Journal
of Mathematical
Economics
22 (1993) 373-397.
North-Holland
Order indifference and rank-dependent
probabilities
Uzi Segal*
University of Toronto, Toronto, Ont., Canada
Submitted
August
1989, accepted
June 1992
This paper presents
a new axiomatization
of the anticipated
utility (also known
as the
rank-dependent)
model for decision making under uncertainty.
This axiomatization
is based on
the analysis of two-stage
lotteries with the compound
independence
axiom but without the
reduction of compound
lotteries axiom. The major new axiom assumes indifference to the order
at which uncertainty
is resolved in some simple two-stage lotteries.
1. Introduction
Recent years saw the emergence of several axiomatized generalizations of
expected utility theory. One of the most widely used of these new theories is
Quiggin’s (1982) anticipated utility theory, also known as expected utility
with rank-dependent probabilities. Let the lottery X =(x1, pl;. . . ; x,, p,,) yield
$xi with probability pi, 1 = 1,. . . , n and assume, without loss of generality, that
05x,5 ... sx,. Denote p. =O. According to anticipated utility theory, the
value of the lottery X is given by
(1)
where the utility function u is strictly increasing, continuous, and unique up
to positive linear transformations and the probability transformation function g: [0, l]-[0, l] is strictly increasing, continuous, onto, and unique.’
Since its introduction, its theoretic foundations and behavioral implications
were intensively investigated [see, for example, Yaari (1987), Chew et al.
(1987), R6ell (1987), Lute (1988), Segal (1984, 1987a, b, 1989, 1990), Chew
Correspondence to: Uzi Segal, Department
of Economics,
University
of Toronto,
150 St.
George Street, Toronto, Ontario M5S lA1, Canada.
*I am grateful to Chew Soo Hong, Larry Epstein, Kim Border, and two anonymous
referees
for suggestions and comments.
‘See section 2 for a discussion of the two different forms in which this functional
appears in
the literature. Note that when the function g is linear the anticipated
utility functional reduces to
expected utility: cpiu(xi).
03044068/93/$06.00
0
1993-Elsevier
Science Publishers
B.V. All rights reserved
374
U. &gal, Order indifference and rank-dependent probabilities
and Epstein (1990), Chew (1989), and Karni and Safra (1987)]. Several sets of
axioms for this theory were suggested, and this paper is to offer another such
set. Each of the different axiomatizations
reveals another aspect of the
theory, and usually leads to new and different applications.
Most axiomatizations of this theory, as those of other alternatives to
expected utility theory, seek to weaken the mixture independence axiom, that
for all three lotteries X, Y, and Z, and tl~(O, 11, CYX
+( 1 - cc)Z is preferred to
cry +(l -a)Z if and only if X is preferred to Y. [See Quiggin (1982) and
Chew (1989).] Yaari (1987) and Roe11 (1987), on the other hand, replaced the
mixture independence axiom by its dual presentation: For all lotteries
X=(xr,~r;...;x,,,~~),
Y=(Y,,P~;...;Y,,P,),
and Z=(z,,p,;...;z,,p,)
and
a~(O,l], the lottery (ax,+(l-~)z,,p,;...;ax,+(l--)z,,p,)
is preferred to
the lottery (ay,+(l-cl)z,,p,;...;
a~,,+(1 -c()z,,p,) if and only if X is preferred to Y. This axiom leads to the above functional with a linear utility
function U.
A different approach is suggested in Segal (1989). Although the anticipated
utility model is a generalization of expected utility theory, it is yet a special
case of a wider set of functionals, those that assign to a lottery a measure of
its epigraph. This family of functionals is defined as follows. Let 9 be a
positive measure on [0, co) x [0, l] such that for a bounded set S, S(S) < co.
Denote as before pO=O, let X=(x,,p,;...;x,,p,)
where O~X,~*..IX,,
and
define
( x[~Pjy$oPj]).
WV= i 9 [O‘xL
.]
i=l
It is easy to verify that anticipated utility is the most general case in which
this measure can be decomposed as a product measure (see section 2 below).
As proved in Segal (1984, 1989, 1991), Green and Jullien (1988) and Chew
and Epstein (1990), measure representations can be obtained by axiomatizing
preference relations over simple lotteries. For axioms that further guarantee
that the preference relation can be represented by a product measure, and is
therefore anticipated utility, I suggest here, as in Segal (1990), to enlarge the
space of lotteries by letting in two-stage lotteries.
The rejection of the mixture independence axiom by most recent writers is
justified, as experimental data shows that it has very little behavioral appeal
- the paradox of Allais and the common ratio effect are just two of these
examples.’ However, in an intertemporal model, where the set of lotteries
includes lotteries over lotteries, this axiom can be obtained from two more
*By the paradox
of allais, (5,000,000,0.1;0,0.9)
is preferred
to (1,000,000,0.11;0.0.89),
but
(l,OOO,OOO,1) is preferred
to (S,OOO,OOO,O.l; 1,000,000,0.89;0,0.01).
By the common ratio effect,
(3,000,l) is preferred to (4,000,0.8;0,0.2),
but (4,000,0.2;0,0.8)
is preferred to (3,000,0.25;0,0.75).
V. Segal, Order indifference and rank-dependent probabilities
375
fundamental axioms, the reduction of compound lotteries axiom and the
compound independence axiom [see Lute and Narens (1985), Segal (1984,
1990)]. The reduction of compound lotteries axiom suggests that a two-stage
lottery is equally attractive as the simple lottery yielding the same prizes with
the compound probabilities. By the compound independence axiom, the twostage lottery (X,p;Z, 1 -p) is preferred to the two-stage lottery (Yp;Z, 1 -p)
if and only if the simple lottery X is preferred to the simple lottery Y. Lute
(1988) offered an axiomatization of anticipated utility theory based on the
distinction between one- and two-stage lotteries. In Segal (1990) I show how
replacing the reduction axiom by a weaker dominance axiom, while keeping
the compound independence axiom, still maintains expected utility theory,
and how a weaker concept of this dominance axiom yields the anticipated
utility model.3
According to the reduction of compound lotteries axiom, the decision
maker is only interested in the compounded probabilities, rather than in the
probabilities themselves. In particular, this implies that decision makers do
not care for the order in which uncertainties are resolved. A weaker version
of this idea is formalized in section 3 under the name of order indifference,
and together with the compound independence axiom it is proved to imply
that a measure representation is a product measure, hence anticipated utility.
