Abstract
In this chapter we introduce topological structure, and use this to obtain additive representations under continuity assumptions. As before, the set of alternatives is a Cartesian product ПiєIΓi. As always (with Chapter V excepted) I is the finite set {1 ,...,n}. Furthermore, we shall from now on assume that every Γi is a connected and separable topological space. A reader not interested in general topology may simply assume that Γi is a convex subset of a Euclidean space, e.g. Γi is \(\mathbb{R}_ + ^{{m_i}} \) , or \(\mathbb{R}\). Then all topological assumptions in the sequel are satisfied, and can be ignored. ПiєIΓi will always be endowed with the product topology, hence is connected (see Kelley, 1955, Chapter 3, problem 0) and separable too. The condition of topological separability has been added only for simplicity of presentation. It can nearly always be omitted (see Remarks A3.1 and III.7.1). The crucial assumption is connectedness.
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Such a view has been expressed in Aumann(1962, p.446). Some further discussions of the ‘rationality’ of completeness (and other elementary conditions for preferences) can be found in Shafer(1986, p.469), and throughout Hogarth Reder(1987).
Wakker(1988d) gives arguments in favour of the algebraic approach. Further comments will be given here in section III.8. More elaborated topological results are provided in Vind(1986a, Chapter III).
There have been many vivid discussions about ordinal versus cardinal utility. Recent references are Cooter Rappoport(1984, 1985), and Basu(1982). Hence, to avoid unintended connotations, in the literature often the expression ‘unique up to a linear/positive affine transformation’ is used instead of cardinal, and ‘unique up to a strictly increasing transformation’ instead of ordinal. Sometimes ordinal is used instead of continuously ordinal. In psychological literature the terms ordinal scale and interval scale (the latter for cardinal representation) are customary. The reason for our choice of terminology is linguistic. The adjective cardinal is more convenient than the noun (interval) scale, and ‘unique up to a positive affine transformation’ is inconvenient because of its length.
Other usual scale types are ‘nominal scales’, for which only the = and # relations are relevant, and ’ratio-scales’, where in comparison with cardinal scales also a natural ’zero-point’ is given. Also sometimes ’consists of’ in section III.2 has been changed into ’is a subset of the set of’, or ’is a superset of the set or. Although in principle many more kinds of scale types could be thought of, in practice mainly the above-mentioned ones occur. Luce Narens(1987) use ’homogeneity arguments’ to explain this.
Lemma I.3.2 in Vind(1986a) may serve as an illustration of the meaning of topological connectedness.
Recent proofs are provided in Jaffray(1975) and Richter(1980).
Fishburn(1970) gives as well the topological approach, as the algebraic approach, as a `mixture-space-approach’.
There is mainly one exception concerning generality, and that is the result of Gorman(1968). His generalization falls somewhat outside the scope of this monograph; its derivation will not be given. An efficient derivation of it is provided in Theorem V.2.4 in Vind(1986a). Example III.6.8(b) will show two aspects in which our approach is less general than that of KLST.
rf8 For this mainly Blaschke(1928) and Keeney Raiffa(1976)) were inspiring to us. Also Fleming(1952) uses this way of constructing a grid; he uses an infinitesimally small mesh.
and without Keeney Raiffa(1976, Chapter 3) available
In Keeney Raiffa(1976) it is called the corresponding trade-offs condition.
This result was pointed out in Debreu(1960, introduction).
e.g. in Wakker(1988a)
Probably Lemma III.1.5 in Vind(1986a) is the most advanced result to show that topological separability need not be assumed unless exactly one coordinate is essential.
In this we follow Pfanzagl(1968, Definitions 8.5.12 and 8.6.3), Vind(1986a, condition c3), and Fuhrken Richter(1988).
That independence is required only for equal subalternatives of length n - 2, has also been established by Gorman(1968, section 4.4) for the case where his stronger restrictive assumptions are satisfied. The problems of CI with equivalences instead of preferences, and without weak separability (or monotonicity) are addressed in Mak(1988).
Debreu(1960, p.17, lines 2–4) suggests that this will suffice, and uses it in his proof. Also Corollary IV.2.6 in Vind(1986a) shows this, for the Thomsen condition.
A recent reference is Be11(1987).
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© 1989 Springer Science+Business Media Dordrecht
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Wakker, P.P. (1989). Additive Representations. In: Additive Representations of Preferences. Theory and Decision Library C, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7815-8_4
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DOI: https://doi.org/10.1007/978-94-015-7815-8_4
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