A pedestrian review of games on structured populations

Evolutionary games on heterogeneous structures
  • Saptarshi Sinha
  • Susanta Ghosh
  • Soumen RoyEmail author


Understanding the mechanisms of evolution of cooperation and its sustenance has gathered momentum since the last few decades of the twentieth century. However, the complete picture is yet to emerge. Evolutionary game theory aims to model evolutionary dynamics in a population by drawing on the principles of game theory. Spatially restricted interactions, such as in ecological systems, are rather common in nature. When interactions among the individuals in a population are structured, the outcome of the game is significantly different from that of a well-mixed population. In this mini-review, targeted towards a very broad audience of all backgrounds, we summarise some of the critical research by evolutionary biologists, computer scientists, mathematicians and physicists on evolutionary games in structured populations. We also discuss the influence of structure on evolutionary games in diverse scenarios ranging from laboratory environments to multiplex networks. Along the way, we also try to harmonise a few conflicting results in the literature.


Game theory Complex networks Structured populations Evolutionary dynamics 


  1. 1.
    Clutton-Brock, T.H., O’Riain, M.J., Brotherton, P.N., Gaynor, D., Kansky, R., Griffin, A.S., Manser, M.: Selfish sentinels in cooperative mammals. Science 284, 1640–4 (1999)Google Scholar
  2. 2.
    Wilkinson, G.S., Shank, C.C.: Rutting-fight mortality among musk oxen on Banks Island, Northwest Territories. Canada. Anim. Behav. 24, 756–758 (1976)Google Scholar
  3. 3.
    Yurtsev, E.A., Chao, H.X., Datta, M.S., Artemova, T., Gore, J.: Bacterial cheating drives the population dynamics of cooperative antibiotic resistance plasmids. Mol. Syst. Biol 9, 683 (2013)Google Scholar
  4. 4.
    Backhed, F., Ley, R.E., Sonnenburg, J.L., Peterson, D.A., Gordon, J.I.: Host-bacterial mutualism in the human intestine. Science 307, 1915–1920 (2005)Google Scholar
  5. 5.
    Fehr, E., Fischbacher, U.: The nature of human altruism. Nature 425, 785 (2003)Google Scholar
  6. 6.
    Hamilton, W.D.: The genetical evolution of social behaviour. II. J. Theor. Biol. 7, 17–52 (1964)Google Scholar
  7. 7.
    Nowak, M.A., McAvoy, A., Allen, B., Wilson, E.O.: The general form of Hamiltons rule makes no predictions and cannot be tested empirically. Proc. Natl. Acad. Sci. 114, 5665–5670 (2017)Google Scholar
  8. 8.
    Birch, J.: The inclusive fitness controversy: finding a way forward. R. Soc. Open Sci 4, 170335 (2017)Google Scholar
  9. 9.
    Rousset, F.: Regression, least squares, and the general version of inclusive fitness. Evolution 69, 2963–2970 (2015)Google Scholar
  10. 10.
    Gadagkar, R.: Evolution of eusociality: the advantage of assured fitness returns. Philos. Trans. R. Soc. Lond. B 329, 17–25 (1990)Google Scholar
  11. 11.
    Smith, J.M., Price, G.R.: The logic of animal conflict. Nature 246, 15 (1973)zbMATHGoogle Scholar
  12. 12.
    Hamilton, W.D., Hamilton, W.D.: Narrow Roads of Gene Land: Volume 2: Evolution of Sex. Oxford University Press, Oxford (1996)Google Scholar
  13. 13.
    Trivers, R.L.: The evolution of reciprocal altruism. Q. Rev. Biol. 46, 35–57 (1971)Google Scholar
  14. 14.
    Fundenberg, D., Maskin, E.: Evolution and cooperation in noisy repeated games. Am. Econ. Rev. 80, 274–279 (1990)Google Scholar
  15. 15.
    Selten, R., Hammerstein, P.: Gaps in Harley’s argument on evolutionarily stable learning rules and in the logic of tit for tat behavioral and Brain. Science 7, 115–116 (1984)Google Scholar
  16. 16.
    Rockenbach, B., Milinski, M.: The efficient interaction of indirect reciprocity and costly punishment. Nature 444, 718–723 (2006)Google Scholar
  17. 17.
    Hauert, C., Traulsen, A., Brandt, H., Nowak, M.A., Sigmund, K.: Via freedom to coercion: the emergence of costly punishment. Science 316, 1905–1907 (2007)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Santos, F.C., Pacheco, J.M.: Scale-free networks provide a unifying framework for the emergence of cooperation. Phys. Rev. Lett. 95, 098104 (2005)Google Scholar
  19. 19.