The order indifference axiom suggests that for Osx, s ... 5x,, the decision
maker is indifferent between the two-stage lottery A, where with probability
4i he wins a ticket for the lottery (0, 1, -p; xi,p), i= 1,. . . , n and the two-stage
lottery B, where with probability 1 -p he wins zero and with probability p he
wins a ticket for the lottery (xi, ql;. . .; x,,,qJ.
To a certain extent one may
say that the difference between these two lotteries is the stage in which the
decision maker knows that he did or did not win zero.
This axiom has some immediate applications for games in normal forms.
Assume two players. Such games may be interpreted by each of the players
in two different ways. Firstly, each pure strategy of a player results in a
lottery, defined by the other player’s mixed strategy. The player now uses a
mixed strategy, that is, he plays a lottery, the outcome of which determines
which pure strategy he is going to play. He thus faces a two-stage lottery.
Alternatively, he may consider his opponent as the first player. Each pure
strategy of the other player results in a lottery, its probabilities being defined
by the first player’s own mixed strategy. This too results in a (different) twostage lottery. In an extensive form, these two interpretations differ in the
order of the players in the tree. Of course, if both players employ the
reduction of compound lotteries axiom, there is no difference in the lotteries
resulting from these two interpretations. Without this axiom, two different
‘For a discussion
see section 5.
of Lute’s (1988) axioms
and some of the dominance
axioms
in Segal (1990),
376
U. Segal, Order indifference and rank-dependent probabilities
games may be defined. [See Dekel et al. (1991) for more details on these two
interpretations.] Indifference to the order of the two stages, even if only in
some cases, may thus be of some interest.
The paper is organized as follows. Section 2 presents the definitions of
one- and two-stage lotteries, and the axioms leading to a measure representation. The order indifference axiom and the main representation theorem
appear in section 3, and several alternatives to this axiom are discussed in
section 4. Section 5 concludes with some remarks on the literature.
2. Definitions
and axioms
For a positive real number s, let L”, be the set of all the finite lotteries with
outcomes in [O,s]. That is, LS,=((x,,p,;...;x,,p,):
~pi=l,pi~O,
i=
1,..., n,O~x,~...~x,~s}.
Let L1=U,,O L”,, which is the set of all the finite
lotteries with non-negative outcomes. For X EL,, define the cumulative
distribution function F, by F,(x) = Pr(X 5x), and let X0 be the epigraph of
this function, that is, X0 = Cl {(x, p) E [0, co) x [0, 11: p > F,(x)}. The degenerate
lottery (x, 1) is denoted 6,. Let D”= [0, s] x [0, l] c ‘%’ and ds = OS\
((0, l), (s, O)}. F or t>O, let L:’ be the set of lotteries {XE L”,: for
x E [0, t), F,(x) 5 1 -t} and let Q’ be the square [0, t) x (1 -t, 11. Let L: =
u szo L”i’.
Let 2 be a complete and transitive preference relation over L,. Define the
relations > and - by X> Y if and only if X> Y but not YtX, and X - Y
if and only if X2Y and YtX.
We say that the function I/: L,+!K
represents the preference relation 2 if V(X) 2 V( Y)oX> Y. Also, define the
preference relation 2” on L”, by: t/X, YE L”,, X 2” Y if and only if Xt Y, and
define the relations >” and -’ accordingly. Consider the following three
axioms:
(a) Continuity. The preference relation 2” on L; is continuous in the
topology of weak convergence. That is, let X, Y, Y1,Y,, . . . E L”, such that at
each continuity point of z of F,, F,i(z)+F,(z).
If, for every i, X>“x, then
X t” Y. If, for every i, yi t”X, then Y 2” X.
(b) First-order stochastic dominance. Let X, YE L”,. If, for every z, F,(z) 5
Fy(z) and there exists z such that F,(z)< Fy(z), then X>“Y.
(c) Irrelevance.
Let X, Y,X’, Y’E L”, and let S be a finite union of segments
in [O,s]. If VZES, F,(z) = F,(z) and F,.(z) = Fy,(z), and VZE[O,s]\S, F,(z) =
F,.(z) and Fy(z) = FyG(z), then X 2” Y if and only if X’ 2” Y’.
U. Segal, Order indifference
and rank-dependent
probabilities
377
Definition. A curve Cc D” is the continuous
image of a function f: [O, l] -+
D”. The curve C is increasing if (x, p) E C + C n {(y, q): y < x, q > p} = 8.
Let 9” be a countably
additive measure on ds such that for every t ~(0, I],
Q’ is a measurable
set. For t > 0, define the measure ,Y~’ on D” as follows: For
every P-measurable
set B c D’, P’(B) = 9”(B\Q’).
Theorem 1.
(1)
(4
The following two conditions are equivalent:
The preference relation &-” on Ls satisfies the continuity, first-order
stochastic dominance, and irrelevance axioms.
There is a (countably) additive measure 9” on D”, unique up to multiplication by a positive number, satisfying:
(a) For B=[x,y]
x [p,q]cD’
such that x<y and p<q, O<P(B)<co;
(b) If Ccds is an increasing curve, then P(C) =O; and
(c) For every t > 0, the preference relation 2” on L”;’ can be represented by
P’(X) = Isrs9’(X”). That is, for X, YE L;‘,X 2” Y if and only if
P’(X”) 2 W( Y”).
For a proof of this theorem, see Segal (1991).4 Note that the measures 9
are not necessarily bounded. Next, I show how to extend Theorem 1 to the
space Lt. As before, if 9 is a countably
additive measure on [0, cc) x [O, 11,
then for t > 0, 9’ is defined by ,9’(B) = 9(B\Q’).
Lemma I. The following two conditions about the preference relation 2
L, are equivalent:
on
(I) For every s, the restriction of 2 to L”, satisfies axioms (a)+c).
(2) There exists a measure 9 on [0, co) x [0,11, unique up to multiplication by
a positive number, such that for every t >O, the functional Vt(X) = $‘(X”)
represents the preference relation 2 on L:.
Proof. Clearly
(2) a(1).
