Cooperative Games and Coalition Cohesion Indices: The Choquet–Owen Value

1 Cooperative games and coalition cohesion indices: the Choquet-Owen value M. G. Fiestras-Janeiro, J. M. Gallardo, A. Jim´ enez-Losada and M. Mosquera Abstract In a cooperative game with transferable utility, it is usually assumed that all coalitions are equally feasible. However, if we deal with cooperative games with coalition configuration, only some coalitions are a priori feasible, due to the preferences of the agents. In this paper we propose a generalization of games with coalition configuration. In our model, the feasibility of a coalition is determined by the cohesion of its members and, obviously, this cohesion does not have to be equal for all coalitions. The cohesion of each coalition will be determined by a cohesion index. We introduce the class of games with cohesion index and propose an allocation rule, which is characterized by using reasonable properties. The cohesion idea is not only a concept related to social groups. In software design this concept explains the relationships among all the elements of a module. Our value can be applied in this way as we show in the paper. Index Terms game theory, Owen value, coalition configuration, group cohesion, software cohesion J.M. Gallardo and A. Jim´ enez-Losada are in University of Seville, Spain. E-mail: [email protected] M.G. Fiestras-Janeiro and M. Mosquera are in University of Vigo, Spain. 2 I. INTRODUCTION Cooperative game theory studies situations where a set of agents (players) bargain to allocate a common profit obtained from their collaboration. The resulting allocation is given by a vector (payoff vector) whose coordinates are the payoffs assigned to the players. A game on a set of players is modelled by a function (characteristic function) that assigns to each subset of players (coalition) the profit that the members of the coalition can achieve when they decide to cooperate. The Shapley value [1] is one of the most studied solutions for cooperative games. It is a correspondence that assigns a payoff vector to each game based on a set of reasonable conditions (axioms) which allow us to compare this value with alternative allocations. Several variations of the Shapley value have been proposed for situations where some additional information is known. Frequently this additional information is about the interpersonal relationships among the players. This paper focuses on the Owen variation [2]. In that paper, it is considered that there exists a partition of the set of players based on the affinities between them. Owen proposed a Shapley-type solution which takes into account that information in order to get a fair allocation of the profit obtained by the grand coalition. Later on Albizuri et al. [3] introduced coalition configurations as a way of modelling exogenous information. A coalition configuration is a family of coalitions, not necessarily disjoint, whose union is the whole set of players. They defined an Owen-type value for these situations, named the configuration Owen value. The term group cohesion (see Beal et al. [4]) refers to the cumulative effect of all the factors causing members of a group to stay in it, the ”social glue” that binds a group together while working towards a goal or satisfying the needs of its members. Hence, cohesiveness is a measure of the attraction of the group to its members and their resistance to leaving it. A cohesion index assigns a number between 0 and a maximum level to the groups in order to compare their 3 cohesiveness. A coalition configuration can be seen as a family such that the coalitions in the family have the maximum level of cohesiveness whereas the other coalitions are not cohesive at all. Group cohesion has been studied in different fields, for instance, Sociology and Group Psicology (see Carless et al. [5]), sports (see Gammage et al. [6]), software design (see Bieman and Ott [7]), . . . These studies proposed different measures of cohesiveness and studied the impact of cohesiveness on the performance of a task or project showing the importance of the cohesiveness amongst all the agents involved in the project so as to achieve good results. In our paper we go a step further in the following sense. Suppose that several departments of a company are involved in a project. Once the project is finished and some benefits/costs arise, we are interested in assigning a portion of them to each worker, taking into account the cohesiveness of the departments. To do this, we will extend the model of transferable utility cooperative games. As an example of application of our model, we will consider, in software design, the cost of design, development and/or maintenance of a program, where each one of its elementary units (modules) is considered as a player. In this setting, a coalition is a set of modules and its cost is given by the sum of its individual costs if there is no connection among them; if the elements are connected the cost will be determined taking into consideration the cohesiveness of the coalition. The final goal is to split the total cost among the elements by using a cohesion measure. We can also take a reverse perspective and model the benefit obtained from creating some elements together. In this paper, our goal is to provide a framework to deal with those situations. We introduce games with coalition cohesion index. A cohesion index will be given by a function on the family of all coalitions. We also provide and characterize a sharing value for games with coalition index. Our model extends that of games with coalition configuration structure and our value extends 4 the configuration Owen value. The paper is organized as follows. Section 2 is devoted to some preliminaries on cooperative games, coalition configurations and fuzzy sets. Cohesion indices are introduced in Section 3. In Section 4 we define the Choquet-Owen value for games with coalition cohesion index. This value is characterized in Section 5. Finally, some conclusions are presented. II. PRELIMINARIES A. Cooperative games Acooperative game (with transferable utility) is a pair (N, v)where Nis a set of cardinality n∈Nand v: 2N→Ris a function with v(∅)=0. The elements of Nare called players, the subsets S⊆Ncoalitions and v(S)is the worth of S. Let (N, v)be a game. The game is monotone if v(T)≤v(S)for all T⊆S⊆N. A player i∈Nis said to be a null player for (N, v)if it satisfies v(S) = v(S\ {i})for all S⊆Nwith i∈S. Two players i, j ∈Nare symmetric for the game (N, v)if v(S∪ {i}) = v(S∪ {j})for all S⊆N\ {i, j}. If T⊆Nis a non-empty coalition, the unanimity game (N, uT)is given by uT(S)=1if T⊆Sand uT(S)=0 otherwise. The unanimity games can generate all the characteristic functions in the sense that each game (N, v)satisfies that there exist numbers {αT}∅6=T⊆N⊂R(named dividends, see [8]) such that v=X ∅6=T⊆N αTuT. Apayoff vector for the game (N, v)is any x∈RNwhere, for each player i∈N, the number xirepresents the payment of iowing to his cooperation possibilities in the game. A value for cooperative games assigns to each game (N, v)a payoff vector in RN. The Shapley value [1] 5 of a game (N, v)is defined for any player i∈Nas Shi(N, v) = X {S⊆N:i∈S} γs n[v(S)−v(S\ {i})] , where γs n=(n−s)!(s−1)! n!with s=|S|. This value is the only one satisfying the following conditions: (S1) Efficiency. It holds that Pi∈NShi(N, v) = v(N)for every game (N, v). (S2) Linearity. For all games (N, v1),(N, v2)and for all a1, a2∈Rit holds that Sh(N, a1v1+a2v2) = a1Sh(N, v1) + a2Sh(N, v2). (S3) Null player property. If i∈Nis a null player for a game (N, v)then Shi(N, v)=0. (S4) Equal treatment property. If i, j are symmetric players for a game (N, v)then Shi(N, v) = Shj(N, v). B. Coalition configurations Let Nbe a finite set of players in a game situation. Following Albizuri et al. [3], a coalition configuration