242 F
. Philippe Mathematical Social Sciences 40 2000 237 –263
Additionally, since clearly
E
j dV 5 2
E
2 j dv,
6
E E
the dual capacity V of the capacity v satisfies
E
j dV 5
O
yXmax jx.
x [X
X E E
E
If v is moreover convex, the set corev of all probabilities on E, 2 which dominate v is not empty Shapley, 1971, and the above Choquet integral satisfies Choquet, 1953,
54.2
E
j dv 5 inf
E
j dQ. 7
Q [corev
E E
3. The decision criteria
3.1. Cumulative prospect theory
S
Let S be the set of the states of nature, and A 2 an algebra of events. Let X be the
set of consequences, endowed with its powerset algebra; the status quo consequence is denoted by
g. A decision d is an A-measurable finite-ranged mapping: S
→
X, and we
will write d 5 A , c ; . . . ; A , c
1 1
n n
21
when A 5 d
c for 1 i n and
hA , . . . ,A j is a partition of S; a decision d is h j
i i
1 n
n-ranged if n is the cardinality of dS. The set D of the decisions to be considered is
assumed to contain each constant i.e. one-ranged decision S, c, so that preference on D induces preference on X. Both preference relations are denoted by the symbol |
1
and s for their respective symmetric and asymmetric parts. The further symbols X
2
and X stand for the subsets
hc [ X: cg j and hc [ X: g cj respectively, namely
gains and losses in the CPT model; for the sake of interest, we assume in the rest of the
1 2
paper that neither X nor X
reduces to hg j. The consequences of a decision A ,
1
c ; . . . ; A , c are agreed to be indexed in increasing order:
1 n
n
c . . . c . . . c .
n k
1
Under uncertainty, cumulative prospect theory is based on the following frame: consequences being described as deviations with regard to the status quo, the d.m.
expresses simultaneously her information on ambiguous events and its impact on
1 2
preference by means of two capacities v and v
defined on the algebra of events A.
1
Given an event A in A, the value v A could, e.g., be obtained by searching a probabilized event B for which the d.m. shows indifference between a bet on A and a bet
2
on B with the prospect of only gains; likewise, v A could be obtained via bets against
F . Philippe Mathematical Social Sciences 40 2000 237 –263
243
A and B, with the prospect of only losses Sarin and Wakker, 1994. Criterion 8 presented hereafter is derived from several axiomatic systems.
1 2
We assume that there exist two capacities v and v
defined on A, and a bounded function u:X
→
R with ug 5 0, such that preference of the d.m. on the set D of
decisions is representable by the following CPT criterion: for each
d 5 A , c ; . . . ; A , c in D with c . . . c | g . . . c ,
1 1
n n
n k
1 k 21
i i 21
2 2
U d 5
O
v A
2 v A
uc
F S D S DG
j j
i j 51
j 51 i 51
n n
n 1
1
1
O
v A
2 v A
uc . 8
F S D S DG
j j
i j 5i
j 5i 11 i 5k 11
Using the Choquet integral 1, another expression of 8 is
1 1
2 2
U d 5
E
u + d dv 2
E
u + d dv ,
9
S S
1 2
1
where u x 5 max0, ux and u 5 2u .
1 2
1 2
c
Notice that, if v 5 V
5 v i.e. v A 1 v A 5 1 on A, then by 6
1 1
2 2
1 2
E
u + d dv 2
E
u + d dv 5
E
u + d dv 1
E
2 u + d dv,
S S
S S
so that criterion 8 reads U
d 5
E
u + d dv,
10
S
which is precisely the CEU criterion with respect to v and the utility function u. In particular, if v is a probability measure then criterion 10 is the SEU criterion with
respect to v and u. Therefore, the CPT criterion may be thought of as a generalization of the CEU criterion, this last criterion in turn generalizing the SEU criterion.
Conversely, Proposition 1 shows that, faced with probabilized information, the consistency of criterion 8 with the EU criterion with respect to a probability measure P
1 2
forces the identities v 5 v 5 P. Observe that this result is not self-evident since, in the CPT model, capacities are not necessarily additive on probabilized events. A priori, they
only have to be ordinally equivalent with the involved probability.
