Double-Exponential Utility Functions

Ordinal additivity between two attributes X and Y is a property that permits the utility function u(x, y) to be represented as ϕ(v(x) + w(y)) for some functions ϕ, v and w where ϕ may be thought of as a single attribute utility function over v + w. In applications ϕ is usually taken to be either linear (the additive decomposition) or exponential (the multiplicative decomposition). The utility function can be shown to be exactly one of these forms whenever the two attributes are mutually utility independent. In this paper we introduce a relaxation of utility independence and show that it too implies ordinal additivity. We identify the class of utility functions ϕ that are consistent with the relaxed assumptions.

One member of this class, ϕ(z) = −exp(b exp(−cz)), which we call double-exponential, seems particularly appealing, for when b and c are positive it is increasing, risk averse and decreasingly risk averse for all z. The multiattribute decomposition corresponding to it −exp(b exp(−c(v + w))) may be represented as u(x, y) = c exp(cv(x)w(y)) which we call the exponential product.