Axiomatic Utility Theory under Risk pp 1-67 | Cite as

# A Survey

Chapter

## Abstract

This part reviews the developments of axiomatic utility theory under risk beginning with von Neumann and Morgenstern’s (1947) first axiomatic formulation of expected utility. We proceed as follows: First, the general framework and some basic definitions are introduced and then the axioms and the functional representation of expected utility are presented. We also sketch out the empirical evidence concerning the independence axiom of expected utility in order to explain the motivation for further developments.

## Keywords

Risk Aversion Utility Theory Stochastic Dominance Indifference Curve Expect Utility Theory
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## Notes

- 1.Cf. Sugden (1986), (1997), Weber and Camerer (1987), Fishburn (1988a,b,c), (1989), Machina (1983a), (1987b), Kischka and Puppe (1990), Kami and Schmeidler (1991a), and Epstein (1992).Google Scholar
- 2.We define a model as hybrid if it contains rank-dependent utility representations as well as betweenness satisfying utility representations.Google Scholar
- 3.Some evidence contradicting the empirical validity of the reduction of compound lotteries axiom is reported in Carlin (1992), Bernasconi (1992), (1994), Bernasconi and Loomes (1992), and Camerer and Ho (1994). A model of expected utility without this axiom is considered in Segal (1990).Google Scholar
- 4.Cf. Herstein and Milnor (1953, p. 292).Google Scholar
- 5.Cf. Bauer (1968, p. 123).Google Scholar
- 6.A probability measure
*p*has finite support if there exists a finite set*W*⊂*X*with*p*(*W*) = 1. Cf. Fishburn (1970, p. 105).Google Scholar - 7.Cf. Jensen (1967, p. 171).Google Scholar
- 8.Cf. Sen (1970, pp. 8-9).Google Scholar
- 9.Note that some authors label a complete and transitive relation as “weak ordering” or “complete preordering”.Google Scholar
- 10.A strict partial ordering is transitive and asymmetric (
*p*≻*q*⇒ ¬(*q*≻*P*) ∀*p*,*q*∈*P*). Cf. Sen (1970, p. 9).Google Scholar - 11.Cf. Karni and Schmeidler (1991a, p. 1766).Google Scholar
- 12.The irrationality of intransitive preferences, for instance, can be established by money-pump arguments. For a critical survey of these arguments, cf. Machina (1989a, p. 1634).Google Scholar
- 13.The theoretical criticisms are presented in Anand (1987, pp. 190-208), (1993, pp. 55-71 and 87-96). Empirical failures of transitivity are reported in, e.g. May (1954) and Tversky (1969). In addition, the preference reversal phenomenon can be interpreted as a violation of transitivity. A comprehensive analysis of this phenomenon is presented in Seidl (1997).Google Scholar
- 14.Cf. Fishburn (1971b), Aumann (1962), and Kannai (1963) for weakenings of O in the expected utility framework. Non-transitive generalizations of expected utility are mentioned in section 1.4.1.2.Google Scholar
- 15.Cf. Jensen (1967, p. 173).Google Scholar
- 16.Cf. Herstein and Milnor (1953, p. 293).Google Scholar
- 17.Cf. Chew et al. (1991, p. 141).Google Scholar
- 18.Cf. Parthasarathy (1967, p. 40).Google Scholar
- 19.Cf. Chew (1985a, p. 3).Google Scholar
- 20.Cf. Herstein and Milnor (1953, pp. 293-294).Google Scholar
- 21.Cf. Karni and Schmeidler (1991a, p. 1769).Google Scholar
- 22.Cf. Samuelson (1952, pp. 672-673). For a critical evaluation of this argument cf. McClennen (1983).Google Scholar
