A Rank-Dependent Theory for Decision under Risk and Ambiguity
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 $Δ$ of probabilities, a probability weighting function $ψ$, generating the measure $ν_ψ$ by transforming an objective probability measure, and a utility function $ϕ$, such that, for all $\tilde{v},\tilde{u}\in\tilde{V}$, \begin{align*} \tilde{v}\succeq\tilde{u} \Leftrightarrow \min_{Q \in Δ} \left\{\mathbb{E}_Q\left[\intϕ\left(\tilde{v}^{\centerdot}\right)\,\mathrm{d}ν_ψ\right]+c(Q)\right\} \geq \min_{Q \in Δ} \left\{\mathbb{E}_Q\left[\intϕ\left(\tilde{u}^{\centerdot}\right)\,\mathrm{d}ν_ψ\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 $ψ$ is the identity, and is dual to variational preferences when $ϕ$ 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.