of Nis a family of non-empty coalitions C={C1, ..., Cm}satisfying Sm p=1 Cp= N. These coalitions, that we name groups in the paper, represent a priori unions of the players caused by common interests or social relationships. A coalition structure in the sense of Owen [2] is a particular case of coalition configuration C, whenever Cis a partition of N, that is, for all p, q ∈ {1, . . . , m}with p6=qit holds that Cp∩Cq=∅. A game with coalition configuration is a triple (N, C, v)where (N, v)is a cooperative game and Cis a coalition configuration of N. 6 Since SC=Nwe will denote (C, v)instead of (N, C, v). Aconfiguration value assigns a payoff vector to each game with coalition configuration. That is, if ψis a configuration value then ψ(C, v)is a payoff vector in RNfor each game with coalition configuration (C, v). Particularly the (configuration) Owen value [3] is a configuration value based on the Shapley value. It is defined following a heuristic development in two steps of bargaining: the first one among the groups and the second one into each group. Let (C, v)be a game with coalition configuration. If C={C1, ..., Cm}then we denote M={1, ..., m}. The quotient game is the cooperative game (M, vC)given by vC(Q) = v [ p∈Q Cp!for every Q⊆M. (1) Consider p∈M. For each S⊆Cpwe take the coalition configuration Cp(S) = {C1, ..., Cp−1, S, Cp+1, ..., Cm}.(2) A game (Cp, vC p)is defined by vC p(S) = Shp(M, vCp(S))for every S⊆Cp.(3) Finally, the Owen value of (C, v)is given by φi(C, v) = X {p∈M:i∈Cp} Shi(Cp, vC p)for every i∈N. (4) The Owen value satisfies the properties of efficiency (O1), linearity (O2) and null player (O3), which are analogous to (S1), (S2) and (S3) respectively, regardless of the coalition configuration. Albizuri et al. [3] introduced three other properties to axiomatize the Owen value: anonymity, coalitional symmetry and merger. 7 (O4) Anonymity. If πis a permutation of Nsuch that πCp=Cpfor all p∈Mthen φi(C, πv) = φπi(C, v)for each i∈N, with πv(S) = v(πS)for all S⊆N. Anonymity implies that φsatisfies a property similar to the equal treatment property satisfied by the Shapley value. If i, j ∈Nare symmetric players for (N, v)and i∈Cpif and only if j∈Cpwhenever Cp∈ C, then φi(C, v) = φj(C, v). To show this it is enough to consider the permutation π(i) = j,π(j) = iand π(k) = kfor any other player, and apply the anonymity property. Two groups Cp, Cq∈ C are symmetric for (C, v)if v(Sr∈QCr∪Cp) = v(Sr∈QCr∪Cq)for all Q⊆M\ {p, q}. (O5) Coalitional symmetry. For every symmetric groups Cp, Cq∈ C it holds that X i∈Cp φi(C, v) = X i∈Cq φi(C, v). Two players i, j ∈Nare double for (C, v)when: 1) v(S∪ {i}) = v(S∪ {j})for all S⊆N and 2) if there is a coalition S⊆N\ {i, j}with S∪ {i}∈Cthen S∪ {j}, S ∪ {i, j}/∈ C. Double players have the same influence jointly or individually. If i, j are double for (C, v)then Ci −j= (C \ {Cp∈ C :j∈Cp})∪ {(Cp\ {j})∪ {i}:Cp∈ C, j ∈Cp}is a coalition configuration of N\ {j}. Consider (Ci −j, vi −j)where vi −j(S) =        v(S∪ {j})if i∈S, v(S)otherwise. (5) (O6) Merger. If i, j ∈Nare double then φk(C, v) = φk(Ci −j, vi −j)for all k∈N\ {i, j}. 8 C. Fuzzy sets and the Choquet integral The functions minimum and maximum will be denoted by ∧,∨respectively throughout this paper. Fuzzy sets were described by Zadeh [9]. Let Abe a finite set. A fuzzy subset of Ais a mapping ρ:A−→ [0,1] where ρassigns to a∈Aa degree of membership. Given a fuzzy set ρof A, the support of ρis the set supp (ρ) = {a∈A:ρ(a)>0}and the image of ρis the set im (ρ) = {ρ(a) : a∈A}. We will denote the image of ρas im (ρ) = {λ1<· · · < λq}when we want to consider that its elements are written in increasing order. The fuzzy sets ρ, ρ0of A are called comonotone when (ρ(a)−ρ(b)) (ρ0(a)−ρ0(b)) ≥0for all a, b ∈A. For each number t∈[0,1] the t-level set is Aρ t={a∈A:ρ(a)≥t}.