Axiomatic of the Shapley value of a game with a priori unions

Axiomatic of the Shapley value of a game with a priori unions J.M. Alonso-Meijide a , B. Casas-Méndez b , A.M. González-Rueda c , S. Lorenzo-Freire d, ∗ a MODESTYA Research Group, Department of Statistics and Operations Research and Faculty of Sciences, University of Santiago de Compostela, Spain b MODESTYA Research Group, Department of Statistics and Operations Research and Faculty of Mathematics, University of Santiago de Compostela, Spain c Department of Statistics and Operations Research and Faculty of Mathematics, University of Santiago de Compostela, Spain d MODES Research Group, Department of Mathematics and Faculty of Computer Science, University of A Coruña, Spain Abstract In this paper, we dene a modication of the Shapley value for the model of TU games with a priori unions. We provide two characterizations of this value and a new characterization of the Banzhaf-Owen coalitional value. Keywords: cooperative game, Shapley value, a priori unions, coalitional value 2010 MSC: 91A12 Published in Top (2014) Published version available at: https://link.springer.com/journal/11750 DOI: 10.1007/s11750-013-0298-4 1. Introduction A value determines the payos allocated to each player in a cooperative game. Two of the most important values for cooperative games are the Shapley [25] and the Banzhaf [11] values. When information on relationships among players is available, coalitional values are the most appropriate tools to establish the payos. These agreements among players are modeled by a set of a priori unions, i.e., a partition of the set of players. Games with a priori unions were rst considered by Aumann and Drèze [10], who extended the Shapley value to this new framework in such a manner that the game really splits into subgames played by the unions, and every player receives the payo allocated to him by the Shapley value in the subgame he is playing within his union. Two coalitional values were also proposed by Owen in [22] and [23]. Alonso-Meijide et al. [2] presented a comparative study of these two coalitional values. Many other papers on the topic can be found in the literature (Winter [27], Amer and Carreras [8], Vázquez-Brage et al. [26], Hamiache [15], Amer et al. [9], Alonso-Meijide et al. [3], Gómez-Rúa and Vidal-Puga [14], Kamijo [16], or Álvarez-Mozos et al. [6], among others). The coalitional value proposed in [22] (the Owen coalitional value) divides among the players the Shapley value of the so-called quotient game whereas the proposed one in [23] (the Banzhaf-Owen coalitional value) assigns to each player the Banzhaf value of a certain modied game. In this paper, we propose a new coalitional value that assigns to each player the Shapley value of the same modied game. We provide two characterizations of this value and a new characterization of the Banzhaf-Owen coalitional value. The paper is organized as follows. Some preliminaries and the proposed coalitional value are introduced in Section 2. In Section 3 we characterize this value and the Banzhaf-Owen coalitional value. We also show ∗ Corresponding author. Postal code: 15071, A Coruña. Phone: +34 981167000. Fax: +34 981167160 Email addresses: josemaria.alo[email protected] (J.M. Alonso-Meijide), balbina.casa[email protected] (B. Casas-Méndez), angelmanuel[email protected] (A.M. González-Rueda), silvia.lore[email protected] (S. Lorenzo-Freire) Preprint submitted to Elsevier July 31, 2013 that the properties used in the characterizations are logically independent. Finally, in Section 4 we provide the multilinear extension of the new coalitional value. 