Multidimensional inequalities and generalized quantile functions

Multidimensional inequalities and generalized quantile functions Sinem Bas Philippe Bich Alain Chateauneuf Abstract In this paper, we extend the generalized Yaari dual theory for multidimensional distributions, in the vein of Galichon and Henry s paper [6]. We show how a class of generalized quantiles -which encompasses Galichon and Henry s one or multivariate quantile transform [5] [8] [10]- allows to derive a general representation theorem. Moreover, we derive from this representation theorem a more tractable formula which could be applicable to multidimensional measure of inequalities. Keywords: multidimensional distributions, quantile, inequality, optimal coupling This paper forms part of the research project The Multiple dimensions of Inequality (Contract No. ANR 2010 BLANC 1808) of the French National Agency for Research, whose financial support is gratefully acknowledged. Université Catholique de Louvain, CORE - Voie du Roman Pays 34 / L1.03.01 1348 Louvain-la- Neuve. E-mail : sinem.bas@uclouvain.be Paris School of Economics, Centre d Economie de la Sorbonne UMR 8174, Université Paris I Panthéon/ Sorbonne, 106/112 Boulevard de l Hôpital 75647 Paris Cedex 13. E-mail : bich@univparis1.fr IPAG Business School and PSE, Centre d Economie de la Sorbonne UMR 8174, Université Paris I Panthéon/ Sorbonne, 106/112 Boulevard de l Hôpital 75647 Paris Cedex 13. E-mail : Alain.Chateauneuf@univ-paris1.fr 1

1 Introduction In a recent paper, Galichon and Henry [6] generalize Yaari s dual theory [12] to multidimensional distributions, using optimal coupling theory. 1 They prove that the preference relationship of a decision maker confronted with choices on multidimensional prospects can be evaluated with a weighted sum of multidimensional quantiles, if one assumes that this preference relationship can be written as a multidimensional index I which preserves some first order stochastic dominance and which satisfies some comonotonic independence property. Formally: I() = Q.φdν (1) where is any multidimensional distribution, Q is the multidimensional quantile of, φ the multidimensional weight function, ν some well chosen measure, and where. denotes the usual scalar product. One of the main difficulties in extending Yaari s dual theory is to define a multidimensional quantile function. A solution proposed by Galichon and Henry is to use the characterization of the standard univariate quantile as a solution of an optimal coupling problem. Then, the same optimal coupling problem in a multidimensional setting provides a candidate to be a multidimensional quantile function, they call µ-quantile, where µ is some given probability distribution. Yet, the choice of a particular multidimensional quantile may not be completely obvious. In particular, other multidimensional quantiles have been proposed in the literature, for example multivariate quantile transform (see [5], [8], [9] or [10]). In this paper, we propose to extend the generalized Yaari dual theory for multidimensional distributions, in the vein of Galichon and Henry s paper. But the class of quantiles we consider encompasses µ quantile or the multivariate quantile transform: it is simply based on an axiomatic foundation, which is a natural extension of the unidimensional case. In particular, the general setting we consider makes impossible to use the techniques introduced in Galichon and Henry: in their paper, a crucial as- 1. Yaari has extended his theory to multidimensional distributions (see [11]), but his extension does not completely take into account the multidimensional aspect of the problem. 2

sumption is the Fréchet differentiability of the functional I at some point. It permits to write the weight function φ used in the representation formula (1) as the gradient of I at a well chosen. Since we do not have such differentiability assumption, we use techniques from Functional Analysis: Equation (1) is first derived for a particular subset of distributions, and then is extended to any distribution, thanks to Hahn-Banach theorem on lattices. Last, from the main representation theorem, we propose some explicit formula of the multidimensional index, which could be implemented by a policy-maker. The paper is organized as follows. Section 2 recalls the Yaari dual theory of choice for unidimensional distributions (Theorem 2.1). Section 3 introduces our specific framework of multidimensional distributions, offers a general definition of quantiles (Definition 3.1) and states a first representation result (Theorem 3.3). In particular, it is proved that our axiomatic definition of quantiles encompasses Galichon and Henry s µ-quantile (Section 3.4), or multivariate quantiles (Section 3.6). In Section 4, we compare our representation theorem with Galichon and Henry one s. In Section 5, it is shown how all the involved parameters can be elicited by a policy-maker. Finally, in Section 6, - discussion and concluding remarks-, some limitations of the proposed inequality measure are raised. 2 Yaari dual theory of choice for unidimensional distributions In this section, we define a probability space (S, F, P ) by S = [0, 1], P being the Lebesgue measure and F being the σ algebra of Borelian subsets of S. Let V be the set of random variables on (S, F, P ) and V 2 be the set of elements in V with a finite second moment. Two elements (, Y ) V V are said to be equal in law (which is denoted = d Y ) if the probability law of and Y coincide. For every Borelian subset E of S, 1I E denotes the characteristic function of E, that is 1I E (s) = 1 if s E and 1I E (s) = 0 otherwise. For every random variable V, we can define its 3

cumulative distribution function F (x) = P ( x), and its unidimensional quantile F 1, as follows: F 1 (p) = inf{x R : P ( x) p}. The quantile function can also be seen as an element of V (since F 1 is a measurable function defined on S). It can be defined as a solution of an optimization problem, called optimal coupling problem, which we now recall. For every U V 2, define the maximum correlation functional associated to U as follows: V 2, ρ U () := sup U(p) (p)dp V 2, =d [0,1] Choosing U(p) = p (which we now assume in this section), Hardy-Littlewood- Pólya [7] inequality guarantees that F 1 is the unique (almost surely) solution in V 2 of the optimization problem above. In particular, ρ U () := E(F 1.U), and it follows that the quantile possesses the following properties, whose proofs can be found in the appendix: Proposition 2.1. (i) For every, Y V 2 such that = d Y, F 1 surely, and F 1 = d. (ii) For every, Y V 2 and λ 0, F 1 quantiles of F 1 + F 1 Y and of λ. + F 1 Y and λf 1 = F 1 Y almost are themselves the (iii) For every λ R and in V 2, the quantile of +λ is F 1 +λ (almost surely). More generally, univariate quantile exhibits some Invariance to Monotonic Transformations, which encompasses the last properties. In the next section, we will define a generalized class of multidimensional quantiles by assuming similar properties. Now, to state Yaari dual choice theory main representation result, we need to recall comonotonicity concept: Definition 2.2. Two random variables and Y in V are comonotonic if ((s) (t))(y (s) Y (t)) 0 almost surely with respect to (s, t) S S. 4

