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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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Paper provided by David K. Levine in its series Levine's Working Paper Archive with number 506439000000000350.

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Date of creation: 20 Feb 2003
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Handle: RePEc:cla:levarc:506439000000000350

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References

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  1. Ed Hopkins, 2001. "Two Competing Models of How People Learn in Games," NajEcon Working Paper Reviews 625018000000000226, www.najecon.org.
  2. 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.
  3. Glenn Ellison & Drew Fudenberg, 1998. "Learning Purified Mixed Equilibria," Harvard Institute of Economic Research Working Papers 1817, Harvard - Institute of Economic Research.
  4. 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.
  5. Tilman B�rgers & Rajiv Sarin, . "Learning Through Reinforcement and Replicator Dynamics," ELSE working papers 051, ESRC Centre on Economics Learning and Social Evolution.
  6. Colin Camerer & Teck-Hua Ho, 1999. "Experience-weighted Attraction Learning in Normal Form Games," Econometrica, Econometric Society, vol. 67(4), pages 827-874, July.
  7. Martin Posch, 1997. "Cycling in a stochastic learning algorithm for normal form games," Journal of Evolutionary Economics, Springer, vol. 7(2), pages 193-207.
  8. Sandholm, William H, 2002. "Evolutionary Implementation and Congestion Pricing," Review of Economic Studies, Wiley Blackwell, vol. 69(3), pages 667-89, July.
  9. Josef Hofbauer & Ed Hopkins, 2004. "Learning in Perturbed Asymmetric Games," ESE Discussion Papers 53, Edinburgh School of Economics, University of Edinburgh.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. Rustichini, Aldo, 1999. "Optimal Properties of Stimulus--Response Learning Models," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 244-273, October.
  15. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
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Citations

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Cited by:
  1. Hopkins, Ed, 2007. "Adaptive learning models of consumer behavior," Journal of Economic Behavior & Organization, Elsevier, vol. 64(3-4), pages 348-368.
  2. Drew Fudenberg & Satoru Takahashi, 2008. "Heterogeneous Beliefs and Local Information in Stochastic Fictitious Play," Levine's Working Paper Archive 122247000000001695, David K. Levine.
  3. Carlos Oyarzun & Rajiv Sarin, 2012. "Learning and Risk Aversion," Levine's Working Paper Archive 786969000000000572, David K. Levine.
  4. Josephson, Jens, 2001. "A Numerical Analysis of the Evolutionary Stability of Learning Rules," Working Paper Series in Economics and Finance 474, Stockholm School of Economics.
  5. 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.
  6. 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.
  7. Ianni, Antonella, 2014. "Learning strict Nash equilibria through reinforcement," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 148-155.
  8. 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.
  9. Ianni, Antonella, 2011. "Learning Strict Nash Equilibria through Reinforcement," MPRA Paper 33936, University Library of Munich, Germany.
  10. Schuster, Stephan, 2010. "Network Formation with Adaptive Agents," MPRA Paper 27388, University Library of Munich, Germany.
  11. Oyarzun, Carlos & Sarin, Rajiv, 2013. "Learning and risk aversion," Journal of Economic Theory, Elsevier, vol. 148(1), pages 196-225.
  12. Schuster, Stephan, 2012. "Applications in Agent-Based Computational Economics," MPRA Paper 47201, University Library of Munich, Germany.
  13. Leslie, David S. & Collins, E.J., 2006. "Generalised weakened fictitious play," Games and Economic Behavior, Elsevier, vol. 56(2), pages 285-298, August.
  14. 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.
  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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