Some structural properties of a lattice of embedded coalitions

Abstract In this paper we investigate some structural properties of the order on the set of embedded coalitions outlined in the paper de Clippel G. and R. Serrano (2008) “Marginal Contributions and Externalities in the Value.” Econometrica 76: 1413–1436. Besides, we characterize the scalars associated to the basis they proposed of the vector space of partition function form games.


Introduction
A number of partially ordered sets emerge naturally from a finite set. The Boolean lattice and the partition lattice were two remarkable and deeply studied examples. In this paper we consider a poset defined over the set of so-called embedded coalitions. Given a finite set N, an embedded coalition is a pair consisting of a subset of N and a partition of its complement of the subset. The partial order that we consider is indeed a combination of the inclusion and refinement relations of subsets and partitions, respectively. The main goal of our work is to study the Möbius function associated to this poset for its application in Economics.
It is important to point out that we are not the first to study a partially ordered set over the set of embedded coalitions. Myerson (1977) introduced a partial order on this set and Grabisch (2010) studied its associated lattice structure meticulously. Using a number of isomorphisms he derived a Möbius function and applied the results to cooperative game theory. In this paper we aim at following a similar path but from a different starting point. The difference of our paper with respect to Grabisch (2010) lies on the binary relation that we consider. The relation between embedded coalitions that we consider here was implicitly used in de Clippel and Serrano (2008) and formally defined in Alonso-Meijide, Álvarez-Mozos, and Fiestras-Janeiro (2015). According to it, an embedded coalition gets bigger if the subset grows and the partition of its complement gets finer.
The motivation for our study comes from economics. More precisely, from cooperative game theory. A cooperative game is a model that describes situations where a set of agents interact by forming coalitions. Each coalition of agents generates some utility or worth.
CONTACT M. G. Fiestras-Janeiro fiestras@uvigo.es Thus, a game in characteristic function determines the worth of every possible coalition. A more general approach is to allow the worth of a coalition to depend also on how are the rest of agents organized. These situations are covered by games in partition function form. These latter games were first introduced in Thrall and Lucas (1963) and have recently received some attention (see for instance Dutta, Ehlers, and Kar 2010;Álvarez-Mozos and Tejada 2015).
The main subject of study in cooperative game theory is how to share the utility generated from the cooperation. The Shapley value (Shapley 1953) stands as one of the most successful answers to this question. This value was originally defined as the only sharing rule that satisfies a set of reasonable properties or axioms. The problem of extending the Shapley value to games in partition function is intricate and a number of different proposals have been developed to date (Albizuri, Arin, and Rubio 2005;Macho-Stadler, Pérez-Castrillo, and Wettstein 2007;Pham Do and Norde 2007;Dutta, Ehlers, and Kar 2010). We argue that our work may shed some light in understanding the differences between the different proposals and obtaining new characterization results.
It is well known that both, the set of games in characteristic function and the set of games in partition function form, are vector spaces over R. The former vector space is studied in Harsanyi (1959Harsanyi ( , 1963. Harsanyi provided a closed expression for the coefficients of a game (in characteristic function) in a basis, the so-called Harsanyi dividends. These results have contributed to the progress of the theory of cooperative games in a large extent. For games in partition function form de Clippel and Serrano (2008) proposed a basis. In our paper we provide a closed expression for the coefficients of any game in partition function form with respect to this basis.
The rest of the paper is organized as follows. Section 2 introduces some basic concepts and notations. In Section 3, the structure of the poset of embedded coalitions is studied thoroughly. The study of the associated Möbius function is presented in Section 4. Finally, Section 5 presents the application of these results to cooperative games.

