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Re-examining the law of iterated expectations for Choquet decision makers


Yoo (Economic Letters 37:145–149, 1991) argues that the law of iterated expectations must be violated if the probability measure of a Choquet decision maker is non-additive. In this article, we prove the positive result that the law of iterated expectations is satisfied for Choquet decision makers whenever they update their non-additive beliefs in accordance with the Sarin and Wakker (Journal of Risk and Uncertainty 16:223–250, 1998) update rule. The formal key to this result is the act-dependence of the Sarin–Wakker update rule, which does not hold for the update rules considered by Yoo (1991).

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  1. Anscombe F. J., Aumann R. J.: A definition of subjective probability. Annals of American Statistics 34, 199–205 (1963)

  2. Billingsley P.: Probability and measure. Wiley, New York (1995)

  3. Cochrane J. H.: Asset pricing. Princeton University Press, Princeton (2001)

  4. Duffie D.: Dynamic asset pricing theory. Princeton University Press, Princeton (2001)

  5. Eichberger J., Kelsey D.: E-capacities and the Ellsberg paradox. Theory and Decision 46, 107–140 (1999)

  6. Eichberger J., Grant S., Kelsey D.: Updating Choquet beliefs. Journal of Mathematical Economics 43, 888–899 (2006)

  7. Ellsberg D.: Risk, ambiguity and the savage axioms. Quarterly Journal of Economics 75, 643–669 (1961)

  8. Epstein L.G.: A definition of uncertainty aversion. The Review of Economic Studies 66, 579–608 (1999)

  9. Ghirardato P., Marinacci M.: Ambiguity made precise: A comparative foundation. Journal of Economic Theory 102, 251–289 (2002)

  10. Gilboa I.: Expected utility with purely subjective non-additive probabilities. Journal of Mathematical Economics 16, 65–88 (1987)

  11. Gilboa I., Schmeidler D.: Updating ambiguous beliefs. Journal of Economic Theory 59, 33–49 (1993)

  12. Quiggin J. P.: Risk perception and risk aversion among australian farmers. Australian Journal of Agricultural Economics 25, 160–169 (1981)

  13. Quiggin J. P.: A theory of anticipated utility. Journal of Economic Behavior and Organization 3, 323–343 (1982)

  14. Sarin R., Wakker P. P.: A simple axiomatization of nonadditive expected utility. Econometrica 60, 1255–1272 (1992)

  15. Sarin R., Wakker P. P.: Revealed likelihood and Knightian uncertainty. Journal of Risk and Uncertainty 16, 223–250 (1998)

  16. Savage L. J.: The foundations of statistics. Wiley, New York (1954)

  17. Schmeidler D.: Integral representation without additivity. Proceedings of the American Mathematical Society 97, 255–261 (1986)

  18. Schmeidler D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)

  19. Tversky A., Kahneman D.: Advances in prospect theory: Cumulative representations of uncertainty. Journal of Risk and Uncertainty 5, 297–323 (1992)

  20. Tversky A., Wakker P. P.: Risk attitudes and decision weights. Econometrica 63, 1255–1280 (1995)

  21. Wakker P. P.: Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle. Econometrica 69, 1039–1059 (2001)

  22. Wakker P. P.: On the composition of risk preference and belief. Psychological Review 111, 236–241 (2004)

  23. Wakker P. P., Tversky A.: An axiomatization of cumulative prospect theory. Journal of Risk and Uncertainty 7, 147–176 (1993)

  24. Yoo K. R.: The iterative law of expectations and non-additive probability measure. Economics Letters 37, 145–149 (1991)

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Correspondence to Alexander Zimper.

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Zimper, A. Re-examining the law of iterated expectations for Choquet decision makers. Theory Decis 71, 669–677 (2011). https://doi.org/10.1007/s11238-010-9221-8

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  • Non-additive probability measures
  • Choquet expected utility theory
  • Dynamic decision making
  • Asset pricing

JEL Classification

  • C72
  • D81
  • D90