Journal of Risk and Uncertainty, 6: 99-107 (1993)
© 1993 Kluwer Academic Publishers
UZI SEGAL*
Department of Eeonomies. University of Toronto, 150 St. Geoige Street. Toronto. Ontario M5S IA /, Canada
Keywords: 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 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.

100 UZI SEGAL
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 satisty 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 =
A{SO,SM} (8.V is the degenerate lottery yielding^ with probability 1). For every A' £ L, define the cumulative distribution function f x by fA'(-*^) = Vx{X<x).¥ors > 0, letL., =
{X^L: forx E [0, s), Fxix) < \ - s}. Note that if ^ < s', then !,• C L,. For5 > 0^ let
Q, be the square [0, s) x (1 -^,1]. Let D = [0, M] x [0,1], D = D\{(0,1), (M, 1)}, and
Ds = i>Qs. F o r ; f G L , let A " = C\{{{x,p) &D:p> Fx{x)}).
Let L° be the family of all the nonempty closed sets 5 in D satisfying [{x,p) £ S, 0 < _y
< x,p < q < 1] => {y^) G 5. Obviously, for every S E L° there is a unique lottery A' G
L such that A'° = S. The cumulative distribution function of this lottery is given by Fxix)
= min{p:{x,p) G S}. Denote this lottery A'by 5 +.
LetL* be the set of all finite lotteries A" in L of the form ( x i , p i ; . . . •,x,,,pn) and let A
= {[x^] X [p,q] C b:x < y,p < q}. Obviously, ifA' G L*, then A" can be represented as a finite union of elements of A.^ Let ^ be a complete and transitive preference relation overL. Define the relations > and ~ by A" > y if and only if A" ^ ybut not Y > X, and
X~ yifandonlyifA"^ Yand Y > A". LetL C L. We say that the function K: £ ^ 9i represents the preference relation > o n L if for all A',y G L, V{X) > V{Y)ti'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, let A',y,y 1,72, ...EL such that at each continuity pointx of FY, FY.{X) -^ FY{X).
If, for every i, X > y,, then A" > y. If, for every /, y, fc X, then

THE MEASURE REPRESENTATION: A CORRECTION 101
Axiom 2: First-order stochastic dominance. If, for every x, Fx(x) < FY{X) and there existsxsuch i\ia\.Fx{x) < FY{X), then A" > Y.
Axiom 3: Irrelevance. Let A',y, A",y' G L and let 5 be a finite union of segments in
[0, A/]. If on S, Fx(x) = FY{X) and Fx' (x) = F r (x), and on [0, Af| \ S, Fx{x) = Fx' {x)
= Fr{x),t\\Qx\X > yif and only ifA" > Y'.
Definition. A curve C C D is the continuous image of a function/: [0,1] -^ D. The curve C is increasing if {xp) G 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,/j) e A ° : y > x,q < p^
(y,q) e A^}forallA'G L.
Let -9 be a countably additive measure on D such that for every s > 0, Qs Pi D is a measurable set. For s > 0, define the measure ^g on D as follows: For every
^-measurable set S C D,^j(5) =
Theorem 1. The 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 ^ on D satisfying
(a) For S = [a, b] x [p^] C D such that a < b and/? < q,0 < ^(S) < oo;
(b) If C C D is an increasing curve, then i)(C) = 0; and
(c) The preference relation > on Ly can be represented by Vs{X) =
3. There is a measure -9 as in condition 2 satisfying (a), (b), and
( c ' ) F o r e v e r y A ' , y G L , A ' > yif and only if ^ ( A ° \ r ) >
Proof Condition 2 <» condition 3: LetA',y G L. By definition, there exists e > Osuch that
Fx{0), Fy(O) < 1 - 6. Since cumulative distribution functions are continuous from the right, there ise' > Osuch that forz < t',Fx(z),FY(Z) < 1 - e. Defines = min{e,e'}and obtain thatA',y G L,; hence Qs C A° n r . It follows that ^ ( A ^ \ r ) > ^ ( ^ \A°) if and only if ^X°\Q,) > Q{Y°\Q,) if and only if ^,(A") > fi.iY'). (Note that
Condition 2 =» condition 1: Let X,, -^ X. It follows by the first-order stochastic dominance axiom that the condition in the continuity assumption is trivially satisfied if A' E
{8(),8M} (although 8{),8M ^ L). Assume therefore that there exists^ > 0 such that A" G
Ls. Without loss ofgenerality, we may assume that for every AT. A',, G L,. To show that the order > is continuous, one has to prove that V{X,,) - V{X) -^ 0. Let S,, be the symmetric difference between A','; and A°, S,, = (X° U A°)\(A'° n A°), and let T,, = U-^,, 5/. Note that ^(5,,) = ^siSn) ^ ^si^ U A-") < 00. Since ViX) = •9,s(A"), it follows that | (/(A",,)
- V(X) I < f)siS,,) = ^(S,,) < ^(7,,). LetA'be the southeast boundary of A°, that is. A'
= {{x, p) E ?C : y > X, q < p ^ (y, q) ^ X°}. As mentioned above, X is an increasing curve; hence ^(.k) = 0. Moreover, r\^^^T„ C X. Otherwise, let {x,p) E (n,f^,r,,)\Y.
