A numerical approach for a class of risk-sharing problems

Journal of Mathematical Economics 47 (2011) 1–13

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A numerical approach for a class of risk-sharing problems G. Carlier ∗ , A. Lachapelle CEREMADE, UMR CNRS 7534, Université Paris IX Dauphine, Pl. de Lattre de Tassigny, 75775 Paris Cedex 16, France

a r t i c l e

i n f o

Article history: Received 18 January 2010 Received in revised form 27 September 2010 Accepted 1 October 2010 Available online 25 November 2010

a b s t r a c t This paper deals with risk-sharing problems between many agents, each of whom having a strictly concave law invariant utility. In the special case where every agent’s utility is given by a concave integral functional of the quantile of her individual endowment, we fully characterize the optimal risk-sharing rules. When there are many agents, these rules cannot be computed analytically. We therefore give a simple convergent algorithm and illustrate it on several examples. © 2010 Elsevier B.V. All rights reserved.

Keywords: Risk-sharing Comonotonicity Sup-convolution Calculus of variations Numerical approximation

1. Introduction Risk-sharing problems have their roots in the seminal works of Arrow (1963, 1974) and Borch (1962) in insurance and have received a lot of attention since. Starting from the case of twoagents having preferences given by expected utilities, the theory has been developed in recent years in particular to incorporate more general law invariant preferences such as rank-dependent utilities or monetary risk measures in the financial literature (see Dana (2005), Jouini et al. (2008), Carlier and Dana (2006, 2008, 2010) as well as the book by Föllmer and Schied (2004) and the references therein). In the expected utility framework, optimality conditions for efficient risk-sharing rules take the form of state by state equations equalizing weighted marginal utility ratios. By differentiating these conditions, one can perform comparative statics and analyze qualitative properties in a very fine way (including convexity or concavity properties as in Chevallier and Müller (1994) and Hara et al. (2007) also see Benninga and Mayshar (2000) for similar results in an intertemporal setting). Decision theory however teaches us that the traditional expected utility theory of von Neumann and Morgenstern is unable to explain a certain number of puzzles such as Allais’ and Ellsberg’s paradoxes, this criticism gave rise to alternative and broader classes of utili-

∗ Corresponding author. E-mail addresses: [email protected] (G. Carlier), [email protected] (A. Lachapelle). 0304-4068/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2010.10.004

ties such as rank-dependent utilities that are able to take into account richer behaviors such as the distortion of probabilities. Working with these more general utilities, that are not state separable, there is of course a price to pay in terms of mathematical tractability. It is however natural from a positive point, to wonder how departing from the expected utility framework may affect properties of risk-sharing rules. Another motivation for studying how agents with somehow non-standard utilities share risk comes from the literature on risk measures in finance. Indeed, frequently used risk measures such as expected shortfalls have a structure similar to that of Choquet’s expectations. We actually believe that the content of the present paper has possible applications to capital allocation among different financial companies (or departments of the same company) subject to regulatory capital requirements. The fact that comonotonicity is a key property in efficient risk-sharing problems is well-known and has been particularly emphasized by Landsberger and Meilijson (1994) (see Carlier et al. (2010) for an extension to multivariate risks). For suitable law invariant (or quantile-based) and concave utilities, the abstract risk-sharing problem may be brought down to the maximization of some concave functional over the set of comonotone allocations. When utilities are integral functionals of the quantile (which covers the rank-dependent utility case and more generally the so-called rank-linear utility case), this enables one to rewrite the problem as a variational problem subject to a comonotonicity constraint. This reformulation was already a crucial for the analysis and the closed-form solutions obtained of Carlier and Dana (2003, 2006, 2008, 2010), and Jouini et al. (2008) in the two-agents case.

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G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13

The mathematical analysis of the many agents-case is not significantly harder than in the two-agents case. However, using in practice the optimality conditions to find by hand qualitative properties of risk-sharing rules is almost impossible as soon as there are more than two agents. This can be understood in terms of the comonotonicity constraint which in fact involves as many constraints as the number of agents. Typically, optimal risk-sharing rules exhibit ranges of aggregated risk for which some agents are fully insured by the others. This is the economic meaning of monotonicity constraints being binding. In the two-agents case, optimal solutions are combinations of three regimes: agent 1 insures agent 2, agent 2 insures agent 1 or the solution is interior and thus given by some first-order condition. If there are N agents, there are as many cases as the possibilities for the subset of fully insured agents that is 2N−1 . Also note that the subset of fully insured agents depends on the value of the aggregate risk. The difficulty of the many agents case is not in the form of the optimality conditions but in guessing who is fully insured and who is not for different values of the aggregate risk. The search for an efficient computational scheme is therefore not only natural but compulsory.1 The main contribution of this paper is to propose a quite simple and efficient numerical method to compute those risk-sharing rules. This paper is therefore normative in the sense that our algorithm may be helpful to determine how several individuals (or companies or departments of one company) should share risks if they have certain non-standard risk-preferences as perhaps dictated to them by regulation (as is indeed the case, at least for some financial companies). The paper is organized as follows. In Section 2, we reformulate a class of risk-sharing problems as tractable variational problems subject to a comonotonicity constraint. More precisely, some notations and preliminaries are given in Section 2.1, the two-agents case is then addressed in Section 2.2, we finally show a stability by sup-convolution result in Section 2.3 which enables us to reformulate the risk-sharing problem as a comonotonicity constrained variational problem. Optimality conditions for such problems are established in Section 3. Finally, a simple and easy to implement algorithm is described in Section 4 in which convergence is established and various numerical simulations are presented.

∞ × L∞ and • are monotone, i.e. V(X) ≥ V(Y) whenever (X, Y ) ∈ L+ + X ≥ Y a.s., ∞ × L∞ and • law invariant, i.e. V(X) = V(Y) whenever (X, Y ) ∈ L+ + L(X) = L(Y ), • satisfy the following Fatou property:

V (X) ≥ limsupn V (Xn ) ∞ and X converges a.s. to X. whenever (Xn )n is bounded in L+ n ∞ and d + 1 agents, i = 1, . . ., d + 1, Given an aggregate risk X0 ∈ L+ each of whom having a utility Vi in the class C, we are interested in Pareto-efficient risk-sharing rules of X0 . As usual, these Pareto allocations can be obtained by attributing a weight i to each agent and  then by maximizing the  corresponding weighted sum of utilities i i Vi (Xi ) subject to i Xi = X0 . By letting weights or Lagrange multipliers vary (rather than reservation utilities for all agents but one), we obtain a canonical parametrization of efficient allocations. Remarking that C = C for any  > 0, there is no loss of generality in taking i = 1 (but one then has to replace Vi by i Vi which amounts to adopt the convention that Pareto weights are already in the function Vi ). With these conventions in mind,2 we are simply left to solving the sup-convolution problem:



d+1



 Vi

i=1

(X0 ) := sup

 d+1 

∞

Vi (Xi ), Xi ∈ L , +

i=1

d+1 



Xi = X0

.

