The Measure Representation: A Correction. UZI SEGAL*. Department of Economics, University of Toronto, 150 St. George Street, Toronto, Ontario M5S 1A1, ...
Journal of Risk and Uncertainty, 6:99-107 (1993) 9 1993 Kluwer Academic Publishers
The Measure Representation: A Correction UZI SEGAL*
Department of Economics, University of Toronto, 150 St. George Street, Toronto, Ontario M5S 1A1, Canada Key words: anticipated utility, rank-dependent probabilities, measure representation
Abstract 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 conditions concerning the continuity of the measure. This article corrects 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 space of lotteries, although these subsets can become arbitrarily close to the whole space of lotteries. Some additional axioms (Segal, 1989, 1990) implying that the measure is a product 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 X with a cumulative distribution function F is given by
AU(X) = f u(x)df(F(x)),
(1)
where f : [0,1] ~ [0,1] is strictly increasing, continuous, and onto. 1 O n e 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) a n d f (defined on the probabilities axis) such that the measure of the rectangle Ix,y] • [p,q] is [u(y) - u(x)][f(q) - f(p)]. Indeed, let X = (xl,Pl; 9 9 9 ;Xn,Pn) such thatxl -< . . . -< xn. Then equation (1) is reduced to n
i
i-I
AU(X) = Z u(xi)[f(~__flj) - f(ZPJ)] i=1
j=0
(2)
j-0
w h e r e p 0 = 0. If we assume u(0) = 0, then the above expression can be viewed as the i-1 t sum of the measures of the rectangles [0,xi] x [ ~ = 0 P j , ~}=oPj], each with the measure i
i-1
[u(xi) - u(O)l[f(Zj=opj) - f(~,)=oPj)].
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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A natural extension of this model is to represent the preference relation >~ on lotteries by a general (not necessarily product) measure of the lotteries' epigraphs. This functional is suggested and axiomatized in Segal (1984, 1989). It turns out, however, that there are some mistakes in these papers (see Wakker, 1991; Puppe, 1990) concerning the questions what sets have zero measure, and what sets that have zero Lebesgue measure must also have zero measure according to the representation functional. The aim of this article is to answer these concerns. It turns out that lines that can serve as the lower boundary of the epigraph of a lottery (i.e., lines that can be created by connecting up pieces of the graph of a cumulative distribution function) are the only sets that must have zero measure. This result is quite natural--if such a set has positive measure, then the order does not satisfy continuity. However, other lines may have positive measure. In particular, the down-sloping line connecting the points (0,1) and (1,0) may have a positive measure (see Wakker, 1991, for examples). Another issue is whether the representation measure may go to infinity. This leads to the conclusion that the measure representation applies only to subsets of the space of lotteries, although these subsets can become arbitrarily close to the whole space of lotteries. Some additional axioms (Segal, 1989, 1990), implying that the measure is a product measure (and hence anticipated utility), also guarantee that the measure is bounded.
1. Axioms and theorem
Let L be the family of all the real random variables with outcomes in [0, M] and let L = L\{g0,gM}. (gx is the degenerate lottery yieldingx with probability 1). For everyX ~ L, define the cumulative distribution function Fx by Fx(x) = Pr(X _< x). For s > 0, let Ls = {X ~ L : forx E [0, s), Fx(x) -< 1 - s}. Note that ifs < s', then Ls' CC_Ls. For s > 0, let Qs be the square [0, s) x (1 -s,1]. Let D = [0, M] x [0, 1],/) = D\{(0, 1), (M, 1)}, and Ds = D'xQs. F o r X E L, letX ~ = Cl({(x,p) ~ D :p > Fx(x)}). Let L ~be the family of all the nonempty closed sets S in D satisfying [(x,p) E S, 0 _ Yif and only ifX >~ Ybut not Y >~ X, and X - Yif and only i f X >~ Yand Y >~ X. L e t L ___L. We say that the function V: L --~ ~t represents the preference relation >~ on L if for all X , Y E L, V(X) >_ V(Y) ~ X >~ Y. Consider the following three axioms: Axiom 1: Continuity. The preference relation ~ on L is continuous in the topology of
weak convergence. That is, letX, Y, Y1,Y2 . . . . E L such that at each continuity pointx of Fy, F~(x) --~ Fy(x). If, for every i, X ~ Yi, then X ~ Y. If, for every i, ~ ~ X, then Y~X.
