Law-invariant functionals that collapse to the mean: Beyond convexity

We establish general"collapse to the mean"principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the"collapse to the mean"for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the"collapse to the mean"to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis.


Introduction
The expression "collapse to the mean" refers to a variety of results about law-invariant functionals defined on spaces of random variables. The common thread of such results lies in the fundamental tension existing between law invariance and suitable "linearity" properties (linearity, affinity, translation invariance). In the context of mathematical finance, insurance, or economics, a random variable typically models the future unknown value of a given financial or economic variable of interest (the payoff of an asset, the return on a portfolio of assets, the net worth of an agent, the capital level of a financial institution). The functional under consideration models the "value" of said variable (a price, a risk measure, a utility index, a capital requirement). In this context, the assumption of law invariance posits that "value" is only sensitive to the distribution of the underlying variables with respect to a given reference E-mail addresses: felix.liebrich@insurance.uni-hannover.de, cosimo.munari@bf.uzh.ch. Date: July 13, 2021.
probability measure, so that statistical tools may be used to perform estimation in concrete situations. The assumption of "linearity" typically captures the presence of a frictionless determinant of "value" (a riskless investment opportunity, a liquidly traded asset without transaction costs). As the term suggests, the "collapse to the mean" is concerned with properties under which the only functionals that are simultaneously law invariant and "linear" are expectations or, more generally, functions of the expectation (with respect to the reference probability measure). These results are an important litmus test because functionals that are fully determined by expectation typically fail to capture "value" in an adequate risk-sensitive way. Avoiding an inadequate representation of "value" would thus force a choice between law invariance and other properties that are often desirable on their own merits. To our knowledge, the earliest "collapse to the mean" is recorded in [8], which proves that the expectation is the only law-invariant Choquet integral defined on the space of bounded random variables that is convex and linear along a nonconstant random variable. This result has natural applications to the literature on Choquet pricing. It shows that the combination of law invariance -a common postulate in insurance pricing -and the existence of a frictionless risky traded asset is only compatible with frictionless markets where prices are determined by expectation (with respect to the physical probability measure) and where, as a consequence, obvious arbitrage opportunities arise. The collapse for Choquet integrals was later extended, again in a bounded setting, to general cash-additive functionals in [16]. Further extensions beyond the bounded setting but retaining the convexity assumption have recently been obtained in [3], to which we refer for further information. In the recent working paper [36], the authors show that a functional defined on bounded random variables is a function of the expectation if and only if it is dependence neutral, i.e., the functional applied to a sum of random variables only depends on their marginal distributions. Notably, [36] does not impose convexity assumptions. A collapse to the mean for conditionally convex maps has been recently obtained in [12]. The goal of this paper is to present general formulations of the "collapse to the mean" that both extend the known results from the literature and can be applied beyond the world of convex functionals. The general "collapse to the mean" principle is stated in Theorem 4.1, which in turn is derived from a sharp version of the Fréchet-Hoeffding bounds recorded in Lemma 3.2. A complementary geometric version of the general principle is stated in Proposition 4.3. We illustrate the versatility of these tools in five case studies.
Collapse for convex functionals. In Section 5.1, we revisit the known "collapse to the mean" for convex functionals. We provide two versions under the assumption that the underlying functional is translation invariant along a nonconstant random variable, see Theorem 5.1 and Theorem 5.2. If the random variable has zero expectation, the functional collapses to a function of the expectation. Otherwise, it collapses to a specific function, namely an affine function, of the expectation. This confirms the results in [3,8,16]. In addition, we provide new dual characterizations of the collapse in terms of weaker translation invariance properties and conjugate functions.
Collapse for quasiconvex functionals. In Section 5.2 we take up the study of quasiconvex functionals. This is an important extension in view of the economic interpretation of quasiconvexity, which is a more elementary mathematical formulation of the diversification principle; see, e.g., [9,14,17,18,23,28] in a risk measure context. We extend both convex versions of the collapse, see Theorem 5.3 and Theorem 5.6, by means of the aforementioned sharp Fréchet-Hoeffding bounds. Moreover, we demonstrate sharpness of our results.
Collapse for consistent risk measures. In Section 5.3, we focus on cash-additive functionals that are monotonic with respect to second-order stochastic dominance. This class of risk measures is named "consistent" in [26] and contains the family of law-invariant convex risk measures, but also functionals that are neither convex nor quasiconvex. The literature on the connection between risk measures and stochastic dominance is rich; see, e.g., [1,11,22,29,30]. The collapse for consistent risk measures is recorded in Theorem 5.10, which is based again on the sharp version of the Fréchet-Hoeffding bounds.
Collapse for Choquet integrals. In Section 5.4, we take one further step beyond convexity and consider Choquet integrals associated with a variety of different law-invariant capacities. In the case of submodular capacities, the Choquet integral is convex and a related collapse to the mean was obtained in [8]. We go beyond submodular capacities and consider the case of coherent as well as Jaffray-Philippe capacities. The corresponding Choquet integrals are neither convex nor quasiconvex and play a natural role in decision theory under ambiguity; see, e.g., [10,20,32]. For a review of capacities and Choquet integrals, we refer to [27] and the references therein. In Theorem 5.14 we use the sharp Fréchet-Hoeffding bounds to derive a collapse result for this general class of Choquet integrals.
