Axiomatic Utility Theory under Risk: Non-Archimedean by Ulrich Schmidt

By Ulrich Schmidt

The first makes an attempt to boost a application concept for selection occasions below probability have been undertaken through Cramer (1728) and Bernoulli (1738). contemplating the recognized St. Petersburg Paradox! - a lottery with an enormous anticipated financial worth -Bernoulli (1738, p. 209) saw that the majority humans wouldn't spend an important sum of money to have interaction in that gamble. To account for this commentary, Bernoulli (1738, pp. 199-201) proposed that the predicted financial worth should be changed via the anticipated software ("moral expectation") because the suitable criterion for choice making below hazard. in spite of the fact that, Bernoulli's 2 argument and especially his collection of a logarithmic software functionality appear to be relatively arbitrary when you consider that they're established totally on intuitively three beautiful examples. no longer till centuries later, did von Neumann and Morgenstern (1947) end up that if the personal tastes of the choice maker fulfill cer­ tain assumptions they are often represented through the anticipated price of a real-valued software functionality outlined at the set of outcomes. regardless of the same mathematical kind of anticipated software, the idea of von Neumann and Morgenstern and Bernoulli's technique have, although, IFor finished discussions of this paradox cf. Menger (1934), Samuelson (1960), (1977), Shapley (1977a), Aumann (1977), Jorland (1987), and Zabell (1987). 2Cramer (1728, p. 212), however, proposed that the software of an sum of money is given by means of the sq. root of this amount.

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Weber (1982, p. 11-13) interprets the value w(x) as a measure of the "conceivability" of the consequence x and proposes a bell-shaped weighting function which is maximal at zero. Furthermore, he shows that this hypothesis is consistent with the preferences displayed in the Allais Paradox. 24 Restricting attention to only three possible consequences, X3, a constant utility level V is given by 76 V= + {1 - Pl - P3)U2 W 2 + P3 U3W 3 PIWI + {1 - PI - P3)W2 + P3 W 3 PI UI WI Solving for PI . 20) yields which is a linear equation.

LlOCf. Fishburn (1988a, p. 52). lllThe following argument is taken from Quiggin (1982, p. 325). 36 although c5 x is stochastically dominated by p. 9(l/n) > lin an analogous argument can be obtained for -E :S Yi < O. "The general 'lesson' here is an important one, namely that in order to avoid ... the necessary ... 31), tries to avoid this problem by introducing an editing phase in which dominated lotteries are removed from considerations before the utility function is maximized. But, as noted by Kahneman and Tversky (1979, p.

The necessary ... 31), tries to avoid this problem by introducing an editing phase in which dominated lotteries are removed from considerations before the utility function is maximized. But, as noted by Kahneman and Tversky (1979, p. 284), indirect violations of dominance are not excluded by this operation and, therefore, intransitivities may occur. Recently, a new variant of prospect theory, termed cumulative prospect theory, has been developed which, like rank-dependent utility, avoids violations of dominance by transforming cumulative probabilities rather than single probabilities.

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