MATHEMATICS OF OPERATIONS RESEARCH
Vol. 24, No. 3, August 1999
Printed in U.S.A.
DECISION MAKING WITH MONOTONE LOWER
PROBABILITIES OF INFINITE ORDER
FABRICE PHILIPPE, GABRIEL DEBS,
AND
JEAN-YVES JAFFRAY
Properties of convex and monotone capacities of infinite order in Polish spaces are studied and
used to justify the representation of certain situations of imprecise risk (imprecisely known
probabilities) by lower probabilities, which are monotone of infinite order. Decision making with
imprecise risk is then modeled, and linear utility theory is shown to be generalizable to the case of
imprecise risk.
1. Introduction. Decision theory generally formalizes problems of choice under
uncertainty by introducing a set of states of nature ᏿, a ␴-algebra of events Ꮽ, which is a
subset of 2 ᏿, and a set of consequences ᐄ, also endowed with a ␴-algebra Ꮽ⬘. Additionally,
each decision is characterized by a measurable mapping ␦ from (᏿, Ꮽ) into (ᐄ, Ꮽ⬘).
In the particular case of uncertainty called risk, the existence of a probability measure P
on (᏿, Ꮽ) makes decision ␦ a random variable with distribution ␦ (P) ⫽ P 䡩 ␦ ⫺1 on (ᐄ, Ꮽ⬘).
Normative models of choice under risk usually ensure that the value of a decision depends
only on its distribution, by defining preference among decisions indirectly through an
ordering Ɑ on the set ᏹ(ᐄ) of all probability measures on (ᐄ, Ꮽ⬘). Additional rationality
requirements, expressable as properties of Ɑ in ᏹ(ᐄ) (weak ordering axiom A1, independence axiom A2, continuity axiom A3), characterize linear utility theory, from which the
expected utility (EU) criterion can be derived, provided the validity of a dominance axiom
A4 is further assumed.
In this paper, we consider a more general case of uncertainty. Such a case arises when
imprecise or incomplete data permit the location of the probability measure P on the events
only in a set ᏼ, hence the location of the distribution of a decision ␦ in the subset ᏼ⬘ ⫽ { ␦ (P)
: P 僆 ᏼ} of ᏹ(ᐄ). The set ᏼ is assumed to be characterized by its lower envelope f, which
is monotone of infinite order on Ꮽ; we call such a situation imprecise risk. Moreover,
requirements on ␦ ensure that these properties of ᏼ and f are inherited by ᏼ⬘ and its lower
envelope f ⬘. Thus the situations of uncertainty in question generalize situations of risk in a
natural way; knowledge concerning the possible consequences of a decision is no longer
representable by an additive measure, but instead by a particular nonadditive set function—a
monotone capacity of infinite order (the term capacity, used in a broad sense here, will be
specified later).
Turning, then, to decision making in this case of uncertainty, we cause preference among
decisions to be dependent on the relevant data by the previously-mentioned device. That is
to say, we define preference indirectly through an ordering Ɑ on the set ⌫ ⬁(ᐄ) of all
monotone capacities of infinite order on (ᐄ, Ꮽ⬘). Axioms A1, A2 and A3 of linear utility
theory still make sense, since ⌫ ⬁(ᐄ), like ᏹ(ᐄ), is a convex set; moreover, it turns out that
traditional rationality arguments, used for justifying these axioms in the case of risk, can still
be put forward in this particular case of uncertainty. Consequently, we apply linear utility
theory and show that, by adding new dominance axioms A5 or A5S, we obtain a
Received November 28, 1997; revised October 10, 1998 and May 19, 1999.
AMS 1991 subject classification. Primary: 62C99.
OR/MS subject classification. Primary: Decision analysis/Risk.
Key words. Upper/lower probability, capacity, utility theory, decision theory.
767
0364-765X/99/2403/0767/$05.00
Copyright © 1999, Institute for Operations Research and the Management Sciences
768
F. PHILIPPE, G. DEBS AND J.-Y. JAFFRAY
representation by criteria all of which are extensions of the EU criterion (to which they
reduce in the particular case of risk).
This axiomatic approach has already been used for deriving decision criteria under
imprecise risk by Jaffray (1989a, 1989b, 1991) and Hendon et al. (1993). However, in all of
the aforementioned papers, the set of states of nature is assumed to be finite and the
expressions of the criteria involve the Möbius transform (see, e.g., Shapley 1953, Rota 1964)
of the lower probability f. Since this concept does not readily generalize to the infinite case,
the extension of such results is all but straightforward. Fortunately, Choquet’s (1954) theory
of capacities can provide appropriate tools that we shall only need to adapt to the case of
Polish spaces, which is now the standard framework of probability theory (see also Gilboa
and Schmeidler 1995, Marinacci 1996, Denneberg 1995 for recent related works).
It must be noted that there is a rather large body of work on uncertainty involving
capacities, starting with Schmeidler’s seminal paper (1989). This paper, as well as Gilboa
(1987) and Wakker (1989), generalizes Savage’s subjective expected utility theory by
relaxing the additivity of probability. These models are quite different in spirit from ours,
since we take lower/upper probability intervals to represent imprecise (but objective) data;
some elements of comparison can be found in Jaffray and Philippe (1997). Finally, subjective
approaches, also resulting in a probability interval representation, have been proposed by
Gilboa and Schmeidler (1989) and Walley (1991).
The paper is organized as follows: Some properties of lower probabilities in Polish spaces
are studied in §2. Special attention is devoted to the cases of convex lower probabilities
(§2.3) or those which are monotone of infinite order (§2.4). The last subsection (2.5) studies
the images of convex or monotone cocapacities of infinite order induced by decisions.
Properties of the set of all monotone cocapacities of infinite order are studied in §3. Finally,
in §4 the decision model is formally defined and axiomatic requirements on preference
among decisions are shown to lead to specific decision criteria.
2.
Lower probabilities as cocapacities on Polish spaces.
2.1. Polish spaces. Given a Hausdorff topological space ᏿, we denote by Ᏻ(᏿) (or
simply by Ᏻ when no confusion is possible) the set of the open sets of ᏿, and by Ᏺ(᏿), ᏷(᏿),
and Ꮽ(᏿) (or Ᏺ, ᏷, and Ꮽ) the sets of the closed, compact, and Borel sets of ᏿ respectively.
If X is a topological subspace of ᏿, note that ᏷(X) is also the set {K 僆 ᏷ : K 債 X}.
The set ᏷ will be endowed with the Hausdorff topology generated by the topology of ᏿;
the Hausdorff topology has as its basis the sets ᏷(G) and ᏷(F) c , the complementary set in
᏷ of ᏷(F), for all G in Ᏻ and all F in Ᏺ.
Let X be a topological subspace of ᏿. From the straightforward properties
᏷共G兲 艚 ᏷共X兲 ⫽ ᏷共G 艚 X兲
for all G 僆 Ᏻ,
᏷共F兲 c 艚 ᏷共X兲 ⫽ ᏷共X兲⶿᏷共F 艚 X兲
for all F 僆 Ᏺ,
it follows that ᏷(X), endowed with the Hausdorff topology, is a topological subspace of ᏷.
Consider now the particular case where ᏿ is Polish, i.e., the topology of ᏿ has a countable
basis and can be defined by a complete metric; it is known that the Polish subspaces of a
Polish space are its G ␦ -subsets (intersections of a sequence of open sets), and that ᏿ can be
embedded in a compact metrizable (therefore Polish) space ᏿̂ (see, e.g., Cohn 1980); since
᏿̂ is compact, ᏷(᏿̂) is metrizable (by the Hausdorff metric associated with any distance
metrizing ᏿̂) and compact (see, e.g., Dellacherie 1972, Chapter 3). The equality
᏷
冉艚 冊
Xn ⫽
nⱖ0
艚
nⱖ0
᏷共X n 兲
MONOTONE LOWER PROBABILITIES OF INFINITE ORDER
769
holds for any sequence (X n ) nⱖ0 of subsets of ᏿̂, so that ᏷ is a G ␦ -set in ᏷(᏿̂), and therefore
a Polish subspace of ᏷(᏿̂).
