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A Survey

  • Ulrich Schmidt
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 461)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 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. 2.
    We define a model as hybrid if it contains rank-dependent utility representations as well as betweenness satisfying utility representations.Google Scholar
  3. 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. 4.
    Cf. Herstein and Milnor (1953, p. 292).Google Scholar
  5. 5.
    Cf. Bauer (1968, p. 123).Google Scholar
  6. 6.
    A probability measure p has finite support if there exists a finite set WX with p(W) = 1. Cf. Fishburn (1970, p. 105).Google Scholar
  7. 7.
    Cf. Jensen (1967, p. 171).Google Scholar
  8. 8.
    Cf. Sen (1970, pp. 8-9).Google Scholar
  9. 9.
    Note that some authors label a complete and transitive relation as “weak ordering” or “complete preordering”.Google Scholar
  10. 10.
    A strict partial ordering is transitive and asymmetric (pq ⇒ ¬(qP) ∀ p, qP). Cf. Sen (1970, p. 9).Google Scholar
  11. 11.
    Cf. Karni and Schmeidler (1991a, p. 1766).Google Scholar
  12. 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. 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. 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. 15.
    Cf. Jensen (1967, p. 173).Google Scholar
  16. 16.
    Cf. Herstein and Milnor (1953, p. 293).Google Scholar
  17. 17.
    Cf. Chew et al. (1991, p. 141).Google Scholar
  18. 18.
    Cf. Parthasarathy (1967, p. 40).Google Scholar
  19. 19.
    Cf. Chew (1985a, p. 3).Google Scholar
  20. 20.
    Cf. Herstein and Milnor (1953, pp. 293-294).Google Scholar
  21. 21.
    Cf. Karni and Schmeidler (1991a, p. 1769).Google Scholar
  22. 22.
    Cf. Samuelson (1952, pp. 672-673). For a critical evaluation of this argument cf. McClennen (1983).Google Scholar
  23. 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. 24.
    Chance nodes and choice nodes are denoted by circles and squares, respectively.Google Scholar
  25. 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. 26.
    Dardanoni (1990, p. 226).Google Scholar
  27. 27.
    Detailed discussions of this concept can be found in Hammond (1986), (1995).Google Scholar
  28. 28.
    Cf. Karni and Schmeidler (1991b, p. 404). An analogous result has been obtained by Hammond (1988a, p. 43).Google Scholar
  29. 29.
    Note that dynamic inconsistencies facilitate the construction of money-pump arguments. Cf. Green (1987).Google Scholar
  30. 30.
    Machina (1989a, p. 1642).Google Scholar
  31. 31.
    Cf. McClennen (1989, pp. 156-218) and Seidenfeld (1988a, pp. 277-278), (1988b, pp. 314-315).Google Scholar
  32. 32.
    Dardanoni (1990, p. 231).Google Scholar
  33. 33.
    Another possible response is to maintain consequentialism and to give up the reduction axiom as in Segal (1990).Google Scholar
  34. 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. 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. 36.
    Cf. Chipman (1971a, p. 289). For instance, Herstein and Milnor (1953, p. 293) assumed the following weak form of.Google Scholar
  37. 37.
    See also Fishburn (1988a, p. 11). Note that Theorem 1.2 is not restricted to the set Ps and also valid for the more general concept of a mixture set.Google Scholar
  38. 38.
    Cf. Fishburn (1988a, p. 8).Google Scholar
  39. 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. 40.
    A critical discussion of this concept can be found in Seidl (1997).Google Scholar
  41. 41.
    Cf. Machina (1982a, pp. 303-304).Google Scholar
  42. 42.
    pi denotes the probability of xi for i = 1, 2, 3.Google Scholar
  43. 43.
    The notion of risk aversion in the expected utility framework was developed by Arrow (1963) and Pratt (1964).Google Scholar
  44. 44.
    Cf. Royden (1968, p. 110).Google Scholar
  45. 45.
    More precisely, F is a mean-preserving spread of G iff and. Cf. Puppe (1991, p. 69).Google Scholar
  46. 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. 47.
