  This paper axiomatizes, in a two-stage setup, a new theory for decision under
risk and ambiguity. The axiomatized preference relation $\succeq$ on the space
$\tilde{V}$ of random variables induces an ambiguity index $c$ on the space
$\Delta$ of probabilities, a probability weighting function $\psi$, generating
the measure $\nu_{\psi}$ by transforming an objective probability measure, and
a utility function $\phi$, such that, for all
$\tilde{v},\tilde{u}\in\tilde{V}$, \begin{align*} \tilde{v}\succeq\tilde{u}
\Leftrightarrow \min_{Q \in \Delta}
\left\{\mathbb{E}_Q\left[\int\phi\left(\tilde{v}^{\centerdot}\right)\,\mathrm{d}\nu_{\psi}\right]+c(Q)\right\}
\geq \min_{Q \in \Delta}
\left\{\mathbb{E}_Q\left[\int\phi\left(\tilde{u}^{\centerdot}\right)\,\mathrm{d}\nu_{\psi}\right]+c(Q)\right\}.
\end{align*} Our theory extends the rank-dependent utility model of Quiggin
(1982) for decision under risk to risk and ambiguity, reduces to the
variational preferences model when $\psi$ is the identity, and is dual to
variational preferences when $\phi$ is affine in the same way as the theory of
Yaari (1987) is dual to expected utility. As a special case, we obtain a
preference axiomatization of a decision theory that is a rank-dependent
generalization of the popular maxmin expected utility theory. We characterize
ambiguity aversion in our theory.
