A Banzhaf value for games with a proximity relation among the agents

International Journal of Approximate Reasoning 88 (2017) 192–208 Contents lists available at ScienceDirect International Journal of Approximate Reasoning www.elsevier.com/locate/ijar A Banzhaf value for games with a proximity relation among the agents J.R. Fernández, I. Gallego, A. Jiménez-Losada ∗, M. Ordóñez Escuela Técnica Superior de Ingenieros, Matemática Aplicada II, Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 26 October 2016 Received in revised form 15 March 2017 Accepted 27 May 2017 Available online 12 June 2017 Keywords: Game theory Proximity relation Owen value Banzhaf value Power indices European Parliament The Banzhaf index is a function determining the power or influence in the decision of a set of agents. The extension of this index to the family of the cooperative games is named Banzhaf value. The relationships of closeness among the agents should modify their power. Games with a priori unions study situations where the closeness relations among the agents are taken into account. In this model the agents are organized in an a priori partition where each element of the partition represents a group of agents with close interests or ideas. The power is determined in two steps, first as a problem among the unions and later, inside each one, the power of each agent is determined. Proximity relations extend this model considering leveled closeness among the agents. In this paper we analyze a version of the Banzhaf value for games with a proximity relation and we show the interest of this value by applying it to the allocation of the power of the political groups in the European Parliament. ©2017 Elsevier Inc. All rights reserved. 1. Introduction In decision situations for committees or centers of distributed control the quantification of the power of each member is an important element to analyze the final position and the different treatment of each of them. Simple games are a way from the cooperative game theory to represent these situations and to study the power of their elements. Apower index for simple games is a function determining the power or influence of the agents in each simple game. One of the most known power indices, the Banzhaf index, was introduced by Penrose [1] in 1946 and later by Banzhaf [2] in 1965. In this context the Banzhaf index was generalized for all cooperative game as the Banzhaf value [3]. The Banzhaf value has been studied in different scenarios incorporating new information over the relationships of the agents (coalition structures [4], communication situations [5], hierarchical relations [6], etc.). Owen [7] proposed a different model with an evident interest for simple games. He considered that agents are organized a priori in groups taking into account the closeness of their interests (ideas). So, besides the game he supposed known a partition of the set of agents in a priori unions based in the relations among the agents. These unions are considered as a starting point for further negotiations. This model allows to determine the power of the agents in a simple game taking into account the a priori unions among them. But closeness is usually a leveled property. For instance, political groups can be organized in a priori ideological unions. The Banzhaf value was studied for this model in [8]. Casajus [9] proposed another version of the Owen model considering also information about the internal estructure of the unions. But the Banzhaf value has not been studied for this version. *Corresponding author. E-mail address: a[email protected] (A. Jiménez-Losada). http://dx.doi.org/10.1016/j.ijar.2017.05.010 0888-613X/©2017 Elsevier Inc. All rights