    Cf. Levy (1992, p. 556) who also reviews the applications of the concept of stochastic dominance in decision theory.Google Scholar
  48. 48.
    Cf. Becker and Sarin (1987, p. 1370).Google Scholar
  49. 49.
    Cf. Allais (1953). $m denotes million $.Google Scholar
  50. 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. 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. 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. 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. 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 rs and ŝr.Google Scholar
  55. 55.
    Panel A consists of defining x1 = $5m, x2 = $1m and x3 = $0 and in panel B we have x1 = $y, y2 = $x and x3 = $0.Google Scholar
  56. 56.
    Cf. Seidl and Traub (1996).Google Scholar
  57. 57.
    Cf. Conlisk (1989, pp. 394-396) and Carlin (1990, p. 242), (1992, pp. 221-224 and 226-228).Google Scholar
  58. 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. 59.
    Cf. Keeney (1992, pp. 57-58).Google Scholar
  60. 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. 61.
    Chew (1989, p. 274).Google Scholar
  62. 62.
    Cf. Dekel (1986, p. 306) and Chew (1989, p. 277).Google Scholar
  63. 63.
    Cf. Chew et al. (1991, p. 142).Google Scholar
  64. 64.
    Cf. Green (1987).Google Scholar
  65. 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. 66.
    Cf. Dekel (1989, p. 166).Google Scholar
  67. 67.
    Cf. Karni and Safra (1989a,b). For a further analysis of betweenness see also Safra and Segal (1995).Google Scholar
  68. 68.
    Chew (1989, p. 274).Google Scholar
  69. 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. 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. 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. 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. 73.
    For the uniqueness of u and w see Chew (1983, p. 1072).Google Scholar
  74. 74.
    Cf. Chew (1989, p. 283).Google Scholar
  75. 75.
    Cf. Chew (1985a, p. 6).Google Scholar
  76. 76.
    We have ui:= ui(xi), wi:= w(xi) and pi:= p(xi) for i = 1, 2, 3.Google Scholar
  77. 77.
    Cf. Chew and Waller (1986).Google Scholar
  78. 78.
    See also Chew and MacCrimmon (1979b).Google Scholar
  79. 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. 80.
    Fishburn (1982b) also analyzes a nontransitive variant of implicit weighted utility. See also Fishburn (1986).Google Scholar
  81. 81.
    Cf. Chew (1985a, p. 6).Google Scholar
  82. 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. 83.
    Cf. Fishburn (1983, p. 298) and Chew (1985a, p. 6).Google Scholar
  84. 84.
    Formally, VWS in conjunction with MC implies BT. Cf. Chew (1985a, p. 4).Google Scholar
  85. 85.
    Cf. Chew (1985a, pp. 11-12).Google Scholar
  86. 86.
    Cf. Dekel (1986, pp. 305-306, 308, and 317).Google Scholar
  87. 87.
    Cf. Dekel (1986, p. 316).Google Scholar
  88. 88.
    Cf. Chew (1989), pp. 280 and 297) and Fishburn (1988a, pp. 65-66).Google Scholar
  89. 89.
    Cf. Chew (1989, p. 280).Google Scholar
  90. 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. 91.
    Cf. Chew and Nishimura (1992, p. 298).Google Scholar
  92. 92.
    The presentation in this section follows Gul (1991) with some minor modifications.Google Scholar
  93. 93.
    Gul (1991, p. 668).Google Scholar
  94. 94.
    For the proof cf. Gul (1991, pp. 680-684).Google Scholar
  95. 95.
    Note that disappointment aversion is, in contrast to risk aversion, a global property since β is constant.Google Scholar
  96. 96.
    Cf. Gul (1991, pp. 677-678).Google Scholar
  97. 97.
    Cf. Neilson (1989a).Google Scholar
  98. 98.
    Cf. Gul (1991, p. 680).Google Scholar
  99. 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. 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. 101.
    Cf. Coombs and Huang (1976, pp. 330-332).Google Scholar
  102. 102.
    Chew et al. (1991, p. 140).Google Scholar
  103. 103.
