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The Foundations of Utility

John S. Chipman

Abstract

The mathematical foundations of rational behavior, in the sense of a transitive ordering of alternatives, are developed without making any assumptions about the special character of the set of alternatives from which choices are made. It is shown that the ordering of alternatives may be characterized by a utility function, where utility is represented by a vector with real-valued components, such vectors being ordered lexicographically (like the words in a dictionary). If an axiom permitting comparison of intensities of preference is admitted, such a utility index must be unique up to transformations (such as proportionality transformations) preserving group operations. A purely topological axiom, called the Axiom of Substitution, is shown to imply that utility is real-valued (i.e., that the above vector has only one component).