    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 (2006)Google Scholar
  20. 20.
    Melbinger, A., Cremer, J., Frey, E.: Evolutionary game theory in growing populations. Phys. Rev. Lett. 105, 178101 (2010)Google Scholar
  21. 21.
    Wienand, K., Frey, E., Mobilia, M.: Evolution of a fluctuating population in a randomly switching environment. Phys. Rev. Lett. 119, 158301 (2017)Google Scholar
  22. 22.
    Maciejewski, W., Fu, F., Hauert, C.: Evolutionary game dynamics in populations with heterogenous structures. PLoS Comput. Biol. 10, e1003567 (2014)Google Scholar
  23. 23.
    Nowak, M.A., Sasaki, A., Taylor, C., Fudenberg, D.: Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646 (2004)Google Scholar
  24. 24.
    Traulsen, A., Claussen, J.C., Hauert, C.: Coevolutionary dynamics: from finite to infinite populations. Phys. Rev. Lett. 95, 238701 (2005)Google Scholar
  25. 25.
    Von Neumann, J., Morgenstern, O.: Theory of games and economic behavior. Bull. Am. Math. Soc. 51, 498–504 (1945)MathSciNetGoogle Scholar
  26. 26.
    Taylor, P.D., Jonker, L.B.: Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Nash, J.F.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA 36, 48–49 (1950)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Szabo, G., Hodsagi, K.: The role of mixed strategies in spatial evolutionary games. Physica A, 462 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Szabo, G., Bunth, G.: Social dilemmas in multistrategy evolutionary potential games. Phys. Rev. E 97, 012305 (2018)Google Scholar
  30. 30.
    Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  31. 31.
    Sanchez, A.: Physics of human cooperation: experimental evidence and theoretical models. J. Stat. Mech. Theory Exp. 2018(2), 024001 (2018). Google Scholar
  32. 32.
    Nowak, M.A., May, R.M.: Evolutionary games and spatial chaos. Nature 359, 826 (1992)Google Scholar
  33. 33.
    Nowak, M.A.: Five rules for the evolution of cooperation. Science 314, 1560–1563 (2006)Google Scholar
  34. 34.
    Roy, S.: Systems biology beyond degree, hubs and scale-free networks: the case for multiple metrics in complex networks. Syst. Synth. Biol. 6, 31–34 (2012)Google Scholar
  35. 35.
    Banerjee, S.J., Roy, S.: Key to network controllability. arXiv:1209.3737 (2012)
  36. 36.
    Grewal, R.K., Mitra, D., Roy, S.: Mapping networks of light-dark transition in LOV photoreceptors. Bioinformatics 31, 3608–3616 (2015)Google Scholar
  37. 37.
    Banerjee, S.J., Sinha, S., Roy, S.: Slow poisoning and destruction of networks: edge proximity and its implications for biological and infrastructure networks. Phys. Rev. E 91, 022807 (2015)Google Scholar
  38. 38.
    Banerjee, S.J., Azharuddin, M., Sen, D., Savale, S., Datta, H., Dasgupta, A.K., Roy, S.: Using complex networks towards information retrieval and diagnostics in multidimensional imaging. Sci. Rep. 5, 17271 (2015)Google Scholar
  39. 39.
    Grewal, R.K., Roy, S.: Modeling proteins as residue interaction networks. Protein Pept. Letts. 22, 923–933 (2015)Google Scholar
  40. 40.
    Grewal, R.K., Sinha, S., Roy, S.: Topologically inspired walks on randomly connected landscapes with correlated fitness. Front. Phys. 6, 138 (2018)Google Scholar
  41. 41.
    Dsouza, R.M., Borgs, C., Chayes, J.T., Berger, N., Kleinberg, R.D.: Emergence of tempered preferential attachment from optimization. Proc. Natl. Acad. Sci. USA 104, 6112–6117 (2007)Google Scholar
  42. 42.
    Barabasi, A.-L.: Albert R: emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Price, D.-D.-S.: A general theory of bibliometric and other cumulative advantage processes. J. Assoc. Inf. Sci. Technol. 27, 292–306 (1976)Google Scholar
  44. 44.
    Axelrod, R.: The social structure of cooperation. In: The Evolution of Cooperation. Basic Books (1984)Google Scholar
  45. 45.
    Schelling, T. C.: Sorting and mixing: race and sex. In: Micromotives and Macrobehavior. WW Norton and Company (1978)Google Scholar
  46. 46.
    Nadell, C.D., Foster, K.R., Xavier, J.B.: Emergence of spatial structure in cell groups and the evolution of cooperation. PLOS Comput. Biol. 6, e1000716 (2010)Google Scholar
  47. 47.