Except in explicit cases, by a measure on ᏿ we mean a Borel measure, i.e., a (␴-additive)
measure on (᏿, Ꮽ).
2.2. Upper and lower probabilities. Let ᏿ be a Polish topological space. Denote by
Ꮽ its Borel ␴-algebra, by ᏹ(᏿) (or ᏹ) the set of all probability (Borel) measures on ᏿, and
by Ꮽ u (᏿) the ␴-algebra of all subsets of ᏿ which are universally measurable, i.e., those
which belong to the completion of Ꮽ with respect to any probability measure on ᏿.
It is known (see, e.g., Parthasarathy 1967, Theorems II.6.2 and II.6.4; see also Dellacherie
and Meyer 1975, p. 118) that, once endowed with the weak topology (i.e., the coarsest
topology which makes : P 3 兰 ␾ dP continuous on ᏹ for each bounded continuous
real-valued mapping ␾ defined on ᏿), ᏹ is itself a Polish space. It is also known (see, e.g.,
Parthasarathy 1967, Theorem II.3.2) that every probability measure P on the measurable
space (᏿, Ꮽ) is tight and outer regular:
sup P共K兲 ⫽ P共A兲 ⫽
inf
P共G兲
for each A 僆 Ꮽ.
G僆Ᏻ,G傶A
K僆᏷共A兲
Note that a Polish subset X of ᏿ is an element of Ꮽ, so that Ꮽ(X) is precisely the set {A 僆 Ꮽ
: A 債 X}.
Every nonempty subset ᏼ of ᏹ has a lower envelope f defined on Ꮽ by
f共A兲 ⫽ inf P共A兲,
P僆ᏼ
called the lower probability of ᏼ, and, similarly, an upper envelope F called the upper
probability of ᏼ. It follows from the obvious duality relation, f( A) ⫹ F( A c ) ⫽ 1 for every
A in Ꮽ, that one need not introduce both f and F. Following Shafer (1976), we shall
(arbitrarily) give f precedence over F.
The set core( f ) ⫽ {P 僆 ᏹ : P ⱖ f} ⫽ {P 僆 ᏹ : F ⱖ P ⱖ f} of all probability
measures which setwise dominate f (and are dominated by F) on Ꮽ satisfies ᏼ 債 core( f ).
ᏼ is said to be m-closed when core( f ) ⫽ ᏼ, a property which makes ᏼ characterizable by
f. An m-closed set ᏼ is convex, but convexity alone does not ensure m-closedness. The
following characteristic property is straightforward: A nonempty subset ᏼ of ᏹ is m-closed
if and only if there exist set functions g and h such that ᏼ ⫽ {P 僆 ᏹ : g ⱕ P ⱕ h}.
The lower probability f of a nonempty set ᏼ clearly always satisfies the following
conditions:
(1)
(2)
f共A兲 ⫽ 0,
for A, B in Ꮽ,
f共᏿兲 ⫽ 1,
A 債 B f f共A兲 ⱕ f共B兲,
and inherits the properties of outer continuity and outer regularity of a probability on (᏿, Ꮽ):
(3)
for A n in Ꮽ,
(4)
for any A in Ꮽ,
A n 2A f f共A n 兲2f共A兲,
f共A兲 ⫽
inf
f共G兲.
G僆Ᏻ,G傶A
With certain topological assumptions about ᏼ, f will also satisfy inner regularity properties:
770
F. PHILIPPE, G. DEBS AND J.-Y. JAFFRAY
PROPOSITION 1. Let ᏿ be a Polish topological space, and let f be the lower probability of
a nonempty subset of ᏹ. Then the following assertions are pairwise equivalent:
(i) core( f ) is compact;
(ii) there is a compact ᏼ such that f ⫽ inf ᏼ;
(iii) f satisfies
(5)
for all G in Ᏻ.
f共G兲 ⫽ sup f共K兲,
K僆᏷共G兲
In particular,
(6)
for G and Gn in Ᏻ,
Gn 1G f f共Gn 兲1f共G兲.
PROOF. If (iii) holds, then core( f ) is closed: Given a sequence P n in core( f ) which
(weakly) converges to P, we have, for all K in ᏷,
P共K兲 ⱖ lim P n 共K兲 ⱖ f共K兲,
n
thus P ⱖ f by (5) and (4). Since sup{ f(K) : K 僆 ᏷} ⫽ 1, core( f ) is moreover tight.
Therefore, it is compact by Prohorov’s Theorem.
(i) f (ii) is clear.
Finally assume that (ii) holds; to obtain (iii), we first show there is a convex compact ᏽ
such that f ⫽ inf ᏽ. Let ᏽ be the closed convex hull of ᏼ. On one hand, ᏽ is compact because
it is tight: Given ⑀ ⬎ 0, there is K 僆 ᏷ such that P(K) ⬎ 1 ⫺ ⑀ for each P in ᏼ, thus for
each P in the convex hull of ᏼ; for each Q in ᏽ, consider a sequence (Q n ) in the convex hull
of ᏼ which converges to Q, then
Q共K兲 ⱖ lim Q n 共K兲 ⱖ 1 ⫺ ⑀ .
n
On the other hand, f ⫽ inf ᏽ because each Q in ᏽ is the barycenter of a probability measure
m on the convex compact ᏽ, which is concentrated on the set of all extremal elements of ᏽ,
that is a subset of ᏼ (see, e.g., Meyer 1966, XI.8 sq); therefore,
Q共G兲 ⫽
冕
for each G in Ᏻ,
P共G兲 m共dP兲
ᏼ
since the mapping : P 3 P(G) is lower semi-continuous. Thus Q ⱖ f on Ᏻ, thus, by (4),
Q ⱖ f on the whole of Ꮽ.
We next show that, for any G in Ᏻ, f(G) ⫽ sup{ f(F) : F 僆 Ᏺ, F 債 G}; by Urysohn’s
Lemma, every P in ᏹ satisfies
P共G兲 ⫽ sup
h僆H G
冕
h dP,
᏿
where H G is the set of all continuous functions from ᏿ to [0, 1] such that supp(h), the support
of h, is contained in G. Moreover, the mapping : (h, P) 3 兰 ᏿ h dP defined on H G ⫻ ᏼ
is linear and continuous w.r.t. both its arguments; since ᏼ is compact, and may be assumed
convex, by applying a minimax theorem (Sion 1958, Theorem 4.2⬘) we get
MONOTONE LOWER PROBABILITIES OF INFINITE ORDER
f共G兲 ⫽ inf P共G兲 ⫽ sup inf
P僆ᏼ
h僆H G P僆ᏼ
冕
771
h dP.
᏿
But supp(h) belongs to Ᏺ and 兰 ᏿ h dP ⱕ P(supp(h)); so, as claimed,
f共G兲 ⱕ sup f共supp共h兲兲 ⱕ
h僆H G
sup
f共F兲.
F僆Ᏺ,F債G
Now, given ⑀ ⬎ 0, there is F ⑀ in Ᏺ such that F ⑀ 債 G and f(F ⑀ ) ⬎ f(G) ⫺ ⑀ . Since ᏼ
is compact, by Prohorov’s Theorem, it is tight: There exists K ⑀ in ᏷ such that, for each P in
ᏼ, P(K ⑀ ) exceeds 1 ⫺ ⑀; thus P(F ⑀ 艚 K ⑀ ) ⬎ P(F ⑀ ) ⫺ ⑀ . Therefore, f(F ⑀ 艚 K ⑀ )
ⱖ f(F ⑀ ) ⫺ ⑀ ⬎ f(G) ⫺ 2 ⑀ , and (5) is proved.