reserved. J.R. Fernández et al. / International Journal of Approximate Reasoning 88 (2017) 192–208 193 Considering equal every ideological closeness between two political parties is actually a simplification of the situation. Aubin [10] and Butnariu [11] introduced fuzzy sets to describe leveled participation of the players in the coalitions (fuzzy coalitions). Proximity relations are reflexive and transitive fuzzy binary relations. Fernández et al. [12] introduced proximity relations to explain the relations among the players in a cooperative game, extending the Owen model in a natural way for another known value, the Shapley value. Related works are given in Meng [13] and Meng and Zhang[14], but they consider a game with fuzzy coalitions with a crisp system of a priori unions. Hence this model contains a different approach. Others analyze this kind of relations in a probabilistic way, Calvo et al. [15] or Kaniovski and Das [16], but they do not use the Owen model. Now, we propose to use proximity relations to study the power of an agent in a game by the Banzhaf value. Section 2is dedicated to the preliminaries about cooperative games, a priori unions and fuzzy sets. In section 3we introduce our Banzhaf value for games with a proximity relation among the agents and particularly the Banzhaf–Owen value is extended to the Casajus version of the Owen model. In section 4we propose axioms for the proposed value extending known properties of the classical Banzhaf value to our fuzzy situation. Finally, last section shows an example of application of the model as index, analyzing the power of the political groups in the European Parliament in an ad hoc situation. 2. Preliminaries 2.1. Cooperative games A cooperative game with transferable utility, game from now on, is a pair (N, v)where Nis a finite set and v :2N→Ris a mapping satisfying v(∅) =0. The elements of Nare named players, the subsets of players are named coalitions and the mapping vis the characteristic function of the game. Asimple game represents a decision situation by a cooperative game (N, v)where: 1) v(S) ∈{0, 1}for every S⊆N, 2) v(N) =1, and 3) v(S) ≤v(T)if S⊂T⊆N(monotonicity). A coalition is called winning if v(S) =1 and losing v(S) =0. A value for games is a function ψwhich determines for each game (N, v)a vector ψ(N, v) ∈RNinterpreted as a payoff vector. Values for simple games are named power indices.1In this case the payoffs mean the power or influence of the agents in the decision. This paper focuses on the Banzhaf value. A swing for a player i ∈Nin a simple games is a winning coalition Scontaining player isuch that S\{i}is losing. The Banzhaf index obtains the probability to get a swing among the coalitions containing a determined agent. Owen [3] extended this index to all the cooperative games. The Banzhaf value is a value defined for every (N, v) ∈Gand i ∈Nas βi(N,v)= {S⊂N:i/∈S} 1 2|N|−1[v(S∪{i})−v(S)].(1) 2.2. Communication structures Myerson [5] analyzed the inclusion in a game of information about the communication of the players. Let Nbe a finite set of players and LN={{i, j} ∈N×N:i = j}the set of unordered pairs of different elements in N. We use ij ={i, j} from now on. Each undirected graph (N, L)where the set of vertices is Nand the set of edges L ⊆LNis considered as a communication structure. So, each L ⊆LNis called a communication structure for N. Myerson defines a game with communication structure as a triple (N, v, L)where (N, v)is a game and Lis a communication structure for N. Ausual cooperative game (N, v)is identified with the game with communication structure (N, v, LN). Let (N, v, L)be a game with communication structure. A coalition S⊆Nwhose vertices are connected by the links in