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Non-Archimedean Representations

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

Abstract

According to the aggregated empirical study of Harless and Camerer (1994), the certainty effect and boundary effects are the most prominent observed violations of the independence axiom. If preferences are representable by expected utility for lotteries with the same number of probable consequences, an explicit modelling of these effects requires a weakening of the continuity axiom. In order to analyze these effects, this second part is devoted to models which employ weakened forms of both the independence and the continuity axiom.

Keywords

Utility Function Risk Aversion 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.
    The following argument is based on Krantz et al. (1971, pp. 38-39).Google Scholar
  2. 2.
    The interval is nontrivial because x2 > implies (x1, x2) ≻ (x1), which yields u(x1, x2) > u(x1).Google Scholar
  3. 3.
    Cf. Royden (1968, p. 33).Google Scholar
  4. 4.
    Cf. Chipman (1960, pp. 209-210) and Blume et al. (1989, p. 232). This result was first derived by Cuesta Dutari (1943) and Sierpiński (1949).Google Scholar
  5. 5.
    Further results are discussed in the surveys of Fishburn (1974) and Martínez-Legaz (1997), as well as in the extensive treatment of Skala (1975).Google Scholar
  6. 6.
    Note, however, that, in general, this sequence needs to be neither finite nor countable. Cf. Newman and Read (1961) and Chipman (1971b, pp. 277-279).Google Scholar
  7. 7.
    Cf., e.g. Richter (1971, p. 39) and Fishburn (1988a, p. 11).Google Scholar
  8. 8.
    Cf. Luce and Raiffa (1957, p. 27).Google Scholar
  9. 9.
    Cf. Kreps (1988, pp. 45-46).Google Scholar
  10. 10.
    Cf., e.g. Anand (1987), (1993), and McClennen (1988).Google Scholar
  11. 11.
    Cf. Blume et al. (1989, p. 233).Google Scholar
  12. 12.
    Cf. Fishburn (1982a, p. 35).Google Scholar
  13. 13.
    Cf. Hausner (1954, p. 167) and Thrall (1954, p. 184).Google Scholar
  14. 14.
    Cf. Hausner (1954, pp. 174-180), Fishburn (1971a, pp. 677-678) and Blume et al. (1989, p. 234).Google Scholar
  15. 15.
    A simple graphical proof of this theorem for the case of only three and four possible consequences appears in Chipman (1971a).Google Scholar
  16. 16.
    Note that V1 is unique up to positive linear transformations while the functions Vi for i ≥ 2 are unique up to linear transformations of V1, V2, ⋯ Vi which assign a positive weight to Vi. This result follows from the fact that for two arbitrary lotteries p, qPf either or Vi is irrelevant for the representation of preferences between p and q since there exists a l < i with Vl(p) ≠ Vl(q).Google Scholar
  17. 17.
    Cf. section 1.3.2.Google Scholar
  18. 18.
    Cf. Blume et al. (1989, p. 234).Google Scholar
  19. 19.
    Therefore, a decision maker whose preferences can be represented by lexicographic expected utility is willing to pay an infinite amount for playing the St. Petersburg Game if V1 is unbounded. Cf. Chipman (1971a, pp. 310-311).Google Scholar
  20. 20.
    Cf. Chipman (1971a, p. 301).Google Scholar
  21. 21.
    Cf. Kreps and Ramey (1987).Google Scholar
  22. 22.
    The main results of this section axe already presented in Schmidt (1997).Google Scholar
  23. 23.
    Cf., e.g. Fishburn (1988a, p. 37).Google Scholar
  24. 24.
    Cf. Cohen (1992, p. 101) and the empirical results of Cohen and Jaffray (1988), Conlisk (1989), Harless (1992), and Harless and Camerer (1994).Google Scholar
  25. 25.
    Cf. Parthasarathy (1967, p. 40) and Segal (1989, p. 362).Google Scholar
  26. 26.
    Cf. Grandmont (1972, p. 48).Google Scholar
  27. 27.
    Cf. Grandmont (1972, p. 49).Google Scholar
  28. 28.
    Cf. section 1.4.2.3. More recently, Wakker (1994) and Chateauneuf and Cohen (1994) showed that by transforming cumulative probabilities a separation of marginal utility and risk aversion can be at least partly obtained even if utility is not assumed to be linear on X.Google Scholar