Finally assume G n 1 G; given K in ᏷ with K 債 G, using compactness an integer m can
be found such that K 債 G m , so that f(K) ⱕ f(G n ) ⱕ f(G) for all n ⱖ m. Then (5) implies
(6). 䊐
2.3. Convex cocapacities. A mapping f from Ꮽ to [0, 1] is called a cocapacity on
(᏿, Ꮽ) when, in addition to (1), (2), and (3), it satisfies the topological condition (5) in
Proposition 1. The term “cocapacity” is justified by the fact that, whereas f is not a Choquet
capacity in the strictest sense, its dual F is an abstract capacity (Choquet 1959) on
(᏿, Ꮽ, Ᏺ)—it satisfies (1), (2), and the properties
F共A n 兲1F共A兲
for A n 1A in Ꮽ,
K n 2K f F共K n 兲2F共K兲
for K and K n in Ᏺ.
According to Proposition 1, it stands to reason that if one wishes to be able to characterize
ᏼ by the lower probability f, in cases where f is a cocapacity, ᏼ must be assumed to be both
m-closed and compact.
The study of the finite case in Jaffray (1989a) shows that construction and justification of
the decision model we wish to introduce later is possible under the additional assumption that
the lower probability f is convex (or 2-monotone), i.e.,
(7)
for all A, B in Ꮽ,
f共A 艛 B兲 ⫹ f共A 艚 B兲 ⱖ f共A兲 ⫹ f共B兲.
As shown, e.g., by Huber and Strassen (1973, Lemma 2.5), a convex cocapacity f is the
lower envelope of the probability measures which dominate it on Ꮽ, and core( f ) is never
empty (see also Shapley 1971 for the finite case). Thus, as we just demonstrated, core( f ) is
compact. A direct adaptation of one of Choquet’s fundamental results leads one then to the
following capacitability statement:
THEOREM 1 (CHOQUET).
(8)
Let ᏿ be a Polish space. A convex cocapacity on (᏿, Ꮽ) satisfies
for each A in Ꮽ,
sup f共K兲⫽f共A兲⫽ inf
K僆᏷共A兲
f共G兲.
G僆Ᏻ,G傶A
PROOF. Let f ᏷ be the restriction of f to ᏷; according to (1), (2) and (4), f ᏷ is a capacity
defined on ᏷ as described by Choquet (1954, 15.2). Recall that the outer capacity of a subset
X of ᏿ is
772
F. PHILIPPE, G. DEBS AND J.-Y. JAFFRAY
f *᏷ 共X兲 ⫽
inf
G僆Ᏻ,G傶X
冉
冊
sup f ᏷ 共K兲 .
K僆᏷共G兲
Since f ᏷ is convex, the equality
f *᏷ 共A兲 ⫽ sup f共K兲
K僆᏷共A兲
holds for any A in Ꮽ (Choquet 1954, 38.2). Finally, properties (5) and (4) mean that
f *᏷ ( A) ⫽ f( A) holds for each A in Ꮽ. 䊐
REMARK 1. According to Choquet (1954, 38.2), property (8) more generally holds when
A is co-analytic, i.e., the complement of A is the continuous image of a Polish space. A
well-established consequence, then, is that each co-analytic subset of ᏿ is universally
measurable.
2.4. Monotone cocapacities of infinite order. In this paper, we will actually make a
bolder assumption. The cocapacity f is said to be monotone of order k (k ⱖ 2), k-monotone
for short, when
(9)
f
冉
k
艛
i⫽1
冊
Ai ⱖ
冘
共⫺1兲 兩I兩⫹1 f
A⫽I債兵1, . . . ,k其
冉艚 冊
Ai .
i僆I
We will require f to be monotone of infinite order (for short: ⬁-monotone), namely
k-monotone for any k ⱖ 2. Note that f has in particular to be convex.
The set of the ⬁-monotone cocapacities on ᏿ is denoted by ⌫ ⬁(᏿), and it contains, of
course, ᏹ as a subset.
This strengthening of the convexity assumption could possibly be avoided. However,
⬁-monotonicity is a natural, fairly common, property—lower probabilities generated by
random sets are ⬁-monotone (Dempster 1967, Matheron 1975, Nguyen 1978). In robust
statistics, the ⑀-contamination model (Wasserman and Kadane 1990) only locates the prior
probability in core( f ), with
f ⫽ 共1 ⫺ ⑀ 兲P ⫹ ⑀ e ᏿
(where P 僆 ᏹ, e ᏿ (᏿) ⫽ 1, e ᏿ ( A) ⫽ 0 if A ⫽ ᏿), which is clearly ⬁-monotone. Moreover,
⬁-monotonicity has the great advantage of providing us with a powerful tool presented in
Theorem 2 below, which is a further adaptation of one of Choquet’s results.
We have seen that ᏷(X) is open (resp. closed, compact) when X is an open (resp. closed,
compact) subset of ᏿. For measurable subsets, one can show:
LEMMA 1.
For any A in Ꮽ, ᏷( A) is universally measurable.
PROOF. Let us first observe that the subset E ⫽ {(K, x) : x 僆 K} of ᏷ ⫻ ᏿ is closed:
᏿ being regular, for any (K, x) in ᏷ ⫻ ᏿ such that x ⰻ K there exist G and G⬘ in Ᏻ such
that K 債 G, x 僆 G⬘, and G 艚 G⬘ ⫽ A; hence ᏷(G) ⫻ G⬘ is a neighborhood of (K, x),
the intersection of which with E is empty. Considering now ᏷( A) with A in Ꮽ, its
complementary set ᏷( A) c is obtained by projection on ᏷ of E 艚 (᏷ ⫻ A c ) because K is
in ᏷( A) c if and only if there is an element x of A c such that (K, x) is in E. Since E 艚 (᏷
⫻ A c ) is a Borel set, its projection ᏷( A) c is analytic; therefore, ᏷( A) is universally
measurable (see Remark 1). 䊐
We can then state:
773
MONOTONE LOWER PROBABILITIES OF INFINITE ORDER
THEOREM 2 (CHOQUET). Let ᏿ be a Polish space. Then a set function f : Ꮽ 3 ⺢ is an
⬁-monotone cocapacity on (᏿, Ꮽ) if, and only if, there is a probability measure ␮ on
(᏷, Ꮽ(᏷)) such that, for each A in Ꮽ,
f共A兲 ⫽ ␮ 共᏷共A兲兲,
(10)
᏷( A) denoting the universally measurable set {K 僆 ᏷ : K 債 A} and ␮ still denoting the
(unique) extension of ␮ to Ꮽ u(᏷). Moreover, such a ␮ is unique.
PROOF. According to Choquet (1954, 50.1), there is, for a given ⬁-monotone cocapacity
f on (᏿, Ꮽ), a unique measure ␮ on (᏷, Ꮽ(᏷)) such that the equality f(K) ⫽ ␮ (᏷(K)) holds
for each K in ᏷. It is known (Christensen 1974, Theorem 3.1) that a closed subset C of ᏷
is compact if and only if
KC ⫽
艛
K
K僆C
is a compact subset of ᏿. Since any element C of ᏷(᏷) is contained in the compact set
᏷(K C ), ␮ (᏷) ⫽ 1 follows from (5).
Now, if A is a Borel subset of ᏿, then ᏷( A) is a universally measurable subset of ᏷. Let
g( A) ⫽ ␮ (᏷( A)); we prove (10) by showing that g meets the premises of Theorem 1.
Properties (1), (2) and (3) are easily checked. Furthermore, for any G in Ᏻ, ᏷(G) is open,
and therefore measurable. Thus, C in ᏷(᏷) can be found such that, given any ⑀ ⬎ 0,
C 債 ᏷共G兲
and ␮ 共C兲 ⬎ ␮ 共᏷共G兲兲 ⫺ ⑀ .
