Annals of Operations Research 19(1989)359-373
359
ANTICIPATED UTILITY: A MEASURE REPRESENTATION
APPROACH*
Uzi SEGAL
Department of Economics, University of Toronto, 150 St. George Street, Toronto,
Ontario M5S lAl, Ganada
Abstract
This paper presents axioms which imply that a preference relation over lotteries
can be represented by a measure of the area above the distribution function of each
lottery. A special case of this family is the anticipated utility functional. One additional axiom implies this theory. This functional is then extended for the case of
vectorial prizes.
1.
Introduction
The optima! choice among a set of uncertain alternatives has been discussed
since the early 18th century. Bernoulli [3] suggested that people prefer the lottery
yielding the greatest expected utility rather than the lottery with the highest expected
value. According to this theory, the value of the lottery {x^,p^]. . . ;x^, p^), which
yieids x^ dollars with probabihty p-, i = 1, . . . ,«, is Xp-uix-). Von Neumann and
Morgenstern [19] first presented a formal set of axioms implying maximization of
expected utihty. Some experimental evidence shows, however, that people do not
always behave in accordance with this theory (fora survey of these, see Machina [10]),
thus challenging the descriptive value of this theory.
Recent years saw the emergence of some alternatives to expected utility
theory. One of the most efficient of these new theories is anticipated utUity (also
known as expected utility with rank-dependent probabilities), first presented by
Quiggin [11]. According to this theory, there is an increasing utility function u,
unique up to positive linear transformations.and a probabilities transformation function
/ , such that the value of the lottery (;c,, p , ; . . . ;x^,p^), where Xj < . . . < x^ is
The author is grateful to Chew Soo Hong, Larry Epstein, Joe Ostroy, Joel Sobel, Peter Wakker,
and Menahem Yaari for their comments.
360
U. Segal, Anticipated utility
1=2
/
\
/J
where /(O) = 0 and /(I) = 1. WithoutlossofgeneraHty,onemay assume that w(0) = 0.
Note that when/(;?) = p, this function is reduced to the expected utility representation
function. This theory claims that for every prize x^, the decision maker is interested
not only in the winning probability of this prize, but also in the probability that he
will win more than x-. This functional is not Frechet differetitiable and therefore does
not satisfy Machina's [9] conditions. However, Chew et al. [5] proved that Machina's
results hold true even when the functional is only Gateaux differentiabie. Moreover,
they showed that the anticipated utility functional is Gateaux differentiabie.
This functional is very useful in the unravelling of some of the most famous
paradoxes in uncertainty theory. The AUais paradox is solved in Quiggin [11] and
Sega! [16], the common ratio effect in Segal [16], and the preference reversal
phenomenon in Karni and Safra [8]. It is also useful in analyzing the Ellsberg paradox
(Segal [17]) and the probabilistic insurance problem (Segal [18]). So far, this theory
has received several axiomatizations. Quiggin suggested a set of axioms, necessarily
implying that /(1/2) = / ( l ) / 2 . However, we now know that in this theory the convexity of / is essential for risk aversion behavior (see Chew et al. [5]). Another set
of axioms was suggested by Yaari [20], but his assumptions restrict the utility function u to be linear. For a further discussion of Quiggin and Yaari's axioms, see
Roell [13]. Recently, Chew and Epstein [4] discussed axiomatizations of nonexpected
utility theories, including anticipated utihty.
One possible way to represent preference relations over lotteries is by a measure
of the epigraphs of their cumulative distribution functions. It turns out that expected
value, expected utility, and rank-dependent probabilities are all special cases of this
geuera! family. The first part of this paper, therefore, presents a set of axioms implying
such a general representation, which we suggest to call the measure representation
(theorem 1). It follows that product measures are of a special interest, where the most
general fonn is the anticipated utility functional. One additional axiom implies indeed
that the general measure is a product measure (theorem 2). This functional is then
extended to the case of non-money prizes (section 3).
2.
Representation of I:
Let L be the family of all the real random variables with outcomes in [0,/V/].For
every X E L, define the cumulative distribution function F^^ by /^x(-^') ~ fr(A' < x).