    Cf. Puppe (1991, p. 74).Google Scholar
  104. 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. 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. 106.
    Edwards (1954, p. 395). Note, however, that this results was not confirmed by later experiments. Cf. Seidl (1997).Google Scholar
  107. 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. 108.
    Kahneman and Tversky (1979, p. 280).Google Scholar
  109. 109.
    Cf. Handa (1977, pp. 115-117) and Kahneman and Tversky (1979, pp. 284-285).Google Scholar
  110. 110.
    Cf. Fishburn (1988a, p. 52).Google Scholar
  111. 111.
    The following argument is taken from Quiggin (1982, p. 325).Google Scholar
  112. 112.
    Machina (1983a, p. 98).Google Scholar
  113. 113.
    In contrast to utility functions, value functions are assessed under certainty. Cf. Schoemaker (1982, p. 535).Google Scholar
  114. 114.
    Therefore, prospect theory cannot be regarded as a generalization of expected utlity.Google Scholar
  115. 115.
    Cf. Tversky and Kahneman (1981, p. 454).Google Scholar
  116. 116.
    Machina (1987a, p. 141).Google Scholar
  117. 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. 118.
    For an intrasitive variant of prospective reference theory cf. Bordley (1992).Google Scholar
  119. 119.
    Cf. Viscusi (1989, pp. 252-257).Google Scholar
  120. 120.
    Cf. Viscusi (1989, pp. 249-252).Google Scholar
  121. 121.
    Quiggin (1987, p. 641).Google Scholar
  122. 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. 123.
    Cf. Chew and Epstein (1989, p. 208) and Puppe (1991, pp. 37-39).Google Scholar
  124. 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. 125.
    Cf. Camerer (1989a, p. 77).Google Scholar
  126. 126.
    Cf. Segal (1987, p. 146) and Yaari (1987, p. 113).Google Scholar
  127. 127.
    Cf. Segal (1984), (1987) and Quiggin (1985), (1987) for a further discussion.Google Scholar
  128. 128.
    Cf. Section 1.4.3.2.Google Scholar
  129. 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. 130.
    Cf. Chew (1985b, p. 4).Google Scholar
  131. 131.
    Cf. Karni and Schmeidler (1991a, pp. 1778-1779).Google Scholar
  132. 132.
    Cf. Quiggin (1982, p. 333).Google Scholar
  133. 133.
    For the proof cf. Chew (1985b, pp. 10 and A1-A4).Google Scholar
  134. 134.
    The following argument follows Camerer (1989a, pp. 77-78).Google Scholar
  135. 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. 136.
    Cf. Karni and Safra (1990), Segal (1987), and Quiggin (1985).Google Scholar
  137. 137.
    Cf. Röell (1987, p. 143).Google Scholar
  138. 138.
    Cf. Yaaxi (1987, p. 99).Google Scholar
  139. 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. 140.
    Cf. note 133 and section 1.4.3.2.Google Scholar
  141. 141.
    Weber and Camerer (1987, p. 137).Google Scholar
  142. 142.
    Cf. Yaari (1987, pp. 105-106) and Röell (1987, pp. 155-158).Google Scholar
  143. 143.
    Cf. Fishburn (1988a, p. 60).Google Scholar
  144. 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. 145.
    For an analogous approach in inequality measurement cf. Ebert (1988).Google Scholar
  146. 146.
    Cf. Green and Jullien (1989, p. 119).Google Scholar
  147. 147.
    Puppe (1991, p. 32).Google Scholar
  148. 148.
    Cf. Jullien (1988, pp. 8-9).Google Scholar
  149. 149.
    Cf. Segal (1987, p. 146).Google Scholar
  150. 150.
    Cf. Puppe (1991, p. 29).Google Scholar
  151. 151.
    Cf. Segal (1987, p. 146).Google Scholar
  152. 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. 153.
    The conditions for differentiability of ψ axe stated in Green and Jullien (1988, pp. 359-360).Google Scholar
  154. 154.
    Cf. Green and Jullien (1988, p. 358).Google Scholar
  155. 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. 156.
