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Learning to Play Nash in Deterministic Uncoupled Dynamics

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  • Vivaldo M. Mendes

    ()
    (ISCTE - Department of Economics and UNIDE-ERC)

  • Diana A. Mendes

    ()
    (ISCTE - Department of Quantitative Methods and UNIDE-StatMath)

  • Orlando Gomes

    ()
    (Instituto Politécnico de Lisboa - Escola Superior de Comunicação Social and UNIDE-ERC)

Abstract

This paper is concerned with the following problem. In a bounded rational game where players cannot be as super-rational as in Kalai and Leher (1993), are there simple adaptive heuristics or rules that can be used in order to secure convergence to Nash equilibria, or convergence only to a larger set designated by correlated equilibria? Are there games with uncoupled deterministic dynamics in discrete time that converge to Nash equilibrium or not? Young (2008) argues that if an adaptive learning rule follows three conditions – (i) it is uncoupled, (ii) each player’s choice of action depends solely on the frequency distribution of past play, and (iii) each player’s choice of action, conditional on the state, is deterministic – no such rule leads the players’ behavior to converge to Nash equilibrium. In this paper we present a counterexample, showing that there are simple adaptive rules that secure convergence, in fact fast convergence, in a fully deterministic and uncoupled game. We used the Cournot model with nonlinear costs and incomplete information for this purpose and also illustrate that this convergence can be achieved with or without any coordination of the players actions.

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File URL: http://bru-unide.iscte.pt/RePEc/pdfs/ERCwp1808.pdf
File Function: First version, 2008
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Bibliographic Info

Paper provided by ISCTE-IUL, Business Research Unit (BRU-IUL) in its series Working Papers Series 1 with number ercwp1808.

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Length: 31 pages
Date of creation: 15 Jun 2008
Date of revision:
Handle: RePEc:isc:iscwp1:ercwp1808

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Web page: http://bru-unide.iscte.pt/
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Related research

Keywords: Uncoupled Deterministic Dynamics; Nash Equilibrium; Bounded Learning; Convergence.;

References

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  1. Peyton Young, 2002. "Learning Hypothesis Testing and Nash Equilibrium," Economics Working Paper Archive 474, The Johns Hopkins University,Department of Economics.
  2. Sergiu Hart & Andreu Mas-Colell, 2000. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Econometrica, Econometric Society, vol. 68(5), pages 1127-1150, September.
  3. Ehud Kalai & Ehud Lehrer, 1990. "Rational Learning Leads to Nash Equilibrium," Discussion Papers 895, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  4. Foster, Dean P. & Young, H. Peyton, 2006. "Regret testing: learning to play Nash equilibrium without knowing you have an opponent," Theoretical Economics, Econometric Society, vol. 1(3), pages 341-367, September.
  5. D. Foster & R. Vohra, 2010. "Regret in the On-line Decision Problem," Levine's Working Paper Archive 569, David K. Levine.
  6. D. Foster & R. Vohra, 2010. "Asymptotic Calibration," Levine's Working Paper Archive 468, David K. Levine.
  7. Foster, Dean P. & Vohra, Rakesh, 1999. "Regret in the On-Line Decision Problem," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 7-35, October.
  8. Drew Fudenberg & David K. Levine, 1996. "The Theory of Learning in Games," Levine's Working Paper Archive 624, David K. Levine.
  9. Foster, Dean P. & Vohra, Rakesh V., 1997. "Calibrated Learning and Correlated Equilibrium," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 40-55, October.
  10. D. Foster & R. Vohra, 2010. "Calibrated Learning and Correlated Equilibrium," Levine's Working Paper Archive 568, David K. Levine.
  11. Sergiu Hart & Andreu Mas-Colell, 2003. "Uncoupled Dynamics Do Not Lead to Nash Equilibrium," American Economic Review, American Economic Association, vol. 93(5), pages 1830-1836, December.
  12. Cox, James C. & Walker, Mark, 1998. "Learning to play Cournot duopoly strategies," Journal of Economic Behavior & Organization, Elsevier, vol. 36(2), pages 141-161, August.
  13. Fabrizio Germano & Gábor Lugosi, 2004. "Global Nash convergence of Foster and Young's regret testing," Economics Working Papers 788, Department of Economics and Business, Universitat Pompeu Fabra.
  14. Young, H. Peyton, 2004. "Strategic Learning and its Limits," OUP Catalogue, Oxford University Press, number 9780199269181.
  15. Leslie, David S. & Collins, E.J., 2006. "Generalised weakened fictitious play," Games and Economic Behavior, Elsevier, vol. 56(2), pages 285-298, August.
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