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.- 1.4.1 Weighted Linear Utility Theory.- 1.4.2 Theories with the Betweenness Property.- 1.4.3 Anticipated Utility Theory.- 1.4.4 The Dual Theory.- 1.4.5 The General Rank-Dependent Utility Model.- 1.4.6 Implicit Rank-Linear Utility Theory.- 2 A Rank-Dependent Utility Model with Prize-Dependent Distortion of Probabilities.- 2.1 Rank-Dependent Utility Theory Reconsidered.- 2.1.1 The Generalized Utility Function.- 2.1.2 Absolute Continuity.- 2.1.3 The Set of Elementary Lotteries.- 2.2 Homogeneity on Elementary Lotteries.- 2.2.1 The Common Ratio Effect.- 2.2.2 The Allais Paradox.- 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.
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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.