Lis called connected. The maximal connected coalitions correspond to the sets of vertices of the connected components of the graph (N, L)and we denote them as N/L. This family N/Lis actually a partition of N. If S⊆Nis a coalition then LS={ij ∈L :i, j ∈S}and (S,v,LS)represents the restriction to Sof the characteristic function of the game and the communication structure. We use S/L =S/LS. Given (N, v, L), Myerson introduces a new game (N,v/L)incorporating the information of the communication structure, v/L(S)= T∈S/L v(T)∀S⊆N. This model supposes then that non-connected coalitions do not obtain extra profits with regard to their components and so they are irrelevant. The Banzhaf value was extended for games with communication structure in [18]. The graph-Banzhaf value is a function defined by η(N,v,L)=β(N,v/L).(2) 1This is a cardinal notion of power rather than other ordinal ones, see [17]. 194 J.R. Fernández et al. / International Journal of Approximate Reasoning 88 (2017) 192–208 2.3. A priori unions A game with a priori unions [7] is a triple (N, v, P)where (N, v) ∈Gand P={N1, ..., Nm}is a partition of N. Players in Nkfor each khave similar interests in the game and they bargain as a block to get a fair payoff. It is supposed that players are interested in the grand coalition Nbut considering the a priori unions as bargaining elements, so the unions can cooperate among them obtaining profits.2The classical situation without a priori unions is identified with the individual partition Pind ={{i} :i ∈N}. The Owen model is a procedure to define values for games with a priori unions in two steps. Let (N, v, P)with P={N1, ...., Nm}. The quotient game is a game (M, vP)with set of players M={1, ..., m}defined by vP(Q)=v⎛ ⎝ q∈Q Nq⎞ ⎠,∀Q⊆M. Let k ∈M. For each S⊂Nkthe partition PSof N\(Nk\S)is to replace Nkwith S. Given a classical value for games ψ1, we define a game over each union (Nk,vk)as vk(S) =ψ1 kM,vPS, ∀S⊆Nk. Finally we solve the game in every union using another value ψ2. So, for each player i ∈Nif k(i)is such that i ∈Nk(i)then the value is i(N, v, P) = ψ2 iNk(i),vk(i). Owen [8] defined an extension of the Banzhaf value, named Banzhaf–Owen value and denoted as βow, in the sense that βow(N, v, Pind) =β(N, v). This extension uses the Owen model with ψ1=ψ2=β, namely for every i ∈N βow i(N,v,P)=βi(Nk(i),vk(i))and vk(S)=βk(Nk,vPS), ∀S⊆Nk.(3) Casajus [9] proposed a variation of the Owen problem. He considers a partition in a priori unions for the agents, but also information about the bilateral relations which defined the unions. These internal relationships introduce an asymmetry among the players inside a union. In order to represent these situations he used a graph (N, L), as in [5], but now the connected components N/Lare the unions and the links inside each component are the bilateral relations determining each union. Hence Casajus uses a value ψ2for games with communication structure in the second step in the Owen model taking into account the asymmetry inside the unions. Each triple (N, v, L)is named now game with a cooperation structure.3 But there is no literature about the Banzhaf value for the Casajus version. 2.4. Fuzzy sets and proximity relations A fuzzy set of a finite set Nis a function τ:N→[0, 1]. The support of τis the set supp(τ) ={i ∈N:τ(i) = 0}. The image of τis the ordered set of the non-null images of the function, im(τ) =λ1<···<λ p={λ ∈(0, 1] :∃i ∈N, τ(i) =λ}. For each t∈(0, 1]the t-cut of the fuzzy set τis [τ]t={i ∈N:τ(i) ≥t}. A (signed) capacity over Nis a set function f:2N→R satisfying that f(∅) =0, namely a game. The (signed) Choquet integral [20,21] of τ∈[0, 1]Nwith respect to a capacity fis defined by τdf = p  k=1λk−λk−1f[τ]λk,(4) where im (τ)=λ1<···<λ pand λ0=0. The following properties of the Choquet integral are known: (C1) eSdf =f(S), for all S⊆N, and eS(i) =1if i ∈Sand eS(i) =0otherwise. (C2) tτdf =tτdf, for all t∈[0,1]. (C3) τd (a1f1+a2f2)=a1τdf1+a2τdf2, when a1, a2∈R. (C4) τdf =a i∈Nτ(i)if f([τ]t) =afor all t∈(0, 1]. (C5) τSdf=τdf if S⊆Nsatisfies f([τ]t) =f([τ]t∩S)for all t∈(0, 1]. (C6) τdf =p q=1(tk−tk−1)f([τ]tk)for all set im(τ) ⊆{t1<···<tp} ⊂(0, 1]and t0=0. A bilateral fuzzy relation over N, see [22], is a function ρ:N×N→[0, 1]satisfying the condition ρ(i, j) ≤ρ(i, i) ∧ ρ(j, j). Aproximity relation over N, is a fuzzy relation ρsatisfying the properties: (Reflexivity) ρ(i, i) =1for all i ∈N, and (Symmetry) ρ(i, j) =ρ(j, i)for all i, j ∈N. If S⊆Nthen the proximity relation ρrestricted to Sis ρS, a new proximity relation over Swith ρS(i, j) =ρ(i, j)for all i, j ∈S. Aproximity relation ρover Ncan be also seen as fuzzy sets over the set LN=LN∪{{ii} :i ∈N}with 1 ∈im(ρ)and {{ii} :i ∈N} ⊆[ρ]1. So, we will use ρ(ij)instead ρ(i, j). Every cooperation structure L ∈LNcan be seen as a crisp proximity relation (adding the vertices), moreover the cuts of a proximity relation 2This idea is different in the communication model of Myerson, see the previous section. 3Actually (N, v, L)is a game communication structure but seen in another way. J.R. Fernández et al. / International Journal of Approximate Reasoning 88 (2017) 192–208 195 Fig. 1. Fuzzy graph representing a proximity relation. are cooperation structures. Each set function fover LNwill be identified to a signed capacity with the same letter fover LNdefined for all A ⊆LNby f(A)=f(L), if A=L∪{{ii}:i∈N} 0,otherwise. 3. A Banzhaf value for games with a proximity relation Fernández et al. [12] studied the Shapley value with proximity relations. The goal of this paper is to define a Banzhaf value for games with a proximity relation among the agents. Definition 1. A game with a proximity relation is a triple (N, v, ρ)where (N, v)is a game and ρis a proximity relation over N. Suppose a game (N, v). Think first about a crisp relation Las in Casajus [9]. If ij ∈Lthen we understand that player j is close to i. If ij, jk ∈Land ik /∈Lthen jis close to iand close to kbut in different sense because iis not close to k. We suppose then that jworks as a valid intermediary between iand k, moderating the position of all of them and forming a union. Now, we consider a proximity relation ρ. For each pair of agents i, jnumber ρ(ij)means the level of closeness of the interests or ideas between both of them. This closeness ρ(ij)is also the cohesion level or confidential level of the coalition {i, j}. For three players i, j, k ∈Nwe take ρ(ij) ∧ρ(jk)as the maximal confidential level of coalition {i, j, k}taking into account the moderation power of j. We can define unions in this context fixing a level of cohesion. So, if we take t0∈(0, 1] (we think that this is the minimal reasonable level to get a union) then Sis a union if the proximity relation connects Sat this level t0and Sis maximal in this sense. For each t∈(0, 1]the cut [ρ]trepresents a cooperation structure in the Casajus sense and it explains the situation in order of increasing the required level of relation to consider a union. Suppose for instance a committee formed by five members N={1, 2, 3, 4, 5}such that the a priori bilateral relations among them are well known. Obviously these relationships are not the same and we level them using a proximity relation ρ(12) =ρ(13) =ρ(45) =0.7, ρ(23) =0.4, ρ(34) =0.2 and ρ(ij) =0otherwise. We represent the proximity relation ρby a graph with leveled links (see Fig. 1) with ρ. Link ij is not in the graph if ρ(ij) =0. The decision in the committee is taken using a simple game (N, u{2,3,4})where u{2,3,4}(S) =1if S⊇{2, 3, 4}and u{2,3,4}(S) =0otherwise. 