Research Articles Least convex capacities w

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1 Economic Theory 13, 263±286 (1999) Research Articles Least convex capacities w Larry G. Epstein and Jiankang Zhang Department of Economics, University of Rochester, Harkness Hall, Rochester, NY , USA ( epstein@uxmail.ust.hk) 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 von- Neumann 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! R 1, where X R n 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

2 264 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

3 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 U f ˆ / u f dh m ; f 2 F : 1:2 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! 0; 1Š is called a capacity if (i) m ; ˆ 0; m S ˆ1; 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. Itisadditive, 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 1 x 2R for each outcome x. Assume that X R n 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! R 1. The second function is a convex capacity m. Utility over acts is de ned by: Z Z 1 U f ˆ u f dm m fs 2 S : u f s tg dt ; 2:2 0 where the rst integral is a Choquet integral, de ned by equality with the second (Riemann) integral. If f yields outcomes x 1 x n on the events A 1 ;...; A n, then U f ˆXn 1 iˆ1 u x i u x i 1 Šm [ i 1 A j u x n :

4 266 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 x 1 ; A 1 ;...;x n ; A n the act that yields outcome x i on the (measurable) event A i, i ˆ 1;...; n, where by convention x 1 x n. For all 1 ` n 1 and x 2 X, f ; [ i6ˆ`;` 1 A i ; 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 1 and x 2 X, f ; [ i6ˆ`;` 1 A i ; x; A` [ A` 1 1 f ˆ) f ; [ i6ˆ`;` 1 A i ; 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 : 3:2 1 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.

5 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 u i and m i the vnm index and capacity corresponding to i in the sense of (2.2), i ˆ 1; 2. Suppose that u 2 ˆ G u 1 for some increasing concave G on Range u 1 and m 2 ˆ g m 1 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 m 1 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 u 2 ˆ G u 1 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, m 2 ˆ g m 1, 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 men- 2 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.

6 268 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 that we have attached to the components u, m and /; h. If u i ; m i ; / i; h i are two iˆ1;2 such tuples, then u 1 ; m 2 ; / 2; h 1 also corresponds to a CEU utility functional satisfying our regularity conditions, because h 1 m 2 is a convex capacity and / 2 u 1 is a concave vnm index.4 Similarly for u 2 ; m 1 ; / 1; h 2 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 m 0 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 ˆr. IfB 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 V cr. If m 2 V cr, then the ordinally equivalent capacity g m is also in V cr for any g : 0; 1Š! 0; 1Š that is increasing, continuous and convex. 6 Restricting attention to capacities in V cr, Theorem 3.1 motivates de nition of the partial ordering cvx on V cr, where m 2 cvx m 1 () m 2 ˆ h m 1 on R ; 4:1 4 That is, h 1 m 2 is a convex capacity because h 1 is increasing and convex and / 2 u 1 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).

7 Least convex capacities 269 for some convex and increasing function h : 0; 1Š! 0; 1Š. 7 Say that m 1 is less convex than m 2 if m 2 cvx m 1. 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 ˆfp 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 m 1 m 2 on R () core m 1 core m 2 : On the other hand, m 2 cvx m 1 ˆ) m 1 m 2 on R ; because any convex, increasing and surjective function h on the unit interval must satisfy h x x on 0; 1Š : Conclude that m 2 cvx m 1 ˆ) core m 1 core m 2 : 4:4 Consequently, the least convex capacity is the capacity in V cr 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 t 0 > t ˆ) h t 0 > 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.

8 270 L. G. Epstein and J. Zhang where m is the dual or conjugate capacity de ned by m A 1 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, orv cr 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 V cr. (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 A ua m ˆfA2R : m A m A c ˆ1g : Then A ua 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 m ja is convex-ranged. This imposes a form of richness of the set ua m A ua m. Then it is immediate that m is the least convex capacity in V cr.

9 Least convex capacities 271 One interpretation is that A ua 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 A ua 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 ˆj if 9A 2 A ua m ; p A ˆj 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 1 and m A ˆmaxf` A ;` A dg : 4:8 Then m is a convex capacity on S; R and has the core core m ˆfp 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 core m ˆfp : dp ˆ w dl; w w w a.e. lš; w dl ˆ 1g ; 4:9 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 L 1 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 ap ;` ˆ bp ; 4:10 in which case p represents ` and the decision-maker is probabilistically sophisticated. (This is Example 1 with g t ˆmaxfat; bt b 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).