Since C 債 ᏷(K C ) 債 ᏷(G), property (5) is satisfied. Finally, for all A i ’s in Ꮽ and any
integer k ⱖ 2,
冉艛 冊
冉艛
k
g
Ai ⫺
i⫽1
k
ⱖ␮
i⫽1
冘
共⫺1兲 兩I兩⫹1 g
冊
᏷共A i 兲 ⫺
冉艚 冊
冉艚
Ai
i僆I
A⫽I債兵1, . . . ,k其
冘
A⫽I債兵1, . . . ,k其
共⫺1兲 兩I兩⫹1 ␮
i僆I
冊
᏷共A i 兲 ⫽ 0
by the inclusion-exclusion formula. Therefore, g is ⬁-monotone, thus convex, and (8)
establishes the equality of f and g. 䊐
The one-to-one mapping from the set ⌫ ⬁(᏿) of the ⬁-monotone cocapacities on ᏿ onto the
set ᏹ(᏷) of the probability measures on ᏷, defined by relation (10), is denoted by ␥ in the
sequel.
REMARK 2. If P is an element of ᏹ, it can be easily verified that ␥ (P) equals the image
␫ (P) of P under the one-to-one mapping ␫ : x 3 { x} from ᏿ to ᏷. Note that ␫ is continuous
since the equality
D共兵x其, 兵y其兲 ⫽ d共x, y兲
holds for any d metrizing ᏿ and the associated Hausdorff metric D on ᏷.
As shown by Strassen in the compact case (1964, Satz 4.3), the existence of ␮ leads to a
useful characterization of the set core( f ) of all probability measures dominating f. Recall that
774
F. PHILIPPE, G. DEBS AND J.-Y. JAFFRAY
a (Markov) kernel from ᏷ to ᏿ is a family (P K ) K僆᏷ of elements of ᏹ such that, for each A
in Ꮽ, the mapping : K 3 P K ( A) is Ꮽ(᏷)-measurable.
THEOREM 3 (STRASSEN). Let ᏿ be a Polish space, let f be an ⬁-monotone cocapacity on
᏿ and ␮ ⫽ ␥(f ) the associated probability measure on ᏷. The following statements are
equivalent:
(i) P 僆 core( f ).
(ii) There is a kernel (P K) K僆᏷ from ᏷ to ᏿ such that
P⫽
冕
P K ␮ 共dK兲
and P K 共K兲 ⫽ 1.
᏷
PROOF.
Assume (ii) holds. Let A be in Ꮽ, then
P共A兲 ⫽
冕
P K 共A兲 ␮ 共dK兲 ⱖ
᏷
冕
P K 共A兲 ␮ 共dK兲 ⫽ ␮ 共᏷共A兲兲 ⫽ f共A兲.
᏷共A兲
Assume (i) now. Let us consider ᏿ as a Polish subspace of a compact metrizable space ᏿̂;
then ᏷ is a Polish subspace of the compact metrizable space ᏷(᏿̂), as we have seen in the
introduction. Identify (using canonical injection) probability measures P and ␮ on ᏿ and ᏷
with measures P̂ and ␮ˆ on ᏿̂ and ᏷(᏿̂) respectively concentrated on ᏿ and ᏷.
It can be easily verified that relation f̂( A) ⫽ f( A 艚 ᏿) for each A in Ꮽ(᏿̂) defines an
⬁-monotone cocapacity f̂ on ᏿̂, which is associated by (10) with ␮ˆ , and that f̂ ⱕ P̂.
According to Strassen’s result, which was previously quoted, there is a kernel (P K ) K僆᏷(᏿ˆ )
from ᏷(᏿̂) to ᏿̂ such that
P̂ ⫽
冕
P K ␮ˆ 共dK兲
and P K 共K兲 ⫽ 1.
᏷共᏿̂兲
Since ␮ˆ is concentrated on ᏷, we may write P ⫽ 兰 ᏷ P K ␮ (dK). Finally, for each A in Ꮽ
(which is a subset of Ꮽ(᏿̂)) the mapping : K 3 P K ( A) is Ꮽ(᏷(᏿̂))-measurable, thus its
restriction to ᏷ is Ꮽ(᏷)-measurable, and (P K ) K僆᏷ satisfies (ii). 䊐
2.5. Images of convex and ⴥ-monotone cocapacities. Given two Polish spaces ᏿ and
ᐄ and a convex cocapacity f defined on ᏿, every (Borel-) measurable mapping ␦ from ᏿ to
ᐄ induces a set function ␦ ( f ) on ᐄ, defined by the relation
␦ 共 f 兲共A⬘兲 ⫽ f共 ␦ ⫺1 共A⬘兲兲
for each A⬘ 僆 Ꮽ共ᐄ兲.
Properties (1), (2), (3), and (9) are clearly inherited from f by ␦ ( f ). Moreover, ␦ ( f )
satisfies (8) (and (5)) if ␦ maps any compact subset of ᏿ onto a compact subset of ᐄ, because
␦ 共 f 兲共A⬘兲 ⫽
sup
␦ 共K兲債A⬘,K僆᏷
f共K兲 ⱕ
sup
␦ 共K兲債A⬘,K僆᏷
␦ 共 f 兲共 ␦ 共K兲兲 ⱕ sup
␦ 共 f 兲共K⬘兲.
K⬘僆᏷共A⬘兲
For short, a mapping with the latter property will be called a compact preserving mapping,
and ␦ is assumed to be both measurable and compact preserving in the sequel. It must be
noted that, even though continuous mappings possess these properties, a stronger continuity
MONOTONE LOWER PROBABILITIES OF INFINITE ORDER
775
assumption would not be appropriate in the context of decision theory, where simple
mappings, i.e., measurable and finite-ranged, are often used.
With the above assumption, ␦ ( f ) is a cocapacity on ᐄ, with the same order of
monotonicity as f. When f is, in particular, ⬁-monotone, measures ␮ ⫽ ␥ ( f ) and ␮ ⬘
⫽ ␥ ( ␦ ( f )) can be associated respectively with f and ␦ ( f ) by (10), and ␮⬘ can be expressed
as follows:
LEMMA 2. Let ⌬ denote the mapping from ᏷ to ᏷(ᐄ) defined by ⌬(K) ⫽ ␦(K). Then ⌬
is universally measurable, and ␦(f ) is associated by (10) with ⌬(␮).
PROOF. For each subset X⬘ of ᐄ, the relation ᏷( ␦ ⫺1 (X⬘)) ⫽ ⌬ ⫺1 (᏷(X⬘)) is straightforward. As a result, ⌬ ⫺1 (᏷( A⬘)) is universally measurable when A⬘ is open or closed. Since
Ꮽ(᏷(ᐄ)) is generated by {᏷( A⬘) : A⬘ 僆 Ᏻ(ᐄ) 艛 Ᏺ(ᐄ)}, ⌬ is measurable with respect to
Ꮽ u (᏷) and Ꮽ(᏷(ᐄ)). Finally, the equalities
␦ 共 f兲共A⬘兲 ⫽ f共 ␦ ⫺1 共A⬘兲兲 ⫽ ␮ 共᏷共 ␦ ⫺1 共A⬘兲兲兲 ⫽ ␮ 共⌬ ⫺1 共᏷共A⬘兲兲兲
for each A⬘ in Ꮽ共ᐄ兲
ensure that ␮⬘ ⫽ ⌬(␮) since ␮⬘ is unique. 䊐
Recall that the convexity of f implies that f is the lower probability of the set core( f ). Let
us now turn to the set core( ␦ ( f )) of all the probability measures of ᏹ(ᐄ) which dominate
␦ ( f ) on Ꮽ(ᐄ), along with the set ␦ (core( f )) of the images under ␦ of the elements of
core( f ).
THEOREM 4. Let ᏿ and ᐄ be two Polish spaces, ␦ a measurable and compact preserving
mapping from ᏿ to ᐄ, and f an ⬁-monotone cocapacity defined on ᏿. Then ␦(f ) is an
⬁-monotone cocapacity on ᐄ and satisfies
core共 ␦ 共 f兲兲 ⫽ ␦ 共core共 f兲兲.
