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Some Comparisons among Quadratic, Spherical, and Logarithmic Scoring Rules

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  • J. Eric Bickel

    () (Department of Industrial and Systems Engineering, Texas A&M University, 236-B Zachry Engineering Center, 3131 TAMU, College Station, Texas 77843-3131)

Abstract

Strictly proper scoring rules continue to play an important role in probability assessment. Although many such rules have been developed, relatively little guidance exists as to which rule is the most appropriate. In this paper, we discuss two important properties of quadratic, spherical, and logarithmic scoring rules. From an ex post perspective, we compare their rank order properties and conclude that both quadratic and spherical scoring perform poorly in this regard, relative to logarithmic. Second, from an ex ante perspective, we demonstrate that in many situations, logarithmic scoring is the method least affected by a nonlinear utility function. These results suggest that logarithmic scoring is superior when rank order results are important and/or when the assessor has a nonlinear utility function. In addition to these results, and perhaps more important, we demonstrate that nonlinear utility induces relatively little deviation from the optimal assessment under an assumption of risk neutrality. These results provide both comfort and guidance to those who would like to use scoring rules as part of the assessment process.

Suggested Citation

  • J. Eric Bickel, 2007. "Some Comparisons among Quadratic, Spherical, and Logarithmic Scoring Rules," Decision Analysis, INFORMS, vol. 4(2), pages 49-65, June.
  • Handle: RePEc:inm:ordeca:v:4:y:2007:i:2:p:49-65
    DOI: 10.1287/deca.1070.0089
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    File URL: http://dx.doi.org/10.1287/deca.1070.0089
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    References listed on IDEAS

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    Cited by:

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      • 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, 2008. "From the Editors..," Decision Analysis, INFORMS, vol. 5(4), pages 173-176, 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.
    4. Corona Francisco & Wiper Michael Peter & Horrillo Juan de Dios Tena, 2017. "On the importance of the probabilistic model in identifying the most decisive games in a tournament," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 13(1), pages 11-23, 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. Elena Verdolini & Laura Díaz Anadón & Erin Baker & Valentina Bosetti & Lara Aleluia Reis, 2018. "Future Prospects for Energy Technologies: Insights from Expert Elicitations," Review of Environmental Economics and Policy, Association of Environmental and Resource Economists, vol. 12(1), pages 133-153.
    7. David J. Johnstone & Victor Richmond R. Jose & Robert L. Winkler, 2011. "Tailored Scoring Rules for Probabilities," Decision Analysis, INFORMS, vol. 8(4), pages 256-268, December.
    8. 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.
    9. L. Robin Keller, 2009. "From the Editor..," Decision Analysis, INFORMS, vol. 6(3), pages 121-123, September.
    10. David V. Budescu & Eva Chen, 2015. "Identifying Expertise to Extract the Wisdom of Crowds," Management Science, INFORMS, vol. 61(2), pages 267-280, February.
    11. Di, Chen & Dimitrov, Stanko & He, Qi-Ming, 2019. "Incentive compatibility in prediction markets: Costly actions and external incentives," International Journal of Forecasting, Elsevier, vol. 35(1), pages 351-370.
    12. Goessling, Marc, 2017. "LogitBoost autoregressive networks," Computational Statistics & Data Analysis, Elsevier, vol. 112(C), pages 88-98.
    13. Edgar C. Merkle & Mark Steyvers, 2013. "Choosing a Strictly Proper Scoring Rule," Decision Analysis, INFORMS, vol. 10(4), pages 292-304, December.
    14. 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.
    15. Jianqing Chen & Lizhen Xu & Andrew Whinston, 2010. "Managing Project Failure Risk Through Contingent Contracts in Procurement Auctions," Decision Analysis, INFORMS, vol. 7(1), pages 23-39, March.
    16. J. Eric Bickel, 2010. "Scoring Rules and Decision Analysis Education," Decision Analysis, INFORMS, vol. 7(4), pages 346-357, December.
    17. L. Robin Keller, 2008. "From the Editor..," Decision Analysis, INFORMS, vol. 5(3), pages 113-115, September.
    18. 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.
    19. Victor Richmond R. Jose & Robert F. Nau & Robert L. Winkler, 2008. "Scoring Rules, Generalized Entropy, and Utility Maximization," Operations Research, INFORMS, vol. 56(5), pages 1146-1157, October.
    20. Ali E. Abbas, 2009. "A Kullback-Leibler View of Linear and Log-Linear Pools," Decision Analysis, INFORMS, vol. 6(1), pages 25-37, March.
    21. Chambers, Christopher P. & Healy, Paul J. & Lambert, Nicolas S., 2019. "Proper scoring rules with general preferences: A dual characterization of optimal reports," Games and Economic Behavior, Elsevier, vol. 117(C), pages 322-341.

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