10 272 L. G. Epstein and J. Zhang This example is not covered by Example 2. The algebra A ua m of unambiguous events, as de ned in Example 2, is given by A ua m ˆfA2R : ` A ˆ` A or ` A c ˆ` A c g : Therefore, if ` A <` A for all nonempty A, then A ua 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 m e on S; R by m e A ˆ 1 A ˆ S 1 e p A A 6ˆ S ; where p is a given probability measure on S; R. Then m e is a convex capacity (see [27]). At the extremes, one obtains m 0 ˆ p and m 1 ˆ complete ignorance. For intermediate values of e, the nature of m e is elucidated by observing its core, core m e ˆ 1 e p em S; R ; which shrinks with as the `contaminating weight' e falls. Observe that each m e, 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 ˆp 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 m e m on R for all e > 0. Letting e! 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 m e 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, m 1 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 com- 9 Given (4.11), A ` B but A [ SnB ` B [ SnB ˆS, violating the additivity property of a qualitative probability.

11 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 V cr, there exists (under a strengthening of convex-range) a minimally convex capacity. Then, under additional assumptions, we show that V cr 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: m 2 dom m 1 if m 1 m 2 on R : 5:1 If the natural meaning is attached to dom -minimality, then the latter implies cvx -minimality because m 2 cvx m 1 ˆ) m 2 dom m 1 : Theorem 5.1. Let m 2 V cr be strongly convex-ranged. Then there exists a dom -minimal convex capacity in V cr mš fbm 2 V cr : m dom bmg and therefore also in V cr. See Appendix B for a proof. Some informal clarifying comments are o ered here. The hypothesis that m 2 V cr 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 V cr, it follows that V cr ˆfg m : g m convex; g : 0; 1Š! 0; 1Š; onto, increasingg : Consequently, if some capacity in V cr is scr then every capacity in V cr 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 fg a g. Therefore, some work is required to show that Zorn's Lemma applies. Turn to the existence of a least convex capacity in V cr. We specify assumptions on `, that is, ordinally invariant assumptions on capacities, that are su cient to deliver the existence of a least convex capacity.

12 274 L. G. Epstein and J. Zhang Consider rst the following property for `, referred to as continuity at certainty: For all B and A n 2 R, If A n % S and S ` B; then A n ` 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 A n %1 whenever A n % S; A n 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 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 d implies m A > 1 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 V cr ; 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 A n %m A whenever A n % 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 ˆfm 2 M S; R : dm ˆ w dl; w 2 H L 1 S; R; l g ; 5:4

13 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 l- continuity 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 d ˆ) m SnE > 1 e ˆ) m SnE > 1 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 l- continuity 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 L 1 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 1 An %U 1 S whenever A n % 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].

14 276 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 SnA n as small in any meaningful sense and where therefore lim U 1 An < U 1 S 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 m 0 : m 0 ˆ h m ; m 0 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 f ; [ i6ˆ`;` 1 A i ; x; A` [ A` 1 1 f ; or equivalently, writing E ˆ[` 1 1 A j, that u 1 x m 1 E A` A` 1 m 1 E Š u 1 x` m 1 E A` m 1 E Š u 1 x` 1 m 1 E A` A` 1 m 1 E A` Š : Because u 2 is more concave than u 1, the inequality is preserved if u 1 is replaced by u 2. It su ces to consider the case where m 1 E A` A` 1 m 1 E > 0. Then the corresponding statement for m 2 is true as well: We have shown that u 2 x u 2 x` m 1 E A` m 1 E Š u 2 x` 1 m 1 E A` A` 1 m 1 E A` Š : m 1 E A` A` 1 m 1 E A:1

15 Least convex capacities 277 Because m 2 ˆ g m 1,withg convex, it follows that m 1 E A` m 1 E m 1 E A` A` 1 m 1 E m 2 E A` m 2 E m 2 E A` A` 1 m 2 E : Thus (A.1) continues to hold if m 1 is replaced by m 2. This proves f ; [ i6ˆ`;` 1 A i ; 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 8x 1 < x 2 in j e; j e \ 0; 1Š there exist A; B satisfying m A \ B ˆx 1 ; m A [ B ˆx 2 ; m A ˆm B ˆ x 1 x 2 =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 e; j e \ 0; 1Š, g x 2 g x 1 2g x 1 x 2 =2 for all x 1 ; x 2 in the interval. Therefore, g is convex on [0, 1]. ( Say that m : R! 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 a 1 ; b 1 and a 2 ; b 2,witha 2 < b 1, then m is locally additive on a 1 ; b 2. Turn to the speci cs of the interval beliefs capacity. De ne Observe that D ˆf ` A ; ` A : A 2 Rg I ˆfA2R : ` A d ` A g II ˆfA2R : ` A d ` A g D I ˆf ` A ; ` A : A 2 Ig D II ˆf ` A ; ` A : A 2 IIg : D I \ D II ˆf ` A ; ` A : A 2 I \ IIg : If either f ` A ; ` A : ` A >` A dg or f ` A ; ` A : ` A > ` A dg is empty, then m ˆ ` 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 ], 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) D I \ D II is a nonsingleton. The rst alternative implies (4.10). Assume, therefore, that D I \ D II is a nonsingleton.

