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Independence in Utility Theory with Whole Product Sets

Independence in Utility Theory with Whole Product Sets One of the most important concepts in value theory or utility theory is the notion of independence among variables or additivity of values. Its importance stems from numerous multiple-criteria procedures used for rating people, products, and other things. Most of these rating procedures rely on the notion of independence (often implicitly) for their validity. However, a satisfactory definition of independence (additivity), based on multi-dimensional consequences and hypothetical gambles composed of such consequences, has not appeared. This paper therefore presents a definition of independence for cases where the set of consequences X is a product set X1 × X2 × ⋯ × Xn, each element in X being an ordered n-tuple (x1, x2, …, xn). The definition is stated in terms of indifference between special pairs of gambles formed from X. It is then shown that if the condition of the definition holds, the utility of each (x1, x2, …, xn) in X can be written in the additive form φ(x1, x2, …, xn) = φ1(x1) + φ2(x2) + ⋯ + φn(xn), where φi is a real-valued function defined on the set Xi, i = 1, 2, …, n. The development is free of any specific assumptions about φ (e.g., continuity, differentiability) except that it be a von Neumann-Morgenstern utility function, and places no restrictions on the natures of the Xi. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Operations Research INFORMS

Independence in Utility Theory with Whole Product Sets

Fishburn, Peter C.
Operations Research , Volume 13 (1): 18 – Feb 1, 1965
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References (9)

Publisher
INFORMS
Copyright
Copyright © INFORMS
Subject
Research Article
ISSN
0030-364X
eISSN
1526-5463
DOI
10.1287/opre.13.1.28
Publisher site
See Article on Publisher Site

Abstract

One of the most important concepts in value theory or utility theory is the notion of independence among variables or additivity of values. Its importance stems from numerous multiple-criteria procedures used for rating people, products, and other things. Most of these rating procedures rely on the notion of independence (often implicitly) for their validity. However, a satisfactory definition of independence (additivity), based on multi-dimensional consequences and hypothetical gambles composed of such consequences, has not appeared. This paper therefore presents a definition of independence for cases where the set of consequences X is a product set X1 × X2 × ⋯ × Xn, each element in X being an ordered n-tuple (x1, x2, …, xn). The definition is stated in terms of indifference between special pairs of gambles formed from X. It is then shown that if the condition of the definition holds, the utility of each (x1, x2, …, xn) in X can be written in the additive form φ(x1, x2, …, xn) = φ1(x1) + φ2(x2) + ⋯ + φn(xn), where φi is a real-valued function defined on the set Xi, i = 1, 2, …, n. The development is free of any specific assumptions about φ (e.g., continuity, differentiability) except that it be a von Neumann-Morgenstern utility function, and places no restrictions on the natures of the Xi.

Journal

Operations Research – INFORMS

Published: Feb 1, 1965

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