2. Preliminaries 2.1. TU Games A cooperative game with transferable utility (briey, a TU game ) is a pair (N, v) dened by a nite set of players N , usually N={1, . . . , n} , and a function v: 2N→R , that assigns to each coalition of players S⊆N the real number v(S) . Moreover, it is assumed that v(∅) = 0. We denote by G the set of all TU games. Two players i, j ∈N are symmetric in a TU game (N, v) if v(S∪ {i}) = v(S∪ {j}) for all S⊆N\ {i, j} . A player i∈N is a dummy in a TU game (N, v) if v(S∪ {i}) = v(S) + v({i}) for all S⊆N\ {i} . Given a TU game (N, v) , take i, j ∈N with i=j . Let the set {i, j} be considered as a new player p∈ N and let us denote Nij = (N\ {i, j})∪ {p} . The {ij} -reduced game Nij, vij∈G of (N, v) is dened for every S⊆Nij by vij(S) = v((S\ {p})∪ {i, j}) if p∈S and vij(S) = v(S) otherwise. The {ij} -reduced game was dened by Lehrer [17]. It considers that a pair of players merges into a new player p . A value is a map f that allocates to each TU game (N, v)∈G a vector f(N, v)∈RN , where fi(N, v) is the payo of each player i∈N . Denition 2.1 The Shapley value φ is the value that assigns to each (N, v)∈G and i∈N the real number φi(N, v) = X S⊆N\{i} s! (n−s−1)! n![v(S∪ {i})−v(S)] , where n=|N| and s=|S| . Denition 2.2 The Banzhaf value β is the value that assigns to each (N, v)∈G and i∈N the real number βi(N, v) = 1 2n−1X S⊆N\{i} [v(S∪ {i})−v(S)] . 2.2. TU Games with a priori unions Let us consider a nite set of players N . A system of a priori unions P={P1, . . . , Pm} over N is a partition of N , that is, Sm k=1Pk=N and Pk∩Ph=∅ when k=h . Along this paper, we will use the trivial sytems of a priori unions over N where: i) each union is a singleton, given by Pn={{i}i∈N} or ii) a unique union is formed, given by PN={N}. If S⊆N, we will denote by PS the system of a priori unions whose members are S and all {i} such that i /∈S. Note that if S=N, PS=PN and if S={i}, PS=Pn . This particular class of systems of unions was used by Carreras et al. [12] to compare the models of games with a priori unions and games with partnerships. Given i∈N , P(i) denotes the family of a priori unions over N where {i} is a singleton union, that is, P∈P(i) if and only if {i} ∈ P . For each i, j ∈Pk∈P , Pij =nPij 1, . . . , Pij mo denotes the partition obtained from P by replacing players i and j with a new player p , i.e., Pij k= (Pk\ {i, j})∪ {p} and Pij h=Ph for h=k. For each i∈Pk∈P , P−i denotes the partition of P(i) obtained from P when player i leaves the union Pk and becomes isolated, i.e., P−i={Ph∈P:h=k}∪{Pk\ {i},{i}} . A TU game with a priori unions is a triple (N, v, P) where (N, v)∈G and P is a partition over N . We denote by Gcs the set of all TU games with a priori unions. 2 Given (N, v, P)∈Gcs with P={P1, . . . , Pm} , the associated quotient game (M, vP) is the TU game played by the unions. From now on, we will assume that player k∈M represents union Pk . So, we will consider M={1, . . . , m} and vP(R) = vSk∈{1,...,m}:k∈RPk for all R⊆M . A coalitional value is a map g that assigns to each TU game with a priori unions (N, v, P)∈Gcs a vector g(N, v, P)∈RN , where gi(N, v, P) is the payo of each player i∈N . Given a value f on G , a coalitional value g on Gcs is called a coalitional f -value when g(N, v, Pn) = f(N, v) for all (N, v)∈G . One of the