Finite-Degree Utility Independence

When u is a von Neumann-Morgenstern utility function on X × Y, Y is “utility independent” of X if u can be written as u(x, y) = f(x)g(y) + a(x) with f positive. This paper introduces a fundamental extension of utility independence that is based on induced indifference relations over gambles on one factor when the level of the other factor is fixed. It is proved that Y is “degree-n utility independent” of X if and only if u can be written as u(x, y) = f_t(x)g_t(y) + ⋯ + f_n(x)g_n(y) + a(x) and cannot be written in a similar way with fewer than n products of single-factor functions. A similar theorem holds when the roles of Y and X are interchanged, it follows that if Y is degree-n utility independent of X, then X is degree-m utility independent of Y for some m ∈ (n − 1, n, n + 1); it is then shown that u can be represented in terms of n conditional utility functions on Y, m conditional utility functions on X, and at most (n + 1)(m + 1) scaling constants.