(6) Aset function over Ais any function f: 2A→R. Observe that if f(∅) = 0 then (A, f)is a game. The Choquet integral [10] was introduced for monotone games over A(capacities). Later on Schmeidler [11] studied this integral for all the set functions. If fis a set function and ρa fuzzy set of A, then the Choquet integral of ρwith respect to fis the continuous operator Zρ df = r X k=1 (λk−λk−1)fAρ λk,(7) where im (ρ) = {λ1<· · · < λr}and λ0= 0. The following properties of the Choquet integral are known: (C1) For all non-empty B⊆A,R1Bdf =f(B)with 1B(a)=1if a∈Band 1B(a)=0 otherwise. (C2) If f(B) = Kfor all non-empty B⊆A, then Rρ df =KWa∈Aρ(a). (C3) Rtρ df =tRρ df, for t∈[0,1] . 9 (C4) Rρ d (a1f1+a2f2) = a1Rρ df1+a2Rρ df2. (C5) R(ρ+ρ0)df =Rρ df +Rρ0df, when ρ(a) + ρ0(a)≤1and ρ, ρ0are comonotone. III. COALITION COHESION INDICES Let Nbe a set of agents with |N|=n∈N. Suppose that a social analysis of the cohesiveness for each coalition, taking into consideration all the relevant facets of the situation, has been done and we describe the results of the study by a cohesion index where the most cohesive coalition has level 1. The index of a singleton coalition means the capacity of the agent to keep his positions without the support of the others. Definition 1. A coalition cohesion index over Nis a function ρ: 2N→[0,1] such that ρ(∅)=0 and there exists S⊆Nwith ρ(S)=1. This definition corresponds to that of cooperation index, introduced by Amer et al. [12], but the sense is totally different. In a cooperation index ρthe number ρ(S)is interpreted as the probability that coalition Sis formed, and then the Owen-type value proposed in that case is just an expected payoff; it is not an allocation of the total profit from cooperation. A coalition cohesion index ρover Nis also a fuzzy set of 2N(or a fuzzy hypergraph for N, see [13]) with special conditions, thus we can use all the notations and definitions from that context. The coalitions in supp(ρ)are the cohesive coalitions, which are called groups, and im(ρ)is the set of the different levels of cohesiveness. Every non-empty family of non-empty coalitions C ⊂ 2Ncan be identified with the coalition cohesion index ρCover Nwith ρC(S)=1 if S∈ C and ρC(S) = 0 otherwise. That is, the coalitions in the family are well cohesive and the others are not cohesive. In particular, we can identify any coalition configuration Con Nwith its corresponding coalition index ρC. Moreover, if we have a coalition cohesion index ρover N 16 index. Efficiency. For all (N, ρ, v)game with coalition cohesion index it holds that X i∈N Ψi(N, ρ, v) = v(N). We consider that buster players should have null payoff. Buster player property. If i∈Nis a buster player for the coalition cohesion index ρthen Ψi(N, ρ, v) = 0 for any game (N, v). Linearity. For all games (N, ρ, v1),(N, ρ, v2)and numbers a1, a2∈Rit holds that Ψ(N, ρ, a1v1+a2v2) = a1Ψ(N, ρ, v1) + a2Ψ(N, ρ, v2). The maximal cohesion degree of a player in a coalition cohesion index ρover Nis defined as ρi=_ {S⊆N:i∈S} ρ(S). A game with a coalition cohesion index (N, ρ, v)is a-null with a∈(0,1] if v(S) = 0, for every S⊆Nwith Wi∈Sρi< a. In particular, v(S) = 0 for every S⊆B(ρ). Given a null player in (N, v), his payoff in an a−null game (N, v, ρ)should be proportional to his payoff in the simplification to the interval [a, 1]. Fuzzy null player property. If i∈Nis a null player in (N, v)and (N, v, ρ)is an a-null game with coalition cohesion index, then Ψi(N, ρ, v) = (1 −a)Ψi(N, ρ1 a, v). If two symmetric players are also symmetric for the index from certain cohesiveness level, then 17 the difference between the payoffs obtained in a game with a coalition cohesion index should be proportional to the difference of the payoffs in the simplification to the interval [0, a]. Fuzzy equal treatment property. Let i, j be two symmetric players for the game (N, v)and ρa coalition cohesion index. If there exists a∈[0,1) such that for every coalition Swith ρ(S)> a it holds that i∈Sif and only if j∈S, then Ψi(N, ρ, v)−Ψj(N, ρ, v) = a[Ψi(N, ρa 0, v)−Ψj(N, ρa 0, v)] . The