Preliminaries
Let (L, ≤) be a partially ordered set with L a finite set and x, y ∈ L. The supremum x ∨ y is an element of L such that x, y ≤ x ∨ y and if z ∈ L satisfies z ≥ x, y, then z ≥ x ∨ y. 1 The infimum x ∧ y is an element of L such that x ∧ y ≤ x, y and if z ∈ L satisfies z ≤ x, y, then z ≤ x ∧ y. 2 A finite lattice is a finite partially ordered set (L, ≤) such that there is a supremum x ∨ y ∈ L and an infimum x ∧ y ∈ L, for every x, y ∈ L. From now on, we assume that (L, ≤) is a finite lattice. Let1 ∈ L be such that x ≤1 for every x ∈ L. We say that1 is the top element. Similarly, the bottom element0 is an element of L such that0 ≤ x for every x ∈ L. A complement of x is an elementx ∈ L, such that x ∨x =1 and x ∧x =0. We say that x is covered by y or y covers x if x < y and there is no z ∈ L \ {x, y} such that A (irreducible) chain C is a totally ordered subset of L, i.e. C = {x 0 , x 1 , . . . , x k } such that x l+1 covers x l , for every l = 0, . . . , k − 1.
• If x, y ∈ L and x ≤ y, we denote by [x, y] L the set of elements z ∈ L such that x ≤ z ≤ y. If no confusion arises, we set [x, y]. Notice that [x, y] is also a lattice. • (L, ≤ ) satisfies the Jordan-Dedekind condition if all chains between any pair of elements have the same length. This common length is called the rank. The height of an element x is the rank of the chains that start at the bottom element and finish at x. The height of the lattice is the rank of every chain that joins the bottom and the top elements.
Apart from the lattice of subsets of a finite set, denoted by (B(N), ⊆), we need to recall some notions related to the partition lattice. Let N be a finite set, |N| = n, and (N) the family of partitions of the set N. Let S ⊆ N and P ∈ (N). We denote by P −S the partition of N \ S given by P −S = {T \ S : T ∈ P} and by P \ R = P \ {R}, for every R ∈ P. If 1 ≤ k ≤ n, the total number of partitions of N with k subsets is the Stirling number of second kind, S n,k . A well-known partial order on (N) is the following: P Q if and only if for every S ∈ P there is some T ∈ Q such that S ⊆ T.
We denote this ordered set by ( (N), ). It is known that ( (N), ) is a lattice. If P, Q ∈ ( (N), ), we denote by P Q the infimum of P and Q; the supremum of P and Q is denoted by P Q. An embedded coalition of N is a pair (S; P) with ∅ = S ⊆ N and P a partition of N \ S, i.e. P ∈ (N \S). If we have the embedded coalition (T; Q) with T = N then, Q = {∅} and we take |Q| = 0. For simplicity we denote by (S; N \S) the embedded coalition (S; {N \S}), for every S ⊆ N. We consider the family of all embedded coalitions of the set N union an additional element ⊥. This set is denoted by EC N . Several partial orders are considered on the family of embedded coalitions of a finite set N. One of them has been studied in

