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Learning Strict Nash Equilibria through Reinforcement

  • Antonella Ianni

This paper studies the analytical properties of the reinforcement learning model proposed in Erev and Roth (1998), also termed cumulative reinforcement learning in Laslier et al. (2001). The stochastic model of learning accounts for two main elements: the Law of Effect (positive reinforcement of actions that perform well) and the Power Law of Practice (learning curves tend to be steeper initially). The paper establishes a relation between the learning process and the underlying deterministic replicator equation. The main results show that if the solution trajectories of the latter converge su¢ ciently fast, then the probability that all the realizations of the learning process over a given spell of time, possibly infinite, becomes arbitrarily close to one, from some time on. In particular, the paper shows that the property of fast convergence is always satisfied in proximity of a strict Nash equilibrium. The results also provide an explicit estimate of the approximation error that could prove to be useful in empirical analysis.

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Paper provided by European University Institute in its series Economics Working Papers with number ECO2007/21.

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Date of creation: 2007
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Handle: RePEc:eui:euiwps:eco2007/21
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  1. Borgers, Tilman & Sarin, Rajiv, 1997. "Learning Through Reinforcement and Replicator Dynamics," Journal of Economic Theory, Elsevier, vol. 77(1), pages 1-14, November.
  2. Martin Posch, 1997. "Cycling in a stochastic learning algorithm for normal form games," Journal of Evolutionary Economics, Springer, vol. 7(2), pages 193-207.
  3. 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..
  4. Colin Camerer & Teck-Hua Ho, 1999. "Experience-weighted Attraction Learning in Normal Form Games," Econometrica, Econometric Society, vol. 67(4), pages 827-874, July.
  5. 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.
  6. Ed Hopkins & Martin Posch, 2004. "Attainability of Boundary Points under Reinforcement Learning," ESE Discussion Papers 79, Edinburgh School of Economics, University of Edinburgh.
  7. Ritzberger, Klaus & Weibull, Jörgen W., 1993. "Evolutionary Selection in Normal Form Games," Working Paper Series 383, Research Institute of Industrial Economics.
  8. 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.
  9. Beggs, A.W., 2005. "On the convergence of reinforcement learning," Journal of Economic Theory, Elsevier, vol. 122(1), pages 1-36, May.
  10. Rustichini, Aldo, 1999. "Optimal Properties of Stimulus--Response Learning Models," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 244-273, October.
  11. 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.
  12. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
  13. Ed Hopkins, 2001. "Two Competing Models of How People Learn in Games," Levine's Working Paper Archive 625018000000000226, David K. Levine.
  14. Antonella Ianni, 2007. "Learning Strict Nash Equilibria through Reinforcement," Economics Working Papers ECO2007/21, European University Institute.
  15. 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.
  16. Michel BenaÔm & J–rgen W. Weibull, 2003. "Deterministic Approximation of Stochastic Evolution in Games," Econometrica, Econometric Society, vol. 71(3), pages 873-903, 05.
  17. 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.
  18. Cross, John G, 1973. "A Stochastic Learning Model of Economic Behavior," The Quarterly Journal of Economics, MIT Press, vol. 87(2), pages 239-66, May.
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