Comonotonicity can be characterized using maximal correlation functional as follows (see Proposition 2.8 in [2]), where U(p) = p: Proposition 2.3. Two random variables and Y in V 2 are comonotonic if and only if ρ U ( + Y ) = ρ U () + ρ U (Y ). In this paper, all distributions V 2 can be interpreted as a income distributions. In the following theorem (and more generally throughout this paper), I will denote an inequality index: we follow the convention that for every distributions and Y, I(Y ) I() means, for the policy maker, that income distribution is no more unequal than income distribution Y. Theorem 2.1. (Yaari main representation theorem) Let I : V 2 R. The two assertions below are equivalent: 2 (1) The functional I satisfies: 1. (Normalization) I(1I S ) = 1 2. (Anonymity) for every (, Y ) V 2 V 2, = d Y I() = I(Y ). 3. (Inequality Aversion) for every (, Y ) V 2 V 2, I( + Y ) I() + I(Y ). 4. (Additive comonotonicity) for every (, Y ) V 2 V 2 comonotonic, I( +Y ) = I() + I(Y ). (2) There exists a unique, convex and non-decreasing function f : [0, 1] [0, 1], f(0) = 0, f(1) = 1, such that: I() = 1 0 f(1 F (p))dp = 1 0 F 1 (p)f (1 p)dp = d(f P ) S where d(f P ) is the Choquet integral of with respect to capacity f P. S 2. In this theorem, recall that f P is defined on the set of Borelian subsets of S as follows: for every Borelian E S, f P (E) = f(p (E)). The function obtained is no longer a probability (it may not be additive), but it is a capacity. Then, Choquet integral is a way to extend the standard expectation with respect to a probability to the case where the probability is replaced by a capacity. 5

3 A first representation result in the multidimensional case 3.1 Framework In this section, we extend the previous framework to a multidimensional one. Let (S, F, P ) be a probability space. For every subset A of R n, denote B A the Borelian σ-algebra on A. Let E(.) be the expectation operator with respect to the probability P. Let V be the set of random n-dimensional vectors on (S, F, P ), i.e. V = { : (S, F, P ) (R n, B R n) : is measurable}. For every V, we write = (,..., n ), where for every s S, we define (s) = ( (s),..., i (s),..., n (s)). The probability distribution of on (R n, B R n) is denoted P. We let V 2 be the set of elements in V with finite second moments, i.e. V 2 = { V : i {1, 2,..., n}, E( 2 i ) < + }. For every and Y in V, we denote = d Y if and Y have the same distribution, that is P = P Y. Recall the definition of the standard scalar product <.,. > on V 2 : for every = (,..., n ) and Y = (Y 1,..., Y n ), <, Y >= n E( iy i ) = n S i(s)y i (s)dp (s). Hereafter,.Y simply denotes n i.y i. 3.2 A new definition of quantiles We now provide an axiomatic definition of multidimensional quantile, generalizing some natural properties that are satisfied in the unidimensional case. Let us explain why this is natural to define the quantile of a n-dimensional random vector as a function from [0, 1] n to R n. Let be a non atomic one-dimensional income distribution. For every p [0, 1], the familiar quantile F 1 (p) is the income level x such that 100p% of incomes are below x. Let Y be another distribution independent from (say, for example, a health level, normalized between 0 and 1). It is natural to define the following two-dimensional quantile Q,Y (p 1, p 2 ) = (F 1 (p 1), F 1 (p 2)), i.e. Q,Y (p 1, p 2 ) indicates the income x and the health level y such that 100p 1 % of incomes are below x and 100p 2 % of health levels are below y. The general quantile operator is a way to extend this naive definition to any random vectors, its components being independent or not. Thus, it could be seen as a way to normalize (in [0, 1] n ) and to 6 Y

rank some multivariate distributions. Throughout this paper, we will allow the quantile operator to be defined on a subset V 2 of V 2 (for example, we will sometimes consider V 2 to be the set of comonotonic random vectors). It is a way, for example, to take into account some possible dependency relationship between the components of V 2. In the following, for every sets E and F, F(E, F ) denote the set of functions from E to F. Definition 3.1. Consider a n-dimensional random vector U V 2 whose support is included in [0, 1] n. Let V 2 be a convex cone of V 2 containing the constant random variables. An operator Q : V 2 F([0, 1] n, R n ) is a U-quantile operator (or simply a quantile operator when U is implicit) if it satisfies the first three following properties: 3 1. (Law of quantiles) For every (, Y ) V 2 V 2 such that = d Y, Q = Q Y almost surely, Q (U) V 2 and Q (U) = d. 2. (Sum of quantiles) For every λ 0 and (, Y ) V 2 V 2, Q + Q Y and λq are the quantiles of Q (U) + Q Y (U) and λ. 3. (Translation unvariance) For every λ R and in V 2, the quantile of + λ(1,..., 1) is Q + λ(1,..., 1). Sometimes, we will assume one of the three additional assumptions: 4. For every strictly positive reals λ 1,..., λ n, the function p [0, 1] n (λ 1 p 1,..., λ n p n ) R n is the quantile of some random vector in V 2. 5. If n 2, for every symmetric definite positive n-matrix A, the function p [0, 1] n Ap R n is the quantile of some random vector in V 2. 6. (Independence) If = (,..., n ) V 2, where the i s are mutually independent, then one has Q (p) = (F 1 (p 1 ),..., F 1 i (p i ),..., F 1 n (p n )) for almost every p = (p 1,..., p i,..., p n ) [0, 1] n, where each F 1 i denotes the usual onedimensional quantile of i. 3. Hereafter, Q() is also denoted Q, and is called a quantile. Sometimes, to make more explicit the parameter U, we will call Q a U-quantile, and will denote it Q U. 7

Remark 3.1. From Proposition 2.1, for n = 1, the standard unidimensional quantile defines a quantile operator by V 2 Q = F 1 F([0, 1], R). Remark 3.2. The mapping U in the definition of quantile operator allows to connect Ω, the set of individuals, to [0, 1] n, the set of proportions. The economic interpretation of U and its probability measure, is similar to the one given in [6] (Section 5), and is reminded in Section 3.7. In the three next subsections, we give particular examples of quantile operators illustrating Definition 3.1. 3.3 Example 1: multidimensional quantile on the set of comonotonic vectors Consider V 2 the class of comonotonic random vectors = (,..., n ) V 2, in the sense that for every (i, j) {1,..., n} 2, i and j are comonotonic. Fix U a random vector whose components are independent and uniformly distributed with values in [0, 1]. Then a quantile operator Q can be defined as follows: = (,..., N ) V 2, Q (p 1,..., p n ) = (F 1 (p 1 ),..., F 1 n (p n )). Importantly, this may not define a quantile operator on whole V 2, because when is not comonotonic, the law of Q (U) and the law of could be different. particular, the first requirement in Definition 3.1 may not be satisfied if Q is defined on V 2. We now check that this operator satisfies the requirements of Definition 3.1. First, V 2 is a convex cone and contains the constant random vectors. Second, in Definition 3.1, Point 1. is satisfied, that is for every (, Y ) V 2 V 2 such that = d Y, Q = Q Y almost surely, and Q (U) = d. The second equality is a consequence of comonotonicity (see [3]: for comonotonic random vectors, equality can be checked component by component), and the first one is true because this is true in the unidimensional case. Third, Point 2. in Definition 3.1 is true, because for every λ 0 and, Y in V 2, Q + Q Y = (F 1 + F 1 Y 1,..., F 1 n + F 1 Y n ) and Q λ = (λf 1,..., λf 1 n ) are In 8

