Maxime Thieffry

Alexandre Kruszewski

Thierry-Marie Guerra

Christian Duriez

Reduced Order Control of Soft Robots with Guaranteed Stability

This work offers the ability to design a closedloop strategy to control the dynamics of soft robots. A numerical model of a robot is obtained using the Finite Element Method, which leads to work with large-scale systems that are difficult to control. The main contribution is a reduced order model-based control law, that consists in two main features: a reduced state feedback tunes the performance while a Lyapunov function guarantees the stability of the large-scale closed-loop systems. The method is generic and usable for any soft robot, as long as a FEM model is obtained. Simulation results show that we can control and reduce the settling time of the soft robot and make it converge faster without oscillations to a desired position.

I. INTRODUCTION

Soft robots -robots made of deformable materialspromise disruptive advances in many areas and bring transversal challenges, among which are dynamical modelling and control, see [START_REF] Rus | Design, fabrication and control of soft robots[END_REF], [START_REF] Majidi | Soft robotics: A perspective-current trends and prospects for the future[END_REF] and [START_REF] Kim | Soft robotics: A bioinspired evolution in robotics[END_REF]. Being lighter and more compliant than rigid ones, vibrations issues arise when dealing with soft robots dynamics. We propose a feedback control design to handle these issues. Designing such a feedback law brings different challenges such as dynamical modelling, large scale control and stability preservation.

Modelling soft robots analytically is hard to achieve as this new type of robot has a theoretical infinite number of degrees of freedoms. Several approaches has been proposed so far: piece-wise constant curvature (PCC), Cosserat rod theory and finite element method (FEM). Depending on the robot geometry, the constant curvature is not always valid. Moreover, the equations coming from the Cosserat model are not often suitable for controller design. Recent work has been done to deal with these issues in [START_REF] Renda | Discrete Cosserat approach for multi-section soft robots dynamics[END_REF] and [START_REF] Falkenhahn | Dynamic modeling of bellows-actuated continuum robots using the Euler-Lagrange formalism[END_REF] to study continuum manipulators or beam-like soft robots. The goal of the present method is to be as generic as possible concerning the geometry of the robot, therefore, the finite element method is used.

This spatial discretization gives rise to large scale systems, which are abounding in many fields of research, such as control theory. Standard control theory tools, like Linear Matrix Inequalities (LMI) or Lyapunov equation solvers, can not deal with a too large number of decision variables. Numerical efficiency for control applications is also an active field of research, see [START_REF] Benner | Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey[END_REF], [START_REF] Koo | Decentralized fuzzy observerbased output-feedback control for nonlinear large-scale systems: an LMI approach[END_REF] or [START_REF] Chang | H∞ fuzzy control synthesis for a largescale system with a reduced number of LMIs[END_REF].

If classical tools of automatic control have to be applied, model order reduction must be considered. The methods used in this work are rather standard and we refer the reader to [START_REF] Benner | Model Reduction and Approximation : Theory and Algorithms[END_REF] for more details.

Recent work has been done on soft robots control: [START_REF] Marchese | Dynamics and trajectory optimization for a soft spatial fluidic elastomer manipulator[END_REF] provides an open-loop stategy for dynamics and trajectory optimization, [START_REF] Lismonde | Trajectory planning of soft link robots with improved intrinsic safety[END_REF] provides an open-loop trajectory planning approach based on FEM model and [START_REF] Zhang | Kinematic modeling and observer based control of soft robot using real-time finite element method[END_REF] provides a closedloop controller based on FEM model but restricted to kinematics. Recently, space reduction has also been used to study continuum manipulators in [START_REF] Sadati | Control space reduction and real-time accurate modeling of continuum manipulators using Ritz and Ritz-Galerkin methods[END_REF].

This paper aims at providing a closed-loop controller that fixes the dynamics of a reduced order system while guaranteeing the stability of the full order model using Lyapunov framework. The remainder of this document is organised as follows. Section II presents a dynamic model of the robot while the main contribution of this work is presented in section III and section IV presents tracks to improve this result. Simulation results are provided along with the theoretical results to illustrate and show the effectiveness of the methodology.

II. SOFT ROBOT MODELLING

A. FEM model

Modelling soft robots relies on both continuum mechanics theory and numerical approaches to solve the underlying equations. Here, a corotated FEM allows us to define position and velocity vectors, respectively q(t) ∈ R n and v(t) ∈ R n whose dimension n is proportional to the size of the FEM mesh used to model the robot. The more nodes the mesh has, the more the model tends to be accurate and, for soft robots application, the size N of the FEM mesh goes from hundreds to thousands of variables. The dimension n of the previous vectors is made of 3 × N variables, as the position and velocity vectors are given in the 3 dimension of space.

Using the SOFA framework, the method proposed in [START_REF] Coevoet | Software toolkit for modeling, simulation, and control of soft robots[END_REF] describes concretely how to use FEM to model soft robots, in the particular context of real-time simulation.

The non-linear equation of motion of the robots is given by the second law of Newton:

M (q) v = P (q) -F (q, v) + H T (q)u(t) (1)
where M (q) is the mass matrix, H T (q)u is the actuators contribution : H T (q) contains the direction of the actuators forces and u their amplitude. The matrix F (q, v) represents the internal forces and P (q) gathers all the known external forces. As we consider only the gravity field, P (q) is constant and P (q) = P . Let q 0 ∈ R n be a stable equilibrium point induced by P and u(t) = u 0 , i.e. q 0 is solution to

0 = P -F (q 0 , 0) + H T (q 0 )u 0 (2)
Equation ( 1) can also be written as:

M (q) v = P -F (q, v) + H T (q)u(t) -P + F (q0, 0) -H T (q0)u0 ⇔ M (q) v = F (q0, 0) -F (q, v) + H T (q)u(t) -H T (q0)u0 (3)
We can approximate the internal forces F with a first order Taylor expansion around this equilibrium point:

F (q, v) ≈f (q0, 0) + ∂F (q, v) ∂q q=q 0 (q -q0) + ∂F (q, v) ∂v v=0 v (4)
where ∂F (q,v) ∂q = K(q, v) is the compliance matrix, and ∂F (q,v) ∂v = D(q, v) is the damping matrix. By definition, mass, compliance and damping matrices are positive definite. With these notations, equation (3) becomes:

M (q) v ≈ -K(q0, 0)(q-q0)-D(q0, 0)v+H T (q)u(t)-H T (q0)u0 (5)
Let d be the displacement vector defined by:

d = q -q 0 (6)
The equation of motion around an equilibrium point q 0 is thus given by the following relation:

M (q) v ≈ -K(q 0 , 0)d-D(q 0 , 0)v +H T (q)u(t)-H T (q 0 )u 0 (7)

B. State-space Equation

Without loss of generality, considering u 0 = 0 allows us to define the following non-linear state-space equation:

         ẋ = -M (x) -1 D(x) -M (x) -1 K(x) I 0 A(x) x + M (x) -1 H T (x) 0 B(x) u y = Cx (8)
where x = v d , x ∈ R 2n and where system matrices are large-scale sparse non-linear matrices, i.e.

A(x) ∈ R 2n×2n , B(x) ∈ R 2n×m , C ∈ R p×2n
, where m is the number of actuators and p the number of outputs.

The results showed in this paper are obtained on simulation experiments, where the non-linear model is used to simulate the robot. For control application, it is more direct to work on a linear model to design the control law and around the equilibrium point q 0 , system (8) can be approximated by the following linear representation:

   ẋ = -M -1 D -M -1 K I 0 x + M -1 H T 0 u y = Cx (9)
where M = M (q 0 , 0), D = D(q 0 , 0) and K = K(q 0 , 0).

Remark 1:

The FEM model of the robot allows us to compute its energy in real-time. The kinetic energy of a soft robot is defined as:

E k (x) = 1 2 v T M v (10)
and its potential energy:

E p (x) = 1 2 d T Kd (11)
The total energy of the robot is then:

E(x) = 1 2 v d T M 0 0 K v d (12)
As M and K are positive definite, i.e. M > 0 and K > 0, this energy function is positive definite:

E(x) > 0 (13)
and

E(x) = 0 ⇔ (q, v) = (0, 0) (14)
III. LARGE SCALE CONTROL DESIGN The general objective of this paper is to compute a state feedback control law

u = Lx (15)
that guarantees the performance of the system (9) in closedloop. Yet, using pole placement or LMI approaches, the dimension of x implies that the computation of the matrix L cannot be done with numerical efficiency. Moreover, this controller could guarantee the closed-loop stability of the large scale model, but to be usable it would require the measurement of the whole state, which is of course impossible in practice due to the large number of variables to measure. The objective of this work is also to reduce the number of parameters in the controller. The use of Lyapunov stability and LMI constraints to study the stability and design the controller permits to optimize the performance of the system.

In many cases, energy functions can be used as Lyapunov functions. Without actuation the studied soft robot converges to a natural equilibrium point where its energy is zero, this energy function is also a Lyapunov function for the system in open-loop. This allows us to design a Lyapunov function based on the system matrices and limits the complexity in the choice of the Lyapunov function. However, this does not reduce the complexity of the controller design, as the matrix L still contains a lot of variables to tune.

The contribution of this paper is a method to deal with this issue: relying on model reduction techniques, a reduced order state feedback control law

u = L r ξ r (16)
is computed and a proof of stability for the original large scale system is given, using large-scale Lyapunov functions.

