Aloisio Araujo
email: aloisio@impa.br

Jean-Marc Bonnisseau
email: jean-marc.bonnisseau@univ-paris1.fr

Alain Chateauneuf
email: alain.chateauneuf@univ-paris1.fr

Rodrigo Novinski
email: rodrigo.novinski@ibmecrj.br

Optimal Risk Sharing with Optimistic and Pessimistic Decision Makers

We prove that under mild conditions individually rational Pareto optima will exist even in presence of non-convex preferences. We consider decision makers dealing with a countable flow of payoffs or choosing among financial assets whose outcomes depend on the realization of a countable set of states of the world. Our conditions for the existence of Pareto optima can be interpreted as a requirement of impatience in the first context and of some pessimism or not unrealistic optimism in the second context. A non-existence example is provided when, in the second context, some decision maker is too optimistic.

We furthermore show that at an individually rational Pareto optimum at most one strictly optimistic decision maker will avoid ruin at each state or date. Considering a risky context this entails that even if risk averters will share risk in a comonotonic way as usual, at most one classical strong risk lover will avoid ruin at each state or date. Finally some examples illustrate circumstances when a risk averter could take advantage of sharing risk with a risk lover rather than with a risk averter.

Introduction

Our aim is to study how the interaction between optimism and pessimism (and also risk propensity and risk aversion) could affect the allocation of economic resources. In order to do this, we will analyze which are the properties of the socially desired allocations, in the sense of being Pareto optimal (PO) for which any decision maker (DM) of the economy is not worse than with the initial endowments of goods. One should remark that the presence of non-convex preferences can cause technical issues to ensure the existence of general equilibrium for the respective economy, but even in these cases, it is possible to ask about the existence of individually rational Pareto optimal (IRPO) -which is a lesser demanding concept than equilibrium, as it does not require the existence of a price system -and on the properties that should be verified by such allocations.

We will model a pure exchange economy as did [START_REF] Bewley | Existence of equilibria in economies with infinitely many commodities[END_REF], with a finite number of DMs and an infinite dimensional consumption space. More specifically, the consumption space is the set of non-negative real bounded sequences ∞ + , and, thus, we can encompass the analysis of a decision problem with infinitely many states or with an infinite flow of payments along the time.

This paper has two main results. The first is about the existence of IRPO allocations for economies without the usual requirements as completeness or convexity over decision makers' preferences. A central assumption for this result is that preferences display weak* sequential upper semi-continuity. Actually, we show that in the particular case of ∞ the standard result in the literature of [START_REF] Aliprantis | Locally solid Riesz spaces with applications to economics[END_REF] ensuring the existence of an IRPO allocation remains valid by replacing their assumption of weak* upper semi-continuity by our weaker one. Furthermore, we show that by merely adding the simple needed upper norm-continuity usual condition our central assumption is equivalent to the important notion of upper semi-myopia which was introduced in [START_REF] Brown | Myopic economic agents[END_REF]. It turns out that both in the case of an infinite flow of payments or in the case of assets whose outcomes depend on the realization of a countable set of states, upper semimyopia prevents cases of unreasonable optimism. We give several examples of preferences (e.g. Choquet, à la Yaari, expected utility) that satisfy this assumption and illustrate that its lack (i.e. the presence of some extreme optimistic DM) may lead to economies with no IRPO allocation. Indeed, [START_REF] Araujo | Lack of Pareto optimal allocations in economies with infinitely many commodities: The Need for Impatience[END_REF] provides examples of economies for which there is no IRPO due to the lack of impatience1 .

We then consider the presence of reasonably strictly optimistic DM. Our second main result, Theorem 2, shows that at IRPO allocations at most one such strictly optimistic DM will always avoid ruin (in the sense of having an interior outcome plan).

Then specializing in the context of risk, we obtain that as in the standard case with finite states and without non-convexities studied by [START_REF] Chateauneuf | Optimal risk-sharing rules and equilibria with Choquet-expected-utility[END_REF], at an IRPO, the allocations of the risk averters are comonotonic. The main feature of risk-sharing between risk averters and risk lovers as settled in Theorem 3, is that for classical strong risk lovers, i.e. those with strictly concave preferences, again ruin could be avoided by at most one of them. We finally illustrate through some examples that a risk averter under some circumstances could take advantage of sharing risk with a risk lover rather than with a risk averter.

The paper is structured as follows: In Section 2 we present sufficient conditions in order to guarantee that there are IRPO and provide the proof of existence. In Section 3 we relate the condition of weak* sequentially upper semi continuity of preferences and the notions of myopia and impatience. Section 4 gives some examples of weak* sequentially upper-semicontinuous preferences. In section 5 we analyze the optimal allocations of strictly optimistic DM and then specialize to the context of risk. Finally, Section 6 aims to give some intuition about circumstances when it is beneficial for a risk averter to share risk with a risk lover.

Existence of IRPO allocations under weak* sequentially upper-semicontinuous preferences

There are some standard results in the literature on the existence of an IRPO allocation for the economies on ∞ (and other infinite dimensional consumption spaces).

For instance, [START_REF] Yannelis | The core of an economy without ordered preferences[END_REF] presents a related result 2 for non-ordered preferences on very general consumption spaces under the requirement of a weak form of convexity on these preferences (in the sense that any plan x can not belong to the convex hull of its strict upper contour set). [START_REF] Aliprantis | Locally solid Riesz spaces with applications to economics[END_REF]) shows the existence even when preferences on ∞ may be non-convex under the assumption of weak* upper semicontinuity. In Theorem 1 below, we show that this assumption can be replaced by the weaker assumption of weak* sequential upper semi-continuity 3 . The motivation for this assumption is given in Proposition 1 at Section 3, where it is proved by merely adding simple usual conditions that it is equivalent to the important notion of upper semi-myopia introduced in Brown and Lewis (1981)4 .5 Some readers might wonder why we do not attempt to prove existence of IRPO allocations for the usually greatest class of Mackey sequentially upper-semicontinuous preferences. In fact, it turns out that in the particular case of ∞ the weak* σ( ∞ , 1 ) sequential convergence and the Mackey τ ( ∞ , 1 ) sequential convergence coincide, which is an immediate application of the corollary in appendix of [START_REF] Hervés-Beloso | Blocking efficacy of small coalitions in myopic economies[END_REF] that states that "the Mackey topology and the weak* topology 2 More precisely, the Theorem 4.1 of this work shows sufficient conditions for the existence of extreme α-core allocations, a notion that in our framework coincides with the definition of IRPO allocations when preferences are complete.

3 One will find in Appendix A some brief recalls concerning ∞ the Banach space of real bounded sequences, and also the weak* topology and the Mackey topology, which are respectively the coarsest and the finest topology on ∞ for which the dual is 1 the Banach space of absolutely convergent real sequences.

coincide on bounded subsets of ∞ ". This justifies our choice of stating our result with the equivalent and more well-known weak* σ( ∞ , 1 ) sequential convergence6 . For sake of completeness we additionally give in Appendix B a direct and elementary proof of this "Folk Theorem" about equivalence on ∞ of both sequentially convergences mentioned.

Assumptions

(1) For every i = 1, . . . , m, i is a transitive reflexive preorder on ∞ + .

(2) For every i, every

y i ∈ ∞ + , {x i ∈ ∞ + | x i i y i } is weak* sequentially closed. Let w i ∈ ∞ + be the initial endowments of individual i = 1, ..., m. Let A = {(x 1 , ..., x i , ..., x m ) ∈ ( ∞ + ) m : m i=1
x i = w where w := m i=1 w i } be the set of feasible allocations.

Definition 1. A feasible allocation (x 1 , ..., x i , ..., x m ) is said to be:

1. Pareto optimal (PO) if there is no other allocation (z 1 , ..., z m ) satisfying z i i x i ∀i and z k k x k for some k.

2. Individually rational Pareto optimal (IRPO) if it is PO and x i i w i ∀i.

3. The set of all individually rational allocations is denoted A r , i.e. ,

A r = {(x 1 , ..., x m ) ∈ A : x i i w i for each i}.

By a special case of Theorem 8.12 in [START_REF] Aliprantis | Locally solid Riesz spaces with applications to economics[END_REF] (i.e., L = ∞ , X = 1 ) we have that an IRPO allocation exists if (1) holds and preferences are weak* upper semicontinuous on ∞ + . One can easily observe that the proof of this result also works if we assume that preferences satisfy the corresponding continuity assumption only on A r . This observation is of great importance in our model, since due to weak* metrizability of the unit ball in ∞ it follows that preferences which satisfy assumption (2) are weak* upper semicontinuous on A r . The details follow in the next proof.

Theorem 1. Under Assumptions (1) and (2), IRPO allocations exist.

Proof: By the Banach-Alaoglu Theorem, the Edgeworth box [0, ω] = {x ∈ ∞ : 0 ≤ x ≤ ω} is weak* compact in ∞ . Thus by applying Theorem 8.10 in [START_REF] Aliprantis | Locally solid Riesz spaces with applications to economics[END_REF], for L = ∞ and X = 1 we have that

A is P = σ(( ∞ ) m , ( 1 ) m )-compact. Let π = sup n∈N w(n) and note that A r = A ∩ m i=1 ({x i ∈ ∞ + , x i i w i } ∩ B(0, π)).
Since the weak* topology on B(0, π) is metrizable from Theorem 6.30 in [START_REF] Aliprantis | Infinite dimensional analysis: a hitchhiker's guide[END_REF], it turns out that {x

i ∈ ∞ + , x i i w i } ∩ B(0, π) is weak* compact since {x i ∈ ∞ + , x i i w i } is weak* sequentially closed. Therefore m i=1 ({x i ∈ ∞ + , x i i w i } ∩ B(0, π)) is P-closed, hence A r is P-closed in A. Thus A r is P-compact.
The end of the proof is similar to Aliprantis and Burkinshaw's one while using the weaker assumption of weak* sequentially upper semicontinuity. Let us prove now that individually rational Pareto efficient allocations exist.

Let us define a partial preorder on A r by: (x 1 , ..., x i , ..., x m ) (y 1 , ..., y i , ..., y m ) if

x i i y i , ∀ i = 1, ..., m.
So an individually rational Pareto efficient allocation exists if and only if A r has a maximal element.

From Zorn's lemma, we know that the preordered set (A r , ) will have a maximal element if every totally preordered subset of (A r , ) is majorized.

Let (ζ α ) be a totally preordered subset of (A r , ) and define, for each α, C α = {t ∈ A r , t ζ α }. From weak* sequential upper-semicontinuity of each preference i, it turns out as above that C α is a P-closed subset of the P-compact set A r , hence C α is P-compact.

Since (ζ α ) is totally preordered, it comes that the intersection of a finite number of C α is non-empty. Consequently, C = α C α is non-empty, and any t ∈ C majorizes (ζ α ), so a maximal element does exist and consequently there exist IRPO allocations.

allocations under upper semi-myopic preferences

We now recall the central notion introduced by [START_REF] Brown | Myopic economic agents[END_REF] (see also [START_REF] Araujo | Lack of Pareto optimal allocations in economies with infinitely many commodities: The Need for Impatience[END_REF]) under the denomination of strong myopia (or in a sequential context, strong impatience) and quoted also in [START_REF] Araujo | General equilibrium, wariness and efficient bubbles[END_REF] under the denomination of upper semi-myopia. In order to do that, let us state a bit of notation:

given x ∈ ∞ + and E ⊂ N, x E denotes the sequence such that x E (s) = x(s) if s ∈ E and x E (s) = 0 otherwise. Moreover, E n is the set of all natural numbers greater than n.

Lastly, 1 1 1 is the real sequence whose terms are all equal to one.

Definition 2. A preference relation i on ∞ + , is upper semi-myopic if ∀(x, y) ∈ ∞ + × ∞ + , x i y, z ∈ ∞
+ , implies that there exists n large enough such that x i y + z En .

Note that in case of preferences over a countable flow of payoffs, the previous definition clearly fits to the notion of upper semi-myopia, while in case of preferences over financial assets whose outcomes depend on the realization of a countable set of states of the world the previous definition would model neglecting gains on "small events" i.e. some pessimism or not unreasonable optimism.

In other words upper semi-myopia prevents unreasonable optimism since it prevents overweighting outcomes arising in a very far future, in case of a flow of payments, and arising on "small events", in the case of assets whose outcomes depend on the realization of a countable set of states.