U. Segal, Anticipated utility
Let A'" = Q\{{x,p)&
fig. 1).
361
R^x [0, I] : p > F ^ ( x ) L w h e r e C\A is the closure of A (see
Fig. I.
Let Z,*^ be the family of al! the non-empty closed sets
satisfy the following conditions;
(1)
(2)
X°,Q<y
<x,3ndp
<q<
° in R^ x [0, 1] that
I, then (>',
Forx
(3)
Obviously, there is a one-to-one correspondence between L and L^.
Let L* be the set of all the elements of L for which the range of Fy is
finite. Elements of Z,*, called prospects, will be denoted by vectors of the fonn
( x ^ . p j ; . . . ;x^, p^), where Xj < . . . < x^ and S/;. = 1. Such a vector represents
a lottery yielding Xj dollars with probability p^, i = 1, . . . ,n. Obviously, if
X < Xi
X > X„ .
362
U. Segal, Anticipated utility
On L (and L°), we assume that there exists a binary relation h. L^i X "^ Y
mean X | : r and Yi^X, and XttX'^Y mean X t 7 but not 7 1 : X. We assume that
^ i s complete and transitive, and that it also satisfies the following assumptions:
(a)
Continuity: h is continuous in the topology of weak convergence. That
is, if X, Y, Yi, Y2,. . -E L, such that at each continuity point x of Fy,
Fy.{x) -* Fyix), and if for all i^XtYi,
then X^Y.
Similarly, if for all (,
y,. |:X,then
A lottery X is said to dominate lottery Y by first-order stochastic dominance if for every x, Fj^(x) < Fyix), and there exists x for which Fj^(x) < Fyix).
(b)
First-order stochastic dominance axiom: If X dominates Y by first-order
stochastic dominance, then X ^ Y.
According to this axiom, the decision maker is interested in the probability
of receiving more (or less) than every possible outcome x. It is, therefore, a natural
extension of this axiom to assume that whenever he compares X and Y, the decision
maker examines those prizes x for which Pr(X < x) 7^ Pr(K < x). Fonnally, we
suggest the following "irrelevance" axiom.
(c)
Irrelevance: Let X, Y, X', Y' E. L, and let 5 be a finite union of segments. If
on S,Fx = FY.F^. = Fy,
and on {^,M]\S, Fyr = F^', Fy^ Fy.
then
This axiom suggests that decision makers first eliminate all the points x for
which the probabihty of receiving more than x is the same under X and Y. Then,
they compare X and Y by their corresponding cumulative distribution functions at
those points where these probabilities are not equal. Furthermore, they do it independently of the probabilities that are equal. This axiom resembles Savage's surething principle (Savage [21]), but it is much weaker. Savage suggested the following
axiom as a basic rule to be used in uncertain situations:
Sure-thing principle: Let 5 be an event and let X, Y. X', and / ' be lotteries.
If GnS,X= Y, X' = Y', and on "not S". X = X'. Y= Y\ then X |: y iff X' I: Y'.
Among other things, this axiom implies that the evaluation of the prizes
available if S happens does not depend on the prizes available if "not S" happens. In
particular, it does not depend on whether the prizes at "not S" are larger or smaller
than those of S. It is well known that this axiom is unacceptable from a descriptive
point of view, apparently because of this last objection. The irrelevance axiom, on the
other hand, restricts the sure-thing principle to those cases where the order of the
This axiom is equivalent to the cancellation axiom in Segal [14].
U. Segal, Anticipated utility
363
Y'
X'
=
X'
X =Y
X =
X'
Y = Y'
S
l-ig.2.
prizes is not reversed. For example, the irrelevance axiom agrees with the sure-thing
principle that (5,0.1; 0,0.01; 0,0.89) I: (I.O.I; 1, 0.01; 0, 0.89) < - > (5, 0.1; 0, 0.01;
-1,0.89) I: (1,0.1; 1,0.01; -1,0.89), but it does not agree that (5,0.1,0,0.01;
0,0.89) t (1,0.1; 1, 0.01; 0, 0.89) < = > (5, 0.1; 0, 0.01; 1, 0.89) ^ (1,0.1; 1,0.01;
1,0.89). Therefore, the irrelevance axiom does not rule out Allais-type behavioral*
patterns, as does the sure-thing principle. Indeed, the reason decision makers violate
the sure-thing principle through the Allais paradox is that replacing (0,0.89) by
(1,0.89) makes (0,0.01) strictly worse than all other prizes in one lottery only. For
a further discussion of these issues, see Quiggin [12] and Gilboa [7].
