris DH, Levin SA (2022) A PDE model for protocell evolution and the origin of chromosomes via multilevel selection. Bull Math Biol 84(10):109
  21. Cooney DB, Levin SA, Mori Y, Plotkin JB (2023) Evolutionary dynamics within and among competing groups. Proc Natl Acad Sci 120(20):e2216186120
  22. Crowley PH, Baik KH (2010) Variable valuations and voluntarism under group selection: an evolutionary public goods game. J Theor Biol 265(3):238���244
  23. Czuppon P, Traulsen A (2021) Understanding evolutionary and ecological dynamics using a continuum limit. Ecol Evol 11(11):5857���5873
  24. Dawson DA (2018) Multilevel mutation-selection systems and set-valued duals. J Math Biol 76:295���378
  25. Duong MH, Han TA (2021) Cost efficiency of institutional incentives for promoting cooperation in finite populations. Proc R Soc A 477(2254):20210568
  26. Efferson C, Bernhard H, Fischbacher U, Fehr E (2024) Super-additive cooperation. Nature 626:1034���1041
  27. Epstein CL, Mazzeo R (2010) Wright-fisher diffusion in one dimension. SIAM J Math Anal 42(2):568���608
  28. Fehr E, G��chter S (2000) Cooperation and punishment in public goods experiments. Am Econ Rev 90(4):980���994
  29. Fehr E, G��chter S (2002) Altruistic punishment in humans. Nature 415(6868):137���140
  30. Fowler JH (2005) Altruistic punishment and the origin of cooperation. Proc Natl Acad Sci 102(19):7047���7049
  31. Fujimoto Y, Sagawa T, Kaneko K (2017) Hierarchical prisoner���s dilemma in hierarchical game for resource competition. New J Phys 19(7):073008
  32. Garc��a J, van den Bergh JC (2011) Evolution of parochial altruism by multilevel selection. Evol Hum Behav 32(4):277���287
  33. Gavrilets S (2015) Collective action problem in heterogeneous groups. Philos Trans R Soc B Biol Sci 370(1683):20150016
  34. Gavrilets S, Richerson PJ (2017) Collective action and the evolution of social norm internalization. Proc Natl Acad Sci 114(23):6068���6073
  35. Gavrilets S, Shrestha MD (2021) Evolving institutions for collective action by selective imitation and self-interested design. Evol Hum Behav 42(1):1���11
  36. Grafen A (1979) The hawk-dove game played between relatives. Anim Behav 27:905���907
  37. Han TA, Duong MH, Perc M (2024) Evolutionary mechanisms that promote cooperation may not promote social welfare. J R Soc Interface 21(220):20240547
  38. Henrich J (2004) Cultural group selection, coevolutionary processes and large-scale cooperation. J Econ Behav Organ 53(1):3���35
  39. Henrich J, McElreath R, Barr A, Ensminger J, Barrett C, Bolyanatz A, Cardenas JC, Gurven M, Gwako E, Henrich N et al (2006) Costly punishment across human societies. Science 312(5781):1767���1770
  40. Henriques GJ, Simon B, Ispolatov Y, Doebeli M (2019) Acculturation drives the evolution of intergroup conflict. Proc Natl Acad Sci 116(28):14089���14097
  41. Hillis V, Bell A, Brandt J, Brooks JS (2018) Applying a cultural multilevel selection framework to the adoption of sustainable management practices in California viticulture. Sustain Sci 13:71���80
  42. Hofbauer J, Sandholm WH (2009) Stable games and their dynamics. J Econ Theory 144(4):1665���1693
  43. Hommes CH, Ochea MI (2012) Multiple equilibria and limit cycles in evolutionary games with logit dynamics. Games Econom Behav 74(1):434���441
  44. Janssen MA, Holahan R, Lee A, Ostrom E (2010) Lab experiments for the study of social-ecological systems. Science 328(5978):613���617
  45. Janssen MA, Manning M, Udiani O (2014) The effect of social preferences on the evolution of cooperation in public good games. Adv Complex Syst 17(03n04):1450015
  46. Lahkar R, Riedel F (2015) The logit dynamic for games with continuous strategy sets. Games Econom Behav 91:268���282
  47. Lerch BA, Abbott KC (2024) A flexible theory for the dynamics of social populations: within-group density dependence and between-group processes. Ecol Monogr 94:e1604
