Stéphane Gonzalez
email: stephane.gonzalez1@malix.univ-paris1.fr

Michel Grabisch
email: michel.grabisch@univ-paris1.fr

Preserving coalitional rationality for non-balanced games

Keywords: cooperative game, core, balancedness, general solution

In cooperative games, the core is one of the most popular solution concept since it ensures coalitional rationality. For non-balanced games however, the core is empty, and other solution concepts have to be found. We propose the use of general solutions, that is, to distribute the total worth of the game among groups rather than among individuals. In particular, the k-additive core proposed by Grabisch and Miranda is a general solution preserving coalitional rationality which distributes among coalitions of size at most k, and is never empty for k ≥ 2. The extended core of Bejan and Gomez can also be viewed as a general solution, since it implies to give an amount to the grand coalition. The k-additive core being an unbounded set and therefore difficult to use in practice, we propose a subset of it called the minimal negotiation set. The idea is to select elements of the k-additive core mimimizing the total amount given to coalitions of size greater than 1. Thus the minimum negotiation set naturally reduces to the core for balanced games. We study this set, giving properties and axiomatizations, as well as its relation to the extended core of Bejan and Gomez. We give a method of computing the minimum bargaining set, and lastly indicate how to eventually get classical solutions from general ones.

Introduction

One of the major challenges of the theory of cooperative games with transferable utility is to propose an allocation of gains obtained by a set N of players. Typically, this distribution (called a solution) is done among the individual players, and many concepts of solutions have been proposed so far, the core being one of the best known solution since it ensures coalitional rationality. However, in many real situations, the distributions of gains is not done to individuals but often to groups of individuals (associations, companies, families, etc.), which can further distribute among their members according to their own rules, which can differ from one group to another. Also, the drawbacks of the classical solutions like the core are well known. For example, the core is often empty, which obliges to take other solution concepts, often violating coalitional rationality.

A natural generalization of the classical view of solutions should be then to distribute among groups rather than among individuals. We call this a general solution, and observe that we can already find in the literature two attempts in this direction. The first one is the k-additive core proposed by Grabisch and Miranda [START_REF] Grabisch | On the vertices of the k-additive core[END_REF][START_REF] Miranda | k-balanced games and capacities[END_REF], which was also proposed earlier by Vassil'ev [START_REF] Vassil'ev | Polynomial cores of cooperative games[END_REF] in a publication in russian. The extended core of Bejan and Gomez [START_REF] Bejan | Core extensions for non-balanced TU-games[END_REF] can also be viewed as a general solution 1 , since it implies to give an amount to the grand coalition. In the k-additive core, the distribution of v(N ) is done among coalitions of size at most k, and coalitional rationality is ensured (thus, the 1-additive core is the classical core). In the G-extended core, the distribution is done among individuals and the grand coalition. When the core is empty, these two general solutions allocate some negative amounts (debts) to some coalitions (the grand coalition in the case of the G-extended core).

The aim of this paper is to propose a subset of solutions of the k-additive core, and to relate it to the G-extended core. Indeed, if the k-additive core has some remarkable features (it is never empty as soon as k ≥ 2, and it preserves coalitional rationality), its main drawback is that it is an unbounded set for any k ≥ 2. We call the subset of the k-additive core we propose the minimum negotiation set, since it is based on the following idea: the less negotiation is necessary, the better the solution. Indeed, a general solution obliges individuals belonging to a group which has received an allocation to bargain among them in order to share the amount received by the group (often a negative amount, if the core is empty). Therefore, it is desirable to focus on solutions minimizing the number of groups and the allocation given to them. This is achieved by minimizing a norm of the vector of allocations to groups. We will show that the minimum negotiation set coincides with the core when the latter is nonempty, it is a singleton for all L p norms with p > 1, and for the L 1 norm, the payments given to individuals are exactly those produced by the G-extended core.

An important feature of general solutions is that they do not define a payment to individuals like classical solutions. Hence, strictly speaking they are not solutions to the game, but rather pre-solutions, which need an additional step where payments given to coalitions (in fact, as we have shown, these are most often debts) have to be further redistributed among the members of these coalitions. The question is then: What do we (l)earn from general solutions if eventually they collapse to classical solutions? A short answer to this question would be that doing so, we main attain solutions which could have never been reached by classical mechanisms. A more refined answer is that by this two-step procedure, we control the result by imposing appropriate rationales at each step. For example, using the k-additive core or subsets of it, we guarantee coalitional rationality in the first step. In the second step, we can choose for each coalition in debt an appropriate sharing rule (and then we get an element of the selectope [START_REF] Derks | The selectope for cooperative games[END_REF]) or, more generally, considering this as a bankruptcy problem [START_REF] Thomson | Axiomatic and game-theoretic analyss of bankruptcy and taxation problems: A survey[END_REF], we can choose an appropriate division rule. Other methods can be used as well, like bargaining procedures. The main aim of the paper is not to develop this second step, which should deserve a whole study, but to focus on the first one and to study properties of general solutions and their relationship. However, we have devoted a section to the second step giving the main ideas.

The paper is organized as follows: Section 2 introduces the concept of general solution and the basic material which is needed. In particular, we present properties for general solutions, and we introduce the k-additive core and the G-extended core. Section 3 focusses on the main achievement of the paper, the minimum negotiation set. It studies its properties, and two particular cases of norms (L 1 and L p for p > 1), and gives axiomatizations. Also, the notion of autonomous coalition is introduced, as well as a method to find them. Section 4 addresses the problem of computing the minimum negotiation set, and lastly we give a brief account on how to derive a classical payoff vector from a general one in Section 5, and finishes the paper with concluding remarks.

General payoff vectors and solutions

Basic definitions and notations

We start by fixing the notation and introducing basic definitions. Let N = {1, . . . , n} ⊂ N be a finite and nonempty set of agents or players. Coalitions of players are subsets of N , denoted by capital letters S, T , and so on. Whenever possible, we will omit braces for singletons and pairs, denoting {i}, {i, j} by i, ij respectively, in order to avoid a heavy notation. A transferable utility (TU) game on N is a pair (N, v) where v is a mapping v : 2 N → R satisfying v(∅) = 0. We will denote by G(N ) the set of all games over N . For any coalition S, v(S) represents the worth of S, i.e., what coalition S could earn regardless of other players.

To every game v ∈ G(N ) we associate its Möbius transform [START_REF] Rota | On the foundations of combinatorial theory I. Theory of Möbius functions[END_REF] (also known as Harsanyi dividends [START_REF] Harsanyi | A simplified bargaining model for the n-person cooperative game[END_REF]) m v : 2 N → R, defined by

m v (S) := T ⊆S (-1) |S\T | v(T ), ∀S ⊆ N, (1)
and m v (∅) = 0. Conversely, v can be recovered from m v by the inverse transform

v(S) = T ⊆S m v (T ), ∀S ⊆ N. ( 2
)
Hence the Möbius transform is a linear bijection on G(N ). Note in particular that

m v ({i}) = v({i}) for all i ∈ N , implying that m v (S) = 0 if |S| > 1 whenever v is additive.
We recall also that the Möbius transform gives the coordinates of a game into the (2 n -1)-dimensional basis of unanimity games {u S } S⊆N,S =∅ , with u S (T ) = 1 if T ⊇ S, and 0 otherwise. A payoff vector is a vector x ∈ R n that assigns agent i the payoff x i . Given x ∈ R n , and S ⊆ N , denote by x(S) the sum i∈S x i with the convention that x(∅) = 0. A payoff vector is efficient for game v if x(N ) = v(N ). We will call preimputation of v each efficient payoff vector of v and we denote by P I(v) the set of preimputations of v.

Coalition S is able to improve upon payoff x if x(S) < v(S). The core C(v) of a game v is the set of efficient payoffs that cannot be improved upon by any coalition, i.e.,

C(v) = {x ∈ R n | ∀S N, x(S) ≥ v(S), and x(N ) = v(N )}.
A game with a nonempty core is said to be balanced. Whenever convenient, we consider the core as a correspondence C : G(N ) ⇒ R n .

A general payoff vector is a vector x ∈ R 2 N \∅ that assigns to a coalition S ⊆ N a payoff x S . Also, 1 S with S ⊆ N indicates the vector with coordinate 1 for S and 0 otherwise.

The notion of efficiency easily generalizes: as for the classical case, the sum of payoffs given to all coalitions should be equal to the total available amount, that is, v(N ). Formally, a general payoff vector is said to be efficient for game v if

∅ =S⊆N x S = v(N ).
Example 1. A payoff vector x ∈ R n defines a general payoff vector by considering x(S) for all nonempty S ⊆ N . Hence, any element of the core induces a general efficient payoff vector.

Example 2. A less trivial example of general payoff vector, which will be central in our study, is the Möbius transform m v of a game v. Note that by (2) m v is efficient for v.

A solution on the set of games G(N ) is a correspondence σ : G(N ) ⇒ R n , i.e., it assigns to any game v ∈ G(N ) a set of payoff vectors, with the additional property that they are all efficient for v. Analogously, a general solution assigns to every game v in G(N ) a set of general payoff vectors efficient for v.

The core is a well-known solution for games. A trivial example of general solution is the Möbius transform m (see Example 2). Although the notion of general solution is not explicitely considered in the literature, there are two recent remarkable examples of general solutions. The first one is the k-additive core [START_REF] Miranda | k-balanced games and capacities[END_REF], where a payoff is given to each coalition of size at most k, while the second example is the extended core [START_REF] Bejan | Core extensions for non-balanced TU-games[END_REF], which can be viewed as a general solution. In the extended core, apart giving a payoff to each player, a payoff is also (virtually) given to the grand coalition N . We will study this concept in Section 2.4. Our paper mainly focusses on the k-additive core, which is presented in Section 2.3. Before that, we introduce some properties for general solutions.

Properties of general solutions

We propose several properties for general solutions, some of them being direct generalization of classical properties of solutions. Let σ be a general solution on G(N ). We say that σ

• is covariant under strategic equivalence (COV) if ∀v ∈ G(N ), ∀α > 0, ∀β ∈ R 2 n -1 : σ(αv + β) = ασ(v) + β. • is an idempotent solution (IDEM) if σ(m -1 • σ) = σ.
The property means that applying the same solution concept to the set of solutions found (after transforming them into games by m -1 ) does not give more solutions.

