Download Research Articles Least convex capacitiesw

Economic Theory 13, 263±286 (1999)
Research Articles
Least convex capacitiesw
Larry G. Epstein and Jiankang Zhang
Department of Economics, University of Rochester, Harkness Hall,
Rochester, NY 14627-0156, USA (e-mail: [email protected])
Received: May 7, 1997; revised version: November 5, 1997
Summary. Debreu proposed the notion of `least concave utility' as a way to
disentangle risk attitudes from the certainty preferences embedded in a vonNeumann Morgenstern index. This paper studies preferences under uncertainty, as opposed to risk, and examines a corresponding decomposition of
preference. The analysis is carried out within the Choquet expected utility
model of preference and is centered on the notion of a least convex capacity.
Keywords and Phrases: Uncertainty, Uncertainty aversion, Choquet expected
utility, Capacity, Convex capacity, Risk aversion, Ambiguity, Non-additive
probability.
JEL Classi®cation Numbers: C69, D81.
1 Introduction
In [3], Debreu proposes the following procedure for separating `certainty
preferences' from risk aversion: Consider concave vNM utility indices
u : X ÿ! R1 , where X Rn is a convex set of outcomes. Each such function
models the decision-maker's ranking of deterministic (vector) outcomes,
through its ordinal properties. In addition, assuming expected utility theory,
then through its concavity u models risk aversion with respect to lotteries
with outcomes in X. The least concave utility function u is de®ned as the
concave function (unique up to ane transformations) having the property
that any concave function u that represents X numerically can be written in
the form
w
Earlier versions of this paper were circulated under the title Beliefs and Capacities. We are
indebted to the Social Sciences and Humanities Research Council of Canada for ®nancial
support and to Zengjing Chen, Chew Soo Hong, Darrell Due, Paolo Ghirardato, Kin Chung
Lo, Mark Machina, Massimo Marinacci and Uzi Segal for valuable discussions and suggestions.
Correspondence to: L. G. Epstein
264
L. G. Epstein and J. Zhang
u ˆ h…u †; for some concave and increasing h :
…1:1†
Debreu proposes u as the canonical representation of `certainty preferences'
and suggests that what is left, namely the transformation h, be identi®ed with
risk aversion. This separation between certainty preferences and risk aversion
is partial, that is, u is also concave (and nonlinear) in general. But the degree of
concavity of u is not a re¯ection of innate risk aversion in as much as it is
implied by certainty preferences. On the other hand, the degree of concavity of h
is `optional' and re¯ects risk aversion that is unrelated to certainty preferences.
Debreu's analysis is formulated within the expected utility framework and
thus presumes that only risk matters to the decision-maker. This is contrary
to evidence, such as the Ellsberg Paradox, that people also dislike `ambiguity'
or `vagueness', and in that case beliefs about the state space cannot be
represented by a probability measure. Use the term uncertainty to refer to the
general choice situation that may include ambiguity. Risk is the special case
of uncertainty where beliefs are probabilistic. Our presumption is that many
choice situations under uncertainty involve ambiguity as well as risk. This
raises the question of a conceptual decomposition of preference (and the
functional form of utility) in this more general environment. This paper
provides such a decomposition.
A three-fold decomposition is proposed. The ranking of deterministic
outcomes forms one component as in Debreu. The second is the decisionmaker's likelihood relation over events, or equivalently, his ranking of binary
acts or bets. The third component is the `willingness to bet', as modeled by the
(conditional) certainty equivalent values attached to bets. This latter component can also be interpreted as uncertainty aversion and is the counterpart for
the present more general setting of the risk aversion component in [3]. The
likelihood relation is only implicit in Debreu's analysis because he adopts the
von Neumann-Morgenstern framework of objective lotteries or probabilities,
which forces all decision-makers to agree about the likelihoods of events.
Our analysis is formulated within the framework of Choquet expected
utility (CEU) theory, due to Schmeidler [24] and Gilboa [8], an axiomatically
based generalization of subjective expected utility theory that was developed
in order to accommodate aversion to ambiguity or uncertainty. Savage's
prior probability measure is replaced in CEU by a (convex) capacity m,
sometimes referred to as a `non-additive probability.' The technical core of
the paper is the proof that (under speci®ed assumptions) there exists a unique
least convex capacity m that is ordinally equivalent to m, where `least convex
capacity' is de®ned in a fashion analogous to Debreu's de®nition of `least
concave utility function'.
A CEU utility function has the form
Z
U …f † ˆ u…f † dm ;
where u is a vNM index, m is a capacity, integration is in the sense of Choquet
and other details are provided later. We assume throughout that u is concave
Least convex capacities
265
and that m is convex. The components of this functional form are related to
the above three conceptual components of preference in the following way:
Denote by u the least concave utility function provided by Debreu and by m
the least convex capacity provided here. By the de®nition of these least elements, there exist monotonic transformations / (concave) and h (convex),
with u ˆ /…u † and m ˆ h…m †. Accordingly, we can write
Z
…1:2†
U …f † ˆ /…u …f †† dh…m †; f 2 F :
Our suggestion is to identify certainty preference with u , the likelihood
relation with m and willingness to bet with the pair …/; h†. As in [3], this
separation between distinct components of preference is partial or incomplete, but it admits intuitive justi®cation similar to that described above for
Debreu's decomposition.
2 Choquet expected utility
Let …S; R† be a measurable space representing states of the world and a
r-algebra of measurable events. A function m : R ÿ! ‰0; 1Š is called a capacity
if
(i) m…;† ˆ 0; m…S† ˆ 1; and
(ii) A B ˆ) m…A† m…B†.
The capacity is called convex if
m…A [ B† ‡ m…A \ B† m…A† ‡ m…B† ;
…2:1†
for all A; B 2 R. It is additive, and therefore a probability measure, if the
inequality is always satis®ed as an equality. The set of probability measures
is denoted M…S; R†. Probability measures are only ®nitely additive unless
explicitly speci®ed otherwise.
Choquet expected utility is de®ned over (simple) acts or uncertain prospects. An act is a function f from S into the outcome set X that has ®nite
range and is measurable in the sense that f ÿ1 …x† 2 R for each outcome x.
Assume that X Rn is a convex set of outcomes; more generally, a convex
subset of any real topological linear space would do. The functional form of
utility has two primitive functions. One is a continuous and concave vNM
utility index u : X ÿ! R1‡ . The second function is a convex capacity m. Utility
over acts is de®ned by:
Z
Z 1
m…fs 2 S : u…f …s†† tg† dt ;
…2:2†
U …f † ˆ u…f † dm 0
where the ®rst integral is a Choquet integral, de®ned by equality with the
second (Riemann) integral. If f yields outcomes x1 xn on the events
…A1 ; . . . ; An †, then
nÿ1
X
‰u…xi † ÿ u…xi‡1 †Šm…[i1 Aj † ‡ u…xn † :
U …f † ˆ
iˆ1
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L. G. Epstein and J. Zhang
Our maintained model of `rationality' is CEU with a convex capacity.
Convexity is widely assumed in applications of CEU. For axiomatizations
see [24] (in an Anscombe-Aumann framework) and [26] (in a Savage
framework). Less formal justi®cation is provided by recalling that CEU with
convex capacity coincides with the intersection of CEU and the related class
of multiple-priors utilities.1 Schmeidler suggests that convexity of a capacity
models aversion to uncertainty or ambiguity. Some readers may ®nd this
interpretation adequate justi®cation for assuming convexity. We prefer not
to rely on it because this interpretation is disputed elsewhere by one of us [6].