- 23.A dynamic choice problem is given if “a decision maker ⋯ must make decisions after the resolution of some uncertainty” [Dardanoni (1990, p. 225)]. “Atemporal” indicates in this context that the time at which uncertainty is resolved is not significant in economic terms. Cf. Karni and Schmeidler (1991a, pp. 1786-1787).Google Scholar
- 24.Chance nodes and choice nodes are denoted by circles and squares, respectively.Google Scholar
- 25.The arguments will be sketched here only in an informal manner. For comprehensive and formal treatments of this issue cf. Hammond (1988a,b,c), (1997), McClennen (1988), (1989), Gul and Lantto (1990), Karni and Safra (1988a,b), Keeney and Winkler (1985), LaValle and Wapmann (1986), and Hazen (1987b). The value of information which is closely related to the issue of dynamic consistency is analyzed in Wakker (1988) and Schlee (1990).Google Scholar
- 26.Dardanoni (1990, p. 226).Google Scholar
- 27.Detailed discussions of this concept can be found in Hammond (1986), (1995).Google Scholar
- 28.Cf. Karni and Schmeidler (1991b, p. 404). An analogous result has been obtained by Hammond (1988a, p. 43).Google Scholar
- 29.Note that dynamic inconsistencies facilitate the construction of money-pump arguments. Cf. Green (1987).Google Scholar
- 30.Machina (1989a, p. 1642).Google Scholar
- 31.Cf. McClennen (1989, pp. 156-218) and Seidenfeld (1988a, pp. 277-278), (1988b, pp. 314-315).Google Scholar
- 32.Dardanoni (1990, p. 231).Google Scholar
- 33.Another possible response is to maintain consequentialism and to give up the reduction axiom as in Segal (1990).Google Scholar
- 34.Cf. Marschak (1950), Samuelson (1952), Herstein and Milnor (1953), Blackwell and Girshick (1954), Luce and Raiffa (1957), Jensen (1967), and Grandmont (1972).Google Scholar
- 35.At the beginning of the fifties some confusion prevailed about the axioms underlying the expected utility representation theorem because axiom I was assumed only implicitly by von Neumann and Morgenstern (1947). Cf. Malinvaud (1952, p. 679) and Samuelson (1952, p. 673, note 3).Google Scholar
- 36.Cf. Chipman (1971a, p. 289). For instance, Herstein and Milnor (1953, p. 293) assumed the following weak form of.Google Scholar
- 37.See also Fishburn (1988a, p. 11). Note that Theorem 1.2 is not restricted to the set
*P*^{s}and also valid for the more general concept of a mixture set.Google Scholar - 38.Cf. Fishburn (1988a, p. 8).Google Scholar
- 39.When we consider lotteries with infinite support, the function
*u*has to be bounded because in the case of an unbounded utility function generalizations of the St. Petersburg Paradox can be constructed which result in an infinite certainty equivalent. Cf. Menger (1934). Therefore, as shown by Arrow (1974, pp. 63-69), an individual with an unbounded utility function violates either the continuity axiom or the completeness condition of axiom O. For a further discussion cf. Ryan (1974), Shapley (1977a,b), Fishburn (1976), Aumann (1977), and Russell and Seo (1978).Google Scholar - 40.A critical discussion of this concept can be found in Seidl (1997).Google Scholar
- 41.Cf. Machina (1982a, pp. 303-304).Google Scholar
- 42.
- 43.The notion of risk aversion in the expected utility framework was developed by Arrow (1963) and Pratt (1964).Google Scholar
- 44.Cf. Royden (1968, p. 110).Google Scholar
- 45.
