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A Geometric Proof of Calibration

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  • Shie Mannor

    ()
    (EE-Technion - Department of Electrical Engineering - Technion - Israel Institute of Technology)

  • Gilles Stoltz

    ()
    (DMA - Département de Mathématiques et Applications - CNRS : UMR8553 - Ecole Normale Supérieure de Paris - ENS Paris, GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - GROUPE HEC - CNRS : UMR2959)

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    Abstract

    We provide yet another proof of the existence of calibrated forecasters; it has two merits. First, it is valid for an arbitrary finite number of outcomes. Second, it is short and simple and it follows from a direct application of Blackwell's approachability theorem to carefully chosen vector-valued payoff function and convex target set. Our proof captures the essence of existing proofs based on approachability (e.g., the proof by Foster, 1999 in case of binary outcomes) and highlights the intrinsic connection between approachability and calibration.

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    File URL: http://hal.archives-ouvertes.fr/docs/00/52/29/11/PDF/Mannor-Stoltz--Geometric-Calibation--Final.pdf
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    Bibliographic Info

    Paper provided by HAL in its series Working Papers with number hal-00442042.

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    Date of creation: 17 Dec 2009
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    Handle: RePEc:hal:wpaper:hal-00442042

    Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00442042/en/
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    1. Foster, Dean P., 1999. "A Proof of Calibration via Blackwell's Approachability Theorem," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 73-78, October.
    2. Chen, Xiaohong & White, Halbert, 1996. "Laws of Large Numbers for Hilbert Space-Valued Mixingales with Applications," Econometric Theory, Cambridge University Press, vol. 12(02), pages 284-304, June.
    3. S. Hart & A. Mas-Collel, 2010. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Levine's Working Paper Archive 572, David K. Levine.
    4. Drew Fudenberg & David K. Levine, 1996. "An Easier Way to Calibrate," Levine's Working Paper Archive 2059, David K. Levine.
    5. 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.
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    Cited by:
    1. Dean Foster & Rakesh Vohra, 2011. "Calibration: Respice, Adspice, Prospice," Discussion Papers 1537, Northwestern University, Center for Mathematical Studies in Economics and Management Science.

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