3.2. Linear utility under imprecise risk Under imprecise risk, the d.m. is able to express her information with the help of an
m-closed set P of probabilities instead of only one under risk defined on the considered algebra of events. In this section and the following one, we assume that this kind of
information is available for certain events of A, which constitute a subalgebra of A that is denoted by A
in the sequel. The additional symbol D will stand for the subset
ULP ULP
of D containing the A -measurable decisions.
ULP
244 F
. Philippe Mathematical Social Sciences 40 2000 237 –263
The subset of A on which lower and upper probabilities f and F coincide i.e. the
ULP
class of unambiguous events with reference to f is denoted by A ; it is closed with
P
respect to complementation and disjoint union a class of sets with such properties is sometimes called a
l-system, because 2 f and F are subadditive. The common
restriction to A of f and F is denoted by P, and P is additive on A . When f is convex,
P P
it is known Jaffray and Philippe, 1997 that A is an algebra, so that P is a probability
P
on A put in other words, A
is an algebra iff the restriction of f to the algebra
P P
generated by A is convex, but A
is not closed under intersection nor union, of
P P
course in general; nevertheless, each finite partition of S composed of elements of A
P
generates an algebra that is contained in A , so that the expression ‘A -measurable
P P
decision’ makes sense. The subset of D containing all such decisions is denoted by
ULP
D . The case of A 5A is not considered in this paper, since imprecise risk reduces
P ULP
P
then to risk; that is why A is now assumed to be strictly contained in A .
P ULP
Several instances of the EUIR model have been axiomatized, all of which requiring that f be at least convex. This requirement is relaxed in the further instance that is
presented in Appendix A, which not only illustrates the concepts and methods used but also fits the assumptions made in Section 4. Each decision
d [ D induces a capacity
ULP 21
f + d
on X, which is denoted, for short, by f in the sequel. Since d is finite-ranged, we
d
¨ may identify f with its restriction to
dS, so that we can consider the Mobius inverse of
d
f with respect to dS. The assumption is made that, under imprecise risk, choice
d
between decisions may be identified with choice between the induced capacities on the set X of consequences. Accordingly, preference, denoted by again, is given on a
convex set G of capacities containing the set
h f : d [ D j, and is consistent with the
d ULP
restriction to D of the preference relation on decisions:
ULP
d d 9 ⇔
f f .
d d 9
Application of EU theory to the convex set G is supported by representation 4; it
leads to the decision criterion H defined on D in the following way. Given a decision
ULP
¨
d [ D
, denote by
f the Mobius inverse of the induced capacity f defined on dS,
ULP d
d
d S
2 . If G contains both sets
h f : d [ D j and he : C dS, d [ D
j, and if there
d ULP
C ULP
exists a linear utility L representing on G, set hC 5 Le and Hd 5 L f to
C d
obtain H
d 5
O
hC f C.
11
d C
d S
Notice that the vNM utility function h is defined on a set of finite subsets of consequences. Ultimately, insofar as H and h derive from the linear utility function
L, both are unique up to a common positive affine transformation.
¨ In particular, for each
d [ D the Mobius inverse f vanishes everywhere but at the
P d
21
singletons, and f
hcj 5 Pd c for all c [ dS. Accordingly, criterion 11 turns
d
into
21
H d 5
O
h hcjPd c,
12
C
d S
F . Philippe Mathematical Social Sciences 40 2000 237 –263
245
which is the EU criterion with respect to the probability P, with a von Neumann utility function under risk given on X by :c∞hhcj.
It must be noted that, commonly, the d.m.’s information is directly expressed with the
S
help of a probability distribution f on 2
S finite, and that the capacity f is a
by-product. As an unsophisticated example, let us consider once again the Ellsberg urn described in the Introduction. The d.m. only knows that the ball may be red with
probability 1 3 and black or yellow with probability 2 3. Denote by S 5 R B Y the
S
S 2
set of the 90 colored balls and by f the probability distribution on 2 , 2 given by
1 2
] ]
fR 5 and
fB Y 5 . 3
3
It can be shown that the true probability on S is only known to belong to the set core f ,
S
where f, defined on S, 2 by
fA 5
O
fB,
S
B [2 , B A
is a belief function see e.g. Nguyen and Walker 1994, or Jaffray and Wakker 1994 for a more general set-up; thus criterion 11 may be seen as the EU criterion with
S
respect to the probability on 2 whose distribution is f.
4. Consistency
