By Clemens Puppe

ISBN-10: 3540542477

ISBN-13: 9783540542476

ISBN-10: 3642582036

ISBN-13: 9783642582035

During the advance of recent chance idea within the seventeenth cen tury it used to be quite often held that the popularity of a big gamble supplying the payoffs :1:17 ••• ,:l: with percentages Pl, . . . , Pn is given via its anticipated n price L:~ :l:iPi. therefore, the choice challenge of selecting between various such gambles - so that it will be referred to as clients or lotteries within the sequel-was regarded as solved through maximizing the corresponding anticipated values. The well-known St. Petersburg paradox posed by means of Nicholas Bernoulli in 1728, notwithstanding, conclusively tested the truth that contributors l ponder greater than simply the anticipated worth. The solution of the St. Petersburg paradox was once proposed independently via 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 line with this argument gambles are evaluated at the foundation of the expression L:~ U(Xi)pi. This speculation -with a a little diverse interpretation of the functionality u - has been given a high-quality axiomatic starting place in 1944 through v. Neumann and Morgenstern and is referred to now because the anticipated application speculation. The ensuing version has served for a very long time because the preeminent conception of selection less than threat, in particular in its fiscal applications.

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An analysis of the Allais paradox and the common ratio effect within the framework of anticipated utility theory can be found in [SEGAL 1987a]. 3. 16) implies V(ct) = u(x)f(p), V(C2) = u(y)f(q), V(cs) = u(x)f(rp) and V(C4) = u(y)f(rq), respectively. It has been shown in [SEGAL 1987a] that a possible explanation for the choice of Cl in the first pair and of C4 in the second pair of the common ratio effect is that the elasticity of f is strictly increasing. Indeed, if the elasticity of f at p, defined by pf'(p)f f(p), is strictly increasing it follows that the function f (rp ) f f (rq) is strictly increasing in r.

1). 12) where u : X -+ R is a utility function and h : [0,1] -+ [0,1] is a strictly increasing, surjective transformation function. 12). Therefore, anticipated utility theory is also known as the theory of expected utility with rank-dependent probabilities. Clearly, the expected utility model corresponds to the case where h is the identity function. The first axiomatization of the anticipated utility model is due to QUIGGIN [1982]. 12) is the following so-called weak certainty equivalent substitution axiom.

This cardinal utility is what we have called the generalized utility function. One may ask then which functions v are admissible as generalized utility functions. Obviously, since for (z,p) E B = (X x {O}) U ({O} x [0,1]) the lotteries (Oj1 - p,Zjp) represent the same distribution giving zero with certainty, v should be constant on B and by normalization equal to zero there. The monotonicity of preferences restricted to nel(x) implies that v should be a strictly increasing function of z for every fixed p 1= 0 and of p for every fixed Z 1= o.