coalitional values which appears in the literature is the Banzhaf-Owen coalitional value. The Banzhaf-Owen coalitional value is introduced in Owen [23] and dened by Ψi(N, v, P) = X R⊆M\{k}X T⊆Pk\{i} 1 2m−1 1 2pk−1hv(Q∪T∪ {i})−v(Q∪T)i for all i∈N and all (N, v, P)∈Gcs , where Pk∈P is the union such that i∈Pk , m=|M| , pk=|Pk| and Q=Sh∈{1,...,m}:h∈RPh . The Banzhaf-Owen coalitional value of a player i∈Pk is a weighted average of the contributions of this player to a certain type of coalitions. This value only considers contributions to coalitions formed by complete unions except the union Pk , which could be not complete. We say that these coalitions are compatible with the system of a priori unions P. In Alonso-Meijide et al. [2] it is shown that the Banzhaf-Owen coalitional value is a coalitional Banzhaf value. Moreover, it is straightforward to prove that, given (N, v, P)∈Gcs and a player i∈Pk∈P , the Banzhaf-Owen coalitional value of i can be interpreted as the Banzhaf value of this player applied to the TU game played by the unions other than Pk and by the players in Pk . Proposition 2.3 For all (N, v, P)∈Gcs and all i∈Pk , Pk∈P , Ψi(N, v, P) = βiM\k∪Pk, v(M\{k})∪Pk where v(M\{k})∪Pk(R∪T) = vSh∈{1,...,m}:h∈RPh∪T for all R⊆M\ {k} and all T⊆Pk . In this paper, we study the coalitional value which assigns to each player the Shapley value of the same modied game. Denition 2.4 The value Γ is the coalitional value that assigns to each (N, v, P)∈Gcs and each i∈N the real number Γi(N, v, P) = X R⊆M\{k}X T⊆Pk\{i} (r+t)!(m+pk−r−t−2)! (m+pk−1)! [v(Q∪T∪ {i})−v(Q∪T)] , where Pk∈P is the union such that i∈Pk , r=|R| , and t=|T| . Proposition 2.5 For all (N, v, P)∈Gcs and all i∈Pk , Pk∈P , Γi(N, v, P) = φiM\k∪Pk, v(M\{k})∪Pk. Taking into account that 1 2m−1 1 2pk−1=1 2m+pk−2 , the coecients of the Banzhaf-Owen coalitional value can be identied with the coecients of the Banzhaf value by exchanging the number of players with the 3 number of compatible coalitions. Thus, the Banzhaf-Owen value of a player is a weighted average of the contributions of this player to this type of coalitions. The value Γ proposed in Denition 2.4 uses the coecients of the Shapley value swapping the number of players for the number of compatible coalitions, and, in a similar way to the Banzhaf-Owen or Owen values, the value Γ of a player is a weighted average of the contributions of this player to the class of compatible coalitions. Moreover, the value Γ is obtained by combining the weighted coecients of the Shapley and Owen values into a new formula. Then, it covers this variation of the Owen`s scheme using the Shapley value, lling this gap in the literature of coalitional values. Example 2.6 Let us dene the simple 1 game where N={1,2,3,4} and the minimal winning coalitions 2 are {1,3} and {2,3,4} . Suppose that the set of a priori unions is given by P={P1, P2} , with P1={1,2} and P2={3,4} , and consider the game (N, v, P) . The computation of the coalitional value Γ for player i= 1 according to its denition is given in Table 1. Q T (r+t)!(m+pk−r−t−2)! v(Q∪T∪ {1})−v(Q∪T) ∅ ∅ 0!(2 + 2 −2)! = 2 0 ∅{2}(0 + 1)!(2 + 2 −0−1−2)! = 1 0 {3,4}∅(1 + 0)!(2 + 2 −1−0−2)! = 1 1 {3,4} {2}(1 + 1)!