Some new

A Generalized Additive Game (GAG) [9] is a Transferable Utility (TU) game ( N, v ) where each player in N is provided with an individual value and the worth v ( S ) of a coalition S ⊆ N is obtained as the sum of the individual values of players in another subset M ( S ) ⊆ N . Based on conditions on the map M (which associates to each coalition S a set of beneﬁcial players M ( S ) not necessarily included in S ), in this paper we characterize classes of GAGs that satisfy properties like monotonicity, superadditivity, (total) balancedness, PMAS-admissibility and supermodularity, for all nonnegative vectors of individual values. We also illustrate the application of such conditions on M over particular GAGs studied in the literature (e.g., glove games [23], generalized airport games [19], ﬁxed tree games [4], link-connection games [18, 16], simple minimum cost spanning tree games [20, 26] and graph coloring games [10, 11]).


Introduction
Generalized additive games (GAGs) are Transferable Utility (TU) games where the worth of any coalition of players can be computed as a sum of individual contributions. As shown in [9], many TU games from the literature can be represented as GAGs, like, for instance, airport games [14,15], connectivity games [2,13], argumentation games [5], centrality games [25,1], peer games [7], games on mountain situations [17], etc.
A basic ingredient for GAGs is the so called coalitional map [9], that specifies the set of friends (or contributors) of a coalition S ⊆ N , where N = {1, . . . , n} is a finite set of players. Given a coalitional map and a vector of n nonnegative real numbers representing the individual contribution of players in N , the associated GAG assigns to every coalition S ⊆ N the sum of the contributions over the set of friends of S. In this paper we study the effect of the combination of four properties for coalitional maps on the associated GAGs corresponding to any vector of nonnegative individual contributions. To be more specific, the first property we introduce is a monotonicity condition: a coalitional map is said monotonic if the friends of any coalition S ⊆ N are also friends of any coalition T containing S. Differently, a coalitional map is said proper if any two disjoint coalitions S, T ⊆ N , S ∩ T = ∅, have no friends in common. The third property deals with the veto-behavior of players. More precisely, a coalitional map is said veto-rich if for any player i ∈ N , either i is not a friend of any coalition, or i is a friend of the grand coalition N and, at the same time, the intersection of coalitions having i as a friend is non-empty. Finally, a coalitional map is supermodular if the intersection of the set of friends of two coalitons S, T ⊆ N coincides with the set of friends of their intersection. We first prove that these properties for coalitional maps characterize some interesting classes of GAGs as follows: 1) monotonicity of a coalitional map is equivalent to monotonicity of the associated GAGs; 2) monotonicity and properness of a coalitional map is equivalent to superadditivity of the associated GAGs; 3) veto-richness of a coalitional map is equivalent to balancedness of the associated GAGs; 4) monotonicity and vetorichness of a coalitional map is equivalent to both total balancedness and admissibility of a population monotonic allocation scheme (PMAS) [24] of the associated GAGs; 5) supermodularity of a coalitional map is equivalent to convexity of the associated GAGs. Then, we use these characterizations to analyze several classes of TU games from the literature, with a particular focus on Operations Research (OR) games [6]. In particular, we consider (weighted) glove games [23,27], (generalized) airport games [19], fixed tree games [4], link-connection games [18,16], simple minimum cost spanning tree games (MCST) [20,26] and (weighted) coloring games [10,11].
A main advantage of representing and studying a TU game as a GAG lies in the combinatorial nature of the four properties proposed for coalitional maps. In all classes of TU games from the literature considered in this paper, it is straightforward to verify whether the properties of monotonicity, properness, veto-richness and supermodularity hold for the associated coalitional maps. So, the method of proof that we propose in this paper is much less time-consuming than the techniques adopted in the related literature. Moreover, some nice properties of certain well-known TU games, can be easily generalized over larger domains, or they can be used to investigate in more details some particular sub-classes of games. As a simple example, consider the case of airport games [14,15], and the corresponding costsaving game where the costs of the airport's runway is supposed to increase with the length of the landing strip, i.e. the cost vector w belongs to the convex cone K 1 ⊂ IR N + defined by K 1 = {w ∈ IR N + : w 1 ≤ w 2 ≤ · · · ≤ w n }. Looking at the corresponding cost-saving game, and using our characterizations, the properties of monotonicity, superadditivity, balancedness and PMAS-admissibility can be extended to airport games associated to any cost vector in IR N + , and in particular to the larger cone K 2 = {w ∈ IR N + : w i ≤ w n for every i ∈ {1, . . . , n − 1}}, whose family of associated GAGs coincides with the class of generalized airport games introduced in [19].
We also stress the fact that all of our results hold for profit games, i.e. for GAGs whose individual contributions can be interpreted as revenues or profits. We adopted this convention for consistency with the interpretation of a coalitional map as a specification of friends of coalitions, and for coherence with the more popular interpretation of TU games as revenue-sharing situations, as presented in the original paper [9]. On the other hand, the majority of games we consider in this paper deal with cost sharing situations. Following a standard approach, we consider an associated cost-saving game, that is a TU game where the worth of each coalition S represents the amount that coalition S saves by cooperation, and is computed as the difference between the sum of the costs of singleton coalitions formed by the members of S minus the total cost of coalition S. In alternative, another standard way to transform a cost game in a profit one, is by means of the associated dual game, where a coalition S gets the rest of the cost of the grand coalition N after the complement of coalition S pays its entire cost in the original game. So, the dual game of a cost game can be interpreted as the opportunity for players in S to fully profit of the contribution of players outside S.
The road-map of the paper is as follows. We start with some preliminary notions and notations on game theory and graph theory in Section 2. In Section 3 we introduce some properties for coalitional maps and we illustrate some characterizations of GAGs using (combinations of) these properties. Then, in Section 4, we analyze several classes of TU games from the literature that can be represented as GAGs and can be studied in view of the results presented in Section 3. Specifically, we consider weighted glove games in Section 4.1, generalized airport games in Section 4.2, fixed tree games in Section 4.3, link connection games in Section 4.4, simple MCST games in Section 4.5 and (weighted) coloring games in Section 4.6. Section 5 concludes.