(2.1)

i=1

d+1

The sup-convolution  Vi is said to be exact if the previous i=1

supremum is attained (and then it is attained at a unique point by strict concavity). Our aim is to reformulate the previous risksharing problem in a more tractable way using the notion of comonotone allocations. Before we do so, we shall need some preliminaries. 2.1. Preliminaries Let X be a bounded random variable on (, F, P) and let FX (t) = P(X ≤ t), t ∈ R denote its distribution function. The generalized inverse of FX (or quantile function of X) is defined by:

2. Reformulation of a class of risk-sharing problems

FX−1 (t) = inf {z ∈ R : FX (z) > t},

Let (, F, P) be a nonatomic probability space i.e. a probability space such that there is no A ∈ F such that P(A) > 0 and P(B) ∈ {0, P(A)} for every B ∈ F with B ⊂ A. The nonatomicity of (, F, P) is well-known to be equivalent to the existence of a uniformly distributed random variable on (, F, P). Given a random variable X on (, F, P), the law of X is denoted L(X). If X and Y are two random variables X on (, F, P), we shall denote by X ∼ Y the fact that L(X) = L(Y ). In the sequel, we will only consider essentially bounded random ∞ the variables – or risks– on (, F, P). We will denote for short by L+ nonnegative cone of L∞ (, F, P). ∞ → R that: Let C be the class of utility functions V: L+

The random variable X and the quantile function FX−1 are then equimeasurable in the sense that for every continuous function ϕ one has:



for all t ∈ (0, 1).

(2.2)

1

ϕ(FX−1 (t))dt.

E(ϕ(X)) = 0

Let us now recall the well known Hardy–Littlewood inequality (see Hardy et al. (1988), Lorentz (1953) and Cambanis et al. (1976)). Proposition 2.1.



∞ then one has Let X and Y be in L+

1

FX−1 (t)FY−1 (t)dt = sup E(XZ)

E(XY ) ≤ 0

Z∼Y

• are strictly concave,

1 We would like to make an analogy with the principal-agent problem with adverse selection and a finite number of types. If there are two types, there are only two incentive-compatibility and two individual rationality constraints and economic intuition often suggests which should be binding which makes the problem solvable by hand. If there are 9 or 10 types, the reader can easily figure out that solving the problem without a computer is generally simply impossible.

2 If utilities Vi are cash invariant (which is excluded by our strict concavity assumption) and if short-selling is allowed constraints), then the (no nonnegativity  weighted maximization problem sup { i Vi (Xi ) : i Xi = X} has a finite value if and only if all weights are equal (in which case adding constants that sum to 0 does not affect the sum of the utilities). This is what happens in the case of monetary utility functions (i.e. the negative of monetary risk measures) solved by Jouini et al. (2008). Our assumptions rule out monetary utility functions, but we will also be able to treat this case numerically by a limiting argument.

G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13

and for every concave u: R → R one has



1

u(FX−1 (t) − FY−1 (t))dt.

E(u(X − Y )) ≤ 0

Definition 2.2. Let X and Y be in L∞ (, F, P), then X dominates Y in the sense of second order stochastic dominance (which we denote by X  Y) if E(u(X)) ≥ E(u(Y )) for every concave increasing function u: R → R. Various characterizations of stochastic dominance are wellknown (see Rothschild and Rothschild (1970)) among which:





t

FX−1 (s)ds ≥

XY ⇔ 0

t

FY−1 (s)ds,

∀t ∈ [0, 1],

0

which is equivalent to





1

∞ , then X  Y if and only if there is Lemma 2.3. Let X and Y be in L+ Nn n Nn n n a sequence Zn of the form Zn =  Y with ni ≥ 0, i=1 i = 1 i=1 i i and each Yin ∼Y , such that Zn converges a.s. to X.

Proof. Assume first that Zn is of the form mentioned above and Zn converges a.s. to X, then for each concave u, one has E(u(Zn )) ≥ E(u(Y )) and one concludes with Lebesgue’s dominated convergence theorem (the Zn ’s being uniformly bounded). Conversely, assume X  Y and let us prove that X ∈ K where K is ∞ the closed convex hull in L1 of the set {Z ∈ L1 , Z ∼ Y} (clearly K ⊂ L+ ∞ ). If X ∈ since Y ∈ L+ / K it follows from Hahn’s-Banach theorem that there exists P ∈ L∞ and ε > 0 such that E(PX) ≥ sup E(PZ) + ε. Z∼Y

With Hardy–Littlewood’s inequality, this yields



1

FP−1 FX−1 0

1

FP−1 FY−1 + ε

≥

Roughly speaking (X1 , . . ., Xd+1 ) is comonotone if all the Xi ’s evolve in the same direction (or coevolve) and we recall that it is well-known that it is equivalent to the fact thateach Xi can be written as a nondecreasing function of the sum i Xi (see for ∞ , the set of allocations instance Denneberg (1994)). Given X ∈ L+ ∞ that sum to X and are comonotone can then be (X1 , . . . , Xd+1 ) ∈ L+ parametrized by Xi : = xi (X) where the xi ’s are nondecreasing function that sum to the identity map (on the range of X but this can be extended to the whole line if necessary). Obviously, each map xi is 1-Lipschitz and therefore, by Ascoli’s theorem, this parametrization of comonotone allocations with a fixed sum is compact. Comonotonicity is well-known to be a key property in risksharing problems as soon as agents have preferences that are compatible with second order stochastic dominance. Indeed, an important result of Landsberger and Meilijson (1994) states that any allocation is dominated (for second order stochastic dominance) by a comonotone one.

1

g(t)FX−1 (t)dt ≤ g(t)FY−1 (t)dt, ∀g bounded nondecreasing. (2.3) 0 The next statement 0 is originally due to Ryff (1967) in the case (, F, P) is [0, 1] endowed with its Borel field and the Lebesgue measure, we believe it is well-known but give a short proof for the sake of completeness:



3

0

which contradicts X  Y because of inequality (2.3) recalled above. We thus deduce that X is the L1 (and thus also a.s. taking a subsequence if necessary) limit of a sequence Zn of the form mentioned in the statement of the lemma.  As a consequence of the previous lemma, we have the following compatibility result which is originally due to Dana (2005): Lemma 2.4. If a utility function V is in the class C then it is compatible with second order stochastic dominance that is V(X) ≥ V(Y) whenever ∞ × L∞ and X  Y. (X, Y ) ∈ L+ + Proof. Assume X  Y, using Lemma 2.3, X is an a.s. limit of a sequence Zn as in Lemma 2.3. Since V is concave and law invariant, V(Zn ) ≥ V(Y) and by the Fatou property, we get V(X) ≥ V(Y). 