To prove that (l)*(2),
define the preference
relation
>-” on L”, as before. By Theorem
1, for every s and t there is a
measure
9” on [O,s] x [O, 11, unique
up to multiplication
by a positive
number, such that the order 2” on L?’ can be represented
by the function
P’(X)=P’(X’).
Since the measure 9” is unique up to multiplication
by a
positive number, one may assume, without loss of generality,
that for sz 1,
P([:, l] x [0,11)= 1. (Since
s”([$, l] x [0, 1])=,!P0.25([i,
l] x [O, 11)
and
p.o.25
represents the order 2” on LS30.25, it follows that P([$, l] x [0,11) is
4Earlier versions of this theorem, with slightly different conditions,
appear in Segal (1984,
1989), Green and Jullien (1988) and Chew and Epstein (1990). As pointed out by Wakker (1990,
1991), these theorems and proofs are erroneous. One of the problems with the earlier versions is
that the measure 9” may be infinite on sets containing (0,l) or ($0).
378
V. Segal, Order indifference and rank-dependent probabilities
finite. Otherwise, since Vs,o.25(X) =P”.25(Xo) represents the order on L”;“.25,
it follows that d2 IV‘Si, a contradiction to the first-order stochastic dominance axiom.) It thus follows that for r Is2 1 and for a given t, the
functionals P’ and I/‘,’ coincide on L”;‘. Of course, this proves that 8,’ is a
bounded measure. Let t 5x, 5 +*.sx,.
For X=(x1, pi;. . . ; x,, p,), define
V(X) = PqX)
= JPqXO).
Define also a measure 9 on [0, co) x [0, l] by S([x, y] x [p, q]) =
JP’([x, y] x [p, q]).5
Obviously,
S’(Cx,~1x CP, 41)= 9y*‘(Cx,
~1x Cp,41). Let
V’(X)=$‘(X’). We have still to prove that the functional I” represents the
preference relation 2 on L\. Note first that as Li consists of finite lotteries
only, V’(X) is always finite. Let X, YE L\. There exists therefore s >O such
that X, YE L? *, hence
x~Yox~sY~vs”(X)~;V”~‘(Y)
0 P’(X”) 2 P’( YO)0 $‘(X”) 2 $‘( Y”) 0 V’(X) 2 V’(Y).
Suppose now that there is another measure $* such that for every X and
Y in L:, Xt Yo$*‘(X”)L9*‘(Y”).
Since both 9’ and $*’ represent the order
on L?’ for every s, it follows that on D”, 9* =K”$, and moreover, that K”
does not depend on s.
Q.E.D.
Let 9 be as in Lemma l(2) and suppose that it is finite on bounded sets
containing (0,l). In that case, the order t can be represented by V(X) =
9(X”). Define G,(x,p) =$([O,x] x [l-p, 11). Obviously, 9 and V can be
reconstructed from Gg, as
S(Cxv
~1x Cl-P, 1- 41)= WY, P) -
Gs(x, P) - Gs(Y, 4) + G,(x, q),
and as, by the properties of distribution functions, X0 is the union of a finite
set of rectangles {Qi> w h ere i # j* Int (Qi) n Int (Qj) = @.Different G functions
thus define different representations of the preference relation 2. Consider
the following four examples. As before, we assume Osx, 5 .*. 5x,:
Expected
Value.
G(x,p)=px,
V(X)=CpiXi,
and S([x,y]x[l-p,l--q])=
(Y-x)(P-q).
Expected
Utility.
G(x,p)=pu(x)
for some (strictly) increasing
V(x) =C Pi”txi)3 and WX,YI x Cl-P, 1 -d)= CU(Y)-WI(P-~.
function
Dual Theory
a.
[Yaari (1987)]. G(x,p) =xf(p)
for some strictly increasing,
continuous, and onto function f: [0, l] -[O, 11.
‘The fact that 9 is a measure
follows immediately
from Theorem
12.5 in Billingsley
(1979).
U. Segal, Order indifference
and rank-dependent
379
probabilities
andWGYI x Cl-_p,1-ql)=(y--W(p)--f(q)].
Anticipated Utility Theory.
[Quiggin (1982)16
functions f and I* as in the last two cases.’
G(x, p) =f(p)u(x)
for some
(2)
and &CX,YI x Cl -_p, l-41)
=C~(Y)--(x)lCf(~)-_(s)l.
In all four examples, the functions G are multiplicative separable and the
corresponding measures 9 are product measures. Anticipated utility theory is
the most general form of a product measure. Indeed, let p be a positive
measure on [0, co) such that the measure of bounded sets is finite, and let v
be a positive bounded measure on [0, 11. Define U(X)=,u([O,x]) and
f(p) = v([l --p, 11). Define a measure 9 on [O, co] x [O, l] as follows. For a
box S = Cx,~1 x CP, 41 = CO,~0) x CO,11, let
W) =P(cx?Yl).
w% 41)= [U(Y) -u(x)lCf(l
-PI - .I-(1 -4)l.
It follows that V(X) in (2) equals the measure 9 of the epigraph of X”.
There is some confusion in the literature as to whether the proper
exposition of the anticipated utility functional should be (1) as in the
introduction or (2). Let g(p) = 1 -f(l -p) and obtain (1) from (2). Note that
in that case f(p)= 1 -g(l -p).
For a further discussion of these two
alternative representations, see Segal (1987a). In the sequel I use form (2) of
the functional.
Later, I use the following Archimedean axiom:
(d) Archimedean.
For every
such that (0,l -E; x, E)2 X.
As stated
in the Introduction,
X E L,
and
for every
E>O, there
the aim of this paper
exists
is to axiomatize
x >O
the
‘%ome authors refer to this theory as ‘expected utility with rank-dependent
probabilities’.
I use
both names throughout
this paper.
‘Quiggin suggested a less general form of this functional
where J(f)=f.
The general form of
this functional
is presented in Segal (1984, 1989). It is useful not to restrict the function f to
satisfy f(p)=p
for some p~(0, l), because risk aversion implies a convex probability
transformation function / [see Chew et al. (1987)].