Collapse in optimisation problems. In Section 5.5 we focus on a general optimisation problem that encompasses a variety of important problems in economics, finance, and insurance, including the maximisation of expected investment returns or expected utility from terminal wealth (von Neumann-Morgenstern utility, rank-dependent utility, Yaari utility, Sshaped utility from prospect theory). More precisely, we study the maximisation of a general law-invariant objective subject to a general law-invariant constraint and a "budget" constraint expressed in terms of a "pricing density". A common intuition for such optimisation problems is that, if a solution exists, then all or some of these solutions have to be antimonotone with the pricing density. This allows to reduce the original problem to an optimisation problem involving quantile functions, which is substantially simpler and for which solution techniques are available; see, e.g., [4,6,19,33,34,37,38]. We provide a slight improvement over the existing results-see in particular [37]-by establishing more general sufficient conditions for the existence of antimonotone solutions. In particular, we highlight some conditions that are often omitted in the literature. In addition, we conduct a careful analysis showing that our result is sharp in the sense that, if any of the conditions is removed, the validity of the result forces the budget constraint to "collapse to the mean": The pricing density is necessarily constant, and the corresponding pricing rule reduces to the expectation with respect to the physical probability measure. This points to an issue in the literature, where the reduction to a quantile formulation is sometimes invoked even though some of the aforementioned conditions are not satisfied. In this situation, the reduction might be illegitimate unless extra analysis of the specific structure of the problem is carried over.
The paper is organised as follows. In Section 2 we describe the underlying setting and introduce the necessary notation. In Section 3 we record our main tool, namely the sharp Fréchet-Hoeffding bounds. In Section 4 we state the general "collapse to the mean" principle and establish a useful geometric counterpart for convex sets. In Section 5 we provide a range of applications to convex and quasiconvex functionals, consistent risk measures, and Choquet integrals. In addition, we discuss a general optimisation problem involving law invariance, provide a result about optimal solutions, and show what can go wrong when passing to its quantile formulation. Appendix A provides a proof of Lemma 3.2.

Setting and notation
Let (Ω, F, P) be an atomless probability space. A Borel measurable function X : Ω → R is called a random variable. By L 0 we denote the set of equivalence classes of random variables with respect to almost-sure equality under P. As is customary, we do not explicitly distinguish between an element of L 0 and any of its representatives. In particular, the elements of R are naturally identified with random variables that are almost-surely constant under P. For two random variables X, Y ∈ L 0 we write X ∼ Y whenever X and Y have the same law with respect to P, i.e., the probability measures P • X −1 and P • Y −1 on the real line agree. The expectation operator under P is denoted by E[·]. The standard Lebesgue spaces are denoted by L p for p ∈ [1, ∞]. We say that a set X ⊂ L 0 is law invariant if X ∈ X for every X ∈ L 0 such that X ∼ Y for some Y ∈ X .
Assumption 2.1. We denote by (X , X * ) a pair of law-invariant vector subspaces of L 1 containing L ∞ . We assume that XY ∈ L 1 for all X ∈ X and Y ∈ X * and denote by σ(X , X * ) the weakest linear topology on X with respect to which, for every Y ∈ X * , the linear functional on X given by X → E[XY ] is continuous. 1 We say that a (nonempty) set C ⊂ X is convex if it contains the convex combination of any of its elements, and σ(X , X * )-closed if it contains the limit of any σ(X , X * )-convergent net of its elements. The (upper) support functional of C is the map σ C : X * → [−∞, ∞] given by We say that ϕ is proper if dom(ϕ) is nonempty. Moreover, the functional ϕ is called: (2) quasiconvex if for all X, Y ∈ X and λ ∈ [0, 1], (3) σ(X , X * )-lower semicontinuous if for all nets (X α ) ⊂ X and X ∈ X , 1 Note that, as X and X * contain L ∞ by assumption, the pairing on X × X * given by (X, is separating. In particular, when equipped with the topology σ(X , X * ), the space X is a locally convex Hausdorff topological vector space. 2 Equivalently, for every m ∈ R the lower level set {X ∈ X ; ϕ(X) ≤ m} is convex. 3 Equivalently, for every m ∈ R the lower level set {X ∈ X ; ϕ(X) ≤ m} is σ(X , X * )-closed. (6) an affine function of the expectation if there exist a, b ∈ R such that, for every X ∈ X , The conjugate of ϕ is the functional ϕ * : The next lemma records the well-known dual representation of convex closed sets and convex lower-semicontinuous functionals, which are direct consequences of the Hahn-Banach theorem; see, e.g., [ Proposition 2.2. Let C ⊂ X be convex and σ(X , X * )-closed. Then, Let ϕ : X → (−∞, ∞] be proper, convex, and σ(X , X * )-lower semicontinuous. Then, One of the guiding threads of this paper is the fact that for many classes of functionals there is a fundamental tension between law invariance and suitable "linearity" properties. A prominent class consists of quasiconvex functionals. In this case, the property of law invariance is equivalent to other well-known properties such as dilatation monotonicity and Schur convexity, to which our results therefore naturally apply. We refer to [2, Theorem 3.6, Proposition 5.6] for a proof in our general setting. Proposition 2.3. Let ϕ : X → (−∞, ∞] be proper, quasiconvex, and σ(X , X * )-lower semicontinuous. Then, the following statements are equivalent: (ii) ϕ is dilatation monotone, i.e., for every X ∈ X and every σ-field G ⊂ F, (iii) ϕ is Schur convex, i.e., for all X, Y ∈ X ,

The key tool: Sharp Fréchet-Hoeffding bounds
In this brief section we record the main tool that is needed to establish our "collapse to the mean" results, which consists of a sharp formulation of the well-known Fréchet-Hoeffding bounds. For any random variable X ∈ L 0 we denote by q X a fixed quantile function of X, i.e., a function q X : (0, 1) → R satisfying for every s ∈ (0, 1) inf{x ∈ R ; P(X ≤ x) ≥ s} ≤ q X (s) ≤ inf{x ∈ R ; P(X ≤ x) > s}.
As the distribution function of X has at most countably many discontinuity points, any two quantile functions of X coincide almost surely with respect to the Lebesgue measure on (0, 1). For X, Y ∈ L 0 we say that X and Y are comonotone if for all x, y ∈ R, Similarly, we say that X and Y are antimonotone if for all x, y ∈ R, In the proof of the sharp version of the Fréchet-Hoeffding bounds and in the sequel, we will repeatedly use the fact that, by nonatomicity, for all X, Y ∈ L 0 we can always find X ′ ∼ X and Y ′ ∼ Y such that X ′ and Y ′ are comonotone. The analogue for anticomonotonicity holds as well. In fact, we have the following stronger result.
