A Monotonic Weighted Banzhaf Value for Voting Games

: The aim of this paper is to extend the classical Banzhaf index of power to voting games in which players have weights representing different cooperation or bargaining abilities. The obtained value does not satisfy the classical total power property, which is justiﬁed by the imperfect cooperation. Nevertheless, it is monotonous in the weights. We also obtain three different characterizations of the value. Then we relate it to the Owen multilinear extension. property, equal treatment of necessary players, and the transfer property.


Introduction
A cooperative n-person game with transferable utility or a TU game is a model for the situations in which players can obtain benefits (measured by real numbers) when cooperating. In a TU game, it is assumed that all players are totally cooperative and that all coalitions are feasible. Formally, a TU game is a pair formed by a set of players and the characteristic function that assigns to each coalition (subset of players) a real number, which is the (transferable) utility that the members of that coalition can obtain by cooperating. Shapley [1] a introduced his well-known allocation rule or point solution for these situations. Another relevant point solution, introduced as a power index for voting games (monotonous games in which coalitions are winning or losing and thus their outcome is 1 or 0, respectively), was suggested by Penrose [2], Banzhaf [3], and Coleman [4], and later generalized and characterized in the space of all TU games by Owen [5,6]. Game-theorists have paid a lot of attention to the Banzhaf value since it was introduced. Different characterizations of the Banzhaf value are given by Roth [7], Dubey and Shapley [8], Lehrer [9], Feltkamp [10], Dragan [11], Nowak [12], Laruelle and Valenciano [13], and Casajus [14], to name a few.
The Banzhaf value has also been defined for games with a priori unions obtaining the so-called Banzhaf-Owen value. It was introduced by Owen [15], and studied by Alonso-Meijide et al. [16] and Lorenzo-Freire [17]. Algaba et al. [18] extended and characterized the Banzhaf value to cooperative games on antimatroids, van den Brink [19,20] to games with a permission structure, Alonso-Meijide and Fiestras-Janeiro [21] to communication situations, Gallego et al. [22] to games with fuzzy communication structure, Tan et al. [23] to cooperative games with interval payoffs, and Fragnelli and Pusillo [24] applied the Banzhaf value to a multicriteria game to a microarray game.
The models in game theory are classically divided into cooperative and non-cooperative. Shapley [25] related cooperative and non-cooperative games by assuming that a TU game may have imperfect cooperation, that is, players can have different cooperation levels. He modeled this situation by associating a positive weight to each player. Then, he introduced the weighted Shapley value extending his rule to TU games [1]. The weighted Shapley value was generalized by Kalai and Samet [26].
The first attempt to obtain and characterize a weighted Banzhaf value was done by Radzik et al. [27] who analyzed cooperative games in which some a priori importance having different cooperative abilities, as well as the definition of the modified game. We also analyze its properties. Then, in Section 4 the new value is introduced. Three different characterizations of it are included in Section 5. Section 6 is devoted to the relationship between multilinear extensions and the defined value. The article ends with a short section of final remarks and an extensive bibliography.