380
U. Segal, Order indifference and rank-dependent probabilities
anticipated utility representation functional through two stage lotteries.* A
two-stage lottery is a lottery at which the outcomes are tickets for lotteries in
L,. Let
be the set of these lotteries. Natural isomorphisms occur between L, and two
subsets of L,. Let ~={((x~,P~;...;x.,P~),~):(x~,P~;...;x~,P~)EL~)
and let
; x,,p,) E L,}. The set n should be
~=(((X1,l),pl;...;(Xn,l),pn):(X1,pl;...
interpreted as consisting of all the lotteries where all the uncertainty is
resolved in the second stage, while the set r should be interpreted as the set
of those lotteries where all the uncertainty is resolved in the first stage. For
X=(x,,p,;...;
x,,p,) EL,, let yx and 1, represent the corresponding lotteries
in r and A, respectively. That is, yx=((xl, l),p,;. . . ;(x., l), p,) and ;1,=
((x,,p,;...; x,,pJ, 1). On L, assume the existence of a complete and transitive
preference relation &. We say that A w2B if and only if A&B and B&A,
and we say that A >213 if and only if A &B but not B&A. Representation
functions of tZ are denoted by W. The relation tz induces preference
relations tr and 2,, on L,, where X&,-Y if and only if yx t2yY and
X 2 ,, Y if and only if 2, & Ar.
Definition.
The preference relation
and t,, satisfy them.
k2 satisfies axioms (a)-(d) if both kr
To the previous four axioms (a)<d) add now the following one:
(e) Time neutrality.
For X EL,, yx -*Ax.
This axiom states that if the decision maker has to participate in the
lottery X E L,, then he does not care whether all the uncertainty is resolved
at the first or at the second stage. If the two stages represent two different
time periods, then this implies that he does not care whether all the
uncertainty is resolved at the first or at the second time period, hence the
term ‘time neutrality’. It is easy to prove [see Segal (1990)] that the
preference relation tZ satisfies the time neutrality axiom if and only if for
every X, YE L,,
A common assumption in decision theory, which this paper does not
assume, is the reduction of compound lotteries axiom, According to this
‘The description
of the space Lz and two-stage
lotteries
follows Segal (1990).
U. Segal, Order indifjkrence and rank-dependent probabilities
381
axiom, a decision maker is indifferent between a two-stage lottery and its
actuarial equivalent one-stage lottery. Formally:
Reduction of compound lotteries.
Xi=(xl,~l;...;xk,,pt,),
and let A=(X,,q,;...;
Let
i=l,...,
m,
X,, q,,J. Define R(A) EL, by
R(A)=(x:,q,p:;...;~,‘~,q,p,‘~;...;x’;,q,p~;...;x~~:_,q,p~~).
Then A
-2YR(A).
If the reduction of compound lotteries axiom is satisfied and the order tr
can be represented by the functional 2,-, then the preference relation k2 can
be represented by the functional W(A) = VdR(A)).9
The reduction of compound lotteries axiom finds for every two-stage
lottery A EL, a corresponding lottery in r, an isomorphic set to L,. This
axiom can therefore be interpreted as a mechanism reducing two-stage
lotteries into one-stage lotteries. Alternatively, decision makers may reduce
two-stage lotteries by using the compound independence axiom:
(f) Compound independence.
Let A =(Z1, ql;. . . ; X, qi;. . . ; Z,, q,,,) and B=
Z,, q,,,) be two lotteries in L,. Then A t2 B if and only if
(Z,,q,;...;Y,qi;...;
X&Y.
For X=(x,,p,;...;
x,,,P”) E L,, let the certainty equivalent of X, denoted
CE(X), be defined implicitly by (CE(X), 1) -X. By the first-order stochastic
dominance axiom, 6maxrxi12X>6,i,1,i1, and by the continuity axiom it thus
follows that the certainty equivalent exists, and moreover, that it is between
min (xi} and max {xi}. Again by the first-order stochastic dominance axiom
it follows that it is unique.
Let CEAX) and CE,(X) be the certainty equivalents of the lottery X with
respect to the orders tr and 2,,, respectively. That is, ((CEAX), l), 1) -2~X
and ((CE,(X), l), 1) - 2;1x. By the conclusion from the time neutrality axiom
it follows that for every X, CEJX)= C_!?,(X). If the preference relation k2
satisfies the compound independence axiom, then
(X,,q,;...;
X,~4rn)~2((C~,4(XJ,
l),q,;. . . ;(CE,W,),
1LafJ.
Note that the right-hand side of this equivalence is an element of r In other
words, the compound independence axiom too can be used to find for each
two-stage lottery an equally attractive lottery in r, an isomorphic set to L,.
‘Note
that V,is defined on L1.
U. Segal, Order indifference and rank-dependent probabilities
382
In that case, if the order t,- can be represented by the functional V,, then
the preference
relation
&
can be represented
by the functional
WX,,q,;...;
x,, 4J = UCE,(X,),
41;. . . ; CE,(X,h
qm).
If the preference relation & satisfies both the reduction of compound
lotteries axiom and the compound independence axiom, then it can be
represented by an expected utility functional [see Samuelson (1952)]. Some
empirical evidence supports the claim that decision makers to not obey the
reduction of compound lotteries axiom. [See, for example, Schoemaker (1980,
1987). See also Segal (1990) for further references.] This may be due to the
fact that risk-averse decision makers prefer to take part in as few lotteries as
possible. This conjecture is indeed proved to be correct within the anticipated utility model - see Segal (1990). In this paper I assume that the
preference relations kr and t,, satisfy axioms (a)-(c) and can therefore be
represented by a measure on the epigraphs of the distribution functions as in
Lemma 1. I assume, furthermore, that the preference relation & satisfies the
compound independence axiom, but not necessarily the reduction of compound lotteries axiom. As is proved in Segal (1990), this does not impose any
new restrictions on the preference relations tr and kn, unless more axioms
are assumed. In the next section I suggest one such additional axiom
implying representability by an anticipated utility functional.
3. Order indifference
Consider the two two-stage lotteries A = ((0, l), l-q; (0,1 - p; x, p), q) and
B = ((0, l), 1 -p; (0,1-q; x, q), p). If the reduction of compound lotteries axiom
is employed, both are equally as attractive as the lottery ((0, l), 1 -Pq; (x, l), Pq),
hence the decision maker is indifferent between them. As mentioned above,
subjects do not necessarily follow the reduction axiom as they may care for
the number of lotteries in which they are to participate. On the other hand,
in both lotteries there are two stages of uncertainty, in one the probability of
success is p and in the other q. Although these uncertainties are resolved in a
different order, it may still be true that the decision maker is indifferent
between A and B, even if he does not follow the reduction of compound
lotteries axiom. Indeed, I suggest indifference to the order at which uncertainty is resolved in a slightly wider range of lotteries:”
(g) Order indijj’krence.