Denote by the implied preference ordering of acts. It delivers a likelihood relation ` over events via
A ` B () …x if A;
x if SnA† …x if B;
x if SnB† ;
…2:3†
for some x x; that is, if the decision-maker prefers to bet on A rather than
on B (for the given stakes). The ranking on the right is independent of the
particular stakes x x chosen because CEU satis®es Savage's axiom P4.
Indeed, we have that
A ` B () m…A† m…B† ;
…2:4†
for all A; B 2 R.
3 Decomposition of preference
One can identify three conceptually distinct components of preference .
The ®rst is preference over sure outcomes or `certainty preference', denoted
by X and de®ned as the restriction of to constant acts, where each act
constant at outcome x is identi®ed with the outcome x 2 X. The second is the
likelihood relation ` , or equivalently, the restriction of to binary acts or
bets as in (2.3). The third is uncertainty aversion or the willingness to bet as
de®ned shortly. These three components provide a complete decomposition,
in the sense that they uniquely determine the entire CEU order .
Denote by f ˆ …x1 ; A1 ; . . . ; xn ; An † the act that yields outcome xi on the
(measurable) event Ai , i ˆ 1; .ÿ. . ; n, where by convention
x1 xn . For
all 1 ` n ÿ 1 and x 2 X, f ; [i6ˆ`;`‡1 Ai ; x; A` [ A`‡1 denotes the act obtained from f by replacing the distinct outcomes x` and x`‡1 by the single
outcome x.
Suppose that two preference orders 1 and 2 satisfy: For all
1 ` n ÿ 1 and x 2 X,
ÿ
ÿ
f ; [i6ˆ`;`‡1 Ai ; x; A` [ A`‡1 1 f ˆ) f ; [i6ˆ`;`‡1 Ai ; x; A` [ A`‡1 2 f : …3:1†
In the special case where A` [ A`‡1 ˆ S, this states that for all binary acts f,
x 1 f ˆ) x 2 f :
1
…3:2†
The latter is axiomatized in an Anscombe-Aumann framework in [9]; we refer here to the
restrictions of these utility functions to the domain of Savage acts.
Least convex capacities
267
Consider the choice between f and the act with certain outcome x. Condition
(3.2) states that whenever the certain outcome is weakly preferred by 1 ,
then it is also weakly preferred by 2 . It is natural to think of 2 as being
more uncertainty averse than 1 when this condition is satis®ed.2 The condition (3.2) involves comparisons with perfect certainty only, while one might
be interested also in the relation between the values attached by 1 and 2 to
a partial reduction in uncertainty. Thus say that 2 is more uncertainty averse
than 1 given the stronger relation between the orders described in (3.1).3
Alternatively, suppose that the ®rst ranking in (3.2) is replaced by indi€erence 1 . Then x is the certainty equivalent of the bet f and thus measures the willingness to pay for the bet. In suitable conditional terms, (3.1)
indicates that 2 exhibits a smaller willingness to pay for bets than does 1 .
The conceptual decomposition of preference leads to the corresponding
decomposition (1.2) of the CEU functional form and to interpretation of
its components. The following theorem is important for justifying such an
interpretation:
Theorem 3.1. Let 1 and 2 be two CEU orderings with common certainty
preferences X and common beliefs ` . Denote by ui and mi the vNM index and
capacity corresponding to i in the sense of (2.2), i ˆ 1; 2. Suppose that
u2 ˆ G…u1 † for some increasing concave G on Range…u1 † and m2 ˆ g…m1 † for
some increasing convex g on ‰0; 1Š. Then 2 is more uncertainty averse than 1 ,
or equivalently, 2 has smaller willingness to pay for bets than does 1 .
See Appendix A for a proof. A similar result appears in [11]. A special case of
the theorem has appeared in the literature on preferences over risky prospects (see [1] and [2, Theorem 3]), as discussed further in the context of
Example 1 below. The converse is also valid if m1 …R† ˆ ‰0; 1Š, as is implied by
the assumption of convex range made in the next section. A proof of the
converse may be constructed along the lines of [1] and [2, Theorem 3].
The theorem shows that ®xing the ordinal properties of u and m, and
hence also X and ` , willingness to bet decreases with the indicated
transformations of the vNM index and capacity. The relation u2 ˆ G…u1 †
de®nes the partial ordering of vNM indices studied by Debreu; his least
concave utility function u is smallest according to this partial ordering. The
corresponding relation for capacities, m2 ˆ g…m1 †, de®nes the partial ordering
of capacities to be studied below; existence of a smallest element m will be
proven under suitable assumptions.
Given the existence of u and m , the decomposition (1.2) is established
and its interpretation supported, because the pair …/; h† models willingness to
bet in a comparative sense. The latter quali®cation is important. As men2
Recall that uncertainty is used in the comprehensive sense and thus includes risk. In particular,
no attempt is being made to distinguish between aversion to risk and aversion to ambiguity. See
[6] for such a distinction.
3
See [15] for parallel conditional and unconditional notions of (comparative) risk aversion in the
theory of preference over lotteries.
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L. G. Epstein and J. Zhang
tioned in the introduction, the least concave utility in [3] is in particular
concave and thus embodies some risk aversion. However, its concavity is
implied by the ordinal properties of the vNM index u and in that sense is
conceptually distinct from the remaining concavity in u, modeled here via the
concavity of /. In the same way, m is convex, but its convexity is necessitated
by the ordinal properties of m, that is, by ` . On the other hand, the convexity
of h may be distinguished because it is `optional' in the sense of not being
necessary either for the numerical representation of likelihood or by the
maintained model of rationality, consisting of (2.2) with a convex capacity.
The decomposition (1.2) of the CEU functional form has an attractive
feature that supports the separate conceptual interpretations
ÿ that we have
,
m
and
…/;
h†.
If
u
;
m
;
…/
;
h
†
are two
attached to the components
u
i
i
i
i
iˆ1;2
ÿ
such tuples, then u1 ; m2 ; …/2 ; h1 † also corresponds to a CEU utility functional
satisfying our regularity conditions, because h1ÿ…m2 † is a convex
capacity and
/2 …u1 † is a concave vNM index.4 Similarly for u2 ; m1 ; …/1 ; h2 † and so on.
4 Less convex
4.1 De®nitions
Given a convex capacity m and the associated likelihood relation ` , denote
by V the class of all convex capacities m0 ordinally equivalent to m and thus
representing the identical likelihood relation ` as in (2.4).
In common with Savage [21], our formal analysis is restricted to `rich'
state spaces and a suitable form of non-atomicity. Say that m is convex-ranged
if for every C A and r 2 …m…C†; m…A††, there exists C B A such that
m…B† ˆ r. If B can always be chosen to satisfy also m…AnB† > 0, then refer to
the capacity as strongly convex-ranged (scr). For a probability measure m,
convex-ranged and scr are equivalent and, in the countably additive case,
they are further equivalent to non-atomicity.5 The Savage axioms deliver a
convex-ranged probability measure. For axiomatizations of CEU that deliver a convex-ranged capacity, see [8, p. 73] and [20, Proposition A.3]. These
axiomatizations can be extended to deliver scr capacities.
The subset of V consisting of convex-ranged capacities is denoted Vcr .
