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

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 comparative dispersion 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 greediness G _u of the utility function u and an index of pessimism P _f of the probability perception function f: the decision maker is monotone risk averse if and only if P_f ≥ G _u. The index of greediness (non-concavity) of u is the supremum of u′(x)/u′(y) taken over y ≤ x. The index of pessimism of f is the infimum of 1-f(v)/1-v/f(v)/v taken over 0 < v < 1. Thus, G_u ≥ 1, with G_u = 1 iff u is concave. If P_f ≥ G _u then P_f ≥ 1, i.e., f is majorized by the identity function. Since P _f = 1 for Expected Utility maximizers, P _f ≥ G_u forces u to be concave in this case; thus, the characterization of risk aversion as P_f ≥ 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 P _f = 1.