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Dynamics of Strategy Distributions in a One-Dimensional Continuous Trait Space for Games with a Quadratic Payoff Function

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  • Georgiy Karev

    (National Center for Biotechnology Information, National Institutes of Health, Bldg. 38A, 8600 Rockville Pike, Bethesda, MD 20894, USA)

Abstract

Evolution of distribution of strategies in game theory is an interesting question that has been studied only for specific cases. Here I develop a general method to extend analysis of the evolution of continuous strategy distributions given a quadratic payoff function for any initial distribution in order to answer the following question—given the initial distribution of strategies in a game, how will it evolve over time? I look at several specific examples, including normal distribution on the entire line, normal truncated distribution, as well as exponential and uniform distributions. I show that in the case of a negative quadratic term of the payoff function, regardless of the initial distribution, the current distribution of strategies becomes normal, full or truncated, and it tends to a distribution concentrated in a single point so that the limit state of the population is monomorphic. In the case of a positive quadratic term, the limit state of the population may be dimorphic. The developed method can now be applied to a broad class of questions pertaining to evolution of strategies in games with different payoff functions and different initial distributions.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jgames:v:11:y:2020:i:1:p:14-:d:327223
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    References listed on IDEAS

    as
    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. Cheung, Man-Wah, 2016. "Imitative dynamics for games with continuous strategy space," Games and Economic Behavior, Elsevier, vol. 99(C), pages 206-223.
    3. Cressman, Ross, 2005. "Stability of the replicator equation with continuous strategy space," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 127-147, September.
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