The structure of (EC N , )
In this section we examine some properties of the partially ordered set (EC N , ). By the definition of the order , it is clear that ⊥ ∨ (S; P) = (S; P) and ⊥ ∧ (S; P) = ⊥, for every (S; P) ∈ EC N \ {⊥}. In the next result we obtain the supremum and the infimum when comparing two embedded coalitions both different from ⊥.
As a consequence of Proposition 1, the top of (EC N , ) is (N; ∅). In the following, we denote by the embedded coalition (N; ∅). Proposition 2: Every (S; P) ∈ (EC N , ) is complemented. In fact, given (S; P) ∈ EC N , any embedded coalition (N \ S; M) ∈ EC N is a complement of (S; P).
Next we illustrate these operators.
(1) If (S; P) = ⊥, the number of embedded coalitions that cover (S; P) is n.
(2) If (S; P) = ⊥, the number of embedded coalitions that cover (S; P) is given by Proof: Let (S; P) ∈ EC N .
(1) Let (S; P) = ⊥. Notice that for every We consider the embedded coalition (T; Q) with T = S ∪ P k and Q = P \ P k . It is clear that (S; P) < (T; Q) and there is no embedded coalition in between. Additionally, if we consider every partition Adding up all cases we obtain Expression 2.  is covered by (S; P). The number of the embedded coalitions of this type is given by |P| 2 in Equation (3). Besides, if |S| > 1, we take i ∈ S and consider (S 1 ; P 1 ) = (S \ {i}; P ∪ {{i}}. Clearly, (S 1 ; P 1 ) is covered by (S; P). The number of embedded coalitions of this type is |S|. The first one is obtained by joining two elements of P. The last two are obtained by isolating an element of S. Proposition 5: The set of join-irreducible embedded coalitions is I 1 ∪ I 2 with Proof: First, we prove that every embedded coalition in In Consequently, any embedded coalition in I 1 ∪ I 2 is join-irreducible. It remains to prove that only the embedded coalitions in I 1 ∪I 2 are join-irreducible. Let (S; P) ∈ EC N \ {⊥} be a join-irreducible embedded coalition such that (S; P) ∈ I 1 ∪ I 2 . Then, |N| ≥ 3 because in case |N| = 2, the non-trivial embedded coalitions are given by I 1 ∪ I 2 . We distinguish two cases. First, we consider that |S| ≥ 2. Let us take i, j ∈ S, i = j and P ∈ (N \ S). We take the embedded coalitions (T; nor (U; M) equals (S; P) and this is a contradiction. Second, we consider that |S| = 1. If |P| < 3, then (S; P) ∈ I 1 ∪ I 2 . Then, |P| ≥ 3, and Then, (S; P) is not join-irreducible and this finishes the proof. Thus, we achieve a contradiction. If |P| < n − |S|, then there is some P k ∈ P with |P k | ≥ 2. We distinguish two cases.

Proposition 6: The set of meet-irreducible embedded coalitions is
• There is some P k ∈ P with |P k | > 2. Let i, j ∈ P k , i = j, and consider • There is some P k ∈ P with |P k | = 2 and |P l | ≤ 2 for every P l ∈ P \ P k . Let us take i ∈ P k and j ∈ P l with P l ∈ P \ P k , (S; In both cases, we get a contradiction and the proof is finished. Proposition 8: Let N be a finite set with |N| ≥ 2. The number of elements of height k in EC N is given by Then, the total number of elements is 2n−2 k=1 (k) + 1.

Proof:
We prove the result considering the following cases.
(1) |N| = 2. In this case, the height of the lattice is 2 and EC N ∼ = 2 N . Then, the result is true. Finally, level k = 4 has a unique embedded coalition that corresponds to and coincides with (4) = n k − n + 2 = 3 3 = 1.
• First we analyze the case |T| = 1. Using Proposition 7, we have |Q| • Second, let (T; Q) be an embedded coalition with |T| = 1. We consider two cases.
(a) k ≤ n. Using Proposition 7 and 1 ≤ |Q| ≤ n − |T|, we have |Q| = Adding up all different types of embedded coalitions obtained above and taking into account the cases of |N| ≤ 3, we compute The proof is concluded.
As a consequence of Proposition 8 we characterize the set of atoms and coatoms of (EC N , ). Additionally, notice that all the atoms are join-irreducible elements and all the coatoms are meet-irreducible elements.  Remark 1: In Table 1 we compare the number of embedded coalitions per level according to the ordering 0 and . For each value of n, the first row contains the number of embedded coalitions using 0 and the second row is obtained through the function . If |N| ≥ 3, the lattice (EC N , ) does not belong to any well-known families of lattices as we see in the following remark. Remark 2: Let N a finite set with |N| ≥ 3.
(1) (EC N , ) is not distributive. For instance, let us take a finite set N with |N| ≥ 3.

