Nothing Special   »   [go: up one dir, main page]
Skip to main content

Thank you for visiting nature.com. 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.

  • Letter
  • Published:

A simple rule for the evolution of cooperation on graphs and social networks

Nature volume 441, pages 502–505 (2006)Cite this article

Abstract

A fundamental aspect of all biological systems is cooperation. Cooperative interactions are required for many levels of biological organization ranging from single cells to groups of animals1,2,3,4. Human society is based to a large extent on mechanisms that promote cooperation5,6,7. It is well known that in unstructured populations, natural selection favours defectors over cooperators. There is much current interest, however, in studying evolutionary games in structured populations and on graphs8,9,10,11,12,13,14,15,16,17. These efforts recognize the fact that who-meets-whom is not random, but determined by spatial relationships or social networks18,19,20,21,22,23,24. Here we describe a surprisingly simple rule that is a good approximation for all graphs that we have analysed, including cycles, spatial lattices, random regular graphs, random graphs and scale-free networks25,26: natural selection favours cooperation, if the benefit of the altruistic act, b, divided by the cost, c, exceeds the average number of neighbours, k, which means b/c > k. In this case, cooperation can evolve as a consequence of ‘social viscosity’ even in the absence of reputation effects or strategic complexity.

Access through your institution
Buy or subscribe

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

Access options

Access through your institution

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

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

Figure 1: The rules of the game.
Figure 2: The simple rule, b/c > k , is in good agreement with numerical simulations.
Figure 3: Some intuition for games on graphs.

Similar content being viewed by others

References

  1. Hamilton, W. D. The genetical evolution of social behaviour. J. Theor. Biol. 7, 1–16 (1964)

    Article  CAS  PubMed  Google Scholar 

  2. Trivers, R. The evolution of reciprocal altruism. Q. Rev. Biol. 46, 35–57 (1971)

    Article  Google Scholar 

  3. Axelrod, R. & Hamilton, W. D. The evolution of cooperation. Science 211, 1390–1396 (1981)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  4. Wilson, E. O. Sociobiology (Harvard Univ. Press, Cambridge, Massachusetts, 1975)

    Google Scholar 

  5. Wedekind, C. & Milinski, M. Cooperation through image scoring in humans. Science 288, 850–852 (2000)

    Article  ADS  CAS  PubMed  Google Scholar 

  6. Fehr, E. & Fischbacher, U. The nature of human altruism. Nature 425, 785–791 (2003)

    Article  ADS  CAS  PubMed  Google Scholar 

  7. Nowak, M. A. & Sigmund, K. Evolution of indirect reciprocity. Nature 437, 1291–1298 (2005)

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  Google Scholar 

  9. Killingback, T. & Doebeli, M. Spatial evolutionary game theory: Hawks and Doves revisited. Proc. R. Soc. Lond. B 263, 1135–1144 (1996)

    Article  ADS  Google Scholar 

  10. Nakamaru, M., Matsuda, H. & Iwasa, Y. The evolution of cooperation in a lattice-structured population. J. Theor. Biol. 184, 65–81 (1997)

    Article  CAS  PubMed  Google Scholar 

  11. van Baalen, M. & Rand, D. A. The unit of selection in viscous populations and the evolution of altruism. J. Theor. Biol. 193, 631–648 (1998)

    Article  CAS  PubMed  Google Scholar 

  12. Mitteldorf, J. & Wilson, D. S. Population viscosity and the evolution of altruism. J. Theor. Biol. 204, 481–496 (2000)

    Article  CAS  PubMed  Google Scholar 

  13. Hauert, C., De Monte, S., Hofbauer, J. & Sigmund, K. Volunteering as red queen mechanism for cooperation in public goods games. Science 296, 1129–1132 (2002)

    Article  ADS  CAS  PubMed  Google Scholar 

  14. Le Galliard, J., Ferriere, R. & Dieckman, U. The adaptive dynamics of altruism in spatially heterogeneous populations. Evolution 57, 1–17 (2003)

    Article  PubMed  Google Scholar 

  15. Hauert, C. & Doebeli, M. Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428, 643–646 (2004)

    Article  ADS  CAS  PubMed  Google Scholar 

  16. Ifti, M., Killingback, T. & Doebeli, M. Effects of neighbourhood size and connectivity on the spatial continuous prisoner's dilemma. J. Theor. Biol. 231, 97–106 (2004)

    Article  MathSciNet  PubMed  Google Scholar 

  17. Santos, F. C. & Pacheco, J. M. Scale-free networks provide a unifying framework for the emergence of cooperation. Phys. Rev. Lett. 95, 098104 (2005)

    Article  ADS  CAS  PubMed  Google Scholar 

  18. Levin, S. A. & Paine, R. T. Disturbance, patch formation, and community structure. Proc. Natl Acad. Sci. USA 71, 2744–2747 (1974)

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  19. Durrett, R. & Levin, S. A. The importance of being discrete (and spatial). Theor. Popul. Biol. 46, 363–394 (1994)

    Article  Google Scholar 

  20. Hassell, M. P., Comins, H. N. & May, R. M. Species coexistence and self-organizing spatial dynamics. Nature 370, 290–292 (1994)

    Article  ADS  Google Scholar 

  21. Skyrms, B. & Pemantle, R. A dynamic model of social network formation. Proc. Natl Acad. Sci. USA 97, 9340–9346 (2000)

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  22. Abramson, G. & Kuperman, M. Social games in a social network. Phys. Rev. E 63, 030901 (2001)

    Article  ADS  CAS  Google Scholar 

  23. Szabó, G. & Vukov, J. Cooperation for volunteering and partially random partnership. Phys. Rev. E 69, 036107 (2004)

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  25. Watts, D. J. & Strogatz, S. H. Collective dynamics of 'small-world' networks. Nature 393, 440–442 (1998)

    Article  ADS  CAS  PubMed  Google Scholar 

  26. Barabasi, A. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  27. Taylor, P. D. & Jonker, L. Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978)

    Article  MathSciNet  Google Scholar 

  28. Hofbauer, J. & Sigmund, K. Evolutionary Games and Population Dynamics (Cambridge Univ. Press, Cambridge, UK, 1998)

    Book  Google Scholar 

  29. 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 

  30. 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 

Download references

Acknowledgements

Support from the John Templeton Foundation, JSPS, NDSEG and Harvard-MIT HST is gratefully acknowledged. The Program for Evolutionary Dynamics at Harvard University is sponsored by Jeffrey Epstein.

Author information

Authors and Affiliations

  1. Department of Biology, Kyushu University, 812-8581, Fukuoka, Japan

    Hisashi Ohtsuki

  2. Program for Evolutionary Dynamics, Department of Organismic and Evolutionary Biology, Department of Mathematics,

    Hisashi Ohtsuki, Christoph Hauert, Erez Lieberman & Martin A. Nowak

  3. Department of Applied Mathematics, Harvard University, Cambridge, Massachusetts, 02138, USA

    Erez Lieberman

Authors
  1. Hisashi Ohtsuki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Hauert
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Erez Lieberman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Martin A. Nowak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin A. Nowak.

Ethics declarations

Competing interests

Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests.

Supplementary information

Supplementary Notes

This file contains the Supplementary Discussion, Supplementary Figure 4 and additional references. (PDF 248 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ohtsuki, H., Hauert, C., Lieberman, E. et al. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505 (2006). https://doi.org/10.1038/nature04605

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature04605

This article is cited by

Search

Advanced search

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