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## Über dieses Buch

During the development of modern probability theory in the 17th cen­ tury it was commonly held that the attractiveness of a gamble offering the payoffs :1:17 ••• ,:l: with probabilities Pl, . . . , Pn is given by its expected n value L:~ :l:iPi. Accordingly, the decision problem of choosing among different such gambles - which will be called prospects or lotteries in the sequel-was thought to be solved by maximizing the corresponding expected values. The famous St. Petersburg paradox posed by Nicholas Bernoulli in 1728, however, conclusively demonstrated the fact that individuals l consider more than just the expected value. The resolution of the St. Petersburg paradox was proposed independently by Gabriel Cramer and Nicholas's cousin Daniel Bernoulli [BERNOULLI 1738/1954]. Their argument was that in a gamble with payoffs :l:i the decisive factors are not the payoffs themselves but their subjective values u( :l:i)' According to this argument gambles are evaluated on the basis of the expression L:~ U(Xi)pi. This hypothesis -with a somewhat different interpretation of the function u - has been given a solid axiomatic foundation in 1944 by v. Neumann and Morgenstern and is now known as the expected utility hypothesis. The resulting model has served for a long time as the preeminent theory of choice under risk, especially in its economic applications.

## Inhaltsverzeichnis

### Introduction

Abstract
During the development of modern probability theory in the 17th century it was commonly held that the attractiveness of a gamble offering the payoffs x1..., xn with probabilities p1, ..., pn is given by its expected valuen1xixi.
Clemens Puppe

### Chapter 1. Axiomatic Utility Theory under Risk

Abstract
This chapter reviews the developments of the theory of choice under risk since V.NEUMANN and MORGENSTERN [1947] presented the first axiomatic formulation of the expected utility model. The purpose of this review is to set the stage for the investigations of the following chapters and to give the context of the problems treated there.
Clemens Puppe

### Chapter 2. A Rank-Dependent Utility Model with Prize-Dependent Distortion of Probabilities

Abstract
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.
Clemens Puppe

### Chapter 3. Risk Aversion

Abstract
This chapter addresses the question under which conditions an RDU maximizer can be said to display risk aversion. Two concepts of risk aversion will be considered here. The first concept defines an individual to be risk averse if the sure gain E(F) of the expectation of a distribution F is always preferred to the distribution itself. An alternative definition of risk aversion, suggested by Rothschild and Stiglitz [1970], requires a risk averse individual to prefer a distribution F to any mean preserving spread of F. Obviously, a risk averter in the second sense is also risk averse in the sense of the first definition. It is well-known that in expected utility theory both concepts of risk aversion are equivalent to the concavity of the v.Neumann-Morgenstern utility function. However, the equivalence of the two notions of risk aversion does not carry over to general non-expected utility theories.
Clemens Puppe

### Conclusion

Abstract
Rank-dependent utility (RDU) theory is a model of transitive preferences over probability distributions where the representation of preferences is based upon a (generalized) utility function defined on the outcome/probability-plane. Several specific forms of the generalized utility function have been examined in the literature, all of which imply multiplicative separability in the outcome/probability space. The most general model of this kind is the anticipated utility model firstly introduced in [Quiggin 1982]. The anticipated utility model has been proved very useful in explaining much of the empirical evidence against expected utility maximizing behavior.2
Clemens Puppe

### Backmatter

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