We now make precise the relationship between weak* sequentially upper semi-continuity and upper semi-myopia.

Definition 3. A preference relation on ∞ + is upper norm-continuous if for every y ∈ ∞ + , x ∈ ∞ + | x y is norm-closed.
Throughout the paper we will assume that the preferences i are complete and transitive (i.e. a total preorder) and monotone (i.e. x, y ∈ ∞ + , x ≥ y ⇒ x y). It turns out that:

Lemma 1. Let on ∞ + satisfying the previous hypotheses then (1) implies (2):

(1) is weak * sequentially upper semi-continuous.

(2) is upper norm-continuous and upper semi-myopic. -→ 0 then, from (1), y + 0 x i.e. y x a contradition, which completes the proof.

Proof: Let x m ∈ ∞ + , where x m tends in norm towards x ∈ ∞ + , denoted x m • -→ x,
For the converse of Lemma 1, let us introduce:

Definition 4. A preference relation on ∞ + is finitely upper norm-continuous if for every y ∈ ∞ + and any sequence (x m ), x m ∈ ∞ + such that x m y ∀m where x m (k) =      x m (k) k ∈ K finite ⊆ N x(k) k ∈ K and ∃ lim m→∞ x m (k) = x(k) ∈ R + ∀k ∈ K one has: x y.
One obtains:

Lemma 2. Let be a monotone total preorder on ∞ + then (ii) ⇒ (i):

(i) is weak * sequentially upper semi-continuous.

(ii) is finitely upper norm-continuous and upper semi-myopic.

Proof: Assume ∃ x ∈ ∞ + , sequence (y m ) such that y m ∈ ∞ + ∀m with y m w * -→ y, y ∈ ∞ +
, where y m x ∀m and x y. Since y m w * -→ y, the sequence is bounded let us say by k ∈ R + , hence ∀n ∈ N: y m ≤ (y m (1), ..., y m (n), k, k, ...), so from monotonicity:

x (y m (1), ..., y m (n), k, k, ...), and for any fixed n, finitely upper norm-continuity implies

x (y 1 , ..., y n , k, k, ...). But upper semi-myopia implies that there exists n ∈ N such that: (y 1 , ..., y n , y n+1 + k, y n+2 + k, ...) ≺ (y 1 , ..., y n , k, k, ...) a contradition with respect to monotonicity which completes the proof.

While it is clear from Lemma 1 and Lemma 2 that weak * sequentially upper semicontinuity is characterized by upper semi-myopia and finitely upper norm-continuity, we nevertheless prefer to state the following proposition which underlines the necessity of upper norm-continuity.

Proposition 1. Let be a monotone total preorder on ∞ + . Then, is weak * sequentially upper semi-continuous if and only if is upper norm-continuous and upper semi-myopic.

Note that even for a monotone total preorder, upper semi-myopia is not sufficient for weak * sequential upper semi-continuity as illustrated by the Example 1 below.

Example 1. Let be the lexicographical order on ∞

+ . For x, y ∈ ∞ + with x y if there exist k o ∈ N such that x(k) = y(k) for 1 ≤ k ≤ k 0 and x(k 0 + 1) > y(k 0 + 1), x ∼ y if x(k) = y(k) ∀k ∈ N.
exhibits upper semi-myopia, actually let x, y ∈ ∞ + with x y as defined above, clearly if z ∈ ∞ + then x y + z En for n ≥ k 0 + 2. is not upper norm-continuous, hence not weak * sequentially upper semi-continuous (see Proposition 1). Actually consider x m y, with x m , y ∈ ∞ + where x m (1) = y(1)

+ 1 m , x m (2) = x(2) < y(2), x m (k) = y(k) ∀k ≥ 3, clearly x m • -→ x where x(1) = y(1) but x ≺ y.
Note also that even for a monotone total preorder upper norm-continuity does not imply weak * sequential upper semi-continuity as illustrated by Example 2 below.

Example 2. Let z be a positive sequence of ∞ + (z k > 0 for all k ∈ N) and define the (utility

) function U from ∞ + to R + by U (x) = sup k∈N (z k • x k ). Define on ∞ + by x, y ∈ ∞ + , x y ⇐⇒ U (x) ≥ U (y). Observe that is upper norm-continuous since U is norm-continuous. In fact, let ε > 0 and take x, y ∈ ∞ + such that x -y ≤ ε. Therefore y k -ε ≤ x k ≤ y k + ε for all k, implying that y k • z k -ε • z k ≤ x k • z k ≤ y k • z k + ε • z k for all k. Hence |sup k (x k z k ) -sup k (y k z k )| ≤ ε z i.e., U (x) -U (y) ≤ ε z .
Let us prove now that if z does not converge to 0, then is not weak * sequential upper semi-continuous.

Consider P = {x ∈ ∞ + : U (x) ≥ 1} and let us prove that P is not weak * -sequentially closed. Since z does not converge to 0, there exists a subsequence (z ϕ(k) ) and t > 0 such that for all k, z ϕ(k) ≥ t. We consider the sequence (

x k ) in ∞ + defined by x k (j) = 0 if j = ϕ(k) and x k (ϕ(k)) = 1 t . Then for all k, u(x k ) = z ϕ(k) t ≥ 1, so x k ∈ P.
But (x k ) is bounded and lim k→∞ x k (j) = 0 for all j ∈ N, so x k w * -→ 0, but 0 / ∈ P which completes the proof.

In accordance with Proposition 1, we now assume throughout the paper that the preferences satisfy Assumption A.1:

Assumption A.1: Preferences i on ∞ + are complete, transitive, monotone and upper norm-continuous.

The Corollary 1 below thus straightforwardly characterizes a large class of preferences for which one can guarantee the existence of IRPO allocations.

Corollary 1. If all the preferences i i = 1, ..., m of the decision makers satisfy A.1 then are equivalent: We first illustrate the fact that our Theorem 1 is a true improvement of [START_REF] Aliprantis | Locally solid Riesz spaces with applications to economics[END_REF] -at least in the case of ∞ -by exhibiting weak * sequentially upper semi-continuous preferences on ∞ + which are not weak * upper-semicontinuous.

1. i is weak * sequentially upper semi-continuous. 2. i is upper semi-myopic.
Example 3. Consider again as in the Example 2 the (utility) function U from ∞ + to R + defined by U (x) = sup k∈N (z k x k ) where z is a positive sequence of ∞ which now converges to 0, but is assumed not to be in 1 (that is, ∞ k=1 z k = +∞). Note that such a utility function is meaningful particularly when valuing flow of payoffs, since it models a decision-maker which discounts "severely" the future (since lim k z k = 0), while expressing some minimal requirement of continuous behaviour ( is upper normcontinuous, from Example 2).

Let us prove first that the monotone total preorder induced by U on ∞ + is weak * sequentially upper semi-continuous. It is enough to show that for a > 0, the set P = {x ∈ ∞ + : U (x) ≥ a} is sequentially weak * -closed. Let (x m ) be a sequence of elements of P weak * converging to x ∈ ∞ + . Then, the sequence (x m ) is bounded and for all k ∈ N, lim m x m k = x k . Let r > 0 be such that x m ∈ B ∞ (0, r) for all m, which means that for all m and for all k,

x m (k) ≤ r. Let k ∈ N be such that r < a 2z k for all k ≥ k + 1. So for all k ≥ k + 1, for all m we have z k x m k < a 2 . For all m, since x m ∈ P, there exists k ≤ k such that z k • x m k ≥ a. So, there exists a subsequence (x ϕ(m) ) of (x m ) and k 0 ≤ k such that for all m, z k 0 • x ϕ(m) k 0 ≥ a. Consequently, since x k 0 = lim m x ϕ(m) k 0 , one concludes that z k 0 •x k 0 ≥ a, hence x ∈ P, which means that P is sequentially weak * -closed.
We now prove that is not weak * upper semi-continuous. It is enough to show that taking a = 1, P is not weak * -closed. This will be performed by showing that 0 belongs to the weak * closure of P whereas 0 ∈ P. Let G be a weak * neighborhood of 0. There exists ε > 0 and a finite family

(α i ) I i=1 of elements of 1 such that V = {x ∈ ∞ | ∀i = 1, ..., I, |α i (x)| < ε} ⊂ G.
Let us prove by contraposition that V ∩ P = ∅, which implies G ∩ P = ∅. For all m ∈ N, let x m defined by x m k = 0 if k = m and x m k = 1 zm . Note that x m ∈ P. If V ∩ P = ∅, x m does not belong to V , so, there exists i ∈ {1, ..., I} such that

|α i m | z m ≥ ε. In others words, for all m ∈ N, sup i=1,...,I { α i m } ≥ εz m .
So, the sequence α = sup i=1,...,I {α i m } m is not in 1 since it is larger for the pointwise order than εz, which is not in 1 . This is in contradiction with the fact that the pointwise supremum of a finite family of sequences in 1 is also in 1 .

Some others examples of weak * sequentially upper-semicontinuous preferences

In order to model preferences under uncertainty or even preferences on flow of payments it has been suggested (see [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF] for the first circumstance and [START_REF] Gilboa | Expectation and variation in multi-period decisions[END_REF] or else [START_REF] Chateauneuf | Some characterizations of non-additive multi-period models[END_REF] in the second case) to use the Choquet integral [START_REF] Choquet | Théorie des capacités[END_REF]).

In our framework every x ∈ ∞ + would be valued through I(x) = x dϑ where ϑ is a capacity on A = 2 N i.e. a set-function ϑ :

A ∈ A → ϑ(A) ∈ [0, 1] such that ϑ(∅) = 0, ϑ(N) = 1 and A, B ∈ A. A ⊂ B ⇒ ϑ(A) ≤ ϑ(B) 7 and the Choquet integral of x w.r.t. to ϑ is defined as x dϑ = +∞ 0 ϑ(x ≥ t) dt. Proposition 2. A Choquet preference on ∞ + is weak* sequentially upper semi-continuous if and only if ϑ is outer-continuous i.e. ∀ A n , A ∈ A, A n ↓ A ⇒ ϑ(A n ) ↓ ϑ(A).
Proof: It is well-known that preferences represented by the Choquet integral satisfy Assumption 1, therefore, from Proposition 1, Choquet preferences i on ∞ + are weak* sequentially u.s.c. iff i is upper semi-myopic. So, let us prove that a Choquet preference

on ∞ + is upper semi-myopic if and only if ϑ is outer-continuous, i.e. ∀ A n , A ∈ A, A n ↓ A ⇒ ϑ(A n ) ↓ ϑ(A).
Outer-continuity is necessary. Actually let A n , A ∈ A be such that A n ↓ A and assume that ϑ(A n ) ↓ k with k > ϑ(A). Therefore we get k 1 1 1 N i 1 1 1 A .

Let B n = A n \A, one gets B n ↓ φ. So one can find an increasing sequence of integers n(m),

increasing with m ∈ N such that 1 1 1 Em ≥ 1 1 1 Bn(m) where E m = {p ∈ N, p ≥ m}. Under upper semi-myopia ∃ m 0 such that k 1 1 1 N i 1 1 1 A + 1 1 1 Em 0 but 1 1 1 Em 0 ≥ 1 1 1 Bn(m 0 ) ⇒ 1 1 1 A + 1 1 1 Em 0 ≥ 1 1 1 A + 1 1 1 A n(m 0 )\A = 1 1 1 An(m 0 )
. Hence by monotonicity of the Choquet integral one would get

k 1 1 1 N i 1 1 1 An(m 0 ) but I 1 1 1 An(m 0 ) = ϑ(A n(m 0 ) ) ≥ k and I k 1 1 1 N ) = k, a contradiction.
Let us prove now that outer-continuity of ϑ is a sufficient condition.

It is enough to see that ∀ y, z ∈ ∞ + , one gets I(y + z En ) → I(y) when n → +∞ one has:

I(y + z En ) = +∞ 0 ϑ(y + z En ≥ t) dt.
Let A n (t) = {y + z En ≥ t} and A(t) = {y ≥ t}. It is straightforward to see that

A n (t) ↓ A(t) ∀ t ∈ R + . From outer-continuity of ϑ, it turns out that ∀ t ∈ R + , f n (t) = ϑ(A n (t)) ↓ f (t) = ϑ(A(t)).
Thus by the Dominated Convergence Theorem I(y + z En ) → I(y).