Game theory
A Transferable Utility (TU) game (or, simply, a game) is a pair (N, v), where N = {1, . . . , n} denotes the set of players and v : 2 N → IR is the characteristic function that maps each element of 2 N (the set of all subsets of N ) to a real number (by convention, v(∅) = 0). For each S ∈ 2 N , v(S) represents the worth or profit of coalition S. In the following, we often identify a game (N, v) with its characteristic function v (if the set N is fixed) , 1} (i.e., the worth of every coalition is either 0 or 1) for each S ∈ 2 N and v(N ) = 1 is said a simple game.
Given a game (N, v), an allocation is a vector x ∈ IR N , and an allocation x ∈ IR N is in the core C(v) of game v if it is efficient (i.e., i∈N x i = v(N )) and stable (i.e., i∈S x i ≥ v(S) for all non-empty coalitions S ∈ 2 N ). So, A game v such that C(v) = ∅ is called balanced. Given a game (N, v) and a coalition S ∈ 2 N , we denote by (S, v |S ) the subgame of v restricted to coalition S such that v |S (R) = v(R) for each R ⊆ S. A game v such that C(v |S ) = ∅ for all S ∈ 2 N is called totally balanced.
A population monotonic allocation scheme or PMAS [24] of the game (N, v) is a scheme {x S,i } S∈2 N \{∅},i∈S with the properties: A PMAS provides an allocation vector for every coalition in a monotonic way, i.e. the value allocated to some player increases if the coalition to which he belongs becomes larger. If a game v has a PMAS, then it is said to be PMAS-admissible. It is easy to check that a PMAS {x S,i } S∈2 N \{∅},i∈S provides a core element for the game and all its subgames, i.e. the allocation (x S,i ) i∈S ∈ C(v |S ) for all S ∈ 2 N , S = ∅. Therefore, a game admitting a PMAS is also totally balanced. We now introduce some basic definitions on GAGs from [9]. A coalitional map is a map M : 2 N → 2 N such that M(∅) = ∅. The elements of M(S), for each S ∈ 2 N , are called friends or contributors of S via M. Definition 2.1 A generalized additive situation (GAS) [9] is a triple (N, M, w), where N is a finite set of players, M is a coalitional map and w ∈ IR N + is a vector of nonnegative real numbers w i ≥ 0, for any i ∈ N . The corresponding generalized additive game (GAG) So, in order to compute the worth of a coalition S ∈ 2 N in a GAG (N, v M,w ), we first select the friends of S, i.e. the players in M(S), and then we add their individual values according to w. Given a coalitional map M on 2 N , the class of all GAGs (N, v M,w ), for any w ∈ IR N + , is denoted by G M . It is straightforward to see that a GAG (N, v M,w ) is a nonnegative combination of simple games where for every i ∈ N the simple game ( for every S ∈ 2 N . So the winning coalitions in (N, v M,i ) are precisely those coalitions that select player i. Note that (N, v M,i ) can also be regarded as the GAG corresponding to coalitional map M and unit weight vector e i ∈ IR N + (e i i = 1 and e i j = 0 if j = i). For each S ∈ 2 N we denote by (S, (v M,w ) |S ) and (S, (v M,i ) |S ) the restriction to S of games (N, v M,w ) and for every S ∈ 2 N . 3

Graph theory
An (undirected) graph is a pair Γ = (V, E), where V is a set of vertices or nodes and E is a set of edges e of the form {i, j} with i, j ∈ V , i = j. The complete graph on a set V of vertices is the graph (V, E V ), where E V = {{i, j}|i, j ∈ V and i = j}. A path between two nodes i, j ∈ V in a graph Γ is a sequence of nodes (i 0 , i 1 , . . . , i k ), where i = i 0 and j = i k , k ≥ 1, such that {i s , i s+1 } ∈ E for each s ∈ {0, ..., k − 1} and such that all these edges Two nodes i, j ∈ V are said to be connected in Γ if i = j or there exists a path between i and j in Γ. A graph Γ is connected if for each i, j ∈ V with i = j there exists a path between i and j in (V, E). A connected component of Γ is a maximal subset of V with the property that any two nodes in this subset are connected. A graph where all paths are without cycles is called a forest, and a forest that is also connected is called a tree.
Given an undirected graph Γ = (V, E), the subgraph Γ |S = (S, E S ) induced by S ∈ 2 V is a graph with set S as set of vertices and where E S = {{i, j} ∈ E : i, j ∈ S}. A clique in Γ is a subset S ∈ 2 V such that Γ |S is complete.

