On ambiguity apportionment

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.

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.