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Multiple equilibria and limit cycles in evolutionary games with Logit Dynamics

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  • Hommes, Cars H.
  • Ochea, Marius I.

Abstract

This note shows, by means of two simple, three-strategy games, the existence of stable periodic orbits and of multiple, interior steady states in a smooth version of the Best-Response Dynamics, the Logit Dynamics. The main finding is that, unlike Replicator Dynamics, generic Hopf bifurcation and thus, stable limit cycles, occur under the Logit Dynamics, even for three-strategy games. We also show that the Logit Dynamics displays another bifurcation which cannot occur under the Replicator Dynamics: the fold bifurcation, with non-monotonic creation and disappearance of steady states.

Suggested Citation

  • Hommes, Cars H. & Ochea, Marius I., 2012. "Multiple equilibria and limit cycles in evolutionary games with Logit Dynamics," Games and Economic Behavior, Elsevier, vol. 74(1), pages 434-441.
  • Handle: RePEc:eee:gamebe:v:74:y:2012:i:1:p:434-441
    DOI: 10.1016/j.geb.2011.05.014
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Kreindler, Gabriel E. & Young, H. Peyton, 2013. "Fast convergence in evolutionary equilibrium selection," Games and Economic Behavior, Elsevier, vol. 80(C), pages 39-67.
    2. Franke, Reiner, 2014. "Aggregate sentiment dynamics: A canonical modelling approach and its pleasant nonlinearities," Structural Change and Economic Dynamics, Elsevier, vol. 31(C), pages 64-72.
    3. Zhang, Boyu & Hofbauer, Josef, 2016. "Quantal response methods for equilibrium selection in 2×2 coordination games," Games and Economic Behavior, Elsevier, vol. 97(C), pages 19-31.
    4. Timothy N. Cason & Daniel Friedman & ED Hopkins, 2014. "Cycles and Instability in a Rock--Paper--Scissors Population Game: A Continuous Time Experiment," Review of Economic Studies, Oxford University Press, vol. 81(1), pages 112-136.
    5. Ochea, Marius-Ionut, 2013. "Evolution of repeated prisoner's dilemma play under logit dynamics," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2483-2499.
    6. Michele Berardi, 2015. "Expectations formation under adaptive learning and evolutionary dynamics," Centre for Growth and Business Cycle Research Discussion Paper Series 206, Economics, The Univeristy of Manchester.
    7. Xu, Bin & Zhou, Hai-Jun & Wang, Zhijian, 2013. "Cycle frequency in standard Rock–Paper–Scissors games: Evidence from experimental economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(20), pages 4997-5005.

    More about this item

    Keywords

    Evolutionary games; Logit dynamics; Hopf bifurcation; Fold bifurcation;

    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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