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Attainability of Boundary Points under Reinforcement Learning

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Abstract

This paper investigates the properties of the most common form of reinforcement learning (the "basic model" of Erev and Roth, American Economic Review, 88, 848-881, 1998). Stochastic approximation theory has been used to analyse the local stability of fixed points under this learning process. However, as we show, when such points are on the boundary of the state space, for example, pure strategy equilibria, standard results from the theory of stochastic approximation do not apply. We offer what we believe to be the correct treatment of boundary points, and provide a new and more general result: this model of learning converges with zero probability to fixed points which are unstable under the Maynard Smith or adjusted version of the evolutionary replicator dynamics. For two player games these are the fixed points that are linearly unstable under the standard replicator dynamics.

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  • Ed Hopkins & Martin Posch, 2003. "Attainability of Boundary Points under Reinforcement Learning," Edinburgh School of Economics Discussion Paper Series 79, Edinburgh School of Economics, University of Edinburgh.
  • Handle: RePEc:edn:esedps:79
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    14. Duffy, John & Hopkins, Ed, 2005. "Learning, information, and sorting in market entry games: theory and evidence," Games and Economic Behavior, Elsevier, vol. 51(1), pages 31-62, April.
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    Cited by:

    1. Beggs, Alan, 2022. "Reference points and learning," Journal of Mathematical Economics, Elsevier, vol. 100(C).
    2. Fudenberg, Drew & Takahashi, Satoru, 2011. "Heterogeneous beliefs and local information in stochastic fictitious play," Games and Economic Behavior, Elsevier, vol. 71(1), pages 100-120, January.
    3. Oyarzun, Carlos & Ruf, Johannes, 2014. "Convergence in models with bounded expected relative hazard rates," Journal of Economic Theory, Elsevier, vol. 154(C), pages 229-244.
    4. Naoki Funai, 2019. "Convergence results on stochastic adaptive learning," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(4), pages 907-934, November.
    5. Ianni, Antonella, 2014. "Learning strict Nash equilibria through reinforcement," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 148-155.
    6. Duffy, John & Hopkins, Ed, 2005. "Learning, information, and sorting in market entry games: theory and evidence," Games and Economic Behavior, Elsevier, vol. 51(1), pages 31-62, April.
    7. Josephson, Jens, 2008. "A numerical analysis of the evolutionary stability of learning rules," Journal of Economic Dynamics and Control, Elsevier, vol. 32(5), pages 1569-1599, May.
    8. Oyarzun, Carlos & Sarin, Rajiv, 2013. "Learning and risk aversion," Journal of Economic Theory, Elsevier, vol. 148(1), pages 196-225.
    9. Mario Bravo & Mathieu Faure, 2013. "Reinforcement Learning with Restrictions on the Action Set," AMSE Working Papers 1335, Aix-Marseille School of Economics, France, revised 01 Jul 2013.
    10. Jacques Durieu & Philippe Solal, 2012. "Models of Adaptive Learning in Game Theory," Chapters, in: Richard Arena & Agnès Festré & Nathalie Lazaric (ed.), Handbook of Knowledge and Economics, chapter 11, Edward Elgar Publishing.
    11. Martino Banchio & Giacomo Mantegazza, 2022. "Artificial Intelligence and Spontaneous Collusion," Papers 2202.05946, arXiv.org, revised Sep 2023.
    12. 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.
    13. Funai, Naoki, 2022. "Reinforcement learning with foregone payoff information in normal form games," Journal of Economic Behavior & Organization, Elsevier, vol. 200(C), pages 638-660.
    14. Schuster, Stephan, 2010. "Network Formation with Adaptive Agents," MPRA Paper 27388, University Library of Munich, Germany.
    15. 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.
    16. Hopkins, Ed, 2007. "Adaptive learning models of consumer behavior," Journal of Economic Behavior & Organization, Elsevier, vol. 64(3-4), pages 348-368.
    17. 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.
    18. Erik Mohlin & Robert Ostling & Joseph Tao-yi Wang, 2014. "Learning by Imitation in Games: Theory, Field, and Laboratory," Economics Series Working Papers 734, University of Oxford, Department of Economics.
    19. Han, Jungsuk & Sangiorgi, Francesco, 2018. "Searching for information," Journal of Economic Theory, Elsevier, vol. 175(C), pages 342-373.
    20. Leslie, David S. & Collins, E.J., 2006. "Generalised weakened fictitious play," Games and Economic Behavior, Elsevier, vol. 56(2), pages 285-298, August.
    21. Antonio Morales, 2005. "On the Role of the Group Composition for Achieving Optimality," Annals of Operations Research, Springer, vol. 137(1), pages 387-397, July.
    22. Pierre Coucheney & Bruno Gaujal & Panayotis Mertikopoulos, 2015. "Penalty-Regulated Dynamics and Robust Learning Procedures in Games," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 611-633, March.
    23. Georgios Chasparis & Jeff Shamma, 2012. "Distributed Dynamic Reinforcement of Efficient Outcomes in Multiagent Coordination and Network Formation," Dynamic Games and Applications, Springer, vol. 2(1), pages 18-50, March.
    24. Conor Mayo-Wilson & Kevin Zollman & David Danks, 2013. "Wisdom of crowds versus groupthink: learning in groups and in isolation," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(3), pages 695-723, August.
    25. Schuster, Stephan, 2012. "Applications in Agent-Based Computational Economics," MPRA Paper 47201, University Library of Munich, Germany.

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    More about this item

    Keywords

    learning in games; reinforcement learning; stochastic approximation; replicator dynamics;
    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
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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