Pareto Optimal Allocations for Law Invariant Robust Utilities

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1 Pareto Optimal Allocations for Law Invariant Robust Utilities on L 1 Claudia Ravanelli Swiss Finance Institute Gregor Svindland University of Munich and EPFL November 2012 Abstract We prove the existence of comonotone Pareto optimal allocations satisfying utility constraints when decision makers have probabilistic sophisticated variational preferences and thus representing criteria in the class of law invariant robust utilities. The total endowment is only required to be integrable. JEL Classifications: D81, D86. MSC 2000 Classifications: 91B16, 91B30, 91B32, 91B50. Keywords: Pareto optimal allocations, variational preferences, robust utility, probabilistic sophistication, law invariance, ambiguity aversion, weighted sup-convolution. 1 Introduction In this paper we prove the existence of Pareto optimal allocations of integrable random endowments when decision makers have probabilistic sophisticated variational preferences. Variational preferences were introduced and axiomatically characterized by Maccheroni, Marinacci, and Rustichini (2006). This broad class of preferences allows to model ambiguity aversion and includes several subclasses of preferences that have been extensively studied in the economic literature. In mathematical finance, variational preferences are known as robust utilities; see Föllmer, Schied, The authors gratefully acknowledge the financial support of the NCCR-FinRisk (Swiss National Science Foundation). 1

2 and Weber (2009) and the references therein. In particular, Föllmer et al. (2009) establish the connection between variational preferences and robust utilities as representing choice criteria of variational preferences on random endowments corresponding to Savage acts; see also Remark 2.2 for a brief summary of these results. Our study focuses on variational preferences which are in addition assumed to be probabilistic sophisticated 1 and thus can be represented by robust utilities which are law invariant. Probabilistic sophistication or law invariance means that the decision maker sees any two random endowments that have the same distribution under a reference probability measure as equivalent. To be consistent both with literature on decision making and (mathematical) finance, we use the term probabilistic sophistication whenever referring to preferences, and law invariance whenever referring to the representing robust utility. We pursue this approach in the spirit of Föllmer et al. (2009) in an attempt to unify the current knowledge on Pareto optimal allocations, and to emphasize the range of the results presented in this paper. The class of law invariant robust utilities which represent probabilistic sophisticated variational preferences are of the following type: (1.1) U(X) = inf (E Q [u(x)] + α(q)), X L 1, Q Q where u : R R { } is a (not necessarily strictly) concave (not necessarily strictly) increasing utility function, Q is a set of probability measures which is closed under densities with the same distribution, and α(q) is a suited law invariant penalization on Q Q; see Definition 2.1 for the details. This broad class nests many well-known choice criteria studied in the economic and finance literature, in particular, the von Neumann and Morgenstern (1947) expected utility, the probabilistic sophisticated maxmin expected utility preferences introduced by Gilboa and Schmeidler (1989), the multiplier preferences introduced by Hansen and Sargent (2000, 2001), and (apart from the sign) the law invariant cash (sub)-additive convex risk measures introduced by Artzner, Delbaen, Eber, and Heath (1999), Föllmer and Schied (2002), and Frittelli and Rosazza Gianin (2005) with the property of cash-invariance, and by El Karoui and Ravanelli (2010) with the generalized property of cash (sub)-additivity. 1 Probabilistic sophisticated preferences were introduced by Machina and Schmeidler (1992), further studied by Marinacci (2002) and by Maccheroni et al. (2006) and Strzalecki (2011) for the case of variational preferences. 2

3 We assume that the decision makers have preferences on a space of future random payoff profiles which we identify with L 1 := L 1 (Ω, F, P), i.e. the space of P-integrable random variables on a fixed non-atomic probability space (Ω, F, P) modulo P-almost sure equality, where P is the reference probability measure. So far, most of the existing literature makes the assumption that payoff profiles are bounded, i.e. in L, or even that the state space is finite. These assumptions are justified in many settings. But for applications in finance where nearly all models involve unbounded distributions, the boundedness assumption is not appropriate. This suggests the model space L 1 and the probabilistic sophistication of the preferences indeed allows for that; see Remark 2.2 or Filipović and Svindland (2012). In this paper we consider n 2 decision makers with probabilistic sophisticated variational preferences on L 1 represented by law invariant robust utilities as in (1.1). All decision makers share the same reference probability. Given the initial endowments W i L 1, i = 1,..., n, of the decision makers, we prove the existence of comonotone Pareto optimal allocations of the aggregate endowment W = W W n which satisfy individual rationality constraints. Indeed we show that under some mild conditions comonotone Pareto optima exist for basically any constraints on the utilities of the allocations as long as there is at least one allocation satisfying these constraints. Note that comonotonicity means that the endowments in the allocation are continuous increasing functions of the aggregate endowment W. The existence of Pareto optimal allocations has so far only been established for a few subclasses of law invariant robust utilities. In case that all decision makers have von Neumann Morgenstern expected utilities, existence results were already proved in the sixties by Borch (1962), Arrow (1963) and Wilson (1968). More recent is the proof of the existence of Pareto optimal allocations when all decision makers apply law invariant convex risk measures on L ; see Jouini, Schachermayer, and Touzi (2008) (and also Acciaio (2007) and Barrieu and El Karoui (2005)) and references therein. Filipović and Svindland (2008) extend this result to integrable (not necessarily bounded) aggregate endowments. Even more recently, Dana (2011) proves the existence of Pareto optimal allocations when the decision makers have choice criteria within a class of law invariant, finitely valued, continuous, concave utility functions on L not necessarily representing variational preferences. In Dana (2011) at least one utility function is required to be cash additive and the others are assumed to be strictly concave. Note that the cash addi- 3

