Quasi-convexity in mixtures for generalized rank-dependent functions

Quasi-convexity in probabilistic mixtures is a common and useful property in decision analysis. We study a general class of non-monotone mappings, called the generalized rank-dependent functions, which include the preference models of expected utilities, dual utilities, and rank-dependent utilities as special cases, as well as signed Choquet integrals used in risk management. As one of our main results, quasi-convex (in mixtures) signed Choquet integrals precisely include two parts: those that are convex (in mixtures) and the class of scaled quantile-spread mixtures, and this result leads to a full characterization of quasi-convexity for generalized rank-dependent functions. Seven equivalent conditions for quasi-convexity in mixtures are obtained for dual utilities and signed Choquet integrals. We also illustrate a conflict between convexity in mixtures and convexity in risk pooling among constant-additive mappings.