A. Model Order Reduction

Before developping the control part, this subsection presents notion of model order reduction that are required to develop the main results. There are two main branches of model order reduction methods: the first one based on optimisation [START_REF] Vuillemin | Poles residues descent algorithm for optimal frequency-limited H 2 model approximation[END_REF] and the other based on projection [START_REF] Gallivan | Sylvester equations and projection-based model reduction[END_REF]. As the implementation of optimisation based model reduction is still challenging for very large scale systems, only projection methods are used here . These methods aim at computing two projectors W r ∈ R n×r and V r ∈ R n×r , with W T r V r = I r , to approximate a large scale system

ẋ = f (x, u) , x ∈ R n (17)
with a reduced order one:

ξr = W T r f (V r ξ r , u), ξ r ∈ R r , r n (18)
where

ξ r = W T r x (19)
is the reduced order state and ξ r is the neglected one:

ξ r = W T r x, ξ r ∈ R r , r = n -r (20)
such that:

x = V r ξ r + V r ξ r (21)
For soft robotics applications, to compute an approximation of the full-order state x = v d , it is interesting to use structure preserving model order reduction. The reduced and neglected states will also keep their initial structure:

ξ r = ξ rv ξ rd ; ξ r = ξ rv ξ rd ; (22)
Concretely, this requires to find projectors for the velocity and the displacement vectors such that equation (21) writes:

v d = V rv 0 V rv 0 0 V rd 0 V rd     ξ rv ξ rd ξ rv ξ rd     ⇔ x = VΞ (23)
where Ξ ∈ R n is the projected state. To find the projectors V and W, state-of-the-art model reduction methods are available, such as Improved Balanced Truncation [START_REF] Benner | An improved numerical method for balanced truncation for symmetric second-order systems[END_REF], Iterative Interpolation Methods [START_REF] Gugercin | An iterative SVD-krylov based method for model reduction of large-scale dynamical systems[END_REF] and Proper Orthogonal Decomposition (POD). The first two methods are for now restricted to the linear case whereas the POD is a projection based method adapted to non-linear systems.

B. Reduced Order Model-Based Controller

To get rid of the whole state measurement to control the large scale system using equation [START_REF] Vuillemin | Poles residues descent algorithm for optimal frequency-limited H 2 model approximation[END_REF], we aim at computing a reduced order state feedback controller. Starting from the Lyapunov function defined previously:

E(x) = x T M 0 0 K x = Ξ T V T M 0 0 K VΞ (24)
In open-loop, with ẋ defined in eq [START_REF] Benner | Model Reduction and Approximation : Theory and Algorithms[END_REF], the derivative of this function is:

Ė(x) = x T -D 0 0 0 x + ( * ) = -2v T Dv (25)
where ( * ) represents the transpose of the matrix preceding it.

The reduced order state ξ r is of reasonable dimension and we can design a reduced order feedback

u = L r ξ r = L rv L rd ξ rv ξ rd (26)
that fixes the performance of the closed loop. With this control law, the derivative of the Lyapunov function becomes:

Ė(x) = x T -D 0 0 0 x + x T H T 0 u + ( * ) (27)
which, in the projected space, is equivalent to:

Ė(Ξ) = Ξ T V T -D 0 0 0 VΞ + Ξ T V T H T 0 u + ( * ) = Ξ T QΞ + ξ r ξ r T V T H T 0 L r 0 ξ r ξ r + ( * ) = Ξ T (Q + R)Ξ (28) where Q =     -V T rv DV rv 0 -V T rv DV rv 0 0 0 0 0 -V T rv DV rv 0 -V T rv DV rv 0 0 0 0 0     + ( * ) (29)
and

R =     V T rv H T L v V T rv H T L d 0 0 0 0 0 0 V T rv H T L v V T rv H T L d 0 0 0 0 0 0     + ( * ) (30)
with Q, R ∈ R 2n×2n . The computation of the matrix L r is done using the framework of LMI constraints. The LMI is defined as:

Ė(x) < -λE(x) ⇔ (Q + R) < -λ M 0 0 K (31)
where λ is a positive scalar that fixes the decay rate of the function E(x). The stability is proven for the large-scale system, even if the number of variables in the LMI is equal to m×r, where m is the number of input and r is the reduced dimension.

C. Simulation Results

The method is tested on a simulated model of a soft robot made of silicone, shown on the following picture: This robot is actuated with 8 cables mounted on the structure as shown on figure 1 so that the robot can deform in any direction of space. Friction between the robot and its cables is neglectable thanks to flexible tubes that guide the cables and that are added in the simulation.

SOFA integrates the CGAL library to compute a FEM mesh from a visual model. In this case, the mesh is made of 1557 nodes and 5157 tetrahedron elements, as shown on figure 2. The size n of both velocity and position vectors is equal to n = 3 × 1557 = 4671, and thus system ( 9) is of dimension 9342. Computing a full-order state feedback, as in [START_REF] Vuillemin | Poles residues descent algorithm for optimal frequency-limited H 2 model approximation[END_REF] would have implied the computation of a matrix L of dimension 8 × 9342, with 74 736 variables to compute. Instead of that, the model reduction step provide us with a reduced system of dimension 6, leading to a feedback matrix L r ∈ R m×r where m = 8 and r = 6. Solving LMI defined in equation ( 31) needs the computation of 48 decision variables.

In simulation experiments the state vector, i.e. both velocity and position vectors, is directly available and we use POD reduction method to obtain the reduced order state. The simulation results are given for the following example: the robot starts from its initial shape (left of figure 2) and converges to its rest position (right of figure 2), where the model has been linearized.

The behaviour of the robot in open-loop is illustrated on figure 3 and4, it shows that the robot converges to its equilibrium point after some oscillations. The goal of the proposed control method is to reduce, or suppress, this oscillations and make the robot converge faster to the desired position.

The control law defined in ( 26) is directly usable in practice, as it only requires the measurement of the state of the robot, i.e. displacement and velocity vectors. In simulation, they are directly available and we reconstruct the reduced state by applying equation ( 19) at each time step. (To test on real robots, an observer should be used to compute the reduced order state from the sensor measurement, this work is not conducted in this paper.) With this feedback controller, the oscillations vanish, as shown on figure 5 and6. This control law allows us to suppress the oscillations in the robot behaviour and the convergence time of the robot is decreased. However, the gain L rd is near zero and does not seem to have an impact on the closed-loop performance. Looking at the definition of matrices Q and R in ( 29) and (30), one can see zero entries on the diagonal corresponding to the displacement quadratic term. This brings conservatism in the choice of matrix L rd , next section proposes an extension of the Lyapunov function (24) to handle this issue.

IV. REDUCTION OF CONSERVATISM A. Choice of Lyapunov function

Adding parameters in the Lyapunov function reduces the conservatism of the results obtained using the previous Lyapunov function. The following result holds: Theorem 1: The following functions are Lyapunov functions for system (9):

V(x) = x T (1 + )M M M (1 + )K + D x (32)
for all scalar such that:

0 < < α 1 -α ( 33
)
where α is the mass-damping coefficient of the material (see appendix for details).

In open-loop, the derivative of V(x) becomes:

V(x) = 2x T -(1 + )D + M 0 0 -K x ( 34
)
By adding parameter in the Lyapunov function, we add a non-zero entry on the diagonal of its derivative. Fig. 5. Results in closed-loop using feedback gains computed thanks to eq (31). Left : Norm of the displacement (cm); Right: Norm of velocity (cm/s 2 ). Fig. 6. Reduced order state in closed-loop using eq (31)

B. Closed-loop algorithm

The derivative of the Lyapunov function according to the trajectories of the closed-loop writes:

V(x) = x T -( 1
+ )D + M 0 0 -K + (1 + )H T H T L x+( * ) (35)
which in the projected space is equivalent to:

Ξ T V T -( 1
+ )D + M 0 0 -K + (1 + )H T H T L VΞ + ( * ) (36)
from which the design of the controller is made possible thanks to the following theorem:

Theorem 2: System (9) with feedback (26) is stable with a decay rate λ if:

(S + T) < -λ (1 + )M M M (1 + )K + D (37)
with

S = V T -(1 + )D + M 0 0 -K V + ( * ) (38)
which also writes

S =     -(1 + )D rv rv + M rv rv 0 -(1 + )D rv rv + M rv rv 0 0 -K rv rv 0 -K rv rv -(1 + )D rv rv + M rv rv 0 -(1 + )D rv rv + M rv rv 0 0 -K rv rv 0 -K rv rv    
where D rv rv = V T rv DV rv , D rv rv = V T rv DV rv and

T = V T (1 + )H T H T LV + ( * ) =     (1 + )V T rv H T Lrv (1 + )V T rv H T L rd 0 0 V T rd H T Lrv V T rd H T L rd 0 0 (1 + )V T rv H T Lrv (1 + )V T rv H T L rd 0 0 V T rd H T Lrv V T rd H T L rd 0 0     + ( * ) (39)
Remark 2: Sketch of the proof. V (x) is a Lyapunov function defined in Theorem 1 and it holds:

(37) ⇔ V(x) < -λV(x) (40)
Moreover, L r = 0 is solution of the previous LMI for λ = 0 which makes it possible to use optimization algorithms. Using the Lyapunov function V(x), more flexibility is given in the choice of the matrix L r than in (31) with the same number of decision variables.

C. Simulation Experiments

The same experiment is done with the robot presented on figure 1 but with the control method of this section, i.e. Theorem 2. The objective remains the same, the oscillations of the open-loop behaviour of figure 3 and4 should be attenuated by the controller. The closed-loop results are presented on figures 7 and 8.

It is clear that the oscillations are removed with this control law, and in this case the tuning of the decay rate of the Lypunov function is easier than with the controller defined in (31). During firsts experiments whose results are shown on figure 5, the displacement vector converges after t = 11s whereas on figure 7, it converges at t = 6s. A more complete control of the robots dynamics is also made possible using the method presented in this section. The large scale Lyapunov function V(x) guarantees the stability of the large scale model but here both reduced velocity and displacement have a direct impact on the closed-loop performance. We also have more flexibility in the tuning of the controller.

Remark 3: The resolution of the LMI (37), with 48 variables and 9342×9342 constraints took 75 minutes on a Intel Core i7 CPU.

V. CONCLUSIONS

We presented a generic method that offers the possibility to control the dynamical behaviour of soft robots. The first benefit of this work is the use of model order reduction methods to provide the user with a reduced order system that models a large-scale accurate model of the robot. Thanks to the reduction of the state variable, the design of a reduced order state feedback is made possible. The second advantage of our work is the use of Lyapunov function to prove the stability of the large-scale closed-loop system. Simulation results provided in this paper show the performance of our approach, which is generic in the sense that it is applicable to any robot with a stable equilibrium point, as long as a FEM mesh of the robot is obtained.

A few steps are still required before testing on real prototypes, as the one presented on figure 1. Our next move is the integration of an observer in the control design to make it possible to reconstruct the reduced state from measurements.