2.2. The two-agents case In this paragraph, we study in details the case of two agents (d = 1). The following proposition is not new (see Carlier and Dana (2006) or Jouini et al. (2008) for similar results), to keep the exposition self-contained we give a proof by a rearrangement argument taken from Carlier et al. (2010). Proposition 2.6. Let V1 and V2 be two utilities in the class C and ∞ , there exists a unique (X ∞ × L∞ such that X ¯ 1 , X¯ 2 ) ∈ L+ ¯ 1 + X¯ 2 = X0 ∈ L+ + X0 and V1 V2 (X0 ) = V1 (X¯ 1 ) + V2 (X¯ 2 ). Moreover (X¯ 1 , X¯ 2 ) is comonotone hence there exist two nondecreasing functions x¯ 1 and x¯ 2 : [FX−1 (0), FX−1 (1)] → R such that x¯ 1 (t) + 0 0 x¯ 2 (t) = t for all t ∈ [F −1 (0), F −1 (1)] and (X¯ 1 , X¯ 2 ) = (¯x1 (X0 ), x¯ 2 (X0 )). X0

X0

Proof. Let us first suppose that X0 has no atom (i.e. FX0 is continu∞ such that ous, or equivalently FX−1 increasing). Let X1 and X2 be in L+ 0 X1 + X2 = X0 , it follows from Jensen’s inequality that E(Xi |X0 )  Xi so that in the sup-convolution, it is enough to maximize over pairs (X1 , X2 ) that are measurable functions of X0 . Let X1 = f(X0 ) and X2 : = X0 − f(X0 ) be such a pair. Since X0 is nonatomic there exists a nondecreasing map (or monotone rearrangement) f˜ such that f˜ (X0 )∼f (X0 ) with 0 ≤ f˜ (X0 ) ≤ X0 (see Ryff (1970)). Then set Y1 := f˜ (X0 ) and Y2 := X0 − f˜ (X0 ). Since Y1 ∼ X1 , V1 (Y1 ) : = V1 (X1 ), we now claim that V2 (Y2 ) ≥ V2 (X2 ) and to prove it, thanks to Lemma 2.4, it is enough to prove that Y2  X2 . Let u be a concave function, we have



1

u(FX−1 (t) − f˜ (FX−1 (t)))dt

E(u(Y2 )) = E(u(X0 − f˜ (X0 ))) =

0

0

0

but since f˜ (FX−1 ) = FX−1 , by Proposition 2.1, we get 0

1



1

u(FX−1 (t) − FX−1 (t))dt = E(u(Y2 )).

E(u(X2 )) = E(u(X0 − X1 )) ≤

0

0

1

A notion that will play an important role in the sequel is that of comonotonicity that we now recall,

All this proves that given an admissible pair (X1 , X2 ) one may find a better one of the form (f(X0 ), X0 − f(X0 )) where f is a nondecreasing function on [FX−1 , FX−1 ] → R such that 0 ≤ f(t) ≤ t for every

Definition 2.5. Let X1 and X2 be two real-valued random variables on (, F, P), then the pair (X1 , X2 ) is called comonotone if

t ∈ [FX−1 (0), FX−1 (1)] → R. In particular, one can find a sequence of 0 0 such maps fn such that

(X1 (ω ) − X1 (ω))(X2 (ω ) − X2 (ω)) ≥ 0

lim V1 (fn (X0 )) + V2 (X0 − fn (X0 )) = V1 V2 (X0 ).

0

for P ⊗ P-a.e. (ω, ω ) ∈ 2 .

A family of random variable (X1 , . . ., Xd+1 ) on (, F, P) is said to be comonotone if (Xi , Xj ) is comonotone for every (i, j) ∈ {1, . . ., d + 1}2 .

0

n

Since the functions fn are all nondecreasing and bounded by ||X0 ||∞ , it follows from Helly’s theorem that there is some (not rela-

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G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13

beled) sequence that converges pointwise to some function f, hence (fn (X0 ), X0 − fn (X0 )) is bounded in L∞ and converges a.s. to (f(X0 ), X0 − f(X0 )). The Fatou property guarantees then that V1 (f (X0 )) + V2 (X0 − f (X0 )) = V1 V2 (X0 ).

∞ be such X ≤ X , since • Monotonicity: assume X0 ≤ Y0 and let X ∈ L+ 0 V1 and V2 are monotone, we have

V1 (X0 − X) + V2 (X) ≤ V1 (Y0 − X) + V2 (X) ≤ V1 V2 (Y0 )

This proves existence of a maximizer in the sup-convolution problem (that is the sup-convolution is exact at X0 ), uniqueness follows from the strict concavity of V1 and V2 . It remains to show that (f(X0 ), X0 − f(X0 )) is comonotone i.e. X0 − f(X0 ) is nondecreasing in X0 . To prove this last claim, we apply the same trick as before: let g be nondecreasing such that 0 ≤ g(t) ≤ t and g(X0 ) ∼ X0 − f(X0 ), then X0 − g(X0 )  f(X0 ) so that V1 (X0 − g(X0 )) + V2 (g(X0 )) ≥ V1 (f (X0 )) + V2 (X0 − f (X0 )) and therefore (X0 − g(X0 ), g(X0 )) is optimal as well and by uniqueness this yields X0 − f(X0 ) = g(X0 ). The supremum is thus uniquely attained at some comonotone pair. ∞ . A theorem Let us now treat the case where X0 is arbitrary in L+ −1 of Ryff (1970) enables us to write X0 = FX (U0 ) with U0 uniformly 0

distributed. For n ∈ N∗ then define X0n := FX−1 (U0 ) + n−1 U0 , since X0n 0 is nonatomic, we deduce from the previous step that there exists, for every n ≥ 0, a pair of nondecreasing 1-Lipschitz functions x1n and x2n summing to the identity such that V1 V2 (X0n ) = V1 (x1n (X0n )) + V2 (x2n (X0n )). By Ascoli’s Theorem, we may extract some converging (and not relabeled) subsequence from (x1n , x2n ) and denote (¯x1 , x¯ 2 ) its limit. Now, using the fact that V1 and V2 are monotone and that X0n ≥ X0 , ∞ such that X ≤ X , we have, for every X ∈ L+ 0

