Theory and Decision

, Volume 73, Issue 1, pp 161–184 | Cite as

Inferring beliefs as subjectively imprecise probabilities

  • Steffen Andersen
  • John Fountain
  • Glenn W. Harrison
  • Arne Risa Hole
  • E. Elisabet Rutström


We propose a method for estimating subjective beliefs, viewed as a subjective probability distribution. The key insight is to characterize beliefs as a parameter to be estimated from observed choices in a well-defined experimental task and to estimate that parameter as a random coefficient. The experimental task consists of a series of standard lottery choices in which the subject is assumed to use conventional risk attitudes to select one lottery or the other and then a series of betting choices in which the subject is presented with a range of bookies offering odds on the outcome of some event that the subject has a belief over. Knowledge of the risk attitudes of subjects conditions the inferences about subjective beliefs. Maximum simulated likelihood methods are used to estimate a structural model in which subjects employ subjective beliefs to make bets. We present evidence that some subjective probabilities are indeed best characterized as probability distributions with non-zero variance.


Subjective risk Subjective beliefs Random coefficients Non-linear mixed logit Experiments 


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  1. Ahn, D., Choi, S., Gale, D., & Kariv, S. (2009). Estimating ambiguity aversion in a portfolio choice experiment. Working paper, Department of Economics, University of California at Berkeley, Feb 2009.Google Scholar
  2. Aitchison J., Begg C.B. (1976) Statistical diagnosis when basic cases are not classified with certainty. Biometrika 63(1): 1–12CrossRefGoogle Scholar
  3. Andersen, S., Fountain, J., Harrison, G. W., & Rutström, E. E. (2010). Estimating subjective probabilities. Working Paper 2010–2006, Center for the Economic Analysis of Risk, Robinson College of Business, Georgia State University.Google Scholar
  4. Andersen, S., Harrison, G. W., Hole, A. R., Lau, & Ruström, E. E. (2011). Non-linear mixed logit. Theory and Decision. doi:10.10071s112.38-011-9277-00.
  5. Andersen S., Harrison G. W., Lau Morten I., Rutström E. E. (2008) Eliciting risk and time preferences. Econometrica 76(3): 583–618CrossRefGoogle Scholar
  6. Binswanger H. P. (1981) Attitudes toward risk: Theoretical implications of an experiment in rural India. Economic Journal 91: 867–890CrossRefGoogle Scholar
  7. de Finetti, B. (1937). La Prévision: Ses Lois Logiques, ses sources subjectives. Annales de l’Institut Henri Poincairé, 7, 1–68; English translation as “Foresight: Its logical laws, its subjective sources,” in H. E. Kyburg & H. E. Smokler (Eds.), Studies in subjective probability. Huntington, NY: Robert E. Krieger, 1980, Second Edition.Google Scholar
  8. de Finetti B. (1970) Logical foundations and measurement of subjective probability. Acta Psychologica 34: 129–145CrossRefGoogle Scholar
  9. Drukker D.M., Gates R. (2006) Generating Halton sequences using mata. Stata Journal 6(2): 214–228Google Scholar
  10. Eckel C. C., Grossman P. J. (2002) Sex differences and statistical stereotyping in attitudes toward financial risk. Evolution and Human Behavior 23(4): 281–295CrossRefGoogle Scholar
  11. Ellsberg D. (1961) Risk, ambiguity, and the savage axioms. Quarterly Journal of Economics 75: 643–669CrossRefGoogle Scholar
  12. Epstein R. A. (1977) The theory of gambling and statistical logic. Academic Press, San DiegoGoogle Scholar
  13. Ergin H., Gul F. (2009) A theory of subjective compound lotteries. Journal of Economic Theory 144(3): 899–929CrossRefGoogle Scholar
  14. Ghirardoto P., Maccheroni F., Marinacci M. (2004) Differentiating ambiguity and ambiguity attitude. Journal of Economic Theory 118: 133–173CrossRefGoogle Scholar
  15. Gilboa I., Postlewaite A. P., Schmeidler D. (2008) Probability and uncertainty in economic modeling. Journal of Economic Perspectives 22(3): 173–188CrossRefGoogle Scholar
  16. Gilboa I., Schmeidler D. (1989) Maxmin expected utility with a non-unique prior. Journal of Mathematical Economics 18: 141–153CrossRefGoogle Scholar
  17. Grant S., Kajii A., Polak B. (1998) Intrinsic preference for information. Journal of Economic Theory 83: 233–259CrossRefGoogle Scholar
  18. Halevy Y. (2007) Ellsberg revisited: An experimental study. Econometrica 75: 503–536CrossRefGoogle Scholar
  19. Harrison G. W., Rutström E. E. (2008) Risk aversion in the laboratory. In: Cox J. C., Harrison G. W. (eds) Risk aversion in experiments. Emerald, Bingley, UKGoogle Scholar