Lemma 3.1. For all X ∈ X and Y ∈ X * there exist X ′ , X ′′ ∼ X such that X ′ and Y are comonotone and X ′′ and Y are antimonotone.
The next result connecting the range of special integrals and quantile functions builds on early work by Fréchet and Hoeffding on joint distribution functions (see [4]) and Chebyshev, Hardy, and Littlewood on rearrangement inequalities (see [25]). Its general formulation in our setting is essentially due to Luxemburg; see [25,Theorem 9.1]. However, as the statements found in the literature contain only portions of the statement we need, we provide a complete proof in our general framework in Appendix A.
Lemma 3.2. For all X ∈ X and Y ∈ X * the functions are both Lebesgue integrable on (0, 1) and The minimum, respectively maximum, is attained by X ′ ∼ X if and only if X ′ and Y are antimonotone, respectively comonotone. Moreover, if both X and Y are nonconstant, 4. The general "collapse to the mean" principle This section contains our prototype version of the "collapse to the mean", which will later be exploited to obtain a variety of results for specific classes of functionals. This general result 5 Equivalently, there are nondecreasing functions f, g : R → R and Z ∈ L 0 with X = f (Z) and Y = g(Z); cf.
shows that the expectation is, up to an affine transformation, the only linear and σ(X , X * )continuous functional that is dominated above by a law-invariant functional which fulfills a suitable local translation invariance property. It should be noted that the result holds for a general law-invariant functional without any additional property.
Then, dom(ϕ * ) ⊂ R. In particular, if there exist c ∈ R and Y ∈ X * such that then Y must be constant.
Proof. If dom(ϕ * ) = ∅, the assertion trivially holds. Hence, suppose we can select Y ∈ dom(ϕ * ). By an affine transformation of ϕ, we can assume without loss of generality that ϕ * (Y ) = 0. For all k ∈ N and Z ′ ∼ Z, we observe that In the same vein, As a result, for every k ∈ N, Letting k → ∞, we infer that As Z is nonconstant, Lemma 3.2 implies that Y has to be constant.
We complement the previous theorem with a geometrical counterpart about convex sets. Recall that the recession cone of a convex set C ⊂ X is defined by The recession cone of C is the set of all directions of recession of the set C. Before stating the announced result, it is useful to highlight the following dual representation of the recession cone of a law-invariant set.
Lemma 4.2. Let C ⊂ X be convex and σ(X , X * )-closed. Then, If C is law invariant, then In particular, C ∞ is law invariant itself.
Proof. To show (4.1), fix an arbitrary U ∈ C. It follows from Proposition 2.2 that To show (4.2), note that law invariance of C together with Lemma 3.2 imply for every This shows that σ C is a law-invariant functional and, thus, dom(σ C ) is a law-invariant set.
As a result, we infer from (4.1) together with Lemma 3.2 that This representation clearly shows that C ∞ is law invariant.
We are now ready to prove the announced geometrical version of the "collapse to the mean", which generalises an earlier result formulated in [24, Proposition 5.10] and provides a simpler proof. It shows that a convex and σ(X , X * )-closed set that is law invariant and admits a nonzero direction of recession with zero expectation must be determined by expectation: Whether or not a random variable belongs to the set depends exclusively on its mean. In particular, the set must contain infinitely many affine spaces. Proof.
Since Z ∈ C ∞ by assumption, Lemma 4.2 implies that, for every Y ∈ dom(σ C ), Note that Z is nonconstant by assumption. If there existed a nonconstant Y ∈ dom(σ C ), then Lemma 3.2 would entail the impossible chain of inequalities This yields dom(σ C ) ⊂ R. By positive homogeneity of σ C , Proposition 2.2 implies This delivers the desired claims and concludes the proof.

5.1.
Collapse to the mean: The convex case. As stated in the introduction, a variety of "collapse to the mean" results have been established in the literature for convex functionals. Early versions of the collapse to the mean were obtained in [8] for convex Choquet integrals and in [16] for convex monetary risk measures. The focus of both papers was on bounded random variables. A general version of the collapse to the mean for convex functionals beyond the bounded setting has recently been established in [3]. To best appreciate the differences with the quasiconvex case, we devote this section to revisiting the most general results from the literature and complementing them with additional conditions. We start by revisiting [3,Theorem 4.7]. This result states that, under convexity and σ(X , X * )-lower semicontinuity, a functional that is law invariant and affine (in particular, linear) along a nonconstant random variable with zero expectation must be, in our terminology, expectation invariant. We provide a self-contained proof of this result and complement it by a number of weak translation invariance conditions and by a dual condition expressed in terms of the conjugate functional.
(v) For every X ∈ X there exists a nonconstant Z X ∈ X with E[Z X ] = 0 such that (vi) There exist X ∈ dom(ϕ) and a nonconstant Z ∈ X with E[Z] = 0 such that Proof. It is straightforward to verify that (ii) implies (iii), which in turn implies (iv), and that (v) implies (vi). Also note that dom(ϕ) ∩ R = ∅ by dilatation monotonicity recorded in Proposition 2.3.
(i) implies (ii): If (i) holds, then Proposition 2.2 yields for every X ∈ X (iv) implies (vii): This is a direct consequence of Proposition 2.2 and Theorem 4.1.
(vii) implies (v): This is a direct consequence of Proposition 2.2.
(vi) implies (i): Let X and Z be as in the assertion of (vi) and consider the nonempty This concludes the proof of the equivalence.