4 tivity assumption is very useful when proving the existence of optimal allocations, because in conjunction with a comonotone improvement result, it immediately allows us to restrain the optimal allocation problem to an essentially compact set. Dropping the cash additivity assumption generates some difficulties which we are able to solve. We also reduce the problem to an optimization over an essentially compact set of comonotone allocations. However, in contrast to the cash additive case, in which this reduction can be simply imposed due to the invariance towards constant re-sharing of sure payoffs, in the concave (non cash additive) case it follows as a necessity from the concavity of the utility functions u in (1.1). Further results on the existence of Pareto optimal allocations are found in Kiesel and Rüschendorf (2008). Here the existence of optimal allocations is proved for convex risk functionals on L which are not necessarily law invariant, however, under the assumption that the aggregate endowment W is in the interior of the domain of the (infimal-)convolution of the convex risk functionals, and that there exists an interior point in the intersection of the domains of the dual functions of these risk functionals. Such interior point conditions are standard when arguing by means of convex duality theory. Unfortunately, such results are not applicable in our case. The reason is that on large model spaces like L 1 the interior of the domains of the robust utilities (1.1), as well as of their (sup-)convolutions, are in general empty. Reducing the model space to e.g. L could solve that problem, but then monotonicity implies that the interior of the domain of the dual function is always empty because it is concentrated on the positive cone of the dual space. Another result in the economic literature is provided by Rigotti, Shannon, and Strzalecki (2008). Here the authors prove the existence of Pareto optimal allocations for variational preferences on the positive cone of L under the strong assumption of mutual absolute continuity, which is in general not satisfied in our case. The paper is organized as follows. In Section 2 we introduce probabilistic sophisticated variational preferences and law invariant robust utilities on L 1, we recall some well-known subclasses of these preferences and some useful properties. The Pareto optimal allocations problem is studied throughout Section 3 in which we state our main results. In Section 4 we illustrate our results by means of some examples. 4

5 2 Setup Throughout this paper (Ω, F, P) is an atom-less probability space, i.e. a probability space supporting a random variable with continuous distribution. All equalities and inequalities between random variables are understood in the P-a.s. sense. Given two random variables X and Y we write X = d Y to indicate that both random variables have the same distribution under the reference probability measure P. The expectation (if well-defined) of a random variable X under a probability measure Q on (Ω, F) will be denoted by E Q [X]. In case Q = P we also write E[X] := E P [X]. We denote by L 1 := L 1 (Ω, F, P) the space of P-integrable random variables modulo P-almost sure equality. 2.1 Probabilistic Sophisticated Variational Preferences on L 1 Variational preferences were introduced by Maccheroni et al. (2006). In the same paper the authors also study the subclass of probabilistic sophisticated variational preferences. Probabilistic sophistication means that X d = Y implies X Y in the preference order. Definition 2.1. (i) A function u : R R { } is a utility function (on R) if it is concave, right-continuous, increasing, dom u := {x R u(x) > }, and not constant in the sense that there exist x, y dom u such that u(x) u(y). (ii) Let denote the set of all probability measures Q on (Ω, F) which are absolutely continuous and have bounded densities with respect to P, i.e. A F, P(A) = 0 Q(A) = 0, and ( ) there exists K > 0 such that P dq dp < K = 1. A set of probability measures Q is closed under densities with the same distribution if Q Q and Q with d Q dp implies that Q Q. d = dq dp (iii) A decision maker has probabilistic sophisticated variational preferences if for all X, Y L 1 : X Y inf (E Q [u(x)] + α(q)) inf (E Q [u(y )] + α(q)) Q Q Q Q where u is a utility function, Q is convex and closed under densities with the same distribution, and α : Q R is a convex and law invariant function on Q in the sense that Q, Q Q with d Q dp d = dq dp implies α( Q) = α(q). In addition α satisfies inf Q Q α(q) >. 5

6 The numerical representation (2.1) U : L 1 R { }, X inf (E Q [u(x)] + α(q)), Q Q is the law invariant robust utility used by the decision maker to quantify the utility of a payoff profile X L 1. The law invariant robust utility U in (2.1) is, clearly, law invariant 2 (X = d Y implies U(X) = U(Y )) and posseses some other useful properties which are collected in Lemma 2.3 below. Due to Jensen s inequality for concave functions, the expectations in (2.1) are all well-defined, possibly taking the value. Note that U(X) = is possible for some X L 1. The interpretation is that the payoff profiles with utility are totally unacceptable. We remark that what we call a utility function in Definition 2.1 (i) satisfies relatively weak requirements and nests the vast majority of utilities proposed in the economic, finance, and insurance literature, including extreme cases such as increasing linear or affine functions. Moreover, by allowing u to take the value we also incorporate the cases when the domain of the utility function u is bounded from below, as e.g. for the power utilities or the logarithmic utilities. Remark 2.2. A standard approach to modeling preferences in presence of model ambiguity (Knightian uncertainty) is to consider preference orders on the set M of all Markov kernels X (ω, dy) from (Ω, F) to (R, B(R)) (where B(R) denotes the Borel-σ-algebra) for which there exists a k > 0 such that X (ω, [ k, k]) = 1 for all ω Ω. It can then be shown under some mild additional assumptions that a preference order on M is in the class of variational preferences if and only if it admits a numerical representation of the form ( (2.2) U(X ) = inf Q C ) u(y)x (ω, dy) dq(ω) + α(q), X M. Here u is a utility function on R, and without loss of generality we may assume that C is the set of all finitely additive normalized measures, and α : C R { } is a convex law invariant function 2 Since U is defined via the law invariant penalization α, it is law invariant. This follows as in Föllmer and Schied (2004) Theorem Conversely, the dual function in the convex duality sense of a law invariant concave function U : L 1 R { }, which in particular can serve as a penalization α in the sense of (2.1), is always law invariant; again see Föllmer and Schied (2004) Theorem

