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Proper Scoring Rules, Dominated Forecasts, and Coherence

Author

Listed:
  • Mark J. Schervish

    (Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213)

  • Teddy Seidenfeld

    (Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213)

  • Joseph B. Kadane

    (Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213)

Abstract
The concept of coherent probabilities and conditional probabilities through a gambling argument and through a parallel argument based on a quadratic scoring rule was introduced by de Finetti (de Finetti, B. 1974. The Theory of Probability. John Wiley & Sons, New York). He showed that the two arguments lead to the same concept of coherence. When dealing with events only, there is a rich class of scoring rules that might be used in place of the quadratic scoring rule. We give conditions under which a general strictly proper scoring rule can replace the quadratic scoring rule while preserving the equivalence of de Finetti's two arguments. In proving our results, we present a strengthening of the usual minimax theorem. We also present generalizations of de Finetti's fundamental theorem of probability to deal with conditional probabilities.

Suggested Citation

  • Mark J. Schervish & Teddy Seidenfeld & Joseph B. Kadane, 2009. "Proper Scoring Rules, Dominated Forecasts, and Coherence," Decision Analysis, INFORMS, vol. 6(4), pages 202-221, December.
    • Handle: RePEc:inm:ordeca:v:6:y:2009:i:4:p:202-221
    DOI: 10.1287/deca.1090.0153
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    References listed on IDEAS

    as
    1. Pearce, David G, 1984. "Rationalizable Strategic Behavior and the Problem of Perfection," Econometrica, Econometric Society, vol. 52(4), pages 1029-1050, July.
    2. Gneiting, Tilmann & Raftery, Adrian E., 2007. "Strictly Proper Scoring Rules, Prediction, and Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 359-378, March.
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    Cited by:

    1. L. Robin Keller & Kelly M. Kophazi, 2011. "From the Editors---Deterrence, Multiattribute Utility, and Probability and Bayes' Updating," Decision Analysis, INFORMS, vol. 8(2), pages 83-87, June.
    2. Tino Werner, 2022. "Elicitability of Instance and Object Ranking," Decision Analysis, INFORMS, vol. 19(2), pages 123-140, June.
    3. Rakesh K. Sarin & L. Robin Keller, 2013. "From the Editors: Probability Approximations, Anti-Terrorism Strategy, and Bull's-Eye Display for Performance Feedback," Decision Analysis, INFORMS, vol. 10(1), pages 1-5, March.
    4. L. Robin Keller, 2011. "Investment and Defense Strategies, Heuristics, and Games: From the Editor ..," Decision Analysis, INFORMS, vol. 8(1), pages 1-3, March.
    5. Rakesh K. Sarin, 2013. "From the Editor ---Median Aggregation, Scoring Rules, Expert Forecasts, Choices with Binary Attributes, Portfolio with Dependent Projects, and Information Security," Decision Analysis, INFORMS, vol. 10(4), pages 277-278, December.
    6. L. Robin Keller & Ali Abbas & Manel Baucells & Vicki M. Bier & David Budescu & John C. Butler & Philippe Delquié & Jason R. W. Merrick & Ahti Salo & George Wu, 2010. "From the Editors..," Decision Analysis, INFORMS, vol. 7(4), pages 327-330, December.
      • L. Robin Keller & Manel Baucells & John C. Butler & Philippe Delquié & Jason R. W. Merrick & Gregory S. Parnell & Ahti Salo, 2008. "From the Editors..," Decision Analysis, INFORMS, vol. 5(4), pages 173-176, December.
      • L. Robin Keller & Manel Baucells & Kevin F. McCardle & Gregory S. Parnell & Ahti Salo, 2007. "From the Editors..," Decision Analysis, INFORMS, vol. 4(4), pages 173-175, December.
      • L. Robin Keller & Manel Baucells & John C. Butler & Philippe Delquié & Jason R. W. Merrick & Gregory S. Parnell & Ahti Salo, 2009. "From the Editors ..," Decision Analysis, INFORMS, vol. 6(4), pages 199-201, December.
    7. L. Robin Keller & Ali Abbas & J. Eric Bickel & Vicki M. Bier & David V. Budescu & John C. Butler & Philippe Delquié & Kenneth C. Lichtendahl & Jason R. W. Merrick & Ahti Salo & George Wu, 2011. "From the Editors ---Probability Scoring Rules, Ambiguity, Multiattribute Terrorist Utility, and Sensitivity Analysis," Decision Analysis, INFORMS, vol. 8(4), pages 251-255, December.
    8. Mark J. Schervish & Teddy Seidenfeld & Rafael B. Stern & Joseph B. Kadane, 2020. "What finite-additivity can add to decision theory," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(2), pages 237-263, June.
    9. López Menéndez, Ana Jesús & Pérez Suárez, Rigoberto, 2017. "Forecasting Performance and Information Measures. Revisiting the M-Competition /Evaluación de Predicciones y Medidas de Información. Reexamen de la M-Competición," Estudios de Economia Aplicada, Estudios de Economia Aplicada, vol. 35, pages 299-314, Mayo.
    10. L. Robin Keller & Ali Abbas & J. Eric Bickel & Vicki M. Bier & David V. Budescu & John C. Butler & Enrico Diecidue & Robin L. Dillon-Merrill & Raimo P. Hämäläinen & Kenneth C. Lichtendahl & Jason R. W, 2012. "From the Editors ---Brainstorming, Multiplicative Utilities, Partial Information on Probabilities or Outcomes, and Regulatory Focus," Decision Analysis, INFORMS, vol. 9(4), pages 297-302, December.
    11. L. Robin Keller, 2011. "From the Editor ---Multiattribute and Intertemporal Preferences, Probability, and Stochastic Processes: Models and Assessment," Decision Analysis, INFORMS, vol. 8(3), pages 165-169, September.
    12. Robert F. Bordley, 2011. "Using Bayes' Rule to Update an Event's Probabilities Based on the Outcomes of Partially Similar Events," Decision Analysis, INFORMS, vol. 8(2), pages 117-127, June.
    13. L. Robin Keller & Kelly M. Kophazi, 2012. "From the Editors ---Copulas, Group Preferences, Multilevel Defenders, Sharing Rewards, and Communicating Analytics," Decision Analysis, INFORMS, vol. 9(3), pages 213-218, September.
    14. Robert F. Bordley & Elena Katok & L. Robin Keller, 2010. "Honoring Michael H. Rothkopf's Legacy of Rigor and Relevance in Auction Theory: From the Editors," Decision Analysis, INFORMS, vol. 7(1), pages 1-4, March.

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    Keywords

    Brier score; finite additivity;

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