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Variation in behaviour promotes cooperation in the Prisoner's Dilemma game

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Abstract

The Prisoner's Dilemma game1,2,3,4 is widely used to investigate how cooperation between unrelated individuals can evolve by natural selection. In this game, each player can either ‘cooperate’ (invest in a common good) or ‘defect’ (exploit the other's investment). If the opponent cooperates, you get R if you cooperate and T if you defect. If the opponent defects, you get S if you cooperate and P if you defect. Here T > R > 0 and P > S, so that ‘defect’ is the best response to any action by the opponent. Thus in a single play of the game, each player should defect. In our game, a fixed maximum number of rounds of the Prisoner's Dilemma game is played against the same opponent. A standard argument based on working backwards from the last round1,5 shows that defection on all rounds is the only stable outcome. In contrast, we show that if extrinsic factors maintain variation in behaviour, high levels of co-operation are stable. Our results highlight the importance of extrinsic variability in determining the outcome of evolutionary games.

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Figure 1: The best response (continuous line) as a function of the variation in the degree of cooperation in the population.
Figure 2: The proportion of each type of individual (as specified by n) in the population at evolutionary stability.
Figure 3: Levels of cooperation in the population, measured as E(n)/N, at evolutionary stability, shown as a function of the mutation parameter ɛ.

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Acknowledgements

We thank A. Dornhaus, M. Enquist, E. Fehr and L.-A. Giraldeau for comments on a previous version of this Letter.Authors' contributions J.M.M. formulated the main ideas as a result of conversations with A.I.H.; J.M.M. also formulated the model, and was responsible for the material in Box 1; Z.B. carried out the computations, and prepared the figures; A.I.H. surveyed the literature, and had the main responsibility for writing the Letter.

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Authors and Affiliations

  1. Department of Mathematics, University of Bristol, University Walk, BS8 1TW, Bristol, UK

    John M. McNamara & Zoltan Barta

  2. School of Biological Sciences, University of Bristol, Woodland Road, BS8 1UG, Bristol, UK

    Alasdair I. Houston

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  1. John M. McNamara
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  2. Zoltan Barta
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  3. Alasdair I. Houston
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Correspondence to John M. McNamara.

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Payoffs and best responses in the game. (DOC 76 kb)

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McNamara, J., Barta, Z. & Houston, A. Variation in behaviour promotes cooperation in the Prisoner's Dilemma game. Nature 428, 745–748 (2004). https://doi.org/10.1038/nature02432

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