taking the supremum with respect to X, we deduce that V1 V2 is monotone. • Law invariance property: assume that X0 ∼ Y0 . We then have by law invariance of V1 and V2 V1 V2 (X0 ) = V1 (¯x1 (Y0 )) + V2 (¯x2 (Y0 )) ≤ V1 V2 (Y0 ), reversing the role of X0 and Y0 then yields V1 V2 (X0 ) = V1 V2 (Y0 ). ∞ that • Fatou property: let (Xn )n be a bounded sequence in L+ converges a.s. to a limit X. Then for every n ≥ 1, there exist nondecreasing and 1 − Lipschitz nonnegative functions x¯ 1n , x¯ 2n such that V1 V2 (Xn ) = V1 (¯x1n (Xn )) + V2 (¯x2n (Xn )), and, by Ascoli’s Theorem, up to a not relabeled subsequence, x¯ 1n and x¯ 2n converge uniformly to some x¯ 1 and x¯ 2 so that x¯ in (Xn ) converges a.s. to x¯ i (X). We thus have limsupn V1 V2 (Xn )

≤ V1 (¯x1 (X)) + V2 (¯x2 (X))

V1 (X0 − X) + V2 (X) ≤ V1 (X0n − X) + V2 (X) ≤ V1 (x1n (X0n )) + V2 (x2n (X0n )). Since (x1n (X0n ), x2n (X0n )) is bounded and converges a.s. (and in fact even in L∞ ) to (X¯ 1 , X¯ 2 ) = (¯x1 (X0 ), x¯ 2 (X0 )), we deduce from the Fatou property that the limsup of the right-hand side is less than V1 (¯x1 (X0 )) + V2 (¯x2 (X0 )). Taking the supremum with respect to X we conclude that the comonotone pair (¯x1 (X0 ), x¯ 2 (X0 )) is the unique maximizer in the sup-convolution problem for X0 .  2.3. Stability by sup-convolution and existence for d + 1 agents

≤ limsupn V1 (¯x1n (Xn )) + limsupn V2 (¯x2n (Xn )) ≤ V1 V2 (X)

from which we deduce the Fatou property for V1 V2 .



Inductively, one immediately deduces from the previous proposition that if V1 , . . ., Vd+1 all belong to the class C then so does d+1

 Vi .

i=1

In this section, our aim is to generalize the existence and characterization result of Proposition 2.6 to the case of d + 1 agents. To do so, let us prove the following stability by sup-convolution result. Proposition 2.7. Let V1 and V2 be two utilities in the class C, then V1 V2 also belongs to C. ∞ × L∞ , then by Proposition 2.6, V V (X ) Proof. Let X0 , Y0 ∈ L+ 1 2 0 + and V1 V2 (Y0 ) are exact that is, there exist nondecreasing functions x¯ 1 , x¯ 2 , and y¯ 1 , y¯ 2 summing to the identity and such that V1 V2 (X0 ) = V1 (¯x1 (X0 )) + V2 (¯x2 (X0 )) and V1 V2 (Y0 ) = V1 (y¯ 1 (Y0 )) + V2 (y¯ 2 (Y0 )). Let us prove step by step that V1 V2 belongs to C.

• Strict concavity: let  ∈ (0, 1), and assume X0 = / Y0 , we thus have (¯x1 (X0 ), x¯ 2 (X0 )) = / (y¯ 1 (Y0 ), y¯ 2 (Y0 )), the strict concavity of V1 and V2 then yields V1 V2 (X0 + (1 − )Y0 )

≥ V1 (¯x1 (X0 ) + (1 − )y¯ 1 (Y0 ))

∞ and V , . . ., V Let X0 ∈ L+ 1 d+1 be in the class C. ∞ )d+1 such that Then there exists a unique X¯ = (X¯ 1 , . . . , X¯ d+1 ) ∈ (L+ d+1 X¯ i = X0 and

Theorem 2.8.

i=1

d+1

 Vi (X0 ) =

i=1

d+1 

Vi (X¯ i ).

i=1

Moreover X¯ is comonotone hence there exist d + 1 nondecreasing

d+1

x¯ (t) = t for all functions x¯ i : [FX−1 (0), FX−1 (1)] → R such that i=1 i 0 0 −1 −1 ¯ t ∈ [F (0), F (1)] and X = (¯x1 (X0 ), . . . , x¯ d+1 (X0 )). X0

X0

Proof. For the sake of simplicity, let us prove the Theorem for d + 1 = 3. Introducing the notation W2 = V2 V3 , the maximization problem reads as:

+V2 (¯x2 (X0 ) + (1 − )y¯ 2 (Y0 )) > [V1 (¯x1 (X0 )) + V2 (¯x2 (X0 ))]

∞ ∞ , Y2 ∈ L+ , X1 + Y2 = X0 }. sup{V1 (X1 ) + W2 (Y2 ); X1 ∈ L+

+(1 − )[V1 (y¯ 1 (Y0 )) + V2 (y¯ 2 (Y0 ))] so that V1 V2 is strictly concave.

By Proposition 2.7, we know that W2 ∈ C. Proposition 2.6 then says ∞× that there exists a uniquesolution (X¯ 1 = x¯ 1 (X0 ), Y¯ 2 = y¯ 2 (X0 )) ∈ L+

G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13 ∞ such that x ¯ 1 , y¯ 2 are nondecreasing, x¯ 1 (X0 ) + y¯ 2 (X0 ) = X0 , and L+

V1 W2 (X0 ) = V1 (¯x1 (X0 )) + W2 (y¯ 2 (X0 )). We then solve the sub-problem

By the same arguments, there exists a unique comonotone solu∞ × L∞ satisfying the same tion (X¯ 2 = z¯ 2 (y¯ 2 (X0 )), X¯ 3 = z¯ 3 (y¯ 2 (X0 )) ∈ L+ + properties as in Proposition 2.6. Finally, it is easy to see that (X¯ 1 = x¯ 1 (X0 ), X¯ 2 = x¯ 2 (X0 ), X¯ 3 = x¯ 3 (X0 )), where x¯ 2 := z¯ 2 ◦ y¯ 2 and x¯ 3 := z¯ 3 ◦ 3

y¯ 2 , is the unique solution of the sup-convolution  Vi (X0 ), with i=1

the desired properties. Inductively, one can prove the theorem for every d.  Remark 2.9. One may wonder if it is really necessary to proceed inductively to prove the previous result. On the one hand, we believe that proving stability of the class C by supremal convolution (that is by aggregation) as we did in Proposition 2.7 has its own economic interest. On the other hand, the rearrangement trick we used in the two-agents case does not generalize to more agents. Remark 2.10. In the previous statements, we have used strict concavity of the elements of C for uniqueness purpose only. Strict concavity is in fact quite restrictive since it rules out the case of monetary risk measures. We now claim that strict concavity, though convenient because it implies uniqueness, is in fact not necessary to obtain the existence of at least one optimal comonotone solution in the supremal convolution problem. To see this, it is enough to proceed by approximation and use the compactness of comonotone allocations. By the way, in the next section, we will treat numerically the case of concave and non-strictly concave utilities by using such strictly concave perturbations.

