IDEAS home Printed from https://ideas.repec.org/p/hal/wpaper/hal-00442042.html
   My bibliography  Save this paper

A Geometric Proof of Calibration

Author

Listed:
  • 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-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - 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.science/hal-00442042v2
    as

    Download full text from publisher

    File URL: https://hal.science/hal-00442042v2/document
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    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, 2013. "A Simple Adaptive Procedure Leading To Correlated Equilibrium," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 2, pages 17-46, World Scientific Publishing Co. Pte. Ltd..
    4. Chen, Xiaohong & White, Halbert, 1996. "Laws of Large Numbers for Hilbert Space-Valued Mixingales with Applications," Econometric Theory, Cambridge University Press, vol. 12(2), pages 284-304, June.
    5. Alvaro Sandroni & Rann Smorodinsky & Rakesh V. Vohra, 2003. "Calibration with Many Checking Rules," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 141-153, February.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Olszewski, Wojciech, 2015. "Calibration and Expert Testing," Handbook of Game Theory with Economic Applications,, Elsevier.
    2. Vianney Perchet, 2015. "Exponential Weight Approachability, Applications to Calibration and Regret Minimization," Dynamic Games and Applications, Springer, vol. 5(1), pages 136-153, March.
    3. Dean Foster & Rakesh Vohra, 2011. "Calibration: Respice, Adspice, Prospice," Discussion Papers 1537, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    4. 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.
    5. Vladimir V'yugin, 2014. "Log-Optimal Portfolio Selection Using the Blackwell Approachability Theorem," Papers 1410.5996, arXiv.org, revised Jun 2015.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Shie Mannor & Gilles Stoltz, 2010. "A Geometric Proof of Calibration," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 721-727, November.
    2. 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.
    3. Wojciech Olszewski & Alvaro Sandroni, 2006. "Strategic Manipulation of Empirical Tests," Discussion Papers 1425, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    4. Sandroni, Alvaro & Smorodinsky, Rann, 2004. "Belief-based equilibrium," Games and Economic Behavior, Elsevier, vol. 47(1), pages 157-171, April.
    5. Foster, Dean P. & Young, H. Peyton, 2003. "Learning, hypothesis testing, and Nash equilibrium," Games and Economic Behavior, Elsevier, vol. 45(1), pages 73-96, October.
    6. Dean Foster & Rakesh Vohra, 2011. "Calibration: Respice, Adspice, Prospice," Discussion Papers 1537, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    7. Olszewski, Wojciech, 2015. "Calibration and Expert Testing," Handbook of Game Theory with Economic Applications,, Elsevier.
    8. Fudenberg, Drew & Levine, David K., 1999. "Conditional Universal Consistency," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 104-130, October.
    9. Kalai, Ehud & Lehrer, Ehud & Smorodinsky, Rann, 1999. "Calibrated Forecasting and Merging," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 151-169, October.
    10. Karl Schlag & Andriy Zapechelnyuk, 2009. "Decision Making in Uncertain and Changing Environments," Discussion Papers 19, Kyiv School of Economics.
    11. Al-Najjar, Nabil I. & Sandroni, Alvaro & Smorodinsky, Rann & Weinstein, Jonathan, 2010. "Testing theories with learnable and predictive representations," Journal of Economic Theory, Elsevier, vol. 145(6), pages 2203-2217, November.
    12. Eddie Dekel & Yossi Feinberg, 2006. "Non-Bayesian Testing of a Stochastic Prediction," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 73(4), pages 893-906.
    13. Emerson Melo, 2021. "Learning in Random Utility Models Via Online Decision Problems," Papers 2112.10993, arXiv.org, revised Aug 2022.
    14. Hanyu Li & Wenhan Huang & Zhijian Duan & David Henry Mguni & Kun Shao & Jun Wang & Xiaotie Deng, 2023. "A survey on algorithms for Nash equilibria in finite normal-form games," Papers 2312.11063, arXiv.org.
    15. Josef Hofbauer & Sylvain Sorin & Yannick Viossat, 2009. "Time Average Replicator and Best Reply Dynamics," Post-Print hal-00360767, HAL.
    16. Dean P. Foster & Sergiu Hart, 2021. "Forecast Hedging and Calibration," Journal of Political Economy, University of Chicago Press, vol. 129(12), pages 3447-3490.
    17. Schlag, Karl H. & Zapechelnyuk, Andriy, 2017. "Dynamic benchmark targeting," Journal of Economic Theory, Elsevier, vol. 169(C), pages 145-169.
    18. Tim Roughgarden, 2018. "Complexity Theory, Game Theory, and Economics: The Barbados Lectures," Papers 1801.00734, arXiv.org, revised Feb 2020.
    19. Feinberg, Yossi & Dekel, Eddie, 2004. "A True Expert Knows which Question Should Be Asked," Research Papers 1856, Stanford University, Graduate School of Business.
    20. Shie Mannor & Nahum Shimkin, 2003. "The Empirical Bayes Envelope and Regret Minimization in Competitive Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 327-345, May.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:wpaper:hal-00442042. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.