7 with the additional property of being the minimal function for which U can be represented as in (2.2). For an axiomatic definition of variational preferences on Markov kernels and the details on their numerical representation (2.2) we refer to Föllmer et al. (2009). Notice that the space of all bounded payoff profiles L is naturally embedded into the space M by identifying each X L with the associated kernel X (ω, dy) = δ X(ω) (dy) where δ x denotes the Dirac measure given x R. The restriction of U to L then takes the form ( (2.3) U(X) = inf Q C ) u(x) dq + α(q), X L, which is a robust utility. In case of probabilistic sophistication/law invarinace, using results in Svindland (2010), it follows that U is σ(l, L )-upper semi continuous and thus we obtain a representation of U as an infimum over σ-additive probability measures in : (2.4) U(X) = inf (E Q [u(x)] + α(q)), X L, Q Q where Q := dom α. Hence, these preferences on L are indeed consistent with our definition of probabilistic sophisticated variational preferences on L 1 ; see Definition 2.1. Moreover, the representation (2.4) shows that the robust utility U and thus the corresponding preference order is canonically extended from L to L Properties of the Law Invariant Robust Utilities In the following Lemma 2.3 we collect some well-known properties of law invariant robust utilities on L 1 which we will make frequently use of. The proofs can be found or easily derived from results in for instance Dana (2005), Föllmer and Schied (2004) and Maccheroni et al. (2006). Lemma 2.3. Consider a law invariant robust utility U as in (2.1). Then U has the following properties: (i) properness: U < and the domain dom U := {X L 1 U(X) > } is not empty. (ii) concavity: U(λX + (1 λ)y ) λu(x) + (1 λ)u(y ) for all λ [0, 1]. (iii) monotonicity: X Y implies U(X) U(Y ). 7

8 (iv) ssd -monotonicity: X ssd Y implies U(X) U(Y ), where X ssd Y E[u(X)] E[u(Y )], for all utility functions u : R R, is the second order stochastic dominance order. (v) upper semi-continuity: for all k R the level sets E k := {X L 1 U(X) k} are closed in L 1 with respect to the topology induced by the norm 1 := E[ ]. Equivalently, if (X n ) L 1 converges to X L 1 (with respect to 1 ), then U(X) lim sup n U(X n ). Probabilistic sophisticated variational preferences do not only preserve second order stochastic dominance (Lemma 2.3 (iv)) but consequently also the concave order, i.e. X c Y implies U(X) U(Y ), where c denotes the concave order, that is (2.5) X c Y E[u(X)] E[u(Y )] for all concave functions u : R R. Clearly, X c Y implies X ssd Y. The property of preserving c is often referred to as Schur concavity of U. Indeed, in case of monotone concave upper semi-continuous functions ssd - monotonicity is equivalent to Schur concavity. See for instance Dana (2005) for more details on the concave order and second order stochastic dominance. 3 Pareto Optimal Allocations for Probabilistic Sophisticated Variational Preferences Consider n 2 decision makers with initial endowments W i L 1. All decision makers are assumed to have probabilistic sophisticated variational preferences on L 1 and corresponding law invariant robust utilities (3.1) U i (X) = inf Q Q i (E Q [u i (X)] + α i (Q)), X L 1, i = 1,..., n, as defined in (2.1). We assume that U i (W i ) > for all i = 1,..., n, and let W := W W n be the aggregate endowment. The aim of this section is to prove the existence of Pareto optimal allocations of W amongst the n agents within a set of reallocations of the aggregate endowment 8

9 W which are accepted by all agents. The set A(W ) of all acceptable allocations of W is (3.2) A(W ) := {(X 1,..., X n ) (L 1 ) n X i = W, U i (X i ) > and U i (X i ) U i (W i ) c i for all i {1,..., n}}, where c i R { }, i = 1,..., n. The constraint (3.3) U i (X i ) U i (W i ) c i expresses the individual (rationality) constraint of decision maker i, specifying which payoff profiles X i in a new re-allocation of W she is willing to accept. Clearly, c i = 0 represents the (classical) case when the decision maker will not accept any allocation which allots her an endowment which is not at least as good as her initial one. An extreme is c i = which means that the decision maker is willing to accept any allocation with finite utility. 3 We also allow for situations in which the decision maker is to some bounded extent willing to accept a worsening as compared to her initial endowment (c i > 0), or requires an improvement (c i < 0). Note that if c i 0 for all i {1,..., n}, then the initial allocation (W 1,..., W n ) is acceptable, so in particular A(W ). However, if some agents demand a strict improvement, it is in general not clear whether the set of acceptable allocations is non-empty. Hence, we make the following assumption. Assumption 3.1. A(W ). We are interested in those acceptable reallocations of W which are Pareto optimal. Definition 3.2. An allocation (X 1,..., X n ) A(W ) is Pareto optimal if (Y 1,..., Y n ) A(W ) and U i (Y i ) U i (X i ) for i = 1,..., n implies that U i (Y i ) = U i (X i ) for all i = 1,..., n. The following stability lemma shows that Pareto optimal allocations given some individual constraints c i on the utility remain Pareto optimal if we loosen these constraints. In other words, the optimality concept is not dependent on these constraints. 4 In particular, in Definition 3.2 we could replace the requirement (Y 1,..., Y n ) A(W ) by simply requiring (Y 1,..., Y n ) (L 1 ) n to be an allocation of W. 3 c i = is understood as the restriction U i(x i) U i(w i) := being redundant. 4 In fact, the constraints only serve as a means to show that we can find Pareto optima which satisfy the given individual constraints on the respective utilities. 9