the quantile of (F 1 (U) + F 1 Y 1 (U),..., F 1 n (U) + F 1 Y n (U)) = Q (U) + Q Y (U) and of (λ,..., λ n ) (see Proposition 2.1). For the last point, just remark that for every λ R and in V 2, the quantile of + λ(1,..., 1) = ( + λ,..., n + λ) is (F 1 1 +λ,..., F ) = Q n+λ + (λ,..., λ). 3.4 Example 2: quantile on the set of anti-comonotonic 2- random vectors In this example, assume n = 2. Recall that a 2-dimensional random vector (, 2 ) is anti-comonotonic if (, 2 ) is comonotonic. Let V 2 be the class of anti-comonotonic random vectors = (, 2 ) V 2. A quantile operator can be defined by V 2, Q (p 1, p 2 ) = (F 1 (p 1 ), F 1 2 (1 p 2 )). Similarly to the previous subsection, it can be proved that Q satisfies the requirements of Definition 3.1. 3.5 Example 3: Galichon and Henry s µ quantile Let U V 2 whose probability law µ is absolutely continuous with respect to Lebesgue measure and has a support included in [0, 1] n. In this subsection, V 2 = V 2. Given V 2 and = d, we recall that the pair (, U) V 2 V 2 is called an optimal coupling if is a solution of the following optimization problem: sup E(.U) = d From Optimal coupling theory (see Appendix 7.1.1), it is possible to prove that the following definition is consistent: Definition 3.2. (Galichon and Henry [6]) For every V 2, there exists a convex and lower semicontinuous function f : [0, 1] n R n such that ( f(u), U) is an optimal coupling, with f(u) = d. Moreover, f : [0, 1] n R n is uniquely determined (almost surely): it is called the µ quantile of (or sometimes, U-quantile of ) 4, and it is denoted Q µ. 4. This quantile depends only on the law of U 9

The following proposition is proved in Appendix 7.2. Proposition 3.3. The function V 2 Q µ defines a U-quantile operator on V 2. 3.6 A fourth example: multivariate quantile transform The multivariate quantile transform was introduced in [5], [8], or [10]. See also [9] for a brief presentation of this concept. We recall the definition of the multivariate quantile transform in the case n = 2, the general case being a straightforward generalization. Let (S, F, P ) be any probability space. Let U 1, U 2 be two independent random variables on S, which are uniformly distributed in [0, 1], and let U = (U 1, U 2 ). Define a U-quantile operator as follows: for every = (, 2 ), a random vector on S, Q = (Q 1, Q2 ) is defined by Q 1 (p 1, p 2 ) = F 1 (p 1 ), where F 1 is the one-dimensional quantile of, and by Q 2 (p 1, p 2 ) = inf{x R : P ( 2 x F 1 (U 1 ) = F 1 (p 1 )) p 2 }, i.e. Q 2 (p 1,.) is the one-dimensional conditional quantile of 2, given Q 1 (U) = Q 1 (p 1, p 2 ). The two following propositions, proved in Appendix 7.3 and Appendix 7.4, show that the multivariate quantile can be obtain as a particular case of a quantile operator, but also that it is not a particular case of Galichon and Henry s quantile: Proposition 3.4. The multivariate quantile transform defines a U-quantile operator on V 2. Proposition 3.5. Denote by Q the multivariate quantile of every V 2. There does not exists some measure µ such that for every V 2, Q = Q µ, where Qµ denotes Galichon and Henry s quantile operator. 10

3.7 The main representation result Fix U V 2 whose probability law is absolutely continuous with respect to Lebesgue measure 5. Let Q be a U-quantile operator defined on V 2, a convex cone of V 2 which contains the set of constant random variables. The preferences of the decision maker on V 2 are now assumed to be represented by a function I : V 2 R. 3.7.1 The main assumptions Throughout this paper, we assume that the function I : V 2 following assumptions: Axioms on I: Axiom 1. Normalization: I(1,..., 1) = 1. Axiom 2. Monotonicity: (, Y ) V 2 V 2, Y I() I(Y ). R satisfies the Axiom 3. Positive homogeneity: for every λ 0 and V 2, we have I(λ) = λi(). Axiom 4. Additivity on Quantiles: for every (, Y ) V 2 V 2, we have I(Q (U) + Q Y (U)) = I(Q (U)) + I(Q Y (U)). Axiom 5. Neutrality: I() only depends of the law of, that is for every (, Y ) V 2 V 2 with the same law, I() = I(Y ). Axiom 6. Inequality Aversion: for every (, Y ) V 2 V 2 such that I() = I(Y ) and every λ [0, 1], we have I(λ + (1 λ)y ) I(). These properties imply the following properties (see the proof in Appendix 7.5). Proposition 3.6. Under the assumptions above, we have: 1. For every V 2 and a 0, I( + a(1,..., 1)) = I() + a. 2. Inequality aversion is equivalent to concavity of I, which is defined as follows: for every (, Y ) V 2 V 2 and λ [0, 1], I(λ +(1 λ)y ) λi()+(1 λ)i(y ). 5. That is, for every borelian subet E of R n whose Lesbegue measure is 0, we have µ(e) = 0. 11

3.7.2 The representation theorem The proof of the following theorem, which generalizes Yaari s representation theorem [12], can be found in Appendix 7.6. To simplify notations, we assume that the support of P U is equal to [0, 1] n. Theorem 3.3. Let Q be a U-quantile operator. The mapping I : V 2 R satisfies axioms 1-6 above if and only if there exists a function φ : [0, 1] n R n, such that the components of φ are non negative (almost surely) and such that: (i) E((1,..., 1).φ(U)) = 1. (ii) For every V 2, I() = Q (U).φ(U)dP = min V2, =d i.e. I(.) is the min correlation risk measure with respect to φ(u)..φ(u)dp, A similar representation theorem, with the first equality in (ii), can be found in Galichon and Henry [6], for the specific class of µ-quantiles (see Section 4 below). Remark 3.4. (Interpretation of Theorem 3.3.) The representation theorem above extends Yaari standard dual choice theory. The formula I() = Q (U).φ(U)dP could be seen as a corrected mean of, the correction being a way to take into account the inequality created by the distribution. Indeed, the quantile Q can be seen as an attempt to re-order the distribution (indeed, for n = 1, the unidimensional quantile corresponds to standard increasing re-ordering). Then, from Q (U).φ(U)dP = min =d.φ(u)dp, the weight φ(u) compensates the effect due to the ordering of Q. Indeed, this equality implies that Q (U) and φ(u) are optimally coupled, which corresponds to the intuition that φ(u) and Q (U) are ordered in opposite directions. In particular, in the unidimensional case, this implies that φ is a decreasing function. Hence, when evaluated by the decision maker, high values of receive lower weights than small values of. Corollary 3.5. In Theorem 3.3: 1. If we additionnaly assume that Q satisfies point 4. in Definition 3.1, then we can write φ(p 1,..., p n ) = g( p 1,..., p n ) almost surely, for some l.s.c. convex function g : R n R. 12