Imagine now that we are in situation of risk, i.e., there is a given σ-additive probability P on (N, 2 N ), with P ({n}) > 0 ∀ n. Let us first consider Yaari's preference [START_REF] Yaari | The dual theory of choice under risk[END_REF]), that is any x ∈ ∞ + is valued through I(x) = x dϑ where ϑ = f • P and f the distortion function: f : [0, 1] → [0, 1] satisfies f (0) = 0, f (1) = 1 is strictly increasing, and continuous.

From Proposition 2, one gets immediately: Proposition 3. A preference à la Yaari on ∞ + is weak* sequentially upper semi-continuous.

Let us finally consider the classical expected utility preferences under risk.

In such a case

I(x) = ∞ n=0 P ({n})u(x(n))
where u is assumed to be strictly increasing and continuous. It is immediate to check that such preferences satisfy Assumption 1.

In fact we obtain:

Proposition 4. A EU (expected utility) preference on ∞ + is weak* sequentially upper semi-continuous. Let us study now the important case of optimistic and pessimistic DMs.

Optimistic and pessimistic DMs

The concepts of pessimism and optimism presented in Definitions 5 and 6 below are mainly motivated by the evidence that such opposite behaviours can be observed specially under uncertainty, hence by the application of Theorem 1 more specifically to the case in which x ∈ ∞ + is interpreted as a plan of outcomes contingent to the realization of one of the (infinitely countable) possible states of nature.

Definition 5. A DM will be said pessimistic if her preferences are convex i.e. ∀ x, y ∈ ∞ + , α ∈ (0, 1) x i y ⇒ αx + (1 -α)y i y. Definition 6. A DM will be said optimistic if her preferences are concave i.e. ∀ x, y ∈ ∞ + , α ∈ (0, 1)

x i y ⇒ x i αx + (1 -α)y.
Note that pessimism expresses that smoothing payoffs can only make the DM better off, while for an optimistic DM this will never be the case. This feature associates the optimism to a willingness to bet. For instance, if x o is an alternative that delivers two units of outcome if the state is odd (and zero otherwise) and x e is the analogous alternative that delivers on even states, an optimistic DM would prefer to bet on x o or x e than receive a certain amount of 1 unit of outcome.

Under uncertainty convexity of preferences is usually called uncertainty aversion [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF], [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF]) while concavity is called uncertainty loving (see also [START_REF] Wakker | Characterizing optimism and pessimism directly through comonotonicity[END_REF] and the seminal survey of [START_REF] Gilboa | Ambiguity and the Bayesian Paradigm[END_REF]). It is well-known that, for Choquet preferences, optimism is equivalent to a convex capacity and pessimism is equivalent to a concave capacity (see Definition 7 below), for Yaari preferences these two notion are respectively equivalent to f convex and f concave, and that for EU DM this amounts to respectively a concave or a convex VNM utility function u.

Definition 7. A capacity ϑ is convex (resp. concave) if ∀ A, B ∈ A ϑ(A∪B)+ϑ(A∩B) ≥ ϑ(A) + ϑ(B) (resp: ϑ(A ∪ B) + ϑ(A ∩ B) ≤ ϑ(A) + ϑ(B)).
Let us recall that for Choquet preferences optimism and pessimism are well motivated by the following properties [START_REF] Schmeidler | Integral representation without additivity[END_REF][START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF]):

If ϑ convex xdϑ = min{E P (x)|P additive probability ≥ ϑ}; If ϑ concave xdϑ = max{E P (x)|P additive probability ≤ ϑ},
where E P (x) denotes the mathematical expectation of x with respect to P .

Existence of IRPO in presence of upper semi myopic pessimistic and reasonably optimistic DMs

Definition 8. An optimistic DM is said to be reasonably optimistic if her preference satisfies A1 and upper semi-myopia.

Thus the reasonably optimistic DMs will not want to exchange all their wealth for a larger income on events with arbitrarily small likelihood or dates arbitrarily far. That prevents excessive optimism, avoiding the difficulties that will be presented in Example 1 of Subsection 5.2.

From the previous developments, the following DMs belong to the class of reasonable optimists:

1) Any optimistic EU (see Proposition 4)

2) Any optimistic Choquet DM such that

A n , A ∈ A, A n ↓ A ⇒ ϑ(A n ) ↓ ϑ(A) (see Proposition 2).
An example of such preferences may be obtained by defining ϑ = f • P where P σ-additive on (N, 2 N ), f concave and right-continuous. Many other less specific examples of such capacities could be exhibited 3) Any optimistic Yaari's DM.

The terminology reasonably optimistic can be justified for a Choquet optimistic DM by the fact that it merely imposes that if an event decreases towards the empty set she reasonably appreciates this event less and less and is valued 0 at the limit as seen in the next lemma.

Lemma 3. For a concave capacity the following assertions are equivalent:

(i) A n , A ∈ A, A n ↓ A ⇒ ϑ(A n ) ↓ ϑ(A) (ii) A n ∈ A A n ↓ ∅ ⇒ ϑ(A n ) ↓ 0
Proof: Such a property is a direct consequence of a dual similar property proved by [START_REF] Rosenmüller | Some properties of convex set functions[END_REF] for the dual case of convex capacities.

So trivially, from Corollary 1, one gets:

Proposition 5. If all decision makers have preferences satisfying A1 and are either upper semi-myopic pessimistic or reasonably optimistic, then individually rational Pareto optimum exists.

Thus, it is possible to ensure the existence of IRPO when the optimistic DMs are also upper semi-myopic. Indeed, as can be seen in Example 4 of the next section, some boundedness on the optimism is required in order to guarantee such existence.

IRPO can fail to exist in the presence of a too optimistic DM

The next example shows an economy for which the unique PO is the trivial one that assigns the whole aggregate endowment for the DM 1.

In this example the nonexistence of an IRPO comes from the presence of an unreasonably optimistic DM. Namely DM 2 exhibits preferences represented by the Choquet integral U 2 (x) = x dϑ where ϑ is the concave capacity defined by ϑ(A) = 1 ∀ A = φ, so clearly condition (ii) of Lemma 3 is not satisfied. Indeed since U 2 (x) = sup s∈N x(s), such a DM is extremely optimistic.

Example 4. Nonexistence of IRPO Consider the consumption space X = ∞ + and the probability measure p on 2 N given by

p s = 1 2 s ∀ s ∈ N.
• The DM 1 is characterized by the utility function U 1 (x) = s∈N x s p s and the initial endowments

ω 1 = (2 -1 s ) s∈N ;
• The DM 2 is characterized by the utility function U 2 (x) = sup s∈N x s and the initial endowments ω 2 = (2 -1 s ) s∈N ;

Affirmation: The unique Pareto optimal allocation is the couple (x 1 , x 2 ) that associates

x 1 = ω 1 + ω 2 to DM 1 and x 2 = 0 to DM 2. Thus, there is no IRPO.

In fact, consider a feasible allocation (x 1 , x 2 ) with x 2 > 0. If sup

s∈N x 2 s < sup s∈N (ω 1 s + ω 2 s ) = 4, take s 0 ∈ N be such that x 2 s 0 > 0. There is s 1 ∈ N large enough such that U 1 (x 1 + x 2 s 0 e s 0 - x 1 Es 1 ) > U 1 (x 1 ). As U 2 (x 2 -x 2 s 0 e s 0 + x 1 Es 1 ) = 4 > U 2 (x 2 ), we have that (x 1 , x 2 ) is not a PO allocation. If sup s∈N x 2 s = 4, there is n 0 ∈ N such that 0 < x 2 n 0 < 4, so we would get U 1 (x 1 + x 2 n 0 e n 0 ) > U 1 (x 1 ) and U 2 (x 2 -x 2 n 0 e n 0 ) = U 2 (x 2
), so, again, (x 1 , x 2 ) is not a PO allocation. Thus, the unique PO allocation assigns x 2 = 0.

At an IRPO at most one strict optimistic will avoid ruin

The results in this section will require a strengthen notion of optimism that is provided in Definition 9. Definition 9. A DM is strictly optimistic if ∀ x, y ∈ ∞ + , x = y and ∀ α ∈ (0, 1)

x i y ⇒ αx + (1 -α)y ≺ i x 8 .
As before (Definition 8), we will prevent extreme forms of optimism: Definition 10. A strictly optimistic DM is said to be reasonably strictly optimistic if her preference satisfies A1 and upper semi-myopia.

In this section as in Section 5.4, when we consider a strictly optimistic DM, we will assume that her preference i satisfies A1 but with the requirement of strict monotonicity that is x, y ∈ ∞ + x > y (i.e. x ≥ y, x = y) ⇒ x i y. It is easy to find examples of such strictly optimistic DM, who would be reasonably optimistic. Indeed an EU DM as defined above in Section 4 with a strictly convex VNM utility would be such an example. Equally a Yaari's DM (see Section 4) with a strictly concave distortion function f would satisfy such a requirement.

The next lemma characterizes strict optimism for Choquet preferences from properties of the respective capacity. Its proof is provided in Appendix C.

Lemma 4. Choquet preferences on ∞

+ are strictly monotone, reasonably strictly optimistic if and only if (a) ϑ is strictly monotone, i.e., A B ⇒ ϑ(A) > ϑ(B) (b) ϑ is strictly concave, i.e., concave and such that:

A ∪ B A B A ∩ B ϑ(A ∪ B) + ϑ(A ∩ B) < ϑ(A) + ϑ(B) (c) ϑ is outer continuous.
Theorem 2 below shows that, when there are two or more strictly optimistic DMs, "corner" plans are a quite general feature of IRPO allocations.

Theorem 2. Consider an individually rational Pareto optimum where two DM i and j satisfy A1, have reasonably strictly optimistic and strictly monotonic preferences. Then, their allocations x i and x j satisfy for any (s, t) ∈ N 2 s = t:

x i (s) • x i (t) • x j (s) • x j (t) = 0. Proof: Let us define x i -(s,t) by x i -(s,t) (u) =      x i (u) if u ∈ N\{s, t} 0 if u ∈ {s, t} , e s ∈ ∞ + by e s (u) =      1 if u = s 0 otherwise and consider ∀ (a, b) ∈ R 2 + x i (a,b) ≡ a e(s) + b e(t) + x i -(s,t) .
Denote ν(s) ≡ x i (s) + x j (s) and ν(t) ≡ x i (t) + x j (t) . We will define

x j (a,b) ≡ (ν(s) - a)e s + (ν(t) -b)e t + x j -(s,t) . Let us show that x i (s) > 0, x i (t) > 0, x j (s) > 0, x j (t) > 0 is impossible.
For any a ∈ [0, x i (s)) by continuity of i and strict monotonicity of , b) . Clearly ȳi + ȳj = x i + x j . Since ȳi ∼ i x i if ȳj j x j IRPO would be contradicted. So necessarily x j j ȳj . Let us choose β ∈ (1, 2) close enough to 1 in order to guarantee that y i ≡ βx i + (1 -β)ȳ i and y j ≡ βx j + (1 -β)ȳ j are such that y i , y j ≥ 0 (this is possible since by hypothesis x i (s) > 0, x i (t) > 0, x i (s) > 0, x j (t) > 0). So y i and y j belong to ∞ + and clearly y i + y j = x i + x j .

i ∃ b(a) > x i (t) such that x i (a,b(a)) ∼ i x i . It is straightforward to check that when a ↑ x i (s), then b(a) ↓ x i (t). Therefore for a = ā sufficiently close to x i (s), b = b(a) is such that x j (ā, b) ≥ 0 i.e. x j (ā, b) ∈ ∞ + . That x i (ā, b) ∈ ∞ + is immediate. Let us set ȳi = x i (ā, b) and ȳj = x j (ā
It is enough to prove that y i i x i and y j j x j which contradicts again IRPO, and then completes the proof. One has

x i = 1 β y i + β -1 β ȳi with y i and ȳi ∈ ∞ + x j = 1 β y j + β -1 β ȳj with y j and ȳj ∈ ∞ +
Let us show first that y i i x i . Otherwise x i i y i hence x i ∼ i ȳi implies ȳi i y i and strict optimism implies ȳi i x i i.e. x i i x i , a contradiction. Now, suppose x j j y j . Note that ȳj = y j otherwise y j = βx j + (1 -β)ȳ j would imply ȳj = x j and this is not the case. So, y j j x j . Thus, either y j j ȳj and y j = ȳj ⇒ y j j x j by strict optimism hence x j j x j , which is impossible. Or ȳj j y j and strict optimism would imply ȳj j x j which contradicts x j j ȳj . Henceforth y j j x j which completes this part of proof. Now we will introduce a notion related to the possible non-interiority of some of the plans designed by an IRPO allocation to DMs.