One possible interpretation of the anticipated utility functional is to look at
it as a measure. Let A' be a random variable. The expected value of X is
M
M
I
The first term in this last expression is the area of the rectangle [0,A/] x [0, 1],
while the second term is the area below F^- The expected value of X, therefore,
equals the area of X°.
Allais [2] found that most people prefer (0, 0.9; 5000000, 0.1) to (0,0.89; 1000000, 0.11). but
(1000000,1) to (0,0.01; 1000000,0.89.5000000,0.1).
364
U. Segal, Anticipated utility
The expected utility functional can be represented as
M
M
Iu
0
M
j
1
j
dpdu{x),
0
wliich is a measure of the area of X''^. Moreover, it is a product measure, where the
measure of [x, y] on the prizes' axis is u(y) - u(x), and the measure on the probabilities' axis is the Lebesgue measure.
Consider now the anticipated utilities functional. Its general form is
M
M
x)) = u{x)f{\ " F^{x))\^
- j f{\ ' F^{x))du{x)
0
M
1
i
- df{{ ~ p)du{x).
This toois a product measure of X°. As before, the measure of [x, y] is u{y) ~ u{x),
but the measure of [p, q] is / ( I - p) - f{l ~ q)- Yaari [20] assumed linear utility
function; in his case, therefore, the measure on the prizes' axis is the Lebesgue measure.
This justifies calhng his theory the dual theory. This discussion indicates the importance
of representing a preference relation by a measure of X^.
THEORI£M 1
| j on I satisfies axioms (a)-(c) iff there is a measure v onV = [0, M] x [0, 1]
mutually absolutely continuous with respect to the Lebesgue measure on f such that
Proof
It is easy to verify that if |^ can be represented by a measure P then it satisfies
axioms (a)-(c). Assume now that ^ satisfies axioms (a)-(c). Let A - {[x, v] x [p, q]
C [0, Af] X [0, 1] : X <- y, p < q] h% the set of all the compact rectangles in F, and
let ^ = {{X.b) e £ X A: Int A'" O IntS = 0, .V° U 6 e L M . The irrelevance axiom
implies thatif (^, 6), {Y,b)e^
,XhQX\ X^ t Y"^ m X° U 5 > K" U 5. Define on A
U. Segal, Anticipated utility
365
partial orders/?_^ by 6, .R^^^S: iff (X, 5,), (X, 5 2 ) €^ a n d X ° Ub^'r^^
U S . . For
every X and Y, R^r and y?y do not contradict each other. That is, if 5[ and 62 can be
compared by both R^ and Ry, then biR^bj iff biRyb2 • Indeed, let X, Y & L and
5 i . 62 e A such that {X, 6,.)^ (3^, 5,0 e * , i = 1. 2 and let Z"^ = X^ r\ r ^
Obviously, Z^L
and (Z. 5 i ) , ( Z , 63) e ^ . There exist 6 ] , . . . ,5^', and 6 j
,5^^
such that
/-I
U U 6^,5;) G ^ ,
fc - 1
/ = 1,2,
'
X° - Z ° U U
fc =
and
f
r° -2"^ u u
It follows that
>:Z° U 6' U . . . U 5' U52 < = > . .. < = > Z ° U 5 ,
1
J
.CZ° U,52 < - > . . . < - > Z ° U 5 j U . . .U5^^ U 5 i
> Z ^ U 6^ U . , . U5,^U 52 < = > 5 i / ^ v 5 2 .
Let R =D Ry. That is, diRSj iff there exists J such that biR;^52. It is
proved in the appendix that R is acyclic. That is, 5iRb2R . . . RS^RS^ imply
5iR6^R. . . R82RS1. Let 5;* be the transitive closure of R : 5i i:* 62 iff there exist
63, . . . , 5j. such that 6,7? 63 /^ . . . /i 5^.^? S^ and S3 is obviously complete and transitive.