    The remainder of this section follows Puppe (1990), (1991, pp. 42-80).Google Scholar
  157. 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. 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. 159.
    Cf. Wakker and Tversky (1993, pp. 159-160).Google Scholar
  160. 160.
    Cf. Sugden (1997, p. 30).Google Scholar
  161. 161.
    Cf. Wakker and Tversky (1993, p. 151).Google Scholar
  162. 162.
    Cf. Tversky and Kahneman (1992, p. 301).Google Scholar
  163. 163.
    This guarantees the consistency with stochastic dominance. See section 1.4.2.2.Google Scholar
  164. 164.
    Cf. Tversky and Kahneman (1992, p. 302).Google Scholar
  165. 165.
    This hypothesis is also supported by the results of Camerer and Ho (1994, p. 191).Google Scholar
  166. 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. 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. 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. 169.
    Machina (1982a, pp. 278-279).Google Scholar
  170. 170.
    See Royden (1968, pp. 111-112).Google Scholar
  171. 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. 172.
    Cf. Machina (1982a, pp. 293 and 314).Google Scholar
  173. 173.
    Cf. Machina (1989b, p. 395).Google Scholar
  174. 174.
    Cf. Machina (1982a, p. 294) and Machina (1983b, p. 268).Google Scholar
  175. 175.
    Cf. Machina (1989b, p. 395).Google Scholar
  176. 176.
    Machina (1982a, p. 294).Google Scholar
  177. 177.
    Cf. Machina (1987b, p. 541).Google Scholar
  178. 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. 179.
    Cf. Machina (1983a, pp. 109-111), (1983b, p. 271).Google Scholar
  180. 180.
    Cf. section 1.3.3.Google Scholar
  181. 181.
    Cf. Machina (1982a, p. 296).Google Scholar
  182. 182.
    See Machina (1989b, pp. 396-402) for details.Google Scholar
  183. 183.
    Since only the weak relation ≽ is employed in FO, expected utility is not ruled out by this hypothesis.Google Scholar
  184. 184.
    Cf. Seidl (1997) for a review.Google Scholar
  185. 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. 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. 187.
    Cf. Kischka and Puppe (1990, pp. 27-28).Google Scholar
  188. 188.
    Cf. Chew and Nishimura (1992, p. 296).Google Scholar
  189. 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. 190.
    Cf. Chew (1983, pp. 1078-1080).Google Scholar
  191. 191.
    Cf. Chew et al. (1987, pp. 377-378).Google Scholar
  192. 192.
    Cf. Chew and Nishimura (1992, p. 298).Google Scholar
  193. 193.
    The presentation in this section mainly follows Chew and Epstein (1989). Some corrections appear in Chew et al. (1993).Google Scholar
  194. 194.
    I.e. consists of k elements and consists of n-k elements.Google Scholar
  195. 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. 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. 197.
    It is assumed that the consequences are arranged in ascending order.Google Scholar
  198. 198.
    The proof is stated in Chew and Epstein (1989, pp. 227-237).Google Scholar
  199. 199.
    The following analysis is based on Chew et al. (1991), (1994).Google Scholar
  200. 200.
    Since SM is demanded, v(x, V(F)) has to be strictly increasing in x.Google Scholar
  201. 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. 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. 203.
    Cf. Chew et al. (1991, pp. 147-149).Google Scholar
  204. 204.
    Cf. Chew et al. (1991, p. 151).Google Scholar
  205. 205.
    An analogous model for the case of uncertainty is developed in Lo (1996).Google Scholar
  206. 206.
    Recall that B is the Borel-algebra of X. See section 1.2.Google Scholar
  207. 207.
    For the proof see Karni and Schlee (1995, pp. 138-141).Google Scholar
  208. 208.
    See Payne et al. (1992).Google Scholar
  209. 209.
    Cf. Harless and Camerer (1994) and Abdellaoui and Munier (1994).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Ulrich Schmidt
    • 1
  1. 1.University of KielInstitut für Finanzwissenschaft und SozialpolitikKielGermany

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