250 F
. Philippe Mathematical Social Sciences 40 2000 237 –263
1 2
2
considered here does not force a , a [ 0, 1 . For example, consider an Ellsberg
f g
urn containing 100 either red or yellow balls, with at least n , 50 balls of each color; the possible events are R a red ball is drawn from the urn and Y. Further consider an
algebra B endowed with a probability P the range of which is 0, 1 such an algebra
f g
can be generated by iterative tosses of a fair coin, and assume that each event in B is independent with both events R and Y. Now, define A
as the algebra generated by B
ULP
and R; it can easily be verified that the available information about A is characterized
ULP
by the lower probability f given for A, A9 in B by
n n
] ]
fA R A9 Y 5 maxPA, PA9 1
S
1 2
D
minPA, PA9. 100
100
2
Moreover, A is the set hA R A9 Y: A, A9 [ B , PA 5 PA9j, and we get, for
P
PA ± PA9, F 1 f
100 PA 1 PA9
]] ]]] ]]]]
A R A9 Y 5 ,
100 2 2n F 2 f
uPA 2 PA9u
1 2
21
so that max u2a 2 1u, u2a 2 1u 1 2 n50 . Therefore, large values of n i.e.
1 2
precise information about the urn enable large gaps between a
and 1 2 or a
and 1 2.
Let us now focus attention on Proposition 3: at first sight, the mapping g described in the latter does not necessarily represent the preference of the d.m. on the finite subsets of
X. As a matter of fact, uniqueness of h refers to a preference defined on the convex hull G of the union
h f : d [ D j he : C dS, d [ D
j,
d ULP
C ULP
but U only represents the preference relation on the set
h f : d [ D j. The equality
d ULP
U 5 H holds, of course, but it does not necessarily entail the equality h 5 g; actually, no decision
d is guaranteed such that f 5 e for a finite C X, unless totally uncertain
d C
events i.e. events on which F 2 f equals 1 are available. Nevertheless, preference of the d.m. on the set
he : C dS, d [ D j could be represented by g without
C ULP
modifying the values of the criterion on D ; in fact, h 5 g seems to hold frequently,
ULP
but let us consider first a negative example. Let Q be a probability on A and let
ULP 2
f 5 Q ; it can easily be verified that f is convex, that 1 2 f 1 F 5 Q, and that, for each ¨
three-ranged decision
d, the Mobius inverse f of f vanishes at dX. If D
only
d d
ULP
contains k-ranged decisions with k 3, we cannot conclude that h 5 g in this case, however, values of h on sets C with
uCu 3 are vain.
Now, let us present a favourable example. Assume for the sake of simplicity that D
ULP
contains every A -measurable decision, and let
d 5 A , c [ D
. Proposition 3
ULP i
i ULP
yields the identity
O
hC 2 gC f C 5 0.
d C
d S
If, for each n 2, A contains a subalgebra A generated by n atoms such that the
ULP n
¨ Mobius inverse
f of f w.r.t. A satisfies f S ± 0, then hC 5 gC holds for each
n n
n
finite subset C of X, as shown by induction on the cardinality of C — use the
F . Philippe Mathematical Social Sciences 40 2000 237 –263
251
above-mentioned identity and remark that h and g coincide at the singletons by 19 and Proposition 1.
Note that the latter example establishes that h and g always coincide at the pairs,
because each event in D \D
generates a fitting algebra A . Also remark that the
ULP P
2
richness assumption about A is conceivable. For instance, algebras A for n 3 may
ULP n
be obtained in the following way: consider an Ellsberg urn that contains n colored balls, n different colors c , . . . ,c being possible but their distribution unknown; then take the
1 n
algebra generated by the n events ‘one c -colored ball is drawn from the urn’. The
i
¨ Mobius inverse of f w.r.t. each A takes the value 1 at S and vanishes elsewhere, just
n
like f itself. Further notice that these n events form a totally uncertain partition of S.
Let us finally examine the case of a convex lower probability f. The next Proposition
states that both consistent criteria coincide on D with a noticeable Hurwicz-like
ULP
criterion. Denote by H the Hurwicz criterion with constant index a:
a
H d 5 a
inf
E
u + d dQ 1 1 2 a
sup
E
u + d dQ.
20
a Q [core f
Q [core f
S S
1
Let d 5 A , c [ D with c . . . c | g . . . c ; we define the decision d
by
i i
n k
1 n
dx if x [
A
1 i 5k 11
i
d x 5 ;
H
g otherwise
2 k 21
likewise, we define the decision d
with A .
i 51 i
Proposition 4. Using the assumptions of Proposition 2: if f is moreover convex, then, for each
d in D ,
ULP 1
2
U d 5 Hd 5 H
d 1 H d .
1 2
a a
Proof. Referring to the proof of Proposition 3 together with 7, we get
1 1
1 1
U d 5 a
inf
E
u dQ 1 1 2 a
sup
E
u dQ
Q [core f
d
Q [core f
d
d S d S
2 2
2 2
2 a
sup
E
u dQ 2 1 2 a
inf
E
u dQ.
Q [core f Q [core f
d d
d S d S
Since f is convex and d is simple, it is known Jaffray and Philippe, 1997 that core f
d 21
is exactly the set of all induced probabilities Q 5 Q + d
with Q [ core f . Then the
d
claimed formula is easily verified. h
The above proof shows that the convexity of f is not a necessary condition for the result to hold, since a convex f satisfying the property core f 5
dcore f suffice.
d d
Furthermore, if hC 5 uM for all finite C then
C
H d 5
E
u + d dF 5 sup
E
u + d dQ;
Q [core f
S S
252 F
. Philippe Mathematical Social Sciences 40 2000 237 –263
likewise, hC 5 um entails
C
H d 5
E
u + d df 5
inf
E
u + d dQ.
Q [core f
S S
Gilboa and Schmeidler 1989 and Chateauneuf 1991 independently justify a model which gives rise to a situation of purely subjective imprecise risk, described by a lower
expectation; the resulting maxmin criterion has the latter form.
5. The reciprocal problem