Some characterizations
We start with the definition of some properties for coalitional maps. Properties in Definition 3.1 are naturally recast in terms of mechanisms aimed to select friends or contributors of coalitions. A coalitional map is said monotonic if it selects friends of coalitions in a monotonic way, i.e. each friend of a coalition is also a friend of any superset of that coalition. A coalitional map is said proper if two disjoint coalitions do not share any friend in common. A coalitional map is said veto-rich if it partitions the elements of N in two particular subsets: the first subset includes those players that are never selected as friends of any coalition, whereas each player of the second subset is a friend of the grand coalition and, in addition, all coalitions with this player as a friend have a non-empty intersection. Finally, a coalitional map is supermodular, if the set of friends in common between any two coalitions coincides with the set of friends selected for their intersection. A natural question to ask is which properties for a coalitional map M induce nice game theoretical properties on the corresponding family of GAGs G M . The following theorems provide necessary and sufficient conditions on a coalitional map M to generate associated GAGs (N, v M,w ) that are, respectively, monotonic, superadditive, (totally) balanced, PMAS-admissible and convex for every weight vector w ∈ IR N + . The first theorem generalizes a result in [9] stating that a monotonic coalitional map induces a monotonic GAG (see Proposition 1 in [9]). Also the following theorem generalizes a result in [9] (see Proposition 2 in [9]).  The next theorem deals with the notion of balancedness of TU games. Here, the veto-richness property for coalitional maps plays a central role. There is a coalition S ∈ 2 N such that i ∈ M(S) and j ∈ N \S.
Since k∈S x k ≥ v M,i (S) = 1 we get k∈N \S x k ≤ 0, so x k = 0 for every k ∈ N \S. In particular we have x j = 0. As this is true for every j ∈ N we have a contradiction. Therefore ∩{S : i ∈ M(S)} = ∅, hence M is veto-rich.
It is well known that the class of TU games that admit a PMAS is a proper subset of the class of totally balanced TU games, i.e. there exist totally balanced games (e.g., glove games) that do not admit a PMAS in general [24]. Given a coalitional map M, the next theorem shows that, whenever all GAGs (N, v M,w ), for all w ∈ IR N + , are totally balanced, then they equivalently admit a PMAS, and this is also equivalent with the fact that M is veto-rich and monotonic.
Theorem 3.6 The following statements are equivalent: I) M is veto-rich and monotonic; Proof "(I) ⇒ (II)" Suppose M is veto-rich and monotonic. Let i ∈ N . If i / ∈ M(S) for every S ∈ 2 N then (N, v M,i ) is the zero game, and hence v M,i admits a PMAS, because the scheme {x S,k } S∈2 N \{∅},k∈S with x S,k = 0 for every S ∈ 2 N \{∅} and k ∈ S is a PMAS.
If i ∈ M(N ) and ∩{S : i ∈ M(S)} = ∅ then v M,i (N ) = 1 and take j ∈ ∩{S : i ∈ M(S)} = ∩{S : v M,i (S) = 1}. So j is a veto player in (N, v M,i ). We are going to show that the scheme x = {x S,k } S∈2 N \{∅},k∈S such that: • x S,k = 1, if k = j, and x S,k = 0, otherwise, for any S with i ∈ M(S) and k ∈ S, • x S,k = 0, for any S with i / ∈ M(S) and k ∈ S, is a PMAS for v M,i . In order to prove it, first notice that for all S, we have k∈S in the definition of PMAS, see Section 2). In order to prove monotonicity of the scheme x (condition (ii) in the definition of PMAS), let S ⊆ T and k ∈ S. If x S,k = 0 then obviously x S,k ≤ x T,k . If x S,k = 1, then i ∈ M(S) and k = j. By monotonicity of M we have that Since (S, v M,i ) admits a PMAS for every i ∈ N , any nonnegative combination of the games (N, v M,i ), i ∈ N , admits a PMAS as well, so (N, v M,w ) has a PMAS for every w ∈ IR N + .
"(II) ⇒ (III)" It follows immediately from the fact that if a game admits a PMAS then it is also totally balanced.
is also balanced for every w ∈ IR N + and hence by Theorem 3.5, M is veto-rich. Moreover, it is well known that every totally balanced game is superadditive. Then, by Theorem 3.4, it directly follows that M is also monotonic.
The next theorem shows that supermodularity of a coalitional map is equivalent with the eponymous property for GAGs on the full domain of vectors of nonnegative contributions.
Moreover, let S, T ∈ 2 N and consider a vector w ∈ IR N + such that w i = 1, if i ∈ M(S) ∩ M(T ), and w i = 0, otherwise. We have that In view of the results provided in this section, we can resume the inclusion relations among families of coalitional maps M and families classes of GAGs G M as shown in Figure 1 (recall that given a coalitional map M on 2 N , the class G M is defined as the set of all GAGs (N, v M,w ) for every w ∈ IR N + ).

Analysis of special classes of GAGs from the literature
In this section, we illustrate some consequences of the characterizations provided in the previous section on some well known classes of TU games. As we mentioned earlier, all of these classes, with the only exception of glove games, are cost games. For such cost games, we will make use of the corresponding cost saving and dual games, which are interpreted as profit games.
First, note that all the definitions provided in Section 2 apply to TU games where v(S), for any coalition S ∈ 2 N , represents a profit, whereas some of the inequality signs should be reversed when v(S) represents a cost. So reversing the inequalities in relations (1) and (2) (or in (3)) we obtain, respectively, the conditions of subadditivity and submodularity (or concavity) for game v and, reversing the inequalities in the definition of core and PMAS previously introduced, we obtain the corresponding definitions of core and PMAS for cost games. for each coalition S ∈ 2 N , where the rest c * (S) obtained from the cost of the grand coalition N after the complement of coalition S pays its entire cost in the original game is also interpreted as a profit of coalition S. We now introduce some useful properties for cost games and the corresponding cost saving games. (N, c) be a cost game and let (N, v c ) be the corresponding cost saving game. Then the following statements hold true:

Proposition 4.1 Let
(iv) c admits a PMAS iff v c admits a PMAS.
Proof Proof of statement (i): We have that for all S, where the second equivalence relation follows from the fact that i∈S c({i}) + i∈T c({i}) = i∈S∩T c({i})+ i∈S∪T c({i}), and where the first and the last inequality are the definition of supermodularity for v c and of submodularity for c, respectively. Proof of statement (ii): It follows from statement (i), considering coalitions S, T ∈ 2 N such that S ∩ T = ∅. Proof of statement (iii): Consider the allocations x, y ∈ IR N such that y i = c({i}) − x i for each i ∈ N . We have that and for each S ∈ 2 N with S = ∅. So, by relations (10) and (11), we have proved that x is efficient and stable for c (equivalently, c is balanced) iff y is efficient and stable for v c (equivalently, v c is balanced). Moreover, for each S ∈ 2 N , S = ∅, we have v c |S = v c |S , i.e. the subgame of v c restricted to S coincides with the cost saving game v c |S corresponding to c |S (which is the subgame of c restricted to S). Then, again from relations (10) and (11), we have that c |S is balanced iff v c |S is balanced, for each S ∈ 2 N , S = ∅ (equivalently, c is totally balanced iff v c is totally balanced). Proof of statement (iv): Consider the schemes {x S,i } S∈2 N \{∅},i∈S and {y S,i } S∈2 N \{∅},i∈S such that y S,i = c({i}) − x S,i for all S ∈ 2 N \{∅} and i ∈ S. Similar to relation (10), it is easy to verify that i∈S x S,i = c(S) iff i∈S y S,i = v c (S) for all S ∈ 2 N \{∅}. To prove the monotonicity condition for the definition of PMAS, notice that for all S, T ∈ 2 N \{∅} and i ∈ N with i ∈ S ⊂ T . So, we have proved that {x S,i } S∈2 N \{∅},i∈S is a PMAS of c iff {y S,i } S∈2 N \{∅},i∈S is a PMAS of v c .
Well-known results for dual games are stated in the following proposition. iii) c is submodular iff c * is supermodular.
Proof A proof of these statements can be found in the literature, for instance, in the book [3].
The remaining of the paper is devoted to the analysis of several classes of TU games from the literature in view of the characterizations provided in the previous section.

Weighted glove games
Given a partition {L, R} of the set of players N and a weight vector w ∈ IR N + (each player i in L owns w i left gloves, each player j in R owns w j right ones), we define a weighted glove game as the TU game (N, v) such that v(S) = min{ i∈S∩L w i , j∈S∩R w j }, representing the profit obtained by members in S selling their pairs of gloves (sold at selling price of 1). Note that players are allowed to have a non-integer number of gloves. If w i = 1 for every i ∈ N the game is a standard glove game [23]. Standard glove games are known to be totally balanced. Generalizing the proof in an obvious way shows that weighted glove games are totally balanced as well: for every S ∈ 2 N \{∅} allocate w i to every i ∈ S ∩ L and 0 to every j ∈ S ∩ R if i∈S∩L w i ≤ j∈S∩R w j , otherwise allocate 0 to every i ∈ S ∩ L and w j to every j ∈ S ∩ R. (Weighted) glove games do not have to be PMAS-admissible: in [24] it is shown that the standard glove game with two players owning a left glove and two players owning a right glove does not have a PMAS.
As suggested in [9], a weighted glove game (N, v) can be described as a GAG using the coalitional map M defined by for all S ∈ 2 N . Note that this coalitional map actually depends on the weight vector w. It is not possible to construct a 'universal' map M that generates all weighted glove games for all possible weight vectors. Such an M would be, according to Theorem 3.6, veto-rich and monotonic and hence, according to the same theorem, all weighted glove games would admit a PMAS. However, we can use the results of Section 3 in order to characterize the subclass of supermodular weighted glove games. We need the following theorem.  N, v) is supermodular if and only if L contains precisely one player l * and w l * ≥ j∈R w j or R contains precisely one player r * and w r * ≥ i∈L w i .
Proof "⇐". Suppose L contains precisely one player l * and w l * ≥ j∈R w j . Let i ∈ N and S ⊂ T ⊆ N \{i}.
. In case R contains precisely one player r * and w r * ≥ i∈L w i the argument is similar. "⇒". Suppose (N, v) is supermodular. First, assume that both L and R have at least two players. Choose a, b ∈ L, a = b, and c, d ∈ R, c = d.
Hence min{w a , w c } = w c . In a similar way we find min{w a , w d } = w d . Now w a ≥ min{w a , w c + w d } ≥ min{w a , w c } + min{w a , w d } = w c + w d . Using a symmetry argument we get w b ≥ w c + w d , w c ≥ w a + w b and w d ≥ w a + w b as well. Hence w a ≥ w c + w d > w c ≥ w a + w b > w a . A contradiction. So |L| = 1 or |R| = 1. Without loss of generality assume that |L| = 1. If |R| = 1 we are done so assume that |R| ≥ 2. Let L = {l * } and let r * ∈ R be such that w r * = min j∈R w j . We have v(N ) − v(N \{r * }) ≥ v({l * , r * }) − v({l * }) so min{w l * , j∈R w j } − min{w l * , j∈R\{r * } w j } ≥ min{w l * , w r * } − 0, so min{w l * , j∈R w j } ≥ min{w l * , j∈R\{r * }wj } + min{w l * , w r * }. Repeating the arguments above we get w l * ≥ min{w l * , j∈R w j } ≥ min{w l * , j∈R\{r * } w j } + min{w l * , w r * } > min{w l * , j∈R\{r * } w j }. So, min{w l * , j∈R\{r * } w j } = j∈R\{r * } w j . In a similar way we find min{w l * , w r * } = w r * . So w l * ≥ min{w l * , j∈R w j } ≥ min{w l * , j∈R\{r * } w j } + min{w l * , w r * } = j∈R\{r * } w j + w r * = j∈R w j .
The if-part of Theorem 4.3 can be shown by using Theorem 3.7 as well. Suppose {L, R} is a partition of the player set N with |L| = 1. Let l * be the unique element of L. Define the coalitional map M : 2 N → 2 N by M(S) = S ∩ R if l * ∈ S and M(S) = ∅ otherwise. It is straightforward to check that M is supermodular. So for every w ∈ IR N + the game v M,w is supermodular, according to Theorem 3.7. The game v M,w is the weighted glove game with weight vector w if w l * ≥ j∈R w j , where l * is the unique element of L.

