# Formal Statement of the Problem

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

## 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 Aa, 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. 1).
For such models we refer to Heinen [1966J, Schmidt-Sudhoff [l967]and Bidlingmayer [1968].Google Scholar
2. 1).
3. 2).
Discussing a quadratic model will make clear the affinity of both methods (section 3.2.).Google Scholar
4. 2).
In the way indicated in footnote of p. 8.Google Scholar
5. 1).
Cf. definition 1.7. of these notes.Google Scholar
6. 2).
7. 3).
The equivalence of the assumptions of the Bernoulli-principle with our objective-concept has been shown in an excursus.Google Scholar
8. 4).
Krümmel [1969, p. 73, footnote 9].Google Scholar
9. 5).
See, e.g. the application in the form of the (μ, σ)-principle within the portfolio selection theory, Markowitz [1957].Google Scholar
10. 6).
\Mosteller and Nogee [1951] have done considerable work on measuring these functions.Google Scholar
11. 1).
The vector maximum problem was first mentioned by Kuhn and Tucker [1951].Google Scholar
12. 2).
Cf. definition 1.4. of these notes.Google Scholar
13. 3).
This condition is referred! to as “functional efficiency” by Charnes and Cooper [1967, Vol. I, p. 321].Google Scholar
14. 1).
The concept of the complete solution is due to Dinkelbach T1971, p. 2].Google Scholar
15. 2).
See the related “condition 3” for a social welfare function given by Arrow [1951, p. 27] or the “axiom 4” given by Harsanyi [1959, p. 330] in order to treat axiomatically Nash’s solution for a cooperative n-person game.Google Scholar
16. 1).
This operation is used in game theory constructing characteristic functions; Aumann [1967, p. 6].Google Scholar
17. 1).
For an extensive view of the literature on multi-objective decision making we may refer to Johnson [1968]. Some recent approaches are to be found in Fandel [1972, pp. 17–50]; a detailed discussion of the lately published concepts by, using the instruments developed in 2.4.1 is presented by Fandel and Wilhelm [1974].Google Scholar
18. 2).
This notion is due to Charnes and Cooper [1967, Vol. I p. 215]. Further approaches to the goal-programming have been suggested by Balderstone [I960] and Ijiri [1965, pp. 34–36 and 43–45]. Similar but a little different is the approach of Sauermann and Selten [1962]; they do not minimize a geometrical distance function, but their solution especially violates (PIIA); the solution technique is somewhat related to those used to solve problems of type (2b): only local information at single points Is required — naturally, not information provided by a preference relation.Google Scholar
19. 3).
20. 4).
21. 5).
Cf. Fandel and Wilhelm [1974]Google Scholar
22. 1).
Under the assumption of linear utility functions and linear restrictions the problem has recently been treated by Zionts and Wallenius [1974].Google Scholar
23. 2).
Cf. Fandel and Wilhelm [1974].Google Scholar
24. 3).
A survey is given by Johnson [1968, pp. 421–434].Google Scholar
25. 1).
26. 2).
For a similar algorithm concerning the chance-constrained programming problem we refer to Naslund [1967, pp. 34–37]. In order to provide the marginal rates of substitution for the algorithm of Geof-frion (p. 64) Dyer [1972, p. 206] and [1973] has suggested a similar approach.Google Scholar
27. 1).
This condition is equivalent to postulate E of Fleming [1952].Google Scholar
28. 1).
Another approach is to be found in Eckenrode [1965, p.l80L From the view of social preference functions Theil [1963] has suggested another method. Lately published approaches are due to Nievergelt [1971] and Zionts and Wallenius [1974].Google Scholar
29. 1).
eff(Y) is always differentiable up to a set of Lebesgue measure zero: Karlin [1962, Vol. I, p. 405].Google Scholar
30. 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
31. 1).
32. 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
33. 1).
Cf. Gutenberg DL968, Vol. I, p. 155]Google Scholar
34. 1).
Bourbaki C1965, chap. 1, §, 9, No 1, Def. 1]Google Scholar
35. 1).
Bourbaki [1965, chap. 1, §, 6, No 1, Def. 1]Google Scholar
36. 1).
Bourbaki [1965, chap. 1, §, 9, No 1, Def. 1]Google Scholar
37. 2).
Bourbaki [1965, chap. 1, § 8, No 1, Prop. 1, Cond. (H)]Google Scholar