• is symmetric (SYM) if ∀v ∈ G(N ), for all permutations π on N such that v(π(S)) = v(S) holds for all S ⊆ N :

(σ(v)) π(S) = (σ(v)) S , ∀∅ = S ⊆ N,
where π(S) = {π(i), i ∈ S}. The property says that if a permutation leaves the game invariant (e.g., i and j can be permuted), then the same permutation leaves the general payoff vectors in the solution invariant too (e.g., S ∪ i and S ∪ j receive the same payoff for all S i, j).

• is coalitionally rational

(CR) if ∀v ∈ G(N ), ∀S ⊆ N , ∀x ∈ σ(v), T ⊆S x T ≥ v(S).
As for efficiency, this is a direct extension of the classical definition. Indeed, the total amount given to coalition S (that is, to all its individuals and all subcoalitions) is precisely T ⊆S x T , which should be at least equal to the worth that S can achieve by itself, that is, v(S).

• gives preimputation (PI) if ∀v ∈ G(N ), ∀S ⊆ N such that |S| ≥ 2 we have:

∀x ∈ σ(v), x S = 0.
Only classical solutions, like the core or the Shapley value, satisfy (PI).

• satisfies the dummy player property (DPP) if for any dummy player i, we have:

∀x ∈ σ(v), x i = v(i),
where as usual a player i is dummy if v(S ∪ i) = v(S) + v(i) for every S ⊆ N \ i.

• is a minimization of the global debt (MGD) if for all nonnegative games v, for all x ∈ σ(v),

S⊆N |S|≥2 min(x S , 0) = -t(v),
where t(v) = min{t ≥ 0 | C(v t ) = ∅}, with v t the game defined by v t (N ) = v(N )+t, and v t (S) = v(S) otherwise (see Section 2.4).

This axiom can be explained as follows. By definition, t(v) is the minimal amount given to the grand coalition so that coalitional rationality with classical payoff vectors is ensured. However, the players in N are in debt of the amount t(v). In terms of general payoff vectors, this means that N has to pay the debt, or more generally, it can be spread among coalitions. The axiom says that the total debt should not exceed the minimal amount t(v), and since it cannot be less without losing coalitional rationality, equality follows.

The set of general solutions is a partially ordered set with the relation defined by:

σ 1 σ 2 ⇐⇒ ∀v ∈ G(N ), σ 1 (v) ⊆ σ 2 (v).
For any set A of general solutions, whenever it exists we denote by (A) the top element of A for the relation , that is, the element satisfying σ (A) for all σ ∈ A. For example, if we denote by P ICR the set of general solutions which satisfy (PI) and (CR), then C = (P ICR).

The k-additive core

A game v ∈ G(N ) is said to be k-additive [START_REF] Grabisch | k-order additive discrete fuzzy measures and their representation[END_REF] if its Möbius transform m v vanishes for subsets of more than k players: m v (S) = 0 if |S| > k, and there exists K ⊆ N , |K| = k, such that m v (K) = 0. Note that a 1-additive game is an additive game in the usual sense.

A game v is said to be at most k-additive if it is q-additive for some q ∈ {1, . . . , k}. We denote by G k (N ) the set of all at most k-additive games.

The k-additive core, introduced by Grabisch and Miranda [7,[START_REF] Grabisch | On the vertices of the k-additive core[END_REF][START_REF] Miranda | k-balanced games and capacities[END_REF], and also by Vassil'ev [START_REF] Vassil'ev | Polynomial cores of cooperative games[END_REF], is the set of efficient and coalitionally rational general payoff vectors, with the restriction that payoffs are limited to coalitions of at most size k.

Definition 1. Let v be a game. The k-additive core 2 of v, denoted by C k (v), is given by C k (v) = x ∈ R 2 N \∅ | T ⊆S x T ≥ v(S), ∀S ⊆ N, T ⊆N x T = v(N ), x T = 0, ∀T ⊆ N, |T | > k .
As before we may consider the k-additive core as a correspondence

C k : G(N ) ⇒ R 2 N \∅ .
Example 3. We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1. [START_REF] Fujishige | Submodular functions and optimization[END_REF] 2

Let a general payoff vector x be defined by

S 1 2 3 12 13 23 123 x S 2 2 3 -1 -2 -2 0
It is easy to check that x satisfies coalitional rationality and efficiency. Moreover, x 123 = 0, hence x ∈ C2 (v). The corresponding 2-additive game φ can be computed through the inverse Möbius transform (Equation ( 2)):

S 1 2 3 12 13 23 123 φ(S) 2 2 3 3 3 3 2
Theorem 1. [START_REF] Miranda | k-balanced games and capacities[END_REF] The k-additive core is a convex polyhedron, nonempty for k ≥ 2.

C k (v) = ∅, ∀v ∈ G(N ), ∀k ≥ 2.
Unlike the core, the k-additive core is always nonempty. However, it is an unbounded set, as shown by the next proposition. Theorem 2. For all k ≥ 2, the k-additive core is unbounded and pointed3 . Moreover, for k = 2, its extremal rays are of the form 1 i -1 ij , for all i = j ∈ N .

Proof. Since for any game v, the inclusion C 2 (v) ⊆ C k (v) holds for any k ≥ 2, it suffices to show that the 2-additive core is unbounded.

From standard results in polyhedra, studying the unboundedness of the 2-additive core amounts to studying its recession cone, i.e., the set of inequalities where the right member is replaced by 0 (see, e.g., [START_REF] Fujishige | Submodular functions and optimization[END_REF]Ch. 1]). If the recession cone is a pointed cone not reduced to {0}, then the corresponding polyhedron has vertices and rays. i.e., it is unbounded. We denote naturally by C 2 (0) the recession cone of C 2 (v).

The system of inequalities defining x ∈ C 2 (0) reads:

x i ≥ 0, ∀i ∈ N x i + x j + x {i,j} ≥ 0, ∀i, j ∈ N, i = j i∈S x i + {i,j}⊆S x {i,j} ≥ 0, ∀S ⊆ N, 2 < |S| < n i∈N x i + {i,j}⊆N
x {i,j} = 0. We claim that any vector of the form r = 1 i -1 {i,j} , i.e., having value 1 for coordinate i, -1 for coordinate {i, j}, and 0 otherwise, is an extremal ray. Indeed, the vector αr satisfies the above system for any α > 0, and it cannot be expressed as the sum of other rays since any ray must have at least 2 nonzero coordinates. Note also that there is no other extremal ray.

As a conclusion, the 2-additive core is unbounded. Let us show that the k-additive core is pointed. This amounts to find in the system of equalities T ⊆S

|T |≤k

x T = 0, S ⊆ N , a subsystem of exactly n + n 2 + • • • + n k equations whose unique solution is the zero vector. For this, it suffices to take those corresponding to all S such that |S| ≤ k. Proposition 1. For all k ∈ {1, . . . , n}, C k satisfies (COV), (IDEM), (SYM), and (CR). Moreover, if we denote by CR the set of general solutions which satisfies (CR), we have:

C n = (CR).
Proof. We prove only (IDEM), the rest being clear. For clarity, we introduce the notation

c k = m -1 (C k ). Then, it amounts to prove c k • c k = c k . Let v be a game on N . Take any φ ∈ c k (v). Then φ ∈ c k (φ), therefore φ ∈ c k (c k (v)) Conversely, if φ ∈ c k (c k (v)), then ∃ψ ∈ c k (v) such that φ ∈ c k (ψ). We have φ(S) ≥ ψ(S) ≥ v(S) ∀S ⊆ N and φ(N ) = ψ(N ) = v(N ). Since φ is k-additive we have φ ∈ c k (v).

The G-extended core

In this section we introduce a new general solution based on the principle of the extended core EC(v) of Bejan and Gomez. Given a game v and t ≥ 0, define its t-expansion v t by v t (S) := v(S) for all S ⊂ N , and v t (N ) := v(N ) + t. Now, to any game v we assign t(v) := min{t ≥ 0 | C(v t ) = ∅}, the minimum amount to be given to the grand coalition in order to ensure balancedness. Given v a game and x a preimputation, we introduce the real number t(v, x) := min{l(N ) | l ∈ R n + and (x + l)(S) ≥ v(S), ∀S ⊆ N }. Bejan and Gomez [START_REF] Bejan | Core extensions for non-balanced TU-games[END_REF] have introduced the concept of extended core of v as follows:

EC(v) := {x ∈ P I(v) | t(v, x) = t(v)}.
The extended core of v coincides with the core whenever C(v) = ∅. It is the set of preimputations x for which there exists

l x ∈ R n + such that (x + l x )(S) ≥ v(S), ∀S ⊆ N and i∈N l x i = t(v).
In other words, the extended core is the set of those preimputations that require a minimal subsidy to the grand coalition.

Since the extended core is a set of preimputations, it is not a general solution as announced in the introduction. However, a general solution is inherent in the way Bejan and Gomez use the extended core: given a tax rule, take any vector in the core of v t(v) (where precisely the amount t(v) has been given to the grand coalition) and tax every player according to the chosen rule, where the tax is t(v). In what follows, we propose a rewriting of the extended core as a general solution, which we call the G-extended core.

Consider z ∈ EC(v) and its associated vector l z ∈ R n + as above. We build from z a general payoff vector x z defined as follows:

x z S =        z i + l z i , if S = {i} -t(v), if S = N 0, otherwise.
Based on this, we define the G-extended core of v as follows.

Definition 2. Let v be a game. Its G-extended core is the set defined by

GEC(v) := {x z ∈ R 2 N \∅ | z ∈ EC(v)},
or equivalently

GEC(v) = {x ∈ R 2 N \∅ | (x i ) i∈N ∈ C(v t(v) ), x N = -t(v), x S = 0 otherwise}. Proposition 2.
The following properties hold for GEC.

(i) GEC(v) is a nonempty convex and compact polyhedral set for all v ∈ G(N ). It is equal to the core4 if the latter is nonempty.

(ii) GEC satisfies (CR). Hence, GEC(v) ⊆ C n (v).

(iii) GEC satisfies (IDEM), (COV), (SYM) and (MGD).

Proof. Most of the properties are clear by definition. We detail only (IDEM). As before, we use the shorthand c for m -1 (C), as well as gec = m -1 • GEC. We suppose first that v is balanced. Therefore, gec(v) = c(v) and the property holds since c • c = c. Suppose then that v is not balanced. Take any φ ∈ gec(v) and let us prove that φ ∈ gec(φ). Since φ ≥ v and φ(N ) = v(N ), we have t(φ) ≥ t(v). We have (φ i ) i∈N ∈ c(φ t(v) ), since for all S = N , i∈S φ i ≥ φ(S), and i∈N φ i = v(N ) + t(v) = φ(N ) + t(v). This proves that t(φ) = t(v), and consequently φ ∈ gec(φ). In conclusion, gec(v) ⊆ gec•gec(v).