Generalized airport games
We recall the definition of airport games [15].
Given an airport game (N, c) associated with N and w the associated cost saving game (N, v c ) is given by for every S ∈ 2 N . Note that this cost saving game in fact is a GAG (N, v M,w ) with coalitional map M defined by for every S ∈ 2 N . Proposition 4.5 Let M be the coalitional map as defined in (16). Then M is monotonic, proper and veto-rich.
Proof Let S, T ∈ 2 N with S ⊆ T . Let i ∈ M(S). Then i ∈ S ⊆ T and i < j(S) ≤ j(T ), so i ∈ M(T ). This shows that M is monotonic. Properness of M follows from the fact that M(S) ⊆ S for every S ∈ 2 N (we refer to Remark 3.2). Veto-richness follows from monotonicity and also from the fact that M(S) ⊆ S for every S ∈ 2 N : if i ∈ M(S) for some S we have, using monotonicity, that i ∈ M(N ) and i ∈ ∩{T : i ∈ M(T )}.
The following example illustrates the fact that the coalitional map M as defined in (16) is not supermodular in case n ≥ 3. The following proposition is a direct consequence of Theorems 3.3, 3.4, 3.5, 3.6 and Proposition 4.5.
Proposition 4.7 Let M be the coalitional map as defined in (16). Then (N, v M,w ) is monotonic, superadditive, (totally) balanced and PMAS-admissible for every w ∈ IR N + .
In case the weight vector w belongs to the cone K 1 ⊂ IR N + defined by K 1 = {w ∈ IR N + : w 1 ≤ w 2 ≤ · · · ≤ w n } then, as remarked before, (N, v M,w ) is the cost saving game corresponding to an airport game and hence Proposition 4.7 provides well-known results concerning airport games. On the somewhat larger cone K 2 = {w ∈ IR N + : w i ≤ w n for every i ∈ {1, . . . , n − 1}} the games (N, v M,w ) coincide with the generalized airport games as introduced in [19].  Proof The proof follows directly from the properties of cost saving games for (generalized) airport games and Proposition 4.1.
Airport games are also known to be submodular (equivalently, by Proposition 4.1, the associated cost saving games are supermodular). However, we are not able to show this result using Theorem 3.7 as M is not supermodular in case n ≥ 3. So, according to Theorem 3.7, we can conclude that not for every w ∈ IR N + the corresponding generalized additive game (N, v M,w ) is convex. This observation was already made in [19] where an example of a generalized airport game was given which is not concave (and hence the corresponding cost saving game is not convex). A natural question now is the following: is it possible to formulate conditions on M such that (N, v M,w ) is convex for every w ∈ K 1 ? The answer is yes. Using (4) and the fact that the cone K 1 is generated by the vectors (1, 1, . . . , 1, 1 In case M is defined by (16)  An alternative way to prove submodularity (as well as monotonicity and balancedness) of airport games, is using the coalitional map for dual fixed tree games as introduced in the next section (see in particular Remark 4.14).

Fixed tree games
A fixed tree situation or maintenance situation [4] is a tuple (N, (V, E), t), where N is a finite set of players; Γ = (N , E) is a rooted tree with N = N ∪ {0} and 0 is the root of the tree; t ∈ IR E + is the cost function. Since nodes in N are connected to the root 0 by exactly one path, a natural orientation of edges is defined by the partial preorder on the vertices such that i j for each i, j ∈ V if and only if the unique path from i to 0 passes through j (with i i). For each i ∈ N we define the set of predecessors of i in Γ as the set P Γ i = {j ∈ N : i j}, and the set of edges among the vertices in P Γ i is denoted by E Γ i = {{j, k} ∈ E : j, k ∈ P Γ i }. The fixed tree game, associated with a fixed tree situation (N, (N , E), t), is the cost game (N, c), where each coalition S ∈ 2 N must support the (minimal) cost of maintaining all paths from players in S to 0, precisely: Given a fixed tree game (N, c) associated with (N, (N , E), t), the corresponding dual game (N, c * ) is defined as: for every S ∈ 2 N . In this case, the dual game c * coincides with the GAG (N, v M,w ) with w i = t {i,p(i)} , where p(i), for each i ∈ N , is the immediate predecessor of i on the unique path from i to 0 (i.e., {i, p(i)} ∈ E Γ i ) and the coalitional map M is defined as follows: for every S ∈ 2 N . Proposition 4.10 Let Γ = (N , E) be a rooted tree and let M be the coalitional map as defined in (20). Then M is monotonic, proper, veto-rich and supermodular.
Proof In order to prove that M is monotonic, consider any two coalitions S, T ∈ 2 E with S ⊆ T . Then, N \ S ⊇ N \ T , implying that the set of predecessors of members in N \ T are included in the set of predecessors of members . Properness of M follows from the fact that M(S) ⊆ S for every S ∈ 2 N (see also Remark 3.2). In fact, let S ∈ 2 N and let j / ∈ S. Then j ∈ N \ S, so j ∈ ∪ i∈N \S P Γ i , and j / ∈ N \ ∪ i∈N \S P Γ i = M(S). This means that if j ∈ M(S), then j ∈ S, and we have proved that M(S) ⊆ S for every S ∈ 2 N .
Veto-richness follows from monotonicity and also from the fact that M(S) ⊆ S for every S ∈ 2 N : if i ∈ M(S) for some S we have, using monotonicity, that i ∈ M(N ) and, from the fact that M(S) ⊆ S, that i ∈ ∩{T : i ∈ M(T )}. We Then, it follows that i ∈ M(S ∩ T ) too, which concludes the proof.

