Probabilistic risk aversion for generalized rank-dependent functions

TL;DR

This paper characterizes probabilistic risk aversion in generalized rank-dependent functions, revealing it depends on the convexity of the distortion function and connecting it to dual utilities and risk management models.

Contribution

It provides a complete characterization of probabilistic risk aversion for a broad class of functions, including new equivalences and conditions for quasi-convexity.

Findings

Probabilistic risk aversion depends on the convexity of the distortion function.

Seven equivalent conditions for quasi-convexity in probabilistic mixtures are established.

Risk aversion notions coincide for dual utilities under continuity.

Abstract

Probabilistic risk aversion, defined through quasi-convexity in probabilistic mixtures, is a common useful property in decision analysis. We study a general class of non-monotone mappings, called the generalized rank-dependent functions, which includes the preference models of expected utilities, dual utilities, and rank-dependent utilities as special cases, as well as signed Choquet functions used in risk management. Our results fully characterize probabilistic risk aversion for generalized rank-dependent functions: This property is determined by the distortion function, which is precisely one of the two cases: those that are convex and those that correspond to scaled quantile-spread mixtures. Our result also leads to seven equivalent conditions for quasi-convexity in probabilistic mixtures of dual utilities and signed Choquet functions. As a consequence, although probabilistic risk aversion is quite different from the classic notion of strong risk aversion for generalized rank-dependent functions, these two notions coincide for dual utilities under an additional continuity assumption.