For now, the control design allows the user to control a studied soft robot from any initial shape to a desired position, where the model has been linearized. An extension of the approach would be interesting to be able to control the robot from any initial shape to any desired position. Further research could focus on the linearization assumption required in the method, removing the linearization step to design a controller for the initial non-linear system (8) could also lead to a controller where the region of convergence is guaranteed.

APPENDIX : PROOF OF THEOREM 1

The continuous function V(x) is a Lyapunov function in open-loop for system (9) if:

1) V (x) > 0 2) V (x) is radially unbounded 3) V (x) < 0 • Proof of 1)

V(x) > 0 ⇔ (1 + )M M M (1 + )K + D > 0
Using Schur complement, it is equivalent to:

(1 + )M > 0 ; (1 + )K + D -

     > - 1 1 + β ⇒ (1 + + β)K > 0 0 < < α 1 -α ⇒ (α - 1 + ) > 0
The following condition is also a sufficient condition for V(x) to be positive definite in open-loop for system (9): 0 < < α 1 -α (41)

• Proof of 2) is trivial.

• Proof of 3)

V (x) < 0 ⇔ -2 (1 + )D -M 0 0 K < 0
With > 0, it directly follows K > 0. One just need (1 + )D -M > 0, for which condition (41) is also a sufficient condition.

Fig. 1 .

 1 Fig. 1. Top: Visual model of the robot Bottom: Design of the robot: slice view (left) and side view (right). The robot is actuated with 8 cables: 4 short cables in red, and 4 long ones in green.

Fig. 2 .

 2 Fig. 2. FEM mesh of the Trunk-like robot, made of 1557 nodes. Left : Deformed position; Right : Rest position.

Fig. 3 .

 3 Fig. 3. Left : Norm of the displacement (cm) in open-loop Right: Norm of velocity (cm/s 2 ) in open-loop.

Fig. 4 .

 4 Fig. 4. Reduced order state in open-loop.

Fig. 7 .

 7 Fig. 7. Results in closed-loop using feedback gains computed thanks to eq (37). Left : Norm of the displacement (cm); Right: Norm of velocity (cm/s 2 ).

Fig. 8 .

 8 Fig.8. Reduced order state in closed-loop using eq (37)

⇔ > - 1 ;

 1 By definition, matrices M, K and D are positive definite. The damping matrix is defined using Rayleigh definition:D = αM + βKwhere α and β are respectively the mass-proportional and the stiffness-proportional damping coefficients of the material. Both are positive scalars lower than one. It holds:V(x) > 0 ⇔ > -1; (1 + )K + (αM + βK) -(1 + + β)K + (α -1 + )M > 0Sufficient conditions are:

Univ. Valenciennes, UMR 8201 -LAMIH F-59313 Valenciennes, France

Univ. Lille, CNRS, Centrale Lille, Inria, UMR 9189 -CRIStAL -Centre de Recherche en Informatique Signal et Automatique de Lille, F-59000 Lille, France


Alain Chateauneuf
email: chateaun@univ-paris1.fr

Rose-Anne Dana
email: dana@ceremade.dauphine.fr

Jean-Marc Tallon
email: jmtallon@univ-paris1.fr

Optimal Risk-Sharing Rules and Equilibria with Choquet-Expected-Utility

Keywords: Choquet expected utility, comonotonicity, risk-sharing, equilibrium

This paper explores risk-sharing and equilibrium in a general equilibrium set-up wherein agents are non-additive expected utility maximizers. We show that when agents have the same convex capacity, the set of Pareto-optima is independent of it and identical to the set of optima of an economy in which agents are expected utility maximizers and have same probability. Hence, optimal allocations are comonotone. This enables us to study the equilibrium set. When agents have different capacities, matters are much more complex (as in the vNM case). We give a general characterization and show how it simplifies when Pareto-optima are comonotone. We use this result to characterize Pareto-optima when agents have capacities that are the convex transform of some probability distribution. comonotonicity of Paretooptima is also shown to be true in the two-state case if the intersection of the core of agents' capacities is non-empty; Pareto-optima may then be fully characterized in the two-agent, two-state case. This comonotonicity result does not generalize to more than two states as we show with a counter-example. Finally, if there is no-aggregate risk, we show that non-empty core intersection is enough to guarantee that optimal allocations are full-insurance allocation. This result does not require convexity of preferences.

Introduction

In this paper, we explore the consequences of Choquet-expected-utility on risk-sharing and equilibrium in a general equilibrium set-up. There has been over the last fifteen years an extensive research on new decision-theoretic models [START_REF] Karni | Utility theory with uncertainty[END_REF] for a survey), and a large part of this research has been devoted to the Choquet-expected-utility model introduced by [START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF]. However, applications to an economy-wide set-up have been relatively scarce. In this paper, we derive the implications of assuming such preference representation at the individuals level on the characteristics of Pareto optimal allocations. This, in turn, allows us to (partly) characterize equilibrium allocations under that assumption.

Choquet-expected-utility (CEU henceforth) is a model that deals with situations in which objective probabilities are not given and individuals are a priori not able to derive (additive) subjective probabilities over the state space. It is well-suited to represent agents' preferences in situation where "ambiguity"(as observed in the Ellsberg experiments) is a pervasive phenomenon1 . This model departs from expected-utility models in that it relaxes the sure-thing principle. Formally, the (subjective) expected-utility model is a particular subclass of the CEU of model. Our paper can then be seen as an exploration of how results established in the von Neumann-Morgenstern (vNM henceforth) case are modified when allowing for more general preferences, whose form rests on sound axiomatic basis as well. Indeed, since CEU can be thought of as representing situations in which agents are faced with "ambiguous events", it is interesting to study how the optimal social risk-sharing rule in the economy is affected by this ambiguity and its perception by agents.

We focus on a pure-exchange economy in which agents are uncertain about future endowments and consume after uncertainty is resolved. Agents are CEU maximizers and characterized by a capacity and a utility index (assumed to be strictly concave).

When agents are vNM maximizers and have the same probability on the state space, it is well-known since [START_REF] Borch | Equilibrium in a reinsurance market[END_REF] that agents' optimal consumptions depend only on aggregate risk, and is a non-decreasing function of aggregate resources : at an optimum, an agent bears only (some of) the aggregate risk. It is easy to fully characterize such Pareto Optima (see Eeck-houdt and Gollier [1995]). More generally, in the case of probabilized risk, [START_REF] Landsberger | Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF] and [START_REF] Chateauneuf | A review of some results related to comonotonicity[END_REF] have shown that Pareto Optima (P.O. henceforth) are comonotone if agents' preferences satisfy second-order stochastic dominance. This, in particular, is true in the rank-dependent-expected-utility case. The first goal of this paper is to provide a characterization of the set of P.O. and equilibria in the rank-dependent-expected-utility case. Our second and main aim is to assess whether the results obtained in the case of risk are robust when one moves to a situation of non-probabilized uncertainty with Choquet-expected-utility, in which there is some consensus.

We first study the case where all agents have the same capacity. We show that if this capacity is convex, the set of P.O. is the same as that of an economy with vNM agents whose beliefs are described by a common probability. Furthermore, it is independent of that capacity. As a consequence, P.O. are easily characterized in this set-up, and depend only on aggregate risk (and utility index). Thus, if uncertainty is perceived by all agents in the same way, the optimal risk-bearing is not affected (compared to the standard vNM case) by this ambiguity. The equivalence proof relies heavily on the fact that, if agents are vNM maximizers with identical beliefs, optimal allocations are comonotone and independent of these beliefs : each agent's consumption moves in the same direction as aggregate endowments. This equivalence result is in the line of a result on aggregation in appendix C of [START_REF] Epstein | Intertemporal asset pricing under Knightian uncertainty[END_REF]. Finally, the information given by the optimality analysis is used to study the equilibrium set. A qualitative analysis of the equilibrium correspondence may be found in [START_REF] Dana | Pricing rules when agents have non-additive expected utility and homogeneous expectations[END_REF].

When agents have different capacities, matters are much more complex. To begin with, in the vNM case, we don't know of any conditions ensuring that P.O. are comonotone in that case. However, in the CEU model, intuition might suggest that if agents have capacities whose cores have some probability distribution in common, P.O. are then comonotone. This intuition is unfortunately not correct in general, as we show with a counterexample. As a result, when agents have different capacities, whether P.O. allocations are comonotone depends on the specific characteristics of the economy. On the other hand, if P.O. are comonotone, they can be further characterized, although not fully. It is also in general non-trivial to use that information to infer properties of equilibrium. This leads us to study cases for which it is possible to prove that P.O. allocations are comonotone.

A first case is when the agents' capacities are the convex transform of some probability distribution. We then know from [START_REF] Chateauneuf | A review of some results related to comonotonicity[END_REF] and [START_REF] Landsberger | Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF] that P.O. are comonotone. Our analysis then enables us to be more specific than they are about the optimal risk-bearing arrangements and equilibrium of such an economy.

The second case is the simple case in which there are only two states (as in simple insurance models à la [START_REF] Mossin | Aspects of rational insurance purchasing[END_REF]). The non-emptiness of the cores' intersection is then enough to prove that P.O. allocations are comonotone, although it is not clear what the actual optimal risk-sharing arrangement looks like. If we specify the model further and assume there are only two agents, the risk-sharing arrangement can be fully characterized. Depending on the specifics of the agents' characteristics, it is either a subset of the P.O. of the economy in which agents each have the probability that minimizes, among the probability distributions in the core, the expected value of aggregate endowments, or the less pessimistic agent insures the other. (This last risk-sharing arrangement typically cannot occur in a vNM setup with different beliefs and strictly concave utility functions.) The equilibrium allocation in this economy can also be characterized.