Since {x,p) £ A', either/? > Fx{x) or/7 < Vmy^.yFxiy). We assumed that (x,p) E

102 UZI SEGAL l~l,r=i^" ~ ^n = \ U,^,,S,; hence there is a subsequence {A",,} such that for every;, (x,/?)
G S,,.. If/? > Fx{x), then {x,p) E X°. Therefore by definition, for every), (x,/?) ^ A7^
Hence, for every), limy^v- Fx,, (y) > p. Since cumulative distribution functions are continuous from the right and increasing, it follows that there exists e > 0 such that for ally G [x,x + e),
^ ^ ^ p + Fx{x)
Since there must be a continuity point ofFx in [x,x + e), it follows that A",, -A A'. If/? < limy_.(- Fxiy), then {x,p) £ X°. Therefore for every;, {x,p) E A7,and limy_.v- Fx,,^ (y)
< /?. As before, it follows that there exists e > 0 such that fory G (x - e, x].
Here too, since there must be a continuity point oiFx in (x - e,x], it follows thatA',, -y^
X. Since n,f^ ,7,, C A'and ^(A^ = 0, it follows that lim ^{T.,) = 0 (see Royden, 1963, p.
192). First-order stochastic dominance follows by condition 2(a), and the irrelevance condition follows by the fact that > on Ls can be represented by a measure.
Condition 1 ^ condition 2: Let ^ = {{X,\) E L x A : Int(A°)n Int(\) = 0 e L°}. The irrelevance axiom implies that if (A',\), {Y,X) E ^ , then A" > yif and only if
(A° U \ ) + ^ ( r U \ ) + . Indeed, for X = [x,y] x [p,q] E A, let 5 = (x,y]. Since (A',\)
(y, \ ) G ^ , it follows that for every z G 5, Fx(z) = FY{Z) = q. Also, for every z E S,
F{x°u\y (z) = F(ru\)+ (z) =/?. Of course, forz ^ (x,y],FA'(z) = F(x^uky (z) and FY{Z)
Define on A a partial order RxbyX\ Rx \2 if and only if (A", Xi), (A", X2) G ^ and (A°
U Xi)""" > (A°U X2) "*". By the irrelevance axiom, we obtain
Fact 1. For every A"] and X2, Rx, and Rx^ do not contradict each other. That is, if X1 and X2can be compared by both/?A'| andRx., then X; Rx, X2 if and only if Xi Rx, X2-
(To see why fact 1 follows from the irrelevance axiom, let X, = [x/,y,] x [/?,, q,], i = 1,2 and defines = (xi,yi] U (X2,y2]).
Let/? = U A'/?A'-That is, X1 R\2 if and only if there exists A'such that Xi Rx^2- Define
X1 / X2 if and only if X ] R X2 and X2 /? X1.
LetXi = [x,,y,] x [pi,qi],i = 1,2. Obviously, Xi and X2 can be compared by/? if and only if eitheryi ^X2and(7i </?2, ory2 < X | a n d ^ 2 ^ / ? i . It thus follows that for every
X|, X2, X3 such that any two of them can be compared by R, there is a lottery A'such that
(A', X,) G ^ , / = 1, 2, 3. Therefore we obtain
Fact 2. If X| / X2, X2 / X3, and Xi and X3 can be compared by R, then X| /X3.
