Pedro Miranda
email: pmiranda@mat.ucm.es

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

k-balanced games and capacities

Keywords: Cooperative Games, k-additivity, balanced games, capacities, core

In this paper we present a generalization of the concept of balanced game for finite games.

Balanced games are those having a nonempty core, and this core is usually considered as the solution of the game. Based on the concept of k-additivity, we define the so-called k-balanced games and the corresponding generalization of core, the k-additive core, whose elements are not directly imputations but k-additive games. We show that any game is k-balanced for a suitable choice of k, so that the corresponding k-additive core is not empty. For the games in the kadditive core, we propose a sharing procedure to get an imputation and a representative value for the expectations of the players based on the pessimistic criterion. Moreover, we look for necessary and sufficient conditions for a game to be k-balanced. For the general case, it is shown

Introduction

One of the main problems of cooperative game theory is to define a solution of a game ν, that is, supposing that all players join the grand coalition N, an imputation to each player represents a sharing of the total worth of the game ν(N ). In the case of finite games of n players, an imputation1 can be written as a n-tuple (x 1 , . . . , x n ) such that n i=1 x i = ν(N ). Of course, some rationality criterion should prevail when defining the sharing.

In this respect, the core is perhaps the most popular solution of a game. It is defined as the set of imputations x on N such that

i∈A x i ≥ ν(A), ∀A ⊆ N, A = ∅ and n i=1 x i = ν(N ).
In this case, (x 1 , . . . , x n ) is a possible satisfactory sharing function because no player can do better, even when forming coalitions with other players.

It is a well known fact that the core is nonempty if and only if the game is balanced [START_REF] Bondareva | Some applications of linear programming to the theory of cooperative games[END_REF].

However, there are games whose core is empty. It is then necessary to give an alternative solution. In this sense, many possibilities have been proposed in the literature, as the dominance core stable sets, Shapley index, Banzhaf index, the -core, the kernel, the nucleolus, etc. (see e.g. [START_REF] Driessen | Cooperative Games[END_REF]).

On the other hand, Grabisch has defined in [START_REF] Grabisch | Alternative representations of discrete fuzzy measures for decision making[END_REF] the concept of k-additive capacities (capacities are monotone games). These capacities can be defined with a reduced number of coefficients and they bring up a model that is both flexible and simple to use. For k-additive capacities, the value of k (varying between 1 and n) denotes the maximum cardinality for which the Möbius transform [START_REF] Rota | On the foundations of combinatorial theory I. Theory of Möbius functions[END_REF] does not vanish, i.e., k represents the maximum cardinality for which subsets can reach importance by their own. The concept of k-additivity extends the concept of additivity in the sense that probability measures are indeed 1-additive capacities. Therefore, k-additive capacities generalize the concept of probability and they fill the gap between probabilities and general capacities. Moreover, as they are defined in terms of the Möbius transform and this transform can be applied to the characteristic function of any game (not necessarily monotone), the concept of k-additivity can be extended to games as well.

The aim of this paper is to provide another solution concept, called the k-additive core, which we think to be a natural generalization of the core. The basic idea is to remark that an imputation is nothing else than an additive game, and if the core is empty, we may allow to search for games more general than additive ones, namely k-additive games, dominating the game. This leads to the concept of k-imputation, that is, an imputation over all coalitions of at most k players. We present a generalization of the concept of balanced games, that we will call k-balanced games; as it will be explained below, these games are those admitting a dominating k-additive game and no dominating (k -1)-additive game. We will see that any game is k-balanced for a suitable choice of k, so that this concept can be applied to any game. By definition of the k-additive core, the total worth assigned to a coalition will be always greater or equal than the worth the coalition can achieve by itself, thus keeping the essential spirit of the notion of core; however, the precise sharing among players has still to be decided (e.g., by some sharing or bargaining process) among each group of at most k players.

Moreover, from a k-additive imputation, we propose a procedure to obtain a representative value (that in general is not an imputation) for each dominating k-additive game based on a pessimistic criterion; for each player, this value represents its minimum payoff. From this, we show that the value of k could be seen as a degree of stability of the game. Hence, players should prefer k as small as possible, which shows that if the core is nonempty, imputations from the core should be taken in priority.

Then, we deal with the problem of deriving a classical imputation from a k-additive imputation. In this sense, we propose a new sharing procedure. Anyway, any procedure to derive an imputation could be applied.

Finally, we look for necessary and sufficient conditions for a game to be k-balanced for a fixed value of k. In particular we show that any game admits a 2-additive game dominating it, so that any game is either balanced or 2-balanced. If we impose monotonicity, that is, if we deal with capacities instead of games, then the situation is more complex.

The paper is organized as follows: In Section 2 we give the basic concepts on games and capacities that we will need throughout the paper. We also introduce the concept of balanced game and give its interpretation. In Section 3 we introduce the concept of k-balanced games; Section 4 is devoted to deriving possible solutions from the k-additive core. In Sections 5 and 6 we look for necessary and sufficient conditions for a given game (resp. capacity) to be k-balanced (resp. k-balanced monotone). We finish with the conclusions and open problems.

Basic concepts

Let us consider a set of n players N = {1, . . . , n}. Coalitions of players are subsets of N, denoted by capital letters A, B, and so on, and also {i 1 , . . . , i j }. Whenever possible, we will omit braces in order to avoid a heavy notation, so {i 1 , . . . , i j } turns into i 1 , . . . , i j , especially for singletons and pairs. The set of all subsets of N is denoted by P(N ), while the set of all subsets of N of cardinality smaller or equal than k is denoted by P k (N ). We start with an example that will play the role of running example for the paper.

Example 1. Three workers share the same office. They need a new bookshelves in order to store their dossiers. However, the number of dossiers increases very quickly so that when the bookshelves arrive, the situation is as follows: The first worker has enough material to fill all the shelves, while the other two workers have enough material to fill 70% of the space each of them.

How should the space be shared?

Definition 1. [13] A game over N is a mapping ν : P(N ) → R (called characteristic func- tion) satisfying ν(∅) = 0.
If, in addition, (i) ν satisfies ν(A) ≤ ν(B) whenever A ⊆ B, the game ν is said to be monotone;

(ii) ν satisfies ν(A ∪ B) = ν(A) + ν(B) whenever A, B ⊆ N , A ∩ B = ∅, the game is said to be additive.
From the point of view of Game Theory, for any A ⊆ N, the value ν(A) represents the minimum asset the coalition of players A will win if the game is played, whatever the remaining players may do, i.e., ν(A) is the payoff that coalition A can guarantee for itself. Definition 2. A non-additive measure [START_REF] Denneberg | Non-additive measures and integral[END_REF] or capacity [START_REF] Choquet | Theory of capacities[END_REF] or fuzzy measure [START_REF] Sugeno | Theory of fuzzy integrals and its applications[END_REF] ν over N is a monotone game with ν(N ) = 1.

Consider a monotone game different from the trivial game defined by ν(A) = 0, ∀A ⊆ N . In this case, we can divide all the values of ν by ν(N ) so that we obtain a new game ν equivalent to ν. Then, ν is a capacity and we conclude that any monotone game can be equivalently represented by a capacity. From now on, we will denote by G(N ) the set of all games over N and by FM(N ) the set of all capacities.