If m 2 Vcr , then the ordinally equivalent capacity g…m† is also in Vcr for
any g : ‰0; 1Š ÿ! ‰0; 1Š that is increasing, continuous and convex.6
Restricting attention to capacities in Vcr , Theorem 3.1 motivates de®nition of the partial ordering cvx on Vcr , where
m2 cvx m1 () m2 …† ˆ h…m1 …†† on R ;
4
…4:1†
That is, h1 …m2 † is a convex capacity because h1 is increasing and convex and /2 …u1 † is a concave
vNM index because /2 is increasing and concave.
5
A probability measure m on …S; R† is non-atomic if m…A† > 0 implies that 0 < m…B† < m…A† for
some measurable B A. In the absence of countable additivity, non-atomicity does not imply
convex range [17, pp. 142±3].
6
In particular, m convex and g a convex function imply that g…m† satis®es the inequality (2.1).
Least convex capacities
269
for some convex and increasing function h : ‰0; 1Š ÿ! ‰0; 1Š.7 Say that m1 is less
convex than m2 if m2 cvx m1 . A capacity that is minimal with respect to cvx is
called minimally convex. A capacity that is smallest according to cvx is called
least convex. Clearly a least convex capacity is unique if it exists.
Some insight into the least convex capacity is provided by examining its
core. For any capacity m, de®ne its core by
core…m† ˆ fp 2 M…S; R† : p…† m…† on Rg :
…4:2†
For any convex capacity, the core is nonempty and
m…† ˆ minfp…† : p 2 core…m†g :
…4:3†
These two relations constitute the essential link between CEU and the
multiple-priors model of preference [9], in which the single prior of Savage is
replaced by a set of priors.
It is easy to see that
m1 …† m2 …† on R () core…m1 † core…m2 † :
On the other hand,
m2 cvx m1 ˆ) m1 …† m2 …† on R ;
because any convex, increasing and surjective function h on the unit interval
must satisfy
h…x† x on ‰0; 1Š :
Conclude that
m2 cvx m1 ˆ) core…m1 † core…m2 † :
…4:4†
Consequently, the least convex capacity is the capacity in Vcr with the
smallest core and in that intuitive sense, `closest' to the Savage single prior
model.
A possible variation of our formal analysis deserves mention. While we
have focussed on ` and thus on bets on an event (see (2.3)), it is seemingly as
intuitive to refer to bets against an event and thus on the binary relation `
de®ned by
A ` B () …x if SnA; x if A† …x if SnB; x if B† () SnA ` SnB ; …4:5†
for x x. The relations ` and ` coincide for qualitative probabilities, but
not in general.8 The counterpart of (2.4) is
A` B () m…A† m…B† ;
7
Increasing is intended in the strict sense, whereby t0 > t ˆ) h…t0 † > h…t†. We use the term
`nondecreasing' to refer to the weak version. Note that (4.1) implies that h is surjective because
each capacity is convex-ranged.
8
See [13, p. 118] for a de®nition of qualitative probability. The key de®ning property is
`additivity', which states that if A ` B and if C is disjoint from A [ B, then A [ C ` B [ C.
Additivity is necessary for representation by a probability measure and is violated by typical
behavior in the Ellsberg paradox.
270
L. G. Epstein and J. Zhang
where m is the dual or conjugate capacity de®ned by
m…A† 1 ÿ m…SnA† :
Convexity of m is equivalent to concavity of m, where concavity is de®ned by
(2.1) with the inequality reversed. Therefore, it is natural to seek the most
concave capacity that represents ` numerically, where `more concave' is
de®ned by the obvious modi®cation of (4.1). It is evident that m is least
convex if and only if its conjugate is most concave. In this sense, least convex
and most concave capacities are equivalent (as opposed to identical) constructs.
4.2 Examples
Example 1 (Probabilistic sophistication). Suppose that ` may be represented
by a probability measure, that is, V contains a convex-ranged probability
measure p, or Vcr is nonempty. In other words, referring to (2.2),
m ˆ g…p† ;
…4:6†
where g is a monotonic and convex. This delivers the rank-dependent
expected utility model that has been studied extensively under the interpretation that p is an objective probability measure (see [1] and the references
therein).
It is well-known that p is necessarily the unique convex-ranged probability measure representing ` . The more general perspective a€orded by this
paper is that p is the unique least convex capacity in Vcr . (This is easily
proven directly. Alternatively, it follows from (4.4).) Therefore, the identi®cation of the likelihood relation with the least convex capacity conforms
with the standard practice of identifying likelihood with a probability measure whenever that is possible.
Because the model consisting of (2.2) and (4.6) is probabilistically
sophisticated in the sense of [16], preference is indi€erent to ambiguity;
for example, a probabilistically sophisticated agent behaves identically to a
subjective expected utility maximizer in the context of Ellsberg-type experiments. Accordingly, the relation (3.1) is more accurately described as saying
that 2 is more risk averse than 1 . The characterization of comparative risk
aversion paralleling Theorem 3.1 is well known ([1] and [2, Theorem 3]).
Example 2 (Rich class of unambiguous events). This example generalizes
Example 1. Given a convex capacity m, de®ne
c
Aua
m ˆ fA 2 R : m…A† ‡ m…A † ˆ 1g :
Then Aua
m is an algebra where m is additive and therefore a probability
measure. The restriction on ` that we impose here is that there exists m 2 V
such that mjAua is convex-ranged. This imposes a form of richness of the set
m
Aua
m . Then it is immediate that m is the least convex capacity in Vcr .
Least convex capacities
271
One interpretation is that Aua
m is a class of events determined by objective randomization, such as the spin of a roulette-wheel, the likelihoods of
which are suciently precise to be represented by a probability measure.
Denote this measure by p. Suitable richness of Aua
m permits the unique
extension of p to a capacity m de®ned on R via calibration, that is, for
any B 2 R,
m …B† ˆ j if 9A 2 Aua
m ; p…A† ˆ j and B ` A :
…4:7†
The capacity m constructed in this way is a natural candidate for identi®cation with likelihood, showing that the suggested use of the least convex
capacity conforms with intuition also in this case.
Example 3 (Interval beliefs). Let ` and ` be two (®nitely additive) nonnegative, convex-ranged measures on …S; R†, such that
` …† ` …†, and 0 < ` …S† < 1 < ` …S† :
(Therefore, ` and ` are not probability measures.) De®ne d ˆ ` …S† ÿ 1 and
m…A† ˆ maxf` …A†; ` …A† ÿ dg :
…4:8†
Then m is a convex capacity on …S; R† and has the core
core…m† ˆ fp 2 M…S; R† : ` …† p…† ` …† on Rg :
(See [27] that also provides references to the robust statistics literature where
this capacity has been used.) The description of the core provides intuition
for m and the reason for its name.
The core has another illuminating representation in the case where `
and ` are countably additive. De®ne the probability measure l by
l…† ˆ ` …†=` …S†. All measures in the core are absolutely continuous with
respect to l. Therefore,
Z
…4:9†
core…m† ˆ fp : dp ˆ w dl; w w w a.e.‰lŠ; w dl ˆ 1g ;
where w ; w and w are the (Radon-Nikodym) densities with respect to l for
` , ` and generic p respectively. The core is thus isomorphic to an `interval'
in L1 …S; R; l†. Uncertainty aversion is modeled through the multiplicity of
admissible densities.