concept of symmetry for players can be extended to groups in the context of coalition cohesion indices. Let (N, ρ, v)be a game with a coalition cohesion index. Two coalitions S, T ∈ supp(ρ)are separable symmetric groups if: •v(S∪R) = v(T∪R)for all R⊆N\(S∪T)and, •S∩R=T∩R=S∩T=∅for all R∈supp(ρ)\ {S, T}. Separable symmetric groups property. Let S, T ⊆Nbe separable symmetric groups for (N, ρ, v)with 0< ρ(T)≤ρ(S). It holds X i∈S [Ψi(N, ρ, v)−[ρ(S)−ρ(T)]Ψi(N, ρρ(S) ρ(T), v)] = X j∈T Ψj(N, ρ, v). Let a∈[0,1). Two players i, j ∈Nare a-double for (N, ρ, v)if: •there are coalitions S, T ⊆N(they can be equal) with ρ(S), ρ(T)> a and i∈S, j ∈T, •v(S∪ {i}) = v(S∪ {j})for all coalition S⊆Nand •if S⊆N\ {i, j}with ρ(S∪ {i})> a then ρ(S∪ {j}), ρ(S∪ {i, j})≤a. If i, j are a-double then ρi −jis another coalition cohesion index over N\ {j}defined for all 18 S⊆N\ {j}as ρi −j(S) =        ρ(S)∨ρ((S\ {i})∪ {j})if i∈S, ρ(S)otherwise. We also consider the game (N\ {j}, vi −j)defined in Expression 5. Merger of players. Let i, j be a-double for (N, ρ, v). It holds that Ψk(N, ρ, v) = aΨk(N, ρa 0, v) + (1 −a)Ψk(N\ {j},(ρ1 a)i −j, vi −j) for every k∈N\ {i, j}. Observe what happens with these axioms when we take Cas a coalition configuration. Efficiency, linearity, fuzzy null player property and merger of players (taking into account that im(ρ) = {0,1}in that case) coincide with the respective axioms in [3]. Moreover, the fuzzy equal treatment property and the separable symmetric groups property coincides with the equal treatment property and the group symmetry property of the Owen value defined in [2], respectively. Theorem 2. The Choquet-Owen value satisfies efficiency, linearity, fuzzy null player property, buster player property, fuzzy equal treatment property, separable symmetric groups and merger of players. Proof. We consider each of the axioms mentioned in the theorem. BUSTER PLAYER PROPERTY. Let (N, ρ, v)be a game with coalition cohesion index. If i∈ B(ρ)then i∈B(Cρ t)for all t∈im(ρ). Hence, (8) implies that φi(N, v)(Cρ t) = 0 for all t∈im(ρ)and, by definition of the Choquet integral (7), Φi(N, ρ, v)=0. EFFICIENCY. We know that the Owen value φis efficient (O1) for coalition configurations. 19 For each family C ⊆ 2Nwe get X i∈N φi(N, v)(C) = X i∈N\B(C) φi(C, v[C]) = v[C](N\B(C)) = v(N). Let (N, ρ, v)be a game with coalition cohesion index. Using (C2) and (C4) we obtain X i∈N Φi(N, ρ, v) = X i∈NZρ dφi(N, v) = Zρ d X i∈N φi(N, v) = v(N). LINEARITY. The linearity of Φfollows from the linearity of the Owen value (O2) over coalition configurations. Let (N, v1),(N, v2)be two games and a1, a2∈R. For each family of coalitions C we get the equality (a1v1+a2v2)[C] = a1v1[C]+a2v2[C]and, then, for every player i∈N\B(C) φi(N, a1v1+a2v2)(C) = φi(C,(a1v1+a2v2)[C]) = a1φi(C, v1[C]) + a2φi(C, v2[C]) =a1φi(N, v1)(C) + a2φi(N, v2)(C). If i∈B(C)then φi(N, a1v1+a2v2)(C) = 0 = a1φi(N, v1)(C) + a2φi(N, v2)(C). So, using again (C4), for a coalition cohesion index ρover N. Φi(N, ρ, a1v1+a2v2) = a1Zρ dφi(N, v1)+a2Zρ dφi(N, v2) = a1Φi(N, ρ, v1)+a2Φi(N, ρ, v2). FUZZY NULL PLAYER PROPERTY. Let (N, ρ, v)be an a-null game with coalition cohesion index and i∈Na null player for (N, v). If i∈B(ρ) = B(ρa 0)⊆B(ρ1 a)we have the equalities Φi(N, ρ, v) = Φi(N, ρa 0, v) = 0 = Φi(N, ρ1 a, v)because Φsatisfies the buster player property. Suppose i /∈B(ρ). We will calculate Φi(N, ρa 0, v). Firstly, B(ρa 0) = B(ρ)and (N, ρa 0, v)is also a-null. Let t∈im(ρa 0)with t<a. Then, B(ρa 0)⊆B(ρa t),ρa 0(B(ρa t)) < t and v(B(ρa t)) = 0. 