# More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model

## Abstract

This paper studies monotone risk aversion, the aversion to monotone, mean-preserving increase in risk (Quiggin [21]), in the Rank Dependent Expected Utility (RDEU) model. This model replaces expected utility by another functional, characterized by two functions, a utility function u in conjunction with a probability-perception function f. Monotone mean-preserving increases in risk are closely related to the notion of comparative dispersion introduced by Bickel and Lehmann [3,4] in Non-parametric Statistics. We present a characterization of the pairs (u,f) of monotone risk averse decision makers, based on an index of greediness G u of the utility function u and an index of pessimism P f of the probability perception function f: the decision maker is monotone risk averse if and only if $P_f\ge G_u$ . The index of greediness (non-concavity) of u is the supremum of $u^{\prime}(x)/u^{\prime}(y)$ taken over $y\leq x$ . The index of pessimism of f is the infimum of ${\frac{{1-f(v)}}{{1-v}}}/ {\frac{{f(v)}}{{v}}}$ taken over 0 > v > 1. Thus, $G_{u}\geq 1$ , with G u =1 iff u is concave. If $P_{f}\geq G_{u}$ then $P_{f}\geq 1$ , i.e., f is majorized by the identity function. Since P f =1 for Expected Utility maximizers, $P_{f}\geq G_{u}$ forces u to be concave in this case; thus, the characterization of risk aversion as $P_{f}\geq G_{u}$ is a direct generalization from EU to RDEU. A novel element is that concavity of u is not necessary. In fact, u must be concave only if P f =1. Copyright Springer-Verlag Berlin/Heidelberg 2005## Download Info

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## Bibliographic Info

Article provided by Springer in its journal Economic Theory.
**Volume (Year):** 25 (2005)

**Issue (Month):** 3 (04)

**Pages:** 649-667

**Handle:**RePEc:spr:joecth:v:25:y:2005:i:3:p:649-667

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Web page: http://link.springer.de/link/service/journals/00199/index.htm

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## Related research

**Keywords:**Risk aversion; Pessimism; Greediness; Rank-dependent expected utility.;

**Other versions of this item:**

- Chateauneuf, A. & Cohen, M. & Meilijson, I., 1997.
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**More Pessimism than Greediness: A Characterization of Monotone Risk Aversion in the Rank-Dependant Expected Utility Model**," Papiers d'Economie MathÃÂ©matique et Applications 97.53, UniversitÃ© PanthÃ©on-Sorbonne (Paris 1).

**Find related papers by JEL classification:**

**D81**- Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty**C60**- Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

## References

**References listed on IDEAS**

Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

- Quiggin John & Wakker Peter, 1994.
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**The Axiomatic Basis of Anticipated Utility: A Clarification**," Journal of Economic Theory, Elsevier, vol. 64(2), pages 486-499, December.

- Quiggin, J. & Wakker, P., 1992.
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**The Axiomatic Basis of Anticipated Utility : A Clarification**," Discussion Paper 1992-3, Tilburg University, Center for Economic Research. - Quiggin, J. & Wakker, P., 1992.
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**The Axiomatic Basis of Anticipated Utility: A Clarification**," Papers 9203, Tilburg - Center for Economic Research.

- Quiggin, J. & Wakker, P., 1992.
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- Quiggin, John, 1982.
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**A theory of anticipated utility**," Journal of Economic Behavior & Organization, Elsevier, vol. 3(4), pages 323-343, December. - Chateauneuf, Alain & Cohen, Michele & Meilijson, Isaac, 2004.
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**Four notions of mean-preserving increase in risk, risk attitudes and applications to the rank-dependent expected utility model**," Journal of Mathematical Economics, Elsevier, vol. 40(5), pages 547-571, August.

- Chateauneuf, Alain & Cohen, Michele, 1994.
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**Risk Seeking with Diminishing Marginal Utility in a Non-expected Utility Model**," Journal of Risk and Uncertainty, Springer, vol. 9(1), pages 77-91, July. - Yaari, Menahem E, 1987.
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**The Dual Theory of Choice under Risk**," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.

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