The capacity (4.8) provides another example where there exists a least
convex capacity m representing the implied ` . The identity of m depends on
whether or not ` is a qualitative probability. One can show that ` is a
qualitative probability if and only if there exist a probability measure p and
constants a, b, such that
` …† ˆ a p…†; ` …† ˆ b p…† ;
…4:10†
in which case p represents ` and the decision-maker is probabilistically
sophisticated. (This is Example 1 with g…t† ˆ maxfat; bt ÿ …b ÿ 1†g.) In this
case, the least convex capacity is p. On the other hand, if m is not a qualitative
probability, then m itself is least convex (see Appendix A).
272
L. G. Epstein and J. Zhang
This example is not covered by Example 2. The algebra Aua
m of unambiguous events, as de®ned in Example 2, is given by
c
c
Aua
m ˆ fA 2 R : ` …A† ˆ ` …A† or ` …A † ˆ ` …A †g :
Therefore, if ` …A† < ` …A† for all nonempty A, then Aua
m is the trivial algebra
f;; Sg and thus is decidedly not rich.
Example 4 (e±contamination). This example illustrates the diculty in
dropping the assumption of convex range. For each e in …0; 1†, de®ne the
capacity me on …S; R† by
me …A† ˆ
1
…1 ÿ e†p…A†
AˆS
A 6ˆ S ;
where p is a given probability measure on …S; R†. Then me is a convex capacity
(see [27]). At the extremes, one obtains m0 ˆ p and m1 ˆ complete ignorance.
For intermediate values of e, the nature of me is elucidated by observing its
core,
core…me † ˆ …1 ÿ e†p ‡ eM…S; R† ;
which shrinks with as the `contaminating weight' e falls.
Observe that each me , for 0 < e < 1, de®nes the identical likelihood relation ` . Moreover, ` is not a qualitative probability if and only if there exist
two disjoint (nonempty) p-null events, or equivalently, if 9
9A 6ˆ B 6ˆ S; A B; p…A† ˆ p…B† :
…4:11†
Assume this condition. Then there does not exist a least convex capacity
representing ` numerically: Any such capacity m would have to satisfy
me …† m …† on R for all e > 0. Letting e ÿ! 0, one obtains p…† m …† m…† for any m in core…m †, implying m ˆ p a contradiction to the fact that `
is not a qualitative probability.
None of the capacities me has convex range, which explains why our result
on the existence of a least convex capacity does not apply. On the other hand,
convex range is not necessary for the existence of a least convex capacity. In
the limiting case of complete ignorance, where e ˆ 1, the likelihood relation
satis®es
S ` A ` ;; for all A 2 R :
Therefore, m1 is the unique capacity representing beliefs numerically and so is
trivially least convex, even though its range is f0; 1g.
5 Existence
This section provides sucient conditions for the existence of a least convex
capacity. Some perspective on the technical analysis is provided by a com9
Given (4.11), A ` B but A [ …SnB† ` B [ …SnB† ˆ S, violating the additivity property of a
qualitative probability.
Least convex capacities
273
parison with Debreu [3] (see the outline in the introduction). In spite of the
super®cial similarity between the two problems, there are substantial differences. For example, the domain R for capacities does not have the linear
structure of a topological vector space. More importantly, there is only an
imperfect parallel between the properties of convexity for capacities on the
one hand and for functions de®ned on a topological vector space on the
other hand. This parallel is emphasized by Rosenmuller [18] and [19]. That it
is imperfect is apparent from (4.3). If one views additive measures as counterparts of linear functions on a Euclidean space, then the fact that a convex
capacity is a lower envelope of measures suggests that there is also a parallel
with concave functions. Another di€erence is that we are looking for a least
convex element, and therefore, for a most (rather than least) concave element. Finally, an important di€erence is that the pointwise maximum of two
convex capacities need not be convex, while convexity for functions is
inherited by a pointwise maximum.
We proceed as follows: First we prove that, given any m in Vcr , there
exists (under a strengthening of convex-range) a minimally convex capacity.
Then, under additional assumptions, we show that Vcr contains only one
minimally convex capacity and that the latter is least convex.
It is convenient to de®ne the alternative partial order dom corresponding
to eventwise dominance:
…5:1†
m2 dom m1 if m1 …† m2 …† on R :
If the natural meaning is attached to dom -minimality, then the latter implies
cvx -minimality because
m2 cvx m1 ˆ) m2 dom m1 :
Theorem 5.1. Let m 2 Vcr be strongly convex-ranged. Then there exists a
dom -minimal convex capacity in Vcr ‰mŠ fbm 2 Vcr : m dom bmg and therefore
also in Vcr .
See Appendix B for a proof. Some informal clarifying comments are
o€ered here. The hypothesis that m 2 Vcr is strongly convex-ranged expresses
the assumption that ` can be represented numerically by at least one convex
and strongly convex-ranged capacity. From the de®nition of Vcr , it follows
that
Vcr ˆ fg…m† : g…m† convex; g : ‰0; 1Š ÿ! ‰0; 1Š; onto, increasingg :
Consequently, if some capacity in Vcr is scr then every capacity in Vcr is scr.
Note further that any g as above is continuous. The existence of a minimally
convex element is nontrivial because the properties `increasing' and `onto' (or
`continuous') are not obviously inherited in passing to the limit of a net of
functions fga g. Therefore, some work is required to show that Zorn's
Lemma applies.
Turn to the existence of a least convex capacity in Vcr . We specify assumptions on ` , that is, ordinally invariant assumptions on capacities, that
are sucient to deliver the existence of a least convex capacity.
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L. G. Epstein and J. Zhang
Consider ®rst the following property for ` , referred to as continuity at
certainty: For all B and An 2 R,
If An % S and S ` B; then An ` B for all sufficently large n :
If ` admits numerical representation by a convex-ranged capacity m,
then this property may be equivalently expressed as the following restriction
on m:
m…An † % 1 whenever An % S; An 2 R :
…5:2†
Refer to such a capacity m as continuous at certainty.
It is advantageous to consider also a weaker property: Given a probability measure l on …S; R†, say that ` (or any representing capacity) is
l-continuous at certainty if for every B, S ` B, there exists d > 0 such that
l…A† > 1 ÿ d implies A ` B; A 2 R :
The corresponding restriction on a representing capacity is that 8e > 0
9d > 0 such that
l…A† > 1 ÿ d implies m…A† > 1 ÿ e :
…5:3†
As an example, the interval beliefs capacity is l-continuous at certainty
where l is the probability measure ` …†=` …S†.
We defer explication of these properties in order to state ®rst our major
result:
Theorem 5.2. Suppose that ` can be represented numerically by some convex
and strongly convex-ranged capacity and that ` is l-continuous at certainty
for some convex-ranged probability measure l. Then there exists a (unique)
least convex capacity m in Vcr ; that is, m cvx m for any convex-ranged m that
represents ` numerically.
A proof is provided in Appendix C. A key to the proof is Theorem C.4,
showing the equivalence between convexity and `local convexity'. This
equivalence mirrors the property of convexity for functions on Euclidean
space whereby such a function is convex on its domain if and only if it is
convex in some open neighborhood of every point in the domain; in other
words, convexity of such functions is a local property. Under (5.3), we show
that convexity of a capacity is also a local property, and then we show that
this implies the uniqueness of the minimal elements delivered by Theorem
5.1.
