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The dual theory of the smooth ambiguity model

Economic Theory volume 56, pages 275–289 (2014)Cite this article

Abstract

This paper studies the “dual” theory of the smooth ambiguity model introduced by Klibanoff et al. (Econometrica 73:1849–1892, 2005). Unlike the original model, we characterize attitudes toward ambiguity captured by second-order probabilities. First, we give a set of axioms to derive a dual representation of the smooth ambiguity model. Second, we present a characterization of ambiguity aversion. Last, as an application of our dual model to a portfolio problem, we conduct comparative static predictions which give sufficient conditions to guarantee that an increase in smooth ambiguity aversion decreases the optimal portfolio.

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Notes

  1. Uncertainty is used as an umbrella term for both risk and ambiguity. Risk is defined as a condition in which the event to be realized is a-priori unknown, but the odds of all possible events are perfectly known. Ambiguity refers to conditions in which not only the event to be realized is a-priori unknown, but the odds of the events are also either not uniquely assigned or are unknown.

  2. Some studies offer different axiomatization of the smooth ambiguity model. See Seo (2009), Nascimento and Riella (2013) and references therein.

  3. For example, Gonzalez and Wu (1996, 1999) conduct experiments to measure subjective probabilities by probability weighting functions. See Wakker (2010) for a more comprehensive survey.

  4. The formulation of ambiguous events in the smooth ambiguity model is discussed in Klibanoff et al. (2011).

  5. Gollier (2011) determine conditions such that the following is satisfied

    $$\begin{aligned}&\sum _{\theta =1}^n q_\theta \phi _1^\prime (U(\alpha _1^*),\theta ) \mathbb { E} \tilde{x}_\theta u^\prime (w_0 + \alpha _1^*\tilde{x}_\theta ) = 0 \\&\quad \Longrightarrow \sum _{\theta =1}^n q_\theta \phi _2^\prime (U(\alpha _1^*),\theta ) \mathbb {E} \tilde{x}_\theta u^\prime (w_0 + \alpha _1^*\tilde{x}_\theta ) \le 0. \end{aligned}$$

    Since \(\phi ^\prime \) is LSPM, by Property 1 we can determine conditions for the above relation by looking for conditions which satisfies the single crossing condition of \(\mathbb {E} \tilde{x}_\theta u^\prime (w_0 + \alpha _1^*\tilde{x}_\theta ) \).

References

  • Athey, S.: Monotone comparative statics under uncertainty. Q J Econ 117, 187–223 (2002)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Fishburn, P.C.: The foundations of expected utility. D. Reidel, Dordrecht (1982)

    Book  Google Scholar 

  • Fishburn, P.C., Porter, R.B.: Optimal portfolios with one safe and one risky asset: effects of changes in rate of return and risk. Manag Sci 22, 1064–1073 (1976)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Gollier, C.: The economics of risk and time. MIT Press, Cambridge (2001)

    Google Scholar 

  • Gollier, C.: Portfolio choices and asset prices: the comparative statics of ambiguity aversion. Rev Econ Stud 78, 1329–1344 (2011)

    Article  Google Scholar 

  • Gonzalez, R., Wu, G.: Curvature of the probability weighting function. Manag Sci 42, 1676–1690 (1996)

    Article  Google Scholar 

  • Gonzalez, R., Wu, G.: On the shape of the probability weighting function. Cogn Sci 38, 129–166 (1999)

    Google Scholar 

  • Grandaunt, J.M.: Continuity properties of a von Neumann–Morgenstern utility. J Econ Theory 4, 45–57 (1972)

    Article  Google Scholar 

  • Jewitt, I.: Risk aversion and the choice between risky prospects: the preservation of comparative statics results. Rev Econ Stud 54, 73–85 (1987)

    Article  Google Scholar 

  • Karlin, S.: Total positivity. Stanford University Press, Stanford (1968)

    Google Scholar 

  • Karlin, S., Novikoff, A.: Generalized convex inequalities. Pac J Math 13, 1251–1279 (1963)

    Google Scholar 

  • Klibanoff, P., Marinacci, M., Mukerji, S.: A smooth model of decision making under ambiguity. Econometrica 58, 1849–1892 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Nascimento, L., Riella, G.: Second-order ambiguous beliefs. Econ Theory 52, 1005–1037 (2013)

    Google Scholar 

  • Rothschild, M., Stiglitz, J.: Increasing risk: I. A definition. J Econ Theory 2, 225–243 (1970)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Segal, U.: The Ellsberg paradox and risk aversion: an anticipated utility approach. IER 28, 175–202 (1987)

    Google Scholar 

  • Seo, K.: Ambiguity and second-order belief. Econometrica 77, 1575–1605 (2009)

    Article  Google Scholar 

  • Wakker, P.: Prospect theory: for risk and ambiguity. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  • Yaari, M.E.: Univariate and multivariate comparison of risk aversion: a new approach. In: Heller, W.P., Starr, R.M., Starrett, D.A. (eds.) Uncertainty, Information, and Communication (Essays in Honor of Kenneth J. Arrow, Vol. 3), pp. 173–187. Cambridge University Press, Cambridge (1986)

  • Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987)

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Business Administration, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto, 603-8555, Japan

    Hideki Iwaki

  2. Faculty of Economics, Osaka Sangyo University, Osaka, Japan

    Yusuke Osaki

Authors
  1. Hideki Iwaki
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  2. Yusuke Osaki
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Corresponding author

Correspondence to Hideki Iwaki.

Additional information

This paper is partly supported by a Grant-in-Aid for Scientific Research (c), a Grant-in-Aid for Young Scientists (B) and the 2010 Zengin Foundation for Studies on Economics and Finance. We acknowledge Harris Schlesinger and an anonymous referee for their valuable comments on an earlier version of this paper.

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Iwaki, H., Osaki, Y. The dual theory of the smooth ambiguity model. Econ Theory 56, 275–289 (2014). https://doi.org/10.1007/s00199-013-0779-6

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  • DOI: https://doi.org/10.1007/s00199-013-0779-6

Keywords

  • Ambiguity
  • Ambiguity aversion
  • Comparative statics
  • Smooth ambiguity model

JEL Classification

  • D800
  • D810