We turn to revisiting [3,Theorem 4.5]. This result states that, under convexity and σ(X , X * )lower semicontinuity, a functional that is law invariant and translation invariant along a nonconstant random variable with nonzero expectation must collapse to the mean up to an affine transformation. We provide a compact proof of this result and complement it by a dual condition expressed in terms of the conjugate functional.
(ii) There exist a ∈ R and a nonconstant Z ∈ X with E[Z] = 0 such that Proof. It is clear that (i) implies (ii), which in turn implies (iii). Now, assume that (iii) holds. By Proposition 2.2 and Theorem 4.
The proof that (iii) implies (iv) is complete. Finally, assume that (iv) holds and let y ∈ R be (the unique scalar) such that ϕ * (y) < ∞. It immediately follows from Proposition 2.2 that This shows that (iv) implies (i) and concludes the proof of the equivalence.

5.2.
Collapse to the mean: The quasiconvex case. In this section we investigate to which extent the collapse to the mean documented above generalises to quasiconvex functionals. It should be noted that, being heavily based on conjugate duality, the proofs in the convex case do not admit a direct adaptation to the quasiconvex case. In fact, we tackle the collapse to the mean in our more general setting by pursuing a different strategy based on the analysis of recession directions and their interaction with law invariance discussed in Section 4. Our first result establishes that Theorem 5.1 continues to hold if we replace convexity with quasiconvexity provided the condition involving conjugate functions is appropriately adapted to a condition involving sublevel sets. In the accompanying remark we show the link between these two conditions. Theorem 5.3. Let ϕ : X → (−∞, ∞] be proper, quasiconvex, σ(X , X * )-lower semicontinuous, and law invariant. Then, the following statements are equivalent: (iii) For every X ∈ X there exists a nonconstant Z X ∈ X with E[Z X ] = 0 such that ϕ(X + tZ X ) ≤ ϕ(X), t ≥ 0.
Proof. It is clear that (i) implies (ii), which in turn implies (iii). Now, assume that (iii) holds. Take m ∈ R and set C m = {ϕ ≤ m}. If C m = ∅, then we have dom(σ Cm ) = ∅. Hence, suppose that C m = ∅ and take any X ∈ C m . By assumption, for every t ≥ 0 we have X + tZ X ∈ C m . This implies that Z X ∈ C ∞ m . It follows from Proposition 4.3 that dom(σ Cm ) ⊂ R, showing that (iii) implies (iv). Finally, assume that (iv) holds. For every m ∈ R set again C m = {ϕ ≤ m}. As dom(σ Cm ) ⊂ R and σ Cm is positively homogeneous, it follows from Proposition 2.2 that As a consequence, we obtain for every X ∈ X In particular, ϕ(X) = ϕ(E[X]) for every X ∈ X . This shows that (iv) implies (i). The next example shows that point (vi) in Theorem 5.1 is specific to the convex case and cannot be added to the equivalent conditions in Theorem 5.3.
Example 5.5. Let the functional ρ : X → R be defined by Note that ρ is convex, σ(X , L ∞ )-lower semicontinuous, and law invariant. Set Define the functional ϕ : X → R by setting We turn to the collapse to the mean established in Theorem 5.2. The next result shows that, if convexity is relaxed to quasiconvexity, then the collapse to the mean continues to hold in the presence of translation invariance (point (ii) in Theorem 5.2).
(iii) There exist a ∈ R and a nonconstant Z ∈ X with E[Z] = 0 such that Proof. It is easy to see that (i) implies (ii) and that (ii) implies (iii). Assume now that (iii) holds. Suppose m ∈ R is such that {ϕ ≤ m} = ∅. By dilatation monotonicity of ϕ recorded in Proposition 2.3, we find x ∈ R such that ϕ(x) ≤ m. Making use of dilatation monotonicity once more, we infer for all t ≥ 0 that for every X ∈ X . That is, ϕ is an affine function of the expectation as stated in (i).
The following example shows that point (iii) in Theorem 5.2 fails to produce a collapse to the mean under mere quasiconvexity. In particular, this observation holds no matter the value of the expectation of the nonconstant random variable along which local translation invariance in the sense of point (iii) in Theorem 5.2 holds. Moreover, the example demonstrates that Theorem 5.6 cannot be improved.
Example 5.7. Consider the setting of Example 5.5, and let the random variable Z be as described there, i.e., P(Z = 2) = 1 − P(Z = −1) = 1 3 . Moreover, let U be a random variable whose distribution is given by P(U = 4) = P(U = 0) = 1 2 . Both Z and U are nonconstant, E[Z] = 0, and E[U ] = 2. We have already observed that ϕ(tZ) = 0, t ≥ 0. One can also verify that ρ(−tZ) = t 2 ≥ 0 which means that ϕ(−tZ) = 0, t ≥ 0. Moreover, for every t ≥ 0, Hence, ϕ satisfies point (iii) in Theorem 5.2 even without the additional condition on the expectation. However, ϕ is neither an affine nor a convex function of the expectation (and not even expectation invariant, as observed in Example 5.5).

5.3.
Collapse to the mean: The case of consistent risk measures. In this and the following section, we establish a collapse to the mean for classes of law-invariant functionals beyond the quasiconvex family. In this section, we focus on functionals that are translation invariant along constants and monotonic with respect to second-order stochastic dominance. Following the terminology in [26], we refer to them as consistent risk measures. This class covers the family of law-invariant convex risk measures but also includes nonconvex functionals, e.g., minima of law-invariant convex risk measures. As translation invariance along constants implies that convexity and quasiconvexity are equivalent, the class of consistent risk measures contains functionals that are not quasiconvex. As a result, we cannot resort to the quasiconvex results in Section 5.2. First, recall that a consistent risk measure is a proper functional ϕ : X → (−∞, ∞] that is: (1) cash-additive, i.e., ϕ(X + m) = ϕ(X) + m for all X ∈ X and m ∈ R.