2. If we additionnaly assume that Q satisfies point 4. and point 5. in Definition 3.1 and n 2, then there exists a R +, b = (b 1,..., b n ) R n, such that φ(u) = (b 1,..., b n ) a.(u 1,..., U n ). Remark 3.6. (Interpretation of Corollary 3.5 when,..., n are mutually independent.) Assume that the quantile operator satisfies point 1-6 in Definition 3.1, and that,..., n are independent. From Corollary 3.5, and since from 6. in Definition 3.1 we have Q 1,..., n (p 1,..., p n ) = (F 1 (p 1 ),..., F 1 n (p n )), we get: I() = n 1 0 F 1 i (t).(b i at)dt (2) For every i = 1,..., n, define f i : [0, 1] [0, 1] by f i (t) = 1 2b i a (at2 + 2(b i a)t). Then f i (0) = 0, f i (1) = 1, f i is increasing (since φ 0 implies b i a), f i is convex and f i(1 t) = b i at α i, where α i = b i a. Taking as a particular case equal to zero 2 except component i equal to one, we get α i = I(0,..., 0, 1, 0,..., 0) because, from point 1. in Quantile s Definition 3.1, we have Q (0,...,0,1,0,...,0) = (0,..., 0, 1, 0,..., 0). Therefore, I() = n I(0,..., 0, 1, 0,..., 0) 1 0 F 1 i (t)f i(1 t)dt. Note that 1 0 F 1 i (t)f i(1 t)dt is the measure of welfare allocated to attribute i when the policy maker applies the inequality distortion evaluation f i, therefore I() can be interpreted as the weighted average of the various attribute evaluations through the weights α i = I(0,..., 0, 1, 0,..., 0) allocated to each attribute i by the policymaker. In the unidimensional case n = 1, we can choose the quantile operator to be equal to the standard 1-dimensional quantile, i.e. Q = F 1. Then we get Corollary 3.7. For n = 1, S = [0, 1], U(p) = p and P be the Lebesgue measure, Theorem 3.3 is equivalent to Yaari Theorem 2.1. Proof. Indeed, from 1. in corollary above, φ(x) = g ( x) for some concave function g defined on [ 1, 0], which can be chosen such that g( 1) = 0. Since φ is non negative, g is also non decreasing. Define f(x) = g(x 1) on [0, 1]. It is convex, non decreasing, f(0) = 0. Moreover, Theorem 3.3 delivers I() = F 1 (p).g ( p)dp = 13

F 1 (p).f (1 p)dp = d(f P ), and finally f(1) = 1 is a consequence of normalization assumption (since I(1) = 1 = F 1 (p).f (1 p)dp = f(1)). 4 The case of µ-quantile: Galichon and Henry revisited In this subsection, we prove that if we consider the particular case where the quantile operator is the µ-quantile of Galichon and Henry (see [6], or also Section 3.5 for a brief presentation of µ-quantile), then Galichon and Henry s representation Theorem 2 is equivalent to Theorem 3.3. Let U V 2 whose probability law µ is absolutely continuous with respect to Lebesgue measure. To make the comparison between our results easier, we first recall the different axioms 6 used in Galichon and Henry (see [6]). Axiom 1. I(1,..., 1) = 1. Axiom 2. The functional I : V 2 R is continuous, and at least at one point its Fréchet derivative exists and is non-zero. To define the third axiom, recall this definition of Galichon et Henry: for every (, Y ) V 2 V 2, µ-first order stochastically dominates Y (resp. µ-first order strictly stochastically dominates Y ) if Q Q Y almost surely for the Lebesgue measure on [0, 1] n (resp. Q > Q Y almost surely). Axiom 3. The functional I preserves µ-first order stochastic dominance, i.e.: (1) If µ-first order stochastically dominates Y, then I() I(Y ). (2) If µ-first order strictly stochastically dominates Y, then I() > I(Y ). 6. The first axiom 1 is not assumed in Galichon and Henry s paper, but it can be added without any loss of generality, since it is only a normalization assumption. 14

For the next axiom, we recall the standard definition of the maximal correlation functional. For every V 2, ρ µ () := sup Ũ V 2,Ũ= du.ũdp. The following definition of µ-comonotonicity is introduced by Galichon et Henry (see also Ekeland, Galichon, and Henry [4]). Definition 4.1. We say that,..., n in V 2 are µ comonotonic if n ρ µ ( i ) = n ρ µ ( i ). Axiom 4. If, Y and Z are µ comonotonic in V 2, then for every α [0, 1], I() I(Y ) implies I(α + (1 α)z) I(αY + (1 α)z). The last axiom is Inequality aversion axiom, which we relabel for simplicity of the presentation: Axiom 5. Inequality Aversion: for every (, Y ) V 2 V 2 such that I() = I(Y ) and every λ [0, 1], we have I(λ + (1 λ)y ) I(). In the following proposition, recall that we consider the particular case of Galichon and Henry s µ-quantile. One of their main representation theorem states a similar representation result as Theorem 3.3, under axioms 1-5 instead of axioms 1-6. Thus, the following proposition proves the equivalence between these two representation theorems for µ quantiles. Proposition 4.2. (Galichon-Henry [6]) The functional I satisfies Axioms 1-5 if and only there exists a functional I : V 2 R satisfying Axioms 1-6 which represents the same preference relationship as I. Proof. From Theorem 2 in [6], I satisfies Axioms 1-5 if and only if there exists a functional I : V 2 R representing the same preference relationship as I, and which can be written V 2, I () = E(Q (U).((b 1,..., b n ) au) 15

for some a R and some b = (b 1,..., b n ) R n such that φ(p 1,..., p n ) = (b 1,..., b n ) a.(p 1,..., p n ) has non negative components for every (p 1,..., p n ) [0, 1] n. From Theorem 3.3 in the particular case of µ-quantile, this is equivalent to I satisfying axioms 1-6. 5 Uniqueness and elicitation of the parameters The following theorem, proved in Appendix 7.6, shows that in our framework, the parameters of the equality mindedness evaluations are unique and depend on the policy maker evaluations of some specific distributions. In the following theorem, let S = [0, 1] n, U(p 1,..., p n ) = (p 1,..., p n ), P be the Lebesgue measure on [0, 1] n, Q be the the associated U-quantile operator when U(p) = p. Theorem 5.1. Let Q a quantile operator defined on V 2, which additionaly satisfies 4., 5. and 6. in Definition 3.1. Let I : V 2 R which satisfies axioms 1-6. Then V 2, I() = a Q (p).p dp + n b i E( i ), and where a = 4 8I(1I {p 1 1 2 },..., 1I {p n 1 2 }) n b i = a 2 + I(0,..., 0, 1I S,..., 0). 6 Discussion and concluding remarks As seen in Corollary 3.7, choosing U(p) = p (p [0, 1]) in Theorem 3.3 allows to recover Yaari s theory for n = 1. But the same choice U(p 1,..., p n ) = (p 1,..., p n ) for n > 1 may lead also to some controversial result, as illustrated below. Hereafter, we concentrate on the µ quantile of Galichon and Henry. The following result is proved in Appendix 7.9. 16

