A Rank-Dependent Theory for Decision under Risk and Ambiguity
Roger J. A. Laeven, Mitja Stadje
TL;DR
The paper introduces a two-stage rank-dependent theory for decision under risk and ambiguity, unifying risk preferences and ambiguity attitudes through a representation $U(\tilde{v}) = \min_{Q} \{ \mathbb{E}_Q[\int \phi(\tilde{v}^{\centerdot}) \\\ d\nu_{\psi}] + c(Q) \}$ that incorporates a probability weighting function $\psi$, a utility transformation $\phi$, an ambiguity index $c$, and a transformed measure $\nu_{\psi}$. It shows how this framework extends Quiggin's rank-dependent utility to risk and ambiguity, reduces to variational preferences when $\psi$ is the identity and is dual to Yaari's dual theory when $\phi$ is affine, thereby providing a unified, dualizable treatment of both risk and ambiguity. The authors develop new axioms—Dual Ambiguity Aversion and Dual Independence variants—ground subjective mixtures of random variables and a diversification preference, leading to a main representation that generalizes both rank-dependent utility and robust maxmin approaches (MEU/VP) as special cases. The theory offers a principled foundation for robust tail-risk measures (e.g., weighted VaR, ESS), links to mean-risk portfolio criteria, and clarifies how ambiguity aversion interacts with diversification, model misspecification, and probabilistic sophistication. Overall, the framework provides a rigorous, axiomatized method to model decision making under uncertainty with explicit separation of wealth, risk, and ambiguity attitudes and yields practical implications for robust risk management and portfolio optimization under model misspecification.
Abstract
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.