Conversely, let φ ∈ gec • gec(v). Then there exists ψ ∈ gec(v) such that φ ∈ gec(ψ). Reasoning as above, we find that t(ψ) = t(v). Then for any S = N we have i∈S φ i ≥ ψ(S) ≥ v(S), and

i∈N φ i = ψ(N ) + t(v) = v(N ) + t(v). Lastly, m φ (N ) = -t(v), which proves that φ ∈ gec(v).
We observe that although the global amount of debt is minimal, it is concentrated only on the grand coalition and therefore does not necessarily optimally express the balance of power of each coalition. By contrast, in the k-additive core, the smallest coalitions are in charge of distributing the debt.

The minimum negotiation set

The principle of the k-additive core is: if a payment x S is given to coalitions S up to size k, that is, if each player accepts to pool a part of his gain with the other players, it is possible to preserve coalitional rationality. However, we can note two important problems of the k-additive core: First, for k ≥ 2, the k-additive core is unbounded, secondly, there could be a conflict between the agents of a coalition S on how the payment x S made by the k-additive core will be allocated to each player. Two situations can occur: either the payment is positive (a benefit has to be shared), or it is negative (a debt has to be shared). Sharing implies that each player in the coalition receives a nonnegative amount in case of benefit, and has to pay a nonnegative amount in case of debt 5 . The more there are coalitions S under this situation with high amounts |x S |, the more the solution x involves negotiation among the agents in N . The idea of the minimum negotiation set is to select those solutions in the k-additive core which involve the "fewest" negotation. Here, "fewest" is defined through a norm of the vector of payments made to coalitions of size at least 2. Taking the family of L p norms, we may consider the three usual cases: the L 1 norm (the negotiation level is the sum of absolute values of the payments), the L 2 norm (sum of squares of payments: high payments are over-weighted), and the L ∞ norm (the maximal payment is the negotiation level).

Another view of the minimal negotiation set is that it is the set of solutions minimizing the distance (defined through the norm) to the set of preimputations. Hence the negotiation level of a payoff of the k-additive core can be seen as a kind of measure of its deviation from additivity.

The set I k (v) of minimal negotiation

Let 2 ≤ k ≤ n be fixed and • be a norm on R 2 n .

We consider the following nonlinear program:

Minimize B(x) := (x S ) S⊆N |S|≥2 subject to x ∈ C k (v).
We call B(x) the negotiation level of x. For a game v ∈ G(N ) and ≥ 0, we define the set

C k (v) by C k (v) := {x ∈ C k (v), B(x) ≤ }
and the set I k (v) by

I k (v) := ≥0 C k (v) =∅ C k (v). I k (v) is nonempty for all games in G(N ) because C k (v) = ∅ implies that there exists 0 such that C k 0 (v) = ∅. Moreover, I k (v)
is the set of optimal solutions of the above nonlinear program.

Theorem 3. Let v ∈ G(N ), 2 ≤ k ≤ n,
and ≥ 0. The following statements hold:

(i) ∀ ≥ 0, C k (v) is a convex and compact set which is equal 6 to C(v) if = 0. (ii) I k (v) is a nonempty convex and compact set, equal to C(v) if C(v) = ∅, otherwise ∃ * > 0 such that I k (v) = C k * (v). Proof. (i) It is obvious that C k 0 (v) = C(v). Also, C k (v) is closed ∀ ≥ 0, and if 0 ≤ 1 ≤ 2 , then C k 1 (v) ⊆ C k 2 (v). Let us prove convexity. If a and b are in C k (v) then ∀t ∈ [0, 1], the payoff vector ta + (1 -t)b is in C k (v) because C k (v) is convex. Furthermore, by convexity of x → x , (ta S + (1 -t)b S ) S⊆N |S|≥2 ≤ t (a S ) S⊆N |S|≥2 ≤ +(1 -t) (b S ) S⊆N |S|≥2 ≤ , from which we deduce that C k (v) is convex. It remains to prove compactness. Suppose a ∈ C k (v).
I k (v) = C(v).

Let us define

* = sup{ ≥ 0 such that C k (v) = ∅} with the convention sup(∅) = 0. It is easily verified that C k * (v) ⊆ I k (v) and, ∀n ∈ N, the set I k (v) satisfies I k (v) ⊆ C k * + 1 n (v). Therefore I k (v) ⊆ ∩ n∈N * C k * + 1 n (v) = C k * (v).
Hence

∃ * ≥ 0 such that I k (v) = C k * (v) and ∀x ∈ I k (v), we have B(x) = * .
The next proposition gathers properties of elements of I k (v).

Proposition 3. Let 2 ≤ k ≤ n be fixed, • be a given norm which is strictly increasing for each coordinate7 , and v ∈ G(N ). For any element x ∈ I k (v), it holds:

(i) x S ≤ 0, ∀S ⊆ N such that |S| ≥ 2.
(ii) The corresponding k-additive game φ = m -1 (x) is concave.

(iii) The vector (x i ) i∈N belongs to C(v B(x) 1 ), where • 1 is the L 1 norm.

(iv)

S⊆N |S|≥2

x S ≥ t(v).

Proof. (i) Let x ∈ I k (v). Suppose that x S > 0 for some S with |S| ≥ 2, then the general payoff vector x eq defined by :

     x eq i = x i + 1 |S| x S , ∀i ∈ S x eq S = 0 x eq T = x T otherwise,
belongs to C k (v), satisfies x eq S = 0 and B(x eq ) < B(x) since • is strictly increasing, which contradicts x ∈ I k (v).

(ii) It is known [START_REF] Chateauneuf | Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion[END_REF] that concavity of φ is equivalent to S|A⊆S⊆B m φ (S) ≤ 0 for every A, B such that |A| = 2 and B ⊇ A, which holds by (i) since m φ (S) = x S .

(iii) Let x ∈ I k (v). By (i), x S ≤ 0 for all S such that |S| ≥ 2. Then, S⊆N

|S|≥2

x S = -B(x) 1 and the vector (x i ) i∈N satisfies

i∈S x i ≥ i∈S x i + T ⊆S |T |≥2 x T ≤0 ≥ v(S) = v B(x) 1 (S), S = N, |S| ≥ 2 i∈N x i =v(N ) - T ⊆N |T |≥2 x T =-B(x) 1 = v B(x) 1 (N ).
(iv) By (iii), C(v B(x) 1 ) is nonempty. Hence by definition of t(v), we have t(v) ≤

B(x) 1 . By (i), B(x) 1 = S⊆N |S|≥2
x S , hence the result.

We comment on these results. (i) shows that only negative or zero payments are given to coalitions. Hence, if the game is non-balanced, the debts implied by the satisfaction of coalitional rationality are to be paid by groups, not individuals, and (iii) shows the exact nature of the payment given to individuals. (iv) shows that the total amount of debts may exceed the minimal amount t(v), hence axiom (MGD) may be not satisfied. We will prove later that for the L 1 norm, (MGD) is always satisfied.

The minimum negotiation set as a general solution

I k (v) being a subset of the k-additive core C k (v), it is a general solution, preserving the desirable properties of the k-additive core without having its drawbacks. Indeed, it preserves coalitional rationality, it is never empty and coincides with the core when the latter is nonempty as shown in Theorem 3, and unlike the k-additive core, it is bounded, and can even be reduced to a singleton when the L p norm with p > 1 is used, as it will be shown in Section 3.5.

In this section and the subsequent ones, we will investigate the properties of this general solution. The next proposition shows that the minimum negotiation set satisfies all properties of general solutions satisfied by the k-additive core. (SYM): Similarly, since C k (v) satisfies (SYM), we have that for all permutations π such that v(π(S)) = v(S) for all S ⊆ N ,

I k (αv + β) = αI k (v) + β. Since C k (v) satisfies (COV), Minimize B(x) := (x S ) S⊆N |S|≥2 subject to x ∈ C k (αv + β)
Minimize B(x) := (x S ) S⊆N |S|≥2 subject to x ∈ C k (v) is equivalent to Minimize B(x) := (x S ) S⊆N |S|≥2 subject to x ∈ C k (v • π).
(IDEM): Here also we use the shorthand c = m -1 (C), and i = m -1 (i). We have to

show that i k • i k = i k . First, if i k (v) = c(v), then for each x ∈ c(v), considering x as an additive game, we have i k (x) = {x}. Therefore i k (c(v)) = c(v). Now, suppose c(v) = ∅ and take any φ ∈ i k (v). Since φ ∈ c k (v), it follows that c k (φ) ⊆ c k (v)
. Therefore, since φ minimizes the negotiation level over c k (v), φ minimizes the negotiation level over c k (φ), which implies φ ∈ i k (φ) and

i k (v) ⊆ i k (i k (v)).
Let φ ∈ i k (i k (v)). Then there exists ψ ∈ i k (v) such that φ ∈ i k (ψ). Observe that ψ ∈ c k (ψ) ⊆ c k (v). Since ψ minimizes the negotiation level over c k (v) and φ minimizes the negotiation level over c k (ψ), it follows that these negotiation levels are equal. Moreover, φ is a k-additive game which satisfies φ(S) ≥ v(S) for all S ⊂ N and φ(N

) = v(N ). Therefore φ ∈ i k (v) and i k (i k (v)) ⊆ i k (v).

Properties of I n 1

In this section, • is the L 1 norm. We denote by I k 1 the solution I k for the L 1 norm. By Proposition 4, we know that I k 1 satisfies (CR), (COV), (SYM) and (IDEM). In addition, we show that I n 1 satisfies (DPP) and (MGD). Proposition 5. I n 1 satisfies (DPP) and (MGD). Proof. (i) We prove (MGD) first. Let v be a game in G(N ), and x ∈ C(v t(v) ). It is clear that the general payoff vector y defined by to C n (v). We have

     y i = x i , ∀i ∈ N y S = 0, ∀S ⊆ N, S = N and |S| ≥ 2 y N = -t(v)
B(y) 1 = S⊆N |S|≥2 |y S | = t(v).
Therefore ∀z ∈ I n 1 (v), we have

S⊆N |S|≥2 |z S | = B(z) 1 ≤ B(y) 1 = t(v).
By Proposition 3 (i),

S⊆N |S|≥2 |z S | = | S⊆N |S|≥2 z S |
and by Proposition 3 (iv),

S⊆N |S|≥2 z S ≥ t(v).
Therefore, I n 1 satisfies (MGD). (ii) We prove now (DPP). By (MGD) and Proposition 3 (i) the negotiation level is equal to t(v), and by Proposition 3 (iii), we deduce that for all y ∈ I n 1 (v), we have

(y i ) i∈N ∈ C(v t(v) ).
Suppose i is dummy for v and take x ∈ C(v t(v) ). Then the vector x * ∈ R n defined by: x * i = v(i) and x * j = x j for j ∈ N \ i satisfies:

x * (N ) ≤ x(N ) x * (S) = x(S) ≥ v(S), if i / ∈ S x * (S) = x(S \ i) + v(i) ≥ v(S \ i) + v(i) = v(S), if i ∈ S. Therefore x * (N ) = x(N ), which entails x i = x * i = v(i), proving that I n 1 satisfies (DPP).
The next theorem is a characterization of I n 1 .