Example 1. (Continued) The situation stated previously can be modelled through the capacity

ν(1) = 1, ν(2) = ν(3) = 0.7, ν(i, j) = 1, ∀i, j, ν(1, 2, 3) = 1.
There are other set functions that can be used to equivalently represent a game. We will need in this paper the so-called Möbius transform. Definition 3. [START_REF] Rota | On the foundations of combinatorial theory I. Theory of Möbius functions[END_REF][START_REF] Harsanyi | A simplified bargaining model for the n-person cooperative game[END_REF] Let ν be a game on N . The Möbius transform (or dividends) of ν is the function defined on P(N ) by

m(A) := B⊆A (-1) |A\B| ν(B), ∀A ⊆ N.
The Möbius transform given, the original characteristic function can be recovered through the Zeta transform [START_REF] Chateauneuf | Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion[END_REF]:

ν(A) = B⊆A m(B).
The value m(A) represents the strength of subset A in any coalition in which it appears. Given a Möbius inverse, necessary and sufficient conditions for its Zeta transform to be a capacity have been investigated. The following can be proved:

Proposition 1. [2] A set of 2 n coefficients m(A), A ⊆ N,
i) m(∅) = 0, A⊆N m(A) = 1, (ii) i∈B⊆A m(B) ≥ 0, for all A ⊆ N, ∀i ∈ A.
Remark that in general m(A) can attain negative values. If we restrict to the case for which the Möbius transform can attain only nonnegative values, we obtain a special subfamily of capacities called belief functions. Belief functions come from the Theory of Evidence developed by Dempster [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF] and Shafer [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF].

Definition 4. Let ν, ν * ∈ G(N ). We say that ν * dominates ν, denoted by ν * ≥ ν, if ν * (A) ≥ ν(A), ∀A ⊆ N, and ν * (N ) = ν(N ).
Let us now recall the basic results on balanced games. Definition 5. Let ν ∈ G(N ). We say that a vector :

x = (x 1 , . . . , x n ) ∈ R n is an imputation for ν if it satisfies n i=1 x i = ν(N ).
Remark 1. For any x ∈ R n , it is convenient to use the notation x(A) := i∈A x i , for all A ⊆ N , with the convention x(∅) = 0. Thus, x identifies with an additive game.

The value x i is the asset player i receives when sharing ν(N ). Several solutions for obtaining imputations have been proposed, the Shapley value [START_REF] Shapley | A value for n-person games[END_REF] being among the most popular. However, it could be the case that some of the players do not consider their corresponding value as a satisfactory asset. Suppose that the imputation satisfies x(A) ≥ ν(A), for all A ⊆ N . Then, no subcoalition of players has interest to form, since they will receive more by the sharing (x 1 , . . . , x n ). In other words, any such (x 1 , . . . , x n ) is a possible satisfactory imputation for all players. Definition 6. [START_REF] Shapley | Cores of convex games[END_REF] Let ν be a game. The core of ν, denoted by C(ν), is defined by

C(ν) := {x ∈ R n | x(A) ≥ ν(A), ∀A ⊆ N, x(N ) = ν(N )}.
Remark 2. Since by Remark 1 any x ∈ R n induces an additive game, the core can be equivalently defined as the set of additive games dominating ν.

When the core is nonempty, it is usually taken as the solution of the game. However, there are games with an empty core. Then, the following definition arises:

Definition 7. [13] A game ν ∈ G(N ) is balanced if C(ν) = ∅.
For the special case of ν being a capacity, if (x 1 , . . . , x n ) is in the core, it follows that (x 1 , . . . , x n ) determines a probability distribution on N dominating ν. Thus, in this case, C(ν) coincides with the set of all probabilities dominating ν, and the concept of balanced games translates to capacities as follows: Definition 8. A capacity ν is said to be balanced if it admits a dominating probability measure.

Note that for the case of the core, giving a dominating additive game, the value x i coincides with m(i).

Let us turn to the concept of k-additivity. In order to define a capacity, 2 n -2 values are necessary. The number of coefficients grows exponentially with n, and so does the complexity of the problem. This drawback reduces considerably the practical use of capacities. Then, some subfamilies of capacities have been defined in an attempt to reduce complexity. In this paper we will use k-additive capacities. This choice is based on two facts: First, as we will see below, kadditive capacities are a generalization of probability distributions and thus, they are suitable to define a concept extending the core. Secondly, they are defined in terms of the Möbius transform and therefore, it seems natural to use this transformation in the sharing procedure. Moreover, the value of k can be interpreted as the maximum size for which a coalition has a non-null dividend and consequently, the maximum cardinality for which a sharing is needed; this will be interesting when interpreting k-balanced games.

Definition 9. [8]

A game ν is said to be k-order additive or k-additive for short if its Möbius transform vanishes for any A ⊆ N such that |A| > k, and there exists at least one subset A of exactly k elements such that m(A) = 0.

Additive games are 1-additive games, and so probability measures are 1-additive capacities (indeed, 1-additive belief functions); thus, k-additive capacities generalize probability measures.

More about k-additive capacities can be found, e.g., in [START_REF] Grabisch | k-order additive discrete fuzzy measures and their representation[END_REF].

The value k represents the maximum cardinality for which subsets can reach importance by their own. We will denote by G k (N ) (resp. FM k (N )) the set of games (resp. capacities) being k -additive, for 1 ≤ k ≤ k.

k-balanced games

Let ν be a game. If the core of ν is empty, it is not possible to find a satisfactory imputation for all players (in terms of coalitional stability; there are other solution concepts for a game that are appealing imputations, as the nucleolus or the Shapley value to cite a few). Since from Remark 2 the core is the set of 1-additive dominating games, when it is empty it is natural to look for 2-additive games dominating ν; if this set is empty too, then look for 3-additive games dominating ν and so on. This leads us to the following definition. In this sense, a balanced game is indeed a 1-balanced game. We will denote by BAL k (N ) the set of all k-balanced games on N . As any game ν is k-additive for a suitable choice of k, then ν is k -balanced, for some value k ≤ k. This implies that the collection {BAL k (N ), k = 1, . . . , n} determines a partition of G(N ) :

BAL 1 (N ) ∪ • • • ∪ BAL n (N ) = G(N ), BAL i (N ) ∩ BAL j (N ) = ∅, i = j.
We will see in Section 5 that in fact this structure simplifies drastically. Definition 11. (i) Given a game ν, we define the k-additive core of ν, denoted by C k (ν), as the set of all k-additive games dominating ν, i.e.,

C k (ν) := {ν * ∈ G k (N ) | ν * (A) ≥ ν(A), ∀A ⊂ N, ν * (N ) = ν(N )}.
(ii) A k-imputation for ν is a sharing of the worth ν(N ) among all coalitions of at most k players, i.e., it is a vector

x ∈ R |P k (N )|-1 satisfying A∈P k (N ),A =∅ x A = ν(N ). Any ν * ∈ C k (ν) defines a k-imputation, which is simply its Möbius transform m * , i.e., x A := m * (A), for all A in P k (N ) \ {∅}.
We stress the fact that any k-order imputation x in the k-additive core of a game ν is such that the total amount given to coalition A, which is ν * (A) := B⊆A|B∈P k (X) x(B), is at least equal to ν(A). This is our generalization of coalitional stability.

As stated before, from a mathematical point of view the set of capacities constitutes a special class of games. In many applications (for example in Imprecise Probabilities [START_REF] Walley | Coherent lower (and upper) probabilities[END_REF]), it is interesting to study the set of probabilities dominating a given capacity; in these frameworks, it makes no sense to remove the condition of monotonicity. Then, it seems interesting to extend the concept of dominance to the k-additive case, but keeping monotonicity. This leads us to the following concept: Definition 12. A capacity ν on N is k-balanced monotone if there exists a k-additive capacity dominating it and no capacity in FM k-1 (N ) dominates ν.