- 46.For further concepts in the theory of risk aversion cf. Diamond and Stiglitz (1974), Ross (1981), Machina (1982b), Machina and Neilson (1987), and section 2.3.3.2.Google Scholar
- 47.Cf. Levy (1992, p. 556) who also reviews the applications of the concept of stochastic dominance in decision theory.Google Scholar
- 48.Cf. Becker and Sarin (1987, p. 1370).Google Scholar
- 49.Cf. Allais (1953). $m denotes million $.Google Scholar
- 50.Cf. Allais (1953, p. 527-529), Morrison (1967, pp. 373-376), MacCrimmon (1968, pp. 8-11), Slovic and Tversky (1974, pp. 369-371), Moskowitz (1974, pp.232-239), MacCrimmon and Larsson (1979, pp. 360-369), Kahneman and Tversky (1979, pp. 265-266), Chew and Waller (1986), MacDonald and Wall (1989, pp. 48-50), Conlisk (1989, pp. 392-394), and Carlin (1990), (1992, pp. 221-224).Google Scholar
- 51.Note that even Savage stated these preferences when he was confronted with the Allais Paradox for the first time. Cf. Savage (1954, p. 103).Google Scholar
- 52.In the study of Conlisk (1989, p. 395), 40% of the subjects violated axiom I, while in the study of Morrison (1967, p. 373, note 3) this proportion was 80%.Google Scholar
- 53.Cf. Allais (1953, pp. 529-530), Tversky (1975), MacCrimmon and Larsson (1979, pp. 350-359), Kahneman and Tversky (1979, pp. 266-267), Hagen (1979, pp. 278-281), Starmer and Sugden (1987, pp. 172-174), Kagel et al. (1990, pp. 917-919), and Carlin (1992, pp. 226-228).Google Scholar
- 54.In the experiment of Carlin (1992, p. 226), for instance, 45% of the subjects violated axiom I and about 90% of this violations consisted of the preference pattern
*r*≻*s*and*ŝ*≻*r*.Google Scholar - 55.Panel A consists of defining
*x*_{1}= $5*m*,*x*_{2}= $1*m*and*x*_{3}= $0 and in panel B we have*x*_{1}= $*y*,*y*_{2}= $*x*and*x*_{3}= $0.Google Scholar - 56.Cf. Seidl and Traub (1996).Google Scholar
- 57.Cf. Conlisk (1989, pp. 394-396) and Carlin (1990, p. 242), (1992, pp. 221-224 and 226-228).Google Scholar
- 58.Note that some authors distinguish between normative and prescriptive theories. In this study, however, we follow the argument of Howard (1992, pp. 51-52) and use the words normative and prescriptive in the same sense.Google Scholar
- 59.Cf. Keeney (1992, pp. 57-58).Google Scholar
- 60.Cf. Slovic and Tversky (1974, p. 370). In this context the results of Moskowitz (1974, pp. 234 and 237-238) also seem to be significant.Google Scholar
- 61.Chew (1989, p. 274).Google Scholar
- 62.Cf. Dekel (1986, p. 306) and Chew (1989, p. 277).Google Scholar
- 63.Cf. Chew et al. (1991, p. 142).Google Scholar
- 64.Cf. Green (1987).Google Scholar
- 65.If preferences are strictly quasiconvex, players have an aversion for mixed strategies and, thus, a Nash equilibrium may not exist. Cf. Crawford (1990), who developed as an alternative to the Nash equilibrium an “equilibrium in beliefs”, which exists even if preferences are strictly quasiconvex. Another possible response is to weaken the reduction axiom as in Dekel et al. (1991).Google Scholar
- 66.Cf. Dekel (1989, p. 166).Google Scholar
- 67.Cf. Karni and Safra (1989a,b). For a further analysis of betweenness see also Safra and Segal (1995).Google Scholar
- 68.Chew (1989, p. 274).Google Scholar
- 69.For a comparison of the models of Bolker, Jeffrey and Chew and MacCrimmon cf. Fishburn (1981, pp. 187-189), (1983, p. 301).Google Scholar
- 70.Originally, Chew and MacCrimmon (1979a, p. 6) employed an additional axiom termed ratio consistency which turned out to be superfluous since it is implied by BT and WS. Cf. Chew (1983, pp. 1086-1087) and Chew (1985a, p. A. 1).Google Scholar
- 71.This equivalence is proved in Fishburn (1988a, pp. 133-135). Further axiomatizations appear in Chew (1982) and Nakamura (1985). A weighted utility model under uncertainty is developed in Hazen (1987a).Google Scholar
- 72.Note that Chew (1983, pp. 1071-1072) additionally employed BT and M. Since C implies MC and MC and WS imply BT we can omit BT in Theorem 1.6. The consequences of M will be explored in section 1.4.3.2. See also Chew (1989, p. 284).Google Scholar
- 73.
- 74.Cf. Chew (1989, p. 283).Google Scholar
- 75.Cf. Chew (1985a, p. 6).Google Scholar
- 76.
