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Regret minimization in repeated matrix games with variable stage duration

  • Mannor, Shie
  • Shimkin, Nahum
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    Regret minimization in repeated matrix games has been extensively studied ever since Hannan's seminal paper [Hannan, J., 1957. Approximation to Bayes risk in repeated play. In: Dresher, M., Tucker, A.W., Wolfe, P. (Eds.), Contributions to the Theory of Games, vol. III. Ann. of Math. Stud., vol. 39, Princeton Univ. Press, Princeton, NJ, pp. 97-193]. Several classes of no-regret strategies now exist; such strategies secure a long-term average payoff as high as could be obtained by the fixed action that is best, in hindsight, against the observed action sequence of the opponent. We consider an extension of this framework to repeated games with variable stage duration, where the duration of each stage may depend on actions of both players, and the performance measure of interest is the average payoff per unit time. We start by showing that no-regret strategies, in the above sense, do not exist in general. Consequently, we consider two classes of adaptive strategies, one based on Blackwell's approachability theorem and the other on calibrated play, and examine their performance guarantees. We further provide sufficient conditions for existence of no-regret strategies in this model.

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    File URL: http://www.sciencedirect.com/science/article/B6WFW-4PR3GBX-4/1/8f4133c256f00856c5758e8314acb646
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    Article provided by Elsevier in its journal Games and Economic Behavior.

    Volume (Year): 63 (2008)
    Issue (Month): 1 (May)
    Pages: 227-258

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    Handle: RePEc:eee:gamebe:v:63:y:2008:i:1:p:227-258
    Contact details of provider: Web page: http://www.elsevier.com/locate/inca/622836

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    1. Lehrer, Ehud, 2003. "A wide range no-regret theorem," Games and Economic Behavior, Elsevier, vol. 42(1), pages 101-115, January.
    2. Rustichini, Aldo, 1999. "Minimizing Regret: The General Case," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 224-243, October.
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    4. Fudenberg, Drew & Levine, David K., 1999. "Conditional Universal Consistency," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 104-130, October.
    5. Hart, Sergiu & Mas-Colell, Andreu, 2001. "A General Class of Adaptive Strategies," Journal of Economic Theory, Elsevier, vol. 98(1), pages 26-54, May.
    6. 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.
    7. Ehud Kalai & Ehud Lehrer & Rann Smorodinsky, 2010. "Calibrated Forecasting and Merging," Levine's Working Paper Archive 584, David K. Levine.
    8. Sergiu Hart, 2004. "Adaptive Heuristics," Levine's Bibliography 122247000000000471, UCLA Department of Economics.
    9. Dean P Foster, 1997. "A proof of Calibration via Blackwell's Approachability Theorem," Levine's Working Paper Archive 591, David K. Levine.
    10. Fudenberg, Drew & Levine, David, 1999. "An Easier Way to Calibrate," Scholarly Articles 3203773, Harvard University Department of Economics.
    11. 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.
    12. 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.
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