An Axiomatization of Linear Cumulative Prospect Theory with Applications to Portfolio Selection and Insurance Demand

The present paper combines loss attitudes and linear utility by providing an axiomatic analysis of corresponding preferences in a cumulative prospect theory (CPT) framework. CPT is one of the most promising alternatives to expected utility theory since it incorporates loss aversion, and linear utility for money receives increasing attention since it is often concluded in empirical research, and employed in theoretical applications. Rabin (2000) emphasizes the importance of linear utility, and highlights loss aversion as an explanatory feature for the disparity of significant small-scale risk aversion and reasonable large-scale risk aversion. In a sense we derive a two-sided variant of Yaari s dual theory, i.e. nonlinear probability weights in the presence of linear utility. The first important difference is that utility may have a kink at the status quo, which allows for the exhibition of loss aversion. Also, we may have different probability weighting functions for gains than for losses. The central condition of our model is termed independence of common increments. The applications of our model to portfolio selection and insurance demand show that CPT with linear utility has more realistic implications than the dual theory since it implies only a weakened variant of plunging.

Diversification Preferences in the Theory of Choice

Diversification represents the idea of choosing variety over uniformity. Within the theory of choice, desirability of diversification is axiomatized as preference for a convex combination of choices that are equivalently ranked. This corresponds to the notion of risk aversion when one assumes the von-Neumann-Morgenstern expected utility model, but the equivalence fails to hold in other models. This paper studies axiomatizations of the concept of diversification and their relationship to the related notions of risk aversion and convex preferences within different choice theoretic models. Implications of these notions on portfolio choice are discussed. We cover model-independent diversification preferences, preferences within models of choice under risk, including expected utility theory and the more general rank-dependent expected utility theory, as well as models of choice under uncertainty axiomatized via Choquet expected utility theory. Remarks on interpretations of diversification preferences within models of behavioral choice are given in the conclusion

Axiomatic structure of k-additive capacities

In this paper we deal with the problem of axiomatizing the preference relations modelled through Choquet integral with respect to a $k$-additive capacity, i.e. whose Möbius transform vanishes for subsets of more than $k$ elements. Thus, $k$-additive capacities range from probability measures ($k=1$) to general capacities ($k=n$). The axiomatization is done in several steps, starting from symmetric 2-additive capacities, a case related to the Gini index, and finishing with general $k$-additive capacities. We put an emphasis on 2-additive capacities. Our axiomatization is done in the framework of social welfare, and complete previous results of Weymark, Gilboa and Ben Porath, and Gajdos.Axiomatic; Capacities; k-Additivity

Fuzzy measures and integrals in MCDA

This chapter aims at a unified presentation of various methods of MCDA based onfuzzy measures (capacity) and fuzzy integrals, essentially the Choquet andSugeno integral. A first section sets the position of the problem ofmulticriteria decision making, and describes the various possible scales ofmeasurement (difference, ratio, and ordinal). Then a whole section is devotedto each case in detail: after introducing necessary concepts, the methodologyis described, and the problem of the practical identification of fuzzy measuresis given. The important concept of interaction between criteria, central inthis chapter, is explained in details. It is shown how it leads to k-additivefuzzy measures. The case of bipolar scales leads to thegeneral model based on bi-capacities, encompassing usual models based oncapacities. A general definition of interaction for bipolar scales isintroduced. The case of ordinal scales leads to the use of Sugeno integral, andits symmetrized version when one considers symmetric ordinal scales. Apractical methodology for the identification of fuzzy measures in this contextis given. Lastly, we give a short description of some practical applications.Choquet integral; fuzzy measure; interaction; bi-capacities

Stronger Utility

Empirical research often requires a method how to convert a deterministic economic theory into an econometric model. A popular method is to add a random error term on the utility scale. This method, however, violates stochastic dominance. A modification of this method is proposed to avoid violations of dominance. The modified model compares favorably to other existing models in terms of goodness of fit to experimental data. The modified model can rationalize the preference reversal phenomenon. An intuitive axiomatic characterization of the modified model is provided. Important microeconomic concept of risk aversion is well-defined in the modified model.Decision Theory, Probabilistic Choice, Stochastic Dominance, Strong Utility, Risk Aversion

Utility Independence of Multiattribute Utility Theory is Equivalent to Standard Sequence Invariance of Conjoint Measurement

Utility independence is a central condition in multiattribute utility theory, where attributes of outcomes are aggregated in the context of risk. The aggregation of attributes in the absence of risk is studied in conjoint measurement. In conjoint measurement, standard sequences have been widely used to empirically measure and test utility functions, and to theoretically analyze them. This paper shows that utility independence and standard sequences are closely related: utility independence is equivalent to a standard sequence invariance condition when applied to risk. This simple relation between two widely used conditions in adjacent fields of research is surprising and useful. It facilitates the testing of utility independence because standard sequences are flexible and can avoid cancelation biases that affect direct tests of utility independence. Extensions of our results to nonexpected utility models can now be provided easily. We discuss applications to the measurement of quality-adjusted life-years (QALY) in the health domain