In the following result, S = [0, 1] n is endowed with Lebesgue measure, and we define U = (U 1,..., U n ) V 2 whose probability law is absolutely continuous with respect to Lebesgue measure. Proposition 6.1. (Most inegalitarian situation with fixed marginals) Let I : V 2 R which satisfies Axioms 1-6. For every V 2, the solution of min I(Y 1,..., Y n ) k = d Y k,k=1,...,n is Y = (Q U 1 (U 1 ),..., Q Un n (U n )), where Q U k k k = 1,..., n. denotes the U k -quantile of k for every In particular, if U(p) = p (or more generally if the components of U = (U 1..., U n ) are mutually independent) then the components of Y = (Y 1,..., Y n ) are independent. It could be intuitive that the most inegalitarian situation occurs when the marginals are comonotonic. The following result shows that capturing such an intuition with µ quantile is not possible. Corollary 6.1. There does not exist U V 2 whose probability law is absolutely continuous with respect to the Lebesgue measure such that for every V 2, is reached at a comonotonic vector. min I(Y 1,..., Y n ) k = d Y k,k=1,...,n Proof. By contradiction: assume such a random vector U exists. Then, take as a particular case = U. Since the U k -quantile of k is identity, the above proposition implies that min I(Y 1,..., Y n ) Y k = d k = d U k,k=1,...,n is reached at Y = U. But if U is comonotonic, the support Supp(µ) of the probability measure µ of U cannot contain any subsets of dimension larger than 1. But then the Lebesgue measure of Supp(µ) should be 0, thus by absolute continuity, µ(supp(µ)) = 0, a contradiction. Thus, it seems an open problem to extend Yaari s theory in order to capture the intuition described above. 17

7 Appendix 7.1 Proof of Proposition 2.1 Property (i) is a consequence of the definition of F 1 in terms of an optimal coupling solution. For Property (ii), and since U(p) = p, the quantile function F 1 characterized by the two following properties: and (1) F 1 is increasing (2) F 1 = d. Indeed, the first condition can be equivalently written: G(t) := t F 1 0 is (p)dp is a continuous and convex function, and condition (2) can be written G = d. These two conditions characterize the (unique almost surely) solution F 1 coupling problem (see Appendix 7.1.1). Now, let characterized by (1) F 1 +Y F 1 sup p. (p)dp V 2, =d [0,1] of the optimal = F 1 and Y = F 1 Y. From above, the quantile of + Y is is increasing (2) F 1 +Y = d + Y. Thus, F 1 +Y = + F 1, because it satisfies the two conditions (1) and (2). This is similar to prove Y that F 1 λ 1 = λf, or to prove Property (iii). 7.1.1 Reminders of optimal transportation and optimal coupling. Consider hereafer a random vector U V 2. Sometimes, we shall assume that the law of U is absolutely continuous with respect to Lebegue measure: this means that for every borelian subset A R n of Lebesgue-measure 0, P U (A) = 0. An important object hereafter is ρ U, the maximal correlation functional with respect to U: this is the real function defined on V 2 by (P) V 2, ρ U () = sup E(.U) = d From Optimal coupling theory, given V 2, we have the following proposition: 18

Proposition 7.1. Assume the law of U is absolutely continuous with respect to Lebesgue measure. Then: (i) Existence and uniqueness: there exists a solution V 2 of (P), unique (almost surely). The pair (, U) is called an optimal coupling. 7 (ii) Form of the solution: a solution V 2 can be written = f(u) for f : R n R convex and lower semicontinuous. Moreover, f in the previous decomposition is unique, which means that if f(u) = g(u) is a solution of (P), where f, g : R n R are convex and l.s.c., then f = g. (iii) Characterization of the solution: if f : R n R convex and l.s.c. function such that f(u) = d, then f(u) is a solution of (P). (iv) Symmetry: sup E(.Ũ) = sup E(.U) = sup = d,ũ= du = d Ũ= d U Remark 7.1. In particular, when the law of U is absolutely continuous with respect to Lebesgue measure, this proves that for every V 2, there exists a convex and l.s.c. function f : R n R such that f exists almost surely, and f(u) has the same law as. Remark 7.2. The equality ρ U () = supũ=d U E(.Ũ) proves that ρ U depends only on the law of U. Thus, one could define equivalently, for every probability measure µ with a finite second moment: ρ µ () = sup E(.U) = d where U is any element in V 2 whose law is µ. With the previous notations, we could note ρ P U = ρ U. 7. Obviously, depends on U, not only on the law of U. 19

7.2 Proof that Galichon et al. µ-quantile defines a quantile operator in Subsection 3.5 We have to prove that 1., 2. and 3. in the definition of U-quantile operator are satisfied (with V 2 = V 2), where U = (U 1,..., U n ). As a matter of fact, we will also show that the three additional properties 4., 5. and 6. are also astisfied (the last one being true in the particular case where S = [0, 1] n, U(p 1,..., p n ) = (p 1,..., p n ) and P is the Lebesgue measure). Point 1. is true by definition of µ-quantile. For Point 2., we use the following lemma: Lemma 7.3. i) The µ-quantile of Q (U) + Q Y (U) is Q + Q Y. ii) For every λ 0, the µ-quantile of λ is λ.q. Proof. i) Use the characterization of µ-quantile: considering any U V 2 such that P U = µ, Q = f and Q Y = g for some convex and l.s.c. functions f, g : R n R, and ( f(u), U) and ( g(u), U) are optimal couplings (see Proposition 7.1, (ii)). Thus, from Proposition 7.1, (iii), (Q + Q Y )(U) = (f + g)(u) is a solution of sup E(.U) = d Q (U)+Q Y (U) (because f + g is convex and l.s.c., and (f + g)(u) = Q (U) + Q Y (U). Thus, by definition, (f + g) = Q + Q Y is the µ-quantile of Q (U) + Q Y (U). ii) straightforward. For Point 3., remark that from optimal coupling theory (see reminders above) sup E(.U) = sup = d Ũ= d U In particular, the µ-quantile of + λ(1,..., 1) satisfies n E(Q +λ(1,...,1) (U).U) = sup E(Ũ.( + λ(1,..., 1)) = sup E(Ũ.) + λ E(U i ). Ũ= d U Ũ= d U This is also equal to sup E(U. )+λ = d n E(U i ) = E(Q (U).U)+λ 20 n E(U i ) = E((Q (U)+λ(1,..., 1)).U).

