Decompositions of Multiattribute Utility Functions Based on Convex Dependence

We describe a method of assessing von Neumann-Morgenstern utility functions on a two-attribute space and its extension to n-attribute spaces. First, we introduce the concept of convex dependence between two attributes, where we consider the change of shapes of conditional utility functions. Then, we establish theorems which show how to decompose a two-attribute utility function using the concept of convex dependence. This concept covers a wide range of situations involving trade-offs. The convex decomposition includes as special cases Keeney's additive/multiplicative decompositions, Fishburn's bilateral decomposition, and Bell's decomposition under the interpolation independence. Moreover, the convex decomposition is an exact grid model which was axiomatized by Fishburn and Farquhar. Finally, we extend the convex decomposition theorem from two attributes to an arbitrary number of attributes.