For every
((O,l),l-p;(x,,q,;...;
(xl,q,;..
.;x,,q,)~L~
and p>O,
~“,q”)~P)-,((~,~-P;~,,P),q,;...;(~,~-PP;~”,P),q,).
[See fig. 1.1
“‘For
Theorem
the possibility of replacing
3 below and Lute (1988).
the common
outcome
‘0’ by other
common
outcomes
see
383
U. Segal, Order indifference and rank-dependent probabilities
1-P
P
A
0
Xl
5,
0
Xl
0
5,
Fig. 1
Theorem 2. Let the preference relation tZ satisfy the compound independence
axiom. Then the following two conditions are equivalent:
(1) The preference relation k2 satisfies axioms (a)-(e) and (g).
(2) The orders kJ- and tn can be represented by the (same) continuous
anticipated utility functional (2) with an unbounded utility function u.
Proof.
. .. 5 x, and let the orders
(2)=(I).
Let 05x,5
represented by the same anticipated utility functional
t,-
and 2,,
be
By the first-order stochastic dominance axiom it follows that CE,(0,
1 -p;
SCCEr(O,l-p;x,,p)=CE,(O,l-p;x,,p).
We
may assume without loss of generality, that u(O)=O, hence the anticipated
utility of the lottery (0,l -p; x, p) is u(x) f (p). Suppose that W: L,-+%
represents the order &. It thus follows by the compound independence
axiom that
x,,p)=CE,(O,l-p;x,,p)~...
=f(p)
c
( 3 f (jzi
i 4.J) -f
(Ir:uxl[
i
(jzi+I
4.J)] + u( xn)f(q n) 1.
384
U. Segal, Order ind$ereerenceand rank-dependent probabilities
It is easy to verify that if the preference relation t2 can be represented by
the functional W, then it satisfies axioms (a)-(c) and (e)-(g). Since the
function u is unbounded, the relation & satisfies axiom (d) as well.”
(l)*(2).
It follows from Lemma 1 that there exists a measure 9 on
[0, co) x [0, I] such that for every t >O, the two orders tr and tn can be
represented on L: by the function V(X) = ,9*(X”). For t >O, let M’=
{(x,,p,;...; x,,p,) E L,:min {xi> 2 t}. Suppose that for every such t, there are
u’ and f’ such that when restricted to M’, the preference relations can be
represented by an anticipated utility functional (2) with a utility function u’
and a probability transformation function f’. Denote this representation by
“V. For 0 <t’ < t, 9’“” and -lr’ represent the same order on M’. Therefore
f” E f’ and u” is an increasing linear transformation of u’. Without loss of
generality assume that for t< 1, u’(x) =ul(x), x= 1,2, and for t> 1, uf(x) =
u’(x), x = t, t+ 1. It thus follows that for all t’ < t, u” and U’coincide on [t, 00).
Define f =f’, and for x>O, define u(x) =ux(x). Obviously, for x2 t, U(X)=
u’(x).
Next I show that lim x+O~(x) > - co. Otherwise, let pi+0 and xi-+0 such
that for every i, u(x,)[l -f( 1 -pi)] <41)-u(2).
It follows that for every i,
b1>(xi,pi;2, 1 -pi). By continuity 6,2d2,
a violation of the first-order
stochastic dominance axiom. It remains to prove the existence of u’ and f’.
To simplify the notations, assume that t = 0. That is, assume that there exists
a measure 9 on [0, co) x [0, l] such that both krand
t,, can be represented
by V(X)=@X’). In particular, 9 is finite on sets of the form [0,x] x [0, 11.”
Let
u(x) =G(x, 1) (as before,
G(x, P) = S(CO,xl x Cl-P, 11)). BY the
continuity axiom it follows that for every x and p there is y such that
(0,l -p; x, p) N r(y, l), hence u(y) = G(x, p) and u- ‘[G(x, p)] is well defined.
Note
uP’[G(x,p)]=CE~O,l-p;x,p)=CE,(O,l-p;x,p).
As
that
((0, l), 1 - 4; (0,l -P; x, ~1, q) N &A U, 1- P; CO,1 - 4; x, d, P), it follows that
G(u - 1 CW,
p)l,q)= G(u- ’ CG(x,41, P),
u-‘CG(u- ‘CG(x,p)l,dl =u- ‘CG(u-‘CW, dl,p)l.
Let
h:[O,co)x[O,l]+[O,oo)
It follows that
be given
by h=u-‘oG,
that
is,
h(x,p)=
u-‘[G(x,p)].
M&k PI, 4) = we,
41, P).
(3)
“The fact that the orders 2,. and >,, can be represented
by the same anticipated
utility
functional does not imply that the order k2 satisfies the compound
independence
axiom. For a
counterexample,
see Segal (1990).
“The only assumption
needing elaboration
is the Archimedean
axiom. Let t>O. Note that
the Archimedean
axiom implies that for every X E L: and for every E> 0, there exists x > t such
that (t, 1 -E;x,E)>X.
U. Segal, Order indifference and rank-dependent probabilities
385
Define H: (0, co) x {CO,1) u (1, co)> -+(O, co) as follows. For p E [0, l),
H(x,p) =h(x, 1 -p). For p> 1, define H(x,p) to be that number y such that
x = h(y, 1 -(l/p)). (Note that by the Archimedean axiom there exists y* such
that (0, (l/p); y*, 1 - (l/p))>,-6,.
Therefore, by the continuity and first-order
stochastic dominance axioms, H is well defined.) Let x > 0 and let p, q E [0, 1).
It follows immediately by (3) that
H(W,
P), q) = H(H(x, q), P).
Next, let p,q> 1. Define y=H(x,p),
z=H(y,q),
By the definition of H and by (3) we obtain
(4)
y’=H(x,q),
and z’=H(y’,p).
and
It follows by first-order stochastic dominance that z=z’. Finally, let p> 1 and
y,z, y', and z’ as before and obtain x=h(y, 1 -(l/p)).