In the remainder of this section, we clarify the scope of the theorem by
elucidating the meaning of the two forms of continuity at certainty. As for
the meaning of (5.2), Schmeidler [24, Proposition 3.15] shows that it implies,
given that m is convex, the seemingly stronger form of continuity whereby
m…An † % m…A† whenever An % A and similarly for monotone decreasing
sequences. He also shows that, again given convexity, (5.2) is equivalent to
the following structure for the core of m:
core…m† ˆ fm 2 M…S; R† : dm ˆ w dl; w 2 H L1‡ …S; R; l†g ;
…5:4†
Least convex capacities
275
where l is a countably additive probability measure on …S; R†, l lies in
core…m†, H is weakly sequentially compact and each w is a Radon-Nikodym
density.10 Consequently, all measures in the core are countably additive and
absolutely continuous with respect to the stated l. Finally, if m is convexranged, then (and only then) l is convex-ranged.
The preceeding can be extended to provide a characterization of l-continuity at certainty and to clarify the relation between the two properties. Say
that a set P of measures is uniformly absolutely continuous with respect to l
if for every e > 0 there exists d > 0 such that …8E 2 R† lE < d implies
mE < e; 8m 2 P .
Theorem 5.3. (a) m is l-continuous at certainty for some l if and only if core…m†
is uniformly absolutely continuous with respect to some l.
(b) m is continuous at certainty if and only if m is l-continuous at certainty
for some countably additive l if and only if core…m† satis®es (5.4).
Under continuity at certainty, all probability measures in the core are
countably additive. The upshot of this theorem is that the weakening to lcontinuity at certainty eliminates the imposition of countable additivity on
the elements of the core, while retaining uniform absolute continuity of the
core. This relaxation of countable additivity means that Theorem 5.2 includes the Savage model, with ®nitely additive probability, as a special case.11
Proof. (a) ˆ) in the ®rst equivalence: Given e, choose d as in (5.3). Then
lE < d ˆ) l…SnE† > 1 ÿ d ˆ) m…SnE† > 1 ÿ e ˆ) m…SnE† > 1 ÿ e for all
m 2 core…m† ˆ) m…E† < e 8m 2 core…m†. For the reverse implication, reverse
the above argument.
(b) By [22], continuity at certainty is equivalent to weak sequential
compactness of core…m† in ca…S; R†. By [5, Theorem IV.9.2], the latter implies
uniform absolute continuity of core…m† with respect to l. It is evident that lcontinuity at certainty for a countably additive l implies continuity at certainty. The ®nal equivalence is due to the isometric isomorphism between
ca…S; R; l† and L1 …S; R; l†: (
The real meanings of the above properties are revealed only by examining
their implications for preference over acts. Consider continuity at certainty;
obvious modi®cations apply to the weaker property. It is immediate from the
Choquet expected utility form (2.2), that continuity at certainty is equivalent
to the following restriction for utility:
U …1An † % U …1S † whenever An % S :
This assumption rules out a categorical distinction between uncertainty, no
matter how `small', and perfect certainty. (See [7, Sect. 2] for elaboration and
10
See [24, Theorems 3.2, 3.10] and [5, Theorem IV.8.9]. Delbaen [4, p. 226] points out that one
can choose l 2 core…m†.
11
Further equivalences in Theorem 5.3 are: (a) core…m† is weakly sequentially compact in
ba…S; R†, and (b) core…m† is weakly sequentially compact in ca…S; R†. See [5, Theorem IV.9.12].
276
L. G. Epstein and J. Zhang
for the connection with lower semi-continuity of U in the Mackey topology.)
Such a distinction may be plausible in a variety of circumstances and is a
characteristic of `complete ignorance', where there is no reason to view SnAn
as small in any meaningful sense and where therefore lim U …1An † < U …1S † is
reasonable. On the other hand, a large number of capacities and preferences,
exhibiting aversion to uncertainty, are consistent with continuity at certainty.
The nonsingleton nature of each set of densities h in (5.4) models such
aversion and a large number of speci®cations for h are possible. One such
speci®cation is (4.9) corresponding to the interval beliefs model.
Conclude with a reinterpretation of our analysis in the context of
co-operative games with transferable utility. Interpret S as a set of players, R
as a class of admissible coalitions and m as a convex characteristic function.
Then core…m† is the standard (multivalued) prediction of the outcome of the
game. However, suppose that the analyst does not know m completely.
Rather, she knows that the game being played is convex and, for all coalitions A and B, she knows which coalition receives a larger payo€. In other
words, she knows m only up to ordinal equivalence. In such a situation, what
is a sensible prediction of the outcome? One suggestion is
\fcore…m0 † : m0 ˆ h…m†; m0 convexg :
Under the conditions of Theorem 5.2, we know that this intersection is
nonempty; it equals core…m †, where m is the least convex capacity (apply
(4.4)).
A. Appendix
Theorem 3.1 and the least convex property asserted for the interval beliefs
example are proven here.
Proof of Theorem 3.1. Let
ÿ
f ; [i6ˆ`;`‡1 Ai ; x; A` [ A`‡1 1 f ;
or equivalently, writing E ˆ [`ÿ1
1 Aj , that
u1 …x†‰m1 …E ‡ A` ‡ A`‡1 † ÿ m1 …E†Š u1 …x` †‰m1 …E ‡ A` † ÿ m1 …E†Š
‡ u1 …x`‡1 †‰m1 …E ‡ A` ‡ A`‡1 † ÿ m1 …E ‡ A` †Š :
Because u2 is more concave than u1 , the inequality is preserved if u1 is replaced by u2 . It suces to consider the case where m1 …E ‡ A` ‡ A`‡1 †
ÿm1 …E† > 0. Then the corresponding statement for m2 is true as well: We have
shown that
u2 …x† u2 …x` †‰m1 …E ‡ A` † ÿ m1 …E†Š ‡ u2 …x`‡1 †‰m1 …E ‡ A` ‡ A`‡1 † ÿ m1 …E ‡ A` †Š
:
m1 …E ‡ A` ‡ A`‡1 † ÿ m1 …E†
…A:1†
Least convex capacities
277
Because m2 ˆ g…m1 †, with g convex, it follows that
m1 …E ‡ A` † ÿ m1 …E†
m2 …E ‡ A` † ÿ m2 …E†
:
m1 …E ‡ A` ‡ A`‡1 † ÿ m1 …E† m2 …E ‡ A` ‡ A`‡1 † ÿ m2 …E†
Thus (A.1) continues to hold if m1 is replaced by m2 . This proves
ÿ
f ; [i6ˆ`;`‡1 Ai ; x; A` [ A`‡1 2 f : (
Now we prove that the interval beliefs capacity (Example 3) is least
convex, if (4.10) is excluded. A preliminary notion is the following: The
capacity m is locally additive if for every j 2 ‰0; 1Š 9e > 0 such that 8x1 < x2 in
…j ÿ e; j ‡ e† \ ‰0; 1Š there exist A; B satisfying
m…A \ B† ˆ x1 ; m…A [ B† ˆ x2 ; m…A† ˆ m…B† ˆ …x1 ‡ x2 †=2 :
…A:2†
Lemma A.1. If m is convex and locally additive, then it is least convex.
Proof. If g…m† is convex, then local additivity implies that given any relatively
open interval …j ÿ e; j ‡ e† \ ‰0; 1Š,
g…x2 † ‡ g…x1 † 2g……x1 ‡ x2 †=2†
for all …x1 ; x2 † in the interval. Therefore, g is convex on [0, 1].
(
Say that m : R ÿ! ‰0; 1Š is locally additive on an interval …a; b† ‰0; 1Š if
the condition (A.2) is satis®ed for every j in …a; b†. Similarly for closed
intervals. A trivial but useful observation is that if m is locally additive on
each of …a1 ; b1 † and …a2 ; b2 †, with a2 < b1 , then m is locally additive on …a1 ; b2 †.