Finally, we consider the situation in which there exists only individual risk, a case first studied by [START_REF] Malinvaud | The allocation of individual risks in large markets[END_REF][START_REF] Malinvaud | Markets for an exchange economy with individual risks[END_REF]. comonotonicity is then equivalent to full-insurance. We show that a condition for optimal allocations to be full-insurance allocations is that the intersection of the core of the agents' capacities is non-empty, a condition that can be intuitively interpreted as minimum consensus. This full-insurance result easily generalizes to the multi-dimensional set-up. Using this result, we show that establish that any equilibrium of particular vNM economies is an equilibrium of the CEU economy. These vNM economies are those in which agents have the same characteristics as in the CEU economy and have common beliefs given by a probability in the intersection of the cores of the capacities of the CEU economy. When the capacities are convex, any equilibrium of the CEU economy is of that type. This equivalence result between equilibrium of the CEU economy and associated vNM economies suggests that equilibrium is indeterminate, an idea further explored in [START_REF] Tallon | Risque microéconomique, aversion à l'incertitude et indétermination de l'équilibre[END_REF] and [START_REF] Dana | Pricing rules when agents have non-additive expected utility and homogeneous expectations[END_REF].

The rest of the paper is organized as follows. Section 2 establishes notation and define the characteristics of the pure exchange economy that we deal with in the rest of the paper. In particular, we recall properties of the Choquet integral. We also recall there some useful information on optimal risk-sharing in vNM economies. Section 3 is the heart of the paper and deals with the general case of convex capacities. In a first sub-section, we assume that agents have identical capacities, while the second sub-section deals with the case where agents have different capacities. Section 4 is devoted to the study of two particular cases of interest, namely the case where agents' capacities are the convex transform of a common probability distribution and the two-state case. The case of no-aggregate risk in a multi-dimensional set-up is studied in section 5.

Notation, definitions and useful results

We consider an economy in which agents make decisions before uncertainty is resolved. The economy is a standard two-period pure-exchange economy, but for agents' preferences.

There are k possible states of the world, indexed by superscript j. Let S be the set of states of the world and A the set of subsets of S. There are n agents indexed by subscript i. We assume there is only one good2 . C j i is the consumption by agent i in state j and C i = (C 1 i , . . . , C k i ). Initial endowments are denoted w i = (w 1 i , . . . , w k i ). w = n i=1 w i is the aggregate endowment.

We will focus on Choquet-Expected-Utility. We assume the existence of a utility index U i : IR + → IR that is cardinal, i.e. defined up to a positive affine transformation. Throughout the paper U i is taken to be strictly increasing and strictly concave. When needed, we will assume differentiability together with the usual Inada condition:

Assumption U1: ∀i, U i is C 1 and U ′ i (0) = ∞.
Before defining CEU (the Choquet integral of U with respect to a capacity), we recall some properties of capacities and their core.

Capacities and the core

A capacity is a set function ν : A → [0, 1] such that ν(∅) = 0, ν(S) = 1, and, for all A, B ∈ A, A ⊂ B ⇒ ν(A) ≤ ν(B). We will assume throughout that the capacities we deal with are such that 1 > ν(A) > 0 for all A ∈ A,

A = S, A = ∅. A capacity ν is convex if for all A, B ∈ A, ν(A ∪ B) + ν(A ∩ B) ≥ ν(A) + ν(B).
The core of a capacity ν is defined as follows

core(ν) =    π ∈ IR k + | j π j = 1 and π(A) ≥ ν(A), ∀A ∈ A   
where π(A) = j∈A π j . Core(ν) is a compact, convex set which may be empty. Since 1 > ν(A) > 0 ∀A ∈ A, A = S, A = ∅, any π ∈ core(ν) is such that π ≫ 0, (i.e., π j > 0 for all j). It is well-known that when ν is convex, its core is non-empty. It is equally well-known that non-emptiness of the core does not require convexity of the capacity. If there are only two states however, it is easy to show that core(ν) = ∅ if and only if ν is convex.

We shall provide an alternative definition of the core in the following sub-section.

Choquet-expected-utility

We now turn to the definition of the Choquet integral of f ∈ IR S :

f dν ≡ E ν (f ) = 0 -∞ (ν(f ≥ t) -1)dt + ∞ 0 ν(f ≥ t)dt Hence, if f j = f (j) is such that f 1 ≤ f 2 ≤ . . . ≤ f k : f dν = k-1 j=1 [ν({j, . . . , k}) -ν({j + 1, . . . , k})] f j + ν({k})f k
As a consequence, if we assume that an agent consumes C j in state j, and that C 1 ≤ . . . ≤ C k , then his preferences are represented by:

v (C) = [1 -ν({2, .., k})] U (C 1 )+... [ν({j, .., k}) -ν({j + 1, .., k})] U (C j )+...ν({k})U (C k )
Observe that, if we keep the same ranking of the states as above, then v (C) = E π U (C), where C is here the random variable giving C j in state j, and the probability π is defined by: π j = ν({j, . . . , k})ν({j + 1, . . . , k}), j = 1, . . . , k -1 and π k = ν({k}).

If U is differentiable and ν is convex, the function v : IR k + → IR defined above is continuous, strictly concave and subdifferentiable. Let

∂v(C) = {a ∈ IR k | v(C) -v(C ′ ) ≥ a(C -C ′ ), ∀C ′ ∈ IR k + } denote the subgradient of the function v at C. In the open set C ∈ IR k + | 0 < C 1 < C 2 < . . . < C k , v is differentiable. If 0 < C 1 = C 2 = . . . = C k then, ∂v(C) is proportional to core(ν).
The following proposition gives an alternative representation of core(ν) that will be useful in section 5.

Proposition 2.1 core(ν) = π ∈ IR k + | k j=1 π j = 1 and v(C) ≤ E π (U (C)) , ∀ C ∈ IR k + Proof: Let π ∈ core(ν) and assume C 1 ≤ C 2 ≤ . . . ≤ C k . Then, v(C) = U (C 1 )+ν({2, . . . , k})(U (C 2 )-U (C 1 ))+. . .+ν({k})(U (C k )-U (C k-1 ))
Hence, since π ∈ core(ν), and therefore ν(A) ≤ π(A) for all events A:

v(C) ≤ U (C 1 )+ k j=2 π j (U (C 2 )-U (C 1 ))+. . .+π k (U (C k )-U (C k-1 )) = E π (U (C))
which proves one inclusion.

To prove the other inclusion, let π

∈ π ∈ IR k + | k j=1 π j = 1, v(C) ≤ E π (U (C)) , ∀ C . Normalize U so that U (C) = 0 and U ( C) = 1 for some C and C. Let A ∈ A and C A = C1 A c + C1 A . Since v(C A ) = ν(A) ≤ E π (U (C A )) = π(A), one gets π ∈ core(ν). A corollary is that if core(ν) = ∅, then v(C) ≤ min π∈core(ν) E π U (C). 3

comonotonicity

We finally define comonotonicity of a class of random variables ( C i ) i=1,...,n . This notion, which has a natural interpretation in terms of mutualization of risks, will be crucial in the rest of the analysis.

Definition 1 A family ( C i ) i=1,...,n of random variables on S is a class of comonotone functions if for all i, i ′ , and for all j, j ′ ,

C j i -C j ′ i C j i ′ -C j ′ i ′ ≥ 0.
An alternative characterization is given in the following proposition (see [START_REF] Denneberg | Non-additive measures and integral[END_REF]):

Proposition 2.2 A family ( C i ) i=1,...,n of non-negative random variables on S is a class of comonotone functions if and only if for all i, there exists a function g i : IR + → IR + , non-decreasing and continuous, such that for all

x ∈ IR + , n i=1 g i (x) = x and C j i = g i n m=1 C j m for all j.
The family ( C i ) i=1,...,n is comonotone if they all vary in the same direction as their sum.

Optimal risk-sharing with vNM agents

We briefly recall here some well-known results on optimal risk-sharing in the traditional vNM case (see e.g. [START_REF] Eeckhoudt | Risk, evaluation, management and sharing[END_REF] or [START_REF] Magill | Theory of incomplete markets[END_REF]). Consider first the case of identical vNM beliefs. Agents have the same probability π = (π 1 , . . . , π k ), π j > 0 for all j, over the states of the world and a utility function defined by

v i (C i ) = k j=1 π j U i (C j i ), i = 1, . . . , n.
The following proposition recalls that the P.O. allocations of this economy are independent of the (common) probability, depend only on aggregate risk (and utility indices), and are comonotone4 . Proposition 2.3 Let (C i ) n i=1 ∈ IR kn + be a P.O. allocation of an economy in which agents have vNM utility index and identical additive beliefs π. Then, it is a P.O. of an economy with additive beliefs π ′ (and same vNM utility index). Furthermore,

(C i ) n i=1 is comonotone.
As a consequence of propositions 2.2 and 2.3, it is easily seen that, at a P.O. allocation, agent i's consumption C i is a non-decreasing function of w.

If agents have different probabilities π j i , j = 1, . . . , k, i = 1, . . . , n, over the states of the world, it is easily seen that P.O. now depend on the probabilities and on aggregate risk. It is actually easy to find examples in which P.O. are not necessarily comonotone (take for instance a model without aggregate risk in which agents have different beliefs : the P.O. allocations are not state-independent and therefore are not comonotone).

3 Optimal risk-sharing and equilibrium with CEU agents: the general convex case

In this section we deal first with optimal risk-sharing and equilibrium analysis when agents have identical convex capacities and then move on to different convex capacities.

Optimal risk-sharing and equilibrium with identical capacities

Assume here that all agents have the same capacity ν over the states of the world and that this capacity is convex. We denote by E 1 the exchange economy in which agents are CEU with capacity ν and utility index U i , i = 1, . . . , n.

Define D ν (w) as follows:

D ν (w) = {π ∈ core(ν) | E π w = E ν w}
The set D ν (w) is constituted of the probabilities that "minimize the expected value of the aggregate endowment". In particular, if w 1 < w 2 . . . < w k , D ν (w) contains only π = (π 1 , . . . , π k ) with π j = ν({j, j + 1, . . . , k})ν({j + 1, . . . , k}) for all j < k and π k = ν({k}). If w 1 = . . . = w k , the set D ν (w) is equal to core(ν).

It is important to note that the Choquet integral of any random variable that is non-decreasing with w is actually the integral of that random variable with respect to a probability distribution in D ν (w). In particular, we have the following lemma.