Preliminaries
A cooperative game with transferable utility (a TU game) is a pair (N, v) in which N = {1, . . . , n} is the set of players and v is the characteristic function defined on 2 N . For each S ⊆ N, v(S) ∈ R is the outcome that the s members of S can obtain if they decide to cooperate. It is supposed that v(∅) = 0.
The family of the unanimity games {(N, u S )} ∅ =S⊆N with u S given by: is a basis of G N , the vector space of all TU games with players set N. Then, for each (N, v) The coefficients ∆ v (S), for S ⊆ N, are known as Harsanyi dividends [34]. They can be written in terms of the value of the coalitions in the following way: The worth of each coalition can also be calculated from Harsanyi dividends as: Given (N, v) ∈ G N and S ⊆ N, the restriction of (N, v) to S is the n-person game (N, v |S ) with characteristic function v |S (T) = v(T ∩ S) for all T ⊆ N.
A map ψ : G N → R n , with ψ i (N, v) representing the outcome for player i in game (N, v), is an allocation rule (point solution) for TU games.
Banzhaf [3] introduced a point solution in the domain of simple games with players set N, SG N , which was mathematically developed later by Dubey and Shapley [8]. They defined a swing or swing for player i as a pair of sets of the form (S, S \ {i}) such that v(S ∪ {i}) − v(S) = 1. They noted η i (v) for the number of swings of player i andη(v) for the total number of swings in the game v ∈ SG N . They named η i as the 'raw' Banzhaf index for player i, and characterized it using the following four axioms: (ii) Null player property: for all (N, v) ∈ SG N and i ∈ N such that η i = 0, ψ i (N, v) = 0. (iii) Anonymity: for any permutation π of N, ψ π(i) (N, πv) = ψ i (N, v). For (N, v) ∈ SG N the characteristic function πv is defined as (πv)(S) = v(π −1 (S)), for all S ⊆ N. (iv) Transfer property: for any (N, v) . For (N, v), (N, w) ∈ SG N , the characteristic functions v ∨ w and v ∧ w are defined as: (v ∨ w)(S) = max{v(S), w(S)} and (v ∧ w)(S) = min{v(S), w(S)}, for all S ⊆ N.
The normalization of the previous index to obtain the proportion of power has been a common practice. The most frequent normalizations are Owen [5] extended the (normalized, dividing by 2 n−1 ) Banzhaf index to G N . Thus, for He used the multilinear extension of a game [33] to calculate it [5], applied this value to the presidential election game [6], and characterized it [5,6].
The Banzhaf value can also be expressed in terms of the dividends, as:

Voting Games
The idea of decision-making by voting is formalized in the following definition.
S is a winner coalition in the voting game (N, v) if v(S) = 1, and it is a losing coalition otherwise. Then, a voting game the grand coalition is necessarily a winning one, and supercoalitions of a winning coalition are also winning coalitions.
A voting game is superadditive (or proper) if, for every S, T ⊆ N, with S ∩ T = ∅, then v(S ∪ T) ≥ v(S) + v(T) holds. As a consequence, in a superadditive voting game two winning coalitions cannot be disjoint; that is, if S, T ∈ 2 N , with v(S) = v(T) = 1, then S ∩ T = ∅.
A player i ∈ N is a dictator in the proper voting game (N, v), if v({i}) = 1. We can have only one dictator in a proper voting game. In the following, we will only consider proper voting games with no dictator. Nevertheless, by misuse of the language, sometimes we will use voting game for a proper voting game with no dictator. G N p will denote the family of these voting games.
In a convex voting game, the winning coalitions cannot be disjoint (as in proper voting games) and moreover, their intersection has to also be a winning coalition.
S is a minimal winning coalition in (N, v) ∈ G N p if v(S) = 1, and for all T S, v(T) = 0. Proposition 1. Ref. [35] If (N, v) is a voting game and MW C(N, v) = {S 1 , S 2 , . . . , S r } is the family of its minimal winning coalitions, we have v = u S 1 ∨ · · · ∨ u S r = r i=1 u S i .

Proposition 2.
A convex voting game is necessarily a unanimity game.
Proof. It is clear that in a voting game there is at least one minimal winning coalition. Suppose (N, v) is a convex voting game and S 1 , S 2 , S 1 = S 2 are both minimal winning coalitions in it. Then, as (N, v) is convex, and S 1 ∩ S 2 must be a winning coalition. Nevertheless, S 1 ∩ S 2 S 1 and S 1 ∩ S 2 S 2 , which contradicts that S 1 and S 2 are minimal. Then, the game (N, v) can have only one minimal winning coalition S, and thus, using the previous proposition, v = u S .
In the following proposition, we obtain an alternative expression for a proper voting game, which allows us to know (in the next corollary) its dividends.
We denote by (N, 1) the game in which v(S) = 1 for all S ⊆ N, S = ∅. That is, the voting game in which all non-empty coalitions are winning. Of course, (N, 1) is not a proper voting game, and thus the previous formula must be understood as an expression of (N, v) in G N .
Then, there is l ∈ {1, 2, . . . , r} such that S l ⊆ S, and thus u S l (S) = 1. As a consequence, On the other hand, if S is a losing coalition, u S i (S) = 0 for i = 1, . . . , r, as S does not contain any minimal winning coalition (in such a case, S is also a winning coalition). Then, , and thus the result is proved.
Proof. It is a direct consequence of the previous proposition that for i, j ∈ {1, . . . , r}, u S i · u S j = u S i ∪S j .