(2 + 2 −1−1−2)! = 2 0 Table 1: Possible congurations for the computation of Γ1(N, v, P ) Then: Γ1(N, v, P) = 2·0+1·0+1·1+2·0 (2 + 2 −1)! =1 3! =1 6. Similarly, we obtain Γ2(N, v, P) = 1 6,Γ3(N, v, P) = 1 2 , and Γ4(N, v, P)=0. Next, we will show that we achieve the same results if we apply the interpretation given in Proposition 2.5. Consider the modied game M\k∪Pk, v(M\{k})∪Pk . If we take into account the possible orders of the players in order to calculate the Shapley value of the modied game, we obtain the payos given in Tables 2 and 3. For i∈P1 , we get the payos according Table 2 and, for i∈P2 , the payos are calculated in Table 3. Thus, φ1M\1∪P1, v(M\{1})∪P1=1 6= Γ1(N, v, P), φ2M\1∪P1, v(M\{1})∪P1=1 6= Γ2(N, v, P), φ3M\2∪P2, v(M\{2})∪P2=1 2= Γ3(N, v, P) , and φ4M\2∪P2, v(M\{2})∪P2= 0 = Γ4(N, v, P). 1 A simple game (N, v) is a TU game such that v(S)=0 or v(S)=1 for all S⊆N ; v(S)≤v(T) for all S⊆T⊆N ; and v(N)=1 . 2 A minimal winning coalition is a coalition S⊆N which satises that v(S)=1 and v(T)=0 for all T⊊S . Note that a simple game can be interpreted as a pair (N, MW ) , where MW is the set of minimal winning coalitions. 4 Coalitions {1} {2} {34} 1234 0 0 1 1342 0 0 1 2134 0 0 1 2341 0 0 1 3412 1 0 0 3421 0 1 0 Payos 1 6 1 6 4 6 Table 2: Payos for i∈ {1,2} Coalitions {12} {3} {4} 1234 0 1 0 1243 0 1 0 3124 1 0 0 3412 1 0 0 4312 1 0 0 4123 0 1 0 Payos 3 6 3 60 Table 3: Payos for i∈ {3,4} Remark 2.7 To illustrate the behaviour of the coalitional value Γ , we obtain its expression for the particular case of the unanimity games. Moreover, we compare it with the expressions of other known coalitional values, such as the Owen and the Banzhaf-Owen coalitional values. If we consider T⊆N , the unanimity game (N, uT) is dened, for all S⊆N , as uT(S) = 1 if T⊆S 0 otherwise. For comparison between our new value and the Shapley and Banzhaf values, we give the following expressions for unanimity games: φi(N, uT) =    X T\{i}⊆S⊆N\{i} s!(n−s−1)! n!=1 |T| if i∈T 0 otherwise and βi(N, uT) =      1 2n−1X T\{i}⊆S⊆N\{i} 1 = 1 2|T|−1 if i∈T 0 otherwise. Let us now consider the coalitional game (N, uT, P) , where P={P1, . . . , Pm} . To obtain the expressions of the three coalitional values, we denote T′ h = Ph∩T for all h∈M , T′=h∈M:T′ h=∅ , and KT=|T′| . Note that if there exists k∈M such that i∈T′ k=Pk∩T , we have that uT(Q∪S∪ {i}) - uT(Q∪S) = 1 if and only if Q=Sh∈{1,...,m}:h∈RPh , where T′\k⊆R⊆M\k and T′ k\ {i} ⊆ S⊆Pk\ {i} . i) The Owen coalitional value. This coalitional value assigns to each (N, v, P)∈Gcs and each i∈N the real number Φi(N, v, P) = X R⊆M\{k}X S⊆Pk\{i} r!(m−r−1)!s!(pk−s−1)! pk!m![v(Q∪S∪ {i})−v(Q∪S)] . Thus, for all i∈Pk , if i∈T then Φi(N, uT, P) = X T′\{k}⊆R⊆M\{k}X T′ k\{i}⊆S⊆Pk\{i} r!(m−r−1)!s!(pk−s−1)! pk!m!=1 KT|Pk∩T| and Φi(N, uT, P)=0 otherwise. ii) The Banzhaf-Owen coalitional value. For all i∈Pk , if i∈T Ψi(N, uT, P) = 1 2m−1 1 2pk−1X T′\{k}⊆R⊆M\{k}X T′ k\{i}⊆S⊆Pk\{i} 1 = 1 2KT+|Pk∩T|−2 and Ψi(N, uT, P) = 0 otherwise. 5 iii) The coalitional value Γ . For all i∈Pk , if i∈T Γi(N, uT, P) = X T′\{k}⊆R⊆M\{k}X T′ k\{i}⊆S⊆Pk\{i} (r+s)!(m+pk−r−s−2)! (m+pk−1)! =1 KT+|Pk∩T| − 1 and Γi(N, uT, P) = 0 otherwise. 3. The axiomatic characterizations Below, we introduce several properties for a coalitional value g . A1. ( Eciency ). For all (N, v)∈G , Pi∈Ngi(N, v, Pn) = v(N). A2. ( Symmetry ). If i, j ∈N are symmetric players in (N, v)∈G , then gi(N, v, Pn) = gj(N, v, Pn). A3. ( Equal marginal contributions ). If (N, v) and (N, w) are TU games with a common player set N , and some player i∈N satises v(S∪ {i})−v(S) = w(S∪ {i})−w(S) for all S⊆N\{i} , then gi(N, v, Pn) = gi(N, w, Pn). A4. ( Neutrality under individual desertion ). If (N, v, P)∈Gcs , Pk∈P , and i, j ∈Pk , i=j , then