The uniqueness of local proper scoring rules: the logarithmic family

  • Jingni YangEmail author


Local proper scoring rules provide convenient tools for measuring subjective probabilities. Savage (J Am Stat Assoc 66(336), 783–801, 1971) has shown that the only local proper scoring rule for more than two exclusive events is the logarithmic family. We generalize Savage (1971) by relaxing the properness and the domain, and provide simpler proof.


Local proper scoring rules Subjective probability Incentive compatibility 



I am grateful to Peter Wakker for illuminative discussions and helpful comments. I thank Drazen Prelec for having raised the question of generalization.


  1. Abdellaoui, M., Bleichrodt, H., Kemel, E., L’Haridon, O. (2017). Measuring beliefs under ambiguity. Working Paper.Google Scholar
  2. Aczél, J., & Pfanzagl, J. (1967). Remarks on the measurement of subjective probability and information. Metrika, 11(1), 91–105.CrossRefGoogle Scholar
  3. Bernardo, J. M. (1979). Expected information as expected utility. Annals of Statistics, 7(3), 686–690.CrossRefGoogle Scholar
  4. Giles, J. (2002). Scientific wagers: Wanna bet? Nature, 420(6914), 354–355.CrossRefGoogle Scholar
  5. Goldstein, D. G., & Rothschild, D. (2014). Lay understanding of probability distributions. Judgment & Decision Making, 9(1), 1–14.Google Scholar
  6. Hollard, G., Massoni, S., & Vergnaud, J. C. (2016). In search of good probability assessors: an experimental comparison of elicitation rules for confidence judgments. Theory and Decision, 80(3), 363–387.CrossRefGoogle Scholar
  7. Johnson, S., Pratt, J. W., & Zeckhauser, R. J. (1990). Efficiency despite mutually payoff-relevant private information: The finite case. Econometrica, 58(4), 873–900.CrossRefGoogle Scholar
  8. Johnstone, D. J. (2007). The value of a probability forecast from portfolio theory. Theory and Decision, 63(2), 153–203.CrossRefGoogle Scholar
  9. Karni, E., & Safra, Z. (1995). The impossibility of experimental elicitation of subjective probabilities. Theory and Decision, 38(3), 313–320.CrossRefGoogle Scholar
  10. Myerson, R. B. (1982). Optimal coordination mechanisms in generalized principal-agent problems. Journal of Mathematical Economics, 10(1), 67–81.CrossRefGoogle Scholar
  11. Offerman, T., Sonnemans, J., Van de Kuilen, G., & Wakker, P. P. (2009). A truth serum for non-bayesians: Correcting proper scoring rules for risk attitudes. The Review of Economic Studies, 76(4), 1461–1489.CrossRefGoogle Scholar
  12. Savage, L. J. (1971). Elicitation of personal probabilities and expectations. Journal of the American Statistical Association, 66(336), 783–801.CrossRefGoogle Scholar
  13. Shuford, E. H., Albert, A., & Edward Massengill, H. (1966). Admissible probability measurement procedures. Psychometrika, 31(2), 125–145.CrossRefGoogle Scholar
  14. van Rooij, A. C., & Schikhof, W. H. (1982). A Second Course on Real Functions. Cambridge University Press.Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Research School of EconomicsAustralian National UniversityCanberraAustralia

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