Turn to the speci®cs of the interval beliefs capacity. De®ne
D ˆ f…` …A†; ` …A†† : A 2 Rg
I ˆ fA 2 R : ` …A† ÿ d ` …A†g
II ˆ fA 2 R : ` …A† ÿ d ` …A†g
DI ˆ f…` …A†; ` …A†† : A 2 Ig
DII ˆ f…` …A†; ` …A†† : A 2 IIg :
Observe that
DI \ DII ˆ f…` …A†; ` …A†† : A 2 I \ IIg :
If either f…` …A†; ` …A†† : ` …A† > ` …A† ÿ dg or f…` …A†; ` …A†† : ` …A† >
` …A† ÿ dg is empty, then m ˆ ` ÿ d or m ˆ ` and m is obviously least convex.
Assume henceforth that neither set is empty. Figure 1 illustrates some of the
following argument.
By the Lyapunov Theorem for ®nitely additive vector measures [17,
Theorem 11.4.9], D is a convex set. It contains the points (0,0) and
…` …S†; ` …S††. It follows that either (a) D equals the line segment joining these
two points, or (b) DI \ DII is a nonsingleton. The ®rst alternative implies
(4.10). Assume, therefore, that DI \ DII is a nonsingleton.
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L. G. Epstein and J. Zhang
Figure 1
Step 1: A 2 I; A B ˆ) B 2 I; A 2 II; B A ˆ) B 2 II.
Step 2: There exist A ; A 2 I \ II such that ` …A † 6ˆ ` …A †. If not, then
DI \ DII would be a singleton.
Step 3: For any A 2 I \ II; m is locally additive on both ‰0; ` …A †Š and
‰` …A †; 1Š: By Step 1, m ˆ ` on fA 2 R : A A g and m ˆ ` ÿ d on
fA 2 R : A Ag. The assertion follows. (Note that ` …A † ˆ ` …A † ÿ d.)
Step 4: m is locally additive on [0,1]: By Step 2, we can pick A 2 I \ II with
` …A † > ` …A †. By Step 3, m is locally additive on ‰0; ` …A †Š. We have
already noted that it is locally additive on ‰` …A †; 1Š. Because these two
intervals overlap, m is locally additive on [0,1].
Lemma A.1 completes the proof.
B. Appendix
This appendix provides a proof of Theorem 5.1.
Lemma B.1. Given a strongly convex-ranged capacity m 2 Vcr , let g : ‰0; 1Š ÿ!
‰0; 1Š be nondecreasing, g…0† ˆ 0; g…1† ˆ 1; g…x† x on [0,1] and suppose that
g…m† is convex. Then: (a) g is increasing, and (b) g is continuous from the right.
Proof. (a) Let 0 x1 < x2 1. By scr, there exist A1 A2 such that
m…Ai † ˆ xi , i ˆ 1; 2, and m…A2 n A1 † > 0. It then follows from the convexity of
g…m† that g…x2 † ÿ g…x1 † ˆ g…mA2 † ÿ g…mA1 † g…m…A2 nA1 †† m…A2 nA1 † > 0.
(b) Suppose to the contrary that there exists x0 2 ‰0; 1† such that
g…x0 ‡† > g…x0 †. As a nondecreasing function, g has at most a countable
number of points of discontinuity. Therefore, there exist x0 < x2 < x1 < 1,
satisfying
Least convex capacities
279
g…x1 † ÿ g…x2 † < g…x0 ‡† ÿ g…x0 † :
By scr, there exist A3 A2 A1 satisfying m…A3 † ˆ x0 , m…A2 † ˆ x2 , m…A1 † ˆ x1
and m…A1 n A2 † > 0. Let
A ˆ A3 [ …A2 n A3 † ˆ A2 ; B ˆ A3 [ …A1 n A2 † :
Thus,
m…A [ B† ˆ m…A1 † ˆ x1 ; m…A \ B† ˆ m…A3 † ˆ x0 ;
m…A† ˆ m…A2 † ˆ x2 > x0 ;
m…B† ˆ m…A3 [ …A1 n A2 †† m…A3 † ‡ m…A1 n A2 † > m…A3 † ˆ x0
and
g…m…A [ B†† ‡ g…m…A \ B†† ˆ g…x1 † ‡ g…x0 †
< g…x2 † ‡ g…x0 ‡†
< g…m…A†† ‡ g…m…B†† :
This contradicts the hypothesis that g…m† is a convex capacity. (
Proof of Theorem 5.1. To show that there exists a dom -minimal element in
mcr ‰mŠ, employ Zorn's Lemma and show that every chain fga …m†g in mcr ‰mŠ has
a lower bound in mcr ‰mŠ. We will show that g …m† is such a lower bound, where
g …x† sup ga …x†; x 2 ‰0; 1Š :
…B:1†
Note that each ga is continuous, increasing and satis®es ga …0† ˆ 0, ga …1† ˆ 1
and ga …x† x on [0,1]. Moreover, fga g is a chain with respect to the usual
ordering of functions.
Claim: There exists a subsequence fgn g of fga g such that, for every x 2 ‰0; 1Š,
g …x† ˆ lim gn …x† :
n
…B:2†
Exploit the claim, deferring its proof. Because g is a pointwise limit, it is
immediate that g …m† is convex, g is nondecreasing and satis®es the other
hypotheses of Lemma B.1. Conclude that g is increasing and continuous
from the right. Because it is de®ned as a pointwise supremum of continuous
functions, g is lower semi-continuous, or equivalently (given that g is
increasing), it is continuous from the left. Finally, (B.1) implies that
ga …m† dom g …m† for all a.
It remains to prove the claim. Let fri g be an enumeration of the rationals
in the unit interval. There exists a subsequence fg1n g satisfying
g1n …r1 † %n g …r1 †. Moreover, because fga g is a chain with respect to the usual
ordering of functions, we can select the subsequence so that g1n …† %n .
Therefore, lim g1n …r2 † exists. If the limit equals g …r2 †, then de®ne g2n ˆ g1n . If
lim g1n …r2 † < g …r2 †, there exists another subsequence fg2n g of fga g such that
g2n …† %n and g2n …r2 † %n g …r2 †. By the chain property, we then have
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L. G. Epstein and J. Zhang
g2n …r1 † %n g …r1 †. (For every g1n …r2 † there exists g2m …r2 † > g1n …r2 †. By the chain
property, g2m …r1 † g1n …r1 †. Therefore, limm g2m …r1 † > limn g1n …r2 † ˆ g …r1 †.) In
this second case, rede®ne g1n ˆ g2n . Proceeding inductively, we construct
fgkn g fga g such that
lim gkn …ri † ˆ g …ri †;
n
k
gn …†
i ˆ 1; . . . ; k; and
%n on ‰0; 1Š; for each k :
Therefore, the sequence fgnn g satis®es
lim gnn …ri † ˆ g …ri †;
n
i ˆ 1; 2; . . .
…B:3†
By Helly's Selection Principle [12, p. 204], there exists fgn g fgnn g such that
limn gn …x† exists everywhere on ‰0; 1Š.12 Denote this limit by g …x†. Then
g …x† g …x† for all x. It remains to show that g …x† ˆ g …x†. If not, then
g …x†_ > g …x† for some x. We noted above that g is lower semi-continuous.
