Distorted Probabilities and Choice under Risk

The book develops a new model of individual choice under risk within the general framework of rank-dependent utility theory. The new approach captures simultaneously an optimistic attitude towards gambling and risk averse behavior. The model is then applied to some of the observed behavioral patterns contradicting the expected utility hypothesis.

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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.

Contenu
1 Axiomatic Utility Theory under Risk.- 1.1 Historical Overview.- 1.2 The Axiomatic Basis of Expected Utility Theory.- 1.3 The Empirical Evidence against the Independence Axiom.- 1.4 Non-Linear Utility Theory under Risk.- 2 A Rank-Dependent Utility Model with Prize-Dependent Distortion of Probabilities.- 2.1 Rank-Dependent Utility Theory Reconsidered.- 2.2 Homogeneity on Elementary Lotteries.- 2.3 Further Evidence for Prize-Dependent Distortions of Probabilities.- 2.4 A Characterization Theorem.- 2.5 Rank-Dependent Utility Theory and Relative Utility.- 2.6 A Generalized Model.- 3 Risk Aversion.- 3.1 Risk Aversion in the General Rank-Dependent Utility Model.- 3.2 Risk Aversion and Homogeneity.- 3.3 Decreasing Risk Aversion.- 3.4 The Friedman-Savage Hypothesis.- Conclusion.- References.