Since the law of Q (U) + λ(1,..., 1) is equal to the law of + λ(1,..., 1), we get that Q +λ(1,...,1) = Q + λ(1,..., 1). Point 4. comes from the fact that f : p n k=1 λ kp 2 k 2 is continuous and convex, thus f(p) = (λ 1 p 1,..., λ n p n ) is the quantile function of f(u) (use Proposition 7.1, (iii)) Point 5. is similar to Point 4. above. The function f : p t pap is continuous and convex, thus f(p) = Ap is the quantile function of f(u). Finally, we prove Point 6. when S = [0, 1] n, U(p 1,..., p n ) = (p 1,..., p n ) and P is the Lebesgue measure. We want to show that if = (,..., n ) where,..., n are independent, then one has Q (p) = (F 1 (p 1 ),..., F 1 i (p i ),..., F 1 n (p n )) for every p = (p 1,..., p i,..., p n ) [0, 1] n. From the characterization of µ quantile from Optimal Coupling Theory, it is enough to prove that there exists a convex l.s.c. f : p [0, 1] n f(p) R such that and Define f(p 1,..., p n ) = n (1) f = (F 1 (p 1 ),..., F 1 i (p i ),..., F 1 n (p n )) (2) (F 1 (p 1 ),..., F 1 i (p i ),..., F 1 n (p n )) = d. pi 0 function F 1 i (t)dt. Clearly, f is convex, continuous, and (1) above is satisfied. Let us check that (2) above is also satisfied. We have P (F 1 (p 1 ) x 1,..., F 1 i (p i ) x i,..., F 1 n (p n ) x n ) = n 1 ν(f i (p i ) x i ) where ν the Lebesgue measure on [0, 1]. Thus, it can also be written n P ( i x i ) = P ( x 1,..., n x n ) because,..., n are independent, and finally (2) above is true. 7.3 Proof of Proposition 3.4 First, from its definition, Q depends only on the distribution of. Moreover, the proof of Q (U) = d can be found in [9], page 14. Thus Point 1. in Definition 3.1 is satisfied. Second, we will prove that the U-quantile of Q (U) + Q Y (U) is Q + Q Y (where = (, 2 ) and Y = (Y 1, Y 2 ) are both 2-dimensional random vectors), which will prove Point 2. in Definition 3.1 (the n-dimensional case is similar). Define U = 21

(U 1, U 2 ). First, by definition of U-quantile, the first component of the U-quantile of Q (U)+Q Y (U) is the standard one-dimensional quantile of F 1 is F 1 + F 1 Y (U 1 )+F 1 Y (U 1 ), which (see Proposition 2.1), i.e. the first component of Q + Q Y. Now, we check the same result for the second component. This is essentially the same proof as for the first component, but with a conditional argument. Let Z = Q (U) + Q Y (U) and define Q Z = (Q 1 Z, Q2 Z ). By definition, Q 2 Z(p 1, p 2 ) = inf{x R : P (Q 2 (U) + Q 2 Y (U) x F 1 (U 1 ) = F 1 (p Q 1 +Q1 Y Q 1 1 )) p 2 }. +Q1 Y But as explained before, the one-dimensional quantile of Q 1 + Q1 Y F 1 Y 1 (U 1 ) is F 1 + F 1 Y 1, thus we get = F 1 (U 1 ) + Q 2 Z(p 1, p 2 ) = inf{x R : P (Q 2 (U) + Q 2 Y (U) x E) p 2 }, where E = {s S : F 1 (U 1 (s)) + F 1 Y 1 (U 1 (s)) = F 1 (p 1 ) + F 1 Y 1 (p 1 )}. This can be written Q 2 Z(p 1, p 2 ) = inf{x R : P E (Q 2 (U) + Q 2 Y (U)) x) p 2 }, where P E denotes P conditionally to the event E. Thus p 1 being fixed, Q 2 Z (p 1,.) is the one-dimensional quantile of Z = Q 2 (U) + Q 2 Y (U) in the new probability space (S, F, P E). We shall use the following lemma to re-inforce this statement: Lemma 7.4. Q 2 Z(p 1, p 2 ) = inf{x R : P E (Q 2 (p 1, U 2 ) + Q 2 Y (p 1, U 2 )) x) p 2 }. Proof. To prove this lemma, we have to prove that the conditional probability allows to replace U 1 (.) in the probability above by p 1. First, remark that E = E 1 E 2 where E 1 = {s S : F 1 (U 1 (s)) = F 1 (p 1 ))} and E 2 = {s S : F 1 Y 1 (U 1 (s)) = F 1 Y 1 (p 1 ))}, which is a consequence of the comonotonicity of F 1 (U 1 ) and F 1 Y 1 (U 1 ). Indeed, clearly, E 1 E 2 E. To prove the other inclusion, let s S such that p 1 = U 1 ( s) (in particular s E), and take s / E 1 E 2 (if it does not exist, E E 1 E 2 = S). In a first case (the other cases being treated similarly), we have F 1 (U 1 )(s ) > F 1 (U 1 ( s)). Then 22

by comonotonicity, F 1 Y 1 (U 1 )(s ) F 1 Y 1 (U 1 ( s)), thus, by summing these inequalities, we get s / E; the other cases are similar. Thus, P E (Q 2 (U 1, U 2 ) + Q 2 Y (U 1, U 2 ) x) is equal to the probability of { s S : inf { x R : P (s S : 2 (s ) x ) U 2 (s) F 1 (U 1 (s )) = F 1 (U 1 (s))) } + inf { x R : P (s S : Y 2 (s ) x ) U 2 (s) F 1 Y 1 (U 1 (s )) = F 1 Y 1 (U 1 (s)) } x conditionally to } E = E 1 E 2 = {s S : F 1 (U 1 (s)) = F 1 (p 1 ) and F 1 Y 1 (U 1 (s)) = F 1 Y 1 (p 1 )}, which is thus the probability of { s S : inf { x R : P (s S : 2 (s ) x ) U 2 (s) F 1 (U 1 (s )) = F 1 (p 1 )) } + inf { x R : P (s S : Y 2 (s ) x ) U 2 (s) F 1 Y 1 (U 1 (s )) = F 1 Y 1 (p 1 )) } x conditionally to E, which is exactly the conclusion of the lemma. The previous lemma proves that p 1 being fixed, Q 2 Z (p 1,.) is the one-dimensional quantile of Z := Q 2 (p 1, U 2 ) + Q 2 Y (p 1, U 2 ) in the new probability space (S, F, P E ). Remark that U 2 is independent from E, thus it is also uniformly distributed in [0, 1] as a random variable of (S, F, P E ). Consequently, Q 2 Z (p 1,.) is characterized by the two following properties: (1) Q 2 Z (p 1, U 2 ) = d Z. (2) It is non decreasing with respect to p 2. Thus, to prove Q 2 Z (p 1, p 2 ) = Q 2 (p 1, p 2 ) + Q 2 Y (p 1, p 2 ), we only have to prove that: (1) Q 2 (p 1, U 2 (p 2 )) + Q 2 Y (p 1, U 2 (p 2 )) = d Z (which is clear by definition), and that: (2) p 2 Q 2 (p 1, p 2 ) + Q 2 Y (p 1, p 2 ) is non decreasing. Let us prove this last property (2). The same proof as the one of Lemma 7.4 above gives that Q 2 (p 1,.) (resp. Q 2 Y (p 1,.)) is the one-dimensional quantile of (resp. of Y ) in (S, F, P E1 ) (resp. in (S, F, P E2 )). In particular, Q 2 (p 1,.) and Q 2 Y (p 1,.) are non decreasing with respect to p 2, thus Q 2 (p 1,.) + Q 2 Y (p 1,.) is non decreasing with respect to p 2, which ends the proof. The last point (the U-quantile of λ is λq ) is straightforward. 23 }