Definition 11. Given an IRPO allocation (x i ) I i=1 , we say that the DM i is going to ruin in state s ∈ N (according to this allocation) when x i (s) = 0; When x i (s) > 0 we will say that the DM i will avoid ruin in state s.

From Definition 11 and Theorem 2, we obtain the second main result of this paper.

Corollary 2. At an IRPO at most one reasonably strictly optimistic will avoid ruin in any state (or equally at any time period).

Proof: Assume that at least two DM i and j are strictly optimistic and satisfy the natural hypotheses of Theorem 2.

Imagine that one of them let us say i has an interior allocation, then at most in one state s the DM j can get x j (s) > 0.

Remark 1. A further conclusion from Theorem 2 is that, under the same hypotheses, if there is an optimistic DM that will avoid ruin in each state, then all the other optimistic DM are going to ruin except in one state. In other words, this extremal willingness to bet could induce an ex post social problem (may be with a very large likelihood).

Optimal risk sharing for risk lovers and risk averters decision makers

Interpret ∞ as the set of all bounded A-measurable mappings x from N to R where A = P(N) and assume that a σ-additive probability P is given on (N, P(N)).9 So any x can now be interpreted as a random variable.

Note that under risk, rational DM are assumed to strictly agree with first order stochastic dominance (FSD), which is the natural extension of monotone dominance under risk.

(b) Yaari's DM are such strict risk lovers or strict risk averters if and only if they are strictly optimistic or strictly pessimistic.

The following Theorem 3 offers two important features of risk sharing between strict risk averters and strict risk lovers.

Feature 1: As in the classical case of risk sharing between only risk averters, the Pareto optimal allocations of risk averters will be comonotonic, but note that this does not entail any longer that these allocations will be necessarily ranked as the aggregate endowment, because nothing in the Theorem guarantees global comonotonicity between risk averters and risk lovers. Example 5 illustrates this last point.

Example 5. IRPO without comonotonicity between risk averters and risk lovers

Let S = {1, 2} and p the probability measure given by p 1 = p 2 = 1 2 . The DM 1 is a strict strong risk averter characterized by the utility function

U 1 (x 1 , x 2 ) = 1 2 (-exp(-2x 1 )) + 1 2 (-exp(-2x 2 )
) and the initial endowment ω 1 = (8, 4). The DM 2 is a strict strong risk lover characterized by the utility function U 2 (y 1 , y 2 ) = 1 2 exp(y 1 ) + 1 2 exp(y 2 ) and the initial endowment ω 2 = (4, 5). So, the aggregate endowment is ω = (12, 9). We intend to show that if x = (4.5, 7.5) and y = (7.5, 1.5), then the couple (x, y) is an IRPO, which will imply that global comonotonicity fails.

It is immediate that (x, y) is a feasible allocation. Furthermore it is straightforward to verify that x 1 ω 1 and y 2 ω 2 . So, it suffices to show that x is the solution of the problem:

max

U 1 (x 1 , x 2 ) s.t. U 2 (12 -x 1 , 9 -x 2 ) = U 2 (y) (x 1 , x 2 ) ∈ [0, 12] × [0, 9]
Since the constraint on U 2 can be written as e -x 2 = e -1.5 + e -7.5 -e 3-x 1 , to solve the above problem it is enough to find a solution z * ∈ argmax{h(z) : 0 ≤ z ≤ 12} for h(z) = -e -2z -(k -e 3-z ) 2 and k = e -1.5 +e -7.5 and make x * 1 = z * , x * 2 = -ln(e -1.5 +e -7.5 +e 3-z * ). It is straightforward to verify that z * = 4.5 (and that this solution is unique), which implies Feature 2: As soon as the strict risk lovers are also strictly optimistic (note that while this is the case for EU or Yaari's DM, this might not be the case for some particular class of strict risk lovers, see the Example 6 below adapted from [START_REF] Chateauneuf | From sure to strong diversification[END_REF] who were dealing with strict risk averters) then at most one of these strict risk lovers will avoid ruin.

(x * 1 , x * 2 ) = (x 1 , x 2 )
Example 6. A strict risk lover with weak* sequentially usc-preferences who is not strictly optimistic

We just exhibit here such a DM. Assume that any x ∈ ∞ + is valued through I(x) = u(x)dP + f ( xdP ), where u : R → R is continuously differentiable, strictly increasing and strictly convex, f : R → R is continuously differentiable and strictly increasing.

Since the preference i represented by I clearly satisfies A1 from Proposition 1, it will be weak* sequentially usc as soon as we prove that I(y + z En ) → I(y) when n → ∞.

But, denoting I 1 (x) = u(x)dP and I 2 (x) = f ( xdP ). From Proposition 4, we have that I 1 (x) satisfies this property and clearly this is also the case for I 2 (x).

Since u and f are strictly increasing it is immediate that i strictly agrees with FSD.

From E P (x) = E P (y), it turns out that x MPS y ⇒ I 2 (x) = I 2 (y). Moreover, u strictly increasing and strictly convex implies that x MPS y ⇒ I 1 (x) ≥ I 1 (y) and x MPS y ⇒ I 1 (x) > I 1 (y). Consequently i exhibits strict risk loving.

Assume now that u(x) = x 2 and f be such that f (2) = 1, f ( 5 2 ) = 5 2 and f (2) = 2. Let B 1 and B 2 be a partition of N such that P (B 1 ) = P (B 2 ) = 1 2 and consider x = 4 1 1 1 B 2 and y = 2 1 1 1 B 1 + 3 1 1 1 B 2 . Simple computations show that I(x) = I(y), that g(α) = I(αx + (1 -α)y) is continuously differentiable on [0, 1], and that g (1) = 3 > 0.

Thus, there exists α * ∈ (0, 1) such that I(αx + (1 -α)y) > I(y) for α ∈ (α * , 1), hence the DM is there locally strictly pessimistic. So i does not exhibit strictly optimism (indeed, does not even exhibit optimism).

Theorem 3 makes precise the statement about comonotonicity of plans of risk averters and emphasizes the tendency of strict risk lovers DMs to go to ruin in some states in IRPO allocations.

Theorem 3. Consider m strict risk averters i = 1, . . . , m satisfying A.1 with weak* sequentially u.s.c. preferences i and n strict risk lovers j = 1, . . . , n satisfying A.1 with weak* sequentially u.s.c. preferences i but exhibiting strict optimism.

Assume initial endowments w i ∈ ∞ ++ , w j ∈ ∞ ++ . Then individual rational Pareto efficient allocations ( in ( ∞ + ) m+n ) exist. For such PO (x i , i = 1, . . . , m; y j , j = 1, . . . n) we have:

1) The allocations of risk averters are pairwise comonotonic, i.e.,

x i 1 (s) -x i 1 (t) x i 2 (s) -x i 2 (t) ≥ 0 ∀(s, t) ∈ N 2 , ∀(i 1 , i 2 ) ∈ {1, . . . , m} 2
2) At most one risk lover will avoid ruin in any state.

Proof:

1) A proof similar as the one of Proposition 4.1 of [START_REF] Chateauneuf | Optimal risk-sharing rules and equilibria with Choquet-expected-utility[END_REF] applies. Actually if (x 1 , x 2 , ..., x m ) is the allocation of the risk averters at an IRPO, the proof of Proposition 4.1 shows that without loss of generality if x 1 and x 2 would not be comonotonic, one could find x 1 and x 2 in ∞ + such that x 1 1 x 1 , x 2 2 x 2 and x 1 + x 2 + m i=3 x i = m i=1 x i , thus contradicting the fact that (x 1 , x 2 , ..., x m , y 1 , ..., y n ) is an IRPO.

2) This is just Corollary 2.

6 Is it beneficial to share risk with a risk lover?

Is it more beneficial for a risk averter to share risk with a risk lover rather than with another risk averter? This section aims at introducing this question through some elementary examples.

We focus on the simple situation of two EU decision makers i = 1, 2 sharing risk in the case of a two-states space S = {s 1 , s 2 } endowed with a probability measure P such that P 0. Let us assume that the initial endowments ω 1 and ω 2 satisfy ω 1 0, ω 2 0 and denote by ω the aggregate endowment.

In order to get tractable formulas, we specialize to the situation where DM1 is a constant absolute risk averse (CARA) DM, i.e., with an EU utility index u 1 (x) = -e -ρ 1 x , ρ 1 > 0, while DM2 is either a CARA DM with a constant coefficient of absolute risk aversion ρ 2 > 0 or a constant absolute risk lover (CARL) with coefficient ρ 2 > 0 of absolute risk loving, i.e., with EU utility index u 2 given by u 2 (x) = e ρ 2 x .

Let us recall that if both DM1 and DM2 are CARA then the interior PO allocations are given by x 1 = a + ρ ρ 1 ω and x 2 = -a + ρ ρ 2 ω, where a ∈ R is chosen in order that x 1 0, x 2 0 and where ρ is the aggregate coefficient of risk aversion defined by

1 ρ = 1 ρ 1 + 1 ρ 2 .
The Proposition 6 (whose proof is available upon request) gives sufficient conditions for interior PO allocations and describes the allocations when DM2 is a CARL DM. Proposition 6. Let r = ρ 1 ρ 2 be the relative level of risk aversion and a r = maxs ω(s) mins ω(s) be the relative level of aggregate risk. If r > a r then there exist interior Pareto optimal allocations and they are given by the formulas x 1 = a + ρ ρ 1 ω and x 2 = -a -ρ ρ 2 ω where a ∈ R is chosen in order that x 1 0, x 2 0 and where ρ defined by 1 ρ = 1 ρ 1 + 1 -ρ 2 extends the usual aggregate coefficient of risk aversion.

Remark 2. Note that in the Proposition 6 ρ is negative so here the allocation of risk averter is anticomonotonic with the aggregate endowment, while the risk lover's one proves to be comonotonic with ω. This indeed contrasts with the situation in which both DM are risk averters, since their PO allocations are comonotonic with ω. Such a fact will matter in the Examples 7 and 8 (in which we assume that ρ 1 = 2).

Example 7. No aggregate risk

Consider the endowments described by Table 1 and assume that P (s 1 ) = P (s 2 ) = 1 2 . applies, so the interior PO allocations which are full insurance satisfy:

x 1 (s) = a -9, x 2 (s) = -a + 18 and 9 < a < 18.

Simple computations show that there exists k 1 ∈ (0, 4.5) such that at any interior IRPO one has x 1 (s) ≤ k 1 < 4.5 ≤ x 2 (s), hence DM1 will get a full insurance always strictly smaller than DM2.

Case 7.b: DM2 is CARA with ρ 2 = 1. In such a case interior PO allocations are indeed again full insurance and satisfy:

x 1 (s) = a + 3, x 2 (s) = -a + 6 and -3 < a < 6.

Here, simple computations show that if a is close enough of 3 2 then the previous PO are IRPO with either

x 1 > x 2 or x 1 < x 2 (in fact, x 1 = x 2 if a = 3
2 ), so DM1 could get a better allocation than DM2. In conclusion in this example of no aggregate risk economy, it appears it would be more beneficial to share risk with a risk averter.

Example 8. Aggregate risk Consider again P (s 1 ) = P (s 2 ) = 1 2 . Now the endowments are described by Table 2. Case 8.a: DM2 is CARL with ρ 2 = 1. By Proposition 6, the interior PO allocations are given by:

x 1 (s 1 ) = a -12, x 1 (s 2 ) = a -9, x 2 (s 1 ) = -a + 24, x 2 (s 2 ) = -a + 18 with a ∈ (12, 18).

Note that since PO allocations of DM1 are anticomonotonic with ω, if this DM would like to transfer wealth from state s 1 to state s 2 "in an anticomonotonic way" for some "strategical" reason, this is a priori possible. It turns out in fact that it is possible in an efficient way, since one can check that if a = 16.5 then the resulting PO is an IRPO, and moreover E P (x 1 ) = E P (ω 1 ), V AR P (x 1 ) < V AR P (ω 1 ).