Theorem 4. Let M GDCR be the set of general solutions satisfying (MGD) and (CR).

Then

I n 1 = (M GDCR).
In words, I n 1 is the largest general solution satisfying coalitional rationality and minimizing the amount of the global debt.

Proof. We already know that I n 1 satisfies (CR), and by Proposition 5, we know that I n 1 satisfies (MGD).

Conversely, let σ be a general solution which satisfies (CR) and (MGD). Let x ∈ σ(v). Since σ satisfies (CR), x ∈ C n (v) because C n (v) = (CR). Moreover, since σ satisfies (MGD), the negotiation level of x is equal to the negotiation level of each element of I n 1 (v). Therefore x minimizes the negotiation level over C n (v).

We conclude that x ∈ I n 1 (v) and ∀v ∈ G(N ), σ(v) ⊆ I n 1 (v).

Remark 1. We have seen that satisfies (MGD) and (CR). Therefore we have the following inclusions:

C(v) ⊆ GEC(v) ⊆ I n 1 (v) ⊆ C n (v)
, for all games v. It follows that GEC inherits the properties of I n 1 .

From the above results, we notice the following important fact.

Corollary 1. Let v ∈ G(N ). We have:

{(x i ) i∈N | x ∈ I n 1 (v)} = C(v t(v) ) = {(x i ) i∈N | x ∈ GEC(v)}.
Proof. We prove the first equality. From the proof of Proposition 5, we see that for all

x ∈ I n 1 (v), it holds (x i ) i∈N ∈ C(v t(v)
). The reverse inclusion holds by Remark 1. Now, the second equality holds by definition of GEC.

These two last results elucidate the nature of I n 1 (v) and its relation to previously introduced notions like the extended core. In a sense, we may say that elements of GEC(v) and I n 1 (v) implement or realize elements of C(v t(v) ), which are not solutions of the game v. Indeed, by GEC(v), an element x of C(v t(v) ) becomes realizable by putting a debt t(v) on the grand coalition N . Now, I n 1 (v) gives more flexibility to realize x by allowing to putting debts on any coalition of size at least 2.

We finish this section by showing that I n 1 is continuous (see the Appendix for necessary definitions and results on continuity of correspondences).

Proposition 6. I n

1 is a continuous correspondence.

Proof. We set i n 1 = m -1 • I n 1 and prove continuity of i n 1 .

• i n 1 is uhc.

Gr(i n 1 ) = {(v, φ) ∈ G(N ) 2 , φ(N ) = v(N ), φ(S) ≥ v(S), ∀S ⊆ N, m φ S ≤ 0 ∀S ⊆ N, |S| ≥ 2,
and

S⊆N |S|≥2 m φ (S) = -t(v)} is a closed subset of G(N ) 2 .
Let B be a compact subset of G(N ). We define w 1 and w 2 two games such that w 1 (S) = inf v∈B v(S) and w 2 (S) = sup v∈B v(S). Let

φ ∈ i n 1 (B) = ∪ v∈B i n 1 (v).
Obviously, for all S ⊆ N , we have w 1 (S) ≤ φ(S). We prove now that for all S ⊆ N , φ(S) ≤ n max S⊆N w 2 (S). Let ψ be an additive game defined by

ψ(i) = max S⊆N w 2 (S) for every i ∈ N . Clearly, ∀v ∈ B, v(S) ≤ i∈S ψ(i), hence, ∀v ∈ B, v(N ) + t(v) ≤ i∈N ψ(i) ≤ n max S⊆N w 2 (S)
. Suppose there exists S ⊆ N such that n max S⊆N w 2 (S) < φ(S). Then there exists v ∈ B such that n max S⊆N w 2 (S) < φ(S) ≤ i∈N φ(i) = v(N ) + t(v) ≤ n max S⊆N w 2 (S), a contradiction. We conclude by Lemma 1 (in Appendix).

• I k is lhc.
G(N ) is a polyhedral convex set, for applying the Lemma 2 (in Appendix), it suffices to prove the convexity of the graph of i n 1 . Let (v 1 , φ 1 ) and (v 2 , φ 2 ) in Gr(i n 1 ). It is

that ∀δ ∈ [0, 1], δφ 1 (S) + (1 -δ)φ 2 (S) ≥ δv 1 (S) + (1 -δ)v 2 (S). Hence, t(δv 1 + (1 -δ)v 2 ) ≤ δ t(v 1 ) + (1 -δ) t(v 2 ). Furthermore, 0 ≥ S⊆N |S|≥2 m δφ 1 +(1-δ)φ 2 (S) = -(δ t(v 1 ) + (1 -δ) t(v 2 )) ≥ -t(δv 1 + (1 -δ)v 2 ), which implies S⊆N |S|≥2 |m δφ 1 +(1-δ)φ 2 (S)| ≤ t(δv 1 + (1 -δ)v 2 ).
The definition of t implies the equality, and we conclude that

δ(v 1 , φ 1 ) + (1 - δ)(v 2 , φ 2 ) ∈ Gr(i n 1 )

Autonomous coalitions

We introduce now the concept of autonomous coalition. Consider a game v and the core C(v t(v) ) of its t-expansion. Suppose it exists a coalition T = N such that, for any payoff

x ∈ C(v t(v) ), x(T ) = v(T ).
It means that the coalition can never receive more than its worth v(T ). Moreover, if the game is not balanced, the debt t(v) > 0 has to be paid anyway by the players, so that it is likely that the members of T will eventually receive in total strictly less than v(T ). Such a game is unstable, because S has nothing to gain from cooperation (and in case v is not balanced, S may even have incentive to leave), and therefore can remain independent of the grand coalition. In this sense, we say that S is autonomous.

In what follows, first we show that there always exists an autonomous coalition when the game is not balanced, and second, we show how to characterize them, and how it is possible to find them in a very easy way by I n 1 (v). We think that it is an important issue to detect autonomous coalitions since they are the source of instability of the game. Definition 3. We say that a coalition T ⊂ N is autonomous if for any payoff vector x of C(v t(v) ), it holds x(T ) = v(T ).

For any coalition T ⊂ N , we denote by v T the restriction of v to the coalition T . As an immediate consequence of the definition, we remark that C(v T ) is nonempty whenever T is autonomous. Indeed, any element

x of C(v t(v) ) is such that (x i ) i∈T ∈ C(v T ).
Proposition 7. Any non-balanced game v has an autonomous coalition T ⊂ N .

Proof. Suppose that v is not balanced, and that ∀T ⊂ N , ∃x ∈ C(v t(v) ) such that x(T ) > v(T ). Then, the convexity of C(v t(v) ) implies that there exists x such that ∀T ⊂ N , x (T ) > v(T ). Let T ∈ arg min T ⊆N (x (T ) -v(T )). The vector x defined by:

xi =    x i -x(T )-v(T ) |T | , if i ∈ T x i , otherwise satisfies x(S) ≥ v(S)

This is not possible by definition of t(v).

It is easy to find all autonomous coalitions. It suffices to compute the coordinates (x i ) i∈N of the barycenter of GEC(v) (or equivalently of the barycenter of I n 1 (v)). By Corollary 1, it is plain that T = N is autonomous if and only if i∈T xi = v(T ).

The next theorem gives another way to detect the greatest autonomous coalitions.

Theorem 5. Let v be a game on N , and S be a coalition of N such that |S| ≥ 2. The following properties are equivalent:

(i) There exists T ⊇ S such that T is autonomous.

(ii) ∀x ∈ I n 1 (v), x S = 0.
(see proof in the appendix)

The above result says that the greatest (in the sense of inclusion) autonomous coalitions correspond to coalitions T such that x T = 0 for all x ∈ I n 1 (v), and any superset of T does not satisfy this property. Combining this with the fact for all x ∈ I n 1 (v), x S ≤ 0 for all S, |S| ≥ 2 (Proposition 3 (i)), the barycenter x of I n 1 (v) permits to detect autonomous coalitions as follows: T is autonomous if xT = 0 and xT∪i < 0 for all i ∈ N \ T .

Properties of I k p with p > 1

In this section, • is the L p norm, with p > 1:

x p = n i=1 |x i | p 1 p .
We denote by I k p the general solution I k for the norm L p .

Theorem 6. Let p > 1. Then I k p (v) = C(v) if C(v) = ∅, otherwise I k p (v)
reduces to the unique element of C k (v) which minimizes the negotiation level.

Proof. Let a and b be two different payoff vectors in I k p (v). From Theorem 3 (ii), there exists * ≥ 0 such that

I k p (v) = C k * (v)
, and, for t ∈]0, 1[, the payoff vector ta + (1 -t)b belongs to I k p (v) by convexity, so in particular for t = 1 2 . If * = 0 then I k p = C(v), otherwise we have, by Proposition 3 (i) : if * = 0 and a = b, we have strict convexity of x → x p on [0, +∞):

(B(a)) p = S⊆N |S|≥2 (-a S ) p = ( * ) p (B(b)) p = S⊆N |S|≥2 (-b S ) p = ( * ) p B a + b 2 p = S⊆N |S|≥2 - 1 2 a S - 1
( * ) p = S⊆N |S|≥2 - 1 2 a S - 1 2 b S p < 1 2 S⊆N |S|≥2 (-a S ) p + 1 2 S⊆N |S|≥2 (-b S ) p = ( * )
(v)) i ) i∈N belongs to C(v ) = {(1, 0, 0)}. This implies in turn      1 + (I 2 2 (v)) 12 ≥ 1 ⇒ (I 2 2 (v)) 12 ≥ 0 1 + (I 2 2 (v)) 13 ≥ 1 ⇒ (I 2 2 (v)) 13 ≥ 0 (I 2 2 (v)) 23 ≥ 0.