Let [0,x] X [0,^] -v* [x,yi] x[p,l]
(see fig. 3), and let i'([O,x]
x[O,p])
~ ^{[x. J'l] X [/?, 1]) = 1. By the continuity assumption, there exist z and w such that
[x,vv'J X [p,\] ^* [0,z] X [0,p] ^'' [w,yr] x [p, ! ] . Define K[j:-vv-] x [p, I ] )
= (^([vy, y, ] X [f, I ] ) = 1/2. This can be repeated again and again for the x as well as
for the p axes. By the first-order stochastic dominance axiom, the areas of all these
rectangles will become smaller and smaller, v can thus be defined as an atomless,
continuous, finitely additive measure on [0,x] x [0,;;] and on [ x . ^ J x [/J, 1 ].
Similarly, it can be defined for the rectangles [.V,-, y^^ j ] x [p, 1] '^* [0, x] x [ 0 , / J ] ,
( - 1, . . . .By the continuity assumption, y^ is finite. Indeed, let lim>'. = y < M. For
aIl/.([O,^] X [ p , I ] ) ^ ( [ O , v J X [p,l])^{i[O,y-_,]
x [/^ 1]) U ( [ 0 , x ] x [Q,p])),
in contradiction to the continuity and the stochastic dominance axioms. This process
U. Segal, Anticipated utility
366
Mg. 3.
defines a finitely additive measure v on [x, M] x [p, 1], which can be extended to
[0,M] X [0,/?] and to [0, x] x [/?, 1], and thusto [0,A/] x [0,1] .Define cj : I -> ^
as follows. If
-
U
where
V/I
U 6^.5,.
\k=\
'
then
Because v is finitely additive, to does not depend on the choice of 5|
Let
U
=
and
=
U
A: = 1
where
such that X "^
that
. Our aim is t o construct t w o sequences { 5 ^ } ^ ^ ^ and { f g l p ^ ^ such
= u
2=I
U. Segal, Anticipated utility
367
and for every / < s', 5- "^* ^'- We will do this by finite induction. If 5i 'V* f^, then
let b[ = 5i and fj = fi. If 5i ^* fi, construct Sj £ A such that 5j 'V* f,,
5; = CC(5j\6;)G A . ( 5 ; . 5 ; ) e vl^, andlet j ; ^ f i - I f f i §-* 5,,construct 5j, f;, and
5*2 similarly. It thus follows that In each step we can reduce the number of nonequivalent elements either in A' or in Y (or in both) by one. The desired representation
will thus be constructed in a finite number of steps. As X ^Y, it follows that t' > s',.
Obviously,
and as r ' > s ' , it follows that co(X) >
Q.E.D.
As indicated above, the anticipated utility function is the most general form
of product measure. To make sure that the epigraph measure u of theorem I is actually
a product measure, one additional assumption is required. For the reasoning behind
this assumption, consider fig. 4.
X
y
z
Fig. 4.
Assume that 65 -v* 5^ and 65 U 65 '^'* 62. That is, X" U 5s 'v X'' U 5i and
X° U 5i U5s U56 -vZ^ U 5 , Ufij.-^" U5s U S e t X ^ U 65,hence 5s U §6 t * 65
and §2 ^ 5i. Since the projections of the rectangles 5i and §2 on the prizes' axis are
the same, the ^* order between them is determined by their projections on the
probabilities axis. The projection of 63 and 64 on the prizes' axis is also the same,
hence the j ; order between them is defined by their projections on the probabilities
axis. Since the probabilities axis projections of 63 and 64 equal those of 5i and §2,
368
U. Segal, Anticipated utility
respectively, ^4 |;* 63 iff 62 k* 5i. This is summarized by the following assumption:
(d)
Projection
independence:
F o r every x < y < z a n d 0<p<q<r<
[y-z] X [p,q] t* [y.z] X [q,r] iff [ x , ; ^ ] x [p.q] h* [x,y] x
1,
lq,r].
THEOREM 2
The following two conditions are equivalent:
(1)
(2)
I; satisfies axioms (a)~(d).