We will denote the set of all k-balanced monotone capacities by BAL k M (N ).

Definition 13. Given a k-balanced monotone capacity ν, we define the k-additive monotone core of ν, denoted by C k M (ν), as the set of all k-additive capacities dominating ν. Any φ ∈ C k M (ν) defines a k-imputation, which is simply its Möbius transform.

As for the general case, {BAL k M (N ), k = 1, . . . , n} forms a partition of FM(N ), i.e.,

BAL 1 M (N ) ∪ • • • ∪ BAL n M (N ) = FM(N ), BAL i M (N ) ∩ BAL j M (N ) = ∅, i = j.
For the special case of belief functions, the following can be shown:

Proposition 2. A necessary and sufficient condition for a capacity ν to have a dominating belief

function is C 1 M (ν) = ∅. Proof: If ν ∈ BAL 1 M (N )
, then it admits a dominating probability measure, and thus, a dominating belief function.

Conversely, suppose that ν admits a dominating belief function ν * with corresponding Möbius transform m * . Consider an order i 1 i 2 • • • i n on N and let us define P by:

P (i j ) := A|i j ∈A⊆{i 1 ,...,i j } m * (A).
It is straightforward to see that P is a probability measure dominating ν * , whence the result.

As a consequence of this result, the following can be proved for k-balanced capacities (k > 1):

Corollary 1. If ν is a capacity such that ν ∈ BAL k M (N ), k ≥ 2, then for any capacity ν * ∈ C k M (ν), there is at least one subset A ∈ P k (N ) such that m * (A) < 0.
This means that if a capacity has an empty core, we necessarily need to allow negative dividends in order to dominate it. can be seen that no additive game can dominate ν. Thus, this game is not balanced. On the other hand, the game ν * whose Möbius transform is given by m * (i) = 2, m * (i, j) = - 5 3 , m * (N ) = 0, dominates ν as next table shows: 4 Building a solution from the k-additive core

{1} {2} {3} {1, 2} {1, 3} {2, 3} {N } m * 2 2 2 -5 3 -5 3 -5 3 0 ν * 2 2 2 7 3 7 3 7 3 1 ν 1 0.7 0.7 1 1 1 1 Thus, ν ∈ BAL 2 (N ).

Defining an imputation from a k-imputation

Suppose ν is a k-balanced game, and take any ν * ∈ C k (ν). We know from Definition 11 that the Möbius transform m * of ν * defines a k-imputation. Since a k-imputation is not an individual allocation to players (i.e., a solution of the game), we have to derive somehow an imputation from m * . A natural way is to proceed as follows. For any A ∈ P k (N ) such that m * (A) = 0 :

• If m * (A) > 0, the value m * (A) is shared among players in coalition A in some way. In this case, all players in coalition A receive a nonnegative quantity.

• If m * (A) < 0, the payment of value -m * (A) must be shared among players in coalition A in some way. In this case, all players in coalition A pay a nonnegative quantity. Indeed, following the interpretation of Möbius transform, we are sharing the dividend of the coalition among its members. Remark that when sharing m * (A), we do not allow that some of the players receive a positive quantity and others a negative value. Following this procedure, we derive an imputation in the classical sense.

In the sequel, we propose two easy ways to derive an imputation from a k-imputation m * . The first one is to give to each player in A the quantity m * (A)

|A| , and we proceed in this way for any A ⊆ N . Then, it is well known that doing so we get the Shapley value of ν * .

Let us introduce another sharing procedure, which we call the proportional sharing. For any A such that m * (A) = 0, the idea is to allocate to player i ∈ A the value m * (A) in proportion of m * (i)/ j∈A m * (j). Specifically, supposing for the moment that m * (i) ≥ 0 for all i ∈ N (which is always true if ν * is monotone):

(i) For any A ∈ P (N ) such that m * (A) = 0, each player i ∈ A receives the amount

x i (A) := m * (i) j∈A m * (j) m * (A), if j∈A m * (j) = 0, otherwise x i (A) := m * (A) |A| .
(ii) The imputation x ∈ R n derived from m * is defined for player i by

x i := A∈P k (N ),A i x i (A).
Clearly, x is a well-defined imputation. In a sense, the values m * (i) could be seen as a kind of decisional power to conduct subsequent sharings for all coalitions in P k (N ) containing i.

If m * (i) < 0 for some i ∈ N , the above procedure may lead to counterintuitive results. Indeed, take A ∈ P k (N ) such that m * (A) > 0 and m * (i) < 0 for all i ∈ A. Then the player i who has the lowest value for m * (i) would receive the biggest part of m * (A). To avoid this, we could proceed as follows.

Consider r := min i∈N m * (i) < 0. We define ν as

ν (A) := ν * (A) -r, ∀A ⊆ N.
Then, if m denotes the Möbius transform of ν , it follows that m (i) ≥ 0, ∀i ∈ N, whence we can apply the proportional sharing. Once the procedure is applied, we subtract r n from the values of all players in order to get an imputation.

The following lemma gives a justification of the proportional sharing.

Lemma 1. Let ν ∈ G(N ). Then there exists ν

* ∈ C n (ν) such that m * (A) = 0, ∀A N , |A| > 1. Proof: We consider m * (i) := max i∈A⊆{1,...,i} {ν(A) - j∈A,j =i m * (j)}, ∀i ∈ N.
Finally, we define m * (N ) := ν(N )-i∈N m * (i); this is necessary in order to have ν * (N ) = ν(N ).

It is straightforward to see that the game ν * whose Möbius transform is m * dominates ν; for if consider A ⊆ N , if A = N, the result holds by construction; otherwise, let i A := max i∈A i.

Then, m * (i A ) ≥ ν(A) - j∈A,j =i A m * (j) ⇔ ν * (A) = i∈A m * (i) ≥ ν(A).
Therefore, the result holds.

Remark that for ν * we only have to share m * (N ) among all the players in N to define an imputation for ν. Now, if ν * satisfies the condition of Lemma 1, then any ν defined through its Möbius transform by

m (i) := λm * (i), ∀i ∈ N, m (N ) = ν(N ) - i∈N m (i),
satisfies it as well for any λ > 1. In a sense, ν * and ν are essentially equivalent (they have the same structure for the Möbius transform) and should lead to the same imputation.

For ν * , let us denote by α i ∈ [0, 1] the proportion of m * (N ) that is given to player i. Then, the allocation for i is

x * i = m * (i) + α i m * (N ).
As ν is essentially the same game, the proportion of m (N ) that is given to i should be again α i . Then, the final value for i is

x i = m (i) + α i m (N ).
On the other hand, note that

m (N ) = ν(N ) -λ i∈N m * (i).
Then, as

x * i = x i , m * (i) + α i ν(N ) - i∈N m * (i) = λm * (i) + α i ν(N ) -λ i∈N m * (i) ,
whence, assuming i∈N m * (i) = 0,

α i = (1 -λ)m * (i) (1 -λ) i∈N m * (i) = m * (i) i∈N m * (i)
.

Example 1. (Continued) In the running example, we have already proved that ν ∈ BAL

Why the k-additive core with smallest k should be preferred

A central question is the following: Why a dominating additive game should be preferred to a dominating 2-additive game? In other words, we need to justify the fact that, for a game ν, we look for a dominating k-additive (monotone) game ν * with k as small as possible. This is important in the sense that we want our solution to be a generalization of the core and not an alternative to it; thus, it is necessary to show that whenever the core is not empty, it still constitutes the solution of the game.

