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Randomized matrix games in a finite population: Effect of stochastic fluctuations in the payoffs on the evolution of cooperation

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  • Li, Cong
  • Lessard, Sabin

Abstract

A diffusion approximation for a randomized 2 × 2-matrix game in a large finite population is ascertained in the case of random payoffs whose expected values, variances and covariances are of order given by the inverse of the population size N. Applying the approximation to a Randomized Prisoner’s Dilemma (RPD) with independent payoffs for cooperation and defection in random pairwise interactions, conditions on the variances of the payoffs for selection to favor the evolution of cooperation, favor more the evolution of cooperation than the evolution of defection, and disfavor the evolution of defection are deduced. All these are obtained from probabilities of ultimate fixation of a single mutant. It is shown that the conditions are lessened with an increase in the variances of the payoffs for defection against cooperation and defection and a decrease in the variances of the payoffs for cooperation against cooperation and defection. A RPD game with independent payoffs whose expected values are additive is studied in detail to support the conclusions. Randomized matrix games with non-independent payoffs, namely the RPD game with additive payoffs for cooperation and defection based on random cost and benefit for cooperation and the repeated RPD game with Tit-for-Tat and Always-Defect as strategies in pairwise interactions with a random number of rounds, are studied under the assumption that the population-scaled expected values, variances and covariances of the payoffs are all of the same small enough order. In the first model, the conditions in favor of the evolution of cooperation hold only if the covariance between the cost and the benefit is large enough, while the analysis of the second model extends the results on the effects of the variances of the payoffs for cooperation and defection found for the one-round RPD game.

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  • Li, Cong & Lessard, Sabin, 2020. "Randomized matrix games in a finite population: Effect of stochastic fluctuations in the payoffs on the evolution of cooperation," Theoretical Population Biology, Elsevier, vol. 134(C), pages 77-91.
    • Handle: RePEc:eee:thpobi:v:134:y:2020:i:c:p:77-91
    DOI: 10.1016/j.tpb.2020.04.006
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    References listed on IDEAS

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    1. Drew Fudenberg & David G. Rand & Anna Dreber, 2012. "Slow to Anger and Fast to Forgive: Cooperation in an Uncertain World," American Economic Review, American Economic Association, vol. 102(2), pages 720-749, April.
    2. Hisashi Ohtsuki & Christoph Hauert & Erez Lieberman & Martin A. Nowak, 2006. "A simple rule for the evolution of cooperation on graphs and social networks," Nature, Nature, vol. 441(7092), pages 502-505, May.
    3. Martin A. Nowak & Akira Sasaki & Christine Taylor & Drew Fudenberg, 2004. "Emergence of cooperation and evolutionary stability in finite populations," Nature, Nature, vol. 428(6983), pages 646-650, April.
    4. Philippe Uyttendaele & Frank Thuijsman & Pieter Collins & Ralf Peeters & Gijs Schoenmakers & Ronald Westra, 2012. "Evolutionary Games and Periodic Fitness," Dynamic Games and Applications, Springer, vol. 2(3), pages 335-345, September.
    5. Christian Hilbe & Štěpán Šimsa & Krishnendu Chatterjee & Martin A. Nowak, 2018. "Evolution of cooperation in stochastic games," Nature, Nature, vol. 559(7713), pages 246-249, July.
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    Cited by:

    1. Kroumi, Dhaker & Martin, Éloi & Lessard, Sabin, 2022. "Evolution of cooperation with respect to fixation probabilities in multi-player games with random payoffs," Theoretical Population Biology, Elsevier, vol. 145(C), pages 1-21.
    2. Lessard, Sabin & Li, Cong & Zheng, Xiu-Deng & Tao, Yi, 2021. "Inclusive fitness and Hamilton’s rule in a stochastic environment," Theoretical Population Biology, Elsevier, vol. 142(C), pages 91-99.
    3. Dhaker Kroumi & Éloi Martin & Cong Li & Sabin Lessard, 2021. "Effect of Variability in Payoffs on Conditions for the Evolution of Cooperation in a Small Population," Dynamic Games and Applications, Springer, vol. 11(4), pages 803-834, December.

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