Case 8.b: DM2 is CARA with ρ 2 = 1 In this case, the interior PO allocations are given by:

x 1 (s 1 ) = a + 4, x 1 (s 2 ) = a + 3, x 2 (s 1 ) = -a + 8, x 2 (s 2 ) = -a + 6 with a ∈ (-3, 6).

Since the PO allocations are comonotonic with ω, there is no hope for DM1 to transfer her wealth from state s 1 to state s 2 in an "anticomonotonic way" through an IRPO.

In conclusion, in case of aggregate risk this example illustrates that for a risk averter who would like to transfer substantially her wealth from state s 1 to state s 2 and in an efficient way, this would only be possible when sharing risk with a risk lover.

Conclusions and Further Research

As the Theorem 1 has shown, the existence of IRPO allocation is a quite general feature of GE models when the space of alternatives is ∞ (or R S ), even in the presence of non-complete preferences or optimistic agents. One of the sufficient requirements is that the DMs should be myopic or impatient for gains, what can be interpreted as a bound on optimism, since, in that case, even the optimistic DMs will not want to exchange all their wealth for a larger income on events with arbitrarily small likelihood or dates arbitrarily far (which avoids the difficulties present in Example 1).

So, one perceives that the concept of IRPO accommodates well the notions of risk and ambiguity propensity (both relevant for Applied Microeconomics and Finance). A further research program is to find the sufficient conditions to ensure the existence of a barter equilibrium (i.e., the non-emptiness of the core of the economy) or an equilibrium with non-linear prices even in the presence of optimistic agents.

At an IRPO allocation of resources, the risk averters will be sharing the individual risk (in the sense that their plans will be 2 × 2 comonotonic), but not necessarily the aggregate risk when there exist risk lovers DMs too. However, when the aggregate risk is very large, maybe the existence of optimistic agents turns the risk averters better than they would be with an aggregate risk sharing as it becomes possible to smooth their consumption plans.

About the properties of IRPO allocation, Theorems 2 and 3 say that is quite likely 27 Documents de Travail du Centre d'Economie de la Sorbonne -2015.71

to observe plans of optimistic DMs leading to ruin at some states. In fact, if there were two optimistic DMs with consumption plans that are bounded away from zero in two states of nature, there would be room for a Pareto improvement. Under the theorems hypotheses, in the case that the consumption plan is interior for one of the optimistic DMs, all the others will concentrate the future wealth in just one of the possible future states, characterizing a very speculative pattern. Thus, a question that naturally arises for future studies is if a policy maker concerned with the ex post aggregate welfare should or not deviate the output allocation of the economy from ex ante IRPO allocations in order to avoid this extreme gambling. And, if the policy maker decides to intervene, which type of instrument should be chosen. Or even, if the incompleteness of markets could weaken the effects of this willingness to gamble future wealth, thereby preventing any need of intervention.

Let us first prove that x ∈ ∞ . Let K ≥ 0 be such that x n ≤ K ∀n. Let ε > 0 be given, Proposition 8. On ∞ the weak* σ( ∞ , 1 ) sequential convergence and the Mackey τ ( ∞ , 1 ) sequential convergence coincide.

Proof: Since σ( ∞ , 1 ) ⊂ τ ( ∞ , 1 ) we just need to prove that for (x n ), x n ∈ ∞ , x n ω * -→ x implies x n τ -→ x.

From Proposition 3.13 in [START_REF] Brézis | Functional analysis, Sobolev spaces and partial differential equations[END_REF] it comes that x n ω * -→ x if and only if the sequence (x n ) is bounded and ∀ i ∈ N lim n

x n (i) = x(i), therefore we may assume without loss of generality that (x n ) is a sequence in the closed unit ball B(0, 1) of ∞ and that x belongs

Furthermore

  if every i satisfy (1) or equivalent (2) then there exist individually Pareto efficient allocations. Proof: Immediate from Proposition 1 and Theorem 1. We now give in Section 4 some examples of suitable preferences for our purpose i.e. of weak* sequentially upper semi-continuous preferences satisfying A.1, either checking directly the required properties above, or using Corollary 1, or else Proposition 1. 4 Weak* sequentially upper-semicontinuous preferences 4.1 Existence of weak * sequentially upper-semicontinuous preferences on ∞ + which are not weak * upper-semicontinuous

13

  Documents de Travail du Centre d'Economie de la Sorbonne -2015.71 Proof: From Proposition 1, it is enough to see that ∀ y, z ∈ ∞ + one gets I(y+z En ) → I(y) when n → +∞ I(y + z En ) = n m=0 P ({m})u(y(m)) + +∞ m=n+1 P ({m})u(y(m) + z(m)). Since y + z is norm bounded and u is monotone, it turns out that u(y + z) is bounded, and since lim n→∞ ∞ n+1 P ({m}) = 0 one gets I(y + z En ) → ∞ n=0 P ({n})u(y(n)) = I(y).

  as desired. 22 Documents de Travail du Centre d'Economie de la Sorbonne -2015.71

  lim n→∞ x n (p) = x(p), then there existn 0 ∈ N such that n ≥ n 0 implies |x(p)| ≤ |x n (p)|+ε hence |x(p)| ≤ K + ε and so x ∈ ∞ since x ≤ K + ε.It remains to show that for all y ∈ 1 we have x n , y → x, y i.e. S n → S whereS n = ∞ p=1 x n (p)y(p) and S = ∞ p=1 x(p)y(p). Let m ∈ N, then: |S n -S| ≤ m p=1 |x n (p) -x(p)| • |y(p)| + ∞ p=m+1 |x n (p) -x(p)| • |y(p)| .Hence,|S n -S| ≤ y 1 • m p=1 |x n (p) -x(p)| + (K + x ) • ∞ p=m+1 |y(p)| . Since y ∈ 1 , there exist m 0 such that ∞ p=m 0 +1 |y(p)| ≤ ε. For such m 0 , x n (p) → x(p) for all 1 ≤ p ≤ m 0 . Then there is n 0 (ε) such that n ≥ n 0 (ε) implies m 0 p=1 |x n (p) -x(p)| ≤ ε. Therefore ∀ε > 0, ∃n 0 (ε) such that |S n -S| ≤ ( y 1 + K + x )ε. Hence x n , y → x, ywhich completes the proof.B Mackey and weak* sequential convergence coincide on ∞For sake of completeness we additionally give a direct and elementary simple proof of Proposition 8 below.

Table 1 :

 1 Endowments of Example 7Case 7.a: DM2 is CARL with ρ 2 = 1: It is straightforward to see that Proposition 6

	s 1 s 2
	ω 1 5 4
	ω 2 4 5
	ω 9 9
	25

Documents de Travail du Centre d'Economie de la Sorbonne -2015.71

Table 2 :

 2 Endowments of Example 8

	s 1 s 2
	ω 1 8 4
	ω 2 4 5
	ω 12 9

Documents de Travail du Centre d'Economie de la Sorbonne -2015.71

It is important to remark that, in this work, as in[START_REF] Brown | Myopic economic agents[END_REF], the term impatience is simply a specialization of the broader concept of myopia. Some authors such as[START_REF] Chateauneuf | Some characterizations of non-additive multi-period models[END_REF] propose alternative definitions for these two concepts.

In this work, they refer to the concept of upper semi-myopia as strong myopia.

In Section 4 one will find an example of preferences which are weak* sequentially upper semicontinuous but not weak* upper semi-continuous. This illustrates that our result strengthens Aliprantis and Burkinshaw's one at least in the ∞ case. 4 Documents de Travail du Centre d'Economie de la Sorbonne -2015.71

Appendix A makes precise that sequential weak* convergence on ∞ is as tractable as the standard convergence in the Euclidean space R m (m ∈ N).

Under uncertainty ϑ(A) is the subjective evaluation of the likelihood of the event A, while when valuing flow of payments, ϑ(A) is the weight given by the DM to the time period A.

Indeed such a DM has non-convex preferences, but actually concave preferences.

For sake of simplicity we will assume that P ({n}) > 0 ∀ n ∈ N. 20 Documents de Travail du Centre d'Economie de la Sorbonne -2015.71

Chateauneuf thanks IMPA for the generous financial support from the "Brazilian-French Network Fronteiras" fellowship. Novinski gratefully acknowledges the financial support from the "Brazilian-French Network in Mathematics" and CERMSEM at the University of Paris 1 for its hospitality. The valuable detailed suggestions of anonymous referees are gratefully acknowledged.

Definition 12. Given x, y ∈ ∞ , we say that:

(i) x dominates y in the FSD sense, which will be denoted by x FSD y, if P (x ≤ u) ≤ P (y ≤ u) ∀u ∈ R;

(ii) x strictly dominates y in the FSD sense if furthermore one of the previous inequalities is strict.

Definition 13. Let i be the preference relation of a DM. We say that i strictly agrees with FSD if x FSD y ⇒ x i y and x FSD y ⇒ x i y.

So, considering in all this section only DM strictly agreeing with FSD, it turns out that the usual definition of strict strong risk aversion coincides with strictly agreeing with second order stochastic dominance in the case of alternatives with the same mean. Definition 14. Let x, y ∈ ∞ be such that E P (x) = E P (y).

(i) x is said to be less risky than y for the second order stochastic dominance or else y is a mean-preserving spread of x, which will be denoted by x MPS y, if:

(ii) x is strictly less risky than y, which will be denoted by x MPS y, if furthermore one of the previous inequalities is strict.

Hence the definitions of strict (strong) risk aversion and of strict (strong) risk loving.

Definition 15. Let i be the preference relation of a DM.

(i) i is a strict (strong) risk averter if x MPS y ⇒ x i y and x MPS y ⇒ x i y.

(ii) i is a strict (strong) risk lover if x MPS y ⇒ x i y and x MPS y ⇒ x ≺ i y.

As examples of weak* sequentially usc strict risk lovers or strict risk averters we can mention:

(a) EU DM are such strict risk lovers or strict risk averters if and only if they are strictly optimistic or strictly pessimistic.

APPENDIX

A The space ∞ N denotes the set of natural numbers {1, 2, 3, ...}.

The space ∞ is the Banach space of real bounded sequences equipped with the norm defined by ||x|| = sup t |x t |. The space 1 is the Banach space set of absolutely convergent real sequences equipped with the norm defined by

When ∞ is endowed with the norm topology, the dual is denoted by ( ∞ ) * . A coarser topology is the Mackey topology, defined as the finest topology on ∞ for which the dual is 1 . Now, (x n ) converges to x in this topology if and only if, for any weakly compact subset A of 1 , x n , y → x, y uniformly on y ∈ A.

The weak* topology is defined as the coarsest topology on ∞ for which the dual is 1 .

By definition a sequence (x n ) converges to x in the weak * topology, denoted x n w * -→ x, if we have (x n ) in ∞ , x ∈ ∞ and x n , y → x, y for any y ∈ 1 . Furthermore (see for instance [START_REF] Brézis | Functional analysis, Sobolev spaces and partial differential equations[END_REF])

It turns out that, in the particular case of ∞ , the simple following tractable propertywhich looks like the standard convergence in the Euclidian space R m (m ∈ N) -characterizes sequential weak * convergence:

Notice that in all this paper when x ∈ ∞ we denote indifferently its p-th component as x(p) or x p .

Proof: Since the necessary condition is immediate, we just prove for sake of completeness the sufficiency property.

So let (x

to B(0, 1). Recall that on B(0, 1) the metric d(x, y) = ∞ i=0 1 2 i |x(i) -y(i)| defines the weak star topology (see e.g. Exercise 4 p.136 in [START_REF] Conway | A course in functional analysis[END_REF]).

Let us recall furthermore that a basis of neighborhoods of 0 for the Mackey topology on ∞ is given by the sets:

sequence of real numbers strictly decreasing towards 0 (see e.g. [START_REF] Brown | Myopic economic agents[END_REF] p.34). Therefore we need to show, letting

Let ϑ (a i ), 0 be a neighborhood of 0 for the Mackey topology τ . From 1 a i ↑ +∞ there

So it remains to prove that if z n ω * -→ 0, there exists

it turns out that z n ∈ ϑ (a i ), 0 ∀ n ≥ n 0 = n 0 (ε), which completes the proof.