10 Lemma 3.3. Let A(W ) be given by the constraints (c i ),...,n (R { }) n as in (3.2), and let (X 1,..., X n ) A(W ) be Pareto optimal. For any constraints ( c i ),...,n (R { }) n which satisfy c i c i, the allocation (X 1,..., X n ) is also Pareto optimal in Ã(W ) where Ã(W ) is the set of acceptable allocations given the individual constraints ( c i ),...,n in (3.2). Proof. Suppose (X 1,..., X n ) is not Pareto optimal in Ã(W ). Then there must be an allocation (Y 1,..., Y n ) Ã(W ) such that U i(y i ) U i (X i ) for all i = 1,..., n and U j (Y j ) > U j (X j ) for at least one j {1,..., n}. However, due to the Pareto optimality of (X 1,..., X n ) with respect to A(W ) we must have (Y 1,..., Y n ) Ã(W ) \ A(W ) (note that A(W ) Ã(W )). But this is a contradiction, because Ã(W ) \ A(W ) is either empty (if c i = c i for all i) or contains allocations such that there is at least one k {1,..., n} for which U k (Y k ) < U k (W k ) c k U k (X k ), so (Y 1,..., Y n ) cannot be an improvement of (X 1,..., X n ). 3.1 Characterization of Pareto Optimal Allocations The following well-known result characterizes Pareto optima as solutions to a weighted supconvolution optimization problem (3.4) Maximize λ i U i (X i ) subject to (X 1,..., X n ) A(W ), where λ i 0, i = 1,..., n, are Negishi weights associated to the Pareto optimal allocation. Note that under Assumption 3.1 the optimization problem (3.4) is well-posed. Proposition 3.4. If (X 1,..., X n ) A(W ) is Pareto optimal, then there exist weights λ i 0, i = 1,..., n, not all equal to zero, such that the allocation (X 1,..., X n ) solves (3.4) with these weights. Conversely, if (X 1,..., X n ) solves (3.4) for some strictly positive weights λ i > 0, i = 1,..., n, then (X 1,..., X n ) is Pareto optimal. It is one of the peculiarities of the characterization of Pareto optimal allocations that the associated Negishi weights in (3.4) are in general only non-negative. So we may have λ j = 0 for some j. 5 As we will see in Section 3.3, our techniques allow us to prove the existence of solutions 5 We think that Pareto optimal allocations that are associated to strictly positive weights are the most interesting ones. Indeed λ j = 0 implies that the decision maker j is not considered in the social welfare maximization problem (3.4), whereas λ j > 0 in many cases will lead to a social welfare that is strictly higher than in case λ j = 0. 10

11 to (3.4) only in case all Negishi weights are strictly positive, i.e. λ i > 0 for all i {1,..., n}; see Theorems 3.11 and So the question may arise when and what kind of Pareto optimal allocations we are neglecting. The following Lemma 3.5 shows that very often there are strictly positive Negishi weights for any Pareto optimum. Lemma 3.5. Let (X 1,..., X n ) A(W ) be Pareto optimal and suppose that for any i = 1,..., n there is some ɛ i > 0 such that U i (X i ɛ i ) dom U i and that (U i (X i ɛ i ), ) m U i (X i + m) is strictly increasing. Then there exists strictly positive Negishi weights associated to (X 1,..., X n ). Proof. Let (X 1,..., X n ) A(W ) be Pareto optimal. Then according to Lemma 3.3 the allocation (X 1,..., X n ) is also Pareto optimal in the set of allocations Ã(W ) where we replace the original constraints c i by the trivial constraints c i = for all i = 1,..., n. Since A(W ) Ã(W ), any Negishi weights (λ 1,..., λ n ) associated to (X 1,..., X n ) with Ã(W ) instead of A(W ) in (3.4) also serve as Negishi weights for the original problem with constraints (c i ),...,n. Assume that for some j {1,..., n} we have λ j = 0 and pick some k {1,..., n} such that λ k > 0. Consider the allocation ( X 1,..., X n ) Ã(W ) where X i = X i for all i j, k and X j = X j ɛ and X k = X k + ɛ for some ɛ (0, ɛ j ). As λ j = 0, we obtain λ i U i ( X i ) = (3.5) > which is a contradiction. λ i U i (X i ) + λ k U k (X k + ɛ) ;i k,j ;i k,j λ i U i (X i ) + λ k U k (X k ) = λ i U i (X i ). k=1 An immediate consequence of Lemma 3.5 and Proposition 3.4 is the following: Corollary 3.6. Suppose that the U i satisfy the following conditions for all i = 1,..., n: dom U i = dom U i + R, R m U i (X + m) is strictly increasing for all X dom U i. Then (X 1,..., X n ) A(W ) is Pareto optimal if and only if it solves (3.4) for some strictly positive weights λ i > 0, i = 1..., n. 11

12 3.2 Comonotone Pareto Optimal Allocations Definition 3.7. We denote by CF the set of all n-tuples (f 1,..., f n ) of increasing functions f i : R R, i = 1,..., n, such that n f i = Id R. These functions f i are necessarily 1-Lipschitzcontinuous. An allocation (Y 1,..., Y n ) (L 1 ) n of W, i.e. n Y i = W, is comonotone if there exists (f i ) n CF such that Y i = f i (W ) for all i = 1,..., n. It is known that if there exists a Pareto optimal allocation, then in particular there is also a comonotone one. This follows from the following Proposition and the ssd -monotonicity of the U i. Proposition 3.8. For any (X 1,..., X n ) A(W ) there is a comonotone allocation (Y 1,..., Y n ) A(W ) such that Y i c X i (and thus Y i ssd X i ) for all i = 1,..., n. Proof. First of all we, recall a result which is often referred to as comonotone improvement: for any allocation (X 1,..., X n ) (L 1 ) n of W there exists a comonotone allocation (Y 1,..., Y n ) (L 1 ) n of W such that Y i c X i for all i = 1,..., n. The proof for the case when W is supported by a finite set goes back to Landsberger and Meilijson (1994). This has been further extended to aggregate endowments W L 1 by Filipović and Svindland (2008), Dana and Meilijson (2011), and Ludkovski and Rüschendorf (2008). Finally, by ssd -monotonicity of the U i (see Lemma 2.3 (iv)) it follows that if (X 1,..., X n ) A(W ), then any comonotone improvement (Y 1,..., Y n ) of (X 1,..., X n ) is acceptable as well, i.e. (Y 1,..., Y n ) A(W ). When proving the existence of solutions to (3.4), and thus of Pareto optimal allocations, we will profit from the fact that we may restrict our attention to the set of comonotone acceptable allocations; see proofs of Theorems 3.11 and 3.12 in Appendix A. Remark 3.9. The comonotone allocations have another desirable property. Suppose that W L p L 1 for some p [1, ], and let (f i (W )) n be a comonotone allocation of W, i.e. (f i ) n CF. Then, by the 1-Lipschitz continuity of the f i, it is easily verified that (f i (W )) n (Lp ) n. Hence, any comonotone Pareto optimal allocation will posses the same integrability/boundedness properties as the aggregate endowment W. In that sense, further restricting the set of acceptable allocations by imposing additional integrability or even bound- 12