4 In our example the cuts (Fig. 2) show that if t∈(0, 0.2]all the agents form a union but they are asymmetric because of their positions in the graph, if t∈(0.2, 0.4]we get a situation with two a priori unions in the Owen sense because the graphs are complete, if t∈(0.4, 0.7]the cooperation structure follows the Casajus version because there is an asymmetry into one of the unions, and finally if t∈(0.7, 1]then there are not any a priori unions because we have the individual partition Pind. Aunion is obtained if there is a level such that this coalition is a component in the corresponding cut. The model in [12] considers a value for games with a priori unions in the Casajus version and the Choquet integral of the proximity relation using this value. But the Banzhaf value in this context has not been studied at the moment. So, first we introduce a Banzhaf value for games with cooperation structure using the Owen model (3) with the graph Banzhaf (2). Definition 2. Let (N, v, L)be a game with L ⊆LNwhere N/L ={N1, ..., Nm}and M={1, ..., m}. If S⊆Nkfor any k ∈M then vk(S) =βkM,v(N/L)S. The graph Banzhaf–Owen value for each player i ∈Nkis βco i(N, v, L) =ηiNk,vk,LNk. Remarks. The proposed graph solution is consistent with the other Banzhaf values in this way. Only from the definitions: •If LS=LSfor all union S∈N/L, namely each component is complete, then βco(N, v, L) =βow(N, v, N/L). •Particularly βco(N, v, ∅) =β(N, v). Observe that L =∅ corresponds to Pind. The graph Banzhaf–Owen value in Definition 2 determines for each player a set function over LN. So, if i ∈Nthen βco i(N,v)(L)=βco i(N,v,L). (5) 4Game (N, uT)with T⊆Na non-empty coalition represents the simple game where all the winning coalitions are those containing T. It is known as unanimity game. 196 J.R. Fernández et al. / International Journal of Approximate Reasoning 88 (2017) 192–208 Fig. 2. Cuts of a proximity relation. Now, we can define our Banzhaf value for games with a proximity relation. We use a Choquet integral (4) of the proximity relation with regard to the above signed capacity. Definition 3. Let (N, v, ρ)be a game with a proximity relation. The prox-Banzhaf value for each agent i ∈Nis defined by Bi(N,v,ρ)=ρdβco i(N,v). The prox-Banzhaf value meets the purpose of being a Banzhaf value in the sense that it coincides with the classical value when we do not have any a priori relation among the players. We denote as ρ=0the trivial proximity relation satisfying 0(ij) =0if i = jand also 0(ii) =1. This proximity relation represents the classical situation without a priori unions among the players. Proposition 1. The prox-Banzhaf value satisfies B(N, v, 0) =β(N, v). Proof. We know that βco verifies that βco(N, v, ∅) =β(N, v). Hence as all the cuts of the trivial proximity relation 0satisfy [0]t=∅ we get that βco i(N, v)([0]t) =βi(N, v)for all t∈(0, 1]. Property (C4) of the Choquet integral and the fact 1 ∈im(ρ) imply Bi(N,v,0)=0dβco i(N,v)=βi(N,v). 2 Suppose our example in Fig. 1 with the game u{2,3,4}. When t∈(0, 0.2]the graph is connected and u{2,3,4}/[ρ]0.2= u{2,3,4}, thus βco(N,u{2,3,4},[ρ]0.2)=η(N,u{2,3,4},[ρ]0.2)=β(N,u{2,3,4})=0,1 4,1 4,1 4,0. If t∈(0.2, 0.4]then we use the Banzhaf–Owen value with a priori unions. There are two unions M={a ={1, 2, 3}, b = {4, 5}}, and va=1 2u{2,3}, vb=1 2u{4}. βco coincides with the Banzhaf value of the above games in each group, βco(N,u{2,3,4},[ρ]0.4)=βow(N,u{2,3,4},N/[ρ]0.4)=0,1 4,1 4,1 2,0. If t∈(0.4, 0.7]we have the same unions M={a, b}but now player 1 is necessary to get a winning coalition. So, va/L = 1 2u{1,2,3}and βco(N,u{2,3,4},[ρ]0.7)=1 8,1 8,1 8,1 2,0. J.R. Fernández et al. / International Journal of Approximate Reasoning 88 (2017) 192–208 197 Finally, if t∈(0.7, 1]we obtain the usual case and also βco(N,u{2,3,4},[ρ]1)=β(N,u{2,3,4})=0,1 4,1 4,1 4,0. The prox-Banzhaf value in this situation is B(N,u{2,3,4},ρ)=0.20,1 4,1 4,1 4,0+0.20,1 4,1 4,1 2,0 +0.31 8,1 8,1 8,1 2,0+0.30,1 4,1 4,1 4,0=3 80,17 80,17 80,30 80,0. 