(11)
PROOF. Let us associate f with ␥ ( f ) ⫽ ␮ by relation (10). We have just seen that ␦ ( f )
is an ⬁-monotone cocapacity on ᐄ, with measure ⌬(␮) associated by (10); since the inclusion
core( ␦ ( f )) 傶 ␦ (core( f )) is clear, we have only to demonstrate that, if P⬘ is an element of
core( ␦ ( f )), then there exists P in core( f ) such that P⬘ ⫽ ␦ (P).
According to Theorem 3, there exists a kernel (P⬘K⬘ ) K⬘僆᏷(ᐄ) from ᏷(ᐄ) to ᐄ such that
P⬘ ⫽
冕
P⬘K ⬘ ⌬共 ␮ 兲共dK⬘兲
and P⬘K ⬘ 共K⬘兲 ⫽ 1.
᏷共ᐄ兲
Then, by the transfer theorem, P⬘ ⫽ 兰 ᏷ P⬘⌬(K) ␮ (dK).
Let E ⫽ {(K, Q) 僆 ᏷ ⫻ ᏹ : Q(K) ⫽ 1, ␦ (Q) ⫽ P⬘⌬(K) }. The mapping : (K, Q) 3
Q(K) is upper semi-continuous. Furthermore, for each open subset O⬘ of ᐄ, the mapping :
K 3 P⬘⌬(K) (O⬘) is Ꮽ u (᏷)-measurable since (P⬘K⬘ ) K⬘僆᏷(ᐄ) is a kernel and ⌬ is measurable with
respect to Ꮽ u (᏷) and Ꮽ(᏷(ᐄ)). On the other hand, the mapping : Q 3 Q( ␦ ⫺1 (O⬘)) is
Ꮽ(ᏹ)-measurable, since : Q 3 Q( A) is measurable when A is a Borel set. Thus E is an
element of Ꮽ u (᏷) ⫻ Ꮽ(ᏹ).
Moreover, for each K in ᏷, K and ␦ (K) are Polish subspaces of ᏿ and ᐄ respectively, and
P⬘⌬(K) is concentrated on ␦ (K). Therefore, there exists a probability measure on K, which may
be identified by canonical injection with a measure on ᏿ concentrated on K, the image of
which under ␦ is P⬘⌬(K) (see, e.g., Dellacherie and Meyer 1975, Chapter 2, 45). Thus
proj ᏷(E) ⫽ ᏷.
In consequence, a section theorem (see, e.g., Cohn 1980, Corollary 8.5.4) gives a function
776
F. PHILIPPE, G. DEBS AND J.-Y. JAFFRAY
p : ᏷ 3 ᏹ, the graph of which is contained in E, and which is measurable with respect to
Ꮽ u (᏷) and Ꮽ(ᏹ).
Hence ( p(K)) K僆᏷ is a kernel from ᏷ to ᏿. Define P ⫽ 兰 ᏷ p(K) ␮ (dK), P is an element
of core( f ) by Theorem 3 and satisfies ␦ (P) ⫽ 兰 ᏷ ␦ ( p(K)) ␮ (dK) ⫽ P⬘. 䊐
3.
The set of ⴥ-monotone cocapacities.
3.1. Induced topology on ⌫ ⴥ(᏿). The Polish space ᏹ(᏷) is metrizable by the
Prohorov distance D, which is associated with an arbitrary given metric on ᏷. According to
Theorem 2, the expression D c ( f, f⬘ ) ⫽ D( ␥ ( f ), ␥ ( f⬘ )) defines a metric D c on ⌫ ⬁(᏿) which
makes ␥ an isometry. The set ⌫ ⬁(᏿) is endowed with the induced topology and is,
consequently, homeomorphic to ᏹ(᏷) endowed with the weak topology. As a first
consequence, one gets:
PROPOSITION 2. The set of the ⬁-monotone cocapacities which are concentrated on a finite
subset of ᏿ is dense in ⌫ ⬁(᏿).
PROOF. The set of the finite subsets of ᏿ is dense in ᏷. Thus the set of the probability
measures the support of which is a finite subset of ᏿ is dense in ᏹ(᏷). Finally, if a measure
␮ 0 of ᏹ(᏷) is concentrated on {{ x j }, 1 ⱕ j ⱕ J}, then
␮ 0 共᏷共兵x j , 1 ⱕ j ⱕ J其兲兲 ⱖ ␮ 0
冉艛
冊
兵兵x j 其其 ⫽ 1;
1ⱕjⱕJ
therefore, the ⬁-monotone cocapacity ␥ ⫺1(␮ 0) is concentrated on the set { x j , 1
ⱕ j ⱕ J}. 䊐
Consider now the mapping e K defined on Ꮽ by the expression e K ( A) ⫽ 1 ᏷( A) (K). This last
equality and Theorem 2 show that, when K is not empty, the mapping e K is the ⬁-monotone
cocapacity on ᏿ associated by (10) with the point mass ␮ K concentrated at {K}; it is called
an elementary cocapacity. Elementary cocapacities provide one with an interesting representation of an ⬁-monotone cocapacity f, since relation (10) may be written as
(12)
f共A兲 ⫽
冕
e K 共A兲 ␮ 共dK兲.
᏷
The following result will be of use later on:
PROPOSITION 3.
The mapping : K 3 e K from ᏷ into ⌫ ⬁(᏿) is one-to-one and continuous.
PROOF. Since e K (K) ⫽ e L (K) implies L 債 K, : K 3 e K is clearly one to one. On the
other hand, if ␮ K denotes the point mass concentrated on {K}, then the continuity of the
mapping: K 3 ␮ K is well known. 䊐
3.2. The convex set ⌫ ⴥ(᏿). Given cocapacities f 1 , f 2 in ⌫ ⬁(᏿) and a real number t in
[0, 1], the mapping f defined on Ꮽ by f ⫽ tf 1 ⫹ (1 ⫺ t) f 2 obviously fulfills conditions (1),
(2), and (3). Moreover, given any G in Ᏻ and ⑀ ⬎ 0, there are K 1 and K 2 in ᏷(G) such that
f i 共K i 兲 ⬎ f共G兲 ⫺ ⑀ ,
i ⫽ 1, 2.
Thus f(K 1 艛 K 2 ) ⬎ f(G) ⫺ ⑀ , in which case f satisfies (5), which in turn means that ⌫ ⬁(᏿)
is convex.
MONOTONE LOWER PROBABILITIES OF INFINITE ORDER
777
Note that ␥ ( f ) equals t ␥ ( f 1 ) ⫹ (1 ⫺ t) ␥ ( f 2 ) because ␥ ( f ) is unique. With respect to the
set of probability measures agreeing with f, Theorem 3 leads to the following result:
PROPOSITION 4.
[0, 1] then
If f 1, f 2 are ⬁-monotone cocapacities and f ⫽ tf 1 ⫹ (1 ⫺ t) f 2 with t in
core共 f兲 ⫽ tcore共 f 1 兲 ⫹ 共1 ⫺ t兲core共 f 2 兲.
PROOF. The set tcore( f 1 ) ⫹ (1 ⫺ t)core( f 2 ) is clearly a subset of core( f ). Conversely
given any P in core( f ), there is, according to Theorem 3, a kernel (P K ) K僆᏷ from ᏷ to ᏿ such
that P ⫽ 兰 ᏷ P K ␮ (dK) and P K (K) ⫽ 1, where ␥ ( f ) is denoted by ␮.