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Axiom 2: First-order stochastic dominance. If, for every x, Fx(x) Y. Axiom 3: Irrelevance. Let X,Y, X',Y' ~ L and let S be a finite union of segments in [0, M]. If on S, Fx(x) = Fr(x) and Fx, (x) = Fy' (x), and on [0, M] \ S, Fx(x) = Fx, (x) a n d F y ( x ) = Fr,(x), t h e n X >~ Y i f a n d only ifX' ~ Y'. Definition. A curve C C D is the continuous image of a f u n c t i o n f : [0,1] ~ D. T h e curve C is increasing if (x~) E C ~ C N {(y, q) : y < x, q > p} = 0. Note that a point in D is an increasing curve, as is the set {(x,p) ~ X ~ :y > x, q < p (y, q) C X ~} for all X ~ L. Let 0 be a countably additive measure o n / ) such that for every s > 0, Qs n D is a measurable set. For s > 0, define the measure a3s on D as follows: For every a%measurable set S C D, a3,(S) = ~(S\Qs). Theorem 1. T h e following three conditions are equivalent: 1. The preference relation >~ on L satisfies the continuity, first-order stochastic dominance, and irrelevance axioms. 2. There is a (countably) additive measure O o n / ) satisfying (a) F o r S = [a,b] x [p,q] C / ) such t h a t a < b a n d p < q, 0 < O(S) < ~ ; (b) If C C / ) is an increasing curve, then O(C) = 0; and (c) T h e preference relation >~ on Ls can be represented by Vs(X) = Os(lC). 3. There is a measure 0 as in condition 2 satisfying (a), (b), and (c') For every X,Y ~ L, x >_ Y if and only if O(X ~ \ Y') _> O(Y ~\X~).
Proof Condition 2 condition 3: LetX, Y ~ L. By definition, there exists e > 0 such that Fx(O), Fy(0) < 1 - e. Since cumulative distribution functions are continuous from the right, there is e' > 0 such that forz _< e', Fx(z), Fy(z) < 1 - e. Define s = min{e,e'} and obtain thatX, Y E Ls; hence Qs _c X ~ n Y~. It follows that O()C \ Y') >_ O(Y~\X ~) if and only if O()C\Qs) >- O(Y~\Qs) if and only if Os(X ~ _> Os(Y~). (Note that )C\Qs =
(X~ \ ~) U ([X~ n 1r \ Qs)). Condition 2 ~ condition 1: Let Xn ~ X. It follows by the first-order stochastic dominance axiom that the condition in the continuity assumption is trivially satisfied i f X E {80,8M} (although 80,8M ~ L). Assume therefore that there exists s > 0 such that X E Ls. Without loss of generality, we may assume that for every n,Xn ~ Ls. To show that the order ~ is continuous, one has to prove that V(Xn) - V(X) ~ O. Let Sn be the s y m m e t r i c difference between X~ and X ~ Sn = (X~ U JC)\(Xn G X~), and let Tn = U i=n ~ Si. Note that O(Sn) = a3s(Sn) -< Os(X ~ U X~) < ~ . Since V(X) = a3s(X~), it follows that [ V(Xn) - V(X)[ _< a3s(Sn) = 0 (Sn) < O(Tn). L e t j 2 b e the southeast boundary o f X ~ that is, J2 = {(x,p) ~ JC :y > x, q < p ~ (y, q) ~ X~}. As mentioned above, J2 is an increasing curve; hence ~(j2) = 0. Moreover, nn~=lrn C 2 . Otherwise, let (x,p) E ( n ~ = l T n ) ~ ( . Since (x,p) ~ J2, e i t h e r p > Fx(x) o r p < limy--,x-Fx(y). We assumed that (x,p)
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n,,\lT,, = Cl n~ = 1 U i=n ~ Si; hence there is a subsequence {Xn/} such that for everyj, (x,p) E Snj. I f p > Fx(x), then (xg~) C X ~ Therefore by definition, for every j, (x, p) ~ ~n~ Hence, for every j, limy-~x- Fx,,~ (y) > p. Since cumulative distribution functions are continuous from the right and increasing, it follows that there exists r > 0 such that for ally @ [x,x + e),
Fx,,, (y) > p >
p + Fx(x) 2 > Fx(y).
Since there must be a continuity point of F x in Ix, x + e), it follows that X~ ~ X. Ifp < limy~x- Fx(y), then (x,p) ~ X ~. Therefore for every j, (x,p) ~ )~]and limy--,x- Fx,,, (y) < p. As before, it follows that there exists e > 0 such that forT ~ ( x - e , x ] ,
Fx,,, (y)
