Abstract
In this chapter we assume that all factor sets Γi in the Cartesian product ∏i∈IΓiare equal; i.e., Γ1 = ... = Γn = Γ for some connected topological space Γ. We study representations of the form
.
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Compare Benson(1987). A discussion of attempts to encompass both freedom and determinism is given in Woolfolk&Sass(1988).
See for instance Winterfeldt&Edwards(1986), first paragraph of section 2.1.
This use of `imaginary choices’ is one of the central topics in Shafer(1986, e.g. p. 500, fourth paragraph). Ramsey(1931, top of p. 170, and p. 172) is very illuminating on this point.
See Sneed(1971, Chapter VI) for a further discussion of classical particle mechanics, many parts of which are relevant to the topics of this monograph. The question as to how far representing notions such as utility may be shown to refer to an a priori specified content and in this sense may be said to be really `existing’, is considered in Bezembinder(1987, section 6).
nonphysical in the terminology of Sneed(1971, p.22). For the context of decision making under uncertainty, comments on this are given at several parts in Drèze(1987, for instance at p.7 lines 9–11, or p.91).
Krantz,Luce,Suppes&Tversky(1971), hereafter abbreviated KLST, state on page 200/201 that it is not evident why the measurement of probability should have been the focus of more philosophic controversy than that of any other scientifically significant attribute. A same opinion can be found in Freudenthal(1965). Further work on the implications and relations between choices, preferences, and representations of these, is provided in Bezembinder(1987). Samuels(1988) ascribes a special normative status to economics, economics being involved in `the complex process through which the economy is made’.
Cf is called a selection function in Basu(1980, p.50). See also Richter(1971, page 31, third paragraph).
Work like Cooke&Draaisma(1984), comparing numbers of arbitrary preference relations to numbers of preference relations with `nice’ properties, can be of use for this. Also Bezembinder(1981), measuring `circularity’ in preferences, can be applied to this problem. A `cycle’ like x r y > z - x then is to be reinterpreted as Cf(x,y) = x, Cf(y,z) = y, Cf(z,x) = z, revealing that C(D) = D for D = (x,y), (y,z), and (z,x) Finally, for predictive applications the preliminary choice problem will be a problem, to yield only the prediction that the decision maker will choose an element from C(D), and not the prediction which element that will be.
Drèze(1987, Chapter 2, end of section 9.1) gives an appreciation of this.
Von Wright(1963) places preference relations between the `anthropological’ (acting) level and the `axiological’ (assessing) level. For more comments and references, see Foxall(1986).
There are many other ways to derive binary relations from choice functions, see Sen(1971). Often first a relation analogous to R above is defined, and then P and E are defined as the asymmetric, respectively symmetric, part of R, see for instance Weddepohl(1970). We have chosen the above definitions to achieve maximal operationality. As soon as we observe x E C(D) and y E D\C(D) for some D E A, we can now conclude xPy. If we had defined xPy by [xRy and not yRx], then for verification of [not yRx] we would have had to find out the choices from all D E A, containing both x and y. This may be an impossible task if most of the involved choice situations are hypothetical (see subsection I.1.3). In the sequel we adapt the results of literature to our deviating definitions. The equivalence (i) <=> (vi) in Theorem I.2.5 will show that one way to characterize the desired representation in (i) there is, with P and R as in Definition 1.2.3, to require that P as derived from our deviating definition of P leads to the same binary relation as the usual definition of P in the literature: the asymmetric part of R.
Richter(1971, Theorems 5 and 8) derived the equivalence of (i), (ii), (iii) and (y).
A graph-theoretic approach to these matters is presented in Wakker(1988c).
The WARP-condition has been introduced in Samuelson(1938a; the term was introduced in Samuelson,1950; see also Samuelson,1938b and 1947, pp. 107–117), and SARP in Houthakker(1950) and Ville(1946). These authors studied the special context of consumer demand theory, the origin of revealed preference theory. There the assumption was often made that C(D) contains exactly one element, for every D E A. Then indeed SARP implies WARP. The extension of these notions to choice functions C with not always I C(D) = 1 is not unique, and has been done in several ways in the literature. In the above way SARP does not imply WARP anymore. A characterization of WARP has been provided in Clark(1988, Corollaries I and II). An early reference to IIA is Nash (1950a, Axiom 3); there it is propagated that utility be considered as a `choice operator, indicating the reaction of an organism to a given set of alternatives’, rather than as an `ordering system’. Nash(1950b), for single-valued choice functions in the context of bargaining game theory, required IIA for every fixed `disagreement point’. Chernoff (1954) uses in fact IIA to criticize the minimax regret criterion. Luce&Raiffa(1957, section 13.3 and other sections) give a discussion of several versions of the condition. Luce(1959, section I.C.1.c) shows that his `choice axiom’, defined for probabilistic choice making, reduces to IIA for single-valued choice functions when considered for the special case of deterministic choice making. Arrow (1959) refers to condition C4 in the unpublished Arrow(1948) as an occurrence of IIA. Arrow himself uses the term IIA for a different condition, in his impossibility theorem for social choice in Arrow(1951a). Ray(1973) and Karni&Schmeidler(1976) comment on the difference between the condition IIA above, and the one from social choice theory. 11.16 For the case where A contains all two-and three-point subsets of X, the equivalence of (i) and (iv) above can also be obtained from the proof in Arrow(1959), which was meant only for the case where A contains all finite subsets of X. Sen(1971, bottom of page 312), noted that Arrow’s proof remains valid in our case. For the case where A is union-closed, the equivalence of (i) and (iv) above is given in Fishburn(1973, Theorem 15.4), or Hansson(1968), or Weddepohl(1970, Theorem 3.9.6, without K5 and K7). For the case where A contains all finite subsets of X, Bandyopadhyay(1988) shows that the statements of Corollary I.2.12 are equivalent to a `sequential path independence’ condition.
See page 48 of Richter(1971).
See Debreu(1954, Lemma II) or KLST,Theorem 2.2, or Jaffray(1975). Further Debreu(1954, 1964) gave simpler sufficient topological conditions for to guarantee the existence of (continuous) representations (see Theorem 1I1.3.6).
This approach is common in consumer demand theory, see for instance Chipman,Hurwicz,Richter,&Sonnenschein(1971). See also Lensberg(1987) and Peters& Wakker(1987).
Luce&Suppes do allow 1/2–1/2 lotteries in algebraic approaches; these may reflect indifference. In our approach neither such lotteries will occur.
Savage’s `small worlds’, intended to avoid the necessity of total formalization of all uncertainty, have often been criticized. A clear account is given in Shafer(1986, section 5).
For a supplement to the latter, see Wakker(1986b).
Bounded rationality is central in Simon(1982).
Fishburn(1982) gives many results for the lottery approach.
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Wakker, P.P. (1989). Continuous Subjective Expected Utility. In: Additive Representations of Preferences. Theory and Decision Library C, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7815-8_5
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DOI: https://doi.org/10.1007/978-94-015-7815-8_5
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