13 edness constraints in the formulation of problem (3.4) will yield the same comonotone solutions as solving the unrestricted problem. 3.3 Main Results Let s i := inf dom u i R { }, i = 1,..., n, and d i H := lim x s i u i (x) (which may be ) and d i L := lim x u i (x)( 0) where u denotes the right-hand-derivative of u. Finally, let N {1,..., n} be the set of all indices such that d i H = di L 6, and M := {1,..., n} \ N the set of all indices such that d i L < di H. Note that N = or M = is possible. Theorem (i) If s i > for all i = 1,..., n, then for any individual constraints (c 1, c 2,..., c n ) (R { }) n in A(W ) there exists a comonotone Pareto optimal allocation of W. (ii) If N 1, then for any (c 1, c 2,..., c n ) (R { }) n in A(W ) there exists a comonotone Pareto optimal allocation of W. (iii) Suppose that N 2. Given any choice of individual constraints (c 1, c 2,..., c n ) (R { }) n in A(W ), if s i = for all i N, and U j ( W ) > for all j N such that c j R, then there exists a comonotone Pareto optimal allocation of W. Moreover, in the situation of (iii), if N = n, then the Pareto optimal allocations have the property of being up to a reallocation of cash, that is if (X 1,..., X n ) is a Pareto optimal allocation of W, then also (X 1 + m 1,..., X n + m n ) is a Pareto optimal allocation whenever the numbers m i R satisfy n m i = 0 and (X 1 + m 1,..., X n + m n ) A(W ). The proof of Theorem 3.10 follows immediately from Proposition 3.4 and Theorems 3.11 and 3.12 below. In these theorems we specify a non-empty set of Negishi weights λ 1,..., λ n for which the associated optimization problem (3.4) admits a solution. Theorem Suppose that s i >, i = 1,..., n. Then for any set of individual constraints c i R { } and every set of strictly positive weights λ i > 0, i = 1,..., n, (3.4) admits a comonotone solution. 6 When d i L = d i H and s i = (i.e. dom u i = R), then the corresponding robust utility U i is cash additive in the sense that U i(x + m) = U i(x) + d i Hm for all m R and X L 1 and thus (if d i H = 1 and multiplied by 1) corresponds to a convex risk measure. 13

14 Theorem Consider the following bounds on the weights λ i, i = 1,..., n, and some δ > 0: (3.6) We consider two cases: λ i = δ d i H λ i d i L < δ < λ id i H for all i N, for all i M. (i) Suppose that N = or N = 1. Then (3.4) admits a comonotone solution for every set of individual constraints c i R { }, i = 1,..., n, and any set of weights λ i > 0, i = 1,..., n, satisfying the constraints (3.6). (ii) Suppose that N 2. If s i = for all i N, and U j ( W ) > for all j N such that c j R, then (3.4) admits a comonotone solution for every set of weights λ i > 0, i = 1,..., n, satisfying the constraints (3.6). In particular, if N = n, the solutions have the property of being up to a reallocation of cash in the sense that if (X 1,..., X n ) is a solution to (3.4) for some given weights, then also (X 1 + m 1,..., X n + m n ) is a solution to (3.4) with that weights whenever the numbers m i R satisfy n m i = 0 and (X 1 + m 1,..., X n + m n ) A(W ). We remark that when N =, the two conditions in (3.6) reduce to the last one. Note that Dana (2011) derives similar bounds on the weights λ i given in (3.6) in her setting. The proofs of Theorems 3.11 and 3.12 are provided in Appendix A. Theorem 3.12 is discussed in several examples in Section 4.2 in which we illustrate that if we drop one of the conditions on the weights stated in (3.6), we cannot in general expect the existence of solutions to (3.4) any longer. Remark If d i H = and di L = 0 for all i = 1,..., n, as for instance when the decision makers utilities u i are chosen amongst the exponential, logarithmic or power utilities, then the bounds in (3.6) are void. Hence, Theorem 3.12 (i) implies the existence of a solution to (3.4) for any set of strictly positive weights λ i > 0, i = 1,..., n. As regards the uniqueness of Pareto optimal allocations, we have the following result. To this end we recall that a function U : L 1 R { } is strictly concave if U(λX + (1 λ)y ) > λu(x) + (1 λ)u(y ) whenever λ (0, 1) and X Y. Corollary Suppose that under the conditions stated in Theorem 3.11 (and Theorem 3.12, respectively) (n 1) among the n law invariant robust utilities U i are strictly concave. Then, for 14