16 278 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 D I \ D II 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 ` d on fa 2 R : A Ag. The assertion follows. (Note that ` A ˆ` A 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 V cr, let g : 0; 1Š! 0; 1Š be nondecreasing, g 0 ˆ0; g 1 ˆ1; 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 x 1 < x 2 1. By scr, there exist A 1 A 2 such that m A i ˆx i, i ˆ 1; 2, and m A 2 n A 1 > 0. It then follows from the convexity of g m that g x 2 g x 1 ˆg ma 2 g ma 1 g m A 2 na 1 m A 2 na 1 > 0. (b) Suppose to the contrary that there exists x 0 2 0; 1 such that g x 0 > g x 0. As a nondecreasing function, g has at most a countable number of points of discontinuity. Therefore, there exist x 0 < x 2 < x 1 < 1, satisfying

17 Least convex capacities 279 g x 1 g x 2 < g x 0 g x 0 : By scr, there exist A 3 A 2 A 1 satisfying m A 3 ˆx 0, m A 2 ˆx 2, m A 1 ˆx 1 and m A 1 n A 2 > 0. Let A ˆ A 3 [ A 2 n A 3 ˆA 2 ; B ˆ A 3 [ A 1 n A 2 : Thus, m A [ B ˆm A 1 ˆx 1 ; m A \ B ˆm A 3 ˆx 0 ; m A ˆm A 2 ˆx 2 > x 0 ; and m B ˆm A 3 [ A 1 n A 2 m A 3 m A 1 n A 2 > m A 3 ˆx 0 g m A [ B g m A \ B ˆ g x 1 g x 0 < g x 2 g x 0 < 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 m cr mš, employ Zorn's Lemma and show that every chain fg a m g in m cr mš has a lower bound in m cr mš. We will show that g m is such a lower bound, where g x sup g a x ; x 2 0; 1Š : B:1 Note that each g a is continuous, increasing and satis es g a 0 ˆ0, g a 1 ˆ1 and g a x x on [0,1]. Moreover, fg a g is a chain with respect to the usual ordering of functions. Claim: There exists a subsequence fg n g of fg a g such that, for every x 2 0; 1Š, g x ˆlim n g n x : 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 g a m dom g m for all a. It remains to prove the claim. Let fr i g be an enumeration of the rationals in the unit interval. There exists a subsequence fg 1 n g satisfying g 1 n r 1 % n g r 1. Moreover, because fg a g is a chain with respect to the usual ordering of functions, we can select the subsequence so that g 1 n % n. Therefore, lim g 1 n r 2 exists. If the limit equals g r 2, then de ne g 2 n ˆ g1 n.if lim g 1 n r 2 < g r 2, there exists another subsequence fg 2 n g of fg ag such that g 2 n % n and g 2 n r 2 % n g r 2. By the chain property, we then have

18 280 L. G. Epstein and J. Zhang g 2 n r 1 % n g r 1. (For every g 1 n r 2 there exists g 2 m r 2 > g 1 n r 2. By the chain property, g 2 m r 1 g 1 n r 1. Therefore, lim m g 2 m r 1 > lim n g 1 n r 2 ˆg r 1.) In this second case, rede ne g 1 n ˆ g2 n. Proceeding inductively, we construct fg k n gfg ag such that lim g k n n r i ˆg r i ; i ˆ 1;...; k; and g k n % n on 0; 1Š; for each k : Therefore, the sequence fg n n g satis es lim g n n n r i ˆg r i ; i ˆ 1; 2;... B:3 By Helly's Selection Principle [12, p. 204], there exists fg n gfg n ng such that lim n g n 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 ˆg 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 ˆg 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 m A m B F 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 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 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 ff a g be a uniformly bounded collection of nondecreasing functions on a; bš. Then there exists a subsequence ff an g that converges at every point in a; bš:

