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The replicator equation in stochastic spatial evolutionary games

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  • Chen, Yu-Ting

Abstract

We study the multi-type stochastic evolutionary game with death–birth updating in expanding spatial populations of size N→∞. The model is a voter model perturbation. For typical eligible populations, we require perturbation strengths satisfying 1/N≪w≪1. Under these conditions, the main results prove that the vector density processes of type obey an extended replicator equation in the limit, and the normalized fluctuations converge to a Gaussian process subject to the Wright–Fisher covariance function. In particular, we obtain a positive resolution of a conjecture from [34] that the replicator equation extends to many non-regular graphs.

Suggested Citation

  • Chen, Yu-Ting, 2023. "The replicator equation in stochastic spatial evolutionary games," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 94-139.
    • Handle: RePEc:eee:spapps:v:157:y:2023:i:c:p:94-139
    DOI: 10.1016/j.spa.2022.11.013
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    References listed on IDEAS

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    1. Benjamin Allen & Gabor Lippner & Yu-Ting Chen & Babak Fotouhi & Naghmeh Momeni & Shing-Tung Yau & Martin A. Nowak, 2017. "Evolutionary dynamics on any population structure," Nature, Nature, vol. 544(7649), pages 227-230, April.
    2. Cox, J. Theodore & Durrett, Rick, 2016. "Evolutionary games on the torus with weak selection," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2388-2409.
    3. Aldous, David J., 1982. "Markov chains with almost exponential hitting times," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 305-310, September.
    4. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, April.
    5. Chen, Yu-Ting & Cox, J. Theodore, 2018. "Weak atomic convergence of finite voter models toward Fleming–Viot processes," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2463-2488.
    6. 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.
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