More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model

Abstract

This paper studies monotone risk aversion, the aversion to monotone, mean-preserving increase in risk (Quiggin [21]), in the Rank Dependent Expected Utility (RDEU) model. This model replaces expected utility by another functional, characterized by two functions, a utility function u in conjunction with a probability-perception function f. Monotone mean-preserving increases in risk are closely related to the notion of comparativedispersion introduced by Bickel and Lehmann [3,4] in Non-parametric Statistics. We present a characterization of the pairs (u,f) of monotone risk averse decision makers, based on an index of greedinessGu of the utility function u and an index of pessimismPf of the probability perception function f: the decision maker is monotone risk averse if and only if \(P_f\ge G_u\). The index of greediness (non-concavity) of u is the supremum of \(u^{\prime}(x)/u^{\prime}(y)\) taken over \(y\leq x\). The index of pessimism of f is the infimum of \({\frac{{1-f(v)}}{{1-v}}}/ {\frac{{f(v)}}{{v}}}\) taken over 0 < v < 1. Thus, \(G_{u}\geq 1\), with Gu = 1 iff u is concave. If \(P_{f}\geq G_{u}\) then \(P_{f}\geq 1\), i.e., f is majorized by the identity function. Since Pf = 1 for Expected Utility maximizers, \(P_{f}\geq G_{u}\) forces u to be concave in this case; thus, the characterization of risk aversion as \(P_{f}\geq G_{u}\) is a direct generalization from EU to RDEU. A novel element is that concavity of u is not necessary. In fact, u must be concave only if Pf = 1.