Computation of the minimum negotiation set

In general the minimum negotiation set I k (v) is difficult to compute because it is a general nonlinear optimization problem. However, in the case of the L 1 norm, the optimization problem reduces to a linear program. We have also some partial result for the case of the L 2 norm. x S under the conditions : Proof. Each x ∈ I k (v) ⊆ C k (v) is such that x satisfies i), ii), and iv). Furthermore Proposition 3 (i) implies that x S ≤ ∀S ⊆ N , |S| ≥ 2, which is iii). Lastly, minimizing belongs to I n 1 (v), and we observe that the optimal value of the objective function is t(v) = -40.

i) S⊆N x S = v(N ) ii) S⊆T x S ≥ v(T ) ∀T ⊆ N iii) x S ≤ 0 ∀S ⊆ N such that |S| ≥ 2. iv) x S = 0 ∀S ⊆ N such that |S| > k.

Computation of I k 2

Giving an exact expression of I k 2 in the general case is quite difficult because it is necessary to solve a quadratic program. However, we can obtain an exact expression of I 2 2 in the case of a symmetric8 and subadditive game. Proposition 9. If v is a nonnegative, symmetric and subadditive game on N then :

(I 2 2 (v)) i = max k∈{1,...,n-1} n -1 k(n -k) v k - k -1 n(n -k) v n , ∀i ∈ N (I 2 2 (v)) ij = 2 n(n -1) (v(N ) -n(I 2 (v)) 1 ) , ∀i, j ∈ N where v k := v(K), |K| = k.
Proof. Subadditivity of v implies C(v) = ∅ or reduced to a singleton. Therefore I 2 2 (v) is a singleton. Now, symmetry of v implies by definition of

I 2 2 that ∀i ∈ N , (I 2 2 (v)) i = (I 2 2 (v)) 1 and ∀i, j ∈ N , (I 2 2 (v)) ij = (I 2 2 (v)) 12 .
Denote by α the value of (I 2 2 (v)) 1 and by β the value of (

I 2 2 (v)) 12 . I 2 2 (v) ⊆ C 2 (v) therefore, ∀k ∈ {1, . . . , n -1}, k(k -1) 2 β + kα ≥ v k and n(n -1) 2 β + nα = v n .
two expressions are equivalent to:

β = 2 v n -nα n(n -1) ( * )
and ∀k = n and n = 1

α ≥ n -1 k(n -k) v k - k -1 n(n -k) v n ( * * )
Furthermore, v is symmetric, therefore, the minimization of B(x) under the condition ∈ C 2 (v) is equivalent to the minimization of (β) 2 under the conditions ( * ) and ( * * ), i.e., the maximization of β since β ≤ 0 by Proposition 3 (i). This amounts to the maximization of v n -nα, hence the of α. The minimum is reached for α

= max k∈{1,...,n-1} n-1 k(n-k) v k -k-1 n(n-k) v n and β = 2 vn+n(n-2)α n(n-1)
. Example 6. We consider a symmetric subadditive game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1. [START_REF] Fujishige | Submodular functions and optimization[END_REF] 2

We have C(v) = ∅. Using the above result, we get:

S 1 2 3 12 13 23 123 (I 2 2 (v)) S 1 1 1 -1 3 -1 3 -1 3 0

From general payoff vectors to classical payoff vectors

As explained in the introduction, the ultimate step when using a general solution is to transform selected general payoff vectors into classical payoff vectors, i.e., sharing among individual players.

A first way to deal with this problem is to consider the selectope [START_REF] Derks | The selectope for cooperative games[END_REF] of the game corresponding to a general payoff vector. Let us first recall some material on the selectope. We consider the set of sharing functions given by:

Q(N ) := {q : 2 N \ {∅} × N → [0, 1] | q(S, i) = 0 if i / ∈ S, i∈S q(S, i) = 1, ∀S ⊆ N, S = ∅}.
For each subset S and each player i in S, the sharing function specifies the proportion of a good (assigned to S) which should be given to i, noting that nothing is given to players outside S. Each sharing function q induces a sharing value x q defined by:

x q : G(N ) → R n such that x q i (v) := S i q(S, i)m v (S), i ∈ N.
Thus, the sharing value shares among players the dividends (Möbius transform) m v (S), S ⊆ N , according to the sharing rule q. The selectope [START_REF] Derks | The selectope for cooperative games[END_REF] of a game v ∈ G(N ), denoted by S(v), is defined as the set of all preimputations which can be obtained by sharing functions applied on m v : S(v) := {x q (v), q ∈ Q(N )}.

Let us come back to our and consider a game v and a general payoff vector x belonging for example to the k-additive core C k (v) or any other general solution of v. The simplest way to transform x into a classical payoff vector is to share each component x S for |S| > 1 the members of S using some sharing rule q ∈ Q(N ). In other words, recalling that general payoff vectors can be seen as games through the Möbius transform, the set of all possible (classical) payoff vectors which can be derived from x is the selectope of the game m -1 (x).

As before, we use as a shorthand

c k = m -1 (C k ), i n 1 = m -1 (I n 1 )
, gec = m -1 (GEC), etc. The following result shows that any preimputation can be attained from the k-additive core by a sharing function.

Theorem 7 (Grabisch and Li, 2010 [7]). For any x q , q ∈ Q(N ) such that q(K, i) > 0 for all K ⊆ N and i ∈ K, for any v ∈ G(N ), for any k ∈ {2, . . . , n}, we have

x q (c k )(v) = P I(v).
where P I(v) is the set of preimputations.

More interestingly, restricting to the minimum negotiation set, we obtain by using all sharing functions the extended core of Bejan and Gomez. Theorem 8. The selectope of i n 1 (v) is given by:

S(i n 1 (v)) = S(gec(v)) = C(v t(v) ) -t(v)∆ n = EC(v) where ∆ n := {η ∈ [0, 1] n | i∈{1,...,n} η i = 1}.
Proof. Take φ ∈ i n 1 (v). Since I n 1 satisfies (MGD), and by Proposition 3 (i), we have (φ(1), . . . , φ(n

)) ∈ C(v t(v) ). Therefore S(φ) ⊆ C(v t(v) ) -t(v)∆ n .
Conversely, since GEC(v) ⊆ I n 1 (v) by Remark 1, we have S(gec(v)) ⊆ S(i n 1 (v)). Let φ ∈ gec(v). We denote by φ + and φ -the games defined by

φ + (T ) := S⊆T m φ (S)>0 m φ (S) φ -(T ) := - S⊆T m φ (S)<0 m φ (S).
By definition of gec(v), we have φ + (T ) = i∈T m φ ({i}) and φ -(T ) = 0 for T = N , and φ

-(N ) = t(v). Therefore C(φ + ) = {φ + } and C(φ -) = t(v)∆ n . Hence S(gec(v)) = C(v t(v) ) -t(v)∆ n .
In summary, we have shown

C(v t(v) ) -t(v)∆ n = S(gec(v)) ⊆ S(i n 1 (v)) ⊆ C(v t(v) ) -t(v)∆ n so that equality holds throughout.
It is well known that the Shapley value can be obtained as a sharing and therefore always belongs to the selectope of the game (it corresponds to the equal sharing). However, observe that, in general, the Shapley value does not belong to the selectope of I n 1 (v), as shown by the following example. A more general way to make a sharing of the amounts given to coalitions is to put the problem into a bankruptcy or tax problem. This was also considered by Bejan and Gomez [START_REF] Bejan | Core extensions for non-balanced TU-games[END_REF], however in their case the problem was simpler since only the grand coalition has debt. We limit ourselves to giving the main ideas, a complete treatment of this topic would deserve a whole study.

Consider some payoff x ∈ I k (v), and the list of coalitions S 1 , . . . , S p with |S| > 1 such that x S < 0, ordered in a way which is compatible with inclusion (i.e., S i ⊂ S j cannot occur for i > j). Take S 1 and define the following bankruptcy problem: the set of agents is S 1 , the estate is φ(S 1 ), where φ = m -1 (x) is the game version of x, and the claims of the agents are simply their individual payoff x i , i ∈ S 1 . This is a bankruptcy problem because the estate is strictly less than the sum of demands:

φ(S 1 ) = T ⊆S 1 x T = i∈S 1 x i + x S < i∈S 1
x i by the ordering on the S i 's and since x S < 0. Taking some division rule (equal award rule [START_REF] Thomson | Axiomatic and game-theoretic analyss of bankruptcy and taxation problems: A survey[END_REF], Talmud rule [START_REF] Aumann | Game theoretic analysis of a bankruptcy problem from the talmud[END_REF], etc.), one ends up with a payment vector (x 1 , . . . , x n ). Also, we modify the game φ since the debt x S 1 has been paid, as follows: φ (S) = T ⊆S,T =S 1 x S , and we set x j = x i for all i ∈ S 1 . Note that x ≤ x and φ ≥ φ.

We take now the next set S 2 in the list and define a new bankruptcy problem with estate φ (S 2 ) and claims x i , i ∈ S 2 . Since x ≤ x and φ ≥ φ, it may happen that the estate is sufficient to pay the claims. If not, we have again a bankruptcy problem, and proceed as above. This leads to vector x and we update the game φ as follows: φ (S) = T ⊆S,T =S 1 ,S 2 x S . We continue the procedure till S p .

Clearly, this way of sharing is more general since the underlying sharing function depends in particular on x. We think for this reason that it is better suited to the problem, although more complicated. Its full study is left for further research.

(i) We have proposed the idea general payoff vectors and general solutions, based on previous works like the k-additive core of Grabisch and Miranda, and the extended core of Bejan and Gomez, which can be seen as a general solution (called here the G-extended core (GEC)).

(ii) We have proposed the minimum negotiation set, as the subset of the k-additive core minimizing the distance to classical preimputations. There many advantages considering this solution: it is compact and convex, never empty, it coincides with the core if the latter is nonempty, guarantees coalitional rationality while minimizing the amounts given to coalitions of size larger than 2. Properties of the minimum negotiation set have been thoroughly investigated.

(iii) If the L 1 norm is chosen for the minimum negotiation set, then one obtains a set of general payoffs which includes GEC, and whose payments to individuals coincide with the core elements of the t-expansion of the game (i.e., elements of C(v t(v) )). Moreover, it is guaranteed that the total debt to be paid by coalitions is minimal. If the L 2 norm is chosen or any L p norm with p > 1, the advantage is to get a single-valued solution, like the nucleolus. Further study is needed to investigate the properties of this solution.