d 

sup J(x), with J(x) : =



Vi (xi (X0 )) +Vd+1

X0 −

d 

i=1



xi (X0 )

(2.4)

i=1

where A := {x ∈ W 1,∞ ([a, b], Rd ) : x˙ ∈  a.e., x(a) ∈ S}, (W 1,∞ ([a, b], Rd ) denotes the space of Lipschitz functions: [a, b] → Rd ) a :=

b :=

FX−1 (1), 0

and  and S are the two simplices:



 :=

u ∈ Rd+ ,

d  i=1

L(t, FX−1 (t))dt.



ui ≤ 1

,

S :=

 x ∈ Rd+ ,

d 

 xi ≤ a

.

i=1

In the definition of the set A above, W 1,∞ ([a, b], Rd ) stands for the set of bounded Lipschitz functions. This reformulation is more tractable than the initial one since maximization is now performed over a more concrete set of functions over an interval, namely the convex and compact (for the uniform norm) set A. As we shall see in the next paragraph, the fact that constraints are given by simplices will be very convenient to express necessary conditions. Of course this reformulation is not very helpful if we keep the previous level of generality on utility functions. This is why we shall

(2.5)

0

This class of utilities is already quite large since it contains expected utilities as well as the Rank-Dependent Utilities defined by Choquet expectations. Utilities in this class are obviously law invariant and satisfy the Fatou property as soon as L is continuous (say). RLU have been studied in details in Carlier and Dana (2008), where it is proved in particular that for VL defined by (2.5) (with a C2 function L to simplify), the following statements are equivalent: 1. VL is compatible with second order stochastic dominance and monotone, 2. ∂x L ≥ 0, ∂xx L ≤ 0 and ∂tx L ≤ 0 on [0, 1] × R 3. VL is concave, monotone and (L∞ (), L1 ()) upper semicontinuous. Now for a utility of the form (2.5), if x is a nondecreasing function and X0 is nonatomic with an increasing distribution function (say), one has





1

L(t, x(FX−1 (t)))dt =

VL (x(X0 )) =

0

0

b

L(FX0 (s), x(s))d 0 (s), a

where 0 is the law of X0 . In this framework, (2.4) takes the form of a simple variational problem with

 b  d a

It follows from the previous results that the risk-sharing problem (2.1) between d + 1 agents, each of whom has a utility in C, and for aggregate risk X0 , can be simply reformulated as

FX−1 (0), 0

1

VL (X) :=

J(x) : =

2.4. Reformulation

x∈A

now further specify the utilities and restrict them to a subclass for which they have a simple expression in terms of quantiles. From now on, we shall restrict utilities to belong to the class of RankLinear Utilities (RLU for short), such utilities are of the form



∞ ∞ sup{V2 (X2 ) + V3 (X3 ); X2 ∈ L+ , X3 ∈ L+ , X2 + X3 = y¯ 2 (X0 )}.

5

i=1

d 

−

Li (FX0 (s), xi (s)) + Ld+1 (FX0 (s), s



xi (s))

d 0 (s).

(2.6)

i=1

The next paragraphs are precisely devoted to the theoretical and numerical study of such problems (under appropriate regularity and concavity assumptions). Let us motivate this detailed study by an informal discussion of the two agents case, due to the comonotonicity constraint there are basically three kinds of subintervals of the state space [a, b]: • those on which x1 is constant: agent 1 is fully insured by agent 2, • those on which x2 is constant: agent 2 is fully insured by agent 1, • those on which no constraint is binding: the risk-sharing is then given by the familiar first-order condition equalizing the marginal utilities of the two agents. Guessing which of these three cases hold according to the values of aggregate risk is often possible by hand (and this is basically what is done in Carlier and Dana (2008, 2010)). Now, imagine that there are three agents, according to the subset of agents that are fully insured, we find 7 different cases. More generally, In the d + 1agents case, there are 2(d+1) − 1 different cases. The complexity explodes with the number of agents which makes the obtention of explicit or almost explicit solutions totally hopeless. This clearly is the bad news side. On the good news side is the fact that a careful inspection of the optimality conditions lead to a simple and efficient numerical algorithm as we shall explain in the next paragraphs.

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G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13

simplex and is therefore explicit. In particular, for x ∈ A and p the associated adjoint variable:

3. Optimality conditions In this section, our aim is to give necessary and sufficient conditions for the problem

 sup J(x) with J(x) := x∈A

∇ x F(s, x(s))ds,

(3.5)

t

F(t, x(t))dt

(3.1)

where we assume that F ∈ C([a, b] × Rd , R) is such that F(t, .) is strictly concave and differentiable for every t ∈ [a, b] and ∇ x F is continuous in its two arguments. Under these assumptions, problem (3.1) admits a unique solution x¯ that is characterized as follows: Theorem 3.1. Problem (3.1) admits a unique solution x¯ that is characterized by the following: • ]¯x ∈ A, • for a.e. t, x¯˙ i (t) = 0 for every i ∈ {1, . . ., d} with i ∈ / I(t) where

this characterizes the set of solutions of the linear programming problem sup J (x) · y.

In particular, a solution of (3.6) is given explicitly by:



u(s)ds, t ∈ [a, b],

(3.7)

a

with

ui (t) =

and p¯ is the adjoint variable:

t

y(t) = y(a) +

 

i=1,...,d

(3.6)

y∈A

yi (a) =

I(t) := {j ∈ {1, . . . , d} : p¯ j (t) = min p¯ i (t)}

a #Ip (a) 0 1 #Ip (t) 0

if i ∈ Ip (a)

(3.8)

otherwise, if i ∈ Ip (t)

(3.9)

otherwise.

and

b

∇ x F(s, x¯ (s))ds.