Letxi = 0, X4 = M, andpk = ^ ^ , k = 1 , . . . , 4. By the continuity and first-order stochastic dominance axioms, there are 0 < X2 < X3 < A/such that ([x/t,Xi + i] x [p/;, p/t + i])/([xf,xc+i] X [pc,pe+\]),k,C E {1,2,3}. Define the strictly increasing sequences
>/•••',;• = 0 , . . . , 2'; /• = 0 , . . . , 00; A: = 1, 2,3, such that

THE MEASURE REPRESENTATION: A CORRECTION 103
2. y^ = yj^'-\j = 0 , . . . , 2 ' - ' ; ( = 1 , . . . , ^;/c = 1, 2, 3;
3. ( [ } j ' ' , } j U X [pk,Pk + i])l{[yy,y^-'+^] X [pe,P(+^]), j J ' = 0 , . . . , 2 ' - 1 ; / =
The only nontrivial requirement is condition 3. By the choice ofx2 andx^, this condition is satisfied for the case / = 0. Suppose j ' | ' , y = 0 , . . . , 2'; / = 0 , . . . , /O; k = ],2,3, satisfy the above three conditions, and construct;/''"•*"',) = 0 , . . . , 2'»+'; /c = 1,2,3, as follows:
Fory = 2m, let j ^ ' " ' " * " ' = jA;'", m = 0,..., 2'"; A: = 1, 2, 3. By the continuity and first-order stochastic dominance assumptions, there are 3^'""''' E {xk,y^"),k = 1,2,3, such that X,,, I • • • I \,,, +1, AT? = 1 , . . . , 4, where y ' ' k = 1,2,3
-3,Pk-2] k= 4,5
By fact 2 it follows that \ i / X.i, X3 / X5, X 1 / X5, and X2 / X4. Also, X5 / Xf, = [y,'""*' \yl'"]
X [/73,/74]. This follows by
' 3'-^' ' 3' -^' 3/
Of course, X,,, I \(,,m = 1,2,4. One can now use X| to define}^'""*"' G {y/-\ ,yj'+\ ),j =
3,5,...,2'"+1 - 1;;t = 2,3,andX3 todefineyM"+' G {y]^,y]j\),j = 3,5,...,2'"+' - 1.
2 2
Condition 3 is clearly satisfied. Moreover, by the definition of / and by the first-order stochastic dominance axiom, for a given /,
J, ' 3'-0: ' 3'-^J> ' 3 / ~ vh' 3'^h' 3'^h' 2,
^i\ + il + 73 ^ i\ + i'l + jy (3)
The next step in the proof is to show that the sequence {}j''},j = 0,... ,2';i = 0,...,
00 is dense in [x*., Xi +1], k = 1, 2, 3. Suppose, for example, that there are no values of
{y-•'} in (a,3) and assume that (a,P) is maximal in that sense. There is a sequence {y/},^() such that70 = O,ji G {2;,_,, 2y,_, + 1} andyj.-^ ^ - yj:' < (^3 - xi) • 2''. By Cantor's lemma, {>'?"'},1() and {v""| i}-'^,, have a common limit; denote ity-. Let m, satisfyy,',;' < a

104 UZI SEGAL
<y,';,;^,,/ = 1 , . . . , oo. By construction, (y;,;;,j;y2-;,,l;x3,^) ~ (y';,;+,, i;}'f, J;A:3,
\). By letting / approach oo, one obtains (a, j ; y^, i ; X3, | ) ~ ((3, ^; y^, j ; X3, ^ ) , a violation of the first-order stochastic dominance axiom.
Define'9([>^-',yJ-^,] x [pk,Pk + i]) = 2-',J = 0 , . . . , 2' - 1;/ = 0,...,^;k = 1,2,
3. F o r x G [xk,Xk + \), let;,(x) be such thaty^J;' <x < y^^.^ + y Define tp^': [A:A,Xi +1] ^ Sft by cp* (x) = lim/^ x ji{x) •2''',k = 1,2,3. By the above argument, 9* is strictly increasing.
