By Clemens Puppe
During the improvement of contemporary likelihood conception within the seventeenth cen tury it used to be usually held that the reputation of a raffle delivering the payoffs :1:17 ••• ,:l: with percentages Pl, . . . , Pn is given via its anticipated n worth L:~ :l:iPi. for that reason, the choice challenge of selecting between assorted such gambles - so one can be known as customers or lotteries within the sequel-was considered solved through maximizing the corresponding anticipated values. The well-known St. Petersburg paradox posed through Nicholas Bernoulli in 1728, despite the fact that, conclusively established the truth that contributors l examine greater than simply the anticipated worth. The solution of the St. Petersburg paradox used to be proposed independently by way of Gabriel Cramer and Nicholas's cousin Daniel Bernoulli [BERNOULLI 1738/1954]. Their argument used to be that during a chance with payoffs :l:i the decisive components are usually not the payoffs themselves yet their subjective values u( :l:i)' in accordance with this argument gambles are evaluated at the foundation of the expression L:~ U(Xi)pi. This speculation -with a slightly various interpretation of the functionality u - has been given an excellent axiomatic origin in 1944 via 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 hazard, specially in its financial applications.
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Extra info for Distorted Probabilities and Choice under Risk
39 Chapter 2 A Rank-Dependent Utility Model with Prize-Dependent Distortion of Probabilities In this chapter a new model of choice under risk within the framework of RDU theory is offered. The suggested model contains the expected utility model as a special case. Compared to anticipated utility theory it is more general in one respect and more restrictive in another. 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.
Let Z and Y denote the random variables underlying the distribution functions F and G, respectively. For every a: E [0,1]' the expression a:F ffi (1 - a:)G refers to the cumulative distribution function corresponding to the random variable a:Z + (1 - a:)Y. The following axiom is known as the dual independence axiom. Axiom 9 (Dual Independence) For all F, G, H E D(X) and all a: E [0,1]' F t G implies a:F EB (1 - a:)H t a:G ffi (1 - a:)H. The representation theorem for the dual theory is proved in [YAARI 1987].
4 a function "p : X X [0,1] -+ R such that represents t on DO(X) where F = (XliPb·... ,xniPn). 18). It will now be shown that there exists a distinct such function "p which allows for an intuitive interpretation. For this purpose define for a given 1jJ satisfying the condition above a function v : X X [0, 1] -+ R by = "p(x,p) - "p(O,p) for (x,p) E X X [0,1]. ,O) = 0, v is continuous in each argument and v(X,p) v(y,q) - v(y,p) - v(x,q) + v(x,p) > 0 for y> x,q > p. 2) 42 and V((Oj 1 - p, Xjp)) = v(x,p) for all (x,p) E X x [0,1].