(2) consistent with second-order stochastic dominance, i.e., for all X, Y ∈ X , (3) normalised, i.e., ϕ(0) = 0. 7 Given its defining properties, a consistent risk measure takes only finite values on L ∞ . Moreover, every consistent risk measure is automatically dilatation monotone and law invariant by property (2). In case X = L ∞ , every normalised, law-invariant, and convex risk measure is a 7 In [26] only the condition ϕ(0) ∈ R is imposed on a consistent risk measure ϕ : L ∞ → R. Applying an affine transformation to ϕ, one can guarantee ϕ(0) = 0 though. consistent risk measure. The same holds for normalised, law-invariant, σ(X , X * )-lower semicontinuous convex risk measures by Proposition 2.3. The next proposition shows that every consistent risk measure on L ∞ can be extended uniquely to a σ(X , X * )-lower semicontinuous consistent risk measure. In particular, a consistent risk measure on L ∞ is automatically σ(L ∞ , X * )-lower semicontinuous.
Proposition 5.8. Let ϕ : L ∞ → R be a consistent risk measure. Then, there is a unique, σ(X , X * )-lower semicontinuous, consistent risk measure ϕ : X → (−∞, ∞] that extends ϕ. Proof. Note that ϕ is dilatation monotone in the sense of [31]. In addition, by [26,Theorem 3.5], ϕ has the Fatou property, i.e., for every uniformly bounded sequence (X n ) ⊂ L ∞ converging to X ∈ L ∞ almost surely, ϕ(X) ≤ lim inf n→∞ ϕ(X n ). Let Π denote the set of finite measurable partitions of Ω. For X ∈ L 1 and π ∈ Π we write E[X|π] := E[X|σ(π)], where σ(π) is the σ-field generated by π. [31,Theorem 4] proves that the functional ϕ ♯ : is a σ(L 1 , L ∞ )-lower semicontinuous, dilatation monotone in the sense of [31], cash-additive extension of ϕ. A fortiori, the restriction of ϕ ♯ to X , denoted by ϕ, is a σ(X , X * )-lower semicontinuous, dilatation monotone in the sense of [31], cash-additive extension of ϕ. It remains to verify consistency of ϕ ♯ , which implies that of ϕ. By [26,Theorem B.3], it suffices to check for dilatation monotonicity in the sense of [26]. To this end, suppose X, Y ∈ L 1 satisfy E[Y |X] = X. Let (π n ) ⊂ σ(X ) be an increasing sequence of finite measurable partitions such that X n = E[X|π n ] → X in L 1 . For all n ∈ N, E[Y |X n ] = E[X|π n ] holds, which entails This is the desired dilatation monotonicity of ϕ ♯ . Uniqueness of ϕ can be seen to be a consequence of the uniqueness statement in [31,Theorem 4].
The following representation result from [26] will play a crucial role in our later analysis. In the terminology of [5], it shows that any consistent risk measure on L ∞ can be expressed as a minimum of adjusted Expected Shortfalls.
Let ϕ : L ∞ → R be a consistent risk measure. Then, for every X ∈ L ∞ , where denotes the acceptance set of ϕ.
Our main result establishes a collapse to the mean for consistent risk measures. We show that linearity along a nonconstant random variable is sufficient to reduce the functional to a standard expectation. In line with our previous result, we also provide an equivalent condition for the collapse in terms of directions of recession and conjugate functions.
(ii) There exist a nonconstant Z ∈ X and a ∈ R such that ϕ(tZ) = at, t ∈ R.
Any of the previous statements implies: Statements (i)-(iv) are equivalent if, additionally, Proof. It is trivial to see that (i) implies (ii). In order to see that (ii) implies (iii), recall first that ϕ is dilatation monotone as observed above. Hence, we may estimate This yields the desired implication. Now, we claim that (iii) implies (i). We first consider the case X = L ∞ and fix an arbitrary X ∈ L ∞ . Using Lemma 5.9, we have As E[U ] = 0 by assumption, Lemma 3.2 implies that ES p (U ) > 0 for every p ∈ (0, 1). Let q ∈ (0, 1) be arbitrary and choose t 0 > 0 such that Moreover, for all p ∈ [q, 1] and t > t 0 , As a result, we get Now, for all p ∈ [0, q],

Combining this inequality with (5.2) and (5.3) yields
We conclude by noting that, by dilatation monotonicity, This shows that ϕ(X) = E[X] whenever X ∈ L ∞ . To conclude the proof of the implication, we consider the case of a general space X . Note that for an arbitrary finite sub-σ-algebra such that E[U |G] ∈ L ∞ is nonconstant, dilatation monotonicity implies The preceding argument shows that ϕ coincides with the expectation under P when restricted to L ∞ . By, e.g., [2,Lemma 4.1], L ∞ is dense in X with respect to σ(X , X * ). Take a net (X α ) ⊂ L ∞ satisfying X α → X with respect to σ(X , X * ). By dilatation monotonicity and σ(X , X * )-lower semicontinuity, This delivers (i). Clearly, (i) implies (iv). We conclude by proving that (iv) implies (iii) under the additional assumption that ϕ(λX) ≤ λϕ(X) for all λ ∈ [0, 1] and X ∈ X . To this end, let A ∈ F satisfy P(A) = 1 2 and set G = {∅, A, A c , Ω}. For every G-measurable, positive, nonconstant Y ∈ L ∞ with E[Y ] = 1 and for every n ∈ N we claim that To see this, observe that At the same time, where we used that E[X|G] ∈ A ϕ holds for every X ∈ A ϕ by dilatation monotonicity. As a consequence, by G-measurability of Y , This delivers (5.4). Now, for n ∈ N define Y n = n−1 n 1 A + n+1 n 1 A c ∈ L ∞ and note that Y n is G-measurable, positive, nonconstant, and satisfies E[Y n ] = 1. It follows from (5.4) that we find a G-measurable X n ∈ {ϕ ≤ 0} with X n ∞ > n and E[X n Y n ] ≥ 1. As E[X n ]E[Y n ] = E[X n ] ≤ ϕ(X n ) ≤ 0 by dilatation monotonicity and cash-additivity, X n cannot be constant by Lemma 3.2. Using compactness of the appropriate unit sphere in R 2 , we can assume without loss of generality that there is a suitable G-measurable U ∈ L ∞ such that U = 0 and X n X n ∞ → U.