C Strictly optimistic Choquet preferences

Proof of Lemma 4

(b) can be found in [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF] (c) comes from Proposition 2.

The necessity of (a) is immediate.

It remains to prove that under (a) and (c) the preferences are strictly monotone.

So take x, x ∈ ∞ + with x > x and let us show that x i x . x > x implies there exists s 0 ∈ N s.t. x(s 0 ) > x (s 0 ). We need to show that

Let t 0 = x(s 0 ), therefore {x ≥ t 0 } {x ≥ t 0 } and (a) implies ϑ(x ≥ t 0 ) > ϑ(x ≥ t 0 ). Let y ∈ ∞ + , and let us see that f (t) = ϑ(y ≥ t) is left-continuous. Take A n = {s ∈ N, y(s) ≥ t n } with t n ∈ R + , t n ↑ t. It is straightforward to see that A n ↓ A = {y ≥ t}, so since from (c) ϑ(A n ) ↓ ϑ(A), f is actually left-continuous. So since t 0 = x(s 0 ) > 0, there exists an interval [t 0 -ε, t 0 ) ⊆ R + such that ∀ t ∈ [t 0 -ε, t 0 ], ϑ(x ≥ t) -ϑ(x ≥ t) > 0, since indeed x ≥ x implies ϑ(x ≥ u) ≥ ϑ(x ≥ u) ∀ u ∈ R + one gets d > 0.


Vuong-Dieu Trinh
email: vuong-dieu.trinh@metz.ensam.fr

Farid Abed-Meraim
email: farid.abed-meraim@metz.ensam.fr

Alain Combescure

New prismatic solid-shell element: Assumed strain formulation and evaluation on benchmark problems

. It is integrated with a set of five Gauss points along a special direction, denoted "thickness", and with only one point in the other in-plane directions. Its discrete gradient is modified in order to attenuate shear and membrane locking. A series of popular linear benchmark problems has been carried out with comparisons to geometrically similar, low-order three-dimensional elements.

Introduction

Accuracy and efficiency are the main features expected in finite element methods. In three-dimensional analysis of structural problems, the development of effective eight-node solid-shell finite elements has been a major objective over the last decade as testified by many recently published contributions [START_REF] Combescure | SHB8PS a new adaptive, assumed-strain continuum mechanics shell element for impact analysis[END_REF][START_REF] Belytschko | Assumed strain stabilization of the eight node hexahedral element[END_REF][START_REF] Hauptmann | A systematic development of solid-shell element formulations for linear and non-linear analyses employing only displacement degrees of freedom[END_REF][START_REF] Legay | Elastoplastic stability analysis of shells using the physically stabilized finite element SHB8PS[END_REF][START_REF] Wall | A deformation dependent stabilization technique, exemplified by EAS elements at large strains[END_REF]. However, to be able to mesh complex geometries and with the advent of free mesh generation tools not generating only hexahedrons, the development of prismatic elements is made necessary. This paper presents the formulation of a six-node solid-shell finite element called SHB6. It represents a thick shell obtained from a purely three-dimensional approach. The assumed strain method is adopted together with an in-plane reduced integration scheme with five Gauss points along the thickness direction. The three-dimensional elastic constitutive law is also modified so that a shell-like behavior is intended for the element and in order to alleviate shear and membrane locking.

Because the reduced integration is known to introduce spurious modes associated with zero energy, an adequate hourglass control is generally needed as proposed by Belytschko et al. [START_REF] Belytschko | Assumed strain stabilization of the eight node hexahedral element[END_REF] with a physical stabilization procedure to correct the rank deficiency of eight-node hexahedral elements. As the SHB6 is also underintegrated, a detailed eigenvalue analysis of the element stiffness matrix is carried out. We demonstrate that the kernel of this stiffness matrix only reduces to rigid body movements and hence, in contrast to the eight-node solid-shell element (SHB8PS), the SHB6 element does not require stabilization.

Numerical evaluations of the SHB6 element showed that its initial version, without modification of its discrete gradient operator, suffered from shear and membrane locking. To attenuate these locking phenomena, several modifications have been introduced into the formulation of the SHB6 element following the assumed strain method adopted by Belytschko et al. [START_REF] Belytschko | Assumed strain stabilization of the eight node hexahedral element[END_REF]. Finally, a variety of popular benchmark problems has been performed and good results have been obtained when compared to other well-established elements in the literature.

Formulation of the SHB6 finite element

The SHB6 finite element is a solid-shell with only three displacement degrees of freedom per node, and it has a special direction called "thickness". It is integrated with a set of five Gauss points along this direction and only one point in the in-plane directions. Figure 1 shows the SHB6 reference geometry as well as its Gauss points. 1

Kinematics and interpolation

The SHB6 is a linear, isoparametric element. Its spatial coordinates i x and displacements are respectively related to the nodal coordinates

∑ = = = 6 1 ) , , ( ) , , (
I I iI I iI i N x N x x ζ η ξ ζ η ξ ∑ = = = 6 1 ) , , ( ) , , (
I I iI I iI i N u N u u ζ η ξ ζ η ξ
Hereafter, unless specified otherwise, the implied summation convention for repeated indices will be adopted. Lowercase indices vary from one to three and represent spatial coordinate directions. Uppercase indices vary from one to six and correspond to element nodes. The tri-linear isoparametric shape functions

i I I N are: ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) [ ] [ ] [ ] 1 , , 1 1 1 1 1 1 1 1 2 0,1 ; 0,1 ; 1,1 I N ξ η ζ ζ ξ η ζ ξ ζ η ζ ξ η ζ ξ ζ η ξ η ξ ζ = - -- - - + -- + + ⎡ ⎤ ⎣ ⎦ = = - =- (2)
Figure 1: Reference geometry of the SHB6 element and its Gauss points

Discrete gradient operator

Using some mathematical derivations, similarly to the procedure of the SHB8PS development reported in [START_REF] Combescure | SHB8PS a new adaptive, assumed-strain continuum mechanics shell element for impact analysis[END_REF], we can explicitly express the relationship between the strain field and the nodal displacements as:

( ) , , , , , , , , , , , , , , , , , , 0 0 0
0 0 0 0 0 0 T T x x x x T T y y y y T T x z z z z s T T T T x y y x y x y x T T T T x z z x z x z x y z z y T T T T z y z y b h u b h u d b h u u d u u b h b h u u b h b h u u b h b h α α α α α α α α α α α α α α α α α α γ γ γ γ γ γ γ γ γ + + + ∇ = = ⋅ + + + + + + + + + ⎡ ⎤ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ y z B d d = ⋅ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (3)
where the h α functions are:

1 2 , h h ηζ ξζ = = ; the i b vectors are: , 0 (0) ; 1, 2, 3 T i i i b N N x i ξ η ζ = = = = =∂ ∂ = and
the constant vectors α γ are given by: ( )

3 1 1, 2 1 ; 2 T i i i h h x b α α α α γ = = = - ⋅ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∑
. In this latter expression, ( )

1 0, 0, 1, 0, 0, 1 ; T h = - ( ) 2 0, 1, 0, 0, 1, 0 T h = -
and the i x vectors denote the nodal coordinates. We can also demonstrate the following orthogonality conditions:

0 T j x α γ ⋅ = , T h β αβ α γ δ ⋅ = (4)

Variational principle

Applying the simplified form of the Hu-Washizu non linear mixed variational principle, in which the assumed stress field is chosen to be orthogonal to the difference between the symmetric part of the displacement gradient and the assumed strain field, we obtain:

( )

, e T T ext V u d V d δπ ε δ ε σ δ 0 f = ⋅ - ⋅ = ∫ (5)
Replacing the assumed strain field, with its expression ( ) ( ) ( )

, x t B x d t ε = ⋅
, in equation ( 5) leads to the following expression for the internal forces:

( )

int e T V f B σ ε = ⋅ ∫ d V (6)
This also leads to the following expression for the element elastic stiffness matrix:

e T e V K B C B d = ⋅ ⋅ ∫ V (7)
Note that for a standard displacement approach, B is simply replaced with B leading to the classical stiffness:

e T e V K B C Bd = ⋅ ⋅ ∫ V (8)

Hourglass mode analysis for the SHB6 element

Hourglass modes are spurious zero-energy modes that are generated by the reduced integration. In static problems, they may lead to singularity of the assembled stiffness matrix for certain boundary conditions; in most cases this results in spurious mechanisms also known as rank deficiencies. Therefore, the analysis of hourglass modes is equivalent to that of the stiffness matrix kernel, namely zero-strain modes d that satisfy:

( ) 0 Gj Gj B d ζ ζ ⋅ = ∀ (9)
To this end, we can build a basis for the discretized displacements by demonstrating that the eighteen column vectors below are linearly independent. Making use of orthogonality conditions (4), we show that only the first six column vectors verify equation ( 9) and correspond to rigid body modes. This means that there are no hourglass modes for the SHB6 element. In other words, the SHB6 element does not require hourglass control.

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - - - 2 1 2 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 h h y x z y x S h h z x y z x S h h z y x z y S (10)
2.4 Assumed strain formulation for the SHB6 discrete gradient operator Among several treatments for alleviating shear and membrane locking, the discrete gradient is appropriately modified. This consists first of decomposing the matrix B into two parts:

2 1 B B B + =
, then of projecting the second part onto an assumed strain operator such that

1 B B B = + 2 .
As a result, the stiffness matrix becomes:

1 1 1 2 2 1 2 2 1 2 3 e e e e T T T T e e e e V V V V K B C B dV B C B dV B C B dV B C B dV K K K K = ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ = + + + ∫ ∫ ∫ ∫ 4 e (11)
The subsequent steps consist of choosing an adequate assumed strain field. This is a key point in the formulation and special care has been exercised in this regard. Finally, the above additive decomposition of the stiffness matrix is calculated using a reduced integration scheme with five Gauss points. Note that the choice of an assumed strain field is mainly guided by the elimination of strain components that are responsible for shear as well as membrane locking. The advantages of this enhanced strain will be shown through benchmark problems.

Numerical results and comparison

Several popular benchmark problems were performed to illustrate the element performance; one of these test problems is given here. This example is frequently used in the literature to test warping effects on shell elements.

Conclusions

This newly developed SHB6 element was implemented into the finite element codes INCA and ASTER. It represents some improvement since it converges relatively well and it performs better than the PRI6 six-node three-dimensional element in all of the benchmark problems tested. Furthermore, it shows very good performances in problems using mixed meshes composed of SHB6 and SHB8PS elements. Thus, we can couple the SHB6 with other finite elements to mesh complex geometries. For the remaining locking modes, exhibited in some test problems, a detailed study revealed that the transverse shear was behind these locking phenomena.

Figure 2 :

 2 Figure 2: Twisted beam. Length =12; width = 1.1; depth = 0.32, twist = 90° (root to tip); E = 29x10 6 ; ν = 0.22. Loading: unit force at tip. The reference vertical displacement of point A at tip is 5.424x10 -3

Table 1 :

 1 Normalized vertical displacement at point A of the twisted cantilever beam problem


Leonardo R Emmendorfer

Neuza T Oro

C Beckel

An application of Hooke and Jeeves Method as a Constructive Heuristic for the Ellipse Packing Problem

Keywords:

. Packed ellipses are tangent to the vertexes of a master ellipse, initially positioned at the center of the polygon. Superposition restrictions are imposed and checked at each stage. R 2 transformations are applied, which raise parametric ellipse equations. Results illustrate that the proposed heuristic can be applied to polygons with varying shapes, either convex or concave, due to the type of search which is performed in the interior of the polygon. The constructive heuristic was able to find the optimal solutions for the polygons tested.