The Möbius function
In this section we characterize the Möbius function of the lattice of embedded coalitions. First, we recall some well known notions and results about the Möbius function of a lattice. Let (L, ≤) be a finite lattice. The dual of (L, ≤) is (L, ≤ * ) with x ≤ * y if and only if y ≤ x, for every x, y ∈ L. The Möbius function of (L, ≤), μ, is given by for every x, y ∈ L with x ≤ y. The direct product of two finite lattices (L 1 , ≤ 1 ), (L 2 , ≤ 2 ) is the partially ordered set (L 1 × L 2 , ≤) with (x 1 , x 2 ) ≤ (y 1 , y 2 ) if and only if x 1 ≤ 1 y 1 and x 2 ≤ 2 y 2 . It holds that (L 1 ×L 2 , ≤) is also a finite lattice. In this section the Möbius function of (B(N), ⊆) and the Möbius function of ( (N), ) play an important role. The Möbius function of (B(N), ⊆) is given byμ 1 (S, T) = ( − 1) |T|−|S| , for every S ⊆ T ⊆ N. The Möbius function of ( (N), ) is given byμ 2 (P, Q) = (−1) |P|−|Q| (m 1 −1)! · · · (m |Q| −1)! with |Q| i=1 m i = |P|, for every P, Q ∈ (N) with P ≺ Q. Next we recall some well-known facts that we use in the proofs of our results.
The second and the third cases follow immediately. This finishes the proof.
Henceforth we omit the subscript corresponding to the set when we consider a direct product of lattices. Next we illustrate the result in Proposition 10.  Next we characterize the Möbius function of (EC N , ). Notice that the embedded coalitions of type (S; N \ S) with S ⊆ N, S = ∅ can be written as Then, μ(⊥, (S; P)) = A⊆{({i};N\{i}): i∈S} ( − 1) |A| = ( − 1) |S| . Thus, the characterization of the Möbius function of (EC N , ) is not a trivial task. We do that in the next result.   It remains to prove Item 3. We check that the meet of the coatoms of the lattice [(S; P), (T; Q)] is different from (S; P) and apply Item 2 in Proposition 9 to derive the result since [(S; P), (T; Q)] is also a lattice with0 = (S; P) and1 = (T; Q). First, notice that any coatom (every embedded coalition covered by (T; Q)) is given by In addition, we have (S; P) (S; H), but the partition H is different from P because there is some i ∈ T \ S such that {i} ∈ P but {i} ∈ H. Then, using Item 2 in Proposition 9, we obtain μ((S; P), (T; Q)) = 0.
Example 6: In this example we will illustrate the proof of Item 3 in Proposition 11. We

Partition function form games
Finally in this section, we use our previous results to characterize the scalars related to the basis proposed in de Clippel and Serrano (2008)  In particular, Example 7: Let us consider |N| = 3. In Figure 5 we depict the lattice of the embedded coalitions for n = 3. Next we obtain the non null values of its Möbius function.    , j, k ∈ N, i = j, k, j = k, α ({i,j};N\{i,j}) We apply this to the example in Grabisch (2010, 486). The game is Grabisch (2010) showed that there is no additive partition function form game in (EC N , 0 ) different from v(S; P) = 0 for every (S; P) ∈ EC N if |N| ≥ 3. On the contrary we show that any additive TU game is also additive in (EC N , ). In lattice theory the concept of a valuation corresponds to the concept of an additive function in the setting of partition function form games. Let (L, ≤) be a finite lattice. A valuation is a real-valued function f on L satisfying for every finite set I and {x i : Proposition 13: There are non-constant monotone valuations on (EC N , ).
If v(⊥) = 0, the valuation defined above can be seen as an additive TU game. We can choose adequate non-null values for v(⊥) and obtain different valuations. The valuation defined above is not strictly monotone because v(S; P) = v(S; Q) for every P, Q ∈ (N \S) and ∅ = S N.

Concluding remarks
We study some structural properties of the set of embedded coalitions endowed with the partial order outlined in de Clippel and Serrano (2008). In particular we prove that this partial ordered set is a lattice. Moreover, we characterize the set of atoms, the set of coatoms, the join or meet-irreducible elements, as well as the number of embedded coalitions that cover any other embedded coalition, the number of embedded coalitions covered by any other embedded coalition, the height of every irreducible chain and the number of embedded coalition per level. Besides, we obtain the Möbius function of this partial ordered set. This finding allows us to calculate explicitly the scalars related to the basis of the vector space of partition function form games that de Clippel and Serrano (2008) used. We expect that our results can contribute to a better understanding of some values proposed in the context of partition function form games. Moreover, we can also define new values in this context using some properties that appear in this paper.