Remark that, depending on the sharing procedure that is applied, the payoffs of the players can vary. If player i is pessimist, he might think that he will receive all the quantities coming from negative Möbius coefficients m * (A), i ∈ A, and nothing at all from the positive Möbius coefficients m * (A), i ∈ A. This provides the minimum value that player i can receive: Definition 14. Let ν be a game, and consider ν * ∈ C k (ν) with Möbius transform m * . We define the minimal value for player i as

x * i := m * (i) + i∈A,m * (A)<0 m * (A). ( 1
)
We will refer to (x * 1 , . . . , x * n ) as the minimal vector associated to ν * .

These minimal values generalize the corresponding sharing function for balanced games (but are not in general an imputation). At this point, remark that we could have considered other representative values that also generalize the sharing procedure of balanced games instead of the pessimistic criterion, as for example the optimistic criterion, an adaptation of the Hurwicz criterion, and so on. However, the pessimistic criterion seems to be in our opinion the most appropriate, in the sense that it is more conservative. We will come again to this point below. Following this pessimistic behavior, let us show that players should prefer a k-additive game with k being as small as possible.

Let us then suppose that a game ν is dominated by a game ν * . From ν * it is possible to derive the corresponding minimal values for all players through Eq. ( 1). In the minimal values we are supposing that we have no information about how m * (B) should be shared (or afforded) among players in coalition B; however, it could be the case that we had the additional information that part of m(B), say r, must be shared or afforded by a subcoalition of players in B, say A. Then, we have to consider this new situation when computing the minimal values. Suppose w.l.o.g. that m * (B) > 0; the natural way to deal with this additional constraint is to add r to m * (A) and skip it from m * (B). This leads us to the following concept.

Definition 15. Let ν be a game on N with Möbius transform m. We say that ν * ∈ G(N ) is a contraction of ν if the Möbius transform m * of ν * can be put under the following form:

m * (A) = B⊇A λ(B, A)m(B), ∀A ∈ P(N ),
where function λ :

P(N ) × P(N ) → [0, 1] is a sharing function such that A⊆B λ(B, A) = 1, ∀B ⊆ N, λ(B, A) = 0, ∀A ∈ P(N ), A ⊆ B.
If ν * is a contraction of ν, then ν * gives more importance (in terms of the Möbius transform) to smaller coalitions than the original ν. Remark that if ν is a k-additive game, then ν * is at most k-additive. Note also that, as λ takes values in [0,1], when multiplying it by m(B), if m(B) > 0 (resp. m(B) < 0), no A ⊆ B can receive a negative (resp. positive) payoff. The Shapley value is just an additive contraction of ν and similarly, the value derived from the procedure we developed previously for a game ν * in C k (ν) is an additive contraction of ν * .

Proposition 3. Let ν be a and consider ν * ≥ ν with Möbius transform m * . Suppose ν is a contraction of ν * such that ν ≥ ν. Then, denoting by x i the minimal value for player i for ν and by x * i the corresponding value for ν * , we have

x i ≥ x * i , ∀i ∈ N.
Proof: Take i ∈ N and consider m * (B) = 0. Take A ⊆ B. Then, applying Eq. ( 1), we have the following:

• If i ∈ B, then x i = x * i . • If i ∈ B, m * (B) > 0 and |A| > 1, then x i = x * i , ∀i ∈ B. If A = {j}, then x i = x * i , ∀i = j and x j = x * j + λ(B, {j})m * (B) ≥ x * j .
• If i ∈ B and m * (B) < 0, then

x i = x * i , ∀i ∈ A and x i = x * i -λ(B, A)m * (B) ≥ x * i , ∀i ∈ B\A.
Therefore, x i ≥ x * i , ∀i ∈ N. Following this process for all B and all A ⊆ B, the result holds.

In particular, Proposition 3 shows that if a k-additive game ν * dominates ν and it is also possible to obtain by contraction of ν * a k -additive dominating game ν with k < k, then all the players should be indifferent or prefer ν instead of ν * in terms of their minimal values.

This shows that k might be seen as a measure of stability, in the sense that the smaller k is, the happier the players are and the corresponding game is more "stable". From an interpretational point of view, the players should prefer k as small as possible, as large coalitions would lead to more conflicts to share (or afford) the value m * (A).

If we had considered the optimistic criterion, then the representative value for player i would be given by i∈A,m * (A)≥0 m * (A).

In this case, players would look for coalitions as large as possible because they are confident about the sharing procedure. This would imply that for balanced games, the core might not provide optimal sharing functions. Indeed, from the point of view of expectations, these games would be quite unstable, as players could feel upset when their expected values could not be attained for all of them.

Suppose that we are dealing with the monotone case and assume that ν * is a k-additive belief function. It can be easily seen that any (k -1)-additive belief function ν obtained by contracting Consider a game ν ∈ G(N and the following linear program with variables m(B), B ∈ P k (N ), which we will call (P1):

min z = B∈P k (N ) m(B) s.t. B⊆A,B∈P k (N ) m(B) ≥ ν(A), ∀A ⊆ N.
Notice that (P1) has always a finite optimal solution, since z is bounded below by ν(N ). Take any ν * ∈ C k (ν), with m * its Möbius transform. Then m * is a feasible solution of (P1), and

z * = ν(N ).
Hence m * is an optimal solution of (P1). Conversely, suppose that (P1) has an optimal solution m * such that z * = ν(N ). Then clearly m * ∈ C k (ν). Hence, we have shown the following.

Theorem 1. Consider ν ∈ G(N ). Then ν has a nonempty k-additive core if and only if the optimal value of the objective function of (P1) is z * = ν(N ).

Let us consider the dual program of (P1), which we shall call (D1):

max q = S⊆N ν(S)y S s.t. S|A⊆S⊆N y S = 1, ∀A ∈ P k (N ), A = ∅ y S ≥ 0, S ⊆ N.
By the strong duality theorem, we deduce that a necessary and sufficient condition such that the k-additive core is nonempty is that the optimal solution y * of (D1) is such that q * = ν(N ).

Hence, we are lead to the following definition and result.

Definition 16. We say that C ⊆ P(N ) is a k-balanced family if there exist positive constants

y S , S ∈ C such that S∈C|S⊇A y S = 1, ∀A ∈ P k (N ), A = ∅.
We call y = {y S } S∈C a k-balanced vector for C, whose components are called k-balanced coefficients.

Theorem 1 can be then rewritten as follows.

hence k 2 = 2 ν(N 1 n(n-1) . Therefore, for any coalition S of s elements:

ν * (S) = sk 1 + s(s -1) 2 k 2 = sk 1 1 - s -1 n -1 + s(s -1) n(n -1) ν(N ).
Thus ν * (S) is a linear increasing function in k 1 , whence ν * (S) can be made as large as necessary to obtain dominance.

k-balanced monotone families

We turn now to the problem of finding sufficient and necessary conditions for the nonemptiness of the k-additive monotone core of some capacity. Since the approach is similar to the previous section and to [START_REF] Owen | Game Theory[END_REF], we omit proofs.