(iv) Ultimately, one should derive classical payoff vector(s) from the general payoff vector chosen by the solution. A simple way to do this is to consider fixed sharing functions, in order to share the debts on coalitions among its members. Taking all possible sharing functions (the selectope) for I n 1 or GEC shows that we recover exactly the extended core of Bejan and Gomez. A more clever way, although more complicated, is to consider the payment of a debt by a coalition as a bankruptcy problem, and use appropriate division rules. The latter proposition seems to be promising and deserves further research.

Proposition 4 .

 4 I k satisfies (CR), (COV), (SYM) and (IDEM) for any norm. Proof. (CR): Clear. (COV): I k satisfies (COV) if and only if ∀v ∈ G(N ), ∀α > 0, ∀β ∈ R 2 n -1 we have:

12

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.22 (Version révisée) is equivalent to Minimize B(x) := S ) S⊆N |S|≥2 subject to x ∈ αC k (v) + β.

13

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.22 (Version révisée)

  , ∀S ⊂ N 16 Documents de Travail du Centre d'Economie de la Sorbonne -2012.22 (Version révisée) and x(N ) < x(N ) v(N ) + t(v).

  * ) p . 17 Documents de Travail du Centre d'Economie de la Sorbonne -2012.22 (Version révisée)

Example 4 . 1 -

 41 p , a contradiction. Therefore * = 0 or a = b.To conclude, if * = 0 thenI k p (v) = C(v), otherwise a = b and I k p (v) = {a}.Unlike I n 1 , the general solution I k p does not satisfy (MGD), as shown by the following example with p = 2. We suppose that k = 2, and consider a game v on N = {1, 2, 3} defined by: Clearly, t(v) = and I 2 2 (v) reduces to a singleton. If (MGD) holds, the minimum negotiation level is , hence by Proposition 3 (iii), ((I 2 2

  This is impossible because by Proposition 3 (i), (I 2 2 (v)) ij ≤ 0 for all {i, j} ⊆ N , and by MGD, (I 2 2 (v)) 12 + (I 2 2 (v)) 13 + (I 2 2 (v)) 23 = -.

4. 1 1 Proposition 8 .

 118 Computation of I k The set I k 1 (v) is the solution of the linear program: maximize S⊆N 2≤|S|≤k

18

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.22 (Version révisée)

Example 5 .

 5 S⊆N 2≤|S|≤k |y S | for y ∈ C k (v) is equivalent to maximize S⊆N 2≤|S|≤k y S under the conditions y ∈ C k (v) and y S ≤ 0, ∀|S| ≥ 2. We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 20 40 30 40 40 40 50We have C(v) = ∅, and ∀t ∈ [0, 1], the payoff vector x defined by: 30 -20t -10t -10t -40(1 -t)

7.

  We consider a game v on N = {1, 2, 3} defined by: (v) = 1 10 and C(v t(v) ) = {(1, 0, 0)}. Therefore the selectope of I n 1 (v) is:S(I n 1 (v)) = {(1 -1 , -η 2 , -η 3 ), η i ≥ 0 ∀i ∈ N, and η 1 + η 2 + η 3

  which implies that i∈N a i is bounded. Since a ∈ C k (v), we have a i ≥ v(i) for all i ∈ N . Hence, each a i is bounded. It follows that each coordinate a S is bounded, therefore C k (v) is compact.(ii) I k (v) is an intersection of compact, convex and nonempty nested sets, therefore I k (v) is a compact, convex and nonempty set. It is easy to see that if C(v) is nonempty then

				By definition,
	(a S ) S⊆N	≤ ,
	|S|≥2
	which implies that each component a S , |S| ≥ 2, is bounded, and so is S⊆N	a S .
				|S|≥2
	Now we have:
	v(N ) =	a i +	a S ,
	i∈N		S⊆N
			|S|≥2

Although in its formulation it is not. We call it G-extended core when rewritten as a general solution.

Documents de Travail du Centre d'Economie de la Sorbonne -2012.22 (Version révisée)

Documents de Travail du Centre d'Economie de la Sorbonne -2012.22 (Version révisée)

The original definition is expressed through k-additive games, and is equivalent to this one. Specifically, viewing classical payoff vectors as additive games, the core can be defined as {φ ∈ G 1 (N ) | φ(S) ≥ v(S), ∀S ⊆ N, φ(N ) = v(N )}. This viewpoint is the current one in decision theory, where coalitional rationality is rather called dominance. Now, the generalization of this definition to k-additive games is straightforward, and leads to the original definition of the k-additive core in[START_REF] Miranda | k-balanced games and capacities[END_REF] (and moreover explains its name):{φ ∈ G k (N ) | φ(S) ≥ v(S), ∀S ⊆ N, φ(N ) = v(N )}.The equivalence is clearly seen by the definition of k-additive games and Equation (2), and we see that the Möbius transform plays the rôle of the generalized payoff vector. Although this view is mathematically more natural and elegant, we have opted for the definition based on payoff vectors, since closer to game theory. 6 Documents de Travail du Centre d'Economie de la Sorbonne -2012.22 (Version révisée)

A polyhedron is pointed if it has vertices (equivalently, if it contains no line).

Up to the natural identification between payoff vectors and general payoff vectors. 8 Documents de Travail du Centre d'Economie de la Sorbonne -2012.22 (Version révisée)

A more general solution would be to allow real-valued amounts, that is, even in the case of benefit, some players would have to pay something. But we do not see any convincing example where this method would be relevant. In game theory, for example when defining the selectope[START_REF] Derks | The selectope for cooperative games[END_REF], sharing functions are assumed to take values in [0, 1]. It is the same for division rules in bankruptcy or tax problems[START_REF] Thomson | Axiomatic and game-theoretic analyss of bankruptcy and taxation problems: A survey[END_REF] (another way to consider the above sharing problem; see Section 5).

Up to the natural identification between payoff vectors and general payoff vectors. The same remark applies to the next item. 10 Documents de Travail du Centre d'Economie de la Sorbonne -2012.22 (Version révisée)

This is the case of all L p norms, excpet L ∞ . However, for the latter a simple modification of the proof makes the statement valid for L ∞ too: it suffices to take for S a maximizer of x S and assume that x S is positive.

A game v is symmetric if v(S) depends only on the cardinality of S for each S.

Concluding remarksWe summarize the main findings of the paper and indicate research perspectives.

Appendix

Continuity of correspondences

We recall the basic definitions and results on the continuity of correspondences (see, e.g., [START_REF] Peleg | Introduction to the theory of cooperative games[END_REF]Ch. 9]).

Let X and Y be two metric spaces, and consider a correspondence ϕ from X to Y , i.e., ∅ = ϕ(x) ⊆ Y for all x ∈ X. The graph of ϕ is defined by Gr(ϕ) := {(x, y) | x ∈ X, and y ∈ ϕ(x)}.

We say that ϕ is closed if Gr(ϕ) is a closed subset of X × Y , and ϕ is bounded if for each compact subset B of X, the image of

Similarly, ϕ is lower hemicontinuous at x ∈ X if for every open subset U of Y such that ϕ(x) ∩ U = ∅, there exists an open subset V of X such that x ∈ V and ϕ(z) ∩ U = ∅ for every z ∈ V . Moreover, ϕ is lower hemicontinuous (lhc) if it is lower hemicontinuous at each x ∈ X. If both upper and lower hemicontinuity hold, then ϕ is continuous. Lemma 1. Let X be a metric space and let Y = R M , with M a finite and nonempty set. If ϕ is a closed and bounded correspondence, then ϕ is uhc.

If ϕ is a bounded set-valued function with a convex graph, then ϕ is lhc.

Proof of Theorem 5

First we need the following result.

Proposition 10. Let v be a game on N . Suppose it exists S ⊂ N , |S| ≥ 2, such that for all x ∈ I n 1 (v), x S = 0. Then the following holds.

(i) For all x ∈ I n 1 (v), for all T ⊆ S, |T | ≥ 2, we have x T = 0.

(ii) There exists

Proof. (i) Suppose there exists x ∈ I n 1 (v) and T ⊆ S such that x S = 0 and x T < 0. Then the payoff x defined by

Observe that t > 0. Indeed, t = 0 would imply that for some T ⊇ S, T = N , K⊆T xK = v(T ). Since x ∈ GEC, its coordinates are 0 for all nonsingleton subsets of T . Therefore, i∈T xi ≥ v(T ) for all T ⊂ T , and i∈T xi = v(T ). It follows that the vector (x i ) i∈N is an element of the core C(v T ), a contradiction.

Then the general payoff vector x defined by

), which yields i∈T x i = v(T ). Therefore if x S < 0, we would have S⊆T x T < v(T ), which contradicts x ∈ I n 1 (v). (ii) ⇒ (i). Define T as a maximal element (in the sense of inclusion) of {L ⊇ S, x L = 0, ∀x ∈ I n 1 (v)}. Suppose T = N . Then for all x ∈ I n 1 (v), x N = 0, which implies that t(v) = 0 since GEC(v) ⊆ I n 1 (v). Therefore C(v) = ∅, and (i) holds for T = N . assume now that T = N . Due to Proposition 10 (i) and the definition of T , for any set L ⊆ T , |L| ≥ 2, we have x L = 0 for all x ∈ I n 1 (v), and for all L ⊃ T , x L = 0 for at least one x ∈ I n 1 (v). It follows that

For the rest of the proof, we introduce the shorthand φ x for m -1 (x), where x is a general payoff vector. From the above equation and Corollary 1, it suffices to show that φ x (T ) = v(T ) for all x ∈ I n 1 (v) in order to prove that y(T ) = v(T ) for all y ∈ C(v t(v) ). Suppose then that there exists x 0 ∈ I n 1 (v) such that φ x 0 (T ) > v(T ). We distinguish two cases. Suppose first that T = N \ i. Suppose there exists

such that x N < 0 (necessarily exists otherwise T = N ), and build the payoff vector x = 1 2 (x + x ). Define x by

Then x ∈ I n 1 (v), although xN\i < 0, a contradiction. Suppose next that T is not of the form N \ i. We know that for all T ⊃ T , T = N , there exists

Then by convexity of I n 1 (v), it follows that x ∈ I n 1 (v), and we have xT < 0, for all T ⊃ T , T = N . Consider the payoff vector x = 1 2 (x 0 + x), which belongs to I n 1 (v) by convexity, and put = min(φ x (T ) -v(T ), min

and choose K which realizes the minimum of min T ⊂K⊂N (-x K ). Build x * as follows:

x

We claim that x * ∈ I n 1 (v), which contradicts the hypothesis x T = 0 for all x ∈ I n 1 (v). Indeed, it suffices to verify coalitional rationality:

(ii) If K ⊇ T and K ⊇ K, then

(iii) If otherwise K ⊆ T , then φ x * (K) = φ x (K) ≥ v(K).


