## Abstract

This paper investigates the notion of changes in ambiguity over loss probabilities in the smooth ambiguity model developed by Klibanoff, Marinacci and Mukerji (Econometrica 73:1849–1892, 2005). Changes in ambiguity over loss probabilities are expressed through the specific concept of stochastic dominance of order *n* defined by Ekern (Econ Lett 6:329–333, 1980). We characterize conditions on the function capturing attitudes towards ambiguity under which an individual always considers one situation to be more ambiguous than another in a model of two states of nature. We propose an intuitive interpretation of the properties of this function in terms of preferences for harms disaggregation over probabilities, also labelled ambiguity apportionment.

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## Notes

i.e. \(u^{G}(w)=u(w)\) and \(u^{B}(w)=u(w-L)\) with \(L>0\) and \(u^{\prime \prime }<0<u^{\prime }\) as in the classical model with a monetary loss

*L*Because risk aversion means that \(E[u(w+\tilde{\epsilon })]<u(w+E(\tilde{ \epsilon }))\).

More precisely, this definition corresponds to the definition (

*I*) proposed by the authors. Jewitt and Mukerji (2014) also propose a second definition based on choices of agents with different levels of ambiguity aversion.We cannot extend this result to the case of a link of stochastic dominance between \(\tilde{\epsilon }_{1}\) and \(\tilde{\epsilon }_{2}\) since successive derivatives of the function

*f*as defined in “Appendix” do not alternate in signs (see “Appendix”).If ambiguity were defined on the good state of nature, contrary to what is usually done in the literature, a greater level of ambiguity would be equivalent to a dominated variable in Ekern’s sense.

Eeckhoudt et al. (2009) results also apply to stochastic dominance.

As suggested by one referee, for \(n=1\) and \(n=2\), changes in ambiguity can be represented through the linear transformation function proposed by Sandmo (1971) \(t(p)=\gamma (p-\overline{p})+\overline{p}+k\), where \(\gamma \) is a multiplicative shift parameter,

*k*is an additive shift parameter, and \( \overline{p}\) is the mean probability. For \(n=1\), an increase in*k*(from \( k=0\) with \(\gamma =1\)) is similar to a first-degree risk improvement. For \( n=2\), an increase in \(\gamma \) (from \(\gamma =1\) with \(k=0\)) is similar to a Rothschild-Stiglitz increase in risk.

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## Appendix

### Appendix

\(V(w,p+\tilde{\epsilon }_{2})\le V(w,p+\tilde{\epsilon }_{1})\) is equivalent to

\(E[\Phi \{(1-(p+\tilde{\epsilon }_{2}))u^{G}(w)+(p+\tilde{\epsilon } _{2})u^{B}(w)\}]\le E[\Phi \{(1-(p+\tilde{\epsilon }_{1}))u^{G}(w)+(p+\tilde{ \epsilon }_{1})u^{B}(w)\}]\), that rewrites as:

\(E[\Phi \{ V_0(w,p) + \tilde{\epsilon }_{2} \Delta u(w) \} ] \le E[\Phi \{ V_0(w,p) + \tilde{\epsilon }_{1} \Delta u(w) \} ]\), with \(\Delta u(w)= u^{B}(w) - u^{G}(w)\).

Let us define the function *f* as follows: \(f(\epsilon )=\Phi \Bigl ( V_{0}(w,p)+\epsilon \Delta u(w)\Bigr )\).

The previous inequality rewrites as: \(E[f(\widetilde{\epsilon }_{2})] \le E[f(\widetilde{\epsilon }_{1})]\).

We obtain:

\(f^{\prime }(\epsilon )=\Delta u(w)\Phi ^{\prime }\Bigl (V_{0}(w,p)+\epsilon (\Delta u(w))\Bigr )\),

\(f^{\prime \prime }(\epsilon )=(\Delta u(w))^{2}\Phi ^{\prime \prime }\Bigl ( V_{0}(w,p)+\epsilon (\Delta u(w))\Bigr )\),

\(f^{(3)}(\epsilon ) = (\Delta u(w))^3 \Phi ^{(3)} \Bigl ( V_0(w,p) + \epsilon (\Delta u(w)) \Bigr )\),

\(f^{(4)}(\epsilon ) = (\Delta u(w))^4 \Phi ^{(4)} \Bigl ( V_0(w,p) + \epsilon (\Delta u(w)) \Bigr )\),

. . . ,

\(f^{(n)}(\epsilon ) = (\Delta u(w))^n \Phi ^{(n)} \Bigl ( V_0(w,p) + \epsilon (\Delta u(w)) \Bigr )\).

As by assumption, \(\Delta u(w) <0\), we obtain:

\(\Phi ^{\prime }(x)>0\) \(\forall x\) \(\Leftrightarrow \) \(f^{\prime }(\epsilon )<0\) \(\forall \epsilon \),

\(\Phi ^{\prime \prime }(x)<0\) \(\forall x\) \(\Leftrightarrow \) \(f^{\prime \prime }(\epsilon )<0\) \(\forall \epsilon \),

\(\Phi ^{(3)}(x)>0\) \(\forall x\) \(\Leftrightarrow \) \(f^{(3)}(\epsilon )<0\) \( \forall \epsilon \),

\(\Phi ^{(4)}(x)<0\) \(\forall x\) \(\Leftrightarrow \) \(f^{(4)}(\epsilon )<0\) \( \forall \epsilon \),

\(\ldots \),

\(\Phi ^{(n)}(x)>0\)
\(\forall x\) when *n* is odd \(\Leftrightarrow \)
\( f^{(n)}(\epsilon )<0\)
\(\forall \epsilon \),

\(\Phi ^{(n)}(x)<0\)
\(\forall x\) when *n* is even \(\Leftrightarrow \)
\( f^{(n)}(\epsilon )<0\)
\(\forall \epsilon \).

Using Ekern (1980), we obtain:

if \(\tilde{\epsilon }_{2}\preceq _{n}\tilde{\epsilon }_{1}\) for *n* even, \(E[f( \widetilde{\epsilon }_{2})]\le E[f(\widetilde{\epsilon }_{1})]\) for all *f* such that \(f^{(n)}(\epsilon )<0\)
\(\forall \epsilon \),

if \(\tilde{\epsilon }_{1}\preceq _{n}\tilde{\epsilon }_{2}\) for *n* odd, \(E[f( \widetilde{\epsilon }_{2})]\le E[f(\widetilde{\epsilon }_{1})]\) for all *f* such that \(f^{(n)}(\epsilon )<0\)
\(\forall \epsilon \), that proves Proposition 1.

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Courbage, C., Rey, B. On ambiguity apportionment.
*J Econ* **118**, 265–275 (2016). https://doi.org/10.1007/s00712-016-0473-9

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DOI: https://doi.org/10.1007/s00712-016-0473-9