  48. Le Veque RJ (2002) Finite volume methods for hyperbolic problems, vol 31. Cambridge University Press, Cambridge
  49. Luo S (2014) A unifying framework reveals key properties of multilevel selection. J Theor Biol 341:41���52
  50. Luo S, Mattingly JC (2017) Scaling limits of a model for selection at two scales. Nonlinearity 30(4):1682
  51. Morgan J, Steiglitz K (2003) Pairwise competition and the replicator equation. Bull Math Biol 65:1163���1172
  52. Nowak MA (2006) Five rules for the evolution of cooperation. Science 314(5805):1560���1563
  53. Nowak MA, Sigmund K (1998) Evolution of indirect reciprocity by image scoring. Nature 393(6685):573���577
  54. Odouard VV, Smirnova D, Edelman S (2023) Polarize, catalyze, and stabilize: how a minority of norm internalizers amplify group selection and punishment. Behav Ecol Sociobiol 77(10):1���18
  55. Ohtsuki H, Iwasa Y (2006) The leading eight: social norms that can maintain cooperation by indirect reciprocity. J Theor Biol 239(4):435���444
  56. Ostrom E, Walker J, Gardner R (1992) Covenants with and without a sword: self-governance is possible. Am Polit Sci Rev 86(2):404���417
  57. Philson CS (2023) Evolution of social position and structure-a multilevel selection perspective. University of California, Los Angeles
  58. Preston EF, Thompson FJ, Ellis S, Kyambulima S, Croft DP, Cant MA (2021a) Network-level consequences of outgroup threats in banded mongooses: grooming and aggression between the sexes. J Anim Ecol 90(1):153���167
  59. Preston EF, Thompson FJ, Kyabulima S, Croft DP, Cant MA ((2021b) The dynamics of social cohesion in response to simulated intergroup conflict in banded mongooses. Ecol Evol 11(24):18662���18675
  60. Puhalskii A, Simon B (2017) A large-population limit for a Markovian model of group-structured populations. arXiv preprint arXiv:1712.09119
  61. Radzvilavicius AL, Stewart AJ, Plotkin JB (2019) Evolution of empathetic moral evaluation. Elife 8:e44269
  62. Rebers S, Koopmans R (2012) Altruistic punishment and between-group competition: evidence from n-person prisoner���s dilemmas. Hum Nat 23:173���190
  63. Reeve HK, H��lldobler B (2007) The emergence of a superorganism through intergroup competition. Proc Natl Acad Sci 104(23):9736���9740
  64. Richerson P, Baldini R, Bell AV, Demps K, Frost K, Hillis V, Mathew S, Newton EK, Naar N, Newson L et al (2016) Cultural group selection plays an essential role in explaining human cooperation: a sketch of the evidence. Behav Brain Sci 39:e30
  65. Rusch H, Gavrilets S (2020) The logic of animal intergroup conflict: a review. J Econ Behav Organ 178:1014���1030
  66. Santos FC, Chalub FA, Pacheco JM (2007) A multi-level selection model for the emergence of social norms. In: European Conference on Artificial Life, Springer, pp 525���534
  67. Santos FP, Santos FC, Pacheco JM (2018) Social norm complexity and past reputations in the evolution of cooperation. Nature 555(7695):242���245
  68. Scheuring I (2009) Evolution of generous cooperative norms by cultural group selection. J Theor Biol 257(3):397���407
  69. Scheuring I (2010) Coevolution of honest signaling and cooperative norms by cultural group selection. Biosystems 101(2):79���87
  70. Schl��ter M, Tavoni A, Levin S (2016) Robustness of norm-driven cooperation in the commons. Proc R Soc B Biol Sci 283(1822):20152431
  71. Schmid L, Chatterjee K, Hilbe C, Nowak MA (2021) A unified framework of direct and indirect reciprocity. Nat Hum Behav 5(10):1292���1302
  72. Sethi R, Somanathan E (1996) The evolution of social norms in common property resource use. Am Econ Rev 86:766���788
  73. Simon B (2010) A dynamical model of two-level selection. Evol Ecol Res 12(5):555���588
  74. Simon B, Pilosov M (2016) Group-level events are catalysts in the evolution of cooperation. J Theor Biol 410:125���136