Therefore,
qE [0,1). Define
Y’ = H(x, q)
=h(h(yJ-$,1-q)
=s z = H( y’, p).
Hence z = z’, and (4) holds for all p, q E [O, 1) u (1, co).
386
U. Segal, Order indifference and rank-dependent probabilities
Invoke now the following corollary from Aczil (1966, p. 273):
If x,9(x, w) E Z,w E P where Z is a real interval and 9(x, w) is continuous
in x and transitive, that is, 9(x, w) =y has at least one solution w for all
pairs x,y~Z, then
with arbitrary
solution of
sq2qx,
continuous
w), a]=9[F(x,
and strictly
monotonic
9 is the general
u), w].
In our case, Z=(O, co) and P= [0, 1) u (1, co). The function H is clearly
continuous in x. It is also transitive. Indeed, let x, y > 0. If x 2 y, then by firstorder stochastic dominance and continuity there exists pi [0,1) such that
+5,,, hence y = h(x, 1 -p) =H(x, p). If y> x, then by the Archi(O,P;x, l-P)medean and continuity axioms it follows that there exists p> 1 such that
(0, (l/p); y, 1 -(l/p)) N ,-SX, hence x = h(y, 1 -(l/p))
and by definition, y =
’
3
Therefore,
Wx, P).
Wx, P) = 9 - 1IX4 + WP)I.
Define &(O,co)-+% by &x)=3(x)
obtain that on (0, co) x (0, 11,
and tj:(O,l]+%
by
$(p)=%(l-p),
and
The value h(x, p) is the certainty equivalent of the lottery (0,l -p; x, p),
hence, by the first-order stochastic dominance axiom, the functions C#Jand II/
are either both increasing or both decreasing. If they are both decreasing,
replace 4 by 4* = -4
and replace + by $* = -$.
[Note
that
c$*-‘(x)=4-i(-x)].
Also,
h(x, 1)=x=$(1)=0.
By the first-order stochastic dominance axiom, the function Ic/ is strictly
increasing, hence for p E (0, l), $(p) -C 0.
Let p, q ~(0,l). By the continuity and first-order stochastic dominance
axioms, for every x*, y* >O there are positive x and y such that
(O,l-p;x*,p)>(O,l-q;y,q)
and
(O,l-q;y*,q)>(O,1-~;~,~).
Let
q>p>O
and let {Xi} and {yi} go down to zero such that for every i, (0,l -p; Xi,p)>
(0,l -q;y,,
q). Since 4 is increasing, so is 4-l. Therefore,
13Note that because
of the transitivity
condition
one cannot
define H for x=0.
U. Segal, Order indifference and rank-dependent probabilities
(‘3l --Pi
Xi, P)>(Q
l -q;
387
yt, q)
=> h(xi2P)>h(Yi3 4)
*
+txi)
+
V+(P) ’
$(Yi) + $(4)
It thus follows by the continuity of 4 that lim,,, 4(x)= -co. Similarly,
lim,,, $(p) = - co.
Consider now the lotteries of the order indifference axiom with 05x, 5
... 5x,. By this axiom
((O,l),l-p;(x,,q,;...;
~“,q”)~P)-_,((~,~-P;~,,P),q,;...;(~,~-P;~,,P),q,),
hence
n-l
=G(u-‘CG(xn,P)l,qn)+
U-‘CG(Xi,P)l,
C
i=l
u-‘CG(xi,P)l,
i
qj
j=i
>
i
qj
j=i+l
.
(5)
By the Archimedean axiom it follows that 9 is unbounded (see below). By
the Archimedean and continuity axioms it follows that for every E>O, OI>0
and n one can choose XT5.. .s xx and qT, . . . , q.* such that x1= 1 qT = 1 and
i=l ,..., n,
and
G(x:,j-$lq+e.
i=l,...,
n-l.
Since this construction is certainly possible for some a> 0, it follows that 9 is
unbounded. (See fig. 2.)
For every x: and p, u- ’ [G(xF,p)] 5x,!: hence
388
U. Segal, Order indifference and rank-dependent probabilities
q; + . . . +
g-1
0
.
=2
=;
..
Fig. 2
u-‘CGW,PII, i
4: 5 6, i=l ,.*.,n-1.
j=i+l
Substituting into (5) one obtains
where Ial, 1~1s(n-
1)~. This equation implies
44 - 1C4(u - 1Gnu
+ 01)+ Ic/(P)l)
n
=
I:( [
u
i=l
Indeed,
qv’
4b*)+$(P)+$
+z.
j=i
)1>
i 4;
(
=;
U. Segal, Order indifference and rank-dependent probabilities
389
=u(h(u_‘[na+a],p))
= 44 - 1c+tu - l Cna+ 01) + VVPU)
and
2 G(ulCGb:,p)l,
i=l
n
u
=
i=l
=
j=i
i q:)+~
5
cc j=i
)I
+r
4;
h h(xT,p),
”
=
b-[
z(
i=l
”
zz
i=l
j=i
11)
j=i
)I)
1 4(4-‘cddx:)+rl/(Pm+II/
U
=
u 6
+$(p)+$
(
Lc
(
i 4T
+z
+-z.
By the definition of ~1,
~~‘(~)=u-‘[G(x:.~~~f)]
=h(x~,~~q:)=g-‘[$(xfl+B(~i4:)
*+(lC1(CL))=4(XT)+$
Hence,
i
i=l
(j=i
>.
i 4;
u(~-l[m(xt)+~(~i4:)+~(P)])
=i~lu(i-1Cl(u-1(a))+9(P)1)
=dC
‘C4(u- ‘(4)++(P)l).
(7)
390
V. Segal, Order indijizrence and rank-dependent probabilities
It follows therefore from (6) and (7) that
U(4-1c+(U-1Cn~+fll)+ wJ)l)=~~(~-‘Cd4~- ‘(4 +vml) +z.
Since this equation holds for all E, it follows by continuity that
44- ‘C4(u-1Cfl~l)+ em
Let w=uo$-l
40-
and w-~=~ou-~
‘(4+&4)
=n44- ‘Cd4u-‘W + 4wl).
(8)
and obtain from (8) that
=n4w-‘(a)
+ Ii/(P)).