Remark 1.
If a proper voting game (N, v) has a dictator (Player 1, for example), then v = u {1} . The simplicity of these situations motivated us to ignore them.

The Modified Game
As mentioned previously, Shapley [25] extended the classical definition of a TU game to include the possibility that players can have different cooperation or bargaining abilities. He also defined a solution, known as the weighted Shapley value, for these situations. With the aim of introducing a weighted Banzhaf value, Radzik et al. [27] also assumed that the players had some measures of importance or weights. In both contributions, an exogenous weight in R + is assigned to each player. Manuel and Martín [28] assigned a vector of parameters λ = (λ 1 , . . . , λ n ) ∈ [0, 1] n to the players in a TU game (N, v). These parameters measure the level of cooperation of the players as in [25], and they can also be seen as a fuzzy coalition [29]. Then, they modified the original game to a new one (N, v λ ) with a characteristic function: Next, we restrict the definition of [28] to voting games.

Definition 2.
A voting game with players having different cooperation abilities is a triple (N, v, λ), in which (N, v) ∈ G N p and λ = (λ 1 , . . . , λ n ) is a fixed vector in [0, 1] n . λ i represents the cooperation ability of i ∈ N. G N p,Λ represents the set of all proper voting games with players having different bargaining abilities. Remark 2. We will identify G N p with the family of (N, v, λ) ∈ G N p,Λ in which λ = (1, . . . , 1).

Remark 3.
As for (N, v, λ) ∈ G N p,Λ , the underlying game is zero-normalized, and (N, v, λ) can be described by a fuzzy communication structure complete by links [30,31].
Regarding [28], we can associate to each voting game with players having different bargaining abilities the modified game in which players in a unanimity game can only retain a fraction of the total unit payoff. In the definition of [28], it is supposed that each singleton coalition can retain its dividend as each player is fully cooperative with himself. In our framework, as dictators are out of the question, we do not need to consider dividends of singletons. In any case, if a dictator is taken into account, of course his dividend must be unchanged.
The underlying idea is that the maximum level of cooperation in a unanimity game must coincide with the corresponding to the less cooperative player. Then we have the following definition:

Remark 4.
It is clear that the modified game will not be, in general, a voting game as v λ (N) can be smaller than one.
Let us consider the Example 3.1 in [28].

Example 1.
In a given parliament, the distribution of seats is 16%, 35%, 9%, and 40%, respectively, for parties 1, 2, 3, 4 which are aligned in a left-right spectrum. To pass a bill, more than 50% of votes are needed. Then, the minimal winning coalitions of this voting game ( In the following proposition, we obtain an alternative expression for v λ (S) when (N, v) is a voting game.
is the set of the minimal winning coalitions of (N, v) that are contained in S. Without loss of generality, let us assume that with the second equality holding as, for i = i 1 , . . . , i l , S i S, and the last one because where we have needed the binomial identities which proves the result.

Example 2. Consider again the voting game in which N
and the bargaining abilities are given by λ = (0.2, 1, 0.8, 0.1).
On the other hand, from Proposition 4, In the following proposition we prove that the modified voting game can be written as a positive lineal combination of voting games. In particular, the obtained expression can be seen as the Choquet extension of the original voting game [36][37][38][39]. (1) Proof. From the previous proposition, we have that, for using again the expression of v | S h+1 ∪···∪Sr for h = 0, 1, . . . , r − 1, in terms of the unanimity games, and the fact that S i l +1 , . . . , S r are the elements in which completes the proof.