20 Secondly, as iis null player, v[Cρa 0 t](S∪ {i}) = v(S∪B(Cρa 0 t)∪ {i}) = v(S∪B(Cρa 0 t)) = v[Cρa 0 t](S) for every S⊆N\B(Cρa 0 t). Then, iis also a null player in (Cρa 0 t, v[Cρa 0 t]) for every t∈im(ρa 0)with t < a. Since the Owen value satisfies the null player property we obtain that φi(N, v)(Cρa 0 t) = 0. Using (7) we conclude Φi(N, ρa 0, v)=0. Finally, since ρ=aρa 0+ (1 −a)ρ1 aand using Lemma 1 item 3 we get Φi(N, ρ, v) = (1 −a)Φi(N, ρ1 a, v). FUZZY EQUAL TREATMENT PROPERTY. Let (N, ρ, v)be a game with a coalition index. Let i, j be symmetric players for the game (N, v)and a∈[0,1) such that if ρ(S)> a, then i∈S iff j∈S. If i, j ∈B(ρ), the result is clearly true because i, j ∈B(ρa 0). Let i, j 6∈ B(ρ). Using that ρ=aρa 0+ (1 −a)ρ1 aand using Lemma 1 item 3, we have Φi(N, ρ, v) = aΦi(N, ρa 0, v) + (1 −a)Φi(N, ρ1 a, v), Φj(N, ρ, v) = aΦj(N, ρa 0, v) + (1 −a)Φj(N, ρ1 a, v). Consider any t∈im(ρ1 a). We have Cρ1 a t=Cρ t(1−a)+a, because ρ1 a(S)≥tiff ρ(S)−a 1−a≥t. Since iand jare symmetric players in (Cρ1 a t, v[Cρ1 a t]), if ρ1 a(S)≥twe have i∈Siff j∈S, and φ satisfies equal treatment of symmetric players, then φi(N, v)(Cρ1 a t) = φj(N, v)(Cρ1 a t). Hence, by (7), Φi(N, ρ1 a, v)=Φj(N, ρ1 a, v)and Φi(N, ρ, v)−Φj(Nρ, v) = a[Φi(N, ρa 0, v)−Φj(N, ρa 0, v)] . SEPARABLE SYMMETRIC GROUPS. Let Cp, Cq∈ C with C={Cp0}m p0=1 ⊆2Na non-empty family of non-empty coalitions and M={1, ..., m}. Suppose Cp, Cqseparable symmetric coalitions for (N, C, v), that is Cp, Cqare disjoint and disjoint with the rest of the groups. 21 We prove that p, q are symmetric players for v[C]C. For each Q⊆M\ {p, q}we have v[C]C(Q∪ {p}) = v [ p0∈Q Cp0∪Cp∪B(C)!=v [ p0∈Q Cp0∪Cq∪B(C)!=v[C]C(Q∪ {q}). From the facts that the Shapley value is efficient (S1) and satisfies the equal treatment property (S4), and using Theorem 2, we obtain X i∈Cp Φi(N, C, v) = X i∈Cp Shi(Cp, vC p) = vC p(Cp) = Shp(M, v0 C) =Shq(M, v0 C) = X j∈Cq Φi(N, C, v). Now, we take S, T separable symmetric groups for (N, ρ, v)with 0< ρ(T)≤ρ(S). Using Lemma 1 item 3, we get for each i∈S, Φi(N, ρ, v) = ρ(T)Φi(N, ρρ(T) 0, v)+(ρ(S)−ρ(T))Φi(N, ρρ(S) ρ(T), v) + (1 −ρ(S))Φi(N, ρ1 ρ(S), v). Since iis a buster player for ρ1 ρ(S)and the Choquet-Owen value satisfies the buster player property, Φi(N, ρ1 ρ(S), v)=0. If j∈Tby the same reasoning we obtain Φj(N, ρ, v) = ρ(T)Φj(N, ρρ(T) 0, v) + (1 −ρ(T))Φj(N, ρ1 ρ(T), v) = ρ(T)Φj(N, ρρ(T) 0, v), because jis a buster player for ρ1 ρ(T). As S, T are separable symmetric groups for Cρρ(T) 0 t, for all t∈im(ρρ(T) 0), it holds X i∈S Φi(N, ρρ(T) 0, v) = X j∈T Φj(N, ρρ(T) 0, v). MERGER OF PLAYERS. Suppose i, j are double players (0-double) for (N, C, v)with C= {Cp}m p=1 ⊆2Na non-empty family of non-empty coalitions and M={1, ..., m}. We denote 22 Qi={p∈M:i∈Cp}. Consider the new family of 2N\{j} Ci −j= (ρC)i −j={C∗ p}m p=1 ={Cp:p∈M\Qj}∪{(Cp\ {j})∪ {i}:p∈Qj}. Observe that |Ci −j|=|C|, because i, j are double players, and then Malso represent the set of subindices of the groups in Ci −j. Moreover, B(Ci −j) = B(C). Let k∈N\ {i, j}. If k∈B(C) then φk(N, v)(C) = 0 = φk(N\ {j}, vi −j)(Ci −j). It is easy to check that v[C]i −j=vi −j[Ci −j]. If k /∈B(C), we get φk(N, v)(C) = φk(Ci −j, v[C]i −j) = φk(Ci −j, vi −j[Ci −j]) = φk(N\ {j}, vi −j)(Ci −j) because the Owen value satisfies merger (O6). Applying (6), we obtain Φk(N, C, v)=Φk(N\ {j},Ci −j, vi −j),for every k∈N\ {i, j}. Now consider ρa coalition cohesion index and i, j a pair of a−double players for some a∈(0,1). By Lemma 1 item 3, we obtain for (N, ρ, v)and k∈N\ {i, j} Φk(N, ρ, v) = aΦk(N, ρa 0, v) + (1 −a)Φk(N, ρ1 a, v). Let t∈im(ρ1 a). We will see that i, j are a-double players for (N, Cρ1 a t, v). For each S⊆N\{i, j} with S∪{i}∈Cρ1 a tthen ρ1 a(S∪{i})≥tiff ρ(S∪ {i})> a. Hence ρ(S∪ {j}), ρ(S∪ {i, j})≤a implies ρ1 a(S∪ {j}) = ρ1 a(S∪ {i, j})=0< t and S∪ {j}, S ∪ {i, j}/∈ Cρ1 a t. We have the equality φk(N, v)(Cρ1 a t) = φk(N\ {j}, vi −j)((Cρ1 a t)i −j)for every k∈N\ {i, j}, because i, j are double players in the game (Cρ1 a t, v[Cρ1 a t)]) and (O6). But (Cρ1 a t)i −j=C(ρ1 a)i −j tas we show next. Let C∈(Cρ1 a t)i −jwith i∈C(otherwise the result is trivial). By the construction of (Cρ1 a t)i −j, we have ρ1 a(C)≥tor ρ1 a(C\ {i} ∪ {j})≥t. Equivalently, (ρ1 a)−j,i(C)≥tiff C∈ C(ρ1 a)i −j t. Then, Φk(N, ρ1 a, v)=Φk(N\ {j},(ρ1 a)i −j, vi −j)for every k∈N\ {i, j}by (7).  