- 77.Cf. Chew and Waller (1986).Google Scholar
- 78.See also Chew and MacCrimmon (1979b).Google Scholar
- 79.Analogous counterparts to SSB utility theory for choice under uncertainty are regret theory [cf. Bell (1982), Loomes and Sugden (1982), (1987), and Sugden (1993)], and the SSA utility theory of Fishburn (1984), (1989).Google Scholar
- 80.Fishburn (1982b) also analyzes a nontransitive variant of implicit weighted utility. See also Fishburn (1986).Google Scholar
- 81.Cf. Chew (1985a, p. 6).Google Scholar
- 82.Additionally,
*w*and*uw*have to be bounded on*X.*Cf. Chew (1985a, p. 11). For the uniqueness of implicit weighted utility cf. Chew (1985a, p. 9).Google Scholar - 83.Cf. Fishburn (1983, p. 298) and Chew (1985a, p. 6).Google Scholar
- 84.Formally, VWS in conjunction with MC implies BT. Cf. Chew (1985a, p. 4).Google Scholar
- 85.Cf. Chew (1985a, pp. 11-12).Google Scholar
- 86.Cf. Dekel (1986, pp. 305-306, 308, and 317).Google Scholar
- 87.Cf. Dekel (1986, p. 316).Google Scholar
- 88.Cf. Chew (1989), pp. 280 and 297) and Fishburn (1988a, pp. 65-66).Google Scholar
- 89.Cf. Chew (1989, p. 280).Google Scholar
- 90.A non-axiomatic model relying on the notions of disappointment and elation is proposed in Bell (1985) and Loomes and Sugden (1986).Google Scholar
- 91.Cf. Chew and Nishimura (1992, p. 298).Google Scholar
- 92.The presentation in this section follows Gul (1991) with some minor modifications.Google Scholar
- 93.Gul (1991, p. 668).Google Scholar
- 94.For the proof cf. Gul (1991, pp. 680-684).Google Scholar
- 95.Note that disappointment aversion is, in contrast to risk aversion, a global property since
*β*is constant.Google Scholar - 96.Cf. Gul (1991, pp. 677-678).Google Scholar
- 97.Cf. Neilson (1989a).Google Scholar
- 98.Cf. Gul (1991, p. 680).Google Scholar
- 99.Cf. Mosteller and Nogee (1951), Luce and Shipley (1962), Becker et al. (1963), Coombs and Huang (1976), Chew and Waller (1986), Conlisk (1987), Camerer (1989a), (1992), Prelec (1990), Battalio et al. (1990), Gigliotti and Sopher (1993), Bernasconi (1994) and Camerer and Ho (1994).Google Scholar
- 100.In Bernasconi (1994, pp. 67-68) violations of betweenness were reduced from 49% to 32% when lotteries were presented in two-stage form. This significant reduction was, however, not confirmed by the experiments of Camerer and Ho (1994, pp. 179-182).Google Scholar
- 101.Cf. Coombs and Huang (1976, pp. 330-332).Google Scholar
- 102.Chew et al. (1991, p. 140).Google Scholar
- 103.Cf. Puppe (1991, p. 74).Google Scholar
- 104.The models considered in this section are not of primary interest for part 1 of this work since they have either not been derived from an axiomatic foundation or violate the continuity and reduction axioms.Google Scholar
- 105.Cf. Lichtenstein (1965, p. 168), Rosett (1971, pp. 489 and 492), Ali (1977, pp. 803-808), Preston and Baratta (1948), Sprowls (1953), Nogee and Liebermann (1960), and Kahneman and Tversky (1979, p. 281).Google Scholar
- 106.Edwards (1954, p. 395). Note, however, that this results was not confirmed by later experiments. Cf. Seidl (1997).Google Scholar
- 107.Special variants of (1.31) are considered in Handa (1977) and Karmarkar (1978), (1979). For a general criticism of these models cf. Fishburn (1978) and Machina (1983a, p. 98).Google Scholar
- 108.Kahneman and Tversky (1979, p. 280).Google Scholar
- 109.Cf. Handa (1977, pp. 115-117) and Kahneman and Tversky (1979, pp. 284-285).Google Scholar