Lemma 3.1 Let ν be a convex capacity, U an increasing function and

C ∈ IR k + be a non-decreasing function of w. Then, v(C) = E π U (C) for any π ∈ D ν (w). Proof: Since C is non-decreasing in w, if w 1 ≤ . . . ≤ w k , then C 1 ≤ . . . ≤ C k . Furthermore, w j = w j ′ implies C j = C j ′ .
The same relationship holds between w j k j=1 and U (C j ) k j=1 , U being increasing. It is then simply a matter of writing down the expression of the Choquet integral to see that

v(C) = E π U (C) for any π ∈ D ν (w). Proposition 3.1 The allocation (C i ) n i=1 ∈ IR kn + is a P.O. of E 1 if
and only if it is a P.O. of an economy in which agents have vNM utility index U i , i = 1, . . . , n and identical probability over the set of states of the world. In particular, P.O. are comonotone.

Proof: Since the P.O. of an economy with vNM agents with same probability are independent of the probability, we can choose w.l.o.g. this probability to be π ∈ D ν (w).

Let (C i ) n i=1 be a P.O. of the vNM economy. Being a P.O., this allocation is comonotone. By proposition 2.2, C i is a non-decreasing function of w. Hence, applying lemma 3.1,

v i (C i ) = E π [U i (C i )], i = 1, . . . , n. If it were not a P.O. of E 1 , there would exist an allocation (C ′ 1 , C ′ 2 . . . C ′ n ) such that v i (C ′ i ) = E ν [U i (C ′ i )] ≥ v i (C i ) = E π [U i (C i )]
for all i, and with at least one strict inequality. Since

E π [U i (C ′ i )] ≥ E ν [U i (C ′ i )] for all i, this contradicts the fact that (C i ) n i=1 is a P.O. of
′ i ) n i=1 such that E π [U i (C ′ i )] ≥ E π [U i (C i )] ≥ v i (C i )
for all i, and with a strict inequality for at least an agent. (C ′ i ) n i=1 being Pareto optimal, it is comonotone and it follows by proposition 2.2 that C ′ i is a non-decreasing function of w. Hence, applying lemma 3.1,

v i (C ′ i ) = E π [U i (C ′ i )], i = 1, . . . , n. This contradicts the fact that (C i ) n i=1 is a P.O. of E 1 .
Note that this proposition not only shows that P.O. allocations are comonotone in the CEU economy, but also completely characterizes them.

We may now also fully characterize the equilibria of E 1 .

Proposition 3.2 (i) Let (p ⋆ , C ⋆ ) be an equilibrium of a vNM economy in which all agents have utility index U i and beliefs given by π

∈ D ν (w), then (p ⋆ , C ⋆ ) is an equilibrium of E 1 . (ii) Conversely, assume U1. If (p ⋆ , C ⋆ ) is an equilibrium of E 1 , then there exists π ∈ D ν (w) such that (p ⋆ , C ⋆
) is an equilibrium of the vNM economy with utility index U i and probability π ∈ D ν (w).

Proof: See [START_REF] Dana | Pricing rules when agents have non-additive expected utility and homogeneous expectations[END_REF].

Corollary 3.1 If U1 is fulfilled and w 1 < w 2 < . . . < w k , then the equilibria of E 1 are identical to those of a vNM economy in which agents have utility index U i , i = 1, . . . , n and same probabilities over states π j = ν{j, j + 1, . . . , k} -ν{j + 1, . . . , k}, j < k and π k = ν({k}). Hence, (w, C ⋆ i , i = 1, ..., n) are comonotone.

To conclude this sub-section, observe that P.O. allocations in the CEU economy inherits all the nice properties of P.O. allocations in a vNM economy with identical beliefs. In particular, P.O. allocations are independent of the capacity. However, the equilibrium allocations in the vNM economy do depend on beliefs, and it is not trivial to assess the relationship between the equilibrium set of a vNM economy with identical beliefs and the equilibrium set of the CEU economy E 1 . Note for instance that E 1 has "as many equilibria" as there are probability distributions in the set D ν (w). If D ν (w) consists of a unique probability distribution, equilibria of E 1 are the equilibria of the vNM economy with beliefs equal to that probability distribution. On the other hand, if D ν (w) is not a singleton, it is a priori not possible to assimilate all the equilibria of E 1 with equilibria of a given vNM economy.

Optimal risk-sharing and equilibrium with different capacities

We next consider an economy in which agents have different convex capacities. Denote the economy in which agents are CEU with capacity ν i and utility index U i , i = 1, . . . , n by E 2 . We first give a general characterization of the set of P.O., when no further restrictions are imposed on the economy. We then show that this general characterization can be most usefully applied when one knows that P.O. are comonotone.

Proposition 3.3 (i) Let (C i ) n i=1 ∈ IR kn + be a P.O. of E 2 such that for all i, C j i = C ℓ i , j = ℓ. Let π i ∈ core(ν i ) be such that E ν i U i (C i ) = E π i U i (C i ) for all i. Then (C i ) n
i=1 is a P.O. of an economy in which agents have vNM utility index U i and probabilities π i , i = 1, . . . , n.

(ii) Let π i ∈ core(ν i ), i = 1, . . . , n and (C i ) n i=1 be a P.O. of the vNM economy with utility index U i and probabilities

π i , i = 1, . . . , n. If E ν i U i (C i ) = E π i U i (C i ) for all i, then (C i ) n i=1 is a P.O. of E 2 . Proof: (i) If (C i ) n i=1
is not a P.O. of a vNM economy, then there exists

(C ′ i ) n i=1 such that E π i U i (C ′ i ) ≥ E ν i U i (C i ) with a strict inequality for some i. Since t i C ′ i + (1 -t i )C i C i , ∀t i ∈ [0, 1]
, by choosing t i small, one may assume w.l.o.g. that C ′ i is ranked in the same order as C i . Hence,

E π i U i (C ′ i ) = E ν i U i (C ′ i ) for all i, which contradicts the fact that (C i ) n i=1 is a P.O. of E 2 .
(ii) Assume there exists a feasible allocation (

C ′ i ) n i=1 such that E ν i U i (C ′ i ) ≥ E ν i U i (C i ) with a strict inequality for at least some i. Then, E π i U i (C ′ i ) ≥ E π i U i (C i )
with a strict inequality for at least some i, which leads to a contradiction.

We now illustrate the implications of this proposition on a simple example.

Example 3.1 Consider an economy with two agents, two states and one good, that thus can be represented in an Edgeworth box. Divide the latter into three zones :

• zone (1), where C 1 1 > C 2 1 and C 1 2 < C 2 2 • zone (2), where C 1 1 < C 2 1 and C 1 2 < C 2 2 • zone (3), where C 1 1 < C 2 1 and C 1 2 > C 2 2
In zone (1), everything is as if agent 1 had probability (ν 1 1 , 1ν 1 1 ) and agent 2, probability (1ν 2 2 , ν 2 2 ). In zone (2), agent 1 uses (1ν 2 1 , ν 2 1 ) and agent 2, (1ν 2 2 , ν 2 2 ), while in zone (3), agent 1 uses (1ν 2 1 , ν 2 1 ) and agent 2, (ν 1 2 , 1ν 1 2 ). In order to use (ii) of proposition 3.3, we draw the three contract curves, corresponding to the P.O. in the vNM economies in which agents have the same utility index U i and the three possible couples of probability. Label (a), (b) and (c) these curves.

One notices that curve (a), which is the P.O. of the vNM economy for agents having beliefs (ν 1 1 , 1ν 1 1 ) and (1ν 2 2 , ν 2 2 ) respectively, does not intersect zone (1), which is the zone where CEU agents do use these probability distributions as well. Hence, no points are at the same time P.O. of that vNM economy and such that

E ν i [U i (C i )] = E π i [U i (C i )], i = 1, 2.
a c b (3) (1) (2) ✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 ✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 contained in zone (2)
. That part constitutes a subset of the set of P.O. that we are looking for. We will show later on that, in order to get the full set of P.O. of the CEU economy, one has to replace the part of curve (b) that lies in zone (3) by the segment along the diagonal of agent 2. ♦ It follows from proposition 3.3 that, without any knowledge on the set of P.O., one has to compute the P.O. of (k!) n -1 economies (if there are k! extremal points in core(ν i ) for all i). Thus, the actual characterization of the set of P.O. of E 2 might be somewhat tedious without further information.

In the comonotone case however, the characterization of P.O. is simpler, even though it remains partial.

Corollary 3.2 Assume w 1 ≤ w 2 ≤ . . . ≤ w k . (i) Let U1 hold and (C i ) n i=1 ∈ IR kn + be a comonotone P.O. of E 2 such that C 1 i < C 2 i < . . . < C k i for all i = 1, . . . , n.
Then, (C i ) n i=1 is a P.O. allocation of the economy in which agents are vNM maximizers with utility index U i and probability π j i = ν i ({j, . . . , k})ν i ({j + 1, . . . , k}) for j < k and π k i = ν i ({k}). (ii) Let (C i ) n i=1 ∈ IR kn + be a P.O. of the economy in which agents are vNM maximizers with utility index U i and probability π j i = ν i ({j, . . . , k})ν i ({j + 1, . . . , k}) for j < k and

π k i = ν i ({k}). If (C i ) n i=1 is comonotone, then it is a P.O. of E 2 .
These results may now be used for equilibrium analysis as follows.

Proposition 3.4 Assume w 1 ≤ w 2 ≤ . . . ≤ w k .

(i) Let (p ⋆ , C ⋆ ) be an equilibrium of E 2 . If 0 < C ⋆1 i < . . . < C ⋆k i for all i, then (p ⋆ , C ⋆ ) is an equilibrium of the economy in which agents are vNM maximizers with utility index U i and probability π j i = ν i ({j, . . . , k})ν i ({j + 1, . . . , k}) for j < k and π k i = ν i ({k}). (ii) Let (p ⋆ , C ⋆ ) be an equilibrium of the economy in which agents are vNM maximizers with utility index U i and probability π j i = ν i ({j, . . . , k})ν i ({j + 1, . . . , k}) for j < k and

π k i = ν i ({k}). If C ⋆ is comonotone, then (p ⋆ , C ⋆ ) is an equilibrium of E 2 .
Proof: (i) Since (p ⋆ , C ⋆ ) is an equilibrium of E 2 , and since v i is differentiable at C ⋆ i for every i, there exists a multiplier λ i such that

p ⋆ = λ i U ′ i (C ⋆1 i )π 1 i , . . . , U ′ i (C ⋆k i )π k i
, where π j i = ν i ({j, . . . , k})ν i ({j + 1, . . . , k}) for j < k and π k i = ν i ({k}) for all i. Hence, (p ⋆ , C ⋆ ) is an equilibrium of the economy in which agents are vNM maximizers with probability π j i for all i, j.

