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- Title
FINITE-DEGREE UTILITY INDEPENDENCE.
- Authors
Fishburn, Peter C.; Farquhar, Peter H.
- Abstract
When u is a von Neumann-Morgenstern utility function on X ⊗ Y. Y is ‘utility independent’ of X if u can be written as u(x, y)=f(x)g(y) + a(x) with f positive. This paper introduces a fundamental extension of utility independence that is base on induced indifference relations over gambles on one factor when the level of the other factor is fixed. It is proved that Y is ‘degree-n utility independent’ of X if and only if u can be written as u(x, y)= f1(x)g1(y)+ … + fn(x)gn(y)+ a(x) and cannot be written in a similar way with fewer than n products of single-factor functions. A similar theorem holds when the roles of Y and X are interchanged. It follows that if Y is degree-n utility independent of X, then X is degree-m utility independent of Y for some m ∈ ¦ n - l . n . n + l ¦; it is then shown that u can be represented in terms of n conditional utility functions on Y . m conditional utility functions on X, and at most (n + l)(m + 1) scaling constants.
- Subjects
UTILITY functions; MATHEMATICAL functions; DIFFERENTIAL equations; MATHEMATICAL analysis; ALGEBRA; MATHEMATICS
- Publication
Mathematics of Operations Research, 1982, Vol 7, Issue 3, p348
- ISSN
0364-765X
- Publication type
Article
- DOI
10.1287/moor.7.3.348