Barış Aykent

Damien Paillot

Frédéric Mérienne

Zhou Fang

Andras Kemeny

STUDY OF THE INFLUENCE OF DIFFERENT WASHOUT ALGORITHMS ON SIMULATOR SICKNESS FOR A DRIVING SIMULATION TASK

Keywords: Driving simulator, virtual immersion, motion sickness

This paper deals with the effects of different washout algorithms used for Stewart platforms on subjective and objective ratings. Washout algorithms are used to represent vehicle dynamics in a restricted spatial place. An adaptive washout algorithm was realized to control the hexapod platform, depending on the specific force error in longitudinal, lateral and vertical directions, in order to compare user"s experience with those in the case of classical algorithm. In this study, the simulator sickness has been evaluated for three algorithms in dynamic driving simulator situation in objective and subjective way.

INTRODUCTION

Driving simulation is a high demanded domain in terms of virtual immersion. Compared to virtual mock-up visualization in a CAVE system, driving simulation induces the motion of the user"s body through the car driving task. In driving simulation, the motion is mainly induced by acting on visual cue (vection process) and vestibular cue (with a motion platform). To improve the virtual immersion in driving simulation task, both cues have to be well correlated. The motion platform is controlled with an algorithm which translates the car simulated motion in a possible platform motion.

The washout algorithms are used to depict the driving dynamics of a real vehicle in a constraint space for a driving experience as realistic as possible (see Figure 2). They were firstly introduced in the mid 60s in the aerospace industry (Stewart, 1965). Afterwards, these techniques have been spread into automotive industry for the development process of the vehicles.

Different algorithms have been developed to control a 6 DOF (Degrees Of Freedom) platform (as a hexapod platform). The following table gives the list of the main common algorithms used in the literature.

Motion control algorithm References

Classical (Stewart, 1965) Adaptive (Parrish et Objectives of the washout algorithms are given below [START_REF]FCS Motion Cueing Model Presentation[END_REF]:

-Main objective for implementation of the washout algorithm is to maximize the contribution of the motion system to the capabilities of the simulator.

-Main limiting factors are the maximum speed and stroke of the actuators.

-Reproduction of motion cues one to one is not possible. Filtering and reducing cues are needed.

Cues at dynamic driving simulators are given in Figure 1. If there is a conflict between the cues, it can cause a phenomenon called "motion/simulator sickness".

The "kinetosis" or "motion sickness", also known as travel sickness, is a situation in which a disagreement exists between visually perceived movement and the perception by the vestibular system.

For the objective evaluation of different washout algorithms on simulator sickness, the tests were realized by using SCANeRstudio ® 1.0 software of OKTAL Company with the "Country Road" scenario.

Results show that an adaptive algorithm provides a less possible opportunity to have simulator sickness, objectively as well as subjectively by using simulator sickness questionnaire.

DRIVING SIMULATORS

Virtual reality environments are used more and more in engineering systems to develop and enhance product design [START_REF] Kolasinski | Simulator Sickness in Virtual Environments[END_REF].

Driving simulators are one of the widespread used types of virtual reality systems. We can distinguish dynamic simulator (with motion platform technology) and static simulator.

As stated in the Figure 1, a dynamic driving simulator is made up of the elements listed below:

-Motion system (6 DOF (degrees of freedom) hexapod platform) -Visual system ( through video projectors) -Vehicle dynamics: vehicle model used in the driving simulation -Motion driving algorithms (motion cueing and washout algorithms) -Control: according to HMI (human machine interface) theory, not only the washout algorithm itself but also the driver has an important role for the control of the driving simulation in the scenario [START_REF] Parrish | Coordinated Adaptive Washout for Motion Simulators[END_REF]. Driver is involved in the simulation process due to his/her reactions which are sourced from his/her "proprioperceptive" and "vestibular system" that also enables him/her as "a second closed-loop controller", apart from the existing adaptive washout algorithm or vehicle dynamics controller (primary control). The correction inputs of the conductor are generally affected of those listed below: The most crucial difficulties that can be encountered in the driving simulator applications, to have a more realistic driving simulation, are:  To embody the dynamics reaction of the driver"s (as it is associated with the secondary closed-loop control)

 To represent the driving dynamics as in real world in the dynamic driving simulator (it is called washout algorithm) in terms of user"s perception.

SIMULATOR SICKNESS

"Simulator sickness" is a phenomenon to define a situation in which the visually perceived movement and the perception of the movement by the vestibular system does not correspond to each other, in other words they disagree.

There are four broadly accepted theories to describe the "simulator sickness" phenomenon [START_REF] Kolasinski | Simulator Sickness in Virtual Environments[END_REF]:

The main objective for implementing the washout algorithm is to use the motion system as best as possible to restitute a good perception of driving situation from user"s point of view. Figure 3 shows the working scheme of the classical algorithm which was developed for our work (for 6 DOF, it consists of the lateral and vertical dynamics and the corresponding inclinations, tilts, as well as the longitudinal shown in Figure 3).

Figure 3. Schematic of the classical washout algorithm

MODELLING OF ADAPTIVE ALGORITHM

Parrish and al [START_REF] Cardullo | Motion cueing algorithms: A human centered approach[END_REF] provided motion cues which can be seen as a classical one where parameters are variable and calculated at each step of simulation time.

Various schemes were proposed to improve the stability of algorithm [START_REF] Telban | A nonlinear human centred approach to motion cueing with neurocomputing solver[END_REF]. Ariel and Sivan [START_REF] Ariel | False Cue Reduction in Moving Flight Simulators[END_REF] include the vestibular system for the lateral false cues reduction. It is based on the minimization of a cost function containing the acceleration error and constraints on the platform displacement. The adaptation is carried out using the "Steepest Descent Method" to resolve the sensitivity equations [START_REF] Ioannou | Robust adaptive control[END_REF][START_REF] Nehaoua | Motion Cueing Algorithms for Small Driving Simulator[END_REF]. The resulting filter is then nonlinear. The operating scheme of that adaptive motion cueing algorithm has been given below in Figure 4.

Figure 4. Schematic of the adaptive washout algorithm

The filter equation is given by: To control the platform motion based on the adaptive washout algorithm, the cost function to be minimized is given in equation ( 2)

pf pf veh pf x b x a x K x           
      0 2 2 2 2 1 i i pi s p s v s veh a P P x x x x J                 ( 2
)
where i  are weighting coefficients, Optimization is realized by the steepest descent method as followings:

i i i P J P        (3)
Copyright © 2011 by ASME Once the weighting of the function cost i  and initial conditions 0 i P are determined, the resolution of sensitivity equation enables the determination of acceleration and position signals to drive the platform.

One problem of this algorithm is the stability of the gradient descent method. This is related strongly with the parameters of adaptation which define the convergence speed of algorithm.

Figure 5 points out the simulation results of adaptive and classical washout algorithms responses together to a step input for longitudinal acceleration of 1 m/s 2 with duration of 4 seconds for a 1 DOF system.

VEHICLE SYSTEM DYNAMICS EMBEDDING IN DRIVING SIMULATOR

Driving simulators nowadays find application areas intensively at both research and production, because of their capability to offer a realistic environment for the driver. In Figure 6, a Man In the Loop (MIL) system has been indicated that had been utilized for these attempts.

Figure 6. Networking of dynamic simulator

The operating structure of the dynamic simulator, given in Figure 6, is made up of five main components: As seen in Figure 7, Host (master) computer contains vehicle types, driving maneuvers, driving environment, animator and vehicle dynamics which are included in SCANeRstudio. In addition to this, the host has access to work PC shown as CS-SIMU-4 in Figure 7 for communication interfaces, for control displays and to specify the vehicle coordinates.

Additionally it includes interfaces for the driving simulator (for Gas/Brake pedals Analog I/O and for steering wheel CAN Bus Interface and also Sound interface for the sound system).

Copyright © 2011 by ASME Motion control computer (PC MOOG in Figure 6) is the element to control the position and orientation of the Stewart platform. Kinematics and dynamics of the platform are manipulated by PC MOOG. This component accomplishes motion algorithms and washout as well as position velocity acceleration (PVA) transformations.

Driving simulator is constituted of four elements and the driver in the loop: 1-Control system: Steering wheel, gas pedal, brake pedal (haptic cues) 2-Sound system (acoustic cues) 3-Motion system: Hexapod platform (inertial cues) 4-View system: 3 channel animator view in the driver"s cabin.(visual cues) 5-Driver (proprio-perceptive,vestibular system)

TESTS ON DRIVING SIMULATOR

In this study, the simulator sickness has been evaluated objectively and subjectively on driver in the dynamic driving simulator SAAM in function of three different motion platform algorithms. The SAAM simulator has been designed and developed by Arts et Métiers ParisTech and RENAULT. For the objective evaluation, we used the data acquisition from the SCANeRstudio ® to collect the data related to commands (steering wheel angle, accelerator, brake pedal force, etc.), dynamics (vehicle dynamics data), engine, motion platform (position in X, Y and Z axis and the angles around X, Y and Z axis). The Table 1 gives the capabilities of the driving simulator SAAM we used. The subjects were asked to drive the three different washout algorithms on the dynamic driving simulator SAAM (Figure 1) at the same scenario of Country Road of SCANeRstudio ® OKTAL (Figure 8 and9) with a constant driving velocity of v= 70 km/h. 7 subjects participated to the experiment. In addition, they had a familiarization drive before each session to avoid misevaluation and to help them assess as objective as possible.

During the testing phase, they drove three times for each motion algorithm. After each attempt they were asked to fill in the regarding questionnaire for the subjective rating of the "simulator sickness" and to evaluate the "simulator fidelity" respectively. Besides, at each essay, the data recorded with the default value of SCANeRstudio ® software for sampling period of 0.1 s. [OKTAL SCANeRstudio User Manual] To rate the simulator sickness, the "simulator sickness questionnaire (SSQ)" (Kennedy 1993), "motion sickness questionnaire (MSQ)" (Kennedy 1992), "biofeedback methods (BFM)" are commonly applied with some other approaches like "motion sickness dose value (MSDV)" (ISO 2631-1, 1997) [START_REF] Griffin | Handbook of Human Vibration[END_REF]).