Set P i ⫽ 兰 ᏷ P K ␮ i (dK) (i ⫽ 1, 2), where ␥ ( f i ) is denoted by ␮ i ; then P i belongs to
core( f i ) and P ⫽ tP 1 ⫹ (1 ⫺ t) P 2 . 䊐
Let f 0 be in ⌫ ⬁(᏿) and ␮ 0 ⫽ ␥ ( f 0 ). If f 0 is concentrated at K 0 ⫽ { x 1 , . . . , x J } then
␮ 0 (᏷(K 0 )) ⫽ 1, so that f 0 ( A) ⫽ ␮ 0 (᏷( A 艚 K 0 )) ⫽ f 0 ( A 艚 K 0 ). This last equality may
be written as
冘
f 0 共A 艚 K 0 兲 ⫽
␮ 0 共兵B其兲,
B僆2 A艚K 0
which establishes an immediate relation between the restriction of ␮ 0 to ᏷(K 0 ) and the
Möbius inverse ␾ 0 (see, e.g., Chateauneuf and Jaffray 1989) of the restriction of f 0 to K 0 :
␾ 0 共K兲 ⫽ ␮ 0 共兵K其兲
(13)
for all K 債 K 0 .
Furthermore,
f 0 共A兲 ⫽
冘
冘
␾ 0 共K兲 ⫽
e K 共A 艚 K 0 兲 ␾ 0 共K兲 ⫽
A⫽K債K 0
K債A艚K 0
冘
e K 共A兲 ␾ 0 共K兲.
A⫽K債K 0
Thus, according to (12), f 0 may be represented by the expressions
(14)
f0 ⫽
冕
e K ␮ 0 共dK兲 ⫽
冘
␾ 0 共K兲e K .
K僆᏷共K 0 兲
᏷共K 0 兲
The second of which introduces f 0 as a convex linear combination of elementary cocapacities,
since
冘 ␾ 共K兲 ⫽ f 共K 兲 ⫽ 1.
0
0
0
K債K 0
REMARK 3. Proposition 2 may thus be reformulated as follows: ⌫ ⬁(᏿) is the closed
convex hull of the set {e K : K 僆 ᏷, K finite}.
At this point, we can get an integral representation of the affine real functions on ⌫ ⬁(᏿)
which meet certain continuity and boundedness requirements:
PROPOSITION 5. Let ⌳ be an affine continuous real mapping defined on ⌳ ⬁(᏿) and assume
that the mapping ␭ : K 3 ⌳(e K) is bounded on ᏷. If f and ␮ are associated by (10) then
⌳共 f兲 ⫽
冕
᏷
␭ d␮.
778
F. PHILIPPE, G. DEBS AND J.-Y. JAFFRAY
PROOF. Take any ⑀ ⬎ 0; since ⌳ is continuous we may choose ␩ satisfying 0 ⬍ ␩ ⬍ ⑀/2
and such that
D c 共f, g兲 ⬍ ␩ f 兩⌳共 f兲 ⫺ ⌳共g兲兩 ⬍
⑀
.
2
Since ␭ is continuous (by Proposition 3) and bounded, the set
再
␯ 僆 ᏹ共᏷兲 :
冏冕
␭ d␯ ⫺
᏷
冕 冏 冎
␭ d␮ ⬍
᏷
⑀
2
is a neighborhood of ␮. According to Proposition 2 there exists an f 0 which is concentrated
on a finite subset K 0 of ᏿ and satisfies D c ( f, f 0 ) ⱕ ␩ . Setting ␮ 0 ⫽ ␥ ( f 0 ) and using (14),
one obtains
⌳共 f 0 兲 ⫽ ⌳
冉冘
冊
␾ 共K兲e K ⫽
K債K 0
冘 ␾共K兲⌳共e 兲 ⫽ 冕
␭ d␮0 ,
K
K債K 0
᏷
and then follows
冏
⌳共 f兲 ⫺
冕 冏
␭ d ␮ ⱕ 兩⌳共 f兲 ⫺ ⌳共 f 0 兲兩 ⫹
᏷
冏冕
␭ d␮0 ⫺
᏷
冕 冏
␭ d␮ ⬍ ⑀.
䊐
᏷
Using Lebesgue approximation, the above continuity assumption may be relaxed to a
measurability assumption on ␭, but a fitting assumption about ⌳ is required. We keep the
notations of Proposition 5, and recall that the support supp(␮) of a bounded regular measure
␮ is defined as the smallest closed set of full ␮-measure.
PROPOSITION 6. Let f, ␮, and ␭ be defined as in Proposition 5. If ␭ is measurable and
bounded on ᏷ and if ⌳ is affine and satisfies
inf ␭ ⱕ ⌳共 f兲 ⱕ sup ␭ ,
supp共 ␮ 兲
supp共 ␮ 兲
for all f in ⌫ ⬁(᏿), then ⌳(f ) ⫽ 兰 ᏷ ␭ d␮.
PROOF. Set ⌿ ⫽ ⌳ 䡩 ␥ ⫺1; since ␥ ⫺1 is affine, ⌿ is affine too on ᏹ(᏷); note that ␭ (K) and
⌿( ␮ K ) are identical, where ␮ K is the point mass concentrated on {K}. Then the proof is
classical (Fishburn 1967). 䊐
4.
The decision model.
4.1. Imprecise risk. The set of the states of nature is denoted by ᏿, and assumed to be
a Polish space; the algebra of events is Ꮽ(᏿), i.e., the Borel ␴-algebra of ᏿.
Imprecise risk is defined as following: A situation in which the information of the decision
maker is expressed by a set ᏼ of probability measures on (᏿, Ꮽ) which can be characterized
by its lower probability f, i.e., ᏼ ⫽ core( f ); as noted in §2.2, the latter property holds if the
information is described by probability intervals. This paper deals with a particular case of
MONOTONE LOWER PROBABILITIES OF INFINITE ORDER
779
imprecise risk, since f is assumed to be an ⬁-monotone cocapacity. Risk is an even more
particular situation, in which ᏼ reduces to a unique probability measure.
The set of outcomes is denoted by ᐄ, also assumed to be Polish, and endowed with its
Borel ␴-algebra Ꮽ(ᐄ). A decision is defined as a measurable compact preserving mapping
from ᏿ to ᐄ; in particular, simple decisions are those which are measurable and finite-ranged.
The decision maker has to choose among a set Ᏸ of decisions, on which a binary preference
relation is supposed to be defined.
We assume that the choice between two of the decisions of Ᏸ depends only on the ⬁-monotone
cocapacities on (ᐄ, Ꮽ(ᐄ)) they induce from f. Theorem 4 affirms that a cocapacity induced from
f characterizes the set of the probability measures induced from ᏼ, so that this hypothesis is
consistent with the information of the decision maker. That is why a binary preference relation
Ɑ on the set ⌫⬁(ᐄ) of the ⬁-monotone cocapacities on (ᐄ, Ꮽ(ᐄ)) is taken as a primitive. The
Ɱ and read “not preferred to.”
complementary relation is denoted by ⬃
Since ⌫ ⬁(ᐄ) is convex, linear utility theory may apply provided that the decision maker
complies with certain axioms about the relation Ɑ. In particular, Ɑ has to be an asymmetric
Ɱ, which
and negatively transitive binary relation (Fishburn 1988). That is to say, the relation ⬃
is reflexive, transitive, and complete, is a weak order. According to Proposition 4, a mixture
of ⬁-monotone cocapacities on ᐄ characterizes the set of the mixtures of probability
measures dominating each cocapacity. Therefore, once again mixture involves no modification of the information of the decision maker about events.
Ɱ on ⌫ ⬁(ᐄ) induces a weak order on the set ᏷(ᐄ) of all compact
Finally, the weak order ⬃
subsets of ᐄ, and a weak order on the set ᐄ of the outcomes itself; both are still denoted by
Ɱ, and they are defined without ambiguity by the formulae
⬃
(15)
Ɱ K⬘ N e K Ɱ e K⬘ ,
K⬃
⬃
(16)
Ɱ x⬘ N 兵x其 ⬃
Ɱ 兵x⬘其 N e 兵x其 Ɱ e 兵x⬘其 .
x⬃
⬃
4.2. Decision criteria. In order to provide one with utility representations (i.e.,
real-valued and order preserving mappings) of the relation Ɑ, some of the following axioms
are used:
Ɱ on ⌫ ⬁(ᐄ) is a weak order.