Therefore, we can ®nd a rational number r < x, such that
g …r† > g …x† g …r† ;
where the last inequality is valid because g is nondecreasing. On the other
hand, g …r† ˆ g …r† by (B.3). This is a contradiction. (
C. Appendix
Theorem 5.2 is proven here. Its hypotheses are adopted throughout.
Write A ‡ F to indicate both the union of the two sets and that F \ A ˆ ;.
Convexity of a capacity m is equivalent to the property of `increasing
di€erences': For all measurable B A and F ,
m…A ‡ F † ÿ m…A† m…B ‡ F † ÿ m…B† :
…C:1†
Say that m is convex on …a; b† ‰0; 1Š if the above inequality is satis®ed
whenever the capacities of the four indicated sets all lie in …a; b†. Similarly for
intervals that are closed or half-closed.
We also employ a notion of local convexity of a capacity.
De®nition C.1. The capacity m is locally convex if for every 0 j 1, there
exists e > 0 such that (C.1) is satis®ed whenever
m…A ‡ F †; m…B ‡ F †; m…A† and m…B† lie in …j ÿ e; j ‡ e† :
…C.2†
Say that m is locally convex on …a; b† if the preceding holds for all j in …a; b†,
where the four capacities in (C.2) are restricted to …j ÿ e; j ‡ e† \ …a; b†.
Similarly for other intervals.
Clearly, convexity implies local convexity. Lemma C.4 below shows that the
reverse implication is valid under suitable assumptions.
12
We use the following special case of the cited selection principle (see [12, Proposition 51]): Let
ffa g be a uniformly bounded collection of nondecreasing functions on ‰a; bŠ. Then there exists a
subsequence ffan g that converges at every point in ‰a; bŠ:
Least convex capacities
281
A property that is intermediate between convexity and local convexity is
the following: Say that m is uniformly locally convex if e in De®nition C.1 can
be chosen independently of j. Uniform local convexity on an interval is
de®ned in the obvious way.
Though seemingly stronger, uniform local convexity is equivalent to local
convexity, as the following preliminary result shows:
Lemma C.2. If m is locally convex, then m is uniformly locally convex.
Proof. Step 1: If m is convex in …a; b† and …c; d† with 0 a < c < b < d 1,
then m is uniformly locally convex in …a; d†. One can verify that ˆ bÿc
3
works.
Step 2: By local convexity, for any j 2 ‰0; 1Š, there exists …j† > 0 such that m
is convex in …j ÿ …j†; j ‡ …j†† \ ‰0; 1Š. Thus, f…j ÿ …j†; j ‡ …j†† :
j 2 ‰0; 1Šg
is
an
open
cover
of
‰0; 1Š.
Let
f…ji ÿ …ji †;
ji ‡ …ji †† : i ˆ 1; 2; . . . ; n, 0 j1 < j2 < . . . < jn 1g be a ®nite subcover.
Apply Step 1 to conclude that m is uniformly locally convex. (
Let l be a convex-ranged measure such that ` is l-continuous at certainty. De®ne the pseudo-metric d on R by
d …A; B† ˆ l…A D B† ;
…C.3†
where D denotes symmetric di€erence. Identifying A and B whenever
l…A D B† ˆ 0 yields a metric d [10, p. 169]. Such an identi®cation is `acceptable' because, by (4.3) and Theorem 5.3,
l…A D B† ˆ 0 ˆ) E [ A ` E [ B 8E :
The metric space …R; d† is path connected: Given A B, there exists
U : ‰0; 1Š ÿ! R continuous, U…0† ˆ A, U…1† ˆ B, U…t† U…t0 † if t < t0 . Path
connectedness follows from [25]; when l is countably additive, see [14,
Lemma 4].
Lemma C.3. The mapping …A; B† ÿ! m…A [ B† is uniformly continuous from
…R R; d d† to ‰0; 1Š. Similarly for …A; B† ÿ! m…A \ B†.
Proof. Given e > 0, choose d such that for all measurable E,
l…E† < d ˆ) m…E† < e for all m 2 core…m† :
This is possible by Theorem 5.3. Let A and B be arbitrary measurable sets,
and choose mA and mB elements of the core such that
m…A† ˆ mA …A†; m…B† ˆ mB …B† :
Because m is the lower envelope of its core, it follows that
ÿmB …B D A† mB …B†ÿmB …A† m…B† ÿ m…A† mA …B† ÿ mA …A† mA …B D A† :
Therefore, l…A D B† < d implies that jm…B† ÿ m…A†j < e. This proves that
A ÿ! m…A† is uniformly continuous. Finally, observe that …A; B† ÿ! A [ B
and …A; B† ÿ! A [ B are each uniformly continuous [10, Theorem 40.A]. (
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L. G. Epstein and J. Zhang
The next result serves as a lemma in our proof of Theorem 5.2. Because it
seems to us to be of independent interest, for the reason indicated in the text,
we call it a theorem.
Theorem C.4. If m is a capacity that is uniformly continuous in the sense of
Lemma C.3, then local convexity of m implies convexity.
The proof requires two lemmas and the preliminary observation that
the stated assumption implies that m is convex-ranged: If A B, then
m…U† : ‰0; 1Š ÿ! ‰m…A†; m…B†Š is a continuous map with connected domain and
thus also a connected range.
Lemma C.5. Under the assumptions of Theorem C.4, for any > 0 there exists
d > 0 such that: For any A; B; F 2 R with A B, F \ B ˆ / and d…A; B† < d,
there exist F0 ; F1 ; . . . ; Fn ; Fn‡1 2 R with / ˆ F0 F1 . . . Fn Fn‡1 ˆ F
and
m…B ‡ Fi † ÿ m…A ‡ Fiÿ1 † < ; i ˆ 1; 2; . . . ; n ‡ 1 :
Proof. By uniform continuity, there exists d > 0 such that m…B† ÿ m…A† < =2,
whenever d…A; B† < d. By path connectedness, there exists a path connecting
A and A ‡ F . It follows that there exist F0 ; F1 ; . . . ; Fn 2 R with
/ ˆ F0 F1 . . . Fn Fn‡1 ˆ F such that
d…A ‡ Fi ; A ‡ Fiÿ1 † < d; i ˆ 1; 2; . . . ; n ‡ 1 ;
implying
m…A ‡ Fi † ÿ m…A ‡ Fiÿ1 † < =2; i ˆ 1; 2; . . . ; n ‡ 1 :
…C.4†
Further, d…A; B† < d implies d…A ‡ Fi ; B ‡ Fi † < d and thus (by selection of d)
that
m…B ‡ Fi † ÿ m…A ‡ Fi † < =2, all i :
Inequalities (C.4) and (C.5) imply that m…B ‡ Fi † ÿ m…A ‡ Fiÿ1 † < e:
…C.5†
(
Lemma C.6. Under the hypotheses of the preceding lemma, there exists d > 0
such that for any A; B and F 2 R with A B, d…A; B† < d and F \ B ˆ /, then
m…B ‡ F † ÿ m…B† m…A ‡ F † ÿ m…A† :
Proof. Recall that local convexity implies uniform local convexity (Lemma C.2). By uniform local convexity, there exists > 0 such that for any
j 2 ‰0; 1Š, m is convex in …j ÿ ; j ‡ † \ ‰0; 1Š. By uniform continuity, there
exists d > 0 such that m…B† ÿ m…A† < , whenever A B and d…A; B† < d. We
show that this d works.
