What if utility functions do not exist?

Abstract

Often the preferences of decision-makers are sufficiently inconsistent so as to preclude the existence of a utility function in the classical sense. Several alternatives for dealing with this situation are discussed. One alternative, that of modifying classical demands on utility functions, is emphasized and described in the context of the theory of measurement developed in recent years by behavioral scientists. The measurement theory approach is illustrated by discussing the concept of the dimension of a partial order. Even if we cannot assign numerical utility or worth values which reflect preferences in the classical utility function sense, from the measurement theory point of view we can still learn a lot about the preferences by finding several measures of worth so that a given alternative x is preferred to an alternative y if and only if x is ranked higher than y on each of the worth scales. If such measures can be found, it follows that the preferences define a partial order, and the smallest number of such scales needed is called the dimension of the partial order. If one-dimensional preferences (those amenable to classical utility assignments) cannot be found, then the next best thing is to search for partially ordered preferences with as small a dimension as possible. Several conditions under which a partial order is two-dimensional are described.