Consider a capacity ν on N. In this case, if ν ∈ FM k (N ) and m denotes its corresponding Möbius transform, the linear programming problem (P1) changes into (P2):

min B∈P k (N ) m (B) = z (4) s.t. B⊆A m (B) ≥ ν(A), ∀A ⊆ N (5) i∈B⊆A,B∈P k (N ) m (B) ≥ 0, ∀i, A, i ∈ A ⊆ N (6)
The first line of constraints is necessary in order to ensure dominance. The second line of constraints in (P2) is needed to ensure monotonicity (Proposition 1).

As for (P1), the following can be shown.

Proposition 6. Given a capacity ν on N , there exists a k-additive (at most) capacity ν * on N dominating ν if and only if the optimal value of the objective function z * of (P2) is such that

z * = 1.
We consider the dual problem of (P2), that we shall name (D2). It is given by:

max A⊆N ν(A)y A = q s.t. A|B⊆A⊆N y A + i,A|i∈B⊆A y i,A = 1, ∀B ∈ P k (N ), B = ∅ • y i = 0.1, y = 0.1, y N = 0.1, y {i},{i} = 0.1, y {i},{i,j} = 0.1, y i,N = 0.3, for all i, j ∈ N , i = j.
This shows that contrarily to the case of games, there is not a single k-balanced monotone family.

On the set of all k-balanced monotone families, it is possible to define the following operation:

C ∪ D = (C 1 ∪ D 1 , C 2 ∪ D 2 ).
We

say that C is contained in D, denoted by C ⊆ D, if C 1 ⊆ D 1 and C 2 ⊆ D 2 .
With these operations, the following can be shown: Proposition 7. The set of k-balanced monotone families is a sup-semilattice [START_REF] Dubreil | Algèbre. Vol. 1. Equivalences, opérations, groupes, anneaux, corps[END_REF], with partial order ⊆ and C ∨ D = C ∪ D.

As a consequence, we have that the union of a finite number of k-balanced monotone families is a k-balanced monotone family. As N is a finite set, the number of k-balanced monotone families is finite. Thus, this implies that the set of all k-balanced monotone families is a complete supsemilattice [START_REF] Dubreil | Algèbre. Vol. 1. Equivalences, opérations, groupes, anneaux, corps[END_REF]. We state now the main result of the section, that gives a necessary and sufficient condition to obtain a k-additive capacity dominating another capacity.

Conclusions

In this paper we have proposed a generalization of the concept of balanced games, called kbalanced games. These games are those admitting a dominant k-additive game. We have shown that k represents the minimum degree of interaction that we have to permit for the existence of a dominating game. We have derived a representative value for each player based on the pessimistic criterion. From these representative values, we have justified that the value of k can be interpreted as a degree of stability of the game. We think this generalization could shed light on games which are not balanced and might give insight in the theory of Cooperative Games.

We have also looked for necessary and sufficient conditions for a game (resp. a capacity) to be at most k-balanced (resp. k-balanced monotone). For the case of games, we have shown that the 2-additive core is never empty, which means that any game is either balanced or 2-balanced.

The situation for the k-additive monotone core is more complex, but any capacity is k-balanced monotone for some 1 ≤ k ≤ n.

Another problem is determining the k-additive core of a given game. This could be interesting in order to give representative values for the players. In the monotone case, this set is a convex polytope, so that it is defined the vertices. A step in this direction for k-additive games can be seen in [START_REF] Grabisch | On the vertices of the k-additive core[END_REF].

We have taken as a basis for our generalization of the core the point of view of coalitional stability. A more fundamental point of view could be to consider non-dominated imputations (see, e.g., Owen [START_REF] Owen | Game Theory[END_REF]). In the classical case, the two approaches coincide if the game is superadditive.

This could be a very interesting topic to investigate in the future.

  corresponds to the Möbius transform of a capacity if and only if

(

Definition 10 .

 10 A game ν on N is called k-balanced if there exists a k-additive game dominating it and no game in G k-1 (N ) dominates ν.

Finally, note that

  if ν is a capacity, we can look for the values k and k such that ν ∈ BAL k (N ) and ν ∈ BAL k M (N ). Obviously k ≤ k holds in general, and next example shows a case where k < k . Example 1. (Continued ) For the game ν modeling the problem of sharing the bookshelves, it

Moreover, there is no 2 -

 2 additive capacity ν * dominating ν; for if such ν * exists, from the definition of dominance and by monotonicity, it would follow that ν * (1) ≥ ν(1) = 1 and ν * (N ) = 1 ≥ ν * (1); therefore, ν * (1) = 1 = m * (1). On the other hand, ν * (1, 2) = 1, but 1 ≥ ν * (1, 2) = m * (1) + m * (2) + m * (1, 2), whence m * (1, 2) = -m * (2); the same can be done for {1, 3}, whence m * (1, 3) = -m * (3).

Finally, 1 =

 1 ν * (2, 3) = m * (2) + m * (3) + m * (2, 3), but as m * (2) = ν * (2) ≥ 0.7 and m * (3) = ν * (3) ≥ 0.7, we conclude that m * (2, 3) ≤ -0.4. As A⊆N m * (A) = 1, we necessarily need m * (N ) > 0. Consequently, ν ∈ BAL 2 M (N ) and this implies that ν ∈ BAL 3 M (N ), as ν is itself a 3-additive monotone game.

Definition 18 .

 18 Given C = (C 1 , C 2 ), and D = (D 1 , D 2 ) two k-balanced monotone families, we define the union of these families as

Proposition 8 .Definition 19 .Proposition 9 .Proposition 11 .

 819911 Let C, D be two k-balanced monotone families satisfying C ⊂ D. Then, there exists a k-balanced monotone family B such that B ∪ C = D and B = D. Moreover, there exist more than one k-balanced monotone vector for D. We say that a k-balanced monotone family C is minimal if it has no proper k-balanced monotone subfamily.Note that from the definition of minimal k-balanced monotone family, these families are atoms in the sup-semilattice of k-balanced monotone families[START_REF] Dubreil | Algèbre. Vol. 1. Equivalences, opérations, groupes, anneaux, corps[END_REF]. Then, as the set of all k-balanced monotone families with the operation ⊆ is a finite sup-semilattice, we have: Any k-balanced monotone family can be put as a finite union of minimal kbalanced monotone families. minimal k-balanced monotone families, the following can be proved: Proposition 10. A k-balanced monotone family has a unique k-balanced monotone vector if and only if it is minimal. The basic feasible solutions of (D2) are the k-balanced monotone vectors of the minimal k-balanced monotone families.

Theorem 2 . 1 y

 21 A necessary and sufficient condition for ν ∈ FM(N ) to be a k-balanced (at most) capacity is that any minimal k-balanced monotone family C = (C 1 , C 2 ) with vector y satisfies S∈C S ν(S) ≤ ν(N ).

The usage in cooperative game theory is to call this a pre-imputation. The distinction being unimportant here since we deal with the core, we keep the name imputation for simplicity.

Acknowledgements

This research has been supported in part by grant MTM2007-61193 and CAM-UCM910707. We gratefully thank Mrs. Tong Li, who pointed out the fact that k-balanced families for games were reduced to a singleton.

* is also a belief function and ν ≥ ν * , so the choice of ν instead of ν * seems rather sensible.

For the optimistic criterion, a belief function would be preferred to a probability, that does not seem natural, as the probability dominates the belief function and is simpler. This is another argument in favor of the pessimistic behavior and against the optimistic.

Note that the result of Proposition 3 only applies when ν is a contraction of ν * . If this is not the case, we can obtain that some players would prefer ν * . Indeed, it is possible that some players would prefer ν * ∈ C k (ν) to any other dominating game in C k (ν), with k < k.

