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Cycling in a stochastic learning algorithm for normal form games

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  • Martin Posch

    () (Institut f, r Medizinische Statistik der Universit, t Wien, Schwarzspanierstra, e 17, A-1090 Vienna, Austria)

Abstract

In this paper we study a stochastic learning model for 2, 2 normal form games that are played repeatedly. The main emphasis is put on the emergence of cycles. We assume that the players have neither information about the payoff matrix of their opponent nor about their own. At every round each player can only observe his or her action and the payoff he or she receives. We prove that the learning algorithm, which is modeled by an urn scheme proposed by Arthur (1993), leads with positive probability to a cycling of strategy profiles if the game has a mixed Nash equilibrium. In case there are strict Nash equilibria, the learning process converges a.s. to the set of Nash equilibria.

Suggested Citation

  • Martin Posch, 1997. "Cycling in a stochastic learning algorithm for normal form games," Journal of Evolutionary Economics, Springer, vol. 7(2), pages 193-207.
  • Handle: RePEc:spr:joevec:v:7:y:1997:i:2:p:193-207
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    Citations

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    Cited by:

    1. M. Keilbach & M. Posch, 1998. "Network Externalities and the Dynamics of Markets," Working Papers ir98089, International Institute for Applied Systems Analysis.
    2. Hopkins, Ed, 1999. "A Note on Best Response Dynamics," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 138-150, October.
    3. Mengel, Friederike, 2012. "Learning across games," Games and Economic Behavior, Elsevier, vol. 74(2), pages 601-619.
    4. Ed Hopkins, 2002. "Two Competing Models of How People Learn in Games," Econometrica, Econometric Society, vol. 70(6), pages 2141-2166, November.
    5. Laslier, Jean-Francois & Topol, Richard & Walliser, Bernard, 2001. "A Behavioral Learning Process in Games," Games and Economic Behavior, Elsevier, vol. 37(2), pages 340-366, November.
    6. Izquierdo, Luis R. & Izquierdo, Segismundo S. & Gotts, Nicholas M. & Polhill, J. Gary, 2007. "Transient and asymptotic dynamics of reinforcement learning in games," Games and Economic Behavior, Elsevier, vol. 61(2), pages 259-276, November.
    7. Werner Güth & Hartmut Kliemt & Bezalel Peleg, 2000. "Co-evolution of Preferences and Information in Simple Games of Trust," German Economic Review, Verein für Socialpolitik, vol. 1(1), pages 83-110, February.
    8. Beggs, A.W., 2005. "On the convergence of reinforcement learning," Journal of Economic Theory, Elsevier, vol. 122(1), pages 1-36, May.
    9. Hopkins, Ed & Posch, Martin, 2005. "Attainability of boundary points under reinforcement learning," Games and Economic Behavior, Elsevier, vol. 53(1), pages 110-125, October.
    10. Cominetti, Roberto & Melo, Emerson & Sorin, Sylvain, 2010. "A payoff-based learning procedure and its application to traffic games," Games and Economic Behavior, Elsevier, vol. 70(1), pages 71-83, September.
    11. Alanyali, Murat, 2010. "A note on adjusted replicator dynamics in iterated games," Journal of Mathematical Economics, Elsevier, vol. 46(1), pages 86-98, January.
    12. Ianni, Antonella, 2014. "Learning strict Nash equilibria through reinforcement," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 148-155.
    13. Giovanni Dosi & Marco Faillo & Luigi Marengo, 2018. "Beyond "Bounded Rationality": Behaviours and Learning in Complex Evolving Worlds," LEM Papers Series 2018/26, Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy.
    14. Windrum, Paul, 1999. "Simulation models of technological innovation: A Review," Research Memorandum 005, Maastricht University, Maastricht Economic Research Institute on Innovation and Technology (MERIT).
    15. Possajennikov, A., 1997. "An Analysis of a Simple Reinforcement Dynamics : Learning to Play an "Egalitarian" Equilibrium," Discussion Paper 1997-19, Tilburg University, Center for Economic Research.
    16. Ianni, A., 2002. "Reinforcement learning and the power law of practice: some analytical results," Discussion Paper Series In Economics And Econometrics 0203, Economics Division, School of Social Sciences, University of Southampton.
    17. Mario Bravo, 2016. "An Adjusted Payoff-Based Procedure for Normal Form Games," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1469-1483, November.
    18. Georgios Chasparis & Jeff Shamma & Anders Rantzer, 2015. "Nonconvergence to saddle boundary points under perturbed reinforcement learning," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(3), pages 667-699, August.
    19. Panayotis Mertikopoulos & William H. Sandholm, 2016. "Learning in Games via Reinforcement and Regularization," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1297-1324, November.
    20. Mario Bravo & Mathieu Faure, 2013. "Reinforcement Learning with Restrictions on the Action Set," AMSE Working Papers 1335, Aix-Marseille School of Economics, Marseille, France, revised 01 Jul 2013.
    21. Darmon, Eric & Waldeck, Roger, 2005. "Convergence of reinforcement learning to Nash equilibrium: A search-market experiment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 355(1), pages 119-130.
    22. Roger Waldeck & Eric Darmon, 2006. "Can boundedly rational sellers learn to play Nash?," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 1(2), pages 147-169, November.
    23. Max Keilbach, 1999. "Network Externalities and the Path Dependence of Markets: Will Bill Gates Make It?," Computing in Economics and Finance 1999 711, Society for Computational Economics.
    24. M. Posch & A. Pichler & K. Sigmund, 1998. "The Efficiency of Adapting Aspiration Levels," Working Papers ir98103, International Institute for Applied Systems Analysis.
    25. Antonio Cabrales & Walter Garcia Fontes, 2000. "Estimating learning models from experimental data," Economics Working Papers 501, Department of Economics and Business, Universitat Pompeu Fabra.

    More about this item

    Keywords

    Evolutionary games ; Learning ; Bounded rationality ; Learning algorithms;

    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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