gi(N, v, P) = gi(N, v, P−j). A5. ( 1-Quotient game property ). If (N, v, P)∈Gcs and P∈P(i) for some i∈N , then gi(N, v, P) = gk(M, vP, Pm), where Pk={i} . A6. ( Neutrality for the reduced game ). If (N, v, P)∈Gcs and i, j ∈Pk∈P with i=j , and l∈N\Pk, then gl(N, v, P) = glNij, vij, Pij . A7. ( Indierence for PS unions ). If (N, v, P)∈Gcs and P=PS for some S⊆N , then gi(N, v, P) = gi(N, v, Pn), if i∈S. A8. ( 2-Eciency ). For all (N, v)∈G and i, j ∈N , i=j , gi(N, v, Pn)+gj(N, v, Pn) = gpNij, vij, Pn−1 , where p replaces the set {i, j} in Nij, vij . A9. ( Dummy player ). If i∈N is a dummy player in (N, v)∈G , then gi(N, v, Pn) = v({i}) . The properties of eciency, symmetry, equal marginal contributions, and dummy player are standard in the literature. We use them in the context of TU games with one of the trivial systems of a priori unions. The properties A4 and A5 were proposed by Alonso-Meijide et al. [2]. It is interesting to note that they propose a new characterization of the Banzhaf-Owen coalitional value where the Banzhaf-Owen coalitional value is the only coalitional Banzhaf value which satises these two properties. On the other hand, adaptations of these two properties were used by Álvarez-Mozos and Tejada [7] to characterize extensions of coalitional values to the model of games with level structure of cooperation and by Meng [18] to extend the Banzhaf-Owen coalitional value to the model of fuzzy games with a priori unions. Property A4 says that the payo of a player i∈Pk does not change when a player j∈Pk decides to leave the union and be alone. This property is related with the balanced contributions for the unions property, which says that the gain (or loss) for a player i when another player j leaves the union both players belong coincides with the gain (or loss) obtained when both players interchange their roles in the process. Property A4 implies the property of balanced contributions for the unions, and it was introduced because, in some cases, the balanced contributions for the unions property is too weak. For instance, let us suppose a game where the players are the workers in a given production area, the partition reects the classication of these workers into the rms they are working for, and the value indicates the salary given to each player. In this context, the balanced contributions for the unions property is too weak and it seems more suitable to assume the stronger hypothesis which says that the salary will not change if a partner leaves the rm. 6 According to the property A5, everybody who does not cooperate with others gets exactly what he produces himself. This property is weaker than the property of quotient game, which says that the sum of the payos of the players in a union coincides with the payo of the union in the quotient game. In some cases, the quotient game property is too demanding, and it is reasonable to use a weak modication, such as A5. Then, this property is the quotient game property but only applied to a priori unions formed by a unique player. A similar property to A6 was dened by Amer et al. [9]. It says that the merger of two players of the same union does not aect to the payos of the players outside this union. For instance, let us consider a parliamentary context and assume that each player represents a political party, the system of a priori unions reects the coalition formation, and the coalitional value measures the power of both parties and unions. In this case, A6 says that if