7.4 Proof of Proposition 3.5 Consider the case n = 2. Let be a random variable uniformly distributed in [0, 1]. By contradiction, assume that there exists some probability measure µ such that for every V 2, Q = Q µ. In particular, since Q (, )(p 1, p 2 ) = (p 1, p 1 ), we should have Q µ (, ) = (p 1, p 1 ). But by definition of Q µ (, ), there is a convex function f : R 2 R such that Q µ (, ) = f. By convexity inequality, we get for every p = (p 1, p 2 ) R 2 and every p = (p 1, p 2) R 2, ( f(p) f(p )).(p p ) 0, i.e., by developping: (p 1 p 1)(p 1 p 1 p 2 + p 2) 0, which is false for example when p = ( 1 2, 1) and p = 0. 7.5 Proof of Proposition 3.6 Proof of 1. + a(1,..., 1) and Q (U) + a(1,..., 1) have the same law from Point 3. in Definition 3.1). Thus, I( + a(1,..., 1)) = I(Q (U) + a(1,..., 1)) = I(Q (U)) + I(a(1,..., 1)) (from additivity on quantiles, because a(1,..., 1) is the quantile of itself) which is also equal to I() + a. Proof of 2. Concavity clearly implies inequality aversion. Conversely, if inequality aversion is true, let us prove concavity. Let (, Y ) V 2 V 2 and λ [0, 1]. Consider = + I(Y )(1,..., 1) V 2, Y = Y + I()(1,..., 1). From Property 1. above, we get I( ) = I(Y ) = I() + I(Y ). Now, from Inequality Aversion at and Y we get I(λ + (1 λ)y ) I( ) = I() + I(Y ) or, equivalently, I(λ + (1 λ)y + (λi(y )(1,..., 1) + (1 λ)i()(1,..., 1)) I() + I(Y ), i.e. from property 1. above, I(λ + (1 λ)y ) + λi(y ) + (1 λ)i() I() + I(Y ) that is I(λ + (1 λ)y ) + λi(y ) + (1 λ)i() I() + I(Y ) which is concavity inequality. 24

7.6 Proof of Theorem 3.3 Step one. First prove that if Q is a quantile operator, and if I satisfies Axioms 1.-6., then there exists φ : [0, 1] n R n, with non negative components, such that (ii) V 2, I() = (i) E((1,..., 1).φ(U)) = 1, Q (U).φ(U)dP = min = d.φ(u)dp. Define W the set of n-random vectors from ([0, 1] n, B [0,1] n, P U ) to (R n, B R n), where P U is the probability measure of U on [0, 1] n, and let W 2 be the set of square-integrable random vectors of W. Remark that in the particular case where U(p) = p, S = [0, 1] n and P is the Lebesgue measure, then W 2 = V 2. Define C = {Q : V 2} the subset of W 2 whose elements are all possible quantiles of random variables in V 2. From the definition of quantiles, C is a cone 8. Indeed, if λ 0, V 2 and Y V 2, then λq and Q + Q Y are the quantile of λ V 2 and Q (U) + Q Y (U) (from Point 2. in the definition of a Quantile operator). Last, 0 C since 0 is the quantile of itself (because if = 0, from Point 1. in the definition of a Quantile operator, Q (U) = d = d 0 thus Q = 0). Let F W 2 be the vector space spanned by C. Since C is a cone, it can be written F = C C = {c c : (c, c ) C C}. Now, define Ĩ : F R by (, Y ) V 2 V 2, Ĩ(Q Q Y ) = I(Q (U)) I(Q Y (U)). It is well defined: indeed, if Q Q Y = Q Q Y for some,, Y and Y in V 2, then Q (U) Q Y (U) = Q (U) Q Y (U), i.e. Q (U) + Q Y (U) = Q Y (U) + Q (U). Consequently, from additivity of I on the set of quantiles, I(Q (U) + Q Y (U)) = 8. Throughout this paper, cone will be used for convex cone containing 0. 25

I(Q (U)) + I(Q Y (U)) = I(Q Y (U) + Q (U)) = I(Q Y (U)) + I(Q (U)), that is I(Q (U)) I(Q Y (U)) = I(Q (U)) I(Q Y (U)), thus I(Q Q Y ) = I(Q Q Y ). To prove linearity of Ĩ, first remark that for every λ 0, we have, from positive homogeneity of I and Point 2. in the definition of quantile operator: Ĩ(λ(Q Q Y )) = Ĩ(Q λ Q λy ) = I(Q λ (U)) I(Q λy U)) = λi(q (U)) I(Q Y U)) = λĩ(q Q Y ). But by definition, Ĩ(Q Q Y ) = Ĩ(Q Y Q ) is clear, thus the last equality holds for λ 0. Moreover, Ĩ(Q + Q Y ) = Ĩ(Q Q (U)+Q Y (U)) = I(Q Q (U)+Q Y (U)(U)) = I(Q (U) + Q Y (U)) = I(Q (U)) + I(Q Y (U)) = Ĩ(Q ) + Ĩ(Q Y ), from additivity of I on quantiles, and from Point 2. of the definition of quantile operators. This proves linearity of Ĩ. Let p : W 2 R be defined by W 2, p() = I( (U)). The following properties hold: First, W 2 and F W 2 are Riesz spaces (by definition, partially ordered vector spaces which also are lattices) for the natural order defined by Y (s) Y (s) for almost every s S. Second, the function p satisfies Monotonicity (because I satisfies Monotonicity), i.e., for every (, Y ) W 2 W 2, Y p() p(y ). Third, p is a sublinear function, which means that the following properties (1) and (2) are true: (1) For every and Y in W 2, p( + Y ) p() + p(y ). (2) p is positively homogeneous. Property (1) is true from Proposition 3.6 and positive homogeneity of I. Property (2) is true because I is positively homogeneous. Fourth, Ĩ is positive on F, that is for every (, Y ) V 2 V 2, Q Q Y 0 implies Ĩ(Q Q Y ) 0, from Monotonicity of I and by definition of Ĩ. Last, we have Ĩ(Q Q Y ) p(q Q Y ) for every (, Y ) V 2 V 2. Indeed, this is equivalent to I(Q Y (U)) I(Q (U)) + I(Q Y (U) Q (U)) which is true 26