It is easy to check that ν * ≥ ν : On the other hand, the probability distribution defined by P (1) = 0.3, P (2) = 0.4, P (3) = 0.3 dominates ν and it is the only probability distribution in these conditions.

The minimal values for ν * are given by x * 1 = 0.4, x * 2 = 0.3, x * 3 = 0.2. Thus, player 1 would prefer to obtain his sharing from ν * instead of the sharing given by the probability. However, the other players would prefer the payoffs from the probability. Then, ν and ν * are not comparable.

k-balanced families

We turn to the problem of finding necessary and sufficient conditions for the nonemptiness of the k-additive core.

Consider ν ∈ G(N ). Then ν has a nonempty k-additive core if and only if every

k-balanced family C with k-balanced vector y satisfies:

The following result shows that there is only one k-balanced family, as soon as k > 1.

Proposition 5. Let 1 < k ≤ n. Then there is a unique k-balanced family, which is {N }, with balanced vector y N = 1.

Proof: We prove the result first for k = 2. A first fact is that clearly {N } is always a k-balanced collection with balanced vector y N = 1.

Take C any 2-balanced family on N , with vector {y S } S∈C , and extend the vector y on R Denoting by m * its Möbius transform, put arbitrarily m * (i) = k 1 and m * ({i, j}) = k 2 for any i, j ∈ N . Since ν * (N ) = ν(N ), we get

The variables y A come from dominance constraints. Variables y i,A come from the monotonicity constraints of Equation ( 6).

Let us denote by q * the optimal value of the objective function of (D2). As before,

Therefore, in order to find conditions for the k-additive core to be nonempty, it suffices to find the conditions for q * = 1.

We are now in position to introduce the concept of k-balanced monotone family:

and C 1 ⊆ P(N ), C 2 ⊆ Q two collections, at least one of them being nonempty. We say that

We say then that vector y = (y S , y (i,T ) ) S∈C 1 ,(i,T )∈C 2 is a k-balanced (monotone) vector for C whose components are called k-balanced coefficients.

Note that k-balanced vectors are the feasible solutions of the dual problem (D2). The elements in C 1 correspond to variables in the dominance constraints, while elements in C 2 are related to monotonicity conditions. • y N = 1, y A = 0, ∀A = N, y i,A = 0, ∀i ∈ A ⊆ N. This is the unique 2-balanced family.


Majd Mayyasi
email: majdm@bu.edu

John Clarke

) Dolon Bhattacharyya

Jean-Yves Chaufray

Jean-Loup Bertaux

Nicholas Schneider

Improving the Detectability of Faint Emissions with MAVEN IUVS Echelle: Applications to Deuterium Lyman- and the D/H Ratio

The Mars Atmosphere and Volatile Evolution (MAVEN) mission's Imaging Ultraviolet Spectrograph (IUVS) high resolution echelle channel can resolve observations of hydrogen and deuterium Lyman-ɑ as well as O triplet FUV emissions. D Lyman-ɑ is too faint to be measured above the detector noise level when Mars is far from perihelion. Lessons learned from 5 years of IUVS echelle observations have been applied to produce a new data reduction pipeline in which the faint D emission can be better resolved.

Introduction

The MAVEN IUVS echelle is designed to optimize observations of resonantly scattered D and H Lyman-, O triplet and forbidden doublet emission lines [START_REF] Mcclintock | The imaging ultraviolet spectrograph (IUVS) for the MAVEN mission[END_REF]. In the ~5 years that MAVEN has been orbiting Mars, the background noise level of the echelle detector was found to limit the detectability of faint emissions to times when they are bright. For deuterium Lymanɑ, this brightening has been observed to be around perihelion [START_REF] Clarke | Variability of D and H in the martian upper atmosphere observed with the MAVEN IUVS echelle channel[END_REF][START_REF] Mayyasi | Seasonal variability of deuterium in the upper atmosphere of Mars[END_REF]. At remaining seasons, the data reduction pipeline developed for signal retrieval showed D brightness levels to drop to the background noise level [START_REF] Mayyasi | IUVS echelle-mode observations of the interplanetary hydrogen: Standard for calibration and reference for cavity variations between Earth and Mars during MAVEN cruise[END_REF].

A new data reduction pipeline has been developed to optimize the signal retrieval for faint emissions in the MAVEN IUVS echelle observations. The new pipeline includes a new calibration using simultaneous IUVS and Hubble Space Telescope observations. The results of the new data reduction pipeline are compared to the previous one to show higher accuracy retrieval of faint emissions [Fig. 1].

H Lyman-ɑ is an optically thick emission that is relatively bright at Mars for most MAVEN observational conditions. Improving the accuracy of MAVEN's D Lyman-ɑ emission retrievals along with the H Lyman-ɑ measurements can provide improved constraints for radiative transfer models to use in order to generate the abundances of D and H in the upper atmosphere of Mars. With D and H densities, the D/H ratio can be determined to provide estimates of water escape flux from Marsa primary objective of the MAVEN mission [START_REF] Jakosky | MAVEN explores the martian upper atmosphere[END_REF].

Figures

Figure 1 :

 1 Figure 1: A comparison of old (red), new (black) and the difference between old and new (purple) data reduction pipelines for the MAVEN IUVS Echelle channel. The H Lyman- brightness (top) varies minimally while the D Lyman- brightness (bottom) shows more pronounced variations. The new pipeline more clearly portrays the variability as well as reliability of faint emissions.


Marc Tabani

War and military matters in Vanuatu: an overview

Two or three things about representations of war and military matters in Vanuatu Since its independence in 1980, Vanuatu has often been described as a paradise. First as a tax haven with one of the least regulated offshore centers. Later on, for tourism purposes, as a tropical paradise. More recently, it has become a textbook case for alternative socio-economic indicators to assess levels of development. Indeed, Vanuatu usually ranks high on the Happy Planet Index. Despite living in of the least developed countries, the ni-Vanuatu people are supposed to be among the happiest in the world. Even after the huge destruction and suffering caused by Cyclone Pam in 2016, the Vanuatu Tourism Board was still able to announce that "Vanuatu is still smiling" because smiles and happiness is guaranteed as part of the indigenous culture.

So taking an interest in how war is portrayed in Vanuatu, and the interest people have in military matters, might seem paradoxical? It would be ignoring the country's history from colonial times to today. In fact, even though Vanuatu has never officially experienced war since independence, some significant armed actions occurred shortly after independence. However, for a long time, these were depicted as a simple and brief rebellion, which was mocked by international media as a "coconut war".

Since the first contacts with Europeans, initiated by the Spaniards in the 17th century and then by the British in the 18th century, the New Hebrides acquired a reputation for savagery, islands where the degree of primitiveness was matched only by widespread violence. The hostility of these islanders discouraged any attempts at colonial conquest until a Franco-British condominium was imposed in the early 20th century. The resurgence of tribal warfare, fueled by the spread of firearms, was also exacerbated by the anarchic settlement of conflicts between settlers, traders and natives. The ultra-coercive policies of certain churches did not hesitate to resort to physical punishment in addition to usual moral violence. In turn, the murder of missionaries contributed to turn this field of evangelization into a land of martyrdom. Finally, even after the establishment of a colonial control, with armed forces engaged into a "gunboat" strategy against rebellious villages, the archipelago was long considered as a frontier line of civilization, a lawless Wild West.

