Abstract
The following simple example of a choice problem will illustrate several questions to be addressed in the sequel.
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Compatibility’ in section 8.2.6 in KLST, formulated in a complicated context, is the condition in the literature most similar to cardinal coordinate independence. Also the ‘invariance-of-standard-sequences property’ in section 6.11.2 in KLST, and `invariance’ in Tversky(1977), are related conditions in making comparisons between `intervals’ on different coordinates.
or Theorem 1 of section 2.1.4 of Aczél(1966), or by (88) of section 3.7 of Hardy, Littlewood,&R1ya(1934)
Grodal(1978, Theorem 4) has given a result for continuous time. KLST sketch a result for finitely many points of time by requiring (for the case of three or more essential points of time) CI and a stationarity condition (in Definition 6.15 there) which is not easily compared with our approach. Also Samuelson(1958) considers the case of finitely many points of time. Zilcha(1988) is a recent survey describing approaches for which finiteness of the set of points of time, faced by a consumer, is essential. 64 See Frisch(1926), Lange(1934), Alt(1936), Scott&Suppes(1958), Debreu(1958), Suppes&Zinnes(1963), Fishburn(1970, Chapter 6), KLST(Chapter 4), Shapley(1975), Basu(1982).
See also Richter(1975), Richter&Shapiro(1978), and Tversky(1967a). 66 Wakker(1981) pointed out some misunderstandings in the literature about this part of Savage’s work.
This was discovered after publication of Savage(1954). See Fishburn(1970, section 14.1), where also a well-organized proof of Savage’s result is given. An original proof of Savage’s theorem, bringing in techniques of additive representations as in Chapter III, is given in Arrow(1971, Essay 2; also published as Chapter 2 in McGuire& Radner,1972). Marschak&Radner(1972) give an appealing presentation of Savage’s theorem, interpreting Savage’s postulates P3 (entailing that the ordering of consequences is the same for all states of nature) and P4 as ‘independence of tastes and beliefs’. Shafer(1986) is a recent discussion of Savage’s work. Cooper(1987) uses evolutionary considerations to argue for the axioms of Savage.
Pratt,Raiffa,&Schlaifer(1964), while claiming not to be innovative, give a very appealing approach, in which consequences in a sense are probabilities.
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© 1989 Springer Science+Business Media Dordrecht
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Wakker, P.P. (1989). From Choice Functions to Binary Relations. In: Additive Representations of Preferences. Theory and Decision Library C, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7815-8_2
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DOI: https://doi.org/10.1007/978-94-015-7815-8_2
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