    Nadell, C.D., Drescher, K., Foster, K.R.: Spatial structure, cooperation and competition in biofilms. Nat. Rev. Microbiol. 14, 589 (2016)Google Scholar
  48. 48.
    Joshi, J., Couzin, I.D., Levin, S.A., Guttal, V.: Mobility can promote the evolution of cooperation via emergent self-assortment dynamics. PLOS Comput. Biol. 13, e1005732 (2017)Google Scholar
  49. 49.
    Menon, S.N., Sasidevan, V., Sinha, S.: Emergence of cooperation as a non-equilibrium transition in noisy spatial games. Front. Phys. 6, 34 (2018)Google Scholar
  50. 50.
    Gomez-Gardenes, J., Campillo, M., Floria, L.M., Moreno, Y.: Dynamical organization of cooperation in complex topologies. Phys. Rev. Lett. 98, 108103 (2007)Google Scholar
  51. 51.
    Roca, C.P., Cuesta, J.A., Sanchez, A.: Effect of spatial structure on the evolution of cooperation. Phys. Rev. E 80, 046106 (2009)Google Scholar
  52. 52.
    Hauert, C., Doebeli, M.: Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428, 643 (2004)Google Scholar
  53. 53.
    Frey, E.: Evolutionary game theory: theoretical concepts and applications to microbial communities. Physica A 389, 4265–4298 (2010)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Lieberman, E., Hauert, C., Nowak, M.A.: Evolutionary dynamics on graphs. Nature 433, 312 (2005)Google Scholar
  55. 55.
    Imhof, L.A., Nowak, M.A.: Evolutionary game dynamics in a Wright-Fisher process. J. Math. Biol. 52, 667–681 (2006)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Doebeli, M., Hauert, C.: Models of cooperation based on the Prisoner’s Dilemma and the Snowdrift game. Ecol. Lett. 8, 748–766 (2005)Google Scholar
  57. 57.
    Szabo, G., Toke, C.: Evolutionary prisoners dilemma game on a square lattice. Phys. Rev. E 58, 69 (1998)Google Scholar
  58. 58.
    Chen, Y.S., Lin, H., Wu, C.X.: Evolution of prisoner’s dilemma strategies on scale-free networks. Physica A 385, 379–384 (2007)Google Scholar
  59. 59.
    Li, M., O’Riordan, C.: The effect of clustering coefficient and node degree on the robustness of cooperation. In: Evolutionary Computation (CEC), 2013 IEEE Congress 2833–2839 (2013)Google Scholar
  60. 60.
    Santos, F.C., Rodrigues, J.F., Pacheco, J.M.: Graph topology plays a determinant role in the evolution of cooperation. Proc. Roy. Soc. Lond. B 273, 51–55 (2006)Google Scholar
  61. 61.
    Li, P., Duan, H.: Robustness of cooperation on scale-free networks in the evolutionary prisoner’s dilemma game. Europhys. Lett. 105, 48003 (2014)Google Scholar
  62. 62.
    Ichinose, G., Tenguishi, Y., Tanizawa, T.: Robustness of cooperation on scale-free networks under continuous topological change. Phys. Rev. E 88, 052808 (2013)Google Scholar
  63. 63.
    Gallos, L.K., Cohen, R., Argyrakis, P., Bunde, A., Havlin, S.: Stability and topology of scale-free networks under attack and defense strategies. Phys. Rev. Lett. 94(18), 188701 (2005)Google Scholar
  64. 64.
    Duran, O., Mulet, R.: Evolutionary prisoner’s dilemma in random graphs. Physica D 208, 257–265 (2005)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Masuda, N.: Participation costs dismiss the advantage of heterogeneous networks in evolution of cooperation. Proc. Roy. Soc. Lond. B 274, 1815–1821 (2007)Google Scholar
  66. 66.
    Szolnoki, A., Perc, M.: Coevolution of teaching activity promotes cooperation. New J. Phys. 10, 043036 (2008)Google Scholar
  67. 67.
    Perc, M., Szolnoki, A., Szabo, G.: Restricted connections among distinguished players support cooperation. Phys. Rev. E 78, 066101 (2008)Google Scholar
  68. 68.
    Eguiluz, V.M., Zimmermann, M.G., Cela-Conde, C.J., Miguel, M.S.: Cooperation and the emergence of role differentiation in the dynamics of social networks. Am. J. Sociol. 110, 977–1008 (2005)Google Scholar
  69. 69.