¯ p(t) =−

b

p(t) = −

b

a





(3.2)

Ip (t) := {j ∈ {1, . . . , d} : pj (t) = min pi (t)}. i=1,...,d

t

For every x ∈ A, set

• ]¯xi (a) = 0, ∀i ∈ {1, . . . , d} with i ∈ / I(a). Proof. Existence and uniqueness follow from standard arguments. Let us denote by x¯ the solution of (3.1), by concavity of J and convexity of A, x¯ is characterized by the variational inequalities



b



∇ x F(t, x¯ (t)) · (y(t) − x¯ (t))dt ≤ 0,

J (¯x) · (y − x¯ ) =

∀y ∈ A. (3.3)

a

Now for y ∈ A, defining p¯ by (3.2) and integrating by parts yields

 J (¯x) · y = −

b

¯ p¯ · y˙ − p(a) · y(a) a

so that (3.3) becomes





b

¯ p¯ · x¯˙ + p(a) · x¯ (a) = inf

b

 ¯ p¯ · y˙ + p(a) · y(a)

∈ argmax ∈ [0,1] J(x + (y − x))}. The set G(x) defines good successors of x: indeed, in the definition of G(x), one first chooses an admissible direction y − x that maximizes the first-order utility variation J (x) · (y − x) (this maximization can be achieved by the closed-form formulas above) and then the point that maximizes J on the segment [x, y] (this is a one-dimensional search that is very cheap in practice). The iterative computation scheme used in the next section will simply consist in generating a sequence of consumptions xk such that xk+1 ∈ G(xk ). By definition of G one has J(z) ≥ J(x) for every x ∈ A and z ∈ G(x), in particular the sequence J(xk ) is monotone nondecreasing. As for convergence, we shall need the following elementary result:

(3.4)

Lemma 3.3. The set valued map x ∈ A → G(x) has a closed graph and x¯ , the solution of (3.1), is characterized by the condition x¯ ∈ G(¯x).

now the rightmost member of the previous identity is easy to compute by pointwise minimization over the simplex and is achieved exactly at those y’s in A such that for a.e. t ∈ [a, b], y˙ i = 0 as soon as / I(a).  i∈ / I(t) and yi (a) = 0 for i ∈

Proof. Let (xn )n ∈ A converge to some x and zn = xn + n (yn − xn ) converge to some z and let us prove that z ∈ G(x). Taking subsequences if necessary, we may assume that n converges to some  ∈ [0, 1] and yn converges uniformly to some y ∈ A so that z = x + (y − x). By the very definition of zn ∈ G(xn ) we have

a

y∈A

,

G(x) : = {x + (y − x), y ∈ argmaxh ∈ A J (x) · h, 

a

Remark 3.2. Let us remark that by definition of the adjoint state: ¯˙ p(t) = ∇ x F(t, x¯ (t)) which is a (state-dependent) vector of marginal utilities. We should also remark that the comonotonicity constraints are expressed in terms of bounds on the derivative of consumption with respect to state t. As usual in control theory, the adjoint has to be interpreted as a vector of multipliers associated with these constraints. Since the constraints involve derivatives of consumption, it is not the adjoint p¯ that has to be interpreted as marginal utilities (or shadow prices) but its derivative. In the algorithm of the next section, we shall take full advantage of the fact that (3.4) amounts to linear programming over the

J (xn ) · (yn − h) ≥ 0, ∀h ∈ A, J(zn ) ≥ J(x + (yn − x)),

∀ ∈ [0, 1].

Using the fact that J is of class C1 enables us to pass to the limit, which gives y ∈ argmaxh ∈ A J (x) · h and { ∈ argmax ∈ [0,1] J(x + (y − x))} i.e. z ∈ G(x). It is obvious that x¯ , the solution of (3.1), is such that x¯ ∈ G(¯x). Now if x ∈ G(x) then one has either  = 0 or x = y for some  ∈ [0, 1] and some y ∈ A as in the definition of G(x). If x = y then J (x) · h is maximized over A for h = x which is exactly the variational inequality characterizing the solution of (3.1). If  = 0 and y = / x then J(x + (y − x)) is maximized over [0, 1] for  = 0 so that

G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13

7

Fig. 1. Numerical convergence for a 20 agents risk-sharing problem. (a) Value of (3.1) at each step of the procedure. (b) Convergence of the approximated solution: ||¯xi+1 − x¯ i ||∞ , represented with a loglog-scale.

J (x) · (y − x) ≤ 0 which again means that J (x) · h is maximized over A for h = x. 

4. Algorithm and numerical results 4.1. Algorithm and convergence

to handle relies again on the fact that linear problems over the simplex are very simple. Our algorithm is defined as follows: • start from x0 ∈ A and define p0 as the corresponding adjoint:



b

∇ x F(s, x0 (s))ds

p0 (t) = − t

The algorithm we propose to solve (3.1) is a simple optimal step gradient ascent-like method. The fact that the constraints are easy

• given (xk , pk ) define yk by

 k

k

t

uk (s)ds, t ∈ [a, b],

y (t) = y (a) + a

with

 yik (a)

=

and



ui (t) =

a #Ipk (t) 0

1 #Ipk (t) 0

if i ∈ Ipk (a) otherwise

if i ∈ Ipk (t) otherwise.

Then set Fig. 2. Risk sharing for 5 agents.

Fig. 3. Risk sharing for 8 agents.

xk+1 := xk + k (yk − xk )

Fig. 4. Risk sharing for 20 agents.

(4.1)

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G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13

Fig. 5. Plots of the solutions for three averages of a Gaussian distributions of the total risk X0 . (a) Gaussian density with = 0.1. (b) Optimal risk sharing for = 0.1. (c) Gaussian density with = 0.5. (d) Optimal risk sharing for = 0.5. (e) Gaussian density with = 0.9. (f) Optimal risk sharing for = 0.9.

Theorem 4.2. The sequence xk generated by the previous algorithm converges to x¯ , the solution of (3.1) as k → ∞.

where k = argmax ∈ [0,1] J(xk + (yk − xk )) which can be computed by simple dichotomy method. Finally let pk+1 be the adjoint state associated to xk+1 :

 k+1

p

b

∇ x F(s, xk+1 (s))ds.

