Abstract
In this chapter we shall assume that П ni=1 Гi, the set of alternatives, is a Cartesian product of ‘mixture spaces’, i.e., spaces endowed with some sort of convex combination operation. Two main examples of mixture spaces are, firstly, convex subsets of Euclidean spaces, and secondly, sets of probability distributions, ‘lotteries’ over a given set of ‘certain outcomes’. Mixture spaces have been introduced in von Neumann& Morgenstern(1944), mainly as generalization of lotteries, and have almost exclusively been studied with the purpose to obtain results useful for lotteriesrf2. We shall study mixture spaces mainly as generalizations of convex subsets of Euclidean spaces, and deal with concavity and convexity of (representing) functions on mixture spaces. To the best of our knowledge concavity and/or convexity of functions on mixture spaces have not yet been studied in the literature, whereas mixture spaces do have the natural structure for the study of these notions.
The first five sections of this chapter closely follow Wakker(1986a). Section 6 closely follows Wakker,Peters&Van Riel(1985, sections 4 and 5). The last section closely follows Wakker(1989).
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References
Fishburn(1982) contains many results. See also Luce&Suppes(1965). Gudder(1977) and Gudder&Schroeck(1980) mention applicability of mixture spaces in quantum mechanics, and colour perception in psychology.
Shubik(1975) remarked that, also without risk or uncertainty, the assumption of concavity of the utility function (to be used in expected utility) is important. Without it, in a Walras allocation the risk-loving agents would ‘create markets for lotteries’. See also Debreu(1976, footnote 1), Drèze(1971), Raiffa(1968, section 4. 13 ).
Arrow(1971, Essay 3, page 96) and Machina(1982b, top of page 1069) comment on the universality of the assumption of nonincreasing risk aversion. See also section 3 in Bernoulli(1738). Risk-aversion in the presence of state-dependent utility functions is studied in Karni(1985).
e.g., see Fishburn(1970, section 8.4)
The same result is given in Gudder(1977, Theorem 4). A related result is given in Krantz(1975).
a generalization of ‘Axiom Q’ in Yaari(1978, p.109)
i.e., the Remark at section 4, and Lemma 2 of section 5 and, by that, the implication of ‘Axiom D’ by ‘Axiom Q’
By Lemma VII.3.4, statement (ii) in Theorem VII.3.5 implies convexity of the preference relation. This in turn implies quasiconcavity of the additive representation V, which exists according to section III.6. In Yaari(1977) it is demonstrated that a quasiconcave additive representing function has all but one of its terms concave. See also Debreu&Koopmans(1982, Theorem 2, and end of section 4). Hence one might conjecture that in (ii) above the revelation of nonincreasing tradeoffs might be replaced by three conditions, as follows. First one uses coordinate independence (and the hexagon condition) to guarantee the existence of additive value functions. Next one uses convexity of ≥ to guarantee quasiconcavity of the sum of the additive value functions, which by the result of Yaari implies concavity of all but one of the additive value functions. Thirdly, one weak condition for ≥ is added to guarantee concavity of the one remaining additive value function. (Figure VII.4.1 (mainly f3 there) will show that such a weak condition cannot be dispensed with.) We have not been able to find such a weak condition, hence we have taken an alternative approach, which does not use Yaari’s results.
This, under the addition of continuity conditions, gives characterizations, alternative to those in Fishburn(1965), Pollak(1967), and Keeney&Raiffa(1976, Theorem 6. 4 ).
The observation that (VII.4.3) does not imply (VII.4.2) for m ≥ 2, is closely related to the observation that quasiconcavity and additivity of a function V do not imply (VII.4.1), i.e., concavity of V. This latter observation has been made several times in the literature. The earliest reference, given in Debreu&Koopmans(1982), is Slutsky(1915).
A derivation of the SEU model with concave utility, using differentiability conditions, is given in Stigum(1972).
Kihlstrom&Mirman(1974) adapted the Pratt-Arrow results to multidimensional quantitative consequences, still using differentiability assumptions. Wakker,Peters&Van Riel(1985, section 3) obtained, for decision making under risk, results for completely general consequence spaces.
Yaari uses a different, but trivially equivalent, formulation by means of ‘acceptance sets’.
Yaari(1969, Remark 1) obtains a related result, using differentiability tools, for the case n=2.
Wakker,Peters&Van Riel(1985, section 3) extended the results, for decision making under risk, to completely general consequence spaces, and completely general utility functions.
or Eichhorn(1978, Theorem 2.5.2)
For the possibility to extend the above results, for the case of constant risk aversion, to multidimensional consequences, Rothblum(1975) may be useful.
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© 1989 Springer Science+Business Media Dordrecht
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Wakker, P.P. (1989). Concavity on Mixture Spaces and Risk Aversion. In: Additive Representations of Preferences. Theory and Decision Library C, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7815-8_8
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DOI: https://doi.org/10.1007/978-94-015-7815-8_8
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