- 110.Cf. Fishburn (1988a, p. 52).Google Scholar
- 111.The following argument is taken from Quiggin (1982, p. 325).Google Scholar
- 112.Machina (1983a, p. 98).Google Scholar
- 113.In contrast to utility functions, value functions are assessed under certainty. Cf. Schoemaker (1982, p. 535).Google Scholar
- 114.Therefore, prospect theory cannot be regarded as a generalization of expected utlity.Google Scholar
- 115.Cf. Tversky and Kahneman (1981, p. 454).Google Scholar
- 116.Machina (1987a, p. 141).Google Scholar
- 117.Framing effects have been observed by Slovic (1969), Payne and Braunstein (1971), Hershey and Schoemaker (1980), Schoemaker and Kunreuther (1979), Kahneman and Tversky (1979), and Tversky and Kahneman (1986).Google Scholar
- 118.For an intrasitive variant of prospective reference theory cf. Bordley (1992).Google Scholar
- 119.Cf. Viscusi (1989, pp. 252-257).Google Scholar
- 120.Cf. Viscusi (1989, pp. 249-252).Google Scholar
- 121.Quiggin (1987, p. 641).Google Scholar
- 122.Note that the concept of rank-dependence had already been used earlier in welfare economics. Cf. Sen (1973), Donaldson and Weymark (1980), and Weymark (1981). For characterization of rank-dependent utility under uncertainty cf. Luce (1988), Schmeidler (1989), Wakker (1990), (1991), (1993), (1996), Chew and Karni (1994), Luce and Fishburn (1991), (1995), Tversky and Kahnemann (1992), Wakker and Tversky (1993), Chew and Wakker (1993), (1996). The relation of rank-dependent utility to two-moment decision models is analyzed in Konrad (1993).Google Scholar
- 123.Cf. Chew and Epstein (1989, p. 208) and Puppe (1991, pp. 37-39).Google Scholar
- 124.Onto or surjective means that for every λ ∈ [0, 1] there exists a
*μ*∈ [0, 1] such that*g*(*μ*) = λ. Cf. Roy den (1968, p. 8). Since*g*is also increasing this implies*g*(0) = 0 and*g*(1) = 1.Google Scholar - 125.Cf. Camerer (1989a, p. 77).Google Scholar
- 126.Cf. Segal (1987, p. 146) and Yaari (1987, p. 113).Google Scholar
- 127.Cf. Segal (1984), (1987) and Quiggin (1985), (1987) for a further discussion.Google Scholar
- 128.Cf. Section 1.4.3.2.Google Scholar
- 129.For further axiomatizations of anticipated utility cf. Segal (1984), (1989), Puppe (1991), Chateauneuf (1990), and Wakker (1994). The approach of Puppe (1991) which is equivalent to the one in Chateauneuf (1990) will be considered in section 1.4.2.4. The axiomatizations of Segal (1984), (1989) and Wakker (1994), on the other hand, may be criticized because of their lack of clear behavioral interpretations in terms of preferences. Cf. Karni and Schmeidler (1991a, p. 1781) and Wakker (1994, pp. 13-14).Google Scholar
- 130.Cf. Chew (1985b, p. 4).Google Scholar
- 131.Cf. Karni and Schmeidler (1991a, pp. 1778-1779).Google Scholar
- 132.Cf. Quiggin (1982, p. 333).Google Scholar
- 133.For the proof cf. Chew (1985b, pp. 10 and A1-A4).Google Scholar
- 134.The following argument follows Camerer (1989a, pp. 77-78).Google Scholar
- 135.This is so because, in the case of a concave function
*g*, the probabilities of the worst consequences are overweighted compared to their untransformed probabilities. Cf. Quiggin (1987) and Section 1.4.3.2.Google Scholar - 136.Cf. Karni and Safra (1990), Segal (1987), and Quiggin (1985).Google Scholar
- 137.Cf. Röell (1987, p. 143).Google Scholar
- 138.Cf. Yaaxi (1987, p. 99).Google Scholar
- 139.For the proof see Yaaxi (1987, pp. 100-101) and Karni and Schmeidler (1991a, p. 1780). Since SM is, in comparison to Theorem 1.9, weakened to M, the function