This last equation holds for all positive integers n and c1>O. Hence, for all
positive integers n and m and for all cc>O,
u(w-l(a)+‘tl/(p))=w
(u-l (y)+*(P))
=nu(u-‘(g+*(P))
=Ao(w-l(a)+~(p))=
u(u-(~)+*~P~)
n
By continuity it now follows that for all ~1,fl> 0,
e4+(B4
+Il/(p)) =/Wo-
‘(4 + ti(P)).
Let S(s,t)=w(w-‘(s)+IC/(t)),B(t)=S(l,t),
S(Pa, p) = BS(a, P)
=>S(s,t)=sfqt)
=>o(t)=o_‘(e(t))-o-‘(l)
and obtain
391
U. Segal, Order indifference and rank-dependent probabilities
The solution of this functional equation is w-‘(x) =alnx+b
(1966, p. 39)], hence o(x) =ecx-b)‘a. By their definitions,
[see AczCl
u- l CW, PII= &G P)= 4 - l C4(4+ +(P)I,
hence
G(x,p)=o(~(~)+Il/(p))=e[“‘““~(~‘-~~’~.
Define
U(x) = etW’ - bl/a
and
f(p) = eIJl(P)l/a
to obtain that G(x, p) =u(x)f(p).
and f(l)= 1.
Q.E.D.
Note that lim,,,
U(X)= 0, lim,,,
f(p) =O,
4. Other versions of order indifference
This section is devoted to a discussion of several potential alternatives to
axiom (g). First I show that if the order indifference axiom is replaced by the
weaker axiom
then Theorem 2 ceases to hold. Indeed, suppose that the preference relation
t can be represented by a measure of the epigraphs of the cumulative
distribution functions with the generating function
G(x, P) =
1
(l/x)-lnp’
xp’o
xp=o
: 0,
Obviously, G(x, 1) =x and by the compound independence axiom
wo, 1191- 4; to,1 =
p;x, P), 4)
Y(O, 11,1- p;(0,l -
q;
x, q),
p) =
1
(l/x)-lnp-lnq’
One has still to show that the function G does not induce a product measure
9. Let
392
U. Segal, Order indifference and rank-dependent probabilities
NCx,~1 x Cl-P, 1- d) = G(Y,PI- G(x, P) - G(Y,d + G(x, d
1
1
1
=(1/y)-lnp
- (l/x)-lnp
- (l/y)-lnq
If 9 is a product
1
+(1/x)-lnq*
then for every x < y < z and r < q < p,
measure,
Let x = 1, y = 2, z = 3, p = 0.6, q = 0.5, and r = 0.3 and obtain that the two sides
of this last equation do not equal each other.
Next, I discuss two possible stronger versions of the order indifference
axiom.
(h) Order indifference
(x,,q,;...;
with respect
to every common outcome.
For
every
and ~20,
x,,q,,)E&,p>O,
((Y,1),1-P;(xl,q,;...;x,,q,),P)
(i) General order indifference.
xi=(xI; ,...,
p=(p1,...,
PA
xi),
For every
i=l ,..., m,
and q=(ql,...,aA
((x:,ql;...;x,l,q,),P,;...;(x~,q,;...;x~,q”),P,)
Definition.
for everyp,
Theorem 3.
dence axiom.
The probability
f(p)+f(l-pp)=l.
transformation
Let the preference
Then the following
function
f is called symmetric if,
relation tz satisfy the compound
two conditions
are equivalent:
indepen-
(1) The preference relation tz satisfies axioms (a)<e) and (h).
(2) The orders kr and t,, can be represented
by the (same) continuous
anticipated
utility functional (2) with an unbounded
utility function u and
a symmetric probability
transformation
function f.
Proof. (2) =>(I).
Let W: L,+%
represent
an order tz and
. ..sx.,.
Assume first that ysCE(x,,q,;...;x,,q,).
...sxksysxk+lJ
for a symmetric function f,
let 0 5x1 5
Then
U. Segal, Order indifference and rank-dependent probabilities
WY,
393
1--Pi x,,p),q,; *. . ;(y, 1--pi %P),4”)
)I
n-l
+i=F+, {“(Y)C1--f(P)l+u(xi)f(P)}
-“ml
+ {U(Y)Cl
[
f
L
,I.%
)
-f
(
j=F+l %
n
+aMP))f(4”)
n-l
= izI {4Y)C1 -f(P)1 + u(xi)f(P)l
f
Kl
j;i 4j
-f
r
j=g
1 4j
11
+ {4YW -f(P)1 +aJf(P)}f(q,)
=u(Y)cl -f(P)1
The proof for the case y> CE(x,, ql;. . . ; x,, q.) is similar.
(l)=>(2). It follows from Theorem 2 that the orders t,- and k,., can be
represented by (the same) anticipated utility functional with an unbounded
utility function U. It remains to prove that f is symmetric. Assume, without
loss of generality, that u(0) =O. First I show that f(f) =f. Let x* > 1 be the
outcome for which u(x*)f(f) = u( 1).14 By axiom (h) it follows that
((0, $; l,M;(x*,k
By the compound
one obtains
2u(I)f(3CI
1,t,,4, ~2((1,l),+;(O,+;x*,t),+).
independence
-ml
axiom and the anticipated
utility functional
+u(x*hm)=41),
hence f(f) = $. Similarly,
((031 -_p; LP),+;(x*,
l-
p; LP),6)“2((1,
aP;(Qf;x*,9,1
hence
‘%ch
an outcome
exists by the continuity
and the Archimedean
axioms.
-PI,
394
V. Segal, Order indlrerence and rank-dependent probabilities
wu-(P)+4Nl-f(l
It thus follows that f(p)+f(l
-PII +u(x*)fU -PII =u(l).
-p)=
1.
Q.E.D.
This proof also shows why one cannot replace the common outcome 0 in
axiom (g) by another common outcome y. Even if the preference relation can
still be represented by an anticipated utility functional, it follows from the
last proof that f(p) + f( 1 -p) = 1. This is so because in the anticipated utility
model the value of an outcome depends not only on its probability, but
also on its position relatively to other possible outcomes.