  20. Hey, J. D., Lotito, G., Maffioletti, A. (2007). Choquet OK? Discussion Paper No. 2007/12, Department of Economics and Related Studies, University of York.Google Scholar
  21. Hey, J. D., Lotito, G., Maffioletti, A. (2008). The descriptive and predictive adequacy of theories of decision making under uncertainty/ambiguity. Discussion Paper No. 2008/04, Department of Economics and Related Studies, University of York.Google Scholar
  22. Hey J. D., Orme C. (1994) Investigating generalizations of expected utility theory using experimental data. Econometrica 62(6): 1291–1326CrossRefGoogle Scholar
  23. Hole A. R. (2007) Fitting mixed logit models by using maximum simulated likelihood. Stata Journal 7(3): 388–401Google Scholar
  24. Holt C. A., Laury S. K. (2002) Risk aversion and incentive effects. American Economic Review 92(5): 1644–1655CrossRefGoogle Scholar
  25. Huber J., Train K. (2001) On the similarity of classical and Bayesian estimates of individual mean partworths. Marketing Letters 12(3): 259–269CrossRefGoogle Scholar
  26. Klibanoff P., Marinacci M., Mukerji S. (2005) A smooth model of decision making under ambiguity. Econometrica 73(6): 1849–1892CrossRefGoogle Scholar
  27. Lesaffre E., Rizopoulos D., Tsonaka R. (2007) The logistic transform for bounded outcome scores. Biostatistics 8(1): 72–85CrossRefGoogle Scholar
  28. Machina M. J. (2004) Almost-objective uncertainty. Economic Theory 24: 1–54CrossRefGoogle Scholar
  29. Mathieson J. E., Winkler R. L. (1976) Scoring rules for continuous probability distributions. Management Science 22(10): 1087–1096CrossRefGoogle Scholar
  30. Nau R. F. (2006) Uncertainty aversion with second-order utilities and probabilities. Management Science 52: 136–156CrossRefGoogle Scholar
  31. Nau R. F. (2007) Extensions of the subjective expected utility model. In: Ward E., Ralph M., von Detlof W. (eds) Advances in decision analysis: From foundations to applications. Cambridge University Press, New YorkGoogle Scholar
  32. Neilson W. S. (2010) A simplified axiomatic approach to ambiguity aversion. Journal of Risk and Uncertainty 41: 113–124CrossRefGoogle Scholar
  33. Oehlert G. W. (1992) A note on the delta method. The American Statistician 46(1): 27–29Google Scholar
  34. Quiggin J. (1993) Generalized expected utility theory: The rank-dependent model. Kluwer Academic, Norwell, MACrossRefGoogle Scholar
  35. Revelt D., Train K. (1998) Mixed logit with repeated choices: Households’ choices of appliance efficiency levels. Review of Economics and Statistics 80: 647–657CrossRefGoogle Scholar
  36. Saha A. (1993) Expo-power utility: A flexible form for absolute and relative risk aversion. American Journal of Agricultural Economics 75(4): 905–913CrossRefGoogle Scholar
  37. Savage L. J. (1971) Elicitation of personal probabilities and expectations. Journal of American Statistical Association 66: 783–801Google Scholar
  38. Segal U. (1987) The Ellsberg paradox and risk aversion: An anticipated utility approach. International Economic Review 28(1): 175–202CrossRefGoogle Scholar
  39. Segal U. (1990) Two-stage lotteries without the independence axiom. Econometrica 58(2): 349–377CrossRefGoogle Scholar
  40. Smith V. L. (1969) Measuring nonmonetary utilities in uncertain choices: The Ellsberg Urn. Quarterly Journal of Economics 83(2): 324–329CrossRefGoogle Scholar
  41. Train K. E. (2003) Discrete choice methods with simulation. Cambridge University Press, New YorkCrossRefGoogle Scholar
  42. Wilcox N. T. (2008) Stochastic models for binary discrete choice under risk: A critical primer and econometric comparison. In: Cox J., Harrison G. W. (eds) Risk aversion in experiments. Emerald, Bingley, UKGoogle Scholar
  43. Wilcox N.T. (2011) ‘Stochastically More Risk Averse:’ A Contextual Theory of Stochastic Discrete Choice Under Risk. Journal of Econometrics 162(1): 87–104CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Steffen Andersen
    • 1
  • John Fountain
    • 2
  • Glenn W. Harrison
    • 3
  • Arne Risa Hole
    • 4
  • E. Elisabet Rutström
    • 5
  1. 1.Department of EconomicsCopenhagen Business SchoolCopenhagenDenmark
  2. 2.Department of EconomicsUniversity of CanterburyChristchurchNew Zealand
  3. 3.Department of Risk Management & Insurance and Center for the Economic Analysis of Risk, Robinson College of BusinessGeorgia State UniversityAtlantaUSA
  4. 4.Department of EconomicsUniversity of SheffieldSheffieldUK
  5. 5.Robinson College of Business and Department of Economics, Andrew Young School of Policy StudiesGeorgia State UniversityAtlantaUSA

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