4. Axioms for the prox-Banzhaf value The authors in [12] introduced the following scaling operations over proximity relations. Let ρbe a proximity relation over N. If a, b ∈[0, 1]with a <bthen ρb ais the interval scaling of ρ, a new proximity relation over Ndefined by ρb a(ij)=⎧ ⎪ ⎨ ⎪ ⎩ 1,if ρ(ij)≥b ρ(ij)−a b−a,if ρ(ij)∈(a,b) 0,if ρ(ij)≤a. (6) Let a, b ∈[0, 1]be numbers with a <band a = 0or b = 1. The dual interval scaling of ρis a new proximity relation over N given by ρb a(ij)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ρ(ij)+a−b 1+a−b,if ρ(ij)≥b a 1+a−b,if ρ(ij)∈(a,b) ρ(ij) 1+a−b,if ρ(ij)≤a. (7) Remarks. 1) If b ∈im(ρ)with b <1then |im(ρb a)| <|im(ρ)|. 2) If b ∈im(ρ)with b <1 and a ∈im(ρ) ∪{0}then |im(ρb a)| <|im(ρ)|. 3) If a =0 and b =1then the dual interval scaling is defined by ρ1 0(ij) =1if ρ(ij) =1 and ρ1 0(ij) =0otherwise. The above scalings allow us to allocate the Choquet integral in particular intervals. This idea is explained in the next lemma. As we said before (see subsection 2.4 in preliminaries) each cut of a proximity relation is identified with a particular element of LN. Lemma 2. (Fernández et al. [12]) Let ρbe a proximity relation over N. For every pair of numbers a, b ∈[0, 1]with a <band for every set function fover LNit holds ρdf =(b−a)ρb adf +(1+a−b)ρb adf. We propose an axiomatization based on classical axioms of the Banzhaf values (for its different versions). The fuzzy information is included into the axioms by the scaling because we can focus on the interval where the axiom is satisfied. Let be a value determining a payoff vector for each game with a proximity relation. It is known that the Banzhaf value satisfies the dummy player axiom (see for instance Casajus [23]). Player iis dummy in a game (N, v)if v(S∪{i}) −v(S) =v({i})for all S⊆N\{i}. The dummy player axiom guarantees the payoff v({i})for a dummy player i. This fact changes if there is asymmetry into the component containing the dummy player. An isolated player in a proximity relation ρis a player isatisfying ρ(ij) =0for all j ∈N\{i}. Dummy isolated player. If i ∈Nis a dummy player in a game (N, v)and iis isolated in a proximity relation ρthen i(N, v, ρ) = v({i}). Next three axioms introduce the delegation or merging situation for proximity relations, classical also in the Banzhaf value (see for instance [23]). Let (N, v)be a game. The amalgamation of players i, j ∈Nconsists of taking the activity of both of them as only one, delegating jto i. The delegation game is (N\{j}, vij)with vij(S) =v(S∪{j})if i ∈S, vij(S) =v(S) 198 J.R. Fernández et al. / International Journal of Approximate Reasoning 88 (2017) 192–208 Fig. 3. Amalgamation in a proximity relation. otherwise. Let ρbe a proximity relation over Nand i, j ∈Ndifferent players with5ρ(ij) =1. The proximity relation changes with the amalgamation, ρij is a new proximity relation over N\{j}given by (see Fig. 3) ρij(kl)=ρ(kl), if k,l= i ρ(ki)∨ρ(kj), otherwise. (8) Next axiom says that the payoff obtained by both of the players in the game is the same that the payoff obtained by the proxy of them while this is possible (until their relation level) plus the payoff of both of them for the rest of the levels. Fuzzy amalgamation. Let (N, v, ρ)be a game with a proximity relation. If i, j ∈Nare different players with ρ(ij) >0then i(N,v,ρ)+j(N,v,ρ)=ρ(ij)iN\{j},vij,ρρ(i,j) 0ij +(1−ρ(ij)) iN,v,ρρ(ij) 0+jN,v,ρρ(ij) 0. If im(ρ) ={1}then ρis identified with a cooperation structure L, and the axiom says for every pair of connected players i, j ∈Nwith ij ∈L, i(N,v,L)+j(N,v,L)=iN\{j},vij,Lij.