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Attainability of Boundary Points under Reinforcement Learning

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  • Ed Hopkins

    ()

  • Martin Posch

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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Bibliographic Info

Paper provided by Edinburgh School of Economics, University of Edinburgh in its series ESE Discussion Papers with number 79.

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Length: 17
Date of creation: Mar 2004
Date of revision:
Handle: RePEc:edn:esedps:79

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  1. Laslier, J.-F. & Topol, R. & Walliser, B., 1999. "A Behavioral Learning Process in Games," Papers 99-03, Paris X - Nanterre, U.F.R. de Sc. Ec. Gest. Maths Infor..
  2. Borgers, Tilman & Sarin, Rajiv, 1997. "Learning Through Reinforcement and Replicator Dynamics," Journal of Economic Theory, Elsevier, vol. 77(1), pages 1-14, November.
  3. Colin Camerer & Teck-Hua Ho, 1999. "Experience-weighted Attraction Learning in Normal Form Games," Econometrica, Econometric Society, vol. 67(4), pages 827-874, July.
  4. Ed Hopkins, 2001. "Two Competing Models of How People Learn in Games," Levine's Working Paper Archive 625018000000000226, David K. Levine.
  5. Martin Posch, 1997. "Cycling in a stochastic learning algorithm for normal form games," Journal of Evolutionary Economics, Springer, vol. 7(2), pages 193-207.
  6. John Duffy & Ed Hopkins, 2004. "Learning, Information and Sorting in Market Entry Games: Theory and Evidence," ESE Discussion Papers 78, Edinburgh School of Economics, University of Edinburgh.
  7. Glenn Ellison & Drew Fudenberg, 1998. "Learning Purified Mixed Equilibria," Harvard Institute of Economic Research Working Papers 1817, Harvard - Institute of Economic Research.
  8. Sandholm, William H, 2002. "Evolutionary Implementation and Congestion Pricing," Review of Economic Studies, Wiley Blackwell, vol. 69(3), pages 667-89, July.
  9. Roth, Alvin E. & Erev, Ido, 1995. "Learning in extensive-form games: Experimental data and simple dynamic models in the intermediate term," Games and Economic Behavior, Elsevier, vol. 8(1), pages 164-212.
  10. Erev, Ido & Roth, Alvin E, 1998. "Predicting How People Play Games: Reinforcement Learning in Experimental Games with Unique, Mixed Strategy Equilibria," American Economic Review, American Economic Association, vol. 88(4), pages 848-81, September.
  11. Hofbauer, Josef & Hopkins, Ed, 2005. "Learning in perturbed asymmetric games," Games and Economic Behavior, Elsevier, vol. 52(1), pages 133-152, July.
  12. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
  13. Arthur, W Brian, 1993. "On Designing Economic Agents That Behave Like Human Agents," Journal of Evolutionary Economics, Springer, vol. 3(1), pages 1-22, February.
  14. Sarin, Rajiv & Vahid, Farshid, 1999. "Payoff Assessments without Probabilities: A Simple Dynamic Model of Choice," Games and Economic Behavior, Elsevier, vol. 28(2), pages 294-309, August.
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Citations

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Cited by:
  1. Ianni, Antonella, 2014. "Learning strict Nash equilibria through reinforcement," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 148-155.
  2. Leslie, David S. & Collins, E.J., 2006. "Generalised weakened fictitious play," Games and Economic Behavior, Elsevier, vol. 56(2), pages 285-298, August.
  3. Ed Hopkins, 2010. "Adaptive Learning Models of Consumer Behaviour," Levine's Working Paper Archive 506439000000000346, David K. Levine.
  4. Schuster, Stephan, 2012. "Applications in Agent-Based Computational Economics," MPRA Paper 47201, University Library of Munich, Germany.
  5. John Duffy & Ed Hopkins, 2010. "Learning, Information and Sorting in Market Entry Games: Theory and Evidence," Levine's Working Paper Archive 506439000000000355, David K. Levine.
  6. Antonella Ianni, 2007. "Learning Strict Nash Equilibria through Reinforcement," Economics Working Papers ECO2007/21, European University Institute.
  7. Schuster, Stephan, 2010. "Network Formation with Adaptive Agents," MPRA Paper 27388, University Library of Munich, Germany.
  8. 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.
  9. Carlos Oyarzun & Rajiv Sarin, 2012. "Learning and Risk Aversion," Levine's Working Paper Archive 786969000000000572, David K. Levine.
  10. 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, vol. 42(3), pages 695-723, August.
  11. Oyarzun, Carlos & Sarin, Rajiv, 2013. "Learning and risk aversion," Journal of Economic Theory, Elsevier, vol. 148(1), pages 196-225.
  12. 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.
  13. 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.
  14. 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.
  15. 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.

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