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Effect of Variability in Payoffs on Conditions for the Evolution of Cooperation in a Small Population

Author

Listed:
  • Dhaker Kroumi

    (King Fahd University of Petroleum and Minerals)

  • Éloi Martin

    (Université de Montréal)

  • Cong Li

    (Université de Montréal
    Northwestern Polytechnical University)

  • Sabin Lessard

    (Université de Montréal)

Abstract

In this paper, we study the effect of stochastic fluctuations in payoffs for two strategies, cooperation and defection, used in random pairwise interactions in a population of fixed finite size with an update according to a Moran model. We assume that the means, variances and covariances of the payoffs are of the same small order while all higher-order moments are negligible. We show that more variability in the payoffs to defection and less variability in the payoffs to cooperation contribute to the evolutionary success of cooperation over defection as measured by fixation probabilities under weak selection. This conclusion is drawn by comparing the probabilities of ultimate fixation of cooperation and defection as single mutants to each other and to what they would be under neutrality. These comparisons are examined in detail with respect to the population size and the second moments of the payoffs in five cases of additive Prisoner’s Dilemmas. The analysis is extended to a Prisoner’s Dilemma repeated a random number of times with Tit-for-Tat starting with cooperation and Always-Defect as strategies. Moreover, simulations with an update according to a Wright–Fisher model suggest that the conclusions are robust.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:dyngam:v:11:y:2021:i:4:d:10.1007_s13235-021-00383-2
    DOI: 10.1007/s13235-021-00383-2
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    References listed on IDEAS

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    1. 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.
    2. Bin Wu & Arne Traulsen & Chaitanya S. Gokhale, 2013. "Dynamic Properties of Evolutionary Multi-player Games in Finite Populations," Games, MDPI, vol. 4(2), pages 1-18, May.
    3. Fudenberg, Drew & Imhof, Lorens A., 2006. "Imitation processes with small mutations," Journal of Economic Theory, Elsevier, vol. 131(1), pages 251-262, November.
    4. 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.
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    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. Zhang, Qinchunxue & Shu, Lan & Jiang, Bichuan, 2023. "Moran process in evolutionary game dynamics with interval payoffs and its application," Applied Mathematics and Computation, Elsevier, vol. 446(C).

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