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

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## 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, where Bε

*A*, μ>_{A}). The set \( \prod\limits_i {{T_i}} \) denoted by T is the set of monotone transformations in R^{n}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 Z^{Γ}the restriction of Z to the situation Γ. Then for each aεA the function Z^{Γ}(a,.): Y_{Γ},_{a}→ R^{n}is measurable and μ_{a}induces on R^{n}a pro-$$ \mu_a^Z(B): = \frac{{{\mu_a}\left( {{Z^{\Gamma }}{{\left( {a,.} \right)}^{{ - 1}}}(B)} \right)}}{{{\mu_a}\left( {{Y_{{\Gamma, a}}}} \right)}}, $$

*B*and*B*is some σ-algebra of R^{n}completely contained in the Borel-σ-algebra.## Keywords

Utility Function Hyperbolic Model Partial Objective Exponential Weight Monotone Transformation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- 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 R
^{n}as individual utilities.Google Scholar - 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
- 1).Arrow, [1965, p. 30]Google Scholar
- 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 [1960].Google Scholar

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© Springer-Verlag Berlin · Heidelberg 1975