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 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 .