It is easy to verify that the general order indifference axiom implies axiom
n, and m=2. It now follows immediately
(h). To see this, let xf=y,j=l,...,
from Theorem 3 that replacing axiom (g) by axiom (i) yields representability
by an expected utility functional. Indeed, let x1 =(a, b,a) and x2 =(a,c,d)
where u(a) =O, u(b) = 1, u(c) = 2, and u(d) = 3, and let p1 =pz =). It follows
from (2) that f(p2+p3)=f(p2)+f(p3),
hence f is linear [see Acztl (1966,
p. 34)], and the anticipated utility functional reduces to expected utility.
5. Some remarks on the literature
Different versions of the anticipated utility (or rank-dependent) functional
have received so far several axiomatic representations [see Quiggin (1982);
Yaari (1987); Roe11 (1987); Lute (1988); Chew (1989); C hew and Epstein
(1990); and Segal (1984, 1989, 1990)]. The approach presented in this paper
obtains the anticipated utility form from axioms made on two-stage lotteries.
In that it is similar to Lute (1988) and to Segal (1990). Instead of weakening
the mixture independence axiom, these approaches suggest to replace the
reduction of compound lotteries axiom by weaker axioms. In Segal (1990)
the reduction axiom is replaced by several possible extensions of the concept
of first-order stochastic dominance to two-stage lotteries. With a little abuse
of notations, for X, YEI,,, let X n Y be the lottery whose epigraph consists
of the intersection of the epigraphs of the cumulative distribution functions
of X and Y, and let X u Y be the lottery whose epigraph consists of the
union of those of the cumulative distribution functions of X and Y. For a
lottery A =(X1,(1/m);. . .; X,,(l/m)) E L,, define
Aij=
(
X,,l;...;XinXj,~;...;Xi"Xj,';...;X,,'
m
m
m>
.
The weak upper compound dominance axiom assumes that A tz Ai,, and the
weak lower compound dominance axiom assumes that Aijtz A. It is then
proved that under assumptions (a)<c) and (e)-(f), the preference relation
satisfies the upper (lower) compound dominance axiom if and only if it can
U. Segal, Order indifference and rank-dependent probabilities
395
be represented by an anticipated utility functional with a convex (concave)
probability transformation function J Note that in Theorem 2 above there is
no restriction on the possible shapes of the function J Of course, axiom (g)
is entirely different from the upper and lower compound dominance axioms,
as it deals with the order at which the uncertainty is resolved and not with
other potential lotteries the decision maker could have won in the second
stage.
Lute (1988) suggests an axiomatization of the rank-dependent model based
on two-stage lotteries with events, rather than probabilities. Although it is in
that respect more general, I prefer to present his axioms in terms of
probabilities, to make it easier to compare them to the model presented in
this paper. His central ‘accounting’ equation is that for every x, y, p and 4,
((Y, l), l-q;(y,
1-P;x,P),q)~Z((Y,
1171 -P;(Y,
l-%X,&P).
(Compare this axiom with the beginning of section 4). This axiom is weaker
than axiom (h) above, as it is restricted to only one possible outcome x.
However, it is not clear whether it is stronger or weaker than the order
indifference axiom. On the one hand it restricts the lottery (x,,p,; . . . ; x,, p,)
to the degenerate lottery 6,. On the other hand, it requires indifference with
respect to all y, not only y = 0.
Lute’s other axiom requires the following. For each vector of probabilities
Pi,..., pn+ 1 such that ~~~~ pi= 1, there are ql,. . . ,qn and q such that
Cy=1qi=l and such that for everyx,s...Sx,+,,
((xl,P,;*..;
X,+1,Pn+1),l)N2((X1,41;...;Xn,4n),4;(Xn+1,1),1-4).
Of course, this second axiom is completely different from the invariance
axiom used above.
Chew (1989) suggests the following axiom. Let x=(x1,. . .,x,) and y=
where xiS...Sx,
and yis...
5 y,.The vectors x and y are called
(Yi,...,yJ
rank preserving if, for every 1< i < II, yi _ 1 5 xi 5 yi + 1 and xi _ i 5 yi 5 xi + I. The
pair (w,z) is a rank preserving rearrangement of (x, y) if w and z are rank
preserving and, for every i, {wi,zi} = {xi,y,}. I5 Chew’s commutative substitution axiom assumes that there exists cr~(O, 1) such that for every vector of
probabilities (pl,. . . , p.) and rank preserving x and y,
C~(C~h,
a; Y,,
1 --a),Pl,.
. .; cm,,
for any rank preserving rearrangement
“For
example, x=(1,3,6),
y=(2,4,5),
w=(1,4,6),
KY,,
1 -f$,P,)
(w, z) of (x, y). This axiom weakens
and z=(2,3,5).
396
U. &gal, Order indifference and rank-dependent probabilities
Quiggin’s (1982) major axiom by assuming that such a just exists, while
Quiggin assumes it for a=$. Together with some other technical assumptions, Chew proves that this axiom implies representability by the anticipated
utility functional. He does not assume that the preference relation can be
represented by a measure of the epigraph of the distribution function, but his
commutative substitution axiom is strictly stronger than the order indifference axiom of this paper. To prove this I show that Theorem 2 does not
hold true without axiom (c). For X=(X~,P~;...;X~,~,,),
let V(X)=max
{v,(X), v,(X)1 w h ere for i = 1,2, y is an anticipated utility functional of the
form (2) with the linear utility function u(x) =x and the probability
transformation function fi.16 It is easy to verify that unless fi and fi do not
intersect each other, this functional is not a transformation of a measure of
the epigraphs of the cumulative distribution functions.” It is therefore not
an anticipated utility functional, hence it does not satisfy the commutative
substitution axiom. This functional satisfies the order indifference axiom. Let
X=(x,,q,;...;%l,q,).
W(O>11,1--Pi
x, p)= w, 1-
p; V(X),
p)
On the other hand,
= V(x,~max(f~(~),f~(~)},q~;...;x,~max(f~(p),f~(P)},q,)
V,(X,
. max
{f,(p), f2(p)), ql;.
. . ; x,
*max{_fl(P),
f2(P)13 4.))
Of course, Chew (and Quiggin) do not consider their axioms as decision
rules for comparing two-stage lotteries, but as weaker versions of the mixture
independence axiom.
16VI and V, are Yaari’s (1987) dual theory functionals.
17V is not an anticipated
utility functional
with the
f(p)=max(f,(p),f,(p)).
probability
transformation
function
U. Segal, Order indifference and rank-dependent probabilities
391
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