23 In the following theorem we prove that the Choquet-Owen value is the only one satisfying the seven axioms above. Theorem 3. The Choquet-Owen value is the only cohesion value satisfying efficiency, linearity, fuzzy null player property, buster player property, fuzzy equal treatment property, separable symmetric groups property and merger of players. Proof. Let Ψbe a cohesion value satisfying the axioms in the theorem. Using linearity, it is enough to prove the result for unanimity games. Let ρbe a coalition cohesion index over N with supp(ρ) = {Cp}m p=1 and M={1, ..., m}. It is clear that Ψi(N, ρ, uT) = Φi(N, ρ, uT), for every i∈B(ρ). It remains to prove the result for non-buster players. We define L(ρ) = [ {p,q∈M:p6=q} Cp∩Cq and l(ρ) = |L(ρ)|. We prove uniqueness by induction on l(ρ). BASE CASE. Let l(ρ)=0. Then, Cp∩Cq=∅, for every p, q ∈M. Let us consider T⊆Nand the unanimity game (N, uT). We prove the result for non-buster players by a second induction on the size of |im(ρ)|. Let MT={p∈M:Cp∩T6=∅}. Notice that if MT=∅, then T⊆B(ρ); otherwise, T\B(ρ)6=∅. BASE SUBCASE. If |im(ρ)|= 1 then Cis a non-empty family of non-empty coalitions and ρ(Cp) = 1, for every p∈M. We consider two cases. •MT=∅. Then, (N, uT)is not a-null game for any a∈(0,1] because uT(T) = 1 and Wi∈Tρi= 0 < a. Every pair of players i, j 6∈ B(ρ)are symmetric in the game (N, uT). Let p∈M. Then, ρ(Cp)=1> a = 0 and applying the fuzzy equal treatment property, we get Ψi(N, ρ, uT)=Ψj(N, ρ, uT) 24 for every i, j ∈Cp. In addition, every pair of coalitions Cp, Cq∈supp(ρ)are separable symmetric groups with ρ(Cp) = ρ(Cq) = 1. Then, |Cp|Ψi(N, ρ, uT) = X l∈Cp Ψl(N, ρ, uT) = X l∈Cq Ψl(N, ρ, uT) = |Cq|Ψj(N, ρ, uT) with i∈Cpand j∈Cq. Finally, if we apply the efficiency property, we obtain Ψi(N, ρ, uT) = 1 m 1 |Cp|= Φi(N, ρ, uT) for every i∈Cpand p∈M. •MT6=∅. Let p∈MTand i0∈T∩Cp. Then, ρi0= 1 and for every coalition S⊆N with Wi∈Sρi<1, we have i06∈ Sand uT(S)=0. Hence, (N, uT)is an 1−null game. Let i6∈ T. Then, Ψi(N, ρ, uT)=0because iis a null player in (N, uT),(N, uT)is an 1−null game and the value Ψsatisfies the fuzzy null player property. Then, ρ(Cp)=1and every pair of players i, j ∈T∩Cpare symmetric players. If we apply the fuzzy equal treatment property to the value Ψ, we obtain Ψi(N, ρ, uT)=Ψj(N, ρ, uT) for every i, j ∈T∩Cpand p∈MT. If MT={p}, then Ψi(N, ρ, uT) = 1 |T∩Cp|= Φi(N, ρ, uT) for every i∈T∩Cp. In other case, let p, q ∈MT. Then, Cpand Cqare separable symmetric groups. Using the separable symmetric groups property, we get |T∩Cp|Ψi(N, ρ, uT) = X l∈Cp Ψl(N, ρ, uT) = X l∈Cq Ψl(N, ρ, uT) = |T∩Cq|Ψj(N, ρ, uT). 25 Finally, if we apply the efficiency property, we obtain Ψi(N, ρ, uT) = 1 m 1 |T∩Cp| for every i∈T∩Cp. GENERAL SUBCASE. Suppose that the result is true when |im(ρ)|< K. Let ρbe a coalition cohesion index with |im(ρ)|=K > 1. We also consider two cases. •MT=∅. Then, (N, uT)is not a-null game for any a∈(0,1] because uT(T) = 1 and Wi∈Tρi= 0 < a. Every pair of players i, j 6∈ B(ρ)are symmetric in the game (N, uT). Let p∈M. Applying the fuzzy equal treatment property, for every i, j ∈Cpwe get Ψi(N, ρ, uT) = Ψj(N, ρ, uT)because ρ(Cp)>0. In addition, every pair of coalitions Cp, Cq∈supp(ρ)are separable symmetric groups. Then, if ρ(Cp)≤ρ(Cq)we obtain Pl∈CqΨl(N, ρ, uT)−Pl∈CpΨl(N, ρ, uT) = |Cq|Ψi(N, ρ, uT)− |Cp|Ψj(N, ρ, uT) = (ρ(Cq)−ρ(Cp)) Pl∈CqΨl(N, ρρ(Cq) ρ(Cp), uT) = (ρ(Cq)−ρ(Cp)) Pl∈CqΦl(N, ρρ(Cq) ρ(Cp), uT) =|Cq|Φi(N, ρ, uT)− |Cp|Φj(N, ρ, uT) (9) with i∈Cqand j∈Cp, where the last equality follows by the induction hypothesis. Let p0∈Mbe such that ρ(Cq)≥ρ(Cp0)for every q∈M. Finally, if we apply the efficiency property and (9) to p0and q∈M, we obtain 1 = m|Cp0|(Ψj(N, ρ, uT)−Φj(N, ρ, uT)) + 1 for every j∈Cp0. Then, Ψj(N, ρ, uT)=Φj(N, ρ, uT)for every j∈Cp0. Finally, using Equality (9), we obtain Ψi(N, ρ, uT) = Φi(N, ρ, uT), for every i∈Cqand q∈M. 32 Sh4(M2, v0 (Cρ λ2)4({2})) = 0! 5! 6! 1 + 1! 4! 6! 3 + 2! 3! 6! 2 + 3! 2! 6! 