*g*is not necessarily strictly increasing.Google Scholar - 140.Cf. note 133 and section 1.4.3.2.Google Scholar
- 141.Weber and Camerer (1987, p. 137).Google Scholar
- 142.Cf. Yaari (1987, pp. 105-106) and Röell (1987, pp. 155-158).Google Scholar
- 143.Cf. Fishburn (1988a, p. 60).Google Scholar
- 144.Wakker (1992) pointed out an error in the approach of Segal (1989) which is corrected in Segal (1993). A further axiomatization of general rank-dependent utility appears in Chew and Epstein (1989). See section 1.4.4.1.Google Scholar
- 145.For an analogous approach in inequality measurement cf. Ebert (1988).Google Scholar
- 146.Cf. Green and Jullien (1989, p. 119).Google Scholar
- 147.Puppe (1991, p. 32).Google Scholar
- 148.Cf. Jullien (1988, pp. 8-9).Google Scholar
- 149.Cf. Segal (1987, p. 146).Google Scholar
- 150.Cf. Puppe (1991, p. 29).Google Scholar
- 151.Cf. Segal (1987, p. 146).Google Scholar
- 152.Cf. Green and Jullien (1988, p. 357-358) and Kischka and Puppe (1990, pp. 23-24). For the proof see Green and Jullien (1988, pp. 378-382).Google Scholar
- 153.The conditions for differentiability of ψ axe stated in Green and Jullien (1988, pp. 359-360).Google Scholar
- 154.Cf. Green and Jullien (1988, p. 358).Google Scholar
- 155.For ψ(
*x*, λ) =*u*(*x*)*f*(λ), (1.45) yields*f*(*x*)[*u*(*x*) —*u*(*y*)] >*f*(*μ*)[*u*(*x*) —*u*(*y*)]. This implies*f*(λ) >*f*(*μ*), since, as a consequence of SM and the fact that V(*δx*) =*u*(*x*), we have*u*(*x*) >*u*(*y*).Google Scholar - 156.The remainder of this section follows Puppe (1990), (1991, pp. 42-80).Google Scholar
- 157.In addition, the conditions
*u*(0) = 0,*u*and*uh*being strictly increasing in*x*, and*h*being non-increasing in*x*, have to be satisfied. For the proof see Puppe (1991, pp. 57-58).Google Scholar - 158.Note that the distortion of probabilities in anticipated utility depends only on the rank-order of consequences but not on the consequences themselves.Google Scholar
- 159.Cf. Wakker and Tversky (1993, pp. 159-160).Google Scholar
- 160.Cf. Sugden (1997, p. 30).Google Scholar
- 161.Cf. Wakker and Tversky (1993, p. 151).Google Scholar
- 162.Cf. Tversky and Kahneman (1992, p. 301).Google Scholar
- 163.This guarantees the consistency with stochastic dominance. See section 1.4.2.2.Google Scholar
- 164.Cf. Tversky and Kahneman (1992, p. 302).Google Scholar
- 165.This hypothesis is also supported by the results of Camerer and Ho (1994, p. 191).Google Scholar
- 166.Cf. Tversky and Kahneman (1992, p. 303). The concept of loss aversion in choice under certainty is analyzed in Tversky and Kahneman (1991) who also review the experimental evidence concerning loss aversion in choice under certainty and uncertainty.Google Scholar
- 167.Cf. Camerer and Ho (1994, p. 186), Hey and Orme (1994, p. 1321), Camerer (1992), and Harless and Camerer (1994, p. 1276).Google Scholar
- 168.Strictly speaking, Wakker et al. (1994) test comonotonic independence which is the analogue to OI in choice under uncertainty. Note that under continuity comonotonic independence is equivalent to OI [cf. Chew and Wakker (1996, remark A1.1)]. Since the test of Wakker et al. (1994) is based on given probabilities, their evidence also applies to OI.Google Scholar
- 169.Machina (1982a, pp. 278-279).Google Scholar
- 170.See Royden (1968, pp. 111-112).Google Scholar
- 171.Cf. Machina (1982a, p. 293). Furthermore, it is assumed that the lower limit of the support equals zero, i.e.