Example 3. Consider again the voting game in which N
and the players have cooperation abilities given by λ = (0.2, 1, 0.8, 0.1). In the Example 2, we and λ S 3 = 0.2 and thus, using the previous proposition, Remark 5. Essentially, this expression (1) coincides with the Choquet one for games with players having different bargaining abilities that is obtained in [28]. In that paper, for each (N, v, λ) the authors define y h , for h = 0, 1, . . . , r, where r ≤ n is the number of different values among the weights of players in (N, v, λ), in the following way: y 0 = 0, and for h = 1, 2, . . . , r, y h = min Applying this expression in the previous example, we have that y As v | N\{1,4} = 0 and v | N\{1,3,4} = 0, we obtain the expression in (2).

Properties of the Modified Game
In the following proposition, we summarize some properties of the modified game. First of all, from Remark 3 in [28], we have that if λ = 1, that is, if all players fully cooperate, the modified game v λ coincides with the original game v; if all players have an equal level of cooperation, the modified game is strategically equivalent to the original one, and finally, if λ = 0, then v λ = 0, that is, if players do not cooperate at all, the modified game becomes inessential. Proposition 6. Given (N, v, λ) ∈ G N p,Λ we have: (i) If λ is such that λ i > 0 for some i ∈ N, and λ j = 0 for j = i, then v λ is the null game. Thus, cooperation cannot emerge if only one player wants to cooperate.
Thus, the lack of cooperation of a player is equivalent to the player leaving the game.
the minimal winning coalitions of (N, v) such that i / ∈ S i l , l = 1, ..., t. Thus, if a player is not willing to cooperate, all the minimal winning coalitions to which he belongs become loser coalitions. (iv) If λ, λ satisfy λ i > λ i and λ j = λ j for j = i then, for S ⊆ N \ {i}. Hence, when increasing the cooperation level of a player, his marginal contributions do not decrease.

Proof. (i) From Proposition 4, given
family of the minimal winning coalitions of (N, v) contained in S (recall that each minimal winning coalition has at least two players as the game has no dictator), where the second equality holds as for j = 1, . . . , l, i / ∈ S i j , given that S i j ⊆ S. Then, for each player in S i j , j = 1, . . . , l, his weight is the same in λ and λ.
the inequality holding as λ i ≥ λ i and as i can eventually be the player with the smaller weight in a minimal winning coalition. This completes the proof.
The following lemma precedes a proposition in which we prove that the modified game of a maximum of voting games is the maximum of its modified games.
Then, suppose that S ⊆ N is such that {S i 1 , . . . , S i t } is the subset of those elements in {S 1 , . . . , S r } such that S i j ⊆ S, j = 1, . . . , t. Let us assume again, without loss of generality, that λ S i 1 ≤ · · · ≤ λ S i t . Then, using (3), and using (4) which completes the proof.
In the following proposition, which needs two previous lemmas, we prove that the modified game of the minimum of the voting games is the minimum of its modified games. (N, u S 1 , λ), (N, u S 2 , λ),..., (N, u S r , λ) ∈ G N p,Λ , then,
Suppose then, S ⊆ N is such that S 1 ⊆ S, . . . , S r ⊆ S. Then which completes the proof.
On the other hand, for i ∈ {1, . . . , r}, j ∈ {1, . . . , m}}, u S i ∧ u T j = u S i ∪T j , and thus, if which is equal to and then the proof is completed.
the second and the seventh equality hold because of Lemma 3, the fourth and the eighth equality using Lemma 1, and the sixth equality because of Lemma 2.

The Banzhaf Value for Voting Games with Players Having Different Cooperation Abilities
First, we define allocation rules in this setting.

