# The dual theory of the smooth ambiguity model

## 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 $$\mathbb {E} \tilde{x}_\theta u^\prime (w_0 + \alpha _1^*\tilde{x}_\theta )$$.

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## Author information

Authors

### Corresponding author

Correspondence to Hideki Iwaki.

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

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