  75. Simon B, Fletcher JA, Doebeli M (2013) Towards a general theory of group selection. Evolution 67(6):1561���1572
  76. Smith D (2020) Cultural group selection and human cooperation: a conceptual and empirical review. Evol Hum Sci 2:e2
  77. Soltis J, Boyd R, Richerson PJ (1995) Can group-functional behaviors evolve by cultural group selection?: An empirical test. Curr Anthropol 36(3):473���494
  78. Szathm��ry E, Smith JM (1995) The major evolutionary transitions. Nature 374(6519):227���232
  79. Tam J, Waring T, Gelcich S, Chan KM, Satterfield T (2021) Measuring behavioral social learning in a conservation context: Chilean fishing communities. Conserv Sci Pract 3(1):e336
  80. Tavoni A, Schl��ter M, Levin S (2012) The survival of the conformist: social pressure and renewable resource management. J Theor Biol 299:152���161
  81. Thompson FJ, Marshall HH, Vitikainen EI, Cant MA (2017) Causes and consequences of intergroup conflict in cooperative banded mongooses. Anim Behav 126:31���40
  82. Traulsen A, Nowak MA (2006) Evolution of cooperation by multilevel selection. Proc Natl Acad Sci 103(29):10952���10955
  83. Traulsen A, Claussen JC, Hauert C (2005) Coevolutionary dynamics: from finite to infinite populations. Phys Rev Lett 95(23):238701
  84. Traulsen A, Pacheco JM, Nowak MA (2007) Pairwise comparison and selection temperature in evolutionary game dynamics. J Theor Biol 246(3):522���529
  85. Traulsen A, Shoresh N, Nowak MA (2008) Analytical results for individual and group selection of any intensity. Bull Math Biol 70:1410���1424
  86. Trivers RL (1971) The evolution of reciprocal altruism. Q Rev Biol 46(1):35���57
  87. Tullock G (2008) Efficient rent seeking. In: 40 Years of Research on Rent Seeking 1, Springer, pp 105���120
  88. Tverskoi D, Senthilnathan A, Gavrilets S (2021) The dynamics of cooperation, power, and inequality in a group-structured society. Sci Rep 11(1):18670
  89. 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���289
  90. Velleret A (2020) Individual-based models under various time-scales. ESAIM Proc Surv 68:123���152
  91. Velleret A (2023) Two level natural selection with a quasi-stationarity approach. Discrete and Continuous Dynamical Systems-Series B
  92. Waring T, Acheson J (2018) Evidence of cultural group selection in territorial lobstering in Maine. Sustain Sci 13:21���34
  93. Waring TM, Goff SH, Smaldino PE (2017) The coevolution of economic institutions and sustainable consumption via cultural group selection. Ecol Econ 131:524���532
  94. Waxman D (2011) Comparison and content of the wright-fisher model of random genetic drift, the diffusion approximation, and an intermediate model. J Theor Biol 269(1):79���87
  95. Wilson DS (1975) A theory of group selection. Proc Natl Acad Sci 72(1):143���146
  96. Wilson DS, Dugatkin LA (1997) Group selection and assortative interactions. Am Nat 149(2):336���351
  97. Wilson DS, Sober E (1994) Reintroducing group selection to the human behavioral sciences. Behav Brain Sci 17(4):585���608
  98. Wilson DS, Ostrom E, Cox ME (2013) Generalizing the core design principles for the efficacy of groups. J Econ Behav Organ 90:S21���S32
  99. Wilson DS, Madhavan G, Gelfand MJ, Hayes SC, Atkins PW, Colwell RR (2023) Multilevel cultural evolution: from new theory to practical applications. Proc Natl Acad Sci 120(16):e2218222120
  100. Zhao L, Yue X, Waxman D (2013) Complete numerical solution of the diffusion equation of random genetic drift. Genetics 194(4):973���985

Grants

  1. Math + X Grant (awarded to the University of Pennsylvania)/Simons Foundation

MeSH Term

Altruism
Punishment
Humans
Cooperative Behavior
Cultural Evolution
Mathematical Concepts
Game Theory
Computer Simulation
Competitive Behavior
Selection, Genetic
Models, Psychological
Journal Article
Article has an altmetric score of 11

Word Cloud

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