Axiomatic Utility Theory under Risk pp 69-121 | Cite as

# Non-Archimedean Representations

Chapter

## 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
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## Notes

- 1.The following argument is based on Krantz et al. (1971, pp. 38-39).Google Scholar
- 2.The interval is nontrivial because
*x*_{2}> implies (*x*_{1},*x*_{2}) ≻ (*x*_{1}), which yields*u*(*x*_{1},*x*_{2}) >*u*(*x*_{1}).Google Scholar - 3.Cf. Royden (1968, p. 33).Google Scholar
- 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.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.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.Cf., e.g. Richter (1971, p. 39) and Fishburn (1988a, p. 11).Google Scholar
- 8.Cf. Luce and Raiffa (1957, p. 27).Google Scholar
- 9.Cf. Kreps (1988, pp. 45-46).Google Scholar
- 10.Cf., e.g. Anand (1987), (1993), and McClennen (1988).Google Scholar
- 11.Cf. Blume et al. (1989, p. 233).Google Scholar
- 12.Cf. Fishburn (1982a, p. 35).Google Scholar
- 13.Cf. Hausner (1954, p. 167) and Thrall (1954, p. 184).Google Scholar
- 14.Cf. Hausner (1954, pp. 174-180), Fishburn (1971a, pp. 677-678) and Blume et al. (1989, p. 234).Google Scholar
- 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.Note that
*V*_{1}is unique up to positive linear transformations while the functions*V*_{i}for*i*≥ 2 are unique up to linear transformations of*V*_{1},*V*_{2}, ⋯*V*_{i}which assign a positive weight to*V*_{i}. This result follows from the fact that for two arbitrary lotteries*p, q*∈*P*^{f}either or*V*_{i}is irrelevant for the representation of preferences between*p*and*q*since there exists a*l*<*i*with*V*_{l}(*p*) ≠*V*_{l}(*q*).Google Scholar - 17.Cf. section 1.3.2.Google Scholar
- 18.Cf. Blume et al. (1989, p. 234).Google Scholar
- 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
*V*_{1}is unbounded. Cf. Chipman (1971a, pp. 310-311).Google Scholar - 20.Cf. Chipman (1971a, p. 301).Google Scholar
- 21.Cf. Kreps and Ramey (1987).Google Scholar
- 22.The main results of this section axe already presented in Schmidt (1997).Google Scholar
- 23.Cf., e.g. Fishburn (1988a, p. 37).Google Scholar
- 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.Cf. Parthasarathy (1967, p. 40) and Segal (1989, p. 362).Google Scholar
- 26.Cf. Grandmont (1972, p. 48).Google Scholar
- 27.Cf. Grandmont (1972, p. 49).Google Scholar
- 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.Cf. section 1.3.3.Google Scholar
- 30.
- 31.Cf. section 1.3.3.Google Scholar
- 32.Cf. Kahneman and Tversky (1979), Neilson (1992a).Google Scholar
- 33.Cf. Kahneman and Tversky (1979, p. 284).Google Scholar
- 34.Cf. Dyer and Saxin (1982), Krzysztofowicz (1983), and Barron et al. (1984).Google Scholar
- 35.Recall that value functions represent preferences under certainty while utility functions are assessed under risk.Google Scholar
- 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.Dyer and Sarin (1979, p. 813).Google Scholar
- 38.Cf. for instance Harsanyi (1978, pp. 226-227).Google Scholar
- 39.Cf. Barron et al. (1984, p. 236).Google Scholar
- 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.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.The contents of this section were already presented in Schmidt (1997).Google Scholar
- 43.Cf. Grandmont (1972, Theorem 2).Google Scholar
- 44.Cf. Mehta (1997).Google Scholar
- 45.Cf. Encarnaciön (1964) and Ferguson (1965).Google Scholar
- 46.Cf. Chipman (1971a, p. 315).Google Scholar
- 47.Cf. Chew and Epstein (1989) and Puppe (1991, pp. 37-39).Google Scholar
- 48.Gilboa (1988) and Jaffray (1988) make weaker continuity assumptions than SC because the choice set is
*P*^{s}in their models. Therefore, it is sufficient to demand axiom AR for every single set*P*_{c}separately.Google Scholar - 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.Cf. Jaffray (1988, p. 174).Google Scholar
- 51.Cf. Theorem 1.5.Google Scholar
- 52.Cf. Kahneman and Tversky (1979, pp. 268-269).Google Scholar
- 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.Cf. Becker and Sarin (1987, p. 1369).Google Scholar
- 55.Cf. section 1.4.3.1.Google Scholar
- 56.
- 57.Cf. Camerer and Ho (1994, p. 181).Google Scholar
- 58.Cf. Kahneman and Tversky (1979, pp. 275 and 282-283).Google Scholar
- 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

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