(t) = − t

Remark 4.1. The economic intuition behind the previous scheme is as follows. Each step can be understood as the requirement that agents for which integrated marginal utility is maximal are those for which the increase of consumption is set to its maximum (i.e. those for which the comonotonicity constraint becomes binding). The convergence of this simple ascent algorithm is then given by:

Proof. Since A is compact (for the uniform topology) and the sequence (xk )k has values in A, it is enough to show that x¯ is the unique cluster point of (xk )k . Let x be such a cluster point, and let xkn be a subsequence converging to x, taking a subsequence if necessary we may also assume that xkn +1 converges to some z. Since G has a closed graph, z ∈ G(x) i.e. z has of the form z = x + (y − x) with y maximizing J (x) · y over A and  maximizing J(x + (y − x)) over [0, 1]. By construction J(xk ) is nondecreasing, consequently one has J(x) = J(z). Now, if x = / x¯ , then by Lemma 3.3, one has x ∈ / G(x) so that  > 0 and J (x) · (y − x) > 0 which implies that J(z) = J(x + (y − x)) > J(x), giving the desired contradiction. This proves that x¯ is the only cluster point of (xk )k and thus that the whole sequence (xk )k converges to x¯ .  Remark 4.3. As already mentioned, it may seem very restrictive to assume strict concavity since it excludes monetary utility functions that are relevant in finance. In fact, as we already explained,

G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13

9

Fig. 6. Plots of the solutions for three different distributions of the total risk X0 . (a) Uniform density. (b) Optimal risk sharing for the uniform density. (c) Exponential density with  = 4. (d) Optimal risk sharing for the exponential density. (e) Quadratic density. (f) Optimal risk sharing for the quadratic density.

strict concavity is by no means essential for existence of an optimal comonotone risk-sharing rule. It is however a crucial assumption for convergence of our algorithm. To deal with problems of the form (3.1) with a concave but not strictly concave J it is enough to consider the perturbed problem sup Jε (x) with Jε (x) := J(x) + ε (x)

tion x¯ ε of the perturbed problem (4.2). Now, it is easy to see that under mild assumptions on the perturbation, x¯ ε converges as ε → 0+ to the unique maximizer of among the maximizers of J on A. This perturbation scheme thus naturally selects a particular optimal risk-sharing rule that can be approximated by the algorithm described above.

(4.2)

x∈A

where ε > 0 is a small perturbation parameter and is a nice (by this, we mean a smooth integral functional) strictly concave perturbation. Then, our algorithm enables us to compute the solu-

Remark 4.4. In the two-agents case, our algorithm enables us to compute the solution of unconstrained risk-sharing problems written as the maximization of a weighted sum of the utilities of

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G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13

Fig. 7. Plots of the risk sharing rules for various Pareto weights. (a)  = 0.45. (b)  = 0.48. (c)  = 0.5. (d)  = 0.55.

the two agents: sup U1 (X1 ) + (1 − )U2 (X2 ) : X1 + X2 = X0 ,

∞ X1 ∈ L+ ,

∞ X2 ∈ L+ .

(4.3) It thus requires to be given the Pareto weights (, 1 − ) in which (1 − )/ may be seen as a Lagrange multiplier for the constrained problem ∞ ∞ sup U1 (X1 ) : U2 (X2 ) ≥ U2 (X20 ), X1 + X2 = X0 , X1 ∈ L+ , X2 ∈ L+

(4.4) Xi0

X10

where is the initial endowment of agent i and X0 = + X20 . Our algorithm computes the solution of (4.3) for given Pareto weights, it may however seem more natural to solve the efficient risk-sharing problem in the form (4.4) (for instance to discuss the influence of initial endowment on the risk-sharing rule). Under mild concavity and strict monotonicity assumptions, there is a one to one correspondence between (4.3) parameterized by  and (4.4) parameterized by u¯ 2 := U2 (X20 ). Therefore if one is interested in the set of all efficient risk-sharing rules, there is no loss of generality in focusing on the form (4.3). More is true, denoting by X = (X1, , X2, ) the solution of (4.3), U2 (X2, ) is decreasing in , so that in

terms of comparative statics studying the qualitative impact of an increase in u¯ 2 is the same as studying the impact of a decrease in  (see example 5 in the next paragraph for a numerical illustration). Finally, if one wants to be more precise, let us remark that X solves (4.4) if and only if U2 (X2, ) = u¯ 2 .

(4.5)

We claim that our algorithm (which is very fast to compute X for given ) can also be used to solve (4.4) for a given value of u¯ 2 , indeed solving (4.4) amounts to find the value of  that solves (4.5) and this can be done by a simple dichotomy method (see example 6 for a numerical illustration). 4.2. Numerical experiments In this final section, we implement the algorithm introduced in Section 4.1 and illustrate it on various examples. The numerical results that we obtain confirm the theoretical convergence established in Theorem 4.2 and show that our algorithm converges really fast on every example we tested. More precisely, we first study three particular cases for a uniform distribution of the total risk X0 on [0, 1]. In a fourth example, we compare the solutions for different distributions of X0 . In the

G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13

11

Fig. 8. Approximation of the solution of the problem for a linear utility. (a) Density of X0 . (b) ε = 1. (c) ε = 0.05. (d) ε = 10−5 .

fifth case, we compare the risk-sharing rule between two agents for different values of  (which, as explained in Remark 4.4 is equivalent to compare it for different values of the utility of the second agent). In the sixth example, we solve a risk sharing problem in the form (4.4) by computing the corresponding value of  by dichotomy as explained in Remark 4.4. The last test aims at showing that our numerical procedure is robust for approaching solutions of risk sharing problems with concave but not strictly concave utilities. But before doing so, we would like to mention that the algorithm gives very good results, in particular very quick convergence. Indeed, it is shown in Fig. 1(a) how fast the convergence of the value is. We can also see in Fig. 1(b) the convergence speed of the difference between two successive iterations. In all our tests, the algorithm numerically converges in very few iterations (between 5 and 15). In the following examples, we always take a = 0 and b = 1 (which is in fact just a matter of normalization). Example 1: RDU agents. In the reformulation (2.6), we take functions of the form

Li (t, x) = (1 − t)˛i (1 + x)ˇi ,

for i = 1, . . ., d + 1 which corresponds to RDU with power distortion and power utility indices.3 It is easy to see that for ˛i ≥ 0 and 0 < ˇi < 1, such functions lead to utilities that belong to the class C. In this first numerical experiment, we look at the case d + 1 = 5 and we choose ˛ : = (2.8, 1.6, 2.2, 1.05, 1.3) and ˇ : = (0.58, 0.52, 0.59, 0.53, 0.57). Fig. 2 shows the optimal risk-sharing for these data and represents the cumulative sharing in the sense that the size of the risk supported by an agent is the difference between her labeled line (the lines are labeled on the right hand side of the picture) and the nearest sub-line. Looking at Fig. 2, one can for instance notice that Agent 4 fully insures the others for any risk with value smaller than 0.34. Then, for risk values smaller than about 0.7, Agents 4 and 5 insure the others and finally, for large values of the risk, every agent bears a part of the risk.

3 The interpretation of the parameters is the following. The concavity of the utility index is measured by the exponent ˇ (the closer to 1 the less risk averse the agent in some sense). Exponent ˛ captures the probability distortion effect (˛ = 0 corresponds to an expected utility maximizer), the higher the ˛ the more the agent underweights the best outcomes and overweights the worst ones.

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G. Carlier, A. Lachapelle / Journal of Mathematical Economics 47 (2011) 1–13

Fig. 9. Distance of the approximations to the numerical limit (loglog-scale), the red line is obtained by the standard least square method.