There exists a continuous and strictly increasing function u : [0,A/] ->• R.
unique up to positive Hnear transformations, and a unique continuous and
strictly increasing function / : [0,1] -^ [0, 1] ,satisfying/(O) = Oand/(I) - 1,
such that X I : riff
M
M
~ ju{x)df{l -F^(x)) > - juix)df[l-Fyix)y
0
(2.1)
0
Proof
Obviously, (2) => (I). Assume now that 5; satisfies axioms (a)-(d). From
theorem 1, it follows that there exists a measure P representing ^ on L. Define
ii:M-^R
by u{x) = v{[Q,x] x [0,1]). Let f{p)^H[^A]
x [1 - p,\\).
Axiom
(d) implies that for 0 < x < >- and p < q, i^{[x,y] x \p, q]) ^ [ii{y) - u{x)]
X [/(I - p) - / ( I - q)]. Theorem 2 thus follows.
Q.E.D.
Gonclusion: On L , the relation |; can be represented by the function
Y u{x,) \f i t p)\ - f i t p)]
L \
= " ( ^ i ) + I [uix,)
i=2
/
\
- w ( x , _ , ) ] / ( YPJ]\j=i
(2.2)
/J
(2.3)
J
Remark: If f{p) = p, then (2.1)-(2.3) reduce to the expected utility representation
function.
U. Segal, Anticipated utility
3.
369
The case of vectorial prizes
The construction of the representation function in section 2 depends on the
assumption that the set of prizes is an ordered set. When the prizes are bundles of
commodities, one must explicitly outline this assumption. For the sake of simphcity,
here we deal only with prospects.
Let L* = {(x,.p, : . . . : x ^ , p j : x , . . . . . x ^ 6 ^ - [ O , i t / ] \ p j . . . . , / ? „ >O.i:p.= \\
and let 1; be a complete and transitive continuous preference relation on L*. Define
on A" a preference relation 1^* by x |f y iff (x, 1) ^ {y, 1), and assume from now
on that if (xi, /3,; . . . ;x^, p^) G / / , then x^ H . . . K Xi. To axioms (a)-(d) add
now the following assumption:
(e)
If X. ^- x;, then (x,, / 3 , ; . . . ;x., p,-;.. . ;.v,,, p^)
Since ?; is continuous, so is i^ . Hence, y can be represented by a real,
order-preserving function (Debreu [6]). For the time being, we arbitrarily choose
one such function, v. Later, we show that the representation of S;; does not depend
on V. By using the function u, elements of L* may be represented as lotteries over
utilities of the form {v{x\),px;. . . ;(j(x^),p^). On such lotteries, assumptions (a)--(d)
of section 2 imply that J; on /,* can be represented by
1=2
Let u = u o 0 and obtain that ^ can be represented by
V = u { x , ) + Y U'('<i)
- "U,-,)]/
I
PJ) .
(3.1)
where /fO) - 0, / ( I ) = \, u and / a r e strictly increasing and continuous, / i s unique,
and u is unique up to positive linear transformations.
THEOREM 3
u and / d o not depend on the choice of u.
Proof
Assume that the relation ^ on L* can be represented by (3.1) and by
1=2
370
U. Segal, Anticipated utility
where f{0)=f*{0) = 0, / ( I ) = / * ( l ) = 1, w(O) = u*(O) = 0. Therefore, there exists
a continuous function h such that K* = ^(K) (see (3.1)~(3.2)). In particular, for
every x and p
u\x)f\p) = h(uix)fip)).
(3.3)
Substitute p = 1 and obtain
u*{x) = h{u{x)).
Substitute x such that u{x) = 1 and obtain
Hence, by (3.3),
h{u(x))h[f{p))
= h{u{x)f(p)l
The solution of this functional equation is h{oi) = aa^
(see Aczei [1]), hence
We now prove that b = 1. Let x ^' y and let X = u{x), Y = u{y) und F = f{p).
There exists z such that (z, 1) '^ (,v, 1 - P',x, p), hence
u{z) = iti^y) + [u{x) - u{y)] f{p) = Y ^ [X - Y]P.
By (3.2), we obtain
u\z)
= u\y)
+ [u\x)
-
u\y)]f\p).