A1 (ordering): ⬃
A2W (weak independance): for all g 1 , g 2 , g in ⌫ ⬁(ᐄ),
g 1 ⬃ g 2 f 12 g 1 ⫹ 12 g ⬃ 12 g 2 ⫹ 12 g.
A2 (independence): for all g 1 , g 2 , g in ⌫ ⬁(ᐄ), and all ␭ in (0, 1),
g 1 Ɑ g 2 f ␭ g 1 ⫹ 共1 ⫺ ␭ 兲g Ɑ ␭ g 2 ⫹ 共1 ⫺ ␭ 兲g.
A3W (weak continuity): for all g 1 , g 2 , g in ⌫ ⬁(ᐄ), there are some ␣, ␤ in (0, 1) such that
g1 Ɑ g Ɑ g2 f
再 g␣gⱭ ⫹␤g共1⫹⫺共1␣兲g⫺ ␤Ɑ兲gg,.
1
2
1
2
A3 (continuity): for each g 1 , g 2 , g in ⌫ ⬁(ᐄ), the following subsets of [0, 1] are closed:
Ɱ ␣ g 1 ⫹ 共1 ⫺ ␣ 兲g 2 其
兵␣ : g ⬃
Ɱ g其.
and 兵 ␣ : ␣ g 1 ⫹ 共1 ⫺ ␣ 兲g 2 ⬃
780
F. PHILIPPE, G. DEBS AND J.-Y. JAFFRAY
A3S (strong continuity): for each g 0 in ⌫ ⬁(ᐄ), the following preference intervals are open:
兵g 僆 ⌫ ⬁ 共ᐄ兲 : g Ɑ g 0 其
and 兵g 僆 ⌫ ⬁ 共ᐄ兲 : g 0 Ɑ g其.
A4 (dominance under risk): for all P in ᏹ(ᐄ), and all x⬘ in ᐄ, if P is concentrated on X
then
Ɱx
x⬘ ⬃
Ɱ x⬘
x⬃
Ɱ P,
for all x 僆 X f e 兵x⬘其 ⬃
Ɱ e 兵x⬘其 .
for all x 僆 X f P ⬃
A5 (dominance under ignorance): for all K in ᏷(ᐄ), and all x⬘ in ᐄ,
Ɱx
x⬘ ⬃
Ɱ x⬘
x⬃
Ɱ K,
for all x 僆 K f 兵x⬘其 ⬃
Ɱ 兵x⬘其.
for all x 僆 K f K ⬃
For a justification of axioms A2, A3W, and A5 in the context of imprecise risk, see Jaffray
(1989a, 1991).
Since ⌫ ⬁(ᐄ) is convex, it is well known (see, e.g., Fishburn 1988) that either Jensen’s
(1967) axioms A1, A2, A3W or Herstein and Milnor’s (1953) axioms A1, A2W, A3 are
necessary and sufficient for the existence of an affine utility function U for Ɑ, which is
unique up to a positive affine transform. The following criterion (17) is obtained, by using
the second representation in (14), when ␦ is simple (Jaffray 1989b, Hendon et al. 1993):
(17)
U共 ␦ 共 f 兲兲 ⫽
冘
V共K兲 ␾ ␦ 共K兲
A⫽K債 ␦ 共᏿兲
with V defined on ᏷(ᐄ) by V(K) ⫽ U(e K ) and ␾ ␦ the Möbius transform of the restriction
of ␦ ( f ) to ␦(᏿).
In the case of risk, axiom A4 is used to provide an expected utility representation of Ɑ on
ᏹ(ᐄ). Note that e { x} is the point mass concentrated on { x}, so that the mapping u defined
on ᐄ by u : x 3 V({ x}) is precisely the von Neumann-Morgenstern utility function under
risk of the decision maker. Assume that all preference intervals { x 僆 ᐄ : x Ɑ x⬘} and
{ x 僆 ᐄ : x⬘ Ɑ x} are elements of Ꮽ(ᐄ), then axioms A1, A2W, A3, and A4 entail that
(18)
U共 ␦ 共 f 兲兲 ⫽
冕
u共x兲 ␦ 共 f 兲 共dx兲
ᐄ
if ␦ ( f ) is a probability measure (see Fishburn 1967).
We aim to generalize both criteria (17) and (18). In the situation of imprecise risk with
⬁-monotone cocapacities, the following representation holds:
THEOREM 5. Let Ɑ be a binary relation on ⌫ ⬁(ᐄ). The two following assertions are
equivalent:
(i) Axioms A1, A2W, and A3S hold.
(ii) There exists a continuous utility function U representing Ɑ on ⌫ ⬁(ᐄ), which satisfies,
for all ␦ in Ᏸ, with V defined on ᏷(ᐄ) by V(K) ⫽ U(e K), and ␮ ⫽ ␥(f ),
MONOTONE LOWER PROBABILITIES OF INFINITE ORDER
U共 ␦ 共 f 兲兲 ⫽
(19)
冕
781
V共 ␦ 共K兲兲 ␮ 共dK兲.
᏷
Moreover, U and V are then bounded, and unique up to a common positive affine transform.
PROOF. (i) f (ii). Let us verify that A3S implies A3. Consider a sequence ( ␣ n ) nⱖ0 in { ␣ :
Ɱ ␣ g 1 ⫹ (1 ⫺ ␣ ) g 2 } that converges to ␭; let ␯ 1 ⫽ ␥ ( g 1 ) and ␯ 2 ⫽ ␥ ( g 2 ), the sequence
g⬃
( ␣ n ␯ 1 ⫹ (1 ⫺ ␣ n ) ␯ 2 ) nⱖ0 weakly converges to the measure ␭␯ 1 ⫹ (1 ⫺ ␭)␯ 2 , so that g
Ɱ ␭ g 1 ⫹ (1 ⫺ ␭ ) g 2 by A3S. Similarly, the set { ␣ : ␣ g 1 ⫹ (1 ⫺ ␣ ) g 2 ⬃
Ɱ g} is closed.
⬃
Axioms A1, A2W, and A3 guarantee the existence of an affine utility function U for Ɑ,
which is unique up to a positive affine transform.
Now, let us verify that, with A3S, U is continuous. Without loss of generality, we may
assume that U is nonconstant. Choose any ⑀ ⬎ 0, any g 0 in ⌫ ⬁(ᐄ). First assume there exists
g 1 in ⌫ ⬁(ᐄ) such that g 1 Ɑ g 0 . Let
␣ ⫽ min 共1, ⑀ 共U共g 1 兲 ⫺ U共g 0 兲兲 ⫺1 兲
and g ⫹ ⫽ ␣ g 1 ⫹ 共1 ⫺ ␣ 兲g 0 ;
then g ⫹ is an element of ⌫ ⬁(ᐄ) and U( g ⫹ ) ⱕ U( g 0 ) ⫹ ⑀ . Similarly, if there exists g 2 in
⌫ ⬁(ᐄ) such that g 0 Ɑ g 2 , then there is an element g ⫺ of ⌫ ⬁(ᐄ) such that U( g ⫺ )
ⱖ U( g 0 ) ⫺ ⑀ . Denoting by O ⫺ and O ⫹ the open preference intervals { g 僆 ⌫ ⬁ (ᐄ) : g
Ɑ g ⫺ } and { g 僆 ⌫ ⬁ (ᐄ) : g ⫹ Ɑ g} respectively, then g 0 僆 O ⫺ 艚 O ⫹ by A2 and each g
in O ⫺ 艚 O ⫹ satisfies
兩U共g兲 ⫺ U共g 0 兲兩 ⬍ ⑀ .