By our additional assumption, for every t > 0 we eventually have t Xn Xn ∞ ∈ A ϕ and, thus, tU ∈ A ϕ or, equivalently, ϕ(tU ) ≤ 0. To prove (iii), it remains to show that E[U ] = 0. To this effect, note that XnYn Xn ∞ → U . As a result, applying dilatation monotonicity again, This concludes the proof.
Remark 5.11. Condition (5.1) means that the risk measure ϕ is star shaped in the sense of [7]. By [7,Proposition 2], the latter is equivalently characterised by the fact that the acceptance set A ϕ is star shaped about 0. Consistent risk measures satisfying (5.1) are characterised in [7,Theorem 11], but we would like to motivate here that, in fact, (5.1) is a very mild constraint. By [26,Theorem 3.3] or Lemma 5.9 above, a consistent risk measure ϕ : L ∞ → R is represented by a family T of convex law-invariant risk measures τ in that If each τ ∈ T is normalised, i.e., τ (0) = 0, then ϕ has property (5.1).

5.4.
Collapse to the mean: The case of Choquet integrals. As mentioned in the introduction, the research on law-invariant functionals and their collapse to the mean was triggered by [8], where the focus was on Choquet integrals associated with special submodular law-invariant capacities. The property of submodularity is equivalent to convexity of the Choquet integral. As such, the collapse to the mean established there can be seen as a special case of the results in Section 5.1. In this section, we extend the collapse to the mean to nonconvex Choquet integrals. To this effect, it should be noted that we cannot resort to the quasiconvex results in Section 5.2 because, for a Choquet integral, quasiconvexity automatically implies convexity in view of translation invariance along constants. We start by recalling some basic notions. A capacity is a function µ : F → [0, 1] such that µ(∅) = 0 and µ(Ω) = 1, and µ(A) ≤ µ(B) for all A, B ∈ F with A ⊂ B. We say that µ is: In what follows, we denote by L ∞ the space of bounded measurable functions X : Ω → R.
The Choquet integral associated with a capacity µ is the functional E µ : L ∞ → R defined by If µ is countably additive, i.e., a probability measure, then the Choquet integral reduces to a standard expectation. The next proposition collects some well-known facts about Choquet integrals. In particular, note that, under a law-invariant capacity, we can unambiguously define the Choquet integral on the space L ∞ as will be tacitly done below.
Proposition 5.12. Let µ be a capacity. Then, the following statements hold: By the classical results in [35], a submodular capacity is automatically coherent. The converse does not hold in general; see, e.g., [21]. We target the extension of Theorem 5.2 to nonconvex Choquet integrals associated with coherent capacities. In fact, we shall go one step further and focus on so-called Jaffray-Philippe (JP) capacities introduced in [20]. A capacity µ is a JP capacity if there is a pair (ν, α) of a coherent capacity ν and α ∈ [0, 1] such that 11 where ν is the dual capacity of ν. JP capacities encompass both submodular and coherent capacities, as well as neo-additive capacities introduced in [10]. 12 A first lemma characterises law invariance of JP capacities. Proof. Law invariance of the capacity ν implies law invariance of the dual capacity ν and thus of µ. Conversely, assume that µ is law invariant. Its dual capacity is given by µ = αν + (1 − α)ν. As α = 1 2 , we may recover ν as As the dual capacity µ is also law invariant, the value of the right-hand side in (5.5) only depends on the P-probability of its argument. This implies law invariance of ν.
We establish the desired collapse to the mean for nonconvex Choquet integrals. Our result encompasses [8, Theorem 3.1], which was established under the assumption of submodularity by means of convex duality. Our proof is direct and solely based on Theorem 4.1.
(i) E µ coincides with the expectation under P.
(ii) There exist a ∈ R and a nonconstant Z ∈ L ∞ such that (iii) There exist a ∈ R and a nonconstant Z ∈ L ∞ such that (iv) There exists a nonconstant Z ∈ L ∞ such that Proof. Clearly, (i) implies (ii) and (iii) implies (iv). As E µ [0] = 0, we also see that (ii) implies (iii). Now, suppose that (iv) holds. By point (i) in Proposition 5.12, the assumption reads . By the polarisation identity in (5.5), . Using point (ii) in Proposition 5.12, we conclude that E ν [tZ] = tE ν [Z] for every t ∈ R. Now, note that ν is law invariant by Lemma 5.13. By coherence and the Radon-Nikodým theorem, there exists a family D ⊂ L 1 of probability densities such that, for every A ∈ F, Note furthermore that each X ∈ L ∞ and each D ∈ D satisfy E ν [X] ≥ E[DX]. By Theorem 4.1, D must be constant. This forces ν = ν = P, and consequently µ = P, that is, (i) holds. The proof of the equivalence is complete.

5.5.
