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**Multiattribute nonlinear utility theory.**
*(English)*
Zbl 0555.90008

In order to avoid the disadvantages exhibited by the linear utility theory and concerning its applicability, Chew and MacCrimmon (1979), and Fishburn (1982), described two nonlinear utility theories, which replace the linearity (independence) axiom, in the von Neumann-Morgenstern linear theory, by other weaker axioms. In this way, the Fishburn theory uses three axioms (continuity, dominance and symmetry) from which a skew- symmetric bilinear functional on the set \(P\times P\) (P being a set of probability distributions on a set of consequences) is obtained for representing preferences. The Chew and MacCrimmon theory adds a transitivity axiom to the last theory, to obtain two linear functionals on P. The von Neumann-Morgenstern theory adds a linearity axiom to the last one, to obtain a linear functional on P.

When the multiattribute context is considered, the author examines in this paper the decompositional effects of his additive-independence axiom on the skew-symmetric bilinear functional, obtained in his nonlinear theory, and on the two linear functionals obtained in the Chew and MacCrimmon nonlinear theory, and pays fundamentally attention to the two- attribute case. In addition, he analyzes the decompositional effects of a new independence axiom, which combines the last one with the Pollak- Keeney form of utility independence (Keeney 1968, Pollak 1967).

When the multiattribute context is considered, the author examines in this paper the decompositional effects of his additive-independence axiom on the skew-symmetric bilinear functional, obtained in his nonlinear theory, and on the two linear functionals obtained in the Chew and MacCrimmon nonlinear theory, and pays fundamentally attention to the two- attribute case. In addition, he analyzes the decompositional effects of a new independence axiom, which combines the last one with the Pollak- Keeney form of utility independence (Keeney 1968, Pollak 1967).

Reviewer: M.A.Gil Alvarez

### MSC:

91B16 | Utility theory |