Definition 4. An allocation rule on G N
p,Λ is a function ψ : G N p,Λ → R n that assigns to each (N, v, λ) ∈ G N p,Λ , ψ(N, v, λ) ∈ R n , satisfying ψ(N, v, 0) = 0. For i = 1, .., n, ψ i (N, v, λ) represents the outcome or payoff for Player i in (N, v, λ). Manuel and Martín [28] introduced (and characterized) the allocation rule for games with players having different bargaining abilities that assigns to every (N, v, λ) the Shapley value of the modified game (N, v λ ). In the following, we will define and study the allocation rule on G N p,Λ that assigns to each (N, v, λ) the Banzhaf value of (N, v λ ).  For voting games with players having different bargaining abilities, the defined value can be calculated as the linear combination of Banzhaf values of appropriate (classical) voting games. The proof of this result is straightforward from Proposition 5, and then it is omitted. (N, v, λ) ∈ G N p,Λ , suppose MW C(N, v) = {S 1 , . . . , S r } is the family of the minimal winning coalitions of (N, v). If λ S i = min j∈S i {λ j } for j = 1, 2, . . . , r, with λ S 1 ≤ λ S 2 ≤ · · · ≤ λ S r , and λ S 0 = 0, then,

Proposition 9. Given
In the next proposition we prove that for convex voting games with players having different bargaining abilities, its Banzhaf value is upper-bounded by the Banzhaf value in the original voting game. Hence, by reducing the cooperation ability, the power is also reduced.
for i ∈ S and zero in another case. As B i (N, v) = 1 2 s−1 for i ∈ S and zero otherwise, the result is proved.
Nevertheless, in superadditive voting games with players having different cooperation abilities, a particular player can increase his Banzhaf value with respect to the one obtained in the original game, as it is shown in the next example.  9 40 , 11 40 ).
The following proposition, whose proof is straightforward from (iv) of Proposition 6, states that the defined Banzhaf value for games with players having different bargaining abilities is monotonous in the weights. (N, v, λ).
As mentioned, Radzik et al. [27] introduced a weighted Banzhaf value, β R,w , that (unlike our proposal) respects the total power but is not monotonous in the weights. According to this linear value β R,w , in a game (N, u S ) with players' weights w = (w 1 , ..., w n ) ∈ (R + ) n , each player i ∈ N receives In terms of the unanimity basis, v can be written as: Suppose that the vector of weights is w = (0.01, 0.3, 0.4). Then, The counterintuitive behavior of β R,w can be seen in this case, since Player 2 who is symmetrical with 3 in the game, and has less weight, gets a higher payoff.
On the other hand, if Player 2's weight decreases from 0.3 to 0.2, keeping the rest of the weights equal, the value in this case is β R,w (N, v) = (0.0319, 0.394, 0.323). It is observed that Player 2, with a smaller weight than before, has greater power.

Characterizations of the Defined Value
First we introduce some properties that allow us to characterize the defined value.

Definition 6. An allocation rule
In the following, we will denote (N, v, λ).

Proposition 12.
The allocation rule on G N p,Λ , β, satisfies the λ-total power.
Proof. By the definition of β, we have (N, v, λ).
In the following proposition, we obtain an alternative expression for TP (N, v, λ) which completes the proof.
In the following proposition we prove that the defined rule, β, satisfies the properties defined above.

