Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Evolutionary games and spatial chaos


MUCH attention has been given to the Prisoners' Dilemma as a metaphor for the problems surrounding the evolution of coopera-tive behaviour1–6. This work has dealt with the relative merits of various strategies (such as tit-for-tat) when players who recognize each other meet repeatedly, and more recently with ensembles of strategies and with the effects of occasional errors. Here we neglect all strategical niceties or memories of past encounters, considering only two simple kinds of players: those who always cooperate and those who always defect. We explore the consequences of placing these players in a two-dimensional spatial array: in each round, every individual 'plays the game' with the immediate neighbours; after this, each site is occupied either by its original owner or by one of the neighbours, depending on who scores the highest total in that round; and so to the next round of the game. This simple, and purely deterministic, spatial version of the Prisoners' Dilemma, with no memories among players and no strategical elaboration, can generate chaotically changing spatial patterns, in which cooperators and defectors both persist indefinitely (in fluctuating proportions about predictable long-term averages). If the starting configurations are sufficiently symmetrical, these ever-changing sequences of spatial patterns—dynamic fractals—can be extraordinarily beautiful, and have interesting mathematical properties. There are potential implications for the dynamics of a wide variety of spatially extended systems in physics and biology.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it


Prices may be subject to local taxes which are calculated during checkout


  1. Axelrod, R. & Hamilton, W. D. Science 211, 1390–1396 (1981).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  2. Axelrod, R. The Evolution of Cooperation (Basic Books, New York, 1984; reprinted by Penguin, Harmondsworth, 1989).

    MATH  Google Scholar 

  3. Nowak, M. A. & Sigmund, K. Nature 355, 250–253 (1992).

    Article  ADS  Google Scholar 

  4. Nowak, M. A. & Sigmund, K. Acta. appl. Math. 20, 247–265 (1990).

    Article  MathSciNet  Google Scholar 

  5. Milinski, M. Nature 325, 434–435 (1987).

    Article  ADS  Google Scholar 

  6. May, R. M. Nature 327, 15–17 (1987).

    Article  ADS  Google Scholar 

  7. Nowak, M. A. & May, R. M. Int. J. Chaos Bifurc. (in the press)

  8. Gardner, M. Sci. Am. 120–123 (October 1970).

  9. Poundstone, W. The Recursive Universe (Morrow, New York, 1985).

    MATH  Google Scholar 

  10. Wolfram, S. Nature 311, 419–424 (1984).

    Article  ADS  Google Scholar 

  11. Langton, C. G. Physica D22, 120–140 (1986).

    MathSciNet  Google Scholar 

  12. Maynard Smith, J. Evolution and the Theory of Games (Cambridge Univ. Press, UK, 1982).

    Book  Google Scholar 

  13. Hamilton, W. D. in Man and Best: Comparative Social Behaviour (eds Eisenberg, J. F. & Dillon, W. S.) (Smithsonian, Washington DC, 1971).

    Google Scholar 

  14. Wilson, D. S. The Natural Selection of Populations and Communities (Benjamin Cummings, Menlo Park, 1980).

    Google Scholar 

  15. Wilson, D. S., Pollock, G. B. & Dugatkin, L. A. Evol. Ecol 6, 331–341 (1992).

    Article  Google Scholar 

  16. Axelrod, R. & Dion, D. Science 242, 1385–1390 (1988).

    Article  ADS  CAS  Google Scholar 

  17. Lombardo, M. P. Science 227, 1363–1365 (1985).

    Article  ADS  CAS  Google Scholar 

  18. Wilkinson, G. S. Nature 308, 181–184 (1984).

    Article  ADS  Google Scholar 

  19. Eigen, M. & Schuster, P. Naturwissenschaften 64, 541 (1977).

    Article  ADS  CAS  Google Scholar 

  20. Maynard Smith, J. Nature 280, 445–446 (1979).

    Article  Google Scholar 

  21. Boeerlijst, M. C. & Hogeweg, P. Physica D48, 17–28 (1991).

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Nowak, M., May, R. Evolutionary games and spatial chaos. Nature 359, 826–829 (1992).

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing