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Distribution-Free Learning

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  • Karl H. Schlag

Abstract

We select among rules for learning which of two actions in a stationary decision problem achieves a higher expected payo¤when payoffs realized by both actions are known in previous instances. Only a bounded set containing all possible payoffs is known. Rules are evaluated using maximum risk with maximin utility, minimax regret, competitive ratio and selection procedures being special cases. A randomized variant of fictitious play attains minimax risk for all risk functions with ex-ante expected payoffs increasing in the number of observations. Fictitious play itself has neither of these two properties. Tight bounds on maximal regret and probability of selecting the best action are included.

Suggested Citation

  • Karl H. Schlag, 2007. "Distribution-Free Learning," Economics Working Papers ECO2007/01, European University Institute.
  • Handle: RePEc:eui:euiwps:eco2007/01
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    References listed on IDEAS

    as
    1. Schlag, Karl H., 1999. "Which one should I imitate?," Journal of Mathematical Economics, Elsevier, vol. 31(4), pages 493-522, May.
    2. Tilman Börgers & Antonio J. Morales & Rajiv Sarin, 2004. "Expedient and Monotone Learning Rules," Econometrica, Econometric Society, vol. 72(2), pages 383-405, March.
    3. Rustichini, Aldo, 1999. "Optimal Properties of Stimulus--Response Learning Models," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 244-273, October.
    4. Schlag, Karl H., 1998. "Why Imitate, and If So, How?, : A Boundedly Rational Approach to Multi-armed Bandits," Journal of Economic Theory, Elsevier, vol. 78(1), pages 130-156, January.
    5. Sergiu Hart & Andreu Mas-Colell, 2000. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Econometrica, Econometric Society, vol. 68(5), pages 1127-1150, September.
    6. Karl Schlag, 2006. "ELEVEN - Tests needed for a Recommendation," Economics Working Papers ECO2006/2, European University Institute.
    7. Jörg Stoye, 2011. "Statistical decisions under ambiguity," Theory and Decision, Springer, vol. 70(2), pages 129-148, February.
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    More about this item

    Keywords

    fictitious play; nonparametric; finite sample; matched pairs; foregone payoffs; minimax risk; ex-ante improving; selection procedure;

    JEL classification:

    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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