A Class of Multiattribute Utility Functions

A function u(z) is a utility function if u′(z) ≥ 0. It is called risk averse if we also have u′′(z) ≤ 0. Some authors, however, require that u(i)(z) ≥ 0 if i is odd and u(i)(z) ≤ 0 if i is even. The notion of a multiattribute utility function can be defined by requiring that it is increasing in each variable and concave as an s-variate function. A stronger condition, similar to the one in case of a univariate utility function, requires that, in addition, all partial derivatives of total order m should be nonnegative if m is odd and nonpositive if m is even. In this paper we present a class of functions in analytic form such that each of them satisfies this stronger condition. We also give sharp lower and upper bounds for E[u(X1, . . . , Xs)] under moment information with respect to the joint probability distribution of the random variables X1, . . . , Xs assumed to be discrete and representing wealths.