General decompositions of multiattribute utility functions with partial utility independence

Abstract

AbstractThe construction of a multiattribute utility function is significantly simplified if it is possible to decompose the function into lower‐order utility assessments. When every attribute is utility independent of its complement, we have a powerful property that reduces the functional form into a multilinear combination of single‐attribute functions. We propose a natural extension to the multilinear form requiring the milder set of independence conditions: every attribute is utility independent of a subset of the attributes in its complement. We introduce a graph, which we call the bidirectional utility diagram, to facilitate the elicitation of these utility independence conditions. We also define a class of bidirectional diagrams, which we refer to as the canonical form. We show how this canonical representation leads to tractable functional forms of utility functions that may appeal to the decision analyst and can be used in practice. We also discuss situations where a canonical form does not exist. We then present an iterative approach to determine the lower‐order utility assessments that are required for a given set of utility independence assertions. We conclude with some extensions of this graph‐based approach to independence relations of the form: a subset of the attributes is utility independent of another subset. Copyright © 2010 John Wiley & Sons, Ltd.