Independence in Utility Theory with Whole Product Sets

Peter C. Fishburn
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Peter C. Fishburn: Research Analysis Corporation, McLean, Virginia

Operations Research, 1965, vol. 13, issue 1, 28-45

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 X 1 × X 2 × ⋯ × X n , each element in X being an ordered n -tuple ( x 1 , x 2 , …, x n ). 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 ( x 1 , x 2 , …, x n ) in X can be written in the additive form φ( x 1 , x 2 , …, x n ) = φ 1 ( x 1 ) + φ 2 ( x 2 ) + ⋯ + φ n ( x n ), where φ i is a real-valued function defined on the set X i , 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 X i .

Date: 1965
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