Proposition 14.
The allocation rule β : G N p,Λ → R n satisfies the transfer and the null player properties, equal treatment of necessary players, balanced contributions, strong monotonicity and marginality.
Proof. (i) To prove that β satisfies the transfer property, consider (N, v, λ), (N, w, λ) ∈ G N p,Λ . Then, we have the first and the last equality hold because of the definition of β, the second equality by the Proposition 7, and the third equality as the Banzhaf value defined on G N satisfies the transfer property. (ii) Suppose that (N, v, λ) ∈ G N p,Λ and i ∈ N is a null player in (N, v). Then, he is also a null player in (N, v λ ). As the Banzhaf value on G N satisfies the null player property, the result is proved. (iii) Suppose that (N, v, λ) ∈ G N p,Λ and i, j ∈ N are necessary players in (N, v). Then, they are also necessary players in (N, v λ ). As the Banzhaf value on G N satisfies equal treatment of necessary players (see [20]), the result is proved. (iv) To prove that β satisfies balanced contributions, let (N, v, λ) Then, an expression which is symmetric in i and j and thus, (v) In [28] (p. 18) it is proved that if the marginal contributions of Player i in (N, v) are greater or equal than in (N, w), then they are also greater or equal in (N, v λ ) than in (N, w λ ). Thus, the marginality of the Banzhaf value is inherited by β.

Remark 7.
We can observe that β satisfies a generalized version of the balanced contributions property: given (N, v, λ) ∈ G N p,Λ with MW C(N, v) = {S 1 , ..., S r }, and i, j ∈ N This is so as Then we provide three characterizations of β. The first one is parallel to the one of Dubey and Shapley [8].
Proposition 15. β is the unique allocation rule on G N p,Λ satisfying λ−total power, the null player property, equal treatment of necessary players, and the transfer property.
Proof. It is already proved that β defined on G N p,Λ satisfies these four properties. Reciprocally, let ψ : G N p,Λ → R n be an allocation rule satisfying λ-total power, the null player property, equal treatment of necessary players and the transfer property. We will prove that ψ(N, v, λ) = β (N, v, λ) for all (N, v, λ) ∈ G N p,Λ by induction on r, the number of minimal winning coalitions of (N, v). If r = 1, and MW C(N, v) = {S 1 }, v = u S 1 and thus all i ∈ S 1 is a necessary player in (N, v) and all i / ∈ S 1 is a null player. Then, for all i / ∈ S 1 , using the null player property, ψ i (N, u S 1 , λ) = 0 and also β i (N, u S 1 , λ) = 0. Using the equal treatment of necessary players, ψ i (N, v, λ) = ψ j (N, v, λ) for all i, j ∈ S 1 . By the λ-total By the induction hypothesis, let us suppose that ψ coincides with β for (N, v, λ) if the number of minimal winning coalitions of (N, v) is k ≤ r, and consider (N, v, λ) with MW C(N, v) = {S 1 , . . . , S r+1 }. Then v = u S 1 ∨ u S 2 ∨ · · · ∨ u S r+1 . As ψ satisfies the transfer property, Since ∧ is distributive with respect to ∨ (see Lemma 3), we have the last equality holds, as for all (N, u S ), (N, u T ) ∈ G N , u S ∧ u T = u S∪T . Then, which, using the induction hypothesis, coincides with Using again the distributive property of ∧ with respect to ∨ and that β satisfies the transfer property, this last expression is equal to (N, v, λ), which completes the proof.
Proposition 16. β is the unique allocation rule on G N p,Λ satisfying the λ-total power and balanced contributions.
Proof. It has been previously proved that β satisfies λ-total power and balanced contributions. Reciprocally, suppose ψ is an allocation rule on G N p,Λ satisfying these two properties. We will prove that ψ (N, v, λ If δ(λ) = 0, then λ = 0, and by the definition of the allocation rule, Suppose δ(λ) = 1, and let i ∈ N be such that λ i > 0. Then, as ψ satisfies balanced contributions, for each j = i, j ∈ N, Since for (N, v, λ) with only one cooperative player the λ-total power is zero, we have ψ i (N, v, λ) = TP(N, v, λ) = 0. Taking into account that β also satisfies λ-total power and balanced contributions, we have the similarly β i (N, v, λ) = 0 for all i ∈ N, and thus both rules coincide.
The following characterization can be considered parallel to that provided by Nowak [12].
Proposition 17. β is the unique allocation rule in G N p,Λ satisfying λ-total power, the equal treatment of necessary players, and marginality. As marginality is a weaker property than strong monotonicity, and β satisfies strong monotonicity, the result also holds when replacing marginality by strong monotonicity.
Proof. It has been previously proved that β satisfies λ-total power, the equal treatment of necessary players' property, and marginality. Reciprocally, suppose ψ is an allocation rule on G N p,Λ satisfying these three axioms. We will prove that ψ = β by induction on r, the number of minimal winning coalitions of (N, v).
If r = 1, then, there exists S ⊆ N such that v = u S . All i ∈ S are necessary players in (N, v), and thus ψ i (N, v, λ) = c for all i ∈ S. On the other hand, (N, 0, λ) for i / ∈ S. By the λ-total power ψ i (N, 0, λ) = 0 for all i ∈ N, and thus, ψ i (N, v, λ) = 0 = β i (N, v, λ) for i / ∈ S. Then, using again the λ-total power, for i ∈ S and thus, ψ i (N, v i , λ) = ψ i (N, v, λ) because of the marginality property.
By induction hypothesis, and thus, in (v − v i ) λ , and the Banzhaf value satisfies the null player property. (N, v, λ).
On the other hand, if i ∈ k t=1 S t , then i is a necessary player in (N, v). All necessary players have equal rewards, and thus using λ-total power, the payoff for all of them is uniquely determined.