15 any given set of weights λ i > 0, i = 1,..., n, (and satisfying the bounds (3.6), respectively), the Pareto optimal allocation which solves the optimization problem (3.4) associated to (λ 1,..., λ n ) is unique and comonotone. Proof. For any given vector of positive weights (λ 1,..., λ n ) the set of solutions to the associated optimization problem (3.4) is convex because the U i are concave. This together with the strict concavity of (n 1) robust utilities implies that the solution to (3.4), if it exists, is unique. 4 Examples 4.1 Yaari Preferences Versus Multiplier Preferences Suppose that decision maker 1 has Yaari (1987) type preferences represented by the robust utility (4.1) U 1 (X) = 1 α α 0 q X (s) ds, X L 1, where α (0, 1) and q X (s) := inf{x : P(X x) s}, s (0, 1), is the quantile function of X. Note that U 1 is the well-known Average Value at Risk (AVaR), that is U 1 (X) = AVaR α (X) = min Q Q 1 E Q [X] = min Q Q q X (s)q dq (1 s) ds dp where Q 1 := {Q P dq dp 1 α }; see e.g. Föllmer and Schied (2004) Theorems 4.47 and As for decision maker 2, her probabilistic sophisticated variational preferences are represented by a law invariant robust utility U 2 as in (2.1) satisfying some additional properties which are listed in Proposition 4.2. Examples of such robust utilities are multiplier preferences or semi-deviation utilities with any strictly increasing utility u 2 : R R. Our case study is inspired by and extends an example in Jouini et al. (2008), Proposition 3.2. In Jouini et al. (2008) U 2 is required to be a monetary utility, i.e. cash additive (u 2 Id R ). Proposition 4.2 below shows that the functional form of the Pareto optimal allocations obtained in Jouini et al. (2008) stays the same also when we allow for a larger class of preferences for the second agent. The proof of Proposition 4.2 is essentially the same as in Jouini et al. (2008), so we omit it here. Before giving the result, we recall the definition of strict risk aversion conditional on lower-tail events. 15

16 Definition 4.1. (i) Let X L 1 and A F with P(A) > 0. The set A is a lower tail-event for X if ess inf A X < ess sup A X ess inf A c X where ess inf A X := sup{m R P(X > m A) = 1} (sup := ) and ess sup A X := inf{m R P(X m A) = 1} (inf := ). (ii) A function U : L 1 R { } is strictly risk averse conditional on lower tail-events if U(X) < U(X1 A c + E[X A]1 A ) for every X dom U and any set A which is a lower tail-event for X. Proposition 4.2. Let decision maker 1 be represented by (4.1) and decision maker 2 by a law invariant robust utility U 2 (X) = inf (E Q [u 2 (X)] + α 2 (Q)) Q Q as in (2.1) with the following additional properties dom U 2 = dom U 2 + R, U 2 is strictly monotone, i.e. X Y and P(X > Y ) > 0 implies U 2 (X) > U 2 (Y ), U 2 is strictly risk averse conditionally on lower-tail events. Given any initial endowments W i dom U i, i = 1, 2, and c 1, c 2 R { } such that A(W ), the comonotone Pareto optimal allocations of the aggregate endowment W = W 1 + W 2 exist and take the following form (4.2) (X 1, X 2 ) = ( (W l) + k, W l k) where l R { } and k = k(l) R with k U 1 (W 1 ) U 1 ( (W l) ) c 1. If U 2 is in addition strictly concave, then according to Corollary 3.14 all Pareto optimal allocations are comonotone and take the form in (4.2). 4.2 Examples Illustrating the Bounds (3.6) Since our examples will only involve two decision makers, in the following we give a version of Theorem 3.12 for two decision makers. Theorem 4.3. Suppose n = 2. We consider two cases: 16

17 (i) Suppose that d i L < di H for at least one i {1, 2}. If the weights λ 1 > 0 and λ 2 > 0 satisfy ( λ 2 d 1 ) (4.3) L λ 1 d 2, d1 H H d 2, L then for any c 1, c 2 R { } there exists a comonotone solution to (3.4). (ii) Suppose that d 1 H = d1 L, d2 H = d2 L, s 1 = s 2 = and U i ( W ) > for any i = 1, 2 such that c i R. If λ i = solution to (3.4). δ, i = 1, 2, for some δ > 0, then there exists a comonotone d i H Example 4.4. Illustration of Theorem 4.3 (i): Let U 1 (X) = E[d L X + d H X ] and U 2 (X) := E[X], X L 1, where 0 < d L < 1 < d H. Moreover, suppose that c 1 = c 2 = and that W 0. If λ 2 λ 1 < d L, consider the allocations (W + k, k) A(W ), k R +. Then λ 1 U 1 (W + k) + λ 2 U 2 ( k) = λ 1 E[d L (W + k)] λ 2 k = λ 1 E[d L W ] + (λ 1 d L λ 2 )k for k because λ 1 d L λ 2 > 0. Hence, (3.4) admits no solution. Analogously, (3.4) admits no solution in case λ 2 λ 1 > d H. However for λ 2 λ 1 comonotone allocation (0, W ). [d L, d H ] we have that (3.4) admits a solution which is the Notice that in Example 4.4 there exists a solution to (3.4) even if λ 2 λ 1 equals one of the bounds given in (4.3). Indeed, for λ 2 /λ 1 = d L any allocation (W + k, k), k 0, is optimal, whereas for λ 2 /λ 1 = d H all allocations ( k, W + k), k 0, are optimal. However, if the robust utility of one of the decision makers is strictly concave, then often (3.4) admits no solution at the interval bounds of (4.3) either. This is illustrated by the following example. Example 4.5. Illustration of Theorem 4.3 (i): Let W = 0, c 1 = c 2 = and consider two utility functions u 1 and u 2 with d i H < and di L > 0, i = 1, 2. Let U i(x) := E[u i (X)], X L 1. Suppose that u 2 (a) = d2 L a for large a > 0 (in particular u 2 does not attain d2 L ) and that u 1 ( a) = d1 H for a > 0. Then for λ 2 λ 1 sup a R = d1 H d 2 L λ 1 U 1 ( a) + λ 2 U 2 (a) = sup a R and some constant k > 0 we have λ 1 u 1 ( a) + λ 2 u 2 (a) sup λ 2 a + k =. a 0 Similar arguments show that in general we cannot expect the existence solutions to (3.4) in case λ 2 λ 1 equals the lower bound d1 L d 2 H either. 17