Introduction

Cutting and packing problems appear in many industrial applications, such as textile, automotive, transportation, among others. Cutting problem consists of cutting smaller units from a substrate, given shapes and sizes, in order to optimize some objective function, such as loss or cost minimization, or profit maximization [START_REF] Brusso | Tecnologia 3D Gemas: otimizac ¸ão do aproveitamento de gemas coradas digitalizadas tridimensionalmente[END_REF]. The insertion of irregular polygons inscribed in ellipses a relevant topic for the Brazilian gem industry. As [START_REF] Brusso | Tecnologia 3D Gemas: otimizac ¸ão do aproveitamento de gemas coradas digitalizadas tridimensionalmente[END_REF] [START_REF] De | Dicionário de mineralogia e gemologia[END_REF], gems are generally natural and inorganic substances which are used for personal adornment. The development of this sector in Brazil led to the creation of multiple lines of research in technologies for the cutting techniques, which could improve the quality of the products composed of found gems in the rough state. Gems 3D Technology [START_REF] Brusso | Tecnologia 3D Gemas: otimizac ¸ão do aproveitamento de gemas coradas digitalizadas tridimensionalmente[END_REF], [START_REF] De | Dicionário de mineralogia e gemologia[END_REF][START_REF] Da | Identificac ¸ão da maior elipse com excentricidade prescrita inscrita em um polígono não convexo através do Continuous GRASP[END_REF][START_REF] Zanatta | Tecnologia 3D Gemas: visualizac ¸ão de projetos virtuais de lapidac ¸ão[END_REF][START_REF] Da | Algoritmo heurístico utilizando Grasp Contínuo para a otimizac ¸ão do aproveitamento volumétrico de gemas de cor[END_REF], has emerged with the intention of improving the quality of the products. This technology is a set of techniques and procedures developed, aiming to assist the process of lapidary faceted of colored gemstones, which consists in applying models of lapidary where the gems are fully constrained by flat surfaces, aiming to find each gem faceting design that results in higher added value, considering the volume and brightness. The insertion of models of lapidary, one of the areas in greater emphasis on lapidary industry, has as a fundamental part of the packaging ellipses or circles that form the basis of the oval and round models of lapidary [START_REF] Beckel | Modelagem Computacional e Otimizac ¸ão de secc ¸ões cônicas em polígonos irregulares de n lados[END_REF] [START_REF] Beckel | An adaptation of the Hooke and Jeeves optimization algorithm for the maximization of the dimensions of an ellipse inscribed in a given non-convex polygon[END_REF], as shown in Figure 1.

This paper presents a constructive heuristic for packing tangent ellipses in a given polygon. We consider a prescribed eccentricity for all packaged ellipses. The Hooke and Jeeves optimization method [START_REF] Beckel | Modelagem Computacional e Otimizac ¸ão de secc ¸ões cônicas em polígonos irregulares de n lados[END_REF][4] [START_REF] Bazaraa | Nonlinear programming: theory and algorithms[END_REF][START_REF] Hooke | Direct Search Solution of Numerical and Statistical Problems[END_REF][START_REF] Al-Fawzan | A tabu search Hooke and Jeeves algorithm for unconstrained optimization[END_REF][START_REF] Babu | Optimizing pulsed current gas tungsten arc welding parameters of AA6061 aluminium alloy using Hooke and Jeeves algorithm[END_REF] was adopted for positioning the initial ellipse inside the polygon. The polygons used for the simulations of the packing ellipses and for the validation of the developed heuristics represent the largest plan that describes a scanned three-dimensionally gem. Tangent ellipse packing in a n-side polygon is modeled as a two-dimensional problem here, where the parameters of each ellipse are the semi-minor and semi-major axes. The Hooke e Jeeves [START_REF] Bazaraa | Nonlinear programming: theory and algorithms[END_REF] is the core optimization method adopted. Translation and rotation of the ellipses are adopted as degrees of freedom.

Packing of Ellipses Tangents at Quadrants Initial Ellipse

The optimal tangent points are obtained using the parametric equation for each ellipse. The packing of ellipses tangent adopted in the developed heuristic is based on the adaptation made in the method used to manufacture of pieces Lego [START_REF] Dreshaper | Soft. 3DReshaper[END_REF]. The following restrictions are considered in the optimization problem:

• restrictions for the stating ellipse;

• restrictions for the tangent ellipses, and tangency conditions;

• additional condition for the convexity test.

First, we make the packing of each ellipse in the polygon P need not be convex. With the initial positioning, is carried out the verification of the inscription of the ellipses and if necessary is applied a process of gradual reduction in semi-major axis (a) the respective ellipse which not completely inscribed in P. The polygon used is composed of a set of vertices {(x 1 , y 1 ), (x 2 , y 2 ), . . . , (x np , y np )} and initial ellipse inscribed in P is described by the following parameters:

• Coordinates of the center of the polygon C(x c , y c ) obtained through software 3DReshaper [START_REF] Dreshaper | Soft. 3DReshaper[END_REF].

• Angle of counterclockwise rotation (θ ) the semi-major axis.

• Semi-major axis (a) of the ellipse, with eccentricity (e) fixed and semi-minor axis (b) defined by b = a 1e 2 ).

The restriction for the optimization of the starting ellipse subject to the restriction of space is:

Maximise a(x c , y c , σ ) subject to:

S = {(x c , y c , σ , a) : (x c + a cos(σ ), y c + a √ 1 -e 2 sin(σ )) ⊂ P, σ ∈ [0, 2π]}
With the initial parameters of the ellipse described, packing is achieved by adapting the method of Hooke and Jeeves [START_REF] Brusso | Tecnologia 3D Gemas: otimizac ¸ão do aproveitamento de gemas coradas digitalizadas tridimensionalmente[END_REF][2] as illustrated in Figure 2.

The points of tangency of every ellipse are obtained, and will be useful for positioning the secondary ellipses around the initial one. The point of tangency of the quadrant is given by the inter-section of the bisectrix with the

x(t) = x c + a cos(t) cos(σ ) -b sin(t) sin(σ ) x(t) = y c + a cos(t) sin(σ ) -b sin(t) cos(σ ) (1)
The heuristic adopted by [START_REF] Godtfred | Toy building brick[END_REF] for the construction of Lego bricks in the mid 1958 is the initial position of the ellipse and tangent. The concept the Lego is to project toy building elements with blocks adapted to be fixed to each other by means of projections extending from faces of the elements and which are arranged to involve protruding portions of an adjacent element when the two elements are assembled, as shown in Figure 3 In the project of the parts [START_REF] Dreshaper | Soft. 3DReshaper[END_REF], each circumference must be inserted so that the maximum area is occupied around the central circumference better fixing the embedded block, in other words, maintaining a tangency point in each quadrant of center circumference. The ideal positioning of the tangents circumferences can be seen in Figure 3(b). The point of tangency in the first quadrant is located in π/4, 3π/4 in the second quadrant, 5π/4 in the third quadrant and 7π/4 in the fourth quadrant. Defined the first point of tangency, the search for the other points is performed using the following symmetries in the plane: reflection around the y-axis, reflection around the origin and reflection around the x-axis.

Exploring Symmetries in the Plane

The heuristic of packing of tangent ellipses to each quadrant of an initial ellipse is based on the use of symmetries in the plane [START_REF] Beckel | Modelagem Computacional e Otimizac ¸ão de secc ¸ões cônicas em polígonos irregulares de n lados[END_REF] to determine the respectives points of tangency. Points are created in the counterclockwise direction of the initial ellipse and the transformations used are described according to the calculated point, considering the following matrix:

A = a cos(t) cos(σ ) -b sin(t) sin(σ ) a cos(t) sin(σ ) -b sin(t) cos(σ ) (2)
where t ∈ [0, 2π]. The reflection about the ordinate axis is described by the transformation T 1(x, y) in the plane, given by:

x y = x c y c + A. -1 0 0 1 , (3)
resulting in the parametric equation shown in Equation ( 4):

x(t) = x c -a cos(t) cos(σ ) -b sin(t) sin(σ ) x(t) = y c -a cos(t) sin(σ ) + b sin(t) cos(σ ) (4)
where (a, b) are respectively, the major and minor semi-axis of the ellipse packed, considering t ∈ [0, 2π]. The transformation T 2(x, y) describes the reflection around the origin given by Equation ( 5):

x y = x c y c + A. -1 0 0 -1 , (5)
and the parametric equation resulting from the reflection around the origin is given by:

x(t) = x c -a cos(t) cos(σ ) + b sin(t) sin(σ ) x(t) = y c -a cos(t) sin(σ ) -b sin(t) cos(σ ) (6)
For determine the point of tangency in the fourth quadrant of the original ellipse is necessary to make a reflection about the y-axis, defined by T 3(x, y) and shown in Equation ( 7):

x y = x c y c + A. 1 0 0 -1 , (7)
resulting in the following parametric equation:

x(t) = x c -a cos(t) cos(σ ) + b sin(t) sin(σ ) x(t) = y c -a cos(t) sin(σ ) -b sin(t) cos(σ ) (8)
In the Figure 4 are shown the symmetries in the plane used for the search of the point of tangency considering the point obtained in the first quadrant of the initial ellipse, positioned in π/4.

The first point of tangency is located in the first quadrant, given in function of bisectrix the quadrant (π/4) assuming the angle of rotation σ and the center C(x c , y c ), we have the parametric equation of the first point of tangency P1(P1x, P1y):

P1x = x c + a cos(π/4) cos(σ ) -b sin(π/4) sin(σ ) P1y = y c + a cos(π/4) sin(σ ) + b sin(π/4) cos(σ ) (9)
The coordinates of each point of tangency obtained are stored and used to determine the length of the major and minor semi-axis of the ellipse tangent. The center of each ellipse tangent is calculated assuming the same search direction of the point of tangency and the distance of initial ellipse center from the point of tangency. The length of the semi-major axis is defined by Equation [START_REF] Hooke | Direct Search Solution of Numerical and Statistical Problems[END_REF]: For the packing of ellipses in the polygon P, it is assumed 80% of the distance so that the ellipses are fully inscribed in the polygon P. In cases in which some points of the ellipses are outside the polygon applies a process of gradual reduction of the semi-major axis. The distance from the center to any point of tangency is equivalent, so it is possible to generalize the calculation of the distance, the value of the scalar λ to the tangent ellipse to the point P1(x, y) is calculated by Equation ( 11):

aw = 0.8 λ , λ ∈ R + (10)
dx = |P1x -x c | dy = |P1y -y c | (11)
with the scalar defined by:

λ = (dx) 2 + (dy) 2 (12)
and the semi-minor axis of the ellipse obtained by: bw = aw 1e 2 [START_REF] Godtfred | Toy building brick[END_REF] where e = 0.593946 is the eccentricity fixed for packaged ellipses. Defined the tangency point in each quadrant, it is used the modified parametric equation to calculate the center of the ellipse being packaged. The center C(x c , y c ) of the ellipse tangent to the point P1(x, y) is obtained by Equation ( 14) assuming t = π/4 and rotation angle σ :

P1x = x c + aw cos(π/4) cos(σ ) -bw sin(π/4) sin(σ ) P1y = y c + aw cos(π/4) sin(σ ) + bw sin(π/4) cos(σ ) (14)
To check the packing becomes necessary to define the vertices, the center and the point of tangency of each packaged ellipse. Considering c = cos(-σ ) and s = sin(-σ ), we have the vertices described by Equations (15-16):

A1 = A1x = x c -aw.c A1y = y c + aw.s B1 = B1x = x c + aw.c B1y = y c -aw.s (15)
Thus, we have established that the major axis of the ellipse is defined by Ā1B1. where the minor axis of the ellipse is given by C1D1. The vertices calculated are shown in Figure 5.

Figure 6 shows the graphical representation of the tangent ellipses, with the vertices of the ellipses are plotted in red and the center points of tangency and blue. The search performed by the method of Hooke and Jeeves [START_REF] Brusso | Tecnologia 3D Gemas: otimizac ¸ão do aproveitamento de gemas coradas digitalizadas tridimensionalmente[END_REF], [START_REF] Beckel | Modelagem Computacional e Otimizac ¸ão de secc ¸ões cônicas em polígonos irregulares de n lados[END_REF] position the initial ellipse near the contour of the polygon in the assumed direction, therefore there is few points of contact with the outside of the polygon. Therefore, it is necessary to define a point in each quadrant that has a likelihood to have points outside the polygon. The point corresponds to intersection of the bisectrix of the quadrant that has outer points to the polygon with the curve that describes the ellipse. Following the heuristic developed and transformations in the plane, with parametric equations and the same search direction of the tangent point and the center is possible to determine the point. Equation 17shows the result for the first quadrant.

PA(x) = P1x + 2aw cos(π/4) cos(σ ) -2bw sin(π/4) sin(σ ) PB(y) = P1y + 2aw cos(π/4) sin(σ ) + 2bw sin(π/4) cos(σ ) (17)
The heuristic for packing ellipses always considers the same search direction from the point of tangency and the center, as shown in Figure 7.