Finally, if either g ⫹ or g ⫺ does not exist, then O ⫺ or O ⫹ will suffice. So, U is then continuous
on ⌫ ⬁(ᐄ), and, by Proposition 3, V is in turn continuous on ᏷(ᐄ).
Now we show that, by A3S and a standard argument, V is bounded on ᏷(ᐄ). If V is not
upper bounded, we may find (K n ) n⬎0 in ᏷(ᐄ) such that V(K n ) ⫽ U(e K n ) ⱖ 2 n for each n;
we may also assume without loss of generality that V(K n ) ⱕ V(K n⫹1 ). Let us consider
g0 ⫽
冘2
⫺n
N ⫽ 兩U共g 0 兲兩,
e Kn ,
n⬎0
g⫽
冘2
冘 1 ⫺2 2
k
⫺n
gk ⫽
e K N⫹n ,
n⬎0
⫺n
⫺k
e K N⫹n .
n⫽1
Ɱ g k for each k ⬎ 0 since
On the one hand, e K 1 ⬃
1
U共g k 兲 ⫽
1 ⫺ 2 ⫺k
冘2
k
⫺n
V共K N⫹n 兲 ⱖ V共K 1 兲;
n⫽1
on the other hand, ( g k ) k⬎0 converges to g, because if ␯ k ⫽ ␥ ( g k ) and ␯ ⫽ ␥ ( g) then we
have, for each real-valued and continuous ␾ defined on ᏷(ᐄ),
1
lim ␯ k 共 ␾ 兲 ⫽ lim
⫺k
k3⬁
k3⬁ 1 ⫺ 2
冘2
k
n⫽1
⫺n
␾ 共K N⫹n 兲 ⫽
冘2
n⬎0
⫺n
␾ 共K N⫹n 兲 ⫽ ␯ 共 ␾ 兲.
782
F. PHILIPPE, G. DEBS AND J.-Y. JAFFRAY
Ɱ g. Thus
Therefore, by A3S, e K 1 ⬃
冘2
N
U共g 0 兲 ⫽
⫺n
V共K n 兲 ⫹ 2 ⫺N U共g兲 ⱖ N ⫹ 2 ⫺N⫹1 ⬎ U共g 0 兲,
n⫽1
which yields the contradiction. A similar reasoning shows that V is lower bounded.
Then Proposition 5 and Lemma 2 result in
U共 ␦ 共 f 兲兲 ⫽
冕
V共K兲 ⌬共 ␮ 兲共dK兲;
᏷共ᐄ兲
whence (19). As a consequence, U is bounded too.
(ii) f (i) is conversely clear. 䊐
REMARK 4. According to Remark 2, the criterion satifies (18) if f is in ᏹ(᏿), and
according to relation (13) it satisfies (17) if ␦ is simple.
Of course, Axiom A2W may be replaced, in the statement of Theorem 5, by Axiom A2.
Although the latter is stronger, it avoids the use of the (controversial) indifference relation.
With the notations of Theorem 5, let u denote the real-valued mapping defined on ᐄ by
u( x) ⫽ V({ x}); if u is assumed to be continuous (which is the case when U, or V, is
continuous), then for each K in ᏷(ᐄ) there exist (at least) two elements x m (K) and x M (K) of
K such that
u共x m 共K兲兲 ⱕ u共x兲 ⱕ u共x M 共K兲兲
for all x in K.
Theorem 5 states that V is bounded. Axiom 5 forces a stronger (but appealing) property:
u( x m (K)) ⱕ V(K) ⱕ u( x M (K)) for each K in ᏷(ᐄ). Consider now the following axiom
(Jaffray 1989a), which clearly implies axiom A5:
Ɱ x m (L) and
A5S (strong dominance under ignorance): for all K, L in ᏷(ᐄ), [ x m (K) ⬃
Ɱ
Ɱ
x M (K) ⬃ x M (L)] f K ⬃ L.
With the latter axiom, V(K) only depends on the (classes of) extremal elements of K. As
a consequence, it expresses through the values of a much simpler utility function, defined on
ᐄ 2 instead of ᏷(ᐄ):
COROLLARY 1. If axioms A1, A2W, A3S, and A5S hold, then there exists an affine utility
function U representing Ɑ on ⌫ ⬁(ᐄ) which satisfies
(20)
U共 ␦ 共 f 兲兲 ⫽
冕
᏷共᏿兲
冕
min max h共x, x⬘兲 ␮ 共dK兲 ⫽
x僆 ␦ 共K兲 x⬘僆 ␦ 共K兲
᏷共᏿兲
max min h共x, x⬘兲 ␮ 共dK兲,
x⬘僆 ␦ 共K兲 x僆 ␦ 共K兲
for all ␦ in Ᏸ, with h defined on ᐄ 2 by h(x, x⬘) ⫽ U(e { x, x⬘}) and ␮ ⫽ ␥(f ); U and h are,
moreover, unique up to a common positive affine transform.
REMARK 5. The Choquet integral of u 䡩 ␦ with respect to the cocapacity f (resp. its dual
capacity F) writes
冕
᏷共᏿兲
min u共x兲 ␮ 共dK兲 ⫽
x僆 ␦ 共K兲
冕
᏿
u䡩 ␦ df
MONOTONE LOWER PROBABILITIES OF INFINITE ORDER
冉 冕
max u共x兲 ␮ 共dK兲 ⫽
resp.
᏷共᏿兲
x僆 ␦ 共K兲
冕
783
冊
u䡩 ␦ dF ;
᏿
the proof given in Gilboa and Schmeidler (1995) holds in our set-up as well. On the other
hand, it is known (see, e.g., Buja 1984, Proposition 2.6) that the Choquet integral of a convex
cocapacity is the lower expectation on its core:
冕
u䡩 ␦ df ⫽ inf
Pⱖf
᏿
冉 冕
冕
u䡩 ␦ dF ⫽ sup
resp.
Pⱖf
᏿
u䡩 ␦ dP
᏿
冕
冊
u䡩 ␦ dP .
᏿
Therefore, if, for some ␣, the utility function h on ᐄ 2 takes the form
(21)
h共x, x⬘兲 ⫽ ␣ u共x兲 ⫹ 共1 ⫺ ␣ 兲u共x⬘兲
with ␣ 僆 关0, 1兴 and u共x兲 ⱕ u共x⬘兲,
then criterion (20) reduces to the celebrated Hurwicz criterion with pessimism (or ambiguity)
index ␣,
U共 ␦ 共 f兲兲 ⫽ ␣ min
Pⱖf
冕
u䡩 ␦ dP ⫹ 共1 ⫺ ␣ 兲 max
Pⱖf
᏿
冕
u䡩 ␦ dP.
᏿
A nontrivial situation in which (21) holds has been studied in Jaffray and Philippe (1997): a
decision maker whose preferences are represented by a CEU criterion, and are consistent
with criterion (19) on a subalgebra of objectively but imprecisely probabilized events.
5. Conclusion. The model presented here extends linear utility theory to a situation of
imprecise risk. The family of criteria that are consistent with the model is larger than the
family of Hurwicz’s criteria.
From studies limited to the finite case (Jaffray 1989a, 1991), it appears that the axiom
system is still acceptable in a more general situation of imprecise risk, where the lower
probability is convex, rather than ⬁-monotone. Moreover, it can be shown that no assumption
on the lower probability is necessary if only bets (2-ranged decisions) are considered. The
extension of our model to the general case is the object of future work.
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F. Philippe: LIP6, Université Pierre et Marie Curie, 4 place Jussieu, F75005 Paris, France and Départment MIAp,
Université Paul Valéry, Route de Mende, 34199 Montpellier Cedex 5, France; e-mail: [email protected]
G. Debs: Equipe d’Analyse, Université Pierre et Marie Curie, 4 place Jussieu, F75005 Paris, France
J.-Y. Jaffray: LIP6, Université Pierre et Marie Curie, 4 place Jussieu, F75005 Paris, France