(9) The amalgamation property is extended also to the isolated players. When two players are isolated one can delegate to the other. Isolated amalgamation. Let (N, v, ρ)be a game with a proximity relation. If i, j ∈Nare different isolated players then i(N,v,ρ)+j(N,v,ρ)=iN\{j},vij,ρij. Players in the group involved in the amalgamation of i, jdifferent to them should not change their payoffs while this situation is working. This fact is the goal of the next axiom, which was introduced for the Banzhaf–Owen value by Amer et al. [24]. Let i, j, l ∈Nbe three different players. Number r{i,j,l}= {∃T∈N/[ρ]t;i,j,l∈T} t represents the maximal level such that this triple of players are in the same group. Fuzzy amalgamation neutrality. If (N, v, ρ)is a game with a proximity relation and i, jdifferent players with ρ(ij) >0then for all l = i, jwith r{i,j,l}<ρ(ij) l(N,v,ρ)=ρ(ij)−r{i,j,l}lN\{j},vij,ρρ(ij) r{i,j,l}ij +1+r{i,j,l}−ρ(ij)lN,v,ρρ(ij) r{i,j,l}. If im(ρ) ={1}then ρis identified with a cooperation structure Land the axiom says l(N, v, L) =lN\{j},vij,Lij. Modified fairness was introduced in [9] as a modification of the classical axiom of fairness [5]. Suppose a crisp situation, namely a cooperation structure L, and ij ∈L. Let S∈N/Lsuch that ij ∈LS. Fairness is not an option because the number of components can change. If we delete ij then we denote by Si ij, Sj ij the components in which Sis divided containing iand jrespectively (they can be the same). Moreover, let Ni ij =(N\S) ∪Si ij and Nj ij =(N\S) ∪Sj ij. Casajus proposed that the 5It is possible to define amalgamation when the level of closeness between the players is not 1 but we will not need it. J.R. Fernández et al. / International Journal of Approximate Reasoning 88 (2017) 192–208 199 difference between the payoffs with and without the link of both of the involved players is the same if we do not consider the action of the new union without each player. Namely, we obtain fairness if we delete the new component with the other player if it exists (otherwise it coincides with fairness) i(N,v,L)−i(Ni ij,v,LNi ij \{ij})=j(N,v,L)−j(Nj ij,v,LNj ij \{ij}). (10) We extended the modified fairness to a fuzzy situation given by a proximity relation in [12]. In this case, we take into account the mere reduction of the relation between two players. So we have to consider that this reduction of level only concerns to the interval between the reduced level and the original one. Let ρbe a proximity relation over a set of players N with im(ρ) ={λ1<···<λ m}and λ0=0. Consider i, j ∈Ntwo different players with ρ(ij) =λk>0 and let ρ∗(ij) =λk−1. If ρis a proximity relation6and ρ(ij) =1then ρ−ij denotes a new proximity relation with ρ−ij(ij) =0 and ρ−ij =ρ otherwise. The proximity relation ρρ(ij) ρ(ij)−t−ij in the following axiom focuses on the relation in the interval where the closeness of ij is reduced. All the cuts of ρρ(ij) ρ(ij)−t−ij use the same Ni ij and Nj ij if t∈(0,ρ(ij)−ρ∗(ij)]because ρρ(ij) ρ(ij)−tis crisp. Modified fuzzy fairness Let (N, v, ρ)be a game with a proximity relation and i, j ∈Nwith ρ(ij) >0. For each t∈(0,ρ(ij)−ρ∗(ij)] it holds i(N,v,ρ)−j(N,v,ρ)=(1−t)iN,v,ρρ(ij) ρ(ij)−t−jN,v,ρρ(ij) ρ(ij)−t +tiNi ij,v,ρρ(ij) ρ(ij)−t−ijNi ij−jNj ij,v,ρρ(ij) ρ(ij)−t−ijNj ij. We prove now that our value satisfies all the axioms described in the above subsection. Theorem 3. The prox-Banzhaf value satisfies dummy isolated player, fuzzy amalgamation, isolated amalgamation, fuzzy amalgamation neutrality and modified fuzzy fairness. Proof. Suppose always L ⊆LNwith N/L ={N1, ..., Nm}and M={1, ..., m}. Let also ρbe a proximity relation. Dummy isolated player. Let i ∈Nbe a dummy player in vand iisolated in L. We take N1={i}. Remember that the Banzhaf value satisfies the dummy player property [23]. Besides set 1is a dummy player for game vN/L, in fact we have for each Q⊆M\{1} vN/L(Q∪{1})−vN/L(Q)=v⎛ ⎝ q∈Q Nq∪{i}⎞ ⎠−v⎛ ⎝ q∈Q Nq⎞ ⎠=v({i})=v(N/L){i}({1}). Therefore v1({i}) =β1(M, vN/L) =vN/L({1}) =v({i}). The graph Banzhaf value satisfies isolated player [19], if iis isolated (dummy or not) in Lthen ηi(N, v, L) =v({i}), thus βco i(N,v,L)=ηi(N1,v1,LN1)=v1({i})=v({i}). Now suppose ρa proximity relation and our dummy player iisolated in ρ. We have {i} ∈N/[ρ]tfor all t∈(0, 1], so βco i(N, v)([ρ]t) =v({i})for each t∈(0, 1]. Using (C4) we get Bi(N, v, ρ) =v({i}). Fuzzy amalgamation. Let ij ∈Lwith i, j ∈N1. We will prove the claim βco i(N,v,L)+βco j(N,v,L)=βco iN\{j},vij,Lij. The graph Banzhaf value satisfies amalgamation [19] for two players in a link, thus βco i(N,v,L)+βco j(N,v,L)=ηiN1,v1,LN1+ηjN1,v1,LN1 =ηiN1\{j},(v1)ij ,LN1ij. On the other hand, βco iN\{j},vij,Lij=ηi(N\{j})1,vij1,LijN1. Since i, j ∈N1, then (N\{j})1=N1\{j}and LN1ij =LijN1. Observe that the amalgamation of two players i, jconnected by a link does not change the number of components of the graph. Namely, if we merge i, j ∈N1we have (N\{j})/Lij ={N1\{j}, N2, ..., Nm}. We will see that 6It is also possible to define this operation when the level is not 1. 200 J.R. Fernández et al. / International Journal of Approximate Reasoning 88 (2017) 192–208 (v1)ij =vij1. For any T⊆N1\{j}we have that (v1)ij(T) =v1(T∪{j})if i ∈Tand (v1)ij(T) =v1(T) =if i /∈T. So, Definition 2 implies (v1)ij (T)= β1M,v(N/L)T∪{ j},if i∈T β1M,v(N/L)T,if i/∈T. Moreover, for all T⊆N1\{j}, vij1(T) =β1M,vij(N\{ j}/Lij)T. We distinguish two cases: 1) If i ∈T, we prove the equality vij(N\{ j}/Lij)T=v(N/L)T∪{ j}. Let Q⊆M. If 1 /∈Qit holds vij !q∈QNq=v !q∈QNq, and if 1 ∈Qthen vij ⎛ ⎝T∪ q∈Q\{1} Nq⎞ ⎠=v⎛ ⎝T∪{j}∪  q∈Q\{1} Nq⎞ ⎠. Thus vij(N\{ j}/Lij)T(Q)=⎧ ⎨ ⎩ v!q∈QNq,if 1 /∈Q vT∪{j}∪!q∈Q\{1}Nq,if 1 ∈Q⎫ ⎬ ⎭ =v(N/L)T∪{ j}(Q). 2) If i /∈T, we prove the equality vij(N\{ j}/Lij)T=v(N/L)T. Let Q⊆M, then if 1 /∈Qit holds vij !q∈QNq=v !q∈QNq, and if 1 ∈Qwe get vij ⎛ ⎝T∪⎛ ⎝ q∈Q\{1} Nq⎞ ⎠⎞ ⎠=v⎛ ⎝T∪⎛ ⎝ q∈Q\{1} Nq⎞ ⎠⎞ ⎠. Thus vij(N\{ j}/Lij)T(Q)=⎧ ⎨ ⎩ v!q∈QNq,if 1 /∈Q, vT∪!q∈Q\{1}Nq,if 1 ∈Q⎫ ⎬ ⎭ =v(N/L)T(Q). Therefore the claim is true. Now let ρbe a proximity relation and ρ(ij) =t>0. Lemma 2 implies Bi(N,v,ρ)+Bj(N,v,ρ)=tρt 0d(βco i(N,v)+βco j(N,v)) +(1−t)%BiN,v,ρt 0+BjN,v,ρt 0&. We denote im ρt 0={λ1<···<λ m}. By the claim we have ρt 0d(βco i(N,v)+βco j(N,v)) = m  k=1λk−λk−1βco iN,v,%ρt 0&λk+βco jN,v,%ρt 0&λk = m  k=1λk−λk−1βco iN\{j},vij,%ρt 0&λkij. Obviously im(ρij) ⊆im(ρ)thus we can write using (C6) Bi(Nij,vij,(ρt 0)ij)= m  k=1λk−λk−1βco iNij,vij,(ρt 0)ijλk. If we prove the equality %(ρt 0)ij&r=%ρt 0&rij for all r∈(0, 1]then the proof is finished. The links in both sets without player iare the same because the amalgamation does not affect. So, let ik ∈%(ρt 0)ij&r. We have ρt 0(ik) ∨ρt 0(jk) ≥rand then one of them, for instance ik satisfies ρt 0(ik) ≥r. But then ik ∈[ρt 0]rand ik ∈%ρt 0&rij. The other inclusion follows in the same way. Isolated amalgamation. Take two players i, j ∈Nwho are isolated in L. Consider N1={i}and N2={j}. Following the proof of dummy isolated player and using that Banzhaf value satisfies amalgamation [25] for any pair of players, we get J.R. Fernández et al. / International Journal of Approximate Reasoning 88 (2017) 192–208 207 Fig. 6. Comparative of Banzhaf indices (III). then B(N, v, ρ) =βow(N, v, N/L). If L =LNthen B(N, v, ρ) =β(N, v). 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