18 Example 4.6. Illustration of Theorem 4.3 (ii): Suppose that u 1 (x) = d 1 x and u 2 (x) = d 2 x for some d 1, d 2 > 0 and suppose that c 1 = c 2 =. If λ 2 λ 1 d1 d 2, then (λ 1 d 1 λ 2 d 2 ) 0 and considering the allocations of type (W + k, k) A(W ) for some constant k yields sup k R λ 1 U 1 (W + k) + λ 2 U 2 ( k) = λ 1 U 1 (W ) + λ 2 U 2 (0) + sup(λ 1 d 1 λ 2 d 2 )k =. k R Hence, (3.4) admits no solution. A Proofs of Theorems 3.11 and 3.12 In the proof of Theorems 3.11 and 3.12 we will apply the following facts about law invariant monetary utilities which apart from the sign are convex risk measures. We omit a proof since most arguments are standard and can for instance be found in Föllmer and Schied (2004). Lemma A.1. Let U(X) = inf (E Q [u(x)] + α(q)), X L 1, Q Q be a law invariant robust utility as (2.1). Define (A.1) so that U( ) = U(u( )). U(X) := inf (E Q [X] + α(q)), X L 1, Q Q Then, U is a proper, law invariant, ssd -monotone, upper semicontinuous, monotone, cash additive (U(X + m) = U(X) + m for all m R), and concave function. Moreover, we have that (A.2) U(X) E[X] + U(0) for all X L 1. Note that (A.2) follows from E[X] ssd X and the cash additivity of U. The next Lemma A.2 is an Arzela-Ascoli type argument which will also play a crucial role in the proof of Theorems 3.12 and It can be derived from Tychonoff s compactness theorem or by a diagonal sequence argument. For a proof see e.g. Filipović and Svindland (2008). Lemma A.2. Let f n : R R, n N, be a sequence of increasing 1-Lipschitz-continuous functions such that f n (0) [ K, K] for all n N where K 0 is a constant. Then there is a subsequence (f nk ) k N of (f n ) n N and an increasing 1-Lipschitz-continuous function f : R R such that lim k f nk (x) = f(x) for all x R. 18

19 Let CFN := {(f i ) n CF f 1(0) =... = f n (0) = 0}. Note that CF = {(f i + a i ) n (f i ) n CFN, a i R, a i = 0}. According to Proposition 3.8 there exists a solution to (3.4) for some given weights (λ 1,..., λ n ) if and only if there is a solution to (A.3) Maximize λ i U i (f i (W ) + a i ) subject to (f i ) n CFN, a i R, a i = 0, (f i (W ) + a i ) n A(W ). Proof of Theorem We will prove the existence of a solution to (A.3). Fix a set of weights (λ 1,..., λ n ) satisfying the conditions (3.6). First of all, we observe that if for some j M the right-hand-derivative u j does not attain the values dj H and/or dj L, we can always find nonnegative numbers d j H and/or d j L in the image of u j such that, for the already given set of weights (λ 1,..., λ n ), the conditions (3.6) still hold true if we replace the d j H and/or dj L by d j H and/or d j H. We assume that all d j H and/or dj L which are not attained by the corresponding u j are replaced as in the described manner, and for the sake of simplicity we keep the notation d j H and dj L. By concavity of the u i there is a constant k such that for all i = 1,..., n the affine functions R x d i L x + k and R x di H x + k both dominate u i 7. Using this, we will show that (A.4) { P := sup λ i U i (f i (W ) + a i ) (f i ) n CFN, a i R, } a i = 0, (f i (W ) + a i ) n A(W ) <, and that this supremum is realized over a bounded set of comonotone allocations where the bound is given by W. More precisely, we will prove that there exists some constant K > 0 depending on W such that (A.5) { P = sup λ i U i (f i (W ) + a i ) (f i ) n CFN, a i [ K, K], } a i = 0, (f i (W ) + a i ) n A(W ). 7 Note that if u is a concave function, then it is always dominated by x u (y)(x y) + u(y). 19

20 To this end, we define the functions U i (X) := inf Q Q i (E Q [X] + α i (Q)), X L 1, i = 1,..., n, as in Lemma A.1. Consider (f i ) n CFN and a i R such that n a i = 0 and (f i (W ) + a i ) n A(W ). Let I := {i {1,..., n} a i < 0} and J := {1,..., n} \ I. By applying monotonicity, cash additivity and finally property (A.2) of the U i (see Lemma A.1) we obtain: λ i U i (f i (W ) + a i ) = (A.6) k λ i U i (u i (f i (W ) + a i )) i I M λ i U i (d i H(f i (W ) + a i ) + k) + λ l U l (d l H(f l (W ) + a l ) + k) l N λ i + i I M λ i U i (d i Hf i (W ))+ j J M λ l U l (d l Hf l (W ))+ λ j U j (d j L f j(w )) + j J M l N ( ) ( ) min λ id i H a i + max λ jd j i I M j J M L a j + δ i I M j J M l N k λ i + E[W + ] λ i d i H + λ i U i (0)+ ( ) min λ id i H i I M i I M Suppose that N =, then we further estimate (A.7) (A.6) E[W + ] λ i d i H + k λ j U j (d j L (f j(w ) + a j ) + k)+ ( ) a i + max λ jd j j J M L a j + δ a l. j J M l N λ i + ( ) λ i U i (0) min λ id i H max λ jd j i I j J L a where a := i J a i ( 0). If N and l N a l < 0, then we estimate (A.8) (A.6) E[W + ] λ i d i H + k λ i + a l ( ) λ i U i (0) δ max λ jd j j J M L ã for ã := i J M a i ( 0) using (3.6). And similarly, if N and l N a l 0, then we estimate (A.9) (A.6) E[W + ] λ i d i H + k λ i + ( ) λ i U i (0) min λ id i H δ â i I M 20

