A Rank-Dependent Theory for Decision under Risk and Ambiguity

Paper: arXiv 2312.05977

Authors: Roger J. A. Laeven, Mitja Stadje

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous axiomatic foundation extending Rank-Dependent Utility to environments with ambiguity, combining elements of variational preferences and Yaari’s dual model. While mathematical density and theoretical novelty are high, it lacks empirical backtesting or real-world data validation typical of practical quantitative finance papers.

Research Flowchart

  flowchart TD
    A["Research Goal: Extend Rank-Dependent Utility (RDU) to incorporate ambiguity"] --> B["Data/Inputs: Two-stage setup with space of random variables Ṽ and probability space Δ"]
    B --> C["Axiomatization: Establish preference relation ≽ on Ṽ"]
    C --> D["Representation Derivation: Derive utility φ, probability weighting ψ, measure ν_ψ, and ambiguity index c(Q)"]
    D --> E["Core Evaluation Process:<br/>min<sub>Q ∈ Δ</sub> { E<sub>Q</sub>[∫ φ(ṽ<sup>•</sup>) dν<sub>ψ</sub>] + c(Q) }"]
    E --> F["Key Outcomes & Findings"]
    F --> G["1. Unifies RDU & Variational Preferences<br/>2. Dual to Variational Preferences (affine φ)<br/>3. Generalizes Maxmin Expected Utility (MEU)<br/>4. Characterizes Ambiguity Aversion"]

Cited References (89)

1. Rank-Dependent Utility Under Multiple Priors (2021) — link
2. Optimizing distortion riskmetrics with distributional uncertainty (2020) — link
3. Risk, Ambiguity, and the Value of Diversification (2020) — link
4. Risk apportionment: The dual story (2017) — link
5. Dual Moments and Risk Attitudes (2016) — link
6. Cumulative Prospect Theory in the Laboratory: A Reconsideration (2016)
7. Hurwicz expected utility and subjective sources (2015) — link
8. Dynamic Portfolio Choice When Risk Is Measured by Weighted VaR (2015) — link

0 of 89 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.