(ii) Let (p ⋆ , C ⋆ ) be an equilibrium of the economy in which agents are vNM maximizers with probability π j i for all i, j. Assume C ⋆ is comonotone. We thus have

p ⋆ C ′ i ≤ p ⋆ w i ⇒ E π i U i (C ′ i ) ≤ E π i U i (C ⋆ i ) Since E ν i U i (C ′ i ) ≤ E π i U i (C ′ i ) and E ν i U i (C ⋆ i ) = E π i U i (C ⋆ i ) , we get E ν i U i (C ′ i ) ≤ E ν i U i (C ⋆ i ) for all i, which implies that (C ⋆ , p ⋆ ) is an equi- librium of E 2 .
Observe that, even though the characterization of P.O. allocations is made simpler when we know that these allocations are comonotone, the above proposition does not give a complete characterization. comonotonicity of the P.O. allocations is also useful for equilibrium analysis. This leads us to look for conditions on the economy under which P.O. are comonotone.

Optimal risk-sharing and equilibrium in some particular cases

In this section, we focus on two particular cases in which we can prove directly that P.O. allocations are comonotone.

Convex transform of a probability distribution

In this sub-section, we show how one can use the previous results when agents' capacities are the convex transform of a given probability distribution. In this case, one can directly apply corollary 3.2 and proposition 3.4 to get a characterization of P.O. and equilibrium. Let π = (π 1 , . . . , π k ) be a probability distribution on S, with π j > 0 for all j.

Proposition 4.1 Assume w 1 ≤ w 2 ≤ . . . ≤ w k . Assume that, for all i, U i is differentiable and ν i = f i • π, where f i is a strictly increasing and convex function from [0, 1] to [0, 1] with f i (0) = 0, f i (1) = 1. Then, at a P.O.,

C 1 i ≤ C 2 i ≤ . . . ≤ C k i for all i.
Proof: Since U i is differentiable, strictly increasing and strictly concave, and f i is a strictly increasing, convex function for all i, it results from corollary 2 in Chew, [START_REF] Chew | Risk aversion in the theory of expected utility with rank dependent preferences[END_REF] that every agent strictly respects second order stochastic dominance. Therefore it remains to show that if every agent strictly respects second order stochastic dominance, then, at a P.O., C 1 i ≤ C 2 i ≤ . . . ≤ C k i for all i. We do so using proposition 4.1 in [START_REF] Chateauneuf | A review of some results related to comonotonicity[END_REF].

Assume (C i ) n i=1 is not comonotone. W.l.o.g., assume that andC ′ 2 be determined by the feasibility condition

C 1 1 > C 2 1 , C 1 2 < C 2 2 , and C 1 1 + C 1 2 ≤ C 2 1 + C 2 2 . Let C ′ be such that: C 1′ 1 = C 2′ 1 = π 1 C 1 1 + π 2 C 2 1 π 1 + π 2 and C j′ 1 = C j 1 , j > 2 Let C ′ i = C i for all i > 2,
C 1 + C 2 = C ′ 1 + C ′ 2 . Hence, C 1′ 2 = C 1 2 + π 2 π 1 + π 2 (C 1 1 -C 2 1 ), C 2′ 2 = C 2 2 - π 1 π 1 + π 2 (C 1 1 -C 2 1 ) and C j′ 2 = C j 2 , j > 2 It may easily be checked that C 2 1 < C 1′ 1 = C 2′ 1 < C 1 1 , and C 1 2 < C 1′ 2 ≤ C 2′ 2 < C 2 2 . Furthermore, π 1 C 1′ 1 + π 2 C 2′ 1 = π 1 C 1 1 + π 2 C 2 1 , and π 1 C 1′ 2 + π 2 C 2′ 2 = π 1 C 1 2 + π 2 C 2 2 . Therefore, C ′ i i = 1
, 2 is a strictly less risky allocation than C i i = 1, 2, with respect to mean preserving increases in risk. It follows that agents 1 and 2 are strictly better off with C ′ , while other agents' utilities are unaffected. Hence, C ′ Pareto dominates C. Thus, any P.O. C must be comonotone, i.e.,

C 1 i ≤ C 2 i ≤ . . . ≤ C k i for all i.
Using corollary 3.2, we can then provide a partial characterization of the set of P.O. Note that such a characterization was not provided by the analysis in [START_REF] Chateauneuf | A review of some results related to comonotonicity[END_REF] or [START_REF] Landsberger | Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF].

Proposition 4.2 Assume w 1 ≤ . . . ≤ w k and that agents are CEU maximizers with ν i = f i • π, f i convex, strictly increasing and such that f i (0) = 0 and f i (1) = 1. Then,

(i) Let (C i ) n i=1 ∈ IR kn + be a P.O. of this economy such that C 1 i < C 2 i < . . . < C k
i for all i = 1, . . . , n. Then, (C i ) n i=1 is a P.O. allocation of the economy in which agents are vNM maximizers with utility index U i and probability

π j i = f i k s=j π s -f i k s=j+1 π s for j = 1, . . . , k -1, and
π k i = f i π k . (ii) Let (C i ) n i=1 ∈ IR kn
+ be a P.O. of the economy in which agents are vNM maximizers with utility index U i and probability

π j i = f i k s=j π s -f i k s=j+1 π s for j = 1, . . . , k -1, and π k i = f i π k . If (C i ) n i=1 is comonotone, then it is a P.O. of the CEU economy with ν i = f i • π. Proof: See corollary 3.2.
The same type of result can be deduced for equilibrium analysis from proposition 3.4, and we omit its formal statement here.

The previous characterization formally includes the Rank-Dependent-Expected-Utility model introduced by [START_REF] Quiggin | A theory of anticipated utility[END_REF] in the case of (probabilized) risk. It also applies to so-called "simple capacities" (see e.g. [START_REF] Dow | Uncertainty aversion, risk aversion, and the optimal choice of portfolio[END_REF]), which are particularly easy to deal with in applications.

Indeed, let agents have the following simple capacities: ν i (A) = (1ξ i )π(A) for all A ∈ A, A = S, and ν i (S) = 1, where π is a given probability measure with 0 < π j < 1 for all j, and 0 ≤ ξ < 1.

These capacities can be written

ν i = f i •π where f i is such that f i (0) = 0, f i (1) = 1,
is strictly increasing, continuous and convex, with:

f i (p) = (1 -ξ i )p if 0 ≤ p ≤ max {π(A)<1} π(A) f i (1) = 1
Hence, ν i is a convex transformation of π, and we can apply the results of this sub-section to characterize the set of P.O. in an economy where all agents have such simple capacities.

The two-state case

We restrict our attention here to the case S = {1, 2}. Agent i has a capacity ν i characterized by two numbers ν i ({1}), ν i ({2}) such that ν i ({1}) ≤ 1ν i ({2}). To simplify notation, we'll denote ν i ({s}) = ν s i . In this particular case, core

(ν i ) = {(π, 1 -π) | π ∈ [ν 1 i , 1 -ν 2 i ]}.
Call E 3 the two-state exchange economy in which agents are CEU maximizers with capacity ν i and utility index U i , i = 1, . . . , n.

Assumption C: ∩ i core(ν i ) = ∅
This assumption is equivalent to

ν 1 i + ν 2 j ≤ 1, i, j = 1, . . . , n, or stated differently, to ∩ i [ν 1 i , 1 -ν 2 i ] = ∅.
Recall that in the two-state case, under C, agents' capacities are convex.

We now proceed to show that this "minimal consensus" assumption is enough to show that P.O. are comonotone.

Proposition 4.3 Let C hold. Then, P.O. are comonotone.

Proof: Assume w 1 ≤ w 2 and C not comonotone. W.l.o.g., assume that

C 1 1 > C 2 1 , C 1 2 < C 2 2 . Let (π, 1 -π) ∈ ∩ i core(ν i
) and C ′ be the feasible allocation defined by

C 1′ 1 = C 2′ 1 = πC 1 1 + (1 -π)C 2 1 and C 1′ 2 and C 2′ 2 are such that C j′ 1 + C j′ 2 = C j 1 + C j 2 , j = 1, 2, i.e. C 1′ 2 = C 1 2 + (1 -π)(C 1 1 -C 2 1 ), C 2′ 2 = C 2 2 -π(C 1 1 -C 2 1 ) One obviously has C 1 2 < C 1′ 2 ≤ C 2′ 2 < C 2 2 . Finally, let C j′ i = C j i , ∀i > 2, j = 1, 2. We now prove that C ′ Pareto dominates C. v 1 (C ′ 1 ) -v 1 (C 1 ) = U 1 (πC 1 1 + (1 -π)C 2 1 ) -ν 1 1 U 1 (C 1 1 ) -(1 -ν 1 1 )U 1 (C 2 1 ) > (π -ν 1 1 ) U 1 (C 1 1 ) -U 1 (C 2 1 ) ≥ 0
since U 1 is strictly concave and π ≥ ν 1 i . Now, consider agent 2's utility:

v 2 (C ′ 2 ) -v 2 (C 2 ) = (1 -ν 2 2 ) U 2 (C 1′ 2 ) -U 2 (C 1 2 ) + ν 2 2 U 2 (C 2′ 2 ) -U 2 (C 2 2 ) Since U 2 is strictly concave and C 1 2 < C 1′ 2 ≤ C 2′ 2 < C 2 2 , we have: U 2 (C 1′ 2 ) -U 2 (C 1 2 ) C 1′ 2 -C 1 2 > U 2 (C 2 2 ) -U 2 (C 2′ 2 ) C 2 2 -C 2′ 2
and hence,

U 2 (C 1′ 2 ) -U 2 (C 1 2 ) 1 -π > U 2 (C 2 2 ) -U 2 (C 2′ 2 ) π . Therefore, v 2 (C ′ 2 ) -v 2 (C 2 ) > (1 -ν 2 2 ) 1 -π π -ν 2 2 U 2 (C 2 2 ) -U 2 (C 2′ 2 ) ≥ 0 since (1 -ν 2 2 )(1 -π) -πν 2 2 = 1 -ν 2 2 -π ≥ 0 and U 2 (C 2 2 ) -U 2 (C 2′ 2 ) > 0. Hence, C ′ Pareto dominates C. Remark: If ν 1 i + ν 2 j < 1, i, j = 1, . . . , n
, which is equivalent to the assumption that ∩ i core(ν i ) contains more than one element, then one can extend proposition 4.3 to linear utilities.