Let fFi gn‡1
iˆ0 be as provided by the preceding lemma. Then,
m…B ‡ F † ÿ m…B† ˆ
n
X
‰m…B ‡ Fi‡1 † ÿ m…B ‡ Fi †Š ;
iˆ0
Least convex capacities
283
and similarly for A. It suces, therefore, to show that for each i,
m…B ‡ Fi‡1 † ÿ m…B ‡ Fi † m…A ‡ Fi‡1 † ÿ m…A ‡ Fi † :
…C.6†
By construction of fFi g, we have, for i ˆ 1; . . . ; n ‡ 1,
m…A ‡ Fiÿ1 † m…B ‡ Fi † < m…A ‡ Fiÿ1 † ‡ m…A ‡ Fiÿ1 † m…A ‡ Fi † m…B ‡ Fi † m…A ‡ Fiÿ1 † ‡ m…A ‡ Fiÿ1 † m…B ‡ Fiÿ1 † m…B ‡ Fi † m…A ‡ Fiÿ1 † ‡ ;
that is, the capacities of each of the sets A ‡ Fiÿ1 , A ‡ Fi , B ‡ Fiÿ1 and B ‡ Fi
lie in the interval …j ÿ e; j ‡ e† where j ˆ m…A ‡ Fiÿ1 †. Therefore, uniform
local convexity implies (C.6). (
Proof of Theorem C.4. Given A; B; F 2 R with A B and B \ F ˆ /, we
must prove that
m…B ‡ F † ÿ m…B† m…A ‡ F † ÿ m…A† :
…C.7†
Let d be as in Lemma C.6. If d…A; B† < d, the desired inequality is proven by
the noted lemma.
Suppose that d…A; B† d. By uniform continuity and path connectedness,
such that A S1 . . . Sn Sn‡1 ˆ B and
there exist fSi gn‡1
iˆ1 ,
d…Si ; Si‡1 † < d. By Lemma C.6,
m…Siÿ1 ‡ F † ÿ m…Siÿ1 † m…Si ‡ F † ÿ m…Si † ;
for each i. Inequality (C.7) follows by combining these inequalities.
(
It remains to prove Theorem 5.2, the ultimate objective of this appendix.
Proof of Theorem 5.2. Step 1: Show that there exists m in Vcr such that
m…† m …† for every m 2 Vcr :
…C.8†
Suppose to the contrary. Then by Theorem 5.1, there exist m and
m0 ˆ h…m†, two distinct dom -minimal elements. Consequently, h : ‰0; 1Š ÿ!
‰0; 1Š is increasing, continuous and onto, and h…x† > x and h…y† < y for some
x and y. There exists an interval …a1 ; a2 †, containing y, such that
h…x† < x on …a1 ; a2 † ;
and that is maximal in this respect. Conclude that h…ai † ˆ ai for each i (see
Fig. 2).
De®ne h …x† ˆ x if x 2 …a1 ; a2 † and ˆ h…x† otherwise. Then h 6ˆ h and
hm dom h m. The proof is completed by showing that h m is convex, contradicting the assumed minimality of hm. By Theorem C.4, it is enough
to prove local convexity. Clearly, the only problematic points are a1 and
a2 . Focus on a1 ; the argument for a2 is similar. We want to verify inequality (C.7) locally for m ˆ h m. Take e < minfa1 ; a2 ÿ a1 ; 1 ÿ a1 g and
denote
284
L. G. Epstein and J. Zhang
Figure 2
L ˆ fC 2 R : a1 ÿ e < m…C† a1 g ;
R ˆ fC 2 R : a1 m…C† < a1 ‡ eg :
(If a1 ˆ 0, we can ignore L and modify the following in the obvious way.)
There are ®ve cases where (C.7) must be veri®ed.
Case 1. A; A ‡ F 2 L and B; B ‡ F 2 R: Because m has convex range, there
exists A A B such that m…A † ˆ a1 . Because m0 ˆ hm is convex, we can
apply (C.7) to m0 , (with A replacing B), to derive
m …A ‡ F † ÿ m …A† ˆ m0 …A ‡ F † ÿ m0 …A† m0 …A ‡ F † ÿ m0 …A † :
Similarly, exploiting the convexity of m derive
m…A ‡ F † ÿ m…A † m…B ‡ F † ÿ m…B† ˆ m …B ‡ F † ÿ m …B† :
Note ®nally that m0 …A ‡ F † ÿ m0 …A † m…A ‡ F † ÿ m…A †, because m…A † ˆ
m0 …A † and A ‡ F lies in R. This completes the proof for this case.
Case 2. A; B 2 L and A ‡ F ; B ‡ F 2 R: Write B ˆ A ‡ G and note that
A; A ‡ G 2 L, and B0 ˆ A ‡ F ; B0 ‡ G 2 R. Apply Case 1.
Case 3. All sets are in the same region. Apply the given convexity of m and m0 .
Case 4. A; A ‡ F ; B 2 L and B ‡ F 2 R: By convexity of m0 and the fact that
h…x† < x on …a1 ; a1 ‡ e†, conclude that
m …A ‡ F † ÿ m …A† ˆ m0 …A ‡ F † ÿ m0 …A†
m0 …B ‡ F † ÿ m0 …B†
m …B ‡ F † ÿ m …B† :
Case 5. A 2 L, A ‡ F ; B and B ‡ F 2 R: De®ne A as above. Then A 2 L \ R.
By Case 2 applied to A; A ; A ‡ F and A ‡ F ,
m …A ‡ F † ÿ m …A† m …A ‡ F † ÿ m …A † :
By Case 3 applied to A ; B; A ‡ F and B ‡ F ,
m …A ‡ F † ÿ m …A † m …B ‡ F † ÿ m …B† :
Least convex capacities
285
Remark: The fact that we use a di€erent argument here than in Case 4 may
seem puzzling in light of the apparent symmetry between the two cases.
However, they are not symmetric, because while h…x† < x on …a1 ; a1 ‡ e† and
in spite of the `well behaved' nature of the graph of h in the Figure, we
cannot rule out the possibility that h…x† ÿ x assumes both positive and negative values on every interval of the form …a1 ÿ d; a1 †.
Step 2: Show that m above is least convex, strengthening (C.8).
If not, then there is a convex capacity m ˆ h…m †, with h increasing,
h…x† x on ‰0; 1Š, but such that h is not convex. Consequently, 9 k 2 …0; 1†
and 0 x < y 1 such that
h…kx ‡ …1 ÿ k†y† > kh…x† ‡ …1 ÿ k†h…y† :
Let L…† be the line through …x; h…x†† and …y; h…y††. Arguing as in Step 1, one
can show that there exist points a1 < a2 such that
h…a1 † ˆ L…a1 †; h…a2 † ˆ L…a2 †;
h…x† L…x†; 8 x 2 ‰a1 ; a2 Š and
h…y 0 † > L…y 0 †; for some a1 < y 0 < a2 :
(See Figure 3.)
De®ne
8
< L…m …A††
0
m …A† ˆ h…m …A††
:
L…m …A††
m …A† a1
a1 m …A† a2
a2 m …A† :
Then, by arguments similar to those used in Step 1, m0 is convex and is
ordinally equivalent to m . However, it need not satisfy m0 …;† ˆ 0 and
m0 …S† ˆ 1. To ensure these normalizing conditions, de®ne
Figure 3
286
L. G. Epstein and J. Zhang
m …A† ˆ
1
‰m0 …A† ÿ L…0†Š :
L…1† ÿ L…0†
Then m 2 Vcr , m …† m …† and m 6ˆ m , contradicting (C.8).
(
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