*A*= 0.Google Scholar - 172.Cf. Machina (1982a, pp. 293 and 314).Google Scholar
- 173.Cf. Machina (1989b, p. 395).Google Scholar
- 174.Cf. Machina (1982a, p. 294) and Machina (1983b, p. 268).Google Scholar
- 175.Cf. Machina (1989b, p. 395).Google Scholar
- 176.Machina (1982a, p. 294).Google Scholar
- 177.Cf. Machina (1987b, p. 541).Google Scholar
- 178.Analogously, an expected utility maximizer with a differentiable utility function
*u*(*x*) ranks differential shifts from a lottery according to the change of expected monetary value. Cf. Samuelson (1960, pp. 34-37).Google Scholar - 179.Cf. Machina (1983a, pp. 109-111), (1983b, p. 271).Google Scholar
- 180.Cf. section 1.3.3.Google Scholar
- 181.Cf. Machina (1982a, p. 296).Google Scholar
- 182.See Machina (1989b, pp. 396-402) for details.Google Scholar
- 183.Since only the weak relation ≽ is employed in FO, expected utility is not ruled out by this hypothesis.Google Scholar
- 184.Cf. Seidl (1997) for a review.Google Scholar
- 185.Cf. Gigliotti and Sopher (1993, p. 98). Similar preference patterns have been observed by Conlisk (1989) and Battalio et al. (1990).Google Scholar
- 186.Cf. Gigliotti and Sopher (1993, p. 97), Harless (1992, pp. 405-406), and Harless and Camerer (1994, p. 1286). Identical results have been obtained by Conslisk (1989) and Hey and Strazerra (1989).Google Scholar
- 187.Cf. Kischka and Puppe (1990, pp. 27-28).Google Scholar
- 188.Cf. Chew and Nishimura (1992, p. 296).Google Scholar
- 189.λ ↓ 0 indicates that A converges to 0 from above. Note that is not defined at λ = 0 since λ ∈ [0, 1].Google Scholar
- 190.Cf. Chew (1983, pp. 1078-1080).Google Scholar
- 191.Cf. Chew et al. (1987, pp. 377-378).Google Scholar
- 192.Cf. Chew and Nishimura (1992, p. 298).Google Scholar
- 193.The presentation in this section mainly follows Chew and Epstein (1989). Some corrections appear in Chew et al. (1993).Google Scholar
- 194.
- 195.TS is not correct in Chew and Epstein (1989, p. 212) since they state rather than Note that this error is not corrected in Chew et al. (1993).Google Scholar
- 196.This is easy to see: Note that for EU
*x*is defined by. Independence now implies, where λ in TS is given by and*q*by.Google Scholar - 197.It is assumed that the consequences are arranged in ascending order.Google Scholar
- 198.The proof is stated in Chew and Epstein (1989, pp. 227-237).Google Scholar
- 199.The following analysis is based on Chew et al. (1991), (1994).Google Scholar
- 200.
- 201.We can define α without loss of generality to be symmetric since an arbitrary α(
*x*,*y*) can always be replaced by [α(*x*,*y*) + α(*y*,*x*)]/2. Cf. Chew et al. (1991, p. 145).Google Scholar - 202.Quadratic Utility has already been considered by Machina (1982a, p. 295), who showed that it is compatible with the fanning out hypothesis. In Epstein and Segal (1992) MS is employed in order to obtain a quadratic social welfare function.Google Scholar
- 203.Cf. Chew et al. (1991, pp. 147-149).Google Scholar
- 204.Cf. Chew et al. (1991, p. 151).Google Scholar
- 205.An analogous model for the case of uncertainty is developed in Lo (1996).Google Scholar
- 206.
- 207.For the proof see Karni and Schlee (1995, pp. 138-141).Google Scholar
- 208.See Payne et al. (1992).Google Scholar
- 209.Cf. Harless and Camerer (1994) and Abdellaoui and Munier (1994).Google Scholar

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