The first indigenous uprisings took the form of murders of Europeans, some accompanied by millenarian hopes against the colonial rule. On the southern island of Tanna, at the end of the 1930s, protest against the missionaries and the colonial order took the form of a politico-religious movement. Its adherents claimed to be the followers of a mysterious spirit called John Frum, a name derived from the English broom, a messiah who was supposed to wipe out the island of the evil foreign influences. The cult leaders of this movement predicted months in advance, with great hope, the extension of the Second World War in the Pacific and the landing of American troops in the New Hebrides. And it was with these troops, many of whom were African Americans, that the islanders of Vanuatu became familiar with modern military affairs.

From 1942 onwards, the New Hebrides were not the scene of any fighting, but they did serve as a rear base for the American army, which was engaged further north in the Solomons. Thousands of Melanesians from across the archipelago were recruited as manpower alongside the hundreds of thousands of GIs who stayed until 1946 in the two naval air bases in Efate and Santo.

After the war, local militias were formed under the command of the French and British authorities. The first Melanesians recruited as modern soldiers, army in Bislama, allowed to handle weapons and sometimes even trained for parachute jumping.

The 1970s started a new period of highly nationalistic demands, this time to claim independence. The tensions between the francophone and Catholic Melanesian communities and the anglophones predominantly Presbyterian, influenced by the disagreements between the condominium authorities, were exacerbated to the point of giving birth to a series of separatist movements on several of the archipelago's larger islands.

In Santo, this crisis occured as a real secessionist attempt, with the Nagriamel movement declaring the sovereignty of its own state. Disregarded as a rebellion by the nationalist leaders, it was mocked in the international press as a folkloric coconut war. This secessionist attempt, which led first to the intervention of French and British troops, ended in a very violent crackdown by Papuan troops on groups opposed to the central power of the Anglophone nationalists. The memory of this repression has left a strong mark on the historical consciousness of the francophone communities.

The aim of my intended paper is to explore these memories of war and the military matters. But also, and above all, to consider the ways in which an older historical memory is now being transcribed from one generation to the next in its vision of war and of martial spirit. From my first fieldwork in the 1990s, I was immediately struck by warmonger considerations, if not fascination, that many of my informants had for martial visions of modern conflicts. From the regret that tensions at independence had not degenerated into open conflict and confrontation between French and British militias, to the enthusiasm among young French speakers for joining the French Foreign Legion, today the range of positive feelings about the value of the military and the idealization of combat is wide.

I'll end this proposal with an anecdote that I think says a great deal about my topic.

During my second research trip to Vanuatu in 1995, I felt particularly anxious and ashamed at the thought of justifying the unjustifiable as a French citizen in relation to the resumption of French nuclear testing in Polynesia. During a group interview with Isak Wan, one of the leaders of the John Frum movement in Tanna, I was told that, on the contrary, I should be proud of these tests. An objection that is often shared among francophones, even when expressed less radically. My reply to Isak was: "But come on, Isak, the nuclear tests were intended to produce bombs, to make war, even though France has no declared enemies"; "Of course", Isak replied with a cynicism I hadn't expected, "France is right, after these tests it will be able to declare war on Australia and then to bomb it!


E P Turtle

M G Trainer

J W Barnes

R D Lorenz

K E Hibbard

D S Adams

P Bedini

W B Brinckerhoff

M L Cable

C Ernst

W B Brinck- Erhoff

C Freissinet

K Hand

A G Hayes

S M Hörst

J R Johnson

E Karkoschka

J W Langelaan

D J Lawrence

A Le Gall

J M Lora

S M Mackenzie

C P Mckay

R S Miller

S Murchie

C D Neish

C E Newman

J Palacios

M P Panning

A M Parsons

P N Peplowski

L C Quick

J Radebaugh

S C R Rafkin

M A Ravine

S Schmitz

J M Soderblom

K S Sotzen

A M Stickle

E R Stofan

C Szopa

T Tokano

C Wilson

R A Yingst

K Zacny

M T Burks

Dragonfly: In Situ Exploration of Titan's Organic Chemistry and Habitability

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction: Titan's abundant complex carbonrich chemistry, interior ocean, and past presence of liquid water on the surface make it an ideal destination to study prebiotic chemical processes and document the habitability of an extraterrestrial environment [e.g., [START_REF] Raulin | Titan's Astrobiology[END_REF][START_REF] Thompson | C. Organic chemistry on Titan: Surface interactions[END_REF][3][4][5][6]. Although pathways for the origin of life as we know it are poorly constrained, there is general agreement that liquid water, essential elements (especially CHNOPS), energetic disequilibrium, and a catalytic surface are required. In addition to the complex organic synthesis that Titan supports today, organic molecules may have interacted with liquid water at the surface in the past (e.g., sites of cryovolcanic activity or impact melt [4]), increasing the potential for oxygenation and chemical processing to progress beyond the compositional functionalities observed in high-altitude organic species. Titan provides an unparalleled opportunity to investigate prebiotic chemistry, as well as search for signatures of potential water-or even hydrocarbon-based life.

The diversity of Titan's surface materials and environments [7] drives a scientific need to sample a variety of locations, thus mobility is key for in situ measurements. Titan's dense atmosphere provides the means for long-range exploration by a vehicle with aerial mobility. The Dragonfly mission concept, under study in NASA's New Frontiers Program, is a rotorcraft lander that would achieve wide-ranging in situ investigation by flying to access different geologic settings 10s -100s of km apart, performing multidisciplinary science measurements at each landing site.

Dragonfly takes advantage of Titan's unique natural laboratory to understand how far chemistry can progress in environments that provide key ingredients for life.

Exploration Strategy: It has long been recognized that Titan's rich organic chemical environment provides a unique opportunity to explore prebiotic chemistry [e.g., 9,10], and development of Titan mobile aerial exploration was identified as a desirable next step after Cassini-Huygens. Although the hydrocarbon seas are an intriguing target [START_REF] Stofan | Proc. Aerospace Conf. IEEE[END_REF], environments that offer the most likely prospects for chemical evolution similar to that on Earth occur on Titan's land [4] and Titan's northern winter precludes direct-to-Earth communication in the 2020-2030s. Furthermore. Moreover, the dune sands may represent a 'grab bag' of materials sourced from all over Titan [START_REF] Leary | Titan Flagship study[END_REF] (similar to Mars Pathfinder's landing site [START_REF] Golombek | [END_REF]) and thus may contain aqueously altered materials. As in Mars exploration, the approach with the highest scientific potential is to obtain samples directly from multiple locations. Given Titan's dense atmosphere (4x that at Earth's surface) and low gravity (1.35 m/s 2 ), heavier-than-air mobility is highly efficient [9,13,14]. Recent developments in autonomous flight enable a lander with aerial mobility to convey a capable instrument suite to explore multiple locations. Modern control electronics make a multi-rotor vehicle [START_REF] Langelaan | Proc. Aerospace Conf. IEEE[END_REF] mechanically simpler than a helicopter, as the recent proliferation of terrestrial drones attests. For a given vehicle mass and rotor diameter, the hover power required on Titan is 38x less than on Earth [9,[START_REF] Langelaan | Proc. Aerospace Conf. IEEE[END_REF]. Flights of up to a few hours are possible using power from a battery recharged via an MMRTG. This strategy also has the advantage of flexibility to adjust to changes in power-source performance or science energy demands, merely taking longer to recharge be- tween flights. Adopting rotors as a substitute for the retrorockets used for landing on other planets, the ability to take off and land elsewhere follows with little incremental cost and tremendous science enhancement.

