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Emergence of cooperation and evolutionary stability in finite populations

Nature volume 428, pages 646–650 (2004)Cite this article

Abstract

To explain the evolution of cooperation by natural selection has been a major goal of biologists since Darwin. Cooperators help others at a cost to themselves, while defectors receive the benefits of altruism without providing any help in return. The standard game dynamical formulation is the ‘Prisoner's Dilemma’1,2,3,4,5,6,7,8,9,10,11, in which two players have a choice between cooperation and defection. In the repeated game, cooperators using direct reciprocity cannot be exploited by defectors, but it is unclear how such cooperators can arise in the first place12,13,14,15. In general, defectors are stable against invasion by cooperators. This understanding is based on traditional concepts of evolutionary stability and dynamics in infinite populations16,17,18,19,20. Here we study evolutionary game dynamics in finite populations21,22,23,24,25. We show that a single cooperator using a strategy like ‘tit-for-tat’ can invade a population of defectors with a probability that corresponds to a net selective advantage. We specify the conditions required for natural selection to favour the emergence of cooperation and define evolutionary stability in finite populations.

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Figure 1: Selection can favour the replacement of AllD by TFT in finite populations.
Figure 2: The 1/3-law of frequency-dependent evolution.
Figure 3: A strategy is ESSN if it is protected by selection against invasion and replacement by another strategy for given N and any w > 0.

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Acknowledgements

The Program for Evolutionary Dynamics is supported by J. Epstein.

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Authors and Affiliations

  1. Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts, 02138, USA

    Martin A. Nowak & Christine Taylor

  2. Department of Organismic and Evolutionary Biology and Department of Mathematics, Harvard University, Cambridge, Massachusetts, 02138, USA

    Martin A. Nowak

  3. Department of Economics, Harvard University, Cambridge, Massachusetts, 02138, USA

    Drew Fudenberg

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

    Akira Sasaki

  5. Department of Mathematics, MIT, Cambridge, Massachusetts, 02139, USA

    Christine Taylor

Authors
  1. Martin A. Nowak
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  2. Akira Sasaki
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  4. Drew Fudenberg
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Correspondence to Martin A. Nowak.

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Nowak, M., Sasaki, A., Taylor, C. et al. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646–650 (2004). https://doi.org/10.1038/nature02414

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