Example 2: logarithmic RLU. In this example, we make a different choice for the functions Li . We take two logarithmic functions h1 (t, x) : = log (t1/2 + x + 1) and h2 (t, x) : = log (t2/3 + x + 1). Note that this RLU example does not correspond to RDU agents. We look at the case of 8 agents, five of them being of type Li = h1 (i = 3, 4, 5, 6, 7) and three of them of type Li = h2 (i = 1, 2, 8). The graph of the optimal risk-sharing is represented in Fig. 3. The continuous lines correspond to agents of type h1 and the discontinuous ones to those of type h2 . We can see that for risk values between 0 and 0.3, the three agents of type h2 fully insure the others. Numerically, agents of the same type support the same proportion of risk which agrees with the theory. Example 3: mixing RLU and RDU agents. We now turn to a case involving more agents than in the previous ones: d + 1 = 20. The functions Li are chosen between Li (t, x) = (1 − t)˛i (1 + x)ˇi , for some ˛i and ˇi satisfying the same assumptions as before, and Li (t, x) = log (t i + x + 1), i ∈ (0, 1). The graph of the solution is represented in Fig. 4. Example 4: different densities for X0 . We now aim to see the impact of the law of the aggregate risk X0 on the optimal individual risks. To do so, we take d + 1 = 4 and functions Li (t, x) = (1 − t)˛i (1 + x)ˇi , with the particular choice ˛ = (1.16, 1.89, 1.21, 2.92) and ˇ = (0.52, 0.58, 0.54, 0.59). We start with a comparison for three truncated and normalized gaussians with variance 2 = 0.01. More precisely, we plot in Fig. 5 the graphs of the density for three different values of the average = 0.1, 0.5, 0.9, and the corresponding solutions. Let us note for instance that Agent 1 fully insures the others for values lower than 0.57 when the average is small (0.1). This phenomenon disappears for larger values of . We can also remark that Agent 4 supports a significant proportion of risk only for high values of X0 and . We have drawn in Fig. 6 the graphs of three other densities with the corresponding optimal risk-sharing. The densities we study are, for t ∈ [0, 1]: • ](a) − (b): fX (t) = 1 (Uniform density), 0 • ](c) − (d): fX (t) = e−t + e− (Exponential-like density), 0 • ](e) − (f): fX (t) = t 2 + 2 (Quadratic density, say). 0 3 Example 5: Impact of varying the Pareto weights. In this case we show how the Pareto weights (or Lagrange multipliers) can affect the risk sharing rules in a two-agents problem, with uniformly distributed total risk X0 . As expalined in Remark 4.4,

increasing  amounts to increase the utility of the second agent. We consider utilities of the form L1 = h1 and L2 = (1 − )h2 , where h1 , h2 are defined in Example 2 of the present section. In Fig. 7 we can observe that the higher is the value of , the higher the risk shared by agent 1. Example 6: Computing the Pareto weights. Here we consider a two-agents risk sharing problem, having utilities Li (t, x) = (1 − t)˛i (1 + x)ˇi , with ˛ = (1/3, 1/4) and ˇ = (2/3, 3/4). The total risk is uniformly distributed on [0, 1] and the initial endowment is x0 (t) = (x10 (t), x20 (t)) = (t/3, 2t/3). Then the initial utility  of agent 2 is L2 (t, x20 (t))dt = u¯ 2 = 0.9736. In order to compute the Pareto weights, we solve the sup-convolution problem and we find by a dichotomy method between L1 and (1 − )L2  L2 (t, x¯ 2 (t))dt = u¯ 2 . We found that the weight  such that for the considered initial endowment, the Pareto weights are (0.5445,0.4555). Example 7: non-stricly concave utilities. To close this series of numerical tests, we aim to show that our algorithm is actually able to treat concave but not strictly concave utilities. This is an important extension which is motivated by Choquet expectations and monetary risk measures such as Expected Shortfalls. The idea is to make a small strictly concave perturbation of the utilities as explained in Remark 4.3. In this example, we look at a case with 5 agents and a gaussian distribution of the aggregate risk X0 (its density is represented in Fig. 8(a)). We then take perturbed utilities of the form Liε (t, x) = (1 + εxˇi + (1 − ε)x)(1 − t)˛i , with 0 < ˇi < 1 and ˛i > 0. We then let ε → 0 to recover the (linear) Choquet expectation. We show the numerical results for ε = 1 (Fig. 8(b)), ε = 0.05 (Fig. 8(c)) and ε = 10−5 (Fig. 8(d)). We have finally computed the uniform distance between the approximations and the numerical limit. If x¯ ε denotes the numerical solution for a fixed ε, we define this quantity by ||¯x10−5 − x¯ ε ||∞ , and we give the results in Fig. 9 for some values of ε (logarithmicscale). We therefore observe a numerical convergence. Acknowledgments The authors wish to express their gratitude to Rose-Anne Dana for interesting discussions and helpful suggestions. References Arrow, K.J., 1963. Uncertainty and the welfare of medical care. American Economic Review 53, 941–973. Arrow, K.J., 1974. Optimal insurance and generalized deductibles. Scandinavian Actuarial Journal, 1–42. Benninga, S., Mayshar, J., 2000. Hetereogeneity and option pricing. Review of Derivatives Research 4, 727. Borch, K., 1962. Equilibrium in a reinsurance market. Econometrica 30, 424– 444. Cambanis, S., Simons, G., Stout, W., 1976. Inequalities for Ek(X, Y) when the marginals are fixed, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36, 285–294. Carlier, G., Dana, R.-A., 2003. Core of convex distortions of a probability. Journal of Economic Theory 113, 199–222. Carlier, G., Dana, R.-A., 2006. Law invariant concave utility functions and optimization problems with monotonicity and comonotonicity constraints. Statistics and Decisions 24, 127–152. Carlier, G., Dana, R.-A., 2008. Two-persons efficient risk-sharing and equilibria for concave law-invariant utilities. Economic Theory 36, 189–223. Carlier, G., Dana, R.-A., 2010. Optimal demand for contingent claims when agents have law invariant utilities. Mathematical Finance, doi:10.1111/j.14679965.2010.00431.x. Carlier, G., Dana, R.-A., Galichon, A., 2010. Pareto efficiency for the concave order and multivariate comonotonicity, preprint available on arxiv. Chevallier, E., Müller, H., 1994. Risk allocation in capital markets: portfolio insurance, tactical asset allocation and collar strategies. ASTIN Bulletin 24 (1), 518. Dana, R.-A., 2005. A representation result for concave Schur concave functions. Mathematical Finance 15 (4), 613–634.

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