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A Geometric Proof of Calibration

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  • Shie Mannor

    () (EE-Technion - Department of Electrical Engineering - Technion [Haïfa] - Technion - Israel Institute of Technology [Haifa])

  • Gilles Stoltz

    () (DMA - Département de Mathématiques et Applications - ENS Paris - ENS Paris - École normale supérieure - Paris - CNRS - Centre National de la Recherche Scientifique, GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique)

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.

Suggested Citation

  • Shie Mannor & Gilles Stoltz, 2009. "A Geometric Proof of Calibration," Working Papers hal-00442042, HAL.
  • Handle: RePEc:hal:wpaper:hal-00442042
    Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00442042v2
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    References listed on IDEAS

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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. Fudenberg, Drew & Levine, David K., 1999. "An Easier Way to Calibrate," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 131-137, October.
    3. Sergiu Hart & Andreu Mas-Colell, 2000. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Econometrica, Econometric Society, vol. 68(5), pages 1127-1150, September.
    4. 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.
    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.
    2. Vladimir V'yugin, 2014. "Log-Optimal Portfolio Selection Using the Blackwell Approachability Theorem," Papers 1410.5996, arXiv.org, revised Jun 2015.
    3. Vianney Perchet, 2015. "Exponential Weight Approachability, Applications to Calibration and Regret Minimization," Dynamic Games and Applications, Springer, vol. 5(1), pages 136-153, March.

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