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No-regret dynamics and fictitious play

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  • Viossat, Yannick
  • Zapechelnyuk, Andriy

Abstract

Potential based no-regret dynamics are shown to be related to fictitious play. Roughly, these are ε-best reply dynamics where ε is the maximal regret, which vanishes with time. This allows for alternative and sometimes much shorter proofs of known results on convergence of no-regret dynamics to the set of Nash equilibria.

Suggested Citation

  • Viossat, Yannick & Zapechelnyuk, Andriy, 2013. "No-regret dynamics and fictitious play," Journal of Economic Theory, Elsevier, vol. 148(2), pages 825-842.
    • Handle: RePEc:eee:jetheo:v:148:y:2013:i:2:p:825-842
    DOI: 10.1016/j.jet.2012.07.003
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    1. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2006. "Stochastic Approximations and Differential Inclusions, Part II: Applications," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 673-695, November.
    2. Jean Tirole, 1988. "The Theory of Industrial Organization," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262200716, April.
    3. Berger, Ulrich, 2005. "Fictitious play in 2 x n games," Journal of Economic Theory, Elsevier, vol. 120(2), pages 139-154, February.
    4. repec:dau:papers:123456789/1014 is not listed on IDEAS
    5. Hofbauer, Josef & Weibull, Jorgen W., 1996. "Evolutionary Selection against Dominated Strategies," Journal of Economic Theory, Elsevier, vol. 71(2), pages 558-573, November.
    6. Mathieu Faure & Gregory Roth, 2010. "Stochastic Approximations of Set-Valued Dynamical Systems: Convergence with Positive Probability to an Attractor," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 624-640, August.
    7. Sergiu Hart & Andreu Mas-Colell, 2013. "A General Class Of Adaptive Strategies," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 3, pages 47-76, World Scientific Publishing Co. Pte. Ltd..
    8. Basu, Kaushik & Weibull, Jorgen W., 1991. "Strategy subsets closed under rational behavior," Economics Letters, Elsevier, vol. 36(2), pages 141-146, June.
    9. Sergiu Hart & Andreu Mas-Colell, 2013. "Regret-Based Continuous-Time Dynamics," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 5, pages 99-124, World Scientific Publishing Co. Pte. Ltd..
    10. 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.
    11. Michel Benaim & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions II: Applications," Levine's Bibliography 784828000000000098, UCLA Department of Economics.
    12. Richard P. Larrick & Jack B. Soll, 2006. "Erratum--Intuitions About Combining Opinions: Misappreciation of the Averaging Principle," Management Science, INFORMS, vol. 52(2), pages 309-310, February.
    13. Sergiu Hart & Andreu Mas-Colell, 2013. "A Simple Adaptive Procedure Leading To Correlated Equilibrium," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 2, pages 17-46, World Scientific Publishing Co. Pte. Ltd..
    14. Fudenberg, Drew & Levine, David K., 1995. "Consistency and cautious fictitious play," Journal of Economic Dynamics and Control, Elsevier, vol. 19(5-7), pages 1065-1089.
    15. Gaunersdorfer Andrea & Hofbauer Josef, 1995. "Fictitious Play, Shapley Polygons, and the Replicator Equation," Games and Economic Behavior, Elsevier, vol. 11(2), pages 279-303, November.
    16. Gilboa, Itzhak & Matsui, Akihiko, 1991. "Social Stability and Equilibrium," Econometrica, Econometric Society, vol. 59(3), pages 859-867, May.
    17. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
    18. Josef Hofbauer & Sylvain Sorin & Yannick Viossat, 2009. "Time Average Replicator and Best-Reply Dynamics," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 263-269, May.
    19. Lehrer, Ehud, 2003. "A wide range no-regret theorem," Games and Economic Behavior, Elsevier, vol. 42(1), pages 101-115, January.
    20. Vijay Krishna & Tomas Sjöström, 1998. "On the Convergence of Fictitious Play," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 479-511, May.
    21. Young, H. Peyton, 2004. "Strategic Learning and its Limits," OUP Catalogue, Oxford University Press, number 9780199269181.
    22. Monderer, Dov & Samet, Dov & Sela, Aner, 1997. "Belief Affirming in Learning Processes," Journal of Economic Theory, Elsevier, vol. 73(2), pages 438-452, April.
    23. Selten, Reinhard, 1995. "An axiomatic theory of a risk dominance measure for bipolar games with linear incentives," Games and Economic Behavior, Elsevier, vol. 8(1), pages 213-263.
    24. Berger, Ulrich, 2007. "Two more classes of games with the continuous-time fictitious play property," Games and Economic Behavior, Elsevier, vol. 60(2), pages 247-261, August.
    25. Timmermann, Allan, 2006. "Forecast Combinations," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 1, chapter 4, pages 135-196, Elsevier.
    26. Michel Benaïm & Mathieu Faure, 2013. "Consistency of Vanishingly Smooth Fictitious Play," Mathematics of Operations Research, INFORMS, vol. 38(3), pages 437-450, August.
    27. Hofbauer, Josef & Sandholm, William H., 2009. "Stable games and their dynamics," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1665-1693.4, July.
    28. G. Elliott & C. Granger & A. Timmermann (ed.), 2006. "Handbook of Economic Forecasting," Handbook of Economic Forecasting, Elsevier, edition 1, volume 1, number 1.
    29. Sparrow, Colin & van Strien, Sebastian & Harris, Christopher, 2008. "Fictitious play in 3x3 games: The transition between periodic and chaotic behaviour," Games and Economic Behavior, Elsevier, vol. 63(1), pages 259-291, May.
    30. Freund, Yoav & Schapire, Robert E., 1999. "Adaptive Game Playing Using Multiplicative Weights," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 79-103, October.
    31. Matsui, Akihiko, 1992. "Best response dynamics and socially stable strategies," Journal of Economic Theory, Elsevier, vol. 57(2), pages 343-362, August.
    32. Richard P. Larrick & Jack B. Soll, 2006. "Intuitions About Combining Opinions: Misappreciation of the Averaging Principle," Management Science, INFORMS, vol. 52(1), pages 111-127, January.
    33. Dean P. Foster & Rakesh V. Vohra, 1993. "A Randomization Rule for Selecting Forecasts," Operations Research, INFORMS, vol. 41(4), pages 704-709, August.
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    5. Leslie, David S. & Perkins, Steven & Xu, Zibo, 2020. "Best-response dynamics in zero-sum stochastic games," Journal of Economic Theory, Elsevier, vol. 189(C).

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    More about this item

    Keywords

    Regret minimization; No-regret strategy; Fictitious play; Best-reply dynamics; Nash equilibrium; Hannan set; Curb set;
    All these keywords.

    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • 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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