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Evolution of cooperation in a finite homogeneous graph

Abstract

Recent theoretical studies of selection in finite structured populations1,2,3,4,5,6,7 have worked with one of two measures of selective advantage of an allele: fixation probability and inclusive fitness. Each approach has its own analytical strengths, but given certain assumptions they provide equivalent results1. In most instances the structure of the population can be specified by a network of nodes connected by edges (that is, a graph)8,9,10, and much of the work here has focused on a continuous-time model of evolution, first described by ref. 11. Working in this context, we provide an inclusive fitness analysis to derive a surprisingly simple analytical condition for the selective advantage of a cooperative allele in any graph for which the structure satisfies a general symmetry condition (‘bi-transitivity’). Our results hold for a broad class of population structures, including most of those analysed previously, as well as some for which a direct calculation of fixation probability has appeared intractable. Notably, under some forms of population regulation, the ability of a cooperative allele to invade is seen to be independent of the nature of population structure (and in particular of how game partnerships are specified) and is identical to that for an unstructured population. For other types of population regulation our results reveal that cooperation can invade if players choose partners along relatively ‘high-weight’ edges.

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Figure 1: Simulation results for the DB model with bi-transitive graphs of degree 3, and population sizes N = 6 to N = 20.

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References

  1. Rousset, F. & Billiard, S. A theoretical basis for measures of kin selection in subdivided populations: finite populations and localized dispersal. J. Evol. Biol. 13, 814–825 (2000)

    Article  Google Scholar 

  2. Taylor, P. D., Irwin, A. J. & Day, T. Inclusive fitness in finite deme-structured and stepping-stone populations. Selection 1, 83–93 (2000)

    Google Scholar 

  3. Proulx, S. R. & Day, T. What can invasion analyses tell us about evolution under stochasticity? Selection 2, 1–16 (2001)

    Article  Google Scholar 

  4. Nowak, M. A., Sasaki, A., Taylor, C. & Fudenberg, D. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646–650 (2004)

    Article  ADS  CAS  PubMed  Google Scholar 

  5. Wild, G. & Taylor, P. D. Fitness and evolutionary stability in game theoretic models of finite populations. Proc. R. Soc. Lond. B 271, 2345–2349 (2004)

    Article  Google Scholar 

  6. Lessard, S. Long-term stability from fixation probabilities in finite populations: new perspectives for ESS theory. Theor. Popul. Biol. 68, 19–27 (2005)

    Article  PubMed  Google Scholar 

  7. Orzack, S. H. & Hines, W. G. S. The evolution of strategy variation: will an ESS evolve? Evolution 59, 1183–1193 (2005)

    Article  PubMed  Google Scholar 

  8. Lieberman, E., Hauert, C. & Nowak, M. A. Evolutionary dynamics on graphs. Nature 433, 312–316 (2005)

    Article  ADS  CAS  PubMed  Google Scholar 

  9. Ohtsuki, H., Hauert, C., Lieberman, E. & Nowak, M. A. A simple rule for the evolution of cooperation on graphs. Nature 441, 502–505 (2006)

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  10. Ohtsuki, H. & Nowak, M. A. Evolutionary games on cycles. Proc. R. Soc. B 273, 2249–2256 (2006)

    Article  PubMed  PubMed Central  Google Scholar 

  11. Moran, P. A. P. Statistical Processes of Evolutionary Theory (Oxford, Clarendon, 1962)

    MATH  Google Scholar 

  12. Hamilton, W. D. The genetical evolution of social behaviour, I and II. J. Theor. Biol. 7, 1–52 (1964)

    Article  CAS  PubMed  Google Scholar 

  13. Wilson, D. S., Pollock, G. B. & Dugatkin, L. A. Can altruism evolve in a purely viscous population? Evol. Ecol. 6, 331–341 (1992)

    Article  Google Scholar 

  14. Taylor, P. D. Altruism in viscous populations – an inclusive fitness model. Evol. Ecol. 6, 352–356 (1992)

    Article  Google Scholar 

  15. Taylor, P. D. Inclusive fitness in a homogeneous environment. Proc. R. Soc. Lond. B 249, 299–302 (1992)

    Article  ADS  Google Scholar 

  16. Wright, S. Isolation by distance. Genetics 28, 114–138 (1943)

    CAS  PubMed  PubMed Central  Google Scholar 

  17. Michod, R. E. & Hamilton, W. D. Coefficients of relatedness in sociobiology. Nature 288, 694–697 (1980)

    Article  ADS  Google Scholar 

  18. Frank, S. A. Kin selection and virulence in the evolution of protocells and parasites. Proc. R. Soc. Lond. B 258, 153–161 (1994)

    Article  ADS  CAS  Google Scholar 

  19. Taylor, P. D. & Frank, S. How to make a kin selection argument. J. Theor. Biol. 180, 27–37 (1996)

    Article  CAS  PubMed  Google Scholar 

Download references

Acknowledgements

We thank D. Gregory for an exchange of ideas, and A. Gardner, J. Pepper and A. Grafen for many comments. This research was funding by grants to P.D.T. and T.D. from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

Author Contributions All authors contributed equally to this work.

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Correspondence to Peter D. Taylor.

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Taylor, P., Day, T. & Wild, G. Evolution of cooperation in a finite homogeneous graph. Nature 447, 469–472 (2007). https://doi.org/10.1038/nature05784

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