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Regret minimization in repeated matrix games with variable stage duration

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  • Mannor, Shie
  • Shimkin, Nahum

Abstract

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.

Suggested Citation

  • Mannor, Shie & Shimkin, Nahum, 2008. "Regret minimization in repeated matrix games with variable stage duration," Games and Economic Behavior, Elsevier, vol. 63(1), pages 227-258, May.
    • Handle: RePEc:eee:gamebe:v:63:y:2008:i:1:p:227-258
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    References listed on IDEAS

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    1. Sergiu Hart, 2013. "Adaptive Heuristics," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 11, pages 253-287, World Scientific Publishing Co. Pte. Ltd..
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    6. 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..
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    Cited by:

    1. Jia Yuan Yu & Shie Mannor & Nahum Shimkin, 2009. "Markov Decision Processes with Arbitrary Reward Processes," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 737-757, August.
    2. Michael Nwogugu, 2020. "Regret Theory And Asset Pricing Anomalies In Incomplete Markets With Dynamic Un-Aggregated Preferences," Papers 2005.01709, arXiv.org.
    3. Andrey Bernstein & Shie Mannor & Nahum Shimkin, 2014. "Opportunistic Approachability and Generalized No-Regret Problems," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1057-1083, November.

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