21 for â := i J N a i ( 0). Consequently we infer that P E[W + ] λ i d i H + k λ i + λ i U i (0) <. Choose any allocation (X 1,..., X n ) A(W ). Letting Then we have that P n λ iu i (X i ) =: k. if N =, or A := min λ id i H max λ jd j,...,n j=1,...,n L {( ) ( )} A := min δ max λ jd j j M L, min λ id i H δ i M if N, we infer from (A.7), (A.8), and (A.9) that the supremum in (A.4) is realized over allocations such that a i k + E[W + ] n λ id i H + k n λ i + n λ iu i (0) =: K A for all i M, and i N a i K too. Note that A > 0 due to the conditions (3.6) on the weights λ i. In the following we argue that in case N > 1 we may also assume that the a i belonging to i N are bounded due to the insensitivity of the cash additive U i, i N, to constant re-sharings of 0 amongst themselves. To this end note that the choice of the λ i and the requirement s i = for i N implies (A.10) i N λ i U i (f i (W ) + a i + m i ) = i N λ i U i (f i (W ) + a i ) + δ i N m i = i N λ i U i (f i (W ) + a i ), whenever m i R such that i N m i = 0. Hence, adding constants m i such that i N m i = 0 to the endowments of the decision makers in N does not affect the contribution of the allocation to P. This immediately implies that we may assume a i K for all i N if c i = for all i N, because in that case we may choose m 1 = i N a i a 1 and m i = a i for all i 1 in (A.10), and the altered allocation satisfies the bound ( m 1 + a 1 = i N a i K, see above). If the set N b N of indices i N such that c i R is not empty, we also need to consider cash amounts that might be needed to make the endowment f i (W ) + a i acceptable. This is the point where the assumption U i ( W ) > for all i N b enters (whenever N > 1). Using the cash additivity (U i (Y + z) = U i (Y ) + d i H z, z R) and monotonicity of U i we obtain that f i (W ) + z is acceptable for decision maker i N b whenever z U i(w i ) U i ( W ) c i d i. H 21

22 Moreover, acceptability of f i (W ) + a i implies (again using cash additivity and monotonicity of U i ) for all i N b : a i U i(w i ) c i U i (f i (W i )) d i H U i(w i ) c i U i (0) d i H E[W + ] where we have used that U i (f i (W i )) U i (W + ) E[d i H W + ]+U i (0) according to (A.2). Therefore we know that there exists a joint constant K > 0 such that the a i, i N b, are bounded from below by K and such that the endowments f(w i ) + a i + m i stay acceptable for m i = [(a i K) 0], i N b. If N b = N, then choosing some j N b and letting m i = [(a i K) 0] for i N b \ {j} and m j = i N\{j} m i, we obtain that i N m i = 0, a i + m i K for all i N b \ {j}, and a j + m j a i + a i + m i K + N b K =: K. i N i N b \{j} Moreover, since a j a j + m j, also f(w j ) + a j + m j is acceptable, i.e. U j (f(w j ) + a j + m j ) U j (W j ) c j. If N u > 1 where N u := N \ N b, then we choose some j N u and let m i = [(a i K) 0] for all i N b and m i = a i for all i N u \ {j}, whereas m j = i N\{j} m i. Again, i N m i = 0, and a i + m i = 0 for all i N u \ {j}, a i + m i K for all i N b, and a j + m j a i + a i + m i K + N b K =: K. i N i N b Hence, (A.4) and (A.5) are proved. By virtue of (A.5) we may choose a sequence ((f p i )n ) p N CF with f p i (0) [ K, K] for all i = 1,..., n and p N such that (f p i (W ))n A(W ) for all p N and P = lim p λ i U i (f p i (W )). According to Lemma A.2 there exists a subsequence, which we for the sake of simplicity also denote by (f p i )n, which converges pointwise to some (f i) n CF. As f p i (W ) W + K for all i = 1,..., n and p N, we may apply the dominated convergence theorem which yields f i (W ) L 1, and lim p E[ f i (W ) f p i (W ) ] = 0 for all i = 1,..., n. By upper semi-continuity of the U i (Lemma 2.3) we have U i (W i ) c i lim sup U i (f p i (W )) U i(f i (W )), p 22

23 and P = lim p λ i U i (f p i (W )) λ i lim sup U i (f p i (W )) p λ i U i (f i (W )). Hence, we infer that (f i (W )) n A(W ) (since P > ) and P = λ i U i (f i (W )). For the last part of (ii) suppose that N = n, and let (X 1,..., X n ) be a solution to (3.4) for the given weights. If m i R such that n m i = 0, then the same computation as in (A.10) yields n λ iu i (X i + m i ) = n λ iu i (X i ). Proof of Theorem Recall (A.3) and let (f i ) n CFN, a i R with n a i = 0 such that (f i (W ) + a i ) n A(W ). Since in particuar U i(f i (W ) + a i ) >, we must have that f i (W ) + a i s i for all i = 1,..., n. Let K := n s i. Then ( W + K) f i (W ) + a i = W ( j i f j (W ) + a j ) W + K. Hence, we deduce that (A.5) holds with K := 2 essinf W + K. The rest of the proof now follows the lines of the proof of Theorem References Acciaio, B., Optimal risk sharing with non-monotone monetary functionals. Finance and Stochastics 11, Arrow, K. J., Uncertainty and the welfare economics of medical care. American Economic Review 5, Artzner, P., Delbaen, F., Eber, J. M., Heath, D., Coherent measures of risk. Mathematical Finance 9, Barrieu, P., El Karoui, N., Inf-convolution of risk measures and optimal risk transfer. Finance and Stochastics 9,

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