By Clemens Puppe
During the advance of contemporary likelihood thought within the seventeenth cen tury it used to be generally held that the recognition of a big gamble delivering the payoffs :1:17 ••• ,:l: with possibilities Pl, . . . , Pn is given through its anticipated n price L:~ :l:iPi. therefore, the choice challenge of selecting between varied such gambles - in order to be referred to as clients or lotteries within the sequel-was considered solved through maximizing the corresponding anticipated values. The recognized St. Petersburg paradox posed by way of Nicholas Bernoulli in 1728, even though, conclusively proven the truth that members l ponder greater than simply the predicted price. The solution of the St. Petersburg paradox was once proposed independently by means of Gabriel Cramer and Nicholas's cousin Daniel Bernoulli [BERNOULLI 1738/1954]. Their argument was once that during a chance with payoffs :l:i the decisive elements usually are not the payoffs themselves yet their subjective values u( :l:i)' in response to this argument gambles are evaluated at the foundation of the expression L:~ U(Xi)pi. This speculation -with a a bit varied interpretation of the functionality u - has been given a great axiomatic starting place in 1944 through 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 thought of selection less than probability, in particular in its fiscal applications.
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Thus, since JL has a density it must be absolutely continuous with respect to the Lebesgue measure. On the other hand, by property l(ii) the density function may be chosen positive on (X x [0,1]) \ B and therefore the Lebesgue measure is also absolutely continuous with respect to JL. 3 may be interpreted as a completeness property of the general RDU model with respect to behavior on the set of lotteries of the form (Oj1- p,Zjp). To explain this idea let nel(X) denote the set consisting of all lotteries with two outcomes one of which is always zero.
It is more general since it allows the probability distortion to depend on the prizes available. But it restricts on the other hand these distortions to be homogeneous in the probabilities. 1 reconsiders the formal structure of the general RDU model. It is shown that there corresponds to every RD U functional V : D(X) - R a distinct function v : X x [0,1] - R unique up to a positive linear transformation such that V is determined by v and v( x, p) can be interpreted as the utility of winning x with probability p (and otherwise nothing).
6) where V12 denotes the derivative of Vl with respect to the second variable. Thus, the function (z,p) 1-+ V12(Z, 1- p) is the Radon-Nikodym density of the measure JL. Before we prove these results in detail an 1 For an approach to this problem without further assumptions but with a considerable amount of technical details see the appendix of [Chew and Epstein 1989]. 2The assumption of absolute continuity is much weaker than the condition of differentiability on basic distributions used in [Green and Jullien 1988].