# Download e-book for iPad: Axiomatic Utility Theory under Risk: Non-Archimedean by Ulrich Schmidt

By Ulrich Schmidt

The first makes an attempt to increase a software concept for selection events lower than threat have been undertaken via 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 almost all humans wouldn't spend an important sum of money to interact in that gamble. To account for this remark, Bernoulli (1738, pp. 199-201) proposed that the predicted financial price should be changed by way of the predicted software ("moral expectation") because the correct criterion for determination making lower than chance. notwithstanding, Bernoulli's 2 argument and especially his collection of a logarithmic application functionality appear to be fairly arbitrary on account that they're dependent solely on intuitively three beautiful examples. no longer until eventually centuries later, did von Neumann and Morgenstern (1947) turn out that if the personal tastes of the choice maker fulfill cer tain assumptions they are often represented via the predicted price of a real-valued software functionality outlined at the set of effects. regardless of the same mathematical type of anticipated application, the speculation of von Neumann and Morgenstern and Bernoulli's procedure have, notwithstanding, IFor accomplished discussions of this paradox cf. Menger (1934), Samuelson (1960), (1977), Shapley (1977a), Aumann (1977), Jorland (1987), and Zabell (1987). 2Cramer (1728, p. 212), nonetheless, proposed that the application of an sum of money is given by way of the sq. root of this amount.

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**Example text**

166). 67Cf. Karni and Safra (1989a,b). 2 Weighted Utility Theory Weighted utility was the first utility theory with the betweenness property to appear in the literature. 69 The central axiom of weighted utility is the following weakened form of I: Weak Substitution (WS): V p, q E P : p '" q::::} V A E ]0, 1[,:3 /-L E ]0,1[, such that AP+ (1- A)r '" /-Lq + (1 - /-L)r V rEP. In contrast to axiom I, WS does not require A and /-L to be identical. However, the value of /-L must be independent of the lottery r.

143) Along the hypotenuse all indifference curves have an identical slope (hypotenuse parallelism) since P2 = 0 yields PI = 1 - P3. Thus, indifference curves in anticipated utility theory do not satisfy the fanning out hypothesis. Nevertheless, anticipated utility can accommodate the common ratio and common consequence effect I36 because indifference curves are not linear as in expected utility theory. 135This is so because, in the case of a concave function g, the probabilities of the worst consequences are overweighted compared to their untransformed probabilities.

The goal of descriptive decision theories is to describe, explain, and predict the actual choice behavior of individuals. In contrast, a prescriptive decision theory is concerned with helping people to make better decisions and, therefore, tries to answer the question how decisions should be made in order to be logicaliy consistent. 59 56Cf. Seidl and Traub (1996). 57 Cf. Conlisk (1989, pp. 394-396) and Carlin (1990, p. 242), (1992, pp. 221-224 and 226-228). 58 Note that some authors distinguish between normative and prescriptive theories.