# Formal Statement of the Problem

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## Abstract

We start with a class α of decision situations under uncertainty. Let (H,Q,g) be the representation of a criterion for *α*. We consider a decision situation \( \Gamma=\left({A, < \Omega, A,\mu {>_A}} \right)\varepsilon \alpha . \) For every aεA the set \( {X_{{\Gamma, a}}}=\left\{{\omega \varepsilon {\Omega_a}\left| {\left({\Gamma, a,\omega} \right)\varepsilon \mathop{\cap}\limits_{{(T,\gamma)\varepsilon H}} \,{E_{{T(\Gamma)}}}} \right.} \right\} \) is defined by (v) of def. 1.6.; this set is an element of A_{a}, it describes the set of environmental responses to a, whose associated final states can be evaluated by the mentioned criterion.

## Keywords

Decision Maker Utility Function Preference Relation Solution Concept Great Element
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## References

- 1).For such models we refer to Heinen [1966J, Schmidt-Sudhoff [l967]and Bidlingmayer [1968].Google Scholar
- 1).Cf. Markowitz [1959].Google Scholar
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- 2).In the way indicated in footnote of p. 8.Google Scholar
- 1).Cf. definition 1.7. of these notes.Google Scholar
- 2).Cf. p. 50Google Scholar
- 3).The equivalence of the assumptions of the Bernoulli-principle with our objective-concept has been shown in an excursus.Google Scholar
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- 5).See, e.g. the application in the form of the (μ, σ)-principle within the portfolio selection theory, Markowitz [1957].Google Scholar
- 6).\Mosteller and Nogee [1951] have done considerable work on measuring these functions.Google Scholar
- 1).The vector maximum problem was first mentioned by Kuhn and Tucker [1951].Google Scholar
- 2).Cf. definition 1.4. of these notes.Google Scholar
- 3).This condition is referred! to as “functional efficiency” by Charnes and Cooper [1967, Vol. I, p. 321].Google Scholar
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- 1).This condition is equivalent to postulate E of Fleming [1952].Google Scholar
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- 1).eff(Y) is always differentiable up to a set of Lebesgue measure zero: Karlin [1962, Vol. I, p. 405].Google Scholar
- 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 - 1).Arrow [1965, p. 30]Google Scholar
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