from Proposition 3.6. By Hahn-Banach Extension Theorem (Th. 8.31 in [1]), and since from above Ĩ is a positive linear function on F, majorized by the monotone sublinear function p on F, Ĩ extends to a positive linear functional Ī on W 2, satifying Ī() p() W 2. Recall in the following that a norm. on W 2 is a lattice norm if Y a.e. Y. In particular, this is the case for. 2. A Banach lattice is, by definition, a Banach space for the lattice norm. Again, this is true for (W 2,. 2 ) (which is even a Hilbert space). Since every positive operator on a Banach lattice is continuous (see Theorem 9.6, p. 350 in [1]), Ī is continuous on W 2, which is a Hilbert space for the scalar product (associated to the norm. 2 ) <, Y >=.Y dp. Thus, from Riesz representation theorem, there is φ W 2 with non negative components on the support of P U i.e. on [0, 1] n (because Ī is positive on W 2) such that W 2, Ī() = (p)φ(p)dp U [0,1] n In particular, for every V 2, we have I() = I(Q (U)) = Ĩ(Q ) = Ī(Q ) = Q (p).φ(p)dp U = [0,1] n S Q (U).φ(U)dP, the first equality being true because I is law invariant, the second by definition of Ĩ, the third because Q F, and the last one is simply a change of variable formula. Note that S (1,..., 1).φ(U)dP = (1,..., 1).φdP U = [0,1] n Q (1,...,1).φdP U = I(1,..., 1) = 1 [0,1] n because the Quantile of (1,..., 1) is (1,..., 1), and because of normalization assumption. Define W 2 = { W 2 : (U) V 2}. For every (, Y ) W 2 W 2 such that (U) = d Y (U), Ī( Y ) = Y.φdP U = [0,1] n S Y (U).φ(U)dP p( Y ) = I(Y (U)) = I((U)) 27

because (U) and Y (U) have the same law, and I is law invariant. Moreover, I((U)) = I(Q (U) (U)) = Ĩ(Q (U)) = Ī(Q (U)) = From the two equations above, we get (, Y ) W 2 W 2 such that Y (U) = d (U), Y (U).φ(U)dP Q (U) (U).φ(U)dP. Q (U) (U).φ(U)dP But given (, Y ) V 2 V 2, there exists W 2 and Y W 2 such that (U) = d and Y (U) = d (from optimal coupling theory, because U is absolutely continuous with respect to Lebesgue measure). Consequently, we get (, Y ) V 2 V 2 such that Y = d, whis ends the proof of (ii). Y.φ(U)dP (3) Q (U).φ(U)dP (4) Step two. Converse implication: assume there exists φ : [0, 1] n R n, with non negative components on [0, 1] n, such that (i) E((1,..., 1).φ(U)) = 1, (ii) for every V 2, I() = Q (U).φ(U)dP = min V2, = d and let us prove I satisfies Axioms 1.-6..φ(U)dP, Axiom 1. is exactly (i) above. Monotonicity, concavity, positive homogeneity and neutrality are a consequence of I() = min V2, =d.φ(u)dp, i.e. I is the min correlation risk measure with respect to φ(u). Indeed, recall that for every Y V 2 with non negative components, the max correlation risk measure Ψ Y is defined by Ψ Y () =.Y dp. This is a coherent (i.e. monotone, positively homogeneous and max =d subadditive) and law invariant risk measure (see [9], p. 192). In particular it is convex. Thus, in the above theorem, I() = Ψ φ(u) ( ) satisfies the desired properties. Moreover, Additivity on quantiles is straightforward. 7.7 Proof of Corollary 3.5 Step one. let us prove that φ(p 1,..., p n ) = g( p 1,..., p n ) (almost surely) for some l.s.c., convex function g : R n R. 28

Indeed, Point (ii) of the main representation Theorem 3.3 implies that for every V 2, I() = min V2, =d.φ(u)dp. Thus, for every V 2, ( Q (U), φ(u)) is an optimal coupling (see Subsection 7.1.1). In particular, since we can take V 2 such that Q (p) = p (from Point 4. in Definition 3.1), we get that ( U, φ(u)) is an optimal coupling. Since P U is absolutely continuous with respect to Lebesgue measure, there exists some l.s.c. convex function g : R n R such that φ(u) = g( U) (see Subsection 7.1.1). This ends the proof, since the support of P U is [0, 1] n. Step two. Let us now assume the quantile operator satisfies additionally 4. in Definition 3.1, and let us prove that φ(p) = (φ 1 (p 1 ),..., φ n (p n )) where each φ i is decreasing. Consider a diagonal n n matrix D with a strictly positive diagonal. From Point 4. in Definition 3.1, p Dp is the quantile of some Y D V 2. Moreover, there exists D W 2 such that D (U) = d Y D. Since quantile operator is law invariant, we get Dp = Q D (U)(p), thus DU = Q D (U)(U) which is also equal (in law) to D (U). Moreover, Equation (ii) in Theorem 3.3 implies that for every Y W 2 whose law is equal to the law of D Y (U).φ(U)dP DU.φ(U)dP (5) From optimal coupling theory, and since the law of DU is absolutely continuous with respect to the Lebesgue measure, there is some l.s.c. convex function f D : R n R such that φ(u) = f D (DU). Thus, since the support of P U is [0, 1] n, we get φ(p) = f D (Dp) for every p [0, 1] n. From Aleksandrof theorem, since f D is convex, f D and φ are almost surely C 1 on ]0, 1[ n. Differentiating the equality above at any p ]0, 1[ n, we get φ(p) = 2 f D (Dp).D for every diagonal n matrix D with strictly positive diagonal, where φ(p) and 2 f(dp) are assimilated to n n matrices. 29

Then, we use the following lemma: Lemma 7.5. Let A be a n n real matrix. i) Assume that for every diagonal n n matrix D with a non negative diagonal, there exists a symmetric non negative matrix B such that A = BD. Then A is diagonal with a non positive diagonal. ii) Moreover, if for every symmetric definite positive matrix D, there exists a symmetric non negative matrix B such that A = BD. Then A = λi n for some λ < 0. Proof. For D = I n, the property i) implies that A is symmetric non negative. Thus, B and D commutes, and taking a diagonal matrix D with distinct elements on the diagonal, this implies that B is diagonal, thus A is diagonal. But A is symmetric non negative, thus A has a negative diagonal. From Point i), Point ii) is straightforward. From the above lemma applied to A = φ(p), B = 2 f D (Dp), we get φ(p) = ( φ 1 (p),..., φ n (p)) is diagonal for almost every p in ]0, 1[ n, with a negative diagonal. Thus φ(p) = (φ 1 (p 1 ),..., φ n (p n )) where each φ i is decreasing. Step three. Let now assume that the quantile operator satisfies additionally Point 4. and Point 5. of Definition 3.1 with n 2, and let us prove that φ(p) = b ap. From Point 5. in Definition 3.1, for every definite positive symmetric n n matrix S, the mapping p Sp is the quantile of some Y S V 2. Moreover, there exists S W 2 such that Y S = d S (U). From law invariance of quantile operator, we get Sp = Q S (U)(p), thus SU = Q S (U)(U). As above, from Equation (ii) in Theorem 3.3, SU = Q S (U)(U) and φ(u) are optimally coupled. From optimal coupling theory, and since the law of Q S (U) (which is also the law of SU from Point 1. in the definition of quantile operator) is absolutely continuous with respect to the Lebesgue measure, there is some l.s.c. convex function f S : R n R such that φ(u) = f S (SU), thus φ(p) = f S (Sp) at every p [0, 1] n (because the support of P U is [0, 1] n ). Differentiating this equality at every interior point p, we get φ(p) = 2 f S (Sp).S 30