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Rational Learning in Imperfect Monitoring Games

  • Mario Gilli


    (Department of Economics, University of Milan-Bicocca)

This paper provides a genera1 framework to analyze rational learning in strategic situations where the players have private information and update their private priors collecting data through optimal experimentation. The theory of statistica1 inference for stochastic processes and of Markovian dynamic programming is applied to study players asymptotic behavior in the context of repeated and recurring games, proving convergence towards Conjectural equilibria, an oyporturie generalization of Nash equilibria for this kind of strategic situations. Since the main bulk of the literature on rational learning regards convergence towards equilibria of repeated games, the main contribution of this paper is to argue for rational learning in recurring games, providing dynamic foundations for equilibria of the one-shot game. The analysis focuses on the problem of non stationary environment and on the problem of the correct specification of the stochastic law which regulates players' observations. In this way the paper shows both the limitations and the possibilities of rational learning models in game theory, in particular explaining when and why consistency rather than merging is the correct notion of learning in games.

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Paper provided by University of Milano-Bicocca, Department of Economics in its series Working Papers with number 46.

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Length: 34 pages
Date of creation: Mar 2002
Date of revision: Mar 2002
Handle: RePEc:mib:wpaper:46
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  1. Battigalli, Pierpaolo, 2003. "Rationalizability in infinite, dynamic games with incomplete information," Research in Economics, Elsevier, vol. 57(1), pages 1-38, March.
  2. Kalai, Ehud & Lehrer, Ehud, 1993. "Rational Learning Leads to Nash Equilibrium," Econometrica, Econometric Society, vol. 61(5), pages 1019-45, September.
  3. Drew Fudenberg & David K. Levine, 1986. "Limit Games and Limit Equilibria," Levine's Working Paper Archive 220, David K. Levine.
  4. Kandori, Michihiro, 2002. "Introduction to Repeated Games with Private Monitoring," Journal of Economic Theory, Elsevier, vol. 102(1), pages 1-15, January.
  5. Nyarko, Y., 1998. "The Truth is in the Eye of the Beholder: or Equilibrium in Beliefs and Rational Learning in Games," Working Papers 98-12, C.V. Starr Center for Applied Economics, New York University.
  6. Gilli, Mario, 1999. "On Non-Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 27(2), pages 184-203, May.
  7. Yaw Nyarko, 1998. "Bayesian learning and convergence to Nash equilibria without common priors," Economic Theory, Springer, vol. 11(3), pages 643-655.
  8. Dekel, Eddie & Fudenberg, Drew & Levine, David K., 2004. "Learning to play Bayesian games," Games and Economic Behavior, Elsevier, vol. 46(2), pages 282-303, February.
  9. Fudenberg, D. & Levine, D.K., 1991. "Self-Confirming Equilibrium ," Working papers 581, Massachusetts Institute of Technology (MIT), Department of Economics.
  10. Sandroni, Alvaro, 1998. "Does Rational Learning Lead to Nash Equilibrium in Finitely Repeated Games?," Journal of Economic Theory, Elsevier, vol. 78(1), pages 195-218, January.
  11. Nyarko, Yaw, 1991. "The Convergence of Bayesian Belief Hierarchies," Working Papers 91-50, C.V. Starr Center for Applied Economics, New York University.
  12. Chakrabarti, Subir K, 1992. "Equilibrium in Behavior Strategies in Infinite Extensive Form Games with Imperfect Information," Economic Theory, Springer, vol. 2(4), pages 481-94, October.
  13. Fudenberg, Drew & Levine, David K, 1993. "Steady State Learning and Nash Equilibrium," Econometrica, Econometric Society, vol. 61(3), pages 547-73, May.
  14. Philippe Aghion & Patrick Bolton & Christopher Harris & Bruno Jullien, 1991. "Optimal Learning by Experimentation," Review of Economic Studies, Oxford University Press, vol. 58(4), pages 621-654.
  15. Gilli, Mario, 2001. "A General Approach to Rational Learning in Games," Bulletin of Economic Research, Wiley Blackwell, vol. 53(4), pages 275-303, October.
  16. Fudenberg, Drew & Kreps, David M., 1995. "Learning in extensive-form games I. Self-confirming equilibria," Games and Economic Behavior, Elsevier, vol. 8(1), pages 20-55.
  17. Jordan J. S., 1995. "Bayesian Learning in Repeated Games," Games and Economic Behavior, Elsevier, vol. 9(1), pages 8-20, April.
  18. Koutsougeras, Leonidas C & Yannelis, Nicholas C, 1994. "Convergence and Approximation Results for Non-cooperative Bayesian Games: Learning Theorems," Economic Theory, Springer, vol. 4(6), pages 843-57, October.
  19. Kalai, Ehud & Lehrer, Ehud, 1994. "Weak and strong merging of opinions," Journal of Mathematical Economics, Elsevier, vol. 23(1), pages 73-86, January.
  20. Jackson, Matthew O. & Kalai, Ehud, 1999. "Reputation versus Social Learning," Journal of Economic Theory, Elsevier, vol. 88(1), pages 40-59, September.
  21. Matthew O. Jackson & Ehud Kalai & Rann Smorodinsky, 1998. "Bayesian Representation of Stochastic Processes under Learning: de Finetti Revisited," Discussion Papers 1228, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  22. Miller, Ronald I. & Sanchirico, Chris William, 1999. "The Role of Absolute Continuity in "Merging of Opinions" and "Rational Learning"," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 170-190, October.
  23. Mario Gilli, 1999. "Adaptive Learning in Imperfect Monitoring Games," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 2(2), pages 472-485, April.
  24. Nyarko, Yaw, 1994. "Bayesian Learning Leads to Correlated Equilibria in Normal Form Games," Economic Theory, Springer, vol. 4(6), pages 821-41, October.
  25. Kalai, Ehud & Lehrer, Ehud, 1993. "Subjective Equilibrium in Repeated Games," Econometrica, Econometric Society, vol. 61(5), pages 1231-40, September.
  26. Alvaro Sandroni & Rann Smorodinsky, 1999. "The speed of rational learning," International Journal of Game Theory, Springer, vol. 28(2), pages 199-210.
  27. Aghion Philippe & Bolton, Patrick & Harris Christopher & Jullien Bruno, 1991. "Optimal learning by experimentation," CEPREMAP Working Papers (Couverture Orange) 9104, CEPREMAP.
  28. Matthew Jackson & Ehud Kalai, 1995. "Social Learning in Recurring Games," Discussion Papers 1138, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  29. Eichberger Jurgen, 1995. "Bayesian Learning in Repeated Normal Form Games," Games and Economic Behavior, Elsevier, vol. 11(2), pages 254-278, November.
  30. Pearce, David G, 1984. "Rationalizable Strategic Behavior and the Problem of Perfection," Econometrica, Econometric Society, vol. 52(4), pages 1029-50, July.
  31. Jordan, J. S., 1991. "Bayesian learning in normal form games," Games and Economic Behavior, Elsevier, vol. 3(1), pages 60-81, February.
  32. Nyarko, Y., 1992. "Bayesian Learning without Common Priors and Convergence to Nash Equilibria," Working Papers 92-25, C.V. Starr Center for Applied Economics, New York University.
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