OBJECTIVE EVALUATION

Objective evaluation refers to an assessment method for the driving simulator applications of which the measured data such as the effective roll, pitch, yaw angles of the hexapod motion platform; positions of the motion platform ; the dynamics data of the vehicle model, etc. are considered. The results from the motion platform (Figure 10, 11 and 12) point out that by increasing the travel scale for longitudinal, lateral and vertical axis of the hexapod; the oscillations of the associated rotation are being more smoothed. It means that the greater the travel scale factor on the adaptive washout algorithms commanded platform dynamics is, the less the undulations will occur.

EFFECT OF THE DRIVER AS A CLOSED LOOP CONTROL

As we discussed above, the subject participating in the experimentation is also a secondary closed loop controller in addition to washout algorithm. In fact, the subject can also be considered as a primary control (we have investigated the primary control as open loop "classical washout" and as closed loop "adaptive washout" in this paper).

The lateral position difference of those which have been obtained from the driving simulator ("driver + washout algorithm" as controller, see blue curve below) and the simulation ("washout algorithm" as controller, see red curve below) results figure out that, even the same scenario with the same washout has been realized, the results have differed to each other which emphasizes the factor of the "driver behavior" that is probably the most difficult part to model, estimate and/or simulate to capture a better fidelity for driving simulator research and development phases.

OBJECTIVATION OF SUBJECTIVE EVALUATION: 'MOTION SICKNESS DOSE VALUE'

"Motion Sickness Dose Value" is a method for the objectification of subjective evaluation which has been defined in accordance of ISO 2631-1 1997 [START_REF] Griffin | Handbook of Human Vibration[END_REF]) which we proposed also in this study for the same purpose. In this work, we propose to use this evaluation method and use the following function to evaluate the motion sickness dose value. The equation above points out the vertical dynamics related to "motion sickness dose value".

              
With this analogy, the MSDV can be expanded for roll and lateral acceleration to assess the motion sickness dose value sourcing from the roll and lateral dynamics within the dynamic driving simulators.

Illness Rating (IR) deduced from MSDV, is the following function:

z MSDV IR   50 1 (5)
According to the yielded results, the subject is prone to be defined in terms of his/her illness scores as "0,1,2,3" as follows: 0 = I felt good, 1 = I felt a mild illness 2 = I felt very bad. 3 = I felt absolutely terrible.

OPTIMIZATION OF SIMULATOR SICKNESS

To prevent and/or avoid the simulator sickness, a method is proposed illustrated in Figure 14.

According to the sickness reduction logic (SRL), three different washout algorithms (classical, adaptive 1, adaptive 2) were embedded in the driving simulation software SCANeRstudio of OKTAL as "dll plugin" which is also capable of configuring as well as changing the motion cueing strategy externally. Copyright © 2011 by ASME Table 3 shows results of evaluation of test data that have been conducted on Country road scenario with classical and adaptive washout algorithms. The tests were done at 70 km / h. Five points were taken into account to compare the behavior of the lateral dynamics of the Stewart platform, and discuss the effect of each. The values of disease ratings for each were selected in Figure 15.

The optimization of simulator sickness was provided at these five points by converting the control algorithm of the platform: instead of conventional algorithm using adaptive algorithm has an improvement of about 21.8%, 23.7%, 29.4%, 15.2% and 17.2% respectively.

SUBJECTIVE EVALUATION TESTS

The driving attempts were accomplished on SAAM driving simulator located at Institute Image, Arts et Métiers ParisTech in Chalon-sur-Saône. The Figure 16 shows the subjects age and driving experience descriptive statistics.

Relating to this subjective assessment of the simulator sickness, the driving simulation experiment used a 6 DOF (degree of freedom) motion system made up of the hexapod system (Stewart platform). The subjective evaluation tests, given in Table 2 In order to assess the driving simulator tests subjectively on behalf of simulator sickness, a simulator sickness questionnaire which consists of twelve questions (notes 0: too little  10: too strong) was proposed to each participant to have an opinion about the influence of different motion washout algorithms on simulator sickness. Figure 18 shows the mean value for the subjective rating of the sickness, whereas the Figure 19, 20 and 21 imply the statistically post-processed sickness values with confidence interval of 95%. According to Figure 18, the means of the rating for adaptive2 and adaptive1 algorithms were acquainted less than those which were obtained for the classical algorithm, as expected. However, for some questions the difference is very weak. In order to come up to a conclusion and to make a stronger statement at those points, it would be advantageous to evaluate also the subjective rating of the dynamics (simulator fidelity questionnaire, Table 6) of the dynamic driving simulator simultaneously.

Figure 1 .Figure 2 .

 12 Figure 1. Structure of the closed loop control of dynamic driving simulator

  pf x : Platform position K , a and b are the parameters of the washout filter.

iP

  with i = 1,2,3 are adaptive parameters and 0 i P , i =1,2,3 are its initial values.

Figure 5 .

 5 Figure 5. Classical and adaptive washout algorithm comparison of a step input for 1 DOF

1 -

 1 Master Computer (Supervision) (CS-SIMU-5) 2-Motion Control Computer (PC MOOG) 3-Driving Simulator 4-View Computers (slave PCs)(CS-SIMU-1, CS-SIMU-2, CS-SIMU-3) 5-Video projectors for the left, centre and right visuals.

Figure 7 .

 7 Figure 7. Connection of the PCs of the simulator

Figure 8 .Figure 9 .

 89 Figure 8. Country scenario OKTAL SCANeRstudio [OKTAL SCANeRstudio User Manual]

Figure 10 .Figure 11 .Figure 12 .

 101112 Figure 10. Effective roll angle of the motion platform at different motion drive algorithms

Figure 13 .

 13 Figure 13. Driver's influence as a second closed loop controller

  sickness dose value for vertical dynamics (ISO 2631-1 1997) [m/s 1.5 ] zwf a : weighted vertical acceleration [m/s 2 ]

Figure 14 .Figure 15 .

 1415 Figure 14. Proposed procedure to reduce the sickness on simulator (Sickness Reduction Logic -SRL-)

  , have been realized with 7 test people aged between 25 and 42 years old and at least 6 years of driving experience. The tests were accomplished in different conditions with Country Road Scenario at constant driving velocity v=70 km/h: -Classical washout algorithm -Adaptive 1 washout algorithm -Adaptive 2 washout algorithm

Figure 16 .

 16 Figure 16. Ages and driving experience of the subjects

Figure 17 . 5 :

 175 Figure 17. Operating of the dynamic driving simulator SAAM Table 5: General information about the participated subjects Age between 25-42 years old Driving experience between 6-22 years old Class of driving licence class A : 1, class B : 6 Earlier experience with static simulator yes : 6, no : 1

Figure 18 .Figure 19 .Figure 20 .Figure 21 .Table 6 :Figure 22 .Figure 23 .

 1819202162223 Figure 18. Subjective evaluation of the different washout algorithms in terms of simulation sickness as mean value

Table 1 :

 1 Software limits for each degree of freedom (DOF) of the dynamic simulator SAAM [MOOG FCS User Manual]

	DOF	Displacement	Velocity	Acceleration
	Pitch	±22 deg	±30 deg/s	±500 deg/s 2
	Roll	±21 deg	±30 deg/s	±500 deg/s 2
	Yaw	±22 deg	±40 deg/s	±400 deg/s 2
	Heave	±0.18 m	±0.30 m/s	±0.5 G
	Surge	±0.25 m	±0.5 m/s	±0.6 G
	Sway	±0.25 m	±0.5 m/s	±0.6 G

Table 2

 2 indicates the experimental conditions of the moving base simulator experiments.

Table 2 :

 2 Conditions of the moving base simulator experiments

Table 3 :

 3 Peak Values of IR by means of motion drive algorithms Peak

values by classical washout algorithm Evaluation steps

		point	point	point	point	point
		1	2	3	4	5
	Time (s)	32.58	61.82	97.11	144.7	152.5
	Illness Rating per	0.8913	0.2196	0.3432	0.4819	0.3853
	lateral axis (Y-
	axis)

Peak values by adaptive1 washout algorithm Evaluation steps point 1 point 2 point 3 point 4 point 5 Time (s)

		32.82	62.96	97.93	143.9	152.8
	Illness Rating per	0.6973	0.1676	0.2424	0.4087	0.3191
	lateral axis (Y-
	axis)

Table 4 :

 4 Optimization of IR by means of motion drive algorithms

		Optimization of illness by steps
	Evaluation steps	point 1	point 2	point 3	point 4	point 5
	Percentage of	≈21.77	≈23.68	≈29.37	≈15.19	≈17.18
	optimization

Copyright © 2011 by ASME

CONCLUSION AND FUTURE WORKS

After the research and implementation stage within the moving base driving simulator SAAM, the use of an adaptive washout algorithm has provided a reduced MSDV. It has been obtained with the help of the correction effect of the closed loop adaptive washout algorithm with respect to the classical algorithm which is an open loop one (see Figure 15, Table 3 and4). More, these good results have been obtained with the same subjective fidelity between the three evaluated algorithms.

According to Q3 in the simulator sickness test, by the adaptive2 algorithm; the subjects reacted less stressfully (cold sweat) to the conditions, whereas they behaved more stressfully in the conditions of the classical algorithm. Regarding the visual sickness (Q5, Q6) of the participants, the adaptive2 algorithm presents the most reasonable situation.

Also concerning the mental pressure, the statistical distribution points out an agreeable experience for adaptive2 algorithm compared to the others. (See Figures 18,[START_REF] Moog Fcs | Motion System 6 DOF 2000E User"s Manual Stewart[END_REF]20 and 21). Based upon the "simulator fidelity assessment" (see Figure 22,23,24 and 25), the most realistic acceleration was perceived by adaptive2 algorithm; where the most unpleasant steering was coincided by classical; whereas the most agreeable condition was experienced by adaptive1 algorithms for perception of the pitch motion severity. The perception on curvature was evaluated the most disagreeably during the attempts with classical washout algorithms.

As a conclusion closed loop control of the platform can reduce effect on simulator sickness. For future work, we will evaluate the simulation sickness with different scenarios, like double lane change maneuvers, with different controllers. Such tests would be interested because of its high dynamic characteristics (instationary behavior). The evaluations were done with 7 subjects. In our future works, we want to improve the obtained results with more subjects. Another part of our work will be focused in the objective evaluation on the subject in order to evaluate simulation sickness. For that, we intend to make some physiological measurements (heart pulse, eye movement, eeg). In parallel to these evaluation research on simulation sickness, we work on technologies for reducing the simulation sickness (improvement of 3D sound coupled with image, vibrations on the seat) as well as studies relative to simulation sickness in static simulator (for low cost systems).