Multilinear Extension of a TU Game with Players Having Different Bargaining Abilities and the Defined Value, β
The multilinear extension of a TU game (N, v) ∈ G N was introduced by Owen [33] as a function f : [0, 1] n → R given by The multilinear extension can also be calculated in terms of the dividends of the game as It is a useful tool when calculating the Shapley value and the Banzhaf value. In particular, Owen [5,33] proved that for i ∈ N . Manuel and Martín [28] generalized the multilinear extensions to the setting of games with players having different bargaining abilities. For (N, v, λ) ∈ G N Λ , the multilinear extension is given by Particularizing this expression to a voting game with players having different bargaining abilities or different levels of cooperation (N, v, λ), in which MW C(N, v) = {S i , . . . , S r }, λ S i = min j∈S i {λ j } with λ S 1 ≤ · · · ≤ λ S r , we have The following proposition gives an expression to calculate the Banzhaf value in G N p,Λ , β, in terms of the multilinear extensions. Evaluating this partial derivative in ( 1 2 , . . . , 1 2 ) we obtain, for i = 1, . . . , n ∂ ∂x i f λ ( 1 2 , . . . , (N, v, λ) which completes the proof.

Conclusions and Final Remarks
In this paper, we have introduced an extension of the classical Banzhaf index of power for voting games to the setting in which players have different cooperation abilities. To do this, the original game is modified to a new one in which the imperfect cooperation reduces the dividend of each coalition, multiplying them by the minimum of members' weights. The Banzhaf value of this modified game can be used to measure the power of players in these situations, generalizes the classical Banzhaf index of power, and admits several characterizations parallel to that existing in the literature and a new one.
The obtained results can be generalized in different ways. For example, by considering the same ideas, we can extend the introduced value to the space of all TU games, but also to the setting of TU games with players having different bargaining abilities and cooperation restricted by a graph [43] or by a priori coalitions. The framework can also be generalized assuming that the weights of players, instead of being fixed, can depend on the coalitions they are involved with, or on the other players. As mentioned, instead of assuming a given weight system, it is possible that a different approach may exist, in the spirit of [26]: to provide axioms that allow to characterize the proposed solution if, and only if a system of weights exists. This problem remains open and deserves attention. Funding: This research has been partially supported by the "Plan Nacional de I+D+i" of the Spanish Government under the project MTM2015-70550-P. D. Martín also wants to thank University Complutense of Madrid and Bank of Santander for his pre-doctoral contract. The authors wish to thank two anonymous referees and the editor for their useful suggestions and comments.