  29. 29.
    Cf. section 1.3.3.Google Scholar
  30. 30.
    Note that, according to NDI, x(p) is independent of q and λGoogle Scholar
  31. 31.
    Cf. section 1.3.3.Google Scholar
  32. 32.
    Cf. Kahneman and Tversky (1979), Neilson (1992a).Google Scholar
  33. 33.
    Cf. Kahneman and Tversky (1979, p. 284).Google Scholar
  34. 34.
    Cf. Dyer and Saxin (1982), Krzysztofowicz (1983), and Barron et al. (1984).Google Scholar
  35. 35.
    Recall that value functions represent preferences under certainty while utility functions are assessed under risk.Google Scholar
  36. 36.
    Axiomatizations of a measurable value function appear in Alt (1936), Suppes and Winet (1955), and Scott and Suppes (1958). Assessment methods are discussed in Farquhar and Keller (1989). However, note that, despite the empirical evidence reported in the literature on psychophysics [cf. Stevens (1975)], some authors [e.g. Machina (1981) and White (1985)] believe that preference intensities cannot be interpreted and measured meaningfully.Google Scholar
  37. 37.
    Dyer and Sarin (1979, p. 813).Google Scholar
  38. 38.
    Cf. for instance Harsanyi (1978, pp. 226-227).Google Scholar
  39. 39.
    Cf. Barron et al. (1984, p. 236).Google Scholar
  40. 40.
    This fact was already recognized and discussed by Ellsberg (1954), Luce and Raiffa (1957) and Baumol (1958). Nevertheless discussion on this issue in the german literature prevailed until 1992 [cf. Kürsten (1992)].Google Scholar
  41. 41.
    The interpretation of v as a measurable value function requires clearly the existence of a quaternary preference relation and some technical assumptions. However, these assumptions are also necessary in order to interpret the von Neumann-Morgenstern function as a measurable value function [cf. Sarin (1982, pp. 986-989)].Google Scholar
  42. 42.
    The contents of this section were already presented in Schmidt (1997).Google Scholar
  43. 43.
    Cf. Grandmont (1972, Theorem 2).Google Scholar
  44. 44.
    Cf. Mehta (1997).Google Scholar
  45. 45.
    Cf. Encarnaciön (1964) and Ferguson (1965).Google Scholar
  46. 46.
    Cf. Chipman (1971a, p. 315).Google Scholar
  47. 47.
    Cf. Chew and Epstein (1989) and Puppe (1991, pp. 37-39).Google Scholar
  48. 48.
    Gilboa (1988) and Jaffray (1988) make weaker continuity assumptions than SC because the choice set is Ps in their models. Therefore, it is sufficient to demand axiom AR for every single set Pc separately.Google Scholar
  49. 49.
    Gilboa (1988, pp. 409-410) notes that there is an intersection between the representation (2.20) and rank-dependent utility representations which would contradict the statements concerning the classification of non-expected utilty models in the preceeding section. But in the example of Gilboa, the distortion function is not continuous and, therefore, his example is not a special case of RDU as defined in section 1.4.2.4.Google Scholar
  50. 50.
    Cf. Jaffray (1988, p. 174).Google Scholar
  51. 51.
    Cf. Theorem 1.5.Google Scholar
  52. 52.
    Cf. Kahneman and Tversky (1979, pp. 268-269).Google Scholar
  53. 53.
    Note that boundary effects may also occur if the probability of a consequence is not equal to zero but very close to zero. This view is treated in section 2.4.4.2 only for the certainty effect but the analysis can be easily applied to boundary effects.Google Scholar
  54. 54.
    Cf. Becker and Sarin (1987, p. 1369).Google Scholar
  55. 55.
    Cf. section 1.4.3.1.Google Scholar
  56. 56.
    u2 denotes the derivative of u with respect to the second argument.Google Scholar
  57. 57.
    Cf. Camerer and Ho (1994, p. 181).Google Scholar
  58. 58.
    Cf. Kahneman and Tversky (1979, pp. 275 and 282-283).Google Scholar
  59. 59.
    To my knowledge the only empirical test of this operation, conducted by Seidl and Traub (1995), refers to choice problems under certainty only.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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