Collapse to the mean in optimisation problems. In this section we focus on a class of optimisation problems involving law invariance at the level of both the objective function and the optimisation domain. We investigate the existence of optimal solutions that are antimonotone with respect to a "pricing density" appearing in the budget constraint under a list of suitable assumptions. We prove sharpness of our existence result in the sense that, if any of the listed assumptions is removed, then the result continues to hold only in the trivial situation where the budget constraint "collapses to the mean". This is relevant in applications because a key monotonicity assumption on the optimisation domain is sometimes omitted in the literature, in which case, contrary to what is sometimes stated, the general result cannot be invoked and one has to proceed case by case. Throughout the entire section we focus on the optimisation problem under the following basic assumptions: The last constraint is typically interpreted as a budget constraint where D plays the role of a "pricing density". We say that the quadruple (ϕ, C, D, p) is feasible if the optimisation problem admits an optimal solution. In this case, we denote by Max(ϕ, C, D, p) the corresponding optimal value. This problem has been extensively studied in the literature, see, e.g., [4,6,19,34,37,38], and the recent overview in [33]. In this literature, one encounters the following two types of statements about optimal solutions: • There exists an optimal solution that is antimonotone with D.
• All optimal solutions are antimonotone with D.
As mentioned in the introduction, these statements are very useful because they allow to reduce the original problem to a deterministic optimisation problem involving quantile functions; see, e.g., [33]. We start by providing a slight extension to the extant results about existence of optimal solutions that are antimonotone with the "pricing density". To this effect, it is convenient to define the following notions: (1) C is increasing if X + m ∈ C for all X ∈ C and m ≥ 0.
The next result shows that antimonotone optimal solutions always exist provided that both C is increasing and ϕ is weakly increasing. If ϕ is also increasing, then every optimal solution must be antimonotone with the "pricing density". (i) If C is increasing and ϕ is weakly increasing, then there exists an optimal solution that is antimonotone with D. (ii) If C is increasing, ϕ is increasing, and Max(ϕ, C, D, p) ∈ R, then all optimal solutions are antimonotone with D.
Proof. Let X ∈ X be an optimal solution. To prove (i), let X ′ ∼ X be antimonotone with D. As X ∈ C, we have X ′ ∈ C by law invariance of C. As C is increasing, In addition, ϕ(X ′ + m) ≥ ϕ(X ′ ) = ϕ(X) because the function ϕ is weakly increasing and law invariant. We conclude that X ′ + m is an optimal solution. It remains to observe that X ′ + m is antimonotone with D by construction.
To establish (ii), assume towards a contradiction that X is not antimonotone with D-which entails in particular that D and X are nonconstant-and take X ′ and m as above. The same argument shows that X ′ + m is an optimal solution. From Lemma 3.2 we derive m > 0. This yields ϕ(X ′ + m) > ϕ(X ′ ) = ϕ(X) because ϕ is increasing and law invariant, and because ϕ(X) ∈ R. However, this contradicts the optimality of X. In conclusion, X and D have to be antimonotone.
The previous result is sometimes stated without the monotonicity assumption on the domain C (see, e.g., [33]) or it is said that the monotonicity assumption on C is made without loss of generality (see, e.g., [37]). 13 The remainder of the section is devoted to showing that all the assumptions in Theorem 5.15, including the monotonicity assumption on C, are necessary for the result to hold. More precisely, we show that, if any of the assumptions is removed, then for every choice of a nonconstant "pricing density" one can find a concrete formulation of the optimisation problem for which the result does not hold. Equivalently, one can preserve the result after discarding any of the preceding assumptions only under a "collapse to the mean": The "pricing density" must be constant, and the "pricing rule" in the budget constraint can be expressed by a standard expectation. (i) ϕ is weakly increasing but no optimal solution is antimonotone with D.
(ii) C is increasing but no optimal solution is antimonotone with D.
(iii) ϕ is increasing and Max(ϕ, C, D, p) ∈ R but there exist optimal solutions that are not antimonotone with D. (iv) C is increasing and Max(ϕ, C, D, p) ∈ R but there exist optimal solutions that are not antimonotone with D.
Proof. Let Z ∈ X be nonconstant and comonotone with D. Note that Z is not antimonotone with D due to Lemma 3.2. Up to an appropriate translation, we can always assume that Clearly, ϕ is both weakly increasing and increasing. Note that (ϕ, C, D, p) is a feasible quadruple and Z is an optimal solution with ϕ(Z) ∈ R. This shows (iii). In addition, by Lemma 3.2, any optimal solution X ∈ X that is antimonotone with D would need to satisfy which is clearly impossible. This shows that (i) holds. Next, consider the law-invariant set C = {Z ′ + m ; Z ′ ∼ Z, m ∈ R} and set for every X ∈ X Clearly, C is increasing. Note that (ϕ, C, D, p) is a feasible quadruple and Z is an optimal solution with ϕ(Z) ∈ R. This shows that (iv) holds. In addition, by Lemma 3.2, any optimal solution X ∈ X that is antimonotone with D would have to satisfy which is clearly impossible. This shows that (ii) holds. 13 We highlight that the result is also typically stated without the finiteness assumption of the optimal value. This is often justified because the special choice of ϕ and C ensures finiteness.
We strengthen the previous result in two ways. In a first step, we show that imposing no condition on the domain C besides law invariance leads to counterexamples independently of the choice of both the "pricing density" D and the objective function ϕ. Proof. To show (i), take any nonconstant Z ∈ X that is comonotone with D and set p = E[DZ]. In addition, set C = {Z ′ ∈ X ; Z ′ ∼ Z}. It is clear that C is law invariant and that (ϕ, C, D, p) is a feasible quadruple with respect to which Z is optimal. If X ∈ X is another optimal solution, then we must have X ∼ Z as well as E[DX] = E[DZ]. As Z is nonconstant, it follows from Lemma 3.2 that X cannot be antimonotone with D. To show (ii), it suffices to repeat the same argument under the additional condition that ϕ(Z) ∈ R, which is possible by assumption.
We reinforce the same message by showing that the monotonicity assumption on C remains critical even if we impose more structure on the set C itself. We illustrate this by focusing on two common choices in the literature, starting from an "interval-like" set. (i) ϕ is weakly increasing but no optimal solution is antimonotone with D.
(ii) ϕ is increasing and Max(ϕ, C, D, p) ∈ R but there exist optimal solutions that are not antimonotone with D.