It is also continuous. Letz,, go down t o z G [xk,Xk+\). For every / there exists« such that
A similar proof holds for the case where 2,, goes up to 2 G (xk, A + i]. It follows by continuity from inequality (3) that fory*,2*^ G [XA,XA + I ] , A^ = 1,2,3,
3 ' ^ ' 3 ' - ^ " ' 3
~ V ' 3 ' ^ ' 3 ' ^ ' 3 /
L e t a ( l ) = 2, o-(2) = 3, and a(3) = 1. By continuity and first-order stochastic dominance, it follows that for every X C X* = [XA.,x^+i] x [pk,Pk+\], there isy G (x^(/.),
x,,(k)+i] such that X / ([x,,(^k),y] x [p,r(A),/J<,(A:)+1]). Define
^ (X) = (p"<'') (y) > 0.
Notethatbyfact2,X/([x,,-i(A),z] x [/?a-i(^),p,,-i(A) + i]), where ip'^**) (y) = (p-^"'**) (2).
T h e set-function ^ satisfies the following condition:
Claim 1. LetX i,X2 C X''. IfXi U X2G A and Int (X]) n Int (X2) = 0, t h e n ^ ( X i U
Proof. Let A' = (X,J(A), J ; X , , - | ( A ) , ^ ; 2 | , ^ I ; . . . ;2,,,^,,) G L such that 2 1 , . . . ,2,, G [XA,
XAM-I] and(A', X|), (A', X| U X.2) G 4^ (assume, without loss ofgenerality, that X| is either above or to the left of X2). It follows by relation (4) and fact 1 that
(A^U Xi U X2) +
~ (A- U X, U ([x^-.(A.),[(p"-'(*)]-'

THE MEASURE REPRESENTATION: A CORRECTION 105
~ (A° U
~ (A° U ( M ) , [ c p - W ] - ' (^(X|) + ^(X,))] X hence the claim. Q
Define il»* : [x;t,A+i] x [pk,Pk+i] -^ 9iby v);'^ (x,p) = -9 ([XA,X] X [/7,/J^t+i]). The function i|i* satisfies the following condition:
Claim 2. For every XA- < X < y < XA:+ l andpA ^ q < P ^ Pk+\, i|i* (x,p) + ili'^ (y,^) - ^'^ (x,^) - i|<^- {y,p) > 0.
Proof. It follows by claim 1 that ^'^ {x,p) + (]i* (y, q) - i];''' (x, q) - i|j''' (y,p) = -9 ([x,y] x b,^])>o. D
By the continuity of > , it follows that i[)^ is continuous. Therefore, {) can be uniquely extended to a countable additive measure on X^ (see Billingsley, 1979, section 12). Moreover, it follows by relation (4) that the order > on D' = {A' = {x\,p\;... ;x.,,p,,) E L : forxG [xk,Xk.^-]),pk ^ Fxix) <pk+\,k = 1,2,3}can be represented by ^ (A^(xi, ^;x2.
•j;x3, |)°). Also, by the continuity of ^ , if C C D' is an increasing curve, then ^ (C) = 0.
The next step in the proof is to extend -0 to D. It follows by continuity and first-order stochastic dominance that for every X3 <y\ < y2 < M a n d | < q\ < qi < \ such that X
= (t}'i.>'2] X [q\,q2])I {[y2,M] x [^2,1]) there is a finite sequence of probabilities j = ri < . . . <r,, <qri such that X/([0,X3] x [/-,,/-,4-i])/?(I0,X3] x [z-,,,^ , ] ) , / = \,...,n-
1.'' Suppose instead that the sequence {/•,} is not finite and let lim r, = r < q\. For every /,
(0,1 - /•,;x3,<7i - rr,yi,q2 - q\;y2,1 - 92)
~ (0,1 - r,+ i;x3,gi - r,;y2,1 - q\)
~ (0,1 -
As / approaches infinity, we obtain that
(0,1 -
~ (0, 1 - /-;x3,^i - ri;y2,q2 - q\;M,\- (72), a contradiction. The measure ^ ean thus be extended to ([0,X2] x [•^, | ] ) U ([0,X3] x
[^ , ^ 1 ]). Since q \ can be taken arbitrarily close to 1, we obtain that the measure ^ ean be extended to ([0,X2] x [1, f]) U ([0,X3] x [f, 1)).
