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By Clemens Puppe

During the improvement of recent chance concept within the seventeenth cen­ tury it used to be normally held that the popularity of a chance providing the payoffs :1:17 ••• ,:l: with chances Pl, . . . , Pn is given by means of its anticipated n price L:~ :l:iPi. therefore, the choice challenge of selecting between diverse such gambles - to be able to be referred to as clients or lotteries within the sequel-was regarded as solved by way of maximizing the corresponding anticipated values. The recognized St. Petersburg paradox posed by way of Nicholas Bernoulli in 1728, notwithstanding, conclusively confirmed the truth that contributors l contemplate greater than simply the predicted worth. The solution of the St. Petersburg paradox used to be proposed independently through Gabriel Cramer and Nicholas's cousin Daniel Bernoulli [BERNOULLI 1738/1954]. Their argument was once that during a bet with payoffs :l:i the decisive components aren't the payoffs themselves yet their subjective values u( :l:i)' in keeping with this argument gambles are evaluated at the foundation of the expression L:~ U(Xi)pi. This speculation -with a a little assorted interpretation of the functionality u - has been given a high-quality axiomatic beginning in 1944 by way of v. Neumann and Morgenstern and is referred to now because the anticipated software speculation. The ensuing version has served for a very long time because the preeminent concept of selection lower than possibility, in particular in its monetary applications.

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Extra resources for Distorted Probabilities and Choice under Risk

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The following integration-by-parts argument shows that E(F) is the area of eF with respect to the Lebesgue measure. {M 10 xdF(x) = M (M xF(x)lo - 10 F(x)dx foM(l - F(x))dx dpdx. {M (l 10 1F(z) Similarily, the expected utility of F can be transformed to show that the expected utility representation equals the area of eF with respect to a product measure where the measure of an interval [x, yJ on the prize axis is given by u(y) - u( x) and the measure on the probability axis is again the Lebesgue measure.

To illustrate this fact it is proved below that the only utility model satisfying both, very weak substitution and ordinal independence, is the expected utility model. Implicit rank-linear utility theory can neither be viewed as a natural extension of implicit weighted utility theory nor as a natural extension of RDU theory. It is, however, surprising that such a general model as the implicit rank-linear utility model still implies the existence of a representation of a specific functional form.

CHEW 1983] or [CHEW AND WALLER 1986]. 2 Theories with the Betweenness Property The weighted linear utility model turned out to be just a special case of a more general class of theories. The models belonging to this class share the so-called betweenness property which asserts that a probability mixture of two distributions is intermediate in preference between the individual distributions. Formally, the betweenness property can be stated as follows. Property 6 (Betweenness) For all F, G E D(X) and all a E (0,1), F »- G implies F»- aF + (1 - a)G »- G.

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