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Imitative dynamics for games with continuous strategy space

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  • Cheung, Man-Wah

Abstract

This paper studies imitative dynamics for games with continuous strategy space. We define imitative dynamics—which include the replicator dynamic as a special case—as evolutionary dynamics that satisfy the imitative property and payoff monotonicity. Our definition of payoff monotonicity, which we use Radon–Nikodym derivatives to define, is weaker than the one proposed in Oechssler and Riedel (2002). We find that Oechssler and Riedel (2002)'s definition is too strong, and our definition is more adequate than theirs. We show that for a broad class of payoff functional dynamics, payoff monotonicity à la Oechssler and Riedel (2002) is equivalent to aggregate monotonicity in the sense of Samuelson and Zhang (1992). We then provide sufficient conditions for imitative dynamics and general evolutionary dynamics to be well-defined. Finally, with our definition of payoff monotonicity, a number of results that are standard for finite games extend to the case of games with continuous strategy space.

Suggested Citation

  • Cheung, Man-Wah, 2016. "Imitative dynamics for games with continuous strategy space," Games and Economic Behavior, Elsevier, vol. 99(C), pages 206-223.
    • Handle: RePEc:eee:gamebe:v:99:y:2016:i:c:p:206-223
    DOI: 10.1016/j.geb.2016.08.003
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    References listed on IDEAS

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    1. Oechssler, Jorg & Riedel, Frank, 2002. "On the Dynamic Foundation of Evolutionary Stability in Continuous Models," Journal of Economic Theory, Elsevier, vol. 107(2), pages 223-252, December.
    2. Ritzberger, Klaus & Weibull, Jorgen W, 1995. "Evolutionary Selection in Normal-Form Games," Econometrica, Econometric Society, vol. 63(6), pages 1371-1399, November.
    3. Hofbauer, Josef & Oechssler, Jörg & Riedel, Frank, 2009. "Brown-von Neumann-Nash dynamics: The continuous strategy case," Games and Economic Behavior, Elsevier, vol. 65(2), pages 406-429, March.
    4. Cheung, Man-Wah, 2014. "Pairwise comparison dynamics for games with continuous strategy space," Journal of Economic Theory, Elsevier, vol. 153(C), pages 344-375.
    5. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
    6. Samuelson, Larry & Zhang, Jianbo, 1992. "Evolutionary stability in asymmetric games," Journal of Economic Theory, Elsevier, vol. 57(2), pages 363-391, August.
    7. Lahkar, Ratul & Riedel, Frank, 2015. "The logit dynamic for games with continuous strategy sets," Games and Economic Behavior, Elsevier, vol. 91(C), pages 268-282.
    8. Cressman, Ross, 2005. "Stability of the replicator equation with continuous strategy space," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 127-147, September.
    9. Friedman, Daniel, 1991. "Evolutionary Games in Economics," Econometrica, Econometric Society, vol. 59(3), pages 637-666, May.
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    Cited by:

    1. Cheung, Man-Wah & Lahkar, Ratul, 2018. "Nonatomic potential games: the continuous strategy case," Games and Economic Behavior, Elsevier, vol. 108(C), pages 341-362.
    2. Lahkar, Ratul & Mukherjee, Saptarshi, 2019. "Evolutionary implementation in a public goods game," Journal of Economic Theory, Elsevier, vol. 181(C), pages 423-460.
    3. Jiabin Wu, 2021. "Stochastic Value Formation," Dynamic Games and Applications, Springer, vol. 11(3), pages 597-611, September.
    4. Cheung, Man-Wah & Wu, Jiabin, 2018. "On the probabilistic transmission of continuous cultural traits," Journal of Economic Theory, Elsevier, vol. 174(C), pages 300-323.
    5. Sarvesh Bandhu & Ratul Lahkar, 2021. "Implementation in Large Population Games with Multiple Equilibria," Working Papers 62, Ashoka University, Department of Economics.
    6. Sarvesh Bandhu & Ratul Lahkar, 2023. "Evolutionary robustness of dominant strategy implementation," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 76(2), pages 685-721, August.
    7. Lahkar, Ratul & Mukherjee, Saptarshi, 2021. "Evolutionary implementation in aggregative games," Mathematical Social Sciences, Elsevier, vol. 109(C), pages 137-151.
    8. Georgiy Karev, 2020. "Dynamics of Strategy Distributions in a One-Dimensional Continuous Trait Space for Games with a Quadratic Payoff Function," Games, MDPI, vol. 11(1), pages 1-12, March.
    9. Lahkar, Ratul & Mukherjee, Sayan & Roy, Souvik, 2023. "The logit dynamic in supermodular games with a continuum of strategies: A deterministic approximation approach," Games and Economic Behavior, Elsevier, vol. 139(C), pages 133-160.
    10. Ratul Lahkar & Vinay Ramani, 2022. "An Evolutionary Approach to Pollution Control in Competitive Markets," Dynamic Games and Applications, Springer, vol. 12(3), pages 872-896, September.
    11. Ratul Lahkar & Vinay Ramani, 2021. "An Evolutionary Approach to Pollution Control in Competitive Markets," Working Papers 68, Ashoka University, Department of Economics.
    12. Wenjun Hu & Haiyan Tian & Gang Zhang, 2019. "Bifurcation Analysis of Three-Strategy Imitative Dynamics with Mutations," Complexity, Hindawi, vol. 2019, pages 1-8, October.
    13. Rabanal, Jean Paul & Lee, Dongwook, 2017. "On the dynamic stability of a price dispersion model using gradient dynamics," Journal of Economic Dynamics and Control, Elsevier, vol. 84(C), pages 32-42.
    14. Ratul Lahkar, 2020. "Convergence to Walrasian equilibrium with minimal information," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 15(3), pages 553-578, July.
    15. Jonathan Newton, 2018. "Evolutionary Game Theory: A Renaissance," Games, MDPI, vol. 9(2), pages 1-67, May.
    16. Jacquot, Paulin & Wan, Cheng, 2022. "Nonatomic aggregative games with infinitely many types," European Journal of Operational Research, Elsevier, vol. 301(3), pages 1149-1165.
    17. Lahkar, Ratul & Mukherjee, Sayan & Roy, Souvik, 2022. "Generalized perturbed best response dynamics with a continuum of strategies," Journal of Economic Theory, Elsevier, vol. 200(C).
    18. Ratul Lahkar & Sayan Mukherjee & Souvik Roy, 2022. "A Deterministic Approximation Approach to the Continuum Logit Dynamic with an Application to Supermodular Games," Working Papers 79, Ashoka University, Department of Economics.
    19. RatulLahkar & Sayan Mukherjee & Souvik Roy, 2021. "Generalized Perturbed Best Response Dynamics with a Continuum of Strategies," Working Papers 51, Ashoka University, Department of Economics.

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    More about this item

    Keywords

    Imitative dynamics; Replicator dynamic; Payoff monotonicity; Continuous strategy space;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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