The Measure Representation: A Correction

نویسنده

  • UZI SEGAL
چکیده

Wakker (1991) and Puppe (1990) point out a mistake in theorem 1 in Segal (1989). This theorem deals with representing preference relations over lotteries by the measure of their epigraphs. An error in the theorem is that it gives wrong eonditions eoneerning the eontinuity of the measure. This artiele eorreets the error. Another problem is that the axioms do not imply that the measure is bounded; therefore, the measure representation applies only to subsets of the spaee of lotteries, although these subsets ean beeome arbitrarily close to the whole space of lotteries. Some additional axioms (Segal, 1989, 1990) implying that the measure is a produet measure (and hence anticipated utility) also guarantee that the measure is bounded. Quiggin's (1982) anticipated utility (or rank-dependent) model for decision making under uncertainty proved itself to be one of the most successful alternatives to expected utility theory. According to this model, the value of a lottery A" with a cumulative distribution function F is given by AU{X) = f u{x)df{F{x)), (1) where/: [0,1] -^ [0,1] isstrictly increasing, continuous, and onto.' One possible interpretation of this model is that the preference relation > over lotteries can be represented by a measure of the epigraphs of the lotteries' cumulative distribution functions, and moreover, that this measure is a product measure. That is, there are two increasing functions u (defined on the outcomes axis) and/(defined on the probabilities axis) such that the measure of the rectangle [x,y] x [p,q] is [u{y) u{x)][f{q) f{p)]. Indeed, let X = (x\,pi;... ;x,,,p,,) such thatxi < . . . < x,,. Then equation (1) is reduced to (2) i=] ; = () 7=0 where/70 = 0. If we assume w(0) = 0, then the above expression can be viewed as the sumof the measures of the rectangles [0,x/] x [S/IoP;, 2!y={)/';], each with the measure [ { d ( ) ] [ / 2 ; 2 ; ; d 2 *I am grateful to Peter Wakker and to C. Puppe for pointing out to me the mistake in my original paper and to Larry Epstein and Peter Wakker for helpful discussions.

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تاریخ انتشار 2005