# Solution Approaches to the Problem of Multi-Objective Decision Making under Uncertainty

• J. Wilhelm
Chapter
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 112)

## Abstract

We now give a short restatement of the problem. α is a class of decision situations under uncertainty and Z a complete system of n partia objectives for α. By Z we also denote a representation of the n-dimensional criterion function of the system. Z is defined as a real vector function on the sets $$\bigcup\limits_{{a\varepsilon A}} {\left\{ a \right\} \times {Y_{{\Gamma, a}}}}$$ where Γ=(A, <Ω, A, μ>A). The set $$\prod\limits_i {{T_i}}$$ denoted by T is the set of monotone transformations in Rn related to the system of objectives. Every representation of the criterion function of Z may be described by Z’=t·Z for an appropriate tεT. For Γεα we denote by ZΓ the restriction of Z to the situation Γ. Then for each aεA the function ZΓ (a,.): YΓ,a → Rn is measurable and μa induces on Rn a pro-
$$\mu_a^Z(B): = \frac{{{\mu_a}\left( {{Z^{\Gamma }}{{\left( {a,.} \right)}^{{ - 1}}}(B)} \right)}}{{{\mu_a}\left( {{Y_{{\Gamma, a}}}} \right)}},$$
where BεB and B is some σ-algebra of Rn completely contained in the Borel-σ-algebra.

## Keywords

Utility Function Hyperbolic Model Partial Objective Exponential Weight Monotone Transformation
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## References

1. 1).
For the following arguments we refer to Harsanyi [1955, p. 3121. The formulation of the theorem given here slightly differs from Harsanyi’s exposition; Harsanyi deals with the problem of a social welfare function. Accordingly, he interprets the coordinates of Rn as individual utilities.Google Scholar
2. Rothenberg, A [1961, chap. 10] argues that the axioms of a v.Neumann-Morgenstern utility function are not as plausible for a social preference system as for individual preferences.Google Scholar
3. 1).
4. 1).
For another axiomatical treatment we refer to Harsanyi [1959, pp. 325–331]. Treating Nash’s solution for the cooperative n-person game Harsanyi introduces two assumptions immediately implying (iii) in place of (iii); he is led to the special case of mutually identical exponential weights. For another use of the Cobb-Douglas utility function we refer to Miller and Starr .Google Scholar