Remark: Although convex capacities can, in the two-state case, be expressed as simple capacities, the analysis of sub-section 4.1 (and in particular proposition 4.1), cannot be used here. Indeed, assumption C does not require that agents' capacities are all a convex transform of the same probability distribution as example 4.1 shows.

Example 4.1 There are two agents with capacity ν 1 1 = 1/3, ν 2 1 = 2/3, and ν 1 2 = 1/6, ν 2 2 = 2/3 respectively. Assumption C is satisfied since π = (1/3, 2/3) is in the intersection of the cores. The only way ν 1 and ν 2 could be a convex transform of the same probability distribution is ν 1 = π and ν 2 = f 2 • π with f 2 (1/3) = 1/6 and f 2 (2/3) = 2/3. But f 2 then fails to be convex. ♦

Intuition derived from proposition 4.3 might suggest that some minimal consensus assumption might be enough to prove comonotonicity of the P.O. However, that intuition is not valid in general, as can be seen in the following example, in which the intersection of the cores of the capacities is non-empty, but where (some) P.O. allocations are not comonotone.

Example 4.2 There are two agents, with the same utility index U i (C) = 2C 1/2 , but different beliefs. The latter are represented by two convex capacities defined as follows:

ν 1 ({1}) = 3 9 ν 1 ({2}) = 3 9 ν 1 ({3}) = 1 9 ν 1 ({1, 2}) = 6 9 ν 1 ({1, 3}) = 6 9 ν 1 ({2, 3}) = 4
The intersection of the cores of these two capacities is non-empty since the probability defined by π j = 1/3, j = 1, 2, 3 belongs to both cores. The endowment in each state is respectively w 1 = 1, w 2 = 12, and w 3 = 13. We consider the optimal allocation associated to the weights (1/2, 1/2) and show it cannot be comonotone. In order to do that, we show that the maximum of v 1 (C 1 ) + v 2 (C 2 ) subject to the constraints C j 1 + C j 2 = w j , j = 1, 2, 3 and C j i ≥ 0 for all i and j, does not obtain for

C 1 i ≤ C 2 i ≤ C 3 i , i = 1, 2. Observe first that if C 1 i ≤ C 2 i ≤ C 3 i , i = 1, 2, then: v 1 (C 1 )+v 2 (C 2 ) = 2 5 9 C 1 1 + 3 9 C 2 1 + 1 9 C 3 1 + 4 9 C 1 2 + 2 9 C 2 2 + 3 9 C 3 2 Call g(C 1 1 , C 2 1 , C 3 1 , C 1 2 , C 2 2 , C 3 2 ) the above expression. Note that v 1 (C 1 ) + v 2 (C 2 ) takes the exact same form if C 1 1 < C 3 1 < C 2 1 and C 1 2 < C 2 2 < C 3 2 .
The optimal solution to the maximization problem:

max g(C 1 1 , C 2 1 , C 3 1 , C 1 2 , C 2 2 , C 3 2 ) s.t. C j 1 + C j 2 = w j j = 1, 2, 3 C j i ≥ 0 j = 1, 2, 3 i = 1, 2 is C 1 1 , C 2 1 , C
0 < C 1 1 < C 3 1 < C 2 1 and 0 < C 1 2 < C 2 2 < C 3 2 . Therefore: v 1 ( C 1 )+v 2 ( C 2 ) > v 1 (C 1 )+v 2 (C 2 ) for all C such that C 1 i ≤ C 2 i ≤ C 3 i , i = 1, 2
and hence the P.O. associated to equal weights for each agent is not comonotone.♦

One may expect that it follows from proposition 4.3 that P.O. of E 3 are the P.O. of the vNM economy in which agents have probability π i = 1-ν 2 i , i = 1, . . . , n. However, it is not so, since as recalled in sub-section 2.4, P.O. of a vNM economy with different beliefs are not in general comonotone. We can nevertheless use proposition 3.3 to provide a partial characterization of the set of P.O.

In this particular case of only two states, we can obtain a full characterization of the set of P.O. if there are only two agents in the economy. This should then be interpreted as a characterization of the optimal risk-sharing arrangement among two parties to a contract (arrangement that has been widely studied in the vNM case).

Figure 2:

✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 (a) ✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 (b)
3 (a), the thin line represents P.O. of the vNM economy in which agent i uses probability (1ν 2 i , ν 2 i ) that are not P.O. of the CEU economy, as they are not comonotone.

If agent 2 has a utility index of the DARA type, it is easy to show that the set of P.O. of the economy in which agents have utility index U i and probability (1ν 2 i , ν 2 i ) crosses agent 1's diagonal at most once, hence preventing the kind of situation represented on figure 3.

When there are only two (types of) agents, we can also go further in the characterization of the set of equilibria. Proposition 4.5 Assume k = 2, n = 2, w 1 ≤ w 2 , C and U1 hold and ν 2 1 < ν 2 2 . Let (p ⋆ , C ⋆ ) be an equilibrium of E. Then there are only two cases:

(i) Either C 1⋆ i < C 2⋆ i , i = 1, 2 and (p ⋆ , C ⋆
) is an equilibrium of a vNM economy in which agents have utility index U i and beliefs given by

π i = 1 -ν 2 i , i = 1, 2. (ii) Or, C 1⋆ 1 = C 2⋆ 1 = C ⋆⋆
and C ⋆⋆ satisfies the following :

(a) (1 -ν 2 2 )(w 1 1 -C ⋆⋆ )U ′ 2 (w 1 -C ⋆⋆ ) + ν 2 2 (w 2 1 -C ⋆⋆ )U ′ 2 (w 2 -C ⋆⋆ ) = 0 (b) ν 2 1 1-ν 2 1 ≤ -(w 1 1 -C ⋆⋆ ) (w 2 1 -C ⋆⋆ )
Proof: It follows from proposition 4.4 that either

C 1⋆ i < C 2⋆ i , i = 1, 2 or C 1⋆ 1 = C 2⋆ 1 = C ⋆⋆ .
The first case follows from proposition 3.4. In the Figure 3:

✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 (a) ✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 (b) ❛ w ❵ ❵ ❵ ❵ ❵ ❇ ❇ ❇ ❇ ❇ ❇
second case, the P.O. allocation is supported by the price (1

-ν 2 2 )U ′ 2 (w 1 - C ⋆⋆ ), ν 2 2 U ′ 2 (w 2 -C ⋆⋆ )
, hence the tangent to agent two's indifference curve at (w 1 -C ⋆⋆ , w 2 -C ⋆⋆ ) has the following equation in the (C 1 , C 2 ) plane:

(1 -ν 2 2 )U ′ 2 (w 1 -C ⋆⋆ )(C 1 -C ⋆⋆ ) + ν 2 2 U ′ 2 (w 2 -C ⋆⋆ )(C 2 -C ⋆⋆ ) = 0
Now C ⋆ is an equilibrium allocation iff (w 1 1 , w 2 1 ) fulfills that equation. Condition (b) follows from condition (a) and condition (ii) from proposition 4.4.

There might therefore exist, for a range of initial endowments, equilibria at which agent 1 is perfectly insured even though agent 2 has strictly convex preferences. Observe also that nothing excludes a priori the possibility of having different kind of equilibria for the same initial endowment (see figure 3 (b)).

Optimal risk-sharing and equilibrium without aggregate risk

We now turn to the study of economies without aggregate uncertainty5 . This corresponds to the case of individual risk first analyzed by [START_REF] Malinvaud | The allocation of individual risks in large markets[END_REF] and [1973]. A particular case is the one of a sunspot economy, in which uncertainty is purely extrinsic and does not affect the fundamentals, i.e., each agent's endowment is independent of the state of the world (see [START_REF] Tallon | Do sunspots matter when agents are Choquet-expectedutility maximizers[END_REF] for a study of sunspot economies with CEU agents). Our analysis of the case of purely individual risk might also yield further insights as to which type of financial contracting (e.g. mutual insurance rather than trade on Arrow securities defined on individual states) is necessary in such economies to decentralize an optimal allocation. It turns out that the economy under consideration possesses remarkable properties: P.O. are comonotone and coincide with full insurance allocations, under the relatively weak condition C. Furthermore, this condition, which is weaker than convexity of preferences, is enough to prove existence of an equilibrium. Finally, the case of purely individual risk lends itself to the introduction of several goods.

We thus move to an economy with m goods, indexed by subscript ℓ. C j iℓ is the consumption of good ℓ by agent i in state j. We have, C j i = (C j i1 , . . . , C j im ), and

C i = (C 1 i , . . . , C k i ). If C j i = C j ′
i for all j, j ′ , then C i will denote both this constant bundle (i.e. C j i ≡ C i ) and the vector composed of k such vectors, the context making it clear which meaning is intended. Let p j ℓ be the price of good ℓ available in state j, p j = (p j 1 , . . . , p j m ), and p = (p 1 , . . . , p k ).

The utility index U i is now defined on IR m + , and is still assumed to be strictly concave and strictly increasing. We will also need a generalization of assumption U1 to the multi-good case, that ensures that the solution to the agent's maximization program is interior.

Assumption Um: ∀i, {x ′ ∈ IR m + | U i (x ′ ) ≥ U i (x)} ⊂ IR m ++ , ∀x ∈ IR m ++ .
Aggregate endowment is the same across states, although its distribution among households might differ in each state. Therefore, we consider a pure exchange economy E 4 with n agents and m goods described by the list: E 4 = v i : IR km + → IR, w i ∈ IR km + , i = 1, . . . , n . We will denote aggregate endowments w, i.e., w = i w i .