    Fu, F., Hauert, C., Nowak, M.A., Wang, L.: Reputation-based partner choice promotes cooperation in social networks. Phys. Rev. E 78, 026117 (2008)Google Scholar
  70. 70.
    Chen, X., Fu, F., Wang, L.: Interaction stochasticity supports cooperation in spatial prisoners dilemma. Phys. Rev. E 78, 051120 (2008)Google Scholar
  71. 71.
    Pestelacci, E., Tomassini, M., Luthi, L.: Evolution of cooperation and coordination in a dynamically networked society. Biol. Theory 3, 139–153 (2008)Google Scholar
  72. 72.
    Antonioni, A., Tomassini, M.: Network fluctuations hinder cooperation in evolutionary games. PLoS One 6, e25555 (2011)Google Scholar
  73. 73.
    Szolnoki, A., Szabo, G.: Cooperation enhanced by inhomogeneous activity of teaching for evolutionary Prisoner’s Dilemma games. Europhys. Lett. 77, 30004 (2007)MathSciNetGoogle Scholar
  74. 74.
    Szolnoki, A., Perc, M.: Promoting cooperation in social dilemmas via simple coevolutionary rules. Eur. Phys. J. B 67, 337–344 (2009)zbMATHGoogle Scholar
  75. 75.
    McNamara, J.M., Barta, Z., Fromhage, L., Houston, A.I.: The coevolution of choosiness and cooperation. Nature 451(7175), 189 (2008)Google Scholar
  76. 76.
    Szolnoki, A., Perc, M., Szabo, G., Stark, H.U.: Impact of aging on the evolution of cooperation in the spatial prisoner’s dilemma game. Phys. Rev. E 80, 021901 (2009)Google Scholar
  77. 77.
    Poncela, J., Gomez-Gardenes, J., Traulsen, A., Moreno, Y.: Evolutionary game dynamics in a growing structured population. New J. Phys. 11, 083031 (2009)Google Scholar
  78. 78.
    O’Toole, G., Kaplan, H.B., Kolter, R.: Biofilm formation as microbial development. Annu. Rev. Microbiol. 54, 49–79 (2000)Google Scholar
  79. 79.
    Watnick, P., Kolter, R.: Biofilm, city of microbes. J. Bacteriol. 182, 2675–2679 (2000)Google Scholar
  80. 80.
    Okabe, S., Hiratia, K., Ozawa, Y., Watanabe, Y.: Spatial microbial distributions of nitrifiers and heterotrophs in mixed-population biofilms. Biotechnol. Bioeng. 50, 24–35 (1996)Google Scholar
  81. 81.
    Xavier, J.B., Foster, K.R.: Cooperation and conflict in microbial biofilms. Proc. Natl. Acad. Sci. USA 104, 876–881 (2007)Google Scholar
  82. 82.
    Czaran, T.L., Hoekstra, R.F., Pagie, L.: Chemical warfare between microbes promotes biodiversity. Proc. Natl. Acad. Sci. USA 99, 786–790 (2002)Google Scholar
  83. 83.
    Pagie, L., Hogeweg, P.: Colicin diversity: a result of eco-evolutionary dynamics. J. Theor. Biol. 196, 251–261 (1999)Google Scholar
  84. 84.
    Kerr, B., Riley, M.A., Feldman, M.W., Bohannan, B.J.: Local dispersal promotes biodiversity in a real-life game of rock paper scissors. Nature 418, 171 (2002)Google Scholar
  85. 85.
    Kirkup, B.C., Riley, M.A.: Antibiotic-mediated antagonism leads to a bacterial game of rock paper scissors in vivo. Nature 428, 412 (2004)Google Scholar
  86. 86.
    Sinha, S., Grewal, R.K., Roy, S.: Modeling bacteria-phage interactions and implications for phage therapy. Adv. App. Microbiol. 103, 103–141 (2018)Google Scholar
  87. 87.
    Samaddar, S., Grewal, R.K., Sinha, S., Ghosh, S., Roy, S., Das Gupta, S.K.: Dynamics of Mycobacteriophage–Mycobacterial host interaction: evidence for secondary mechanisms for host lethality. Appl. Environ. Microbiol. 82, 124–133 (2016)Google Scholar
  88. 88.
    Turner, P.E., Chao, L.: Prisoner’s dilemma in an RNA virus. Nature 398, 441 (1999)Google Scholar
  89. 89.
    Nowak, M.A., Sigmund, K.: Phage-lift for game theory. Nature 398, 367 (1999)Google Scholar
  90. 90.
    May, R.M., Leonard, W.J.: Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29, 243–253 (1975)MathSciNetzbMATHGoogle Scholar
  91. 91.
