Definitions of ambiguous events and the smooth ambiguity model


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.

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  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 

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Correspondence to Peter Klibanoff.

Additional information

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

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Klibanoff, P., Marinacci, M. & Mukerji, S. Definitions of ambiguous events and the smooth ambiguity model. Econ Theory 48, 399–424 (2011).

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

JEL Classification

  • D800
  • D810