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 <Subscript> u </Subscript> of the utility function u and an index of pessimism P <Subscript> f </Subscript> of the probability perception function f: the decision maker is monotone risk averse if and only if <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$P_f\ge G_u$</EquationSource> </InlineEquation>. The index of greediness (non-concavity) of u is the supremum of <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$u^{\prime}(x)/u^{\prime}(y)$</EquationSource> </InlineEquation> taken over <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$y\leq x$</EquationSource> </InlineEquation>. The index of pessimism of f is the infimum of <InlineEquation ID="Equ4"> <EquationSource Format="TEX">${\frac{{1-f(v)}}{{1-v}}}/ {\frac{{f(v)}}{{v}}}$</EquationSource> </InlineEquation> taken over 0 > v > 1. Thus, <InlineEquation ID="Equ5"> <EquationSource Format="TEX">$G_{u}\geq 1$</EquationSource> </InlineEquation>, with G <Subscript> u </Subscript>=1 iff u is concave. If <InlineEquation ID="Equ6"> <EquationSource Format="TEX">$P_{f}\geq G_{u}$</EquationSource> </InlineEquation> then <InlineEquation ID="Equ7"> <EquationSource Format="TEX">$P_{f}\geq 1$</EquationSource> </InlineEquation>, i.e., f is majorized by the identity function. Since P <Subscript> f </Subscript>=1 for Expected Utility maximizers, <InlineEquation ID="Equ8"> <EquationSource Format="TEX">$P_{f}\geq G_{u}$</EquationSource> </InlineEquation> forces u to be concave in this case; thus, the characterization of risk aversion as <InlineEquation ID="Equ9"> <EquationSource Format="TEX">$P_{f}\geq G_{u}$</EquationSource> </InlineEquation> 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 <Subscript> f </Subscript>=1. Copyright Springer-Verlag Berlin/Heidelberg 2005