Link connection games
We recall the definition of link connection games, as introduced in [16] and, as a special case of a broader class of games on matroids in [18]. Let (V, E) be an undirected graph with vertex set V and edge set E and let w ∈ IR E + . The link connection game associated with (V, E) and w is the cost game (E, c), where the set of edges is the set of players, and the cost of a coalition S ∈ 2 E is the minimal cost of a set of edges T ⊆ S such that (V, T ) and (V, S) have the same connected components.
for every S ⊆ E. We will show that this cost saving game is in fact also a GAG (N, v M,w ). First, we construct the coalitional map M : 2 E → 2 E for some undirected graph (V, E). We start with listing the edges in E in some order E = {e 1 , e 2 , e 3 , . . . , e |E| } (not yet having a cost vector w ∈ IR E + in mind). For every S ⊆ E an edge e k ∈ S, (k ∈ {1, . . . , |E|}) is called superfluous in S if it forms a cycle with its predecessors in S, more precisely, if (V, {e 1 , . . . , e k−1 } ∩ S) and (V, {e 1 , . . . , e k } ∩ S) have the same connected components. Now M selects the collection of superfluous edges in any coalition: for every S ⊆ E.

Proposition 4.17
Let M be the coalitional map as defined in (23). Then M is monotonic, proper and veto-rich.
Proof Let S, T ∈ 2 E with S ⊆ T . Let e ∈ M(S). Then e forms a cycle with his predecessors in S, so definitely also a cycle with his predecessors in T . Therefore e ∈ M(T ) which shows that M is monotonic. Properness of M follows from the fact that M(S) ⊆ S for every S ∈ 2 E (see Remark 3.2). Veto-richness follows from monotonicity and also from the fact that M(S) ⊆ S for every S ∈ 2 E : if e ∈ M(S) for some S we have, using monotonicity, that e ∈ M(N ) and e ∈ ∩{T : e ∈ M(T )}.
In the following example we will show that it is possible that the coalitional map M as defined in (23) is supermodular, but that this is not necessarily true. 14 The following proposition is a direct consequence of Theorems 3.3, 3.4, 3.5, 3.6 and Proposition 4.17.

Proposition 4.19
Let (V, E) be an undirected graph and M the coalitional map as defined in (23). Then (E, v M,w ) is monotonic, superadditive, (totally) balanced and PMAS-admissible for every w ∈ IR E + . Now, we will show that the cost saving games (E, v c ) as defined in (22) and corresponding to link connection games as defined in (21), form a subset of GAGs (E, v M,w ) with M specified by (23). Let (E, c) be the link connection game associated with undirected graph (V, E) and cost vector w ∈ IR E + . Choose the ordering of the edges in E according to increasing costs, i.e. E = {e 1 , e 2 , e 3 , . . . , e |E| } such that w e1 ≤ w e2 ≤ · · · ≤ w e |E| , and let M be the corresponding coalitional map. Let S ⊆ E. The graph (V, S) partitions the vertex set V into components. Some components may be singletons, some may be trees and the other components are connected components containing cycles. In order to find a subset T ⊆ S of minimal cost that results in the same partition of V into components we can use the well-known algorithm of Prim: reduce any component in (V, S) with a cycle to a tree by removing edges that form a cycle with the cheaper edges in S. Since we have chosen the order of E with respect to increasing costs this process boils down to removing the superfluous edges in S, i.e. removing the edges in M(S). So an optimal network for coalition S is (V, S\M(S)) and the cost saving, going from (V, S) to (V, S\M(S)), is equal to e∈M(S) w e = v M,w (S). As this is true for every S ⊆ E we get (E, v c ) = (E, v M,w ). As a consequence of Proposition 4.19 we obtain the following one.  Proof The proof follows directly from the properties of cost saving games corresponding to link connection games and by Proposition 4.1.

Simple MCST games
We recall the definition of minimum cost spanning tree (MCST) game, as introduced in [19,26]. First, we need to introduce some graph notions specific for this section. Given a finite set N = {1, . . . , n} and a source denoted by 0, let (N , E N ) be an undirected complete graph with vertex set N = N ∪ {0} and edge set E N and, let z ∈ IR E N + be a vector of nonnegative weights on the edges in E N . An undirected graph (S , T ), T ⊆ E S , is a spanning network on S = S ∪ {0}, S ∈ 2 N , if for every i ∈ S there is a path in (S , T ) from i to the source. For any S ∈ 2 N , it is possible to determine at least one spanning tree on S for z, i.e. a spanning network (S , T ) (without cycles) on S , of minimum cost z(T ) = e∈T z e , which is called an MCST on S for z.  (z, S ). Clearly, the collection of (z, S )-components forms a partition of S which is denoted by W |S . As shown in [8,20,26] we have that c z (S) = n(z, S ) − 1 for every S ∈ 2 N \{∅}. Given a simple MCST game (N, c z ) associated with (N , E N ) and z ∈ {0, 1} E N , the corresponding cost saving for every S ∈ 2 N . Note that this cost saving game (N, v cz ) is in fact a GAG (N, v M,w ) with w i = 1 for each i ∈ N and with coalitional map M defined by where, for every S ∈ 2 N , I S is the set of nodes in S such that z({i, 0}) = 1, and j(T ) = max{j : j ∈ T } for all T ∈ W |S . In fact, it is sufficient to notice that i∈S z {i,0} = |I S | (by definition of I S ), and that all the elements in the (z, S )-connected components not including 0 belong to I S , and then we can obtain v cz (S) just counting the element in I S minus exactly one element of each (z, S )-connected component T ∈ W |S not containing 0 (for instance, the one with largest index j(T )).