Since u*(z) = fl[w(z)]*.it follows that
[Y + {X - Y)P]^ = Y^ + iX^ -
Y^)P^.
Differentiating with respect to X, we obtain, for F =^ 0.
[Y ^{^X
For b^\,
this last equality implies that Y{l-P)
= O, in contradiction to
the assumption that (3.4) holds true for every X, / , and P. It thus follows that
b=[.
Q.E.D.
U. Segal, Anticipated utility
4.
371
Discussion
Section 2 developed a set of axioms implying that the preference relation |;
can be represented by the anticipated utiHty functional. This set of axioms emphasized
the fact that this functional is actually a measure. Indeed, as is claimed in theorem 1,
assumptions (a)—(c) imply that the value of a lottery X is a measure of the area in F
above the graph of F,^ . Given this, one may try to add other axioms which will imply
more specific measures. One such attempt is presented in this paper. As is proved in
theorem 2, axiom (d) guarantees that this measure is a product measure. Other axioms
may imply other functional forms. Alternatively, one may analyze the general
representation function obtained by theorem 1. In that case, it seems essential to
assume strict first-order stochastic dominance, to avoid unreasonable relations Uke
This paper assumed that all lotteries are simple, one-stage lotteries. A possible
source for new axioms may be obtained by extending the space of lotteries to include
two-stage lotteries. For this, see Segal [15].
Appendix
CLAIM
i? = U R^ is acyclic.
Proof
Let b\R82 R . . . R8jR8\
and let 5^^ j - b\. There are Xi, . . . ,X^ such that
Xf U 8.^X9
U fi.^j , i= 1
,r. If J , ^ .. .^ X^, then by the transitivity of
£ and by the irrelevance axiom it follows that X9 U 5,- ^ X^ U bj^^ , i - 1, . . . , / ' ,
hence 5iRb^R . . . R82Rb] and /? is acyclic.
Let 61 Pb2 mean 81R82 but not 82 R8], and let 5,762 mean 8iRb2 and b2R8i.
Suppose now that 8XR82R . . . R8jPb]. Of course, there do not exist Xj, . . . ,Xjas
above. However, in that case it is possible to findf^^, = f i , - - . , S", G A such that
iiR^2R . . .R^^P'^i, and there is Y e L such that Y° U f.-|; /** U f.^,, / = I , . . . ,f,
which contradicts, of course, the transitivity assumption.
Let S E A . { 5 ' , 6 ^ } is a horizontal (vertical) partition of 5 if S ' , 5 ^ G A,
5 ' U 5^ = 5 , int S' n Int 5^ =0, 5 ' is above (to the left of) 5^ and 5, 5 ' , and 5^
1 have the same projection on the horizontal (vertical) axis.
Let biPb2. There are horizontal partitions of 8^ and 62 into S } , 5 j and
62,^2 , respectively, such that 8\Pb^ and b^Pb\. Indeed, there is X such that
X° U 5j U8]^^ X"^ U 8\U bl. Suppose that X^ U 5* U Sj ^ X^ U b\ U S^. It
thus follows that 8]P8\. By the continuity and the stochastic dominance axiom, it
follows that these partitions could be constructed such that 51/^52 as well. The same
372
U. Segal, Anticipated utility
argument holds for vertical partitions. It thus follows that 5 i , . . . ,5^ may be assumed
to be sufficiently small, with no side panels on the boundary of [0, M] x [ 0 , 1 ] , and such
that for / ^ / , 8j O §. = 4>. Obviously, b- and 5^-^, can be compared by the relation R
iff either the lower left-hand corner of 6,-^j is above and to the right of the upper
right-hand corner of 5,-, or if the lower left-hand corner of 6j. is above and to the right
of the upper right-hand corner of 5,-^^. It therefore follows that each 5,- can be replaced
by ^j such that 5.7^,-, / = 1, . . . ,r, and such that the lower left-hand corner of f,-^,
is above and to the right of the upper riglit-hand corner of f,. Moreover, these f i , . . . ,5:,
keep the /^-relation between 5 , , . . . ,5^. The required Kexists, and the contradiction
proves the claim.
Q.E.D.
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