In a similarway, the measured can be extended also to ((0,X2] x [|, 1]) U ([x2,X3] x
( | , 1]). For this, we use as benchmark the sets ([0,y3] x [0, ^3]) / (iy3,y4] x [^3,^4]), where 0 < y3 < y4 < X2 and 0 < q^ < q4 < | . (Both these sets are therefore in X'.) In asimilarway, we can extend the measure to ([x2,X3] x [0, j ] ) U ([X3, 1) x [0, | ] ) a n d t o
([X2, 1] X (0, \]) U ([X3, 1] X [^, | ] ) . The measure d is thus extended to D. It may happen that the measure {) does not represent the order > on L* because it may be

106 UZI SEGAL unbounded (see Wakker, 1991, for an example). Nevertheless, by its construction, -9, and hence ^s, is bounded on Ds. The proof that •Qs represents the order on L* (the set of finite lotteries in Ls) is similar to the proof that •Q represents the order for lotteries A'such that X C D', and so is the proof that if C C D is an increasing curve then {) (C) = 0. The extension for Ls follows by continuity. Q.E.D.
The rank-dependent (or anticipated utility) functional is a product measure. Theorem
2 in Segal (1989) and theorem 9 in Segal (1990) prove that under some further conditions, the measure i) is a product measure. These proofs implicitly assume that ii on D is bounded. Although this is no longer true, the theorems still hold. To see this, observe that ^ is bounded on [e, M - e] x [0,1 ] and on [0, A/] x [e, 1 - e] for all e. Let L^ = {A'
G L : lim;,._>e Fxix) = 0, Fx (M - e ) = 1}, and let L* = {X E L : FxiO) = e, lim;^.^^
= 1 - e}. The above-mentioned theorems thus imply the existence of functions/^
[e, 1 - e]^% and w * : [0, Af] ^ % each of them unique up to positive linear transformations, such that the order > on L^ can be represented by Jj ^ u^dff^iFxix)) and the order > on L* can be represented by J^ u*
x)). For every e > 0 and for every e G (0, e], all these representations cardinally coincide on Li H L j . Hence, without loss if generality, for every/; G [0,1] and e, e' > 0,
ftiP) = ft'ip) '• = fip)- Similarly, for everyx G [0, M] and e,e' > 0, w|(x): = w*'(x): = w(x). Also, \fp G (i, 1 - i) and Ve G (0, l],ftip) = fip) and Vx G (i, M - i) and Ve E
(0, i], Wt(x) = M(X). It follows by the continuity axiom that the order > on L can be
Notes
1. Quiggin's axioms imply/(^) = 5. Yaari (1987) assumes linear utility function 11. The above general form of the rank-dependent model first appeared in Segal (1984).
2. Recently. Tversky and Kahneman (1991) suggested a more general form of this functional, where decision makers use two different distribution transformation functions (for positive and negative outcomes). This too is a special case of the general measure representation.
3. This representation is of course not unique. For A" = (AI./;I; . . . ;.v,,,/;,,) G L*. letpd = O,.V(i = 0, and obtain A-= u;.'^,([O,.v,] x (E;;,lpy, 2;.,,ft|) = u;.'^_,([.v,-,,.v,l x |X;;,',/;y. 1|).
4. Note that (|yi,>'2l x Vl\,'i2\),(b'2,M] x \q2, 1]) C \ \
References
Billingsley, P. (1979). Probability and Measure. New York: John Wiley and Sons.
Puppe, C. (1990). "The Irrelevance Axiom, Relative Utility and Choice Under Risk," Department of Statistics and Mathematical Economics, University of Karlsruhe, Karlsruhe, Germany.
Quiggin, J. (1982). "A Theory of Anticipated Utility," Joumal of Eeonomie Behavior and Organization 3,
323-343.
Royden, H.L. (1963). Real Analysis. New York: MacMillan.
Segal, U. (1984). "Nonlinear Decision Weights with the Independence Axiom," UCLA Working Paper #353.
Segal, U. (1989). "Anticipated Utility: A Measure Representation Approach,"/l«/i«/.s- of Operation Researeh
19,359-373.

THE MEASURE REPRESENTATION: A CORRECTION 107
Segal. U. (1990). "Two-Stage Lotteries Without the Reduction Axiom," Econometrica 58, 349-377.
Tversky, A. and D. Kahneman. (1991). "Cumulative Prospect Theory: An Analysis of Decision Under Uncertainty," mimeo.
Wakker, P. (1991). "Counterexamples to Segal's Measure Representation Theorem," Joumat of Risk and
Uncertainty 6,9]-98.
Yaari, M.E. (1987). "The Dual Theory of Choice Under Risk," Econometrica 55, 95-115.