Definitions of ambiguous events and the smooth ambiguity model

Abstract

We examine a variety of preference-based definitions of ambiguous events in the context of the smooth ambiguity model. We first consider the definition proposed in Klibanoff et al. (Econometrica 73(6):1849–1892, 2005) based on the classic Ellsberg two-urn paradox (Ellsberg Q J Econ 75:643–669, 1961) and show that it satisfies several desirable properties. We then compare this definition with those of Nehring (Math Soc Sci 38(2):197–213, 1999), Epstein and Zhang (Econometrica 69:265–306, 2001), Zhang (Econ Theory 20:159–181, 2002), and Ghirardato and Marinacci (J Econ Theory 102:251–289, 2002). Within the smooth ambiguity model, we show that Ghirardato and Marinacci (J Econ Theory 102:251–289, 2002) would identify the same set of ambiguous and unambiguous events as our definition while Epstein and Zhang (Econometrica 69:265–306, 2001) and Zhang (Econ Theory 20:159–181, 2002) would yield a different classification. Moreover, we discuss and formally identify two key sources of the differences compared to Epstein and Zhang (Econometrica 69:265–306, 2001) and Zhang (Econ Theory 20:159–181, 2002). The more interesting source is that these two definitions can confound non-constant ambiguity attitude and the ambiguity of an event.

This is a preview of subscription content, log in to check access.

References

  1. Amarante M., Filiz E.: Ambiguous events and maxmin expected utility. J Econ Theory 134(1), 1–33 (2007)

    Article  Google Scholar 

  2. Billingsley P.: Probability and Measure. 2nd ed. John Wiley, New York (1986)

    Google Scholar 

  3. Cerreia-Vioglio S., Ghirardato P., Maccheroni F., Marinacci M., Siniscalchi M.: Rational preferences under ambiguity. Econ Theory (2011) (this issue)

  4. Ellsberg D.: Risk, ambiguity, and the savage axioms. Q J Econ 75, 643–669 (1961)

    Article  Google Scholar 

  5. Epstein L.G., Zhang J.: Subjective probabilities on subjectively unambiguous events. Econometrica 69, 265–306 (2001)

    Article  Google Scholar 

  6. Gilboa I., Schmeidler D.: Maxmin expected utility with a non-unique prior. J Math Econ 18, 141–153 (1989)

    Article  Google Scholar 

  7. Ghirardato P., Marinacci M.: Ambiguity made precise: a comparative foundation. J Econ Theory 102, 251–289 (2002)

    Article  Google Scholar 

  8. Klibanoff P., Marinacci M., Mukerji S.: A smooth model of decision making under ambiguity. Econometrica 73(6), 1849–1892 (2005)

    Article  Google Scholar 

  9. Kopylov I.: Subjective probabilities on ‘small’ domains. J Econ Theory 133(1), 236–265 (2007)

    Article  Google Scholar 

  10. Kreps D.: Notes on the Theory of Choice. Westview Press, Boulder and London (1988)

    Google Scholar 

  11. Maccheroni F., Marinacci M., Rustichini A.: Ambiguity aversion, robustness and the variational representation of preferences. Econometrica 74(6), 1447–1498 (2006)

    Article  Google Scholar 

  12. Machina M., Schmeidler D.: A more robust definition of subjective probability. Econometrica 60, 745–780 (1992)

    Article  Google Scholar 

  13. Nehring K.: Capacities and probabilistic beliefs: a precarious coexistence. Math Soc Sci 38(2), 197–213 (1999)

    Article  Google Scholar 

  14. Nehring, K.: Is it possible to define subjective probabilities in purely behavioral terms? A comment on Epstein-Zhang (2001). In: Discussion Paper 0067, Economics Working Papers. Institute for Advanced Study, School of Social Science (2006)

  15. Zhang J.: Qualitative probabilities on λ-systems. Math Soc Sci 38, 11–20 (1999)

    Article  Google Scholar 

  16. Zhang J.: Subjective ambiguity, expected utility and choquet expected utility. Econ Theory 20, 159–181 (2002)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Peter Klibanoff.

Additional information

We thank Simone Cerreia-Vioglio, Paolo Ghirardato, Mark Machina and Marciano Siniscalchi for helpful discussions.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Klibanoff, P., Marinacci, M. & Mukerji, S. Definitions of ambiguous events and the smooth ambiguity model. Econ Theory 48, 399–424 (2011). https://doi.org/10.1007/s00199-011-0641-7

Download citation

Keywords

  • Ambiguity
  • Uncertainty
  • Knightian uncertainty
  • Ambiguity aversion
  • Uncertainty aversion
  • Ellsberg paradox

JEL Classification

  • D800
  • D810