Before dealing with non-additive beliefs, we first recall some known results in the vNM case.

Proposition 5.1 Assume all agents have identical vNM beliefs, π = (π 1 , . . . , π k ).

Then, (i) At a Pareto optimum, C j i = C j ′ i for all i, j, j ′ and C 1 i n i=1 is a P.O. of the static economy (U i , wi = E π (w i ) , i = 1, . . . , n).

(ii) Let Um hold. (p ⋆ , C ⋆ ) is an equilibrium of the vNM economy if and only if there exists q ⋆ ∈ IR m + such that p ⋆ = q ⋆ π 1 , q ⋆ π 2 , . . . , q ⋆ π k and (q ⋆ , C ⋆ ) is an equilibrium of the static economy

(U i , wi = E π (w i ) , i = 1, . . . , n).
It is also easy to see that if agents' beliefs are different, agents will consume state-dependent bundles at an optimum. We now examine to what extent these results, obtained in the vNM case, generalize to the CEU setup, assuming that condition C holds.

Using proposition 2.1 this assumption is equivalent6 to P = ∅, where

P =    π ∈ IR k ++ | k j=1 π j = 1 and v i (C 1 i , . . . , C k i ) ≤ E π (U i (C i )) , ∀ i, ∀ C i   
Recall that assumption C was not enough to prove comonotonicity of P.O. in the general case (though it was sufficient in the two-state case).

We now proceed to fully characterize the set of P.O. Proof: (i) Assume, to the contrary, that there exist an agent (say agent 1) and states j and j ′ such that

C j 1 = C j ′ 1 . Let Cj i ≡ Ci = E π (C i
) for all j and all i where π ∈ P. This allocation is feasible:

i Ci = E π ( i C i ) = w. By definition of P, v i C 1 i , . . . , C k i ≤ E π (U i (C i )) for all i. Now, E π (U i (C i )) ≤ U i (E π (C i )) = U i Ci = v i C1 i , .
. . , Ck i for all i, since U i is concave. This last inequality is strict for agent 1, since C j 1 = C j ′ 1 , π ≫ 0, and U 1 is strictly concave. Therefore, v i C 1 i , . . . , C k i ≤ v i C1 1 , . . . , Ck i for all i, with a strict inequality for agent one, a contradiction to the fact that (C i ) n i=1 is an optimum of E 4 .

(ii) We skip the proof for this part of the proposition for it relies on the same type of argument as that of proposition 3.3. Thus, even with different "beliefs" (in the sense of different capacities), agents might still find it optimal to fully insure themselves : differences in beliefs do not necessarily lead agents to optimally bear some risk as in the vNM case.

We now proceed to study the equilibrium set.

Proposition 5.3 Let C hold.

(i) Let (p ⋆ , C ⋆ ) be an equilibrium of a vNM economy in which all agents have utility index U i and beliefs given by π ∈ P, then (p ⋆ , C ⋆ ) is an equilibrium of E 4 .

(ii) Conversely, assume ν i is convex and U i satisfies Um for all i. Let (p ⋆ , C ⋆ ) be an equilibrium of E 4 , then there exists π ∈ P such that (p ⋆ , C ⋆ ) is an equilibrium of the vNM economy in which all agents have utility index U i and probability π. Furthermore, p ⋆ = q ⋆ π 1 , q ⋆ π 2 , . . . , q ⋆ π k with π ∈ P and q ⋆ = λ i ∇U i (C ⋆ i ), λ i ∈ IR + for all i.

Proof: (i) Let (p ⋆ , C ⋆ ) be an equilibrium of a vNM economy in which all agents have beliefs given by π. We have, C j⋆ i = C j ′ ⋆ i for all j, j ′ and all i. By definition of an equilibrium, i C j⋆ i = i w j i for all j and, for all i:

C ′ i ≥ 0, p ⋆ C ′ i ≤ p ⋆ w i ⇒ E π (U i (C ′ i )) ≤ E π (U i (C ⋆ i )) Now, since π ∈ P, v i (C ′ i ) ≤ E π (U i (C ′ i )). Notice that v i (C ⋆ i ) = E π (U i (C ⋆ i )). Hence, v i (C ′ i ) ≤ v i (C ⋆ i )
, and (p ⋆ , C ⋆ ) is an equilibrium of E 4 . (ii) Let (p ⋆ , C ⋆ ) be an equilibrium of E 4 . Assume ν i is convex and U i satisfy Um for all i. Then C ⋆ i ≫ 0 for all i. From proposition 5.2 and the first theorem of welfare, C j⋆ i = C j ′ ⋆ i for all j, j ′ and all i. From first-order conditions and Um, there exists λ i ∈ IR + for all i, such that p ⋆ ∈ λ i ∂v i (C ⋆ i , . . . , C ⋆ i ). Therefore, p ⋆ = λ i ∇U i (C ⋆ i ) π 1 i , . . . , λ i ∇U i (C ⋆ i ) π k i where π i ∈ core(ν i ). Summing over p ⋆ 's components, one gets:

j λ i ∇U i (C ⋆ i ) π j i = j p j⋆ = j λ i ′ ∇U i ′ (C ⋆ i ′ ) π j i ′ that is: λ i ∇U i (C ⋆ i ) = λ i ′ ∇U i ′ (C ⋆ i ′ ) ∀ i, i ′
Hence, π j i is independent of i for all j, i.e. π i ∈ ∩ i core(ν i ) = P. Let q ⋆ = λ i ∇U i (C ⋆ i ) and π = π i . One gets p ⋆ = q ⋆ π 1 , q ⋆ π 2 , . . . , q ⋆ π k with π ∈ P. It follows from proposition 5.1 that (p ⋆ , C ⋆ ) is an equilibrium of the vNM economy with utility index U i and probability π.

This proposition suggests equilibrium indeterminacy if P contains more than one probability distribution. This road is explored further in Tallon [1997] and [START_REF] Dana | Pricing rules when agents have non-additive expected utility and homogeneous expectations[END_REF]. A direct corollary concerns existence:

Corollary 5.1 Under C, there exists an equilibrium of E 4 .

Hence, since capacities satisfying assumption C need not be convex, convexity of the preferences (which is equivalent, in the CEU setup, to the convexity of the capacity and concavity of the utility index, see [START_REF] Chateauneuf | Diversification, convex preferences and non-empty core[END_REF]) is not necessary to prove that an equilibrium exists in this setup. [START_REF] Malinvaud | Markets for an exchange economy with individual risks[END_REF] noticed that P.O. allocations could be decentralized through insurance contract in a large economy. [START_REF] Cass | Individual risk and mutual insurance[END_REF] showed, in an expected utility framework, how this decentralization can be done in a finite economy: agents of the same type share their risk through (actuarially fair) mutual insurance contract. The same type of argument could be used here in the Choquet expected utility case. It is an open issue whether P.O. allocations can be decentralized with mutual insurance contract where agents in the same pool have different capacities.

  The same is true for curve (c) and zone (3). On the other hand, part of curve (b) is

  Figure 1:

  b

  4 if and only if it is a P.O. of a vNM economy with utility index U i and identical probability over the set of states of the world. Hence, P.O. are independent of the capacities.

	Proposition 5.2 Let C hold. Then,
	(i) Any P.O. (C i ) n i=1 of E 4 is such that C j i is independent of j for all i and C 1 i n i=1 is a P.O. of the static economy in which agents have utility function (U i ) n i=1 .
	(ii) The allocation (C i ) n i=1 ∈ IR kmn

+ is a P.O. of E

See Schmeidler [1989],[START_REF] Ghirardato | Coping with ignorance: unforeseen contingencies and nonadditive uncertainty[END_REF],[START_REF] Mukerji | Understanding the nonadditive probability decision model[END_REF].

In section 5, we will deal with several goods and will introduce the appropriate notation at that time.

It is well-known (see[START_REF] Schmeidler | Integral representation without additivity[END_REF]) that when ν is convex, the Choquet integral of any random variable f is given by f dν = min π∈core(ν) Eπf .

[START_REF] Borch | Equilibrium in a reinsurance market[END_REF] noted that, in a reinsurance market, at a P.O., "the amount which company i has to pay will depend only on (...) the total amount of claims made against the industry. Hence any Pareto optimal set of treaties is equivalent to a pool arrangement." Note that this corresponds to the characterization of comonotone variables as stated in proposition 2.2.

For a study of optimal risk-sharing without aggregate uncertainty and an infinite state space, see[START_REF] Billot | Sharing beliefs: between agreeing and disagreeing[END_REF].

Recall that we are dealing with capacities such that 1 > ν(A) > 0 for all A = ∅ and A = S.

Proposition 4.4 Assume n = 2, w 1 < w 2 and that agents have capacities ν i , i = 1, 2 which fulfill C, and such that ν 2 1 < ν 2 2 . Assume finally U1 and let (C i ) i=1,2 be a P.O. of E 3 . Then, there are only two cases:

,2 is a P.O. of the vNM economy with utility index U i and probabilities

(1)

Proof: It follows from proposition 4.3 that there are three cases :

; hence the right-hand side of (2) is fulfilled and (1) is equivalent to (2). Lastly, the case 0 < C 1 2 = C 2 2 is symmetric. The first-order corresponding conditions imply ν 2 1 ≥ ν 2 2 which contradicts our hypothesis.

We can illustrate the optimal risk-sharing arrangement just derived in an Edgeworth box. Figure 2 (a) represents case (i) and the optimal contract is the same as the one in the associated vNM setup.

However, figure 2 (b) gives a different risk-sharing rule, that can interpreted as follows. The assumption ν 2 1 < ν 2 2 is equivalent to E ν 1 (x) ≤ E ν 2 (x) for all x comonotone with w. But we have just shown that we could restrict our attention to allocations that are comonotone with w. Hence, the assumption ν 2 1 < ν 2 2 can be interpreted as a form of pessimism of agent 1. Under that assumption, agent 2 never insures himself completely, whereas agent 1 might insure do so. This is incompatible with a vNM economy with strictly concave and differentiable utility indices.

Finally, risk-sharing arrangements such as the one represented on figure 3 (a) cannot be excluded a priori, i.e., there is no reason that the contract curve in the vNM economy crosses agent 1's diagonal only once. On figure