Proposition 4.24
Let M be the coalitional map as defined in (27). Then M is monotonic, proper and veto-rich.
Proof Let S, T ∈ 2 N with S ⊆ T . Let i ∈ M(S). Then i ∈ I S ⊆ S and i ∈ I T ⊆ T . This means that there exists C ∈ W |S such that i ∈ C and K ∈ W |T such that i ∈ K. Note that C ⊆ K. If 0 ∈ K, then i ∈ M(T ). Now, suppose that 0 / ∈ K (so 0 / ∈ C). Then i < j(C) ≤ j(K) and it again follows that i ∈ M(T ), and we have shown that M is monotonic. Properness of M follows from the fact that M(S) ⊆ S for every S ∈ 2 N . Veto-richness follows from monotonicity and also from the fact that M(S) ⊆ S for every S ∈ 2 N : if i ∈ M(S) for some S we have, using monotonicity, that i ∈ M(N ) and i ∈ ∩{T : i ∈ M(T )}.
The following proposition is a direct consequence of Theorems 3.3, 3.4, 3.5, 3.6 and Proposition 4.24.
Proposition 4.25 Let (N , E N ) be an undirected complete graph and let M be the coalitional map as defined in (27). Then (N, v M,w ) is monotonic, superadditive, (totally) balanced and PMAS-admissible for every w ∈ IR N + . Proposition 4.25 holds in particular for the case w i = 1 for all i ∈ N . Then, by relation (27) we have the following.  In [8,20,26] it has been shown that every MCST is a nonnegative combination of simple MCST games. Then, by Proposition 4.27 it immediately follows that also MCST games are subadditive, (totally) balanced and PMASadmissible, as already noticed in [20].

