The Distorted Theory of Rank-Dependent Expected Utility

Abstract

This paper re-examines the rank-dependent expected utility theory. Firstly, we follow Quiggin's assumption (Quiggin 1982) that the transformation of cumulative distribution functions is a continuous function of the whole probability distribution to deduce the rank-dependent expected utility formula over lotteries and hence extend it to the case of general random variables. Secondly, we employ an axiomatic approach to build a distorted theory of rank-dependent expected utility. The transformation of the cumulative distribution functions is the decision weights function in the rank-dependent expected utility formula which reflects decision-makers' beliefs. We call it the distortion function which `distorts' the decision-maker's probabilities. Utilizing this distortion function, we propose a distorted independence axiom to prove the representation theorem of rank-dependent expected utility. Finally, we make direct use of the distorted independence axiom to explain the Allais paradox and the common ratio effect. We show that the inconsistency in these examples arises when the distortion function takes specific forms under the framework of the distorted independence axiom.

Keywords: Expected utility, Rank Dependent Expected, Distortion function, Distorted independence axiom, The Allais paradox, The common ratio effect.

JEL Classification: D81