Verification of the inscription of the ellipses in the polygon P

To verify the inscription of ellipses tangent to the initial ellipse is necessary to obtain the matrix shown in Equation ( 18) which contains np-points of each ellipse (vertex, center, and aleatory point). With the matrix of points defined is held a test to verification of each point in relation to the polygon in order to ensure the full inscription of the ellipse, so that:

E i = Ex i Ex i+1 . . .
• P (x,y) (in) → P (x,y) ∈ V x • P (x,y) (∼ in) → P (x,y) / ∈ V x
The points P (x,y) (in) which belong to the polygon are stored in an array E i n as well the points P (x,y) (∼ in) are allocated in an array E (∼in) thereby separating the interiors and exteriors points. If neither point P (x,y) is outside to the polygon then the matrix E (∼in) will be empty. If the matrix E (∼in) have some element, applies the process of refinement of the semi-major axis of the ellipse which being inserted to get an array E (∼in) resultant empty. The matrix used for verification is defined by Equation ( 19):

E = A1x B1x C1x D1x x c PAx A1y B1y C1y D1y y c PAy (19)
containing the vertices, and the center point aleatory of each ellipse. For the verification of matrix of points is used the function inpolygon available in the MatLab [START_REF] Mathworks | Matlab and simulink for technical computing[END_REF] software. The function inpolygon is is structured as follows form:

• IN = inpolygon (X,Y, xv, yv) • [IN ON] inpolygon = (X,Y, xv, yv)
The instruction IN = inpolygon (X,Y, xv, yv) returns an array IN the same size of X and Y . For each element of IN a value from the set 0, 1 is assigned, conditioned to the point (X(p, q),Y (p, q)) be inside the polygonal region whose vertices are specifically defined by the vectors xv and yv, resulting in two cases:

• If (X(p, q),Y (p, q)) is inside the polygonal region or the boundary of the polygon then IN (p, q) = 1;

• If (X(p, q),Y (p, q)) is outside the polygonal region then IN (p, q) = 0.

The instruction [IN ON] inpolygon = (X,Y, xv, yv) returns a second matrix ON the same size of X and Y . For each element of ON is assigned the value of 0 or 1 considering that the point (X(p, q),Y (p, q)) may be in the polygonal region, resulting in two new cases:

• If (X(p, q),Y (p, q)) is on the boundary of the polygonal region then ON (p, q) = 1;

• If (X(p, q),Y (p, q)) is inside or outside the polygonal region then ON (p, q) = 0. According to the heuristic developed for the packing, all ellipses must be fully inscribed in the polygon. Thus, the matrix [IN ON] of each ellipse shall contain only interior points. If return an point outside the polygon, is made the gradual reduction of the semi-major axis and is held again the calculation of semi-minor axis, of the center, coordinates of the vertices and the aleatory point until all points are interior to the polygon.

The new length of the semi-major axis is defined by:

aw = aw.(1 -α) ( 20
)
where α = 0.1 established by the heuristic as the reduction factor. Performed the verification of the inscription all ellipses obtained are fully included in P.

Results and Discussion

The validation of the constructive heuristic for packing of ellipses was performed using both convex and nonconvex polygon. Figure 9 shows the inscription of ellipses tangent to the initial ellipse based on data obtained from Table 1 for a polygon with 122 vertexes. Table 1 summarizes the results. Table 1 shows the computational processing time that is demanded for registration of each ellipse. The simulation time for the verification of the inscription varies according to the number of vertices of the polygon and the number of iterations performed to obtain a viable solution with all ellipses inscribed. With the procedure for the gradual reduction of the semi-major axis of the ellipses partially outside the polygon, the packing of ellipses is total, as can be seen in the Figure 10.

Figure 11 shows the initial positioning of ellipses in another not-necessarily convex polygon with 232 vertexes, whose time of computational processing is equivalent to the processing time of the previous simulation. In the first positioning, as shown in the Figure 11, the ellipses tangent to the initial ellipse feature all the same length of semi-major axis, it is verified on the Table 3 from the data obtained. Applying the procedure of gradual

PT (x, y) 1 C(x c , y c ) 2
PA(x, y) 1. Results obtained with the positioning of the ellipses -simulation 1. 1 Point of tangency, 2 Center of the ellipse, 3 Aleatory point, 4 Total processing time (microseconds), 5 Semi-major axis, 6 Semi-minor axis, 7 Area. reduction of the semi-major axis of the ellipses partially inscribed, we have final positioning of ellipses as shown in Figure 12. PT (x, y) 1 C(x c , y c ) 2 PA(x, y) 2. Results obtained with the positioning of the ellipses and gradual reduction of the semi-major axissimulation 1. 1 Point of tangency, 2 Center of the ellipse, 3 Aleatory point, 4 Total processing time (microseconds), 5 Semi-major axis, 6 Semi-minor axis, 7 Area. Table 3. Results obtained with the positioning of the ellipses -simulation 2. 1 Point of tangency, 2 Center of the ellipse, 3 Aleatory point, 4 Total processing time (microseconds), 5 Semi-major axis, 6 Semi-minor axis, 7 Area.

PT (x, y) 2 Center of the ellipse, 3 Aleatory point, 4 Total processing time (microseconds), 5 Semi-major axis, 6 Semi-minor axis, 7 Area.

With the packing of ellipses tangent to the initial ellipse completed, the parameters are re-evaluated to assure the efficiency of the developed heuristic. The high computational processing time is due to the number of vertices of the polygon and for every check of the inscription of ellipses is necessary to perform a scan for all the vertices of the polygon. Figure 13 shows the complete packing of the ellipses.

Table 5 shows general results, after iteration of the simulation performed for the polygon in Figure 13. Total processing time (tpt), total area of packed ellipses (ta) and remaining area (ra) are computed at each iteration. Figure 14 shows a spline curve fitting for the computed points. The Equation (21) shows performed curve fitting:

y = ae bx (21)
The adjustment of the series of data in Table 5 for parameterized curves shown in Figure 14 and to determine the trend of constructive heuristic generates a function that can be used for the heuristic comparisons in each polygon and a subsequent widespread process. Figure 14 illustrates the exponential behavior of the reduction of remaining area as a function of the computational processing time. The function obtained from curve fitting is:

y = 137.9e -1,403x (22)
With the exponential function described in Equation ( 22), it is possible to determine the percentage remaining area as an elapsed processing time or to establish the time required to obtain the optimal solution. Another factor that should be highlighted in the analysis of the results is the easy adaptation of the heuristics in polygons with large hollows. The heuristic developed performs searches in random directions but not employ any method that uses the gradient vector ∇aw(x c , y c , σ ) and ∇aw(C x ,C y , σ ) information. The behavior of the heuristic for situations with accentuated concavity was also tested. Figure 15 illustrates two of those cases.

Positioning regions vary from iteration to iteration, which illustrates the ability of the heuristic to search for the best positioning, successfully avoiding ellipses to get stuck in the concavities of the polygons.

Conclusion

A constructive heuristic for packing of tangent ellipses in irregular polygons of n-sides was shown to be effective when solving the problem of positioning of ellipses used as basis in the oval gemstones cutting model. In order to make the process viable for to the lapidary industry, the mathematical formulation restricted to the boundary conditions imposed by the problem were taken into consideration. The verification phase of the inscription of ellipses requires more computational time in cases where there are points outside the polygon and subsequently these ellipses go through the process gradual reduction of the semi-major axis. The heuristic developed was shown to be highly adaptive for different types of polygons both convex and no convex. One of the great advantages of using the Hooke and Jeeves method is that this method can be easily modified and adjusted to the problem you want to solve. This change is observed in problems that take into consideration the constraints of the contour or space of possible solutions. Becomes an unconstrained optimization method for a constrained optimization method to ensure that the optimal solution is contained in the polygonal region formed by vertices that describe the contour of the major plane where the initial ellipse is inserted.

Figure 1 .

 1 Figure 1. Packing of models of lapidary on a gem with 3D Technology Gems.

Figure 2 .

 2 Figure 2. Initial ellipse inscribed in P -Base of oval lapidary model.

  (a).

Figure 3 .

 3 Figure 3. (a) Lego blocks, (b) Positioning of the circumferences in each quadrant.

Figure 4 .

 4 Figure 4. Symmetries in the plane and tangency points in the initial ellipse.

Figure 5 .

 5 Figure 5. Coordinates of the center and vertices of the ellipse.

Figure 6 .

 6 Figure 6. Ellipses tangent to the initial ellipse with vertices, points of tangency and center defined.

Figure 7 .

 7 Figure 7. Points with search direction defined.

Figure 8 .

 8 Figure 8. Relative position of the point with respect to the polygon P.

Figure 9 .

 9 Figure 9. Ellipses tangent to the initial ellipse -simulation 1.

Figure 10 .

 10 Figure 10. Ellipses fully inscribed in P -simulation 1.

Figure 11 .

 11 Figure 11. Ellipses tangent to the initial ellipse -simulation 2.

Figure 12 .

 12 Figure 12. Ellipses fully inscribed into P -simulation 2.PT (x, y) 1 C(x c , y c ) 2 PA(x, y) 3 T T P 4 a 5 b 6 A 6 EI -(9.91, 3.22) -974.98 6.67 5.37 112.72 E1 (9.51, -2.81) (9.23, -7.20) (8,94, -11.60) 63.74 4.84 3.9 59.41 E2 (15.90, 4.13) (20.25, 4.79) (24.60, 5.45) 38.19 4.84 3.9 59.41 E3 (10.30, 9.27) (10.59, 13.66) (10.88, 18.05) 38.78 4.84 3.9 59.41 E4 (3.92, 2.32) (-0.43, 1.65) (-4.78, 0.99) 38.56 4.84 3.9 59.41

Figure 13 .

 13 Figure 13. Final packaging ellipses.

Figure 14 .

 14 Figure 14. Simulation with curve fitting.

Figure 15 .

 15 Figure 15. Packaged ellipses restricted to the boundary with concavities.

  Ex np Ey

i Ey i+1 . . . Ey np

(18)

Table

			3	T T P 4	a 5	b 6	A 6
	EI -	(4.12, 11.53) -	1024.03	4.96 3.99 62.31
	E1 (1.16, 14.93) (0.88, 15.27) (0.59, 15.60) 28294.81 0.48 0.39 0.59
	E2 (1.43, 7.91)	(-0.51, 5.29) (-2.47, 2.66) 39.38	3.60 2.90 32.84
	E3 (7.08, 8.13)	(7.75, 7.35)	(8.42, 6.58)	22405.90 1.13 0.91 3.23
	E4 (6.81, 15.14) (7.21, 15.69) (7.61, 16.23) 25919.02 0.74 0.59 1.39

Table 4 .

 4 Results obtained with the positioning of the ellipses -simulation 2.1 Point of tangency,

	1	C(x c , y c ) 2	PA(x, y) 3	T T P 4	a 5	b 6	A 6
	EI -	(9.91, 3.22)	-	1023.52	6.67 5.37 112.72
	E1 (9.51, -2.81) (9.36, -5.15)	(9.21, -7.48)	20572.25 2.57 2.07 16.78
	E2 (15.90, 4.13) (16.63, 4.24)	(17.35, 4.35)	36532.59 0.80 0.65 1.65
	E3 (10.30, 9.27) (10.36, 10.09) (10.41, 10.90) 35731.06 0.89 0.72 2.04
	E4 (3.92, 2.32)	(2.04, 2.03)	(0.17, 1.75)	24995.03 2.08 1.67 11.01

Table 5 .

 5 Results obtained with the constructive heuristic.

	I	Ttp (s)	At (mm 2 ) Ar (mm 2 ) Pr (%)
	0	4.7954	7.8932	207.8757 96.3418
	1	4.7047	16.9199	198.8491 92.1583
	2	19.8998	30.2945	185.4744 85.9597
	3	6.7174	75.8355	139.9335 64.8534
	4	60.8931	112.7322 103.0368 47.7533
	5	119.6117 127.2988 88.4702	41.0023
	6	199.3145 144.9635 70.8055	32.8154
	7	214.8979 150.4614 65.3076	30.2674
	8	225.3092 151.5046 64.2645	29.7839
	9	216.3957 151.2833 64.4857	29.8865
	10 288.3527 159.3059 56.4631	26.1683

Acknowledgments

C. C. Beckel acknowledge financial support from CNPq, Brasilia, DF.