Formal Statement of the Problem

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


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.


Decision Maker Utility Function Preference Relation Solution Concept Great Element 
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  1. 1).
    For such models we refer to Heinen [1966J, Schmidt-Sudhoff [l967]and Bidlingmayer [1968].Google Scholar
  2. 1).
    Cf. Markowitz [1959].Google Scholar
  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).
    Dinkelbach [1971, pp. 7–9]Google Scholar
  20. 4).
    Näslund [1967]Google Scholar
  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).
    Cf. Debreu [1959]Google Scholar
  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).
    Arrow [1965, p. 30]Google Scholar
  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

Copyright information

© Springer-Verlag Berlin · Heidelberg 1975

Authors and Affiliations

  • J. Wilhelm
    • 1
  1. 1.Institut für Gesellschafts- und WirtschaftswissenschaftenUniversität BoonBonnDeutschland

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