0 = 36 120 (Cρ λ2)4({1,2}) = {{1},{2},{3},{1,2},{2,3},{1,2,3}} and Sh4(M2, v0 (Cρ λ2)4({1,2})) = 0! 5! 6! 2 + 1! 4! 6! 6 + 2! 3! 6! 5 + 3! 2! 6! 1 = 76 120 The characteristic function of each group game (Cp, wρ p)are depicted in Table II. T∅ {1} {2} {3} {1,2} {1,3} {2,3}N (C1, wρ 1)01175 1584 (C2, wρ 2)04289 3960 (C3, wρ 3)02172 1980 (C4, wρ 4)065 720 78 720 143 720 (C5, wρ 5)045 720 60 720 105 720 (C6, wρ 6)078 720 88 720 208 720 (C7, wρ 7)0835 7920 1018 7920 1128 7920 1853 7920 1963 7920 2688 7920 3523 7920 TABLE II CHARACTERISTIC FUNCTION OF EACH GROUP GAME. Table III shows the amount that the Shapley value assigns to each player in each group game. 1 2 3 (C1, wρ 1)1175 1584 (C2, wρ 2)4289 3960 (C3, wρ 3)2172 1980 (C4, wρ 4)65 720 78 720 (C5, wρ 5)45 720 60 720 (C6, wρ 6)99 720 109 720 (C7, wρ 7)835 7920 1289 7920 1399 7920 TABLE III SHAPLEY VALUE FOR EACH GROUP GAME. Then, the Choquet-Owen value is obtained by adding up each column in Table III Φ(N, ρ, v) = 1,179 120,181 120. 33 Player 2and player 3are symmetric in the (N, v)but the additional information given by the cohesion function ρbreaks such symmetry. The Choquet-Owen value assigns a different value to each player. By contrast the Shapley value of the game (N, v),Sh(N, v) = (1,1.5,1.5) assigns the same value to player 2and player 3. VII. CONCLUSIONS We can see the model of games with coalition configuration as a class of games with some coalitions having maximal degree of cohesiveness, understanding this value as 1. In this paper we extend this model to situations where the cohesiveness degree of a coalition is not necessarily maximal. We present the model and also provide a value that can be applied to share the total gain taking into account the cohesiveness degree of all the coalitions. This value is inspired by the Owen value. We show by an example how this value can be used in software design. In the future we are going to explore the application of any other value or semivalue to this context. Additionally, we can study set-solutions like the core in this setting. There are some other contexts where our model can be applied. For instance, situations where also uncertainty on payoffs is present (Alparsalan-G¨ ok et al. [14], Gao et al [15], Palanci et al. [16], Branzei et al. [17], Shen and Gao [18]) or additional information like a network is provided (Fujimoto [19]). In this last case, since a network can be represented as a coalition cohesion index (1if the coalition is connected and zero, otherwise), following a similar procedure we can define globally efficient values by contrast to locally efficient values as the Myerson value (Myerson [20]). On the other way around, a coalition cohesion index defines a fuzzy hypergraph that can be seen as a conference situation of Myerson (Myerson [21]). The study of the relationship between both models might deserve some attention in the future. Acknowledgments. This research has been supported by the Ministerio de Econom´ ıa y Competitividad 34 under grants ECO2013-40755-P and MTM2014-53395-C3-3-P, and by the Ministerio de Ciencia e Innovaci´ on under project MTM2011-27731-C03-03. REFERENCES [1] Shapley L.S. (1953). A value for n-person games. In: Kuhn H.W., Tucker A.W. (eds.), Contributions to the theory of games II, Princeton University Press, New Jersey: 307-317. [2] Owen G. (1977). Values of games with a priori unions. In: Henn R., Moeschlin O. (eds.), Essays in Mathematical Economics and Game Theory, Berlin, Springer-Verlag: 76-88. [3] Albizuri M.J., Aurrecoechea J. and Zarzuelo J.M. (2005). Configuration values: extensions of the coalitional Owen value. Games and Economic Behavior 57: 1-17. [4] Beal D.J., Cohen R., Burke, M.J. and McLendon C.L. (2003). Cohesion and perfomance in groups: a meta-analytic clarification of construct relation. Journal of Applied Psichology, 88: 989-1004. [5] Carless S.A. and De Paola C. (2000). 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