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Asymptotically stable equilibrium and limit cycles in the Rock–Paper–Scissors game in a population of players with complex personalities

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  • Platkowski, Tadeusz
  • Zakrzewski, Jan

Abstract

We investigate a population of individuals who play the Rock–Paper–Scissors (RPS) game. The players choose strategies not only by optimizing their payoffs, but also taking into account the popularity of the strategies. For the standard RPS game, we find an asymptotically stable polymorphism with coexistence of all strategies. For the general RPS game we find the limit cycles. Their stability depends exclusively on two model parameters: the sum of the entries of the RPS payoff matrix, and a sensitivity parameter which characterizes the personality of the players. Apart from the supercritical Hopf bifurcation, we found the subcritical bifurcation numerically for some intervals of the parameters of the model.

Suggested Citation

  • Platkowski, Tadeusz & Zakrzewski, Jan, 2011. "Asymptotically stable equilibrium and limit cycles in the Rock–Paper–Scissors game in a population of players with complex personalities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4219-4226.
    • Handle: RePEc:eee:phsmap:v:390:y:2011:i:23:p:4219-4226
    DOI: 10.1016/j.physa.2011.06.041
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    References listed on IDEAS

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    1. Gaunersdorfer Andrea & Hofbauer Josef, 1995. "Fictitious Play, Shapley Polygons, and the Replicator Equation," Games and Economic Behavior, Elsevier, vol. 11(2), pages 279-303, November.
    2. Yuzuru Sato & Eizo Akiyama & J. Doyne Farmer, 2001. "Chaos in Learning a Simple Two Person Game," Working Papers 01-09-049, Santa Fe Institute.
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    Cited by:

    1. Rocha, André Barreira da Silva & Laruelle, Annick & Zuazo Garín, Peio, 2011. "Replicator Dynamics and Evolutionary Stable Strategies in Heterogeneous Games," IKERLANAK info:eu-repo/grantAgreeme, Universidad del País Vasco - Departamento de Fundamentos del Análisis Económico I.
    2. Kukla, Elżbieta & Płatkowski, Tadeusz, 2013. "Onset of limit cycles in population games with attractiveness driven strategy choice," Chaos, Solitons & Fractals, Elsevier, vol. 56(C), pages 77-82.
    3. Wenjun Hu & Haiyan Tian & Gang Zhang, 2019. "Bifurcation Analysis of Three-Strategy Imitative Dynamics with Mutations," Complexity, Hindawi, vol. 2019, pages 1-8, October.
    4. Qin, Shipeng & Zhang, Gang & Tian, Haiyan & Hu, Wenjun & Zhang, Xiaoming, 2020. "Dynamics of asymmetric division of labor game with environmental feedback," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 543(C).

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