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Convergence in models with bounded expected relative hazard rates

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  • Oyarzun, Carlos
  • Ruf, Johannes

Abstract

We provide a general framework to study stochastic sequences related to individual learning in economics, learning automata in computer sciences, social learning in marketing, and other applications. More precisely, we study the asymptotic properties of a class of stochastic sequences that take values in [0,1] and satisfy a property called “bounded expected relative hazard rates.” Sequences that satisfy this property and feature “small step-size” or “shrinking step-size” converge to 1 with high probability or almost surely, respectively. These convergence results yield conditions for the learning models in [13,35,7] to choose expected payoff maximizing actions with probability one in the long run.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jetheo:v:154:y:2014:i:c:p:229-244
    DOI: 10.1016/j.jet.2014.09.014
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    1. Fudenberg Drew & Kreps David M., 1993. "Learning Mixed Equilibria," Games and Economic Behavior, Elsevier, vol. 5(3), pages 320-367, July.
    2. 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 848-881, September.
    3. Tilman Börgers & Antonio J. Morales & Rajiv Sarin, 2004. "Expedient and Monotone Learning Rules," Econometrica, Econometric Society, vol. 72(2), pages 383-405, March.
    4. Hopkins, Ed & Posch, Martin, 2005. "Attainability of boundary points under reinforcement learning," Games and Economic Behavior, Elsevier, vol. 53(1), pages 110-125, October.
    5. Carlos Oyarzun & Johannes Ruf, 2009. "Monotone imitation," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 41(3), pages 411-441, December.
    6. John Huyck & Raymond Battalio & Frederick Rankin, 2007. "Selection dynamics and adaptive behavior without much information," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(1), pages 53-65, October.
    7. Glenn Ellison & Drew Fudenberg, 1995. "Word-of-Mouth Communication and Social Learning," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 110(1), pages 93-125.
    8. Oyarzun, Carlos & Sarin, Rajiv, 2013. "Learning and risk aversion," Journal of Economic Theory, Elsevier, vol. 148(1), pages 196-225.
    9. Venkatesh Bala & Sanjeev Goyal, 1998. "Learning from Neighbours," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 65(3), pages 595-621.
    10. Frank M. Bass, 1969. "A New Product Growth for Model Consumer Durables," Management Science, INFORMS, vol. 15(5), pages 215-227, January.
    11. James J. Choi & David Laibson & Brigitte C. Madrian & Andrew Metrick, 2009. "Reinforcement Learning and Savings Behavior," Journal of Finance, American Finance Association, vol. 64(6), pages 2515-2534, December.
    12. Borgers, Tilman & Sarin, Rajiv, 1997. "Learning Through Reinforcement and Replicator Dynamics," Journal of Economic Theory, Elsevier, vol. 77(1), pages 1-14, November.
    13. Rajiv Sarin & Farshid Vahid, 2004. "Strategy Similarity and Coordination," Economic Journal, Royal Economic Society, vol. 114(497), pages 506-527, July.
    14. H. Peyton Young, 2009. "Innovation Diffusion in Heterogeneous Populations: Contagion, Social Influence, and Social Learning," American Economic Review, American Economic Association, vol. 99(5), pages 1899-1924, December.
    15. James J. Choi & David Laibson & Brigitte C. Madrian & Andrew Metrick, 2009. "Reinforcement Learning and Savings Behavior," Journal of Finance, American Finance Association, vol. 64(6), pages 2515-2534, December.
    16. Schlag, Karl H., 1998. "Why Imitate, and If So, How?, : A Boundedly Rational Approach to Multi-armed Bandits," Journal of Economic Theory, Elsevier, vol. 78(1), pages 130-156, January.
    17. Michel Benaim & Mathieu Faure, 2010. "Stochastic Approximation, Cooperative Dynamics and Supermodular Games," Levine's Working Paper Archive 814577000000000437, David K. Levine.
    18. Beggs, A.W., 2005. "On the convergence of reinforcement learning," Journal of Economic Theory, Elsevier, vol. 122(1), pages 1-36, May.
    19. John G. Cross, 1973. "A Stochastic Learning Model of Economic Behavior," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 87(2), pages 239-266.
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    More about this item

    Keywords

    Hazard rate; Individual learning; Social learning; Two-armed bandit algorithm; Dynamic system; Stochastic approximation; Submartingale; Convergence;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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