Science Objectives: Compositions of the solid materials on Titan's surface are still essentially unknown. So measurements [START_REF] Trainer | [END_REF] in different geologic settings [7,17] will reveal how far organic chemistry has progressed; sites where transient liquid water [4] may have interacted with the abundant photochemical products that litter the surface [START_REF] Thompson | C. Organic chemistry on Titan: Surface interactions[END_REF] are of particular interest.

At each landing site, Dragonfly can answer key scientific questions regarding habitability and prebiotic chemistry and put these measurements in the context of Titan's meteorology and methane cycle, local geologic setting and material properties [18], and geophysical measurements of the subsurface [19]. Dragonfly will:

• analyze chemical components available and processes at work to produce biologically relevant compounds; • measure atmospheric conditions, identify methane reservoirs and determine transport rates [START_REF] Stofan | Proc. Aerospace Conf. IEEE[END_REF]20,[START_REF] Lorenz | Int'l Workshop Instr. Planet. Missions[END_REF]; • characterize geologic features and transport processes, seismic activity, and subsurface structure [22,19]; • constrain processes that mix organics with past surface liquid water or potentially the subsurface ocean; • search for water-or hydrocarbon-based chemical biosignatures.

Since the interior of the lander is maintained at constant benign temperatures using 'waste' heat from the MMRTG, only modest adaptation for Titan's environment (94-K, 1.5-bar nitrogen atmosphere) is needed.

Measuring Titan's Composition: Surface material is sampled with a drill and ingested using a pneumatic transfer system [23] into the mass spectrometer, DraMS [START_REF] Trainer | [END_REF]24]. The sampling system, DrACO, has one drill on each skid for sample diversity and redundancy and provides rotary and rotary-percussive modes. Supported in part by NASA's COLDTech program, the drill has been tested at Honeybee Robotics in a range of cryogenic ices and organic materials chilled with liquid nitrogen. Cuttings are collected by a pneumatic transfer system, which conveys the material rapidly into DraMS for analysis. Pneumatic transfer ensures samples are maintained at near-ambient Titan temperatures, with particular attention to avoiding cross-talk between samples.

DraMS supports both laser desorption (LDMS) and pyrolysis gas chromatography (GCMS) operating modes, with heritage from the MSL SAM and ExoMars MOMA instruments. Phase-A activities have included environmental testing of a scroll pump, which brings the internal atmosphere of the sample delivered by DrACO from 1.5-bar Titan to ~10 mbar, in family with Mars heritage inlet designs.

A novel element of Dragonfly's payload is a neutron-activated gamma-ray spectrometer, DraGNS, to quickly identify bulk elemental composition at landing sites [START_REF] Trainer | [END_REF]25] and inform decisions about sampling and DraMS measurements. DraGNS uses a high-purity germanium (HPGe) detector for superior spectral resolution and a pulsed neutron source to excite gamma-ray emission from surface and near-subsurface material. Previous applications required active cryocoolers, but for Dragonfly Titan's cold, dense atmosphere can be used to passively hold the detector at temperature. Phase-A activities have included demonstration of HPGe passive cooling and operation of a testbed with a commercial neutron source using compositional analogs to simulate Titan organics overlying water ice.

The Dragonfly [START_REF] Lorenz | pdf 50th Lunar and Planetary Science Conference[END_REF] rotorcraft lander is designed to take advantage of Titan's environment to explore dozens of diverse sites, covering 10s -100s km during its >2yr mission, to characterize Titan's habitability and determine how far organic chemistry has progressed in environments providing key ingredients for life.

  Planetary Science Conference 2019 (LPI Contrib. No. 2132)


Michael J Malaska
email: michael.j.malaska@jpl.nasa.gov

Lauren Schurmeier

Rosaly M C Lopes

Ashley M Schoenfeld

Anezina Solomonidou

Alice Le Gall

Jani Radebaugh

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Introduction

Titan is a complex world with a wide diversity of surface terrain classes, including mountains, plains, dunes, labyrinth terrains, hydrocarbon lakes, and rare craters [1]. The labyrinth terrains of Titan are defined as elevated, highly-dissected plateaux with intersecting valleys or remnant ridges of low to medium backscatter [2,[START_REF] Malaska | Lunar and Planetary Science Conference[END_REF]. From radar emissivity, they appear to be large plateaux composed of lowdielectric organic materials [2,[START_REF] Malaska | Lunar and Planetary Science Conference[END_REF].

As part of our global mapping of Titan, we classified several types of labyrinth terrains: valleyed labyrinths, polygonal labyrinth, finely dissected labyrinths, and an outlier labyrinth style in Xanadu -Kronin Labyrinthus. Each of these labyrinths had valley networks that display organized rectangular to dendritic drainage networks in the labyrinth area or in subsections of the labyrinth. During our mapping, we noted several labyrinths that had roughly circular plan form with clear radial valley networks. These we split out as a separate subgroup called radial labyrinths.

Radial labyrinths

The radial labyrinths are roughly circular to oval in planform, usually from 100-200 km across in the longest dimension. They are roughly dome shaped and elevated several hundred meters above the surrounding terrain. Anbus Labyrinthus [39.3°N, 215°W] is a type example shown in Figure 1.

Localization

Radial labyrinths are found in the mid-latitudes, but not in the polar regions, unlike many of the other labyrinth terrain unit types. They are found in two clusters: one is located at [40°N, 205°W] in the Afekan region, and another one near [50°S, 20°W] in the Tseghi region.

Characteristics

Like the other labyrinth terrains, the radial labyrinths have high microwave emissivity, interpreted to be consistent with a composition of low-dielectric organic materials [START_REF] Janssen | [END_REF]. Spectral analysis using the Cassini Visual and Infrared Mapping Spectrometer (VIMS) suggests a thin coating of water ice materials similar to other labyrinth units and the undifferentiated plains [5].

Associations

Radial labyrinths are always found associated with undifferentiated plains. The plains units are found at the downstream terminii of the valleys. Interestingly, the three cross-cut hummocky terrain units found on Titan are associated with the radial labyrinth terrain clusters. The cross-cut hummocky terrain units are both roughly the same size and planform of the radial labyrinths.

In one of the cross-cut hummocky units (Fig. 2), a small section of valley network consistent with a section of labyrinth terrain can be discerned. This observation is consistent with a causal relationship between the two units, with the cross-cut hummocky terrain being the end-stage that followed removal of the central section of labyrinth terrain after uplift as suggested by Schurmeier et al (2018) [START_REF] Schurmeier | American Geophysical Union, Fall Meeting[END_REF]. In this modified scenario, the cross-cut hummocky unit is revealed as a remnant following labyrinth unit uplift, erosion, and plateau removal.

Summary and Conclusions

We describe a subtype of Titan's labyrinth terrain. The terrain units may be part of an evolution sequence involving the cross-cut hummocky terrain in Titan's mid-latitude region.

Figure 1 :

 1 Figure 1: A) Cassini Synthetic Aperture Radar (SAR) image of type radial labyrinth Anbus Labyrinthus. B) Annotated valley trace (black lines) and outer contact with undifferentiated plains.

Figure 2 :

 2 Figure 2: Cross cut hummocky terrain at (38°N, 203.6°W) near the Afekan radial labyrinth cluster. Yellow line shows outline, while white fiduciary arrows show remnant patch of labyrinth terrain.

Acknowledgements

This research was supported by the NASA Astrobiology Institute through its JPL-led team entitled "Habitability of Hydrocarbon Worlds: Titan and Beyond" This work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. Government sponsorship is acknowledged.