19 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: Ifm 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 b c 3 works. Step 2: By local convexity, for any j 2 0; 1Š, there exists j > 0 such that m is convex in j j ; j j \ 0; 1Š. Thus, f j j ; j j : j 2 0; 1Šg is an open cover of 0; 1Š. Let f j i j i ; j i j i : i ˆ 1; 2;...; n, 0 j 1 < j 2 <...< j n 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 ˆl A D B ; C.3 where D denotes symmetric di erence. Identifying A and B whenever l A D B ˆ0 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 ˆ0 ˆ) E [ A ` E [ B 8E : The metric space R; d is path connected: Given A B, there exists U : 0; 1Š!R continuous, U 0 ˆA, U 1 ˆB, U t U t 0 if t < t 0. Path connectedness follows from [25]; when l is countably additive, see [14, Lemma 4]. Lemma C.3. The mapping A; B! m A [ B is uniformly continuous from R R; d d to 0; 1Š. Similarly for A; B! 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 m A and m B elements of the core such that m A ˆm A A ; m B ˆm B B : Because m is the lower envelope of its core, it follows that m B B D A m B B m B A m B m A m A B m A A m A B D A : Therefore, l A D B < d implies that jm B m A j < e. This proves that A! m A is uniformly continuous. Finally, observe that A; B! A [ B and A; B! A [ B are each uniformly continuous [10, Theorem 40.A]. (

20 282 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! 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 >0there 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 F 0 ; F 1 ;...; F n ; F n 1 2 R with / ˆ F 0 F 1... F n F n 1 ˆ F and m B F i m A F i 1 <; i ˆ 1; 2;...; n 1 : Proof. By uniform continuity, there exists d > 0 such that m B 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 F 0 ; F 1 ;...; F n 2 R with / ˆ F 0 F 1... F n F n 1 ˆ F such that implying d A F i ; A F i 1 < d; i ˆ 1; 2;...; n 1 ; m A F i m A F i 1 <=2; i ˆ 1; 2;...; n 1 : C.4 Further, d A; B < d implies d A F i ; B F i < d and thus (by selection of d) that m B F i m A F i <=2, all i : C.5 Inequalities (C.4) and (C.5) imply that m B F i m A F i 1 < e: ( 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 m B m A F 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 ; j \ 0; 1Š. By uniform continuity, there exists d > 0 such that m B m A <, whenever A B and d A; B < d. We show that this d works. Let ff i g n 1 iˆ0 be as provided by the preceding lemma. Then, m B F m B ˆXn iˆ0 m B F i 1 m B F i Š ;

21 Least convex capacities 283 and similarly for A. It su ces, therefore, to show that for each i, m B F i 1 m B F i m A F i 1 m A F i : By construction of ff i g, we have, for i ˆ 1;...; n 1, C.6 m A F i 1 m B F i < m A F i 1 m A F i 1 m A F i m B F i m A F i 1 m A F i 1 m B F i 1 m B F i m A F i 1 ; that is, the capacities of each of the sets A F i 1, A F i, B F i 1 and B F i lie in the interval j e; j e where j ˆ m A F i 1. 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 m B m A F 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, there exist fs i g n 1 iˆ1, such that A S 1... S n S n 1 ˆ B and d S i ; S i 1 < d. By Lemma C.6, m S i 1 F m S i 1 m S i F m S i ; 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 V cr such that m m for every m 2 V cr : C.8 Suppose to the contrary. Then by Theorem 5.1, there exist m and m 0 ˆ h m, two distinct dom -minimal elements. Consequently, h : 0; 1Š! 0; 1Š is increasing, continuous and onto, and h x > x and h y < y for some x and y. There exists an interval a 1 ; a 2, containing y, such that h x < x on a 1 ; a 2 ; and that is maximal in this respect. Conclude that h a i ˆa i for each i (see Fig. 2). De ne h x ˆx if x 2 a 1 ; a 2 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 a 1 and a 2. Focus on a 1 ; the argument for a 2 is similar. We want to verify inequality (C.7) locally for m ˆ h m. Take e < minfa 1 ; a 2 a 1 ; 1 a 1 g and denote

22 284 L. G. Epstein and J. Zhang Figure 2 L ˆfC 2 R : a 1 e < m C a 1 g ; R ˆfC 2 R : a 1 m C < a 1 eg : (If a 1 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 ˆa 1. Because m 0 ˆ hm is convex, we can apply (C.7) to m 0, (with A replacing B), to derive m A F m A ˆm 0 A F m 0 A m 0 A F m 0 A : Similarly, exploiting the convexity of m derive m A F m A m B F m B ˆm B F m B : Note nally that m 0 A F m 0 A m A F m A, because m A ˆ m 0 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, andb 0 ˆ A F ; B 0 G 2 R. Apply Case 1. Case 3. All sets are in the same region. Apply the given convexity of m and m 0. Case 4. A; A F ; B 2 L and B F 2 R: By convexity of m 0 and the fact that h x < x on a 1 ; a 1 e, conclude that m A F m A ˆm 0 A F m 0 A m 0 B F m 0 B m B F 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 m A m A F m A : By Case 3 applied to A ; B; A F and B F, m A F m A m B F m B :