    Szabo, G.: Competing associations in six-species predator prey models. J. Phys. A 38, 6689 (2005)MathSciNetzbMATHGoogle Scholar
  92. 92.
    Harper, J.L., Hawksworth, D.L.: Biodiversity: measurement and estimation. Philos. Trans. R. Soc. Lond. B 345, 5–12 (1994)Google Scholar
  93. 93.
    Claussen, J.C., Traulsen, A.: Cyclic dominance and biodiversity in well-mixed populations. Phys. Rev. Lett. 100, 058104 (2008)Google Scholar
  94. 94.
    Smith, J.M.: Evolution-the games lizards play. Nature 380, 198–199 (1996)Google Scholar
  95. 95.
    Sinervo, B., Miles, D.B., Frankino, W.A., Klukowski, M., DeNardo, D.F.: Testosterone, endurance, and Darwinian fitness: natural and sexual selection on the physiological bases of alternative male behaviors in side-blotched lizards. Horm. Behav. 38, 222–233 (2000)Google Scholar
  96. 96.
    Corl, A., Davis, A.R., Kuchta, S.R., Sinervo, B.: Selective loss of polymorphic mating types is associated with rapid phenotypic evolution during morphic speciation. Proc. Natl. Acad. Sci. USA 107, 4254–4259 (2010)Google Scholar
  97. 97.
    Lotka, A.J.: Analytical note on certain rhythmic relations in organic systems. Proc. Natl. Acad. Sci. USA 6, 410–415 (1920)Google Scholar
  98. 98.
    Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926)zbMATHGoogle Scholar
  99. 99.
    Volterra, V.: Variations and fluctuations of the number of individuals in animal species living together. ICES J. Mar. Sci 3, 3–51 (1928)Google Scholar
  100. 100.
    Reichenbach, T., Mobilia, M., Frey, E.: Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games. Nature 448, 1046 (2007)Google Scholar
  101. 101.
    Szolnoki, A., Mobilia, M., Jiang, L.-L., Szczesny, B., Rucklidge, A.M., Perc, M.: Cyclic dominance in evolutionary games: a review. J. R. Soc. Interface. 11, 20140735 (2014)Google Scholar
  102. 102.
    Wang, Z., Wang, L., Szolnoki, A., Perc, M.: Evolutionary games on multilayer networks: a colloquium. Eur. Phys. J. B 88, 124 (2015)Google Scholar
  103. 103.
    Cassar, A.: Coordination and cooperation in local, random and small world networks: experimental evidence. Games Econ. Behav. 58, 209–230 (2007)MathSciNetzbMATHGoogle Scholar
  104. 104.
    Gracia-Lazaro, C., Ferrer, A., Ruiz, G., Tarancon, A., Cuesta, J.A., Sanchez, A., Moreno, Y.: Heterogeneous networks do not promote cooperation when humans play a Prisoner’s Dilemma. Proc. Natl. Acad. Sci. U.S.A. 109, 12922–12926 (2012)Google Scholar
  105. 105.
    Grujic, J., Fosco, C., Araujo, L., Cuesta, J.A., Sanchez, A.: Social experiments in the mesoscale: Humans playing a spatial prisoner’s dilemma. PloS One 5, e13749 (2010)Google Scholar
  106. 106.
    Requejo, R.J., Camacho, J.: Evolution of cooperation mediated by limiting resources: connecting resource based models and evolutionary game theory. J. Theor. Biol. 272, 35–41 (2011)MathSciNetzbMATHGoogle Scholar
  107. 107.
    Gould, N.E.S.J.: Punctuated equilibria: an alternative to phyletic gradualism. In: Ayala, F.J., Avise, J.C. (eds.) Essential Readings in Evolutionary Biology. JHU Press, Baltimore (1972)Google Scholar
  108. 108.
    Tembine, H., Altman, E., ElAzouzi, R., Sandholm, W.H.: Evolutionary game dynamics with migration for hybrid power control in wireless communications. In: 47th IEEE Conference on Decision and Control (2008)Google Scholar
  109. 109.
    Chastain, E., Livnat, A., Papadimitriou, C., Vazirani, U.: Algorithms, games, and evolution. Proc. Natl. Acad. Sci. of U.S.A. 111, 10620–10623 (2014)MathSciNetzbMATHGoogle Scholar
  110. 110.
    Szabo, G., Fath, G.: Evolutionary games on graphs. Phys. Rep. 446, 97–216 (2007)MathSciNetGoogle Scholar

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© Indian Institute of Technology Madras 2019

Authors and Affiliations

  1. 1.Department of PhysicsBose InstituteKolkataIndia

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