Proof. By assumption on D, we find k ∈ R such that P(D ≤ k) ∈ (0, 1) and E[D1 {D≤k} ] = 0. Define for every X ∈ X Note that ϕ is both weakly increasing and increasing. Indeed, for all X ∈ X and m > 0 we have ϕ(X) ∈ R and ϕ(X + m) = ϕ(X) + m > ϕ(X). Now, set as well as p = E[DZ]. Note that Z is not constant and satisfies ϕ(X) ≤ b = ϕ(Z) for every X ∈ C. As a result, (ϕ, C, D, p) is a feasible quadruple and Z is an optimal solution. Since, by construction, Z is not antimonotone with D, we infer that (ii) holds. In addition, take any optimal solution X ∈ X that is antimonotone with D. From X ≤ b and ϕ(X) = ϕ(Z) = b, we infer that q X (s) = b for almost every s ∈ [P(D ≤ k), 1). Consequently, q X (s) = b holds for almost every s ∈ (0, P(D ≤ k)] as well by antimonotonicity. As a result, we must have X = b, from which we deduce Hence, E[D1 {D≤k} ] = 0, a contradiction to the choice of k. To avoid this contradiction, D has to be constant. This shows that (i) holds.
We conclude by focusing on the situation where C admits a maximum with respect to a suitable preference relation. Recall that a binary relation on X is a preference if it is reflexive and transitive. A preference is compatible with the expectation if for all X, Y ∈ X This weak compatibility property is satisfied by many preference relations encountered in the literature, including the convex order and second-order stochastic dominance.
Proposition 5.19. Let C ⊂ X be law invariant and such that, for a suitable B ∈ C and a preference compatible with the expectation, For every nonconstant D ∈ X * with E[D] > 0 there exists a feasible quadruple (ϕ, C, D, p) such that: (i) ϕ is weakly increasing but no optimal solution is antimonotone with D.
(ii) ϕ is increasing and Max(ϕ, C, D, p) ∈ R but there exist optimal solutions that are not antimonotone with D.
Proof. Let Z ∼ B be comonotone with D. Set p = E[DZ] and define for every X ∈ X ϕ(X) = E[X].
Clearly, ϕ is both weakly increasing and increasing. Note that (ϕ, C, D, p) is a feasible quadruple with respect to which Z is an optimal solution with ϕ(Z) ∈ R. As Z is nonconstant and comonotone with D, it follows from Lemma 3.2 that Z is not antimonotone with D, showing (ii). In addition, take any optimal solution X ∈ X that is antimonotone with D. If X were nonconstant, then we would derive from Lemma   We have equality if and only if P(X ′ > x, Y > y) = min{P(X ′ > x), P(Y > y)}, or equivalently P(X ′ ≤ x, Y ≤ y) = min{P(X ′ ≤ x), P(Y ≤ y)}, for almost all x, y ∈ R with respect to the Lebesgue measure on R × R. By right continuity of distribution functions, this holds if and only if X ′ and Y are comonotone. Note that, by Lemma 3.1, we do find X ′ ∼ X such that X ′ and Y are comonotone. This proves the integrability of q X q Y , the right-hand side equality in (3.1), and the corresponding attainability assertion. In a similar way, we obtain We have equality if and only if P(X ′ > x, Y > y) = max{P(X ′ > x) − P(Y ≤ y), 0}, or equivalently P(X ′ ≤ x, Y ≤ y) = max{P(X ′ ≤ x) + P(Y ≤ y) − 1, 0}, for almost all x, y ∈ R with respect to the Lebesgue measure on R × R. By right continuity of distribution functions, this holds if and only if X ′ and Y are antimonotone. Note that, by Lemma 3.1, we do find X ′ ∼ X such that X ′ and Y are antimonotone. This proves the integrability of q X (1−·)q Y , the left-hand side equality in (3.1), and the corresponding attainability assertion. The statement for general X and Y follows by applying (3.1) and the attainability result to the positive and negative parts of X and Y exploiting the fact that q max{X,0} = max{q X , 0} and q max{−X,0} = max{−q X (1 − ·), 0} almost surely with respect to the Lebesgue measure on (0, 1), and similarly for Y . For the attainability assertion, one observes that X and Y are comonotone if and only if max{X, 0} and max{Y, 0} as well as max{−X, 0} and max{−Y, 0} are comonotone and max{X, 0} and max{−Y, 0} as well as max{−X, 0} and max{Y, 0} are antimonotone, and similarly for antimonotonicity. Now, take general nonconstant X and Y . Observe that The integrand in the last expression is nonnegative. Moreover, we can invoke nonconstancy of X and Y to find some α ∈ (0, 1 2 ) such that q X (t) − q X (s) > 0 and q Y (t) − q Y (s) > 0 for all s < α and t > 1 − α. This shows the right-hand side inequality in (3.2). Repeating the argument by replacing X with −X delivers the left-hand side inequality in (3.2) and concludes the proof.
Remark A.1. The strict inequality in (3.2) is seldom found in the literature and is related to a rearrangement inequality by Chebyshev; see, e.g., [15]. An alternative proof can be obtained from [37,Lemma 8]. Indeed, by nonatomicity of (Ω, F, P) we find two independent random variables U 1 and U 2 with uniform distribution over (0, 1). Hence, X ′ := q X (U 1 ) ∼ X and Y ′ := q Y (U 2 ) ∼ Y are independent as well. Let α ∈ (0, 1 2 ) be such that q X (s) < q X (t) and q Y (s) < q Y (t) for all s ≤ α and t ≥ 1 − α, which is possible as X and Y are not constant.
and note that (P ⊗ P)(R) = α 4 > 0 and that is negative P ⊗ P-almost surely on R. As the random variables X ′ and Y ′ can therefore not be comonotone, we obtain The other inequality follows by exchanging X with −X.