(Weighted) coloring games
Let Γ = (N, E) be an undirected graph and w ∈ IN N a nonnegative integer weight vector. A clique is a set S ∈ 2 N such that {i, j} ∈ E for every i, j ∈ S, i = j. The weight of this clique is i∈S w i . A clique with maximum weight is called a maximum weighted clique and the corresponding weight the weighted clique number of G. This number is denoted by ω w (Γ). For k ∈ IN a k-coloring of graph Γ with respect to weight vector w is a function h that assigns a set of w i different colors to every vertex i ∈ N such that adjacent vertices receive disjoint sets of colors and at most k colors are used. Formally, such a coloring is a map h : N → 2 {1,...,k} such that |h(i)| = w i for every i ∈ N and h(i) ∩ h(j) = ∅ for every {i, j} ∈ E. The minimum number k such that a k-coloring of Γ with respect to w exists is called the weighted chromatic number of Γ with respect to w and denoted as χ w (Γ). The weighted minimum coloring game [12] associates with every coalition the weighted chromatic number of Γ |S with respect to w S . A graph Γ = (N, E) is called perfect if ω(Γ |S ) = χ(Γ |S ) for every S ∈ 2 N , i.e. the clique and chromatic numbers are the same for the full graph and every induced subgraph. Perfect graphs are known to be weighted perfect as well (see, for example [22]): for every weight vector the weighted clique and weighted chromatic numbers are the same for the full graph and every induced subgraph. A graph Γ = (N, E) is called (2K 2 , P 4 )-free if there is no S ∈ 2 N , |S| = 4 such that Γ |S is a graph with two edges that have no vertex in common (2K 2 ) or Γ |S is a line graph with three edges (P 4 ). A graph Γ = (N, E) is called complete multipartite if there is a partition {P 1 , P 2 , . . . , P r } of the vertex set N such that for any two vertices i ∈ P k , j ∈ P l we have {i, j} ∈ E if and only if k = l.
From the literature we know the following results relating properties of graphs to properties of the related (weighted) minimum coloring games: Γ is perfect if and only if c Γ is totally balanced [10], Γ is perfect if and only if c Γ,w is totally balanced for every w ∈ IN N [12], Γ is (2K 2 , P 4 )-free if and only if c Γ has a PMAS [11], Γ is (2K 2 , P 4 )free if and only if c Γ,w has a PMAS for every w ∈ IN N [12], Γ is complete multipartite if and only if c Γ is concave [21] and Γ is complete multipartite if and only if c Γ,w is concave for every w ∈ IN N [12].
Again, we will illustrate that some of these results can be 'reproduced' by considering some classes of (weighted) minimum coloring games as generalized additive games with coalitional maps M satisfying nice properties. First, consider the following example. (so Γ is isomorphic to 2K 2 ) and let w ∈ IN N be such that w i = 1 for every i ∈ N . As Γ is perfect but not (2K 2 , P 4 )-free we know that c Γ is totally balanced but that it does not have a PMAS. Let v c Γ be the cost savings game, corresponding to c Γ . It is readily verified that v c Γ ( Let Γ = (N, E) be a complete multipartite graph and let {P 1 , P 2 , . . . , P r } be the corresponding partition of the vertex set N . We list the nodes in N in some order N = {v 1 , v 2 , v 3 , . . . , v n }, with n = |N | (not yet having a cost vector w ∈ IR N + in mind). Let [n] = {1, . . . , n}. For every v i ∈ N , with i ∈ [n], let k(v i ) ∈ {1, . . . , r} be such that v i ∈ P k(vi) . Define the map M by M(S) = {v i ∈ S|there is a v j ∈ S ∩ P k(vi) with i, j ∈ [n] and j > i} (28) for every S ∈ 2 N . In other words, M(S) is formed by members of S that precede in the order another member of S in the same partition element. One easily verifies the following statement.
Proposition 4.30 Let Γ = (N, E) be a complete multipartite graph and let M be the coalitional map as defined in (28). Then M is monotonic, proper and veto-rich but not necessarily supermodular (this is only true if all partition elements have at most two elements, see Remark 4.31).
Remark 4.31 Let Γ = (N, E) be a complete multipartite graph and let {P 1 , P 2 , . . . , P r } be the corresponding partition of the vertex set N such that |P l | ≤ 2 for l = 1, . . . , r. We want to show that the coalitional map M defined by (28)  The following proposition is a direct consequence of Theorems 3.3, 3.4, 3.5, 3.6 and Proposition 4.30.
Proposition 4.32 Let Γ = (N, E) be a complete multipartite graph and let M be the coalitional map as defined in (28). Then (N, v M,w ) is monotonic, superadditive, (totally) balanced and PMAS-admissible for every w ∈ IR N + .
Now, we will show that the cost savings games v c Γ,w , corresponding to the weighted minimum coloring games c Γ,w on complete multipartite graphs, form a subset of GAGs (N, v M,w ) with M specified by (28). Let (N, c Γ,w ) be the weighted minimum coloring game associated with a complete multipartite graph and cost vector w ∈ IR N + . Choose the ordering of the vertices in N according to increasing costs, i.e. N = {v 1 , v 2 , v 3 , . . . , v n } such that w v1 ≤ w v2 ≤ · · · ≤ w vn , and let M be the corresponding coalitional map according to (28). Define the set K(S) = S\M(S) for every S ∈ 2 N . The set K(S) is a maximum weighted clique in Γ |S with weight i∈[n]:vi∈K(S) w vi , with [n] = {1, . . . , n}, and this is also the weighted chromatic number of Γ |S with respect to w |S since Γ is perfect. Therefore c Γ,w (S) = i∈[n]:vi∈K(S) w vi for every S ∈ 2 N . So indeed we have for every S ∈ 2 N . As a consequence of Proposition 4.32 we obtain the following.  Proof The proof follows directly from the properties of cost saving games corresponding to weighted minimum coloring games on complete multipartite graphs and by Proposition 4.1.
Weighted minimum coloring games corresponding to complete multipartite graphs are known to be concave as well. In case all permutation elements have at most two elements, from Remarks 4.31 and Theorem 3.7 it follows that cost saving games corresponding to weighted minimum coloring games are submodular or concave, but this not sufficient to prove that the corresponding weighted minimum coloring games are supermodular or convex.
Instead, for the unweighted case (w i = 1 for every i ∈ N ) we can also prove that minimum coloring games are concave as follows. For every k ∈ {1, . . . , r} let p k be the element of P k with the smallest index. Define the coalitional map M by M(S) = {p k |k ∈ {1, . . . , r}, P k ⊆ S} (29) for every S ∈ 2 N . One can easily verify the following proposition.
Proposition 4.35 Let Γ = (N, E) be a complete multipartite graph and let M be the coalitional map as defined in (29). Then M is supermodular.
The following proposition is a direct consequence of Theorem 3.7 and Proposition 4.35.
Proposition 4.37 Cost saving games corresponding to minimum coloring games on complete multipartite graphs are convex.
Proof It follows from the fact that c Γ * = v M,w for every w ∈ IR N + , and in particular for the vector w such that w i = 1 for every i ∈ N . Proposition 4.38 Minimum coloring games corresponding to complete multipartite graphs are concave.
Proof The proof follows directly from the properties of convexity of dual games corresponding to minimum coloring games on complete multipartite graphs and by Proposition 4.2.
Finally, let us consider a graph Γ = (N, E) that is (2K 2 , P 4 )-free. Moreover, let w ∈ IN N be such that w i = 1 for every i ∈ N . From [28] and [29] we know that Γ has a rooted forest representation: there is a rooted forest (N, F ) such that for every i, j ∈ N, i = j we have {i, j} / ∈ E if and only if i ∈ D(j) or j ∈ D(i). Here for every i ∈ N the set D(i) denotes the unique path in (N, F ) from i to the root of its tree (including i and the root of the tree). Now define the map M by M(S) = {i ∈ S|there is a j ∈ S, j = i with i ∈ D(j)} for every S ∈ 2 N . One easily verifies the following proposition. It is straightforward to check that c Γ (S) = |S\M(S)| for every S ∈ 2 N . Now, for the cost savings game v c Γ corresponding to c Γ we have v c Γ (S) = i∈S c Γ ({i}) − c Γ (S) = i∈S w i − i∈S\M(S) w i = i∈M(S) w i = v M,w (S) for every S ∈ 2 N . As a consequence of Proposition 4.40 we obtain the following.
In this paper we have introduced some characterizations for large families of GAGs [9], and we have shown how these results can be used to analyze common features among distinct classes of TU games (in particular, within the family of OR games [6]). The results for GAGs provided in Section 3 apply to all vectors of nonnegative contributions. As a consequence, in some cases, the generality of coalitional maps prevent an exhaustive search of properties of the associated games. In this case, as shown for generalized airport games and also for link connection games, conditions for GAGs over subsets of contribution vectors (in particular, convex cones) are more effective, and deserve a deeper understanding.