two political parties from the same coalition merge into a big one it should not aect to the power of parties in other coalitions. If a value satises the property A7, then the payos obtained by the players belonging to the unique possible union with more than one player are the same than the payos obtained with the trivial system of unions. In a coalitional game, it is assumed that each player only collaborates with compatible coalitions. In the two cases, when the partitions are given by PS or Pn , the compatible coalitions of the players who belong to S coincide and it seems reasonable to suppose that the payos of these players should coincide for both systems of a priori unions. It is straightforward to prove that A6 implies A5 whereas A7 is weaker than A4. Finally, the property A8 is applied in Alonso-Meijide et al. [2] to characterize the Banzhaf-Owen coalitional value. This property was also used by Nowak [19] to give an axiomatization of the Banzhaf value without the additivity axiom. According to Nowak, A8 assumes a natural reductionproperty with a natural interpretation. It was originally discussed in Lehrer [17]. Proposition 3.1 A coalitional value g satises A1, A2, and A3 if, and only if, it is a coalitional Shapley value, i.e. gi(N, v, Pn) = φi(N, v) for all i∈N and all (N, v)∈G. Proof . In Young [28] it is proved that the Shapley value is the unique value which satises eciency, symmetry and equal marginal contributions (in the list of properties of the axiomatization proposed by Young [28], another property called strong monotonicity property is included, but the proof uses only equal marginal contributions, a weaker version of strong monotonicity). There is only one dierence between eciency and A1, symmetry and A2, or equal marginal contributions and A3: while the properties considered by Young are applied in the context of values for TU games, A1-A3 are properties to be satised by coalitional values. On the other hand, since A1-A3 should be proved for the trivial system of a priori unions where each union is a singleton, it is not dicult to prove the proposition by adapting the proof of Young. □ Proposition 3.2 The coalitional value Γ satises A1-A7. Proof . 1. Γ satises A1-A3. According to Proposition 3.1, this is equivalent to prove that Γ is a coalitional Shapley value. Let (N, v)∈G . If we consider the trivial system of a priori unions Pn={{i}i∈N} , we obtain that for every Pk∈Pn , the TU game M\k∪Pk, v(M\{k})∪Pk coincides with (N, v) . Thus, by Proposition 2.5 we deduce that Γ(N, v, Pn) = φ(N, v) . 2. Γ satises the property of neutrality under individual desertion (A4). Let (N, v, P)∈Gcs , Pk∈P , and i, j ∈Pk be distinct players. Let us consider P−j={P′ 1, . . . , P′ m+1} and M′=M∪m+ 1, where P′ h=Ph for all h∈ {1, . . . , m} \ {k} , P′ k=Pk\ {j} , P′ m+1 ={j} , Q=Sh∈{1,...,m}:h∈RPh , Q′= Sh∈{1,...,m+1}:h∈RP′ h , |M′|=m′ , and p′ k=|P′ k| . Since m′=m+ 1 and p′ k=pk−1 , 7 Γi(N, v, P−j) =X R⊆M′\{k}X T⊆P′ k\{i} (r+t)!(m′+p′ k−r−t−2)! (m′+p′ k−1)! [v(Q′∪T∪ {i})−v(Q′∪T)] =X R⊆M′\{k}X T⊆P′ k\{i} (r+t)!(m+pk−r−t−2)! (m+pk−1)! [v(Q′∪T∪ {i})−v(Q′∪T)] =X R⊆M′\{k,m+1}X T⊆Pk\{i,j} (r+t)!(m+pk−r−t−2)! (m+pk−1)! [v(Q′∪T∪ {i})−v(Q′∪T)] +X R⊆M′\{k},m+1∈RX T⊆Pk\{i,j} (r+t)!(m+pk−r−t−2)! (m+pk−1)! [v(Q′∪T∪ {i})−v(Q′∪T)] On the other hand, since for all R⊆M′\k such that m+ 1 ∈R , Q′=∪h∈{1,...,m+1}:h∈RP′ h=∪h∈{1,...,m}:h∈RP′ h∪P′ m+1 =∪h∈{1,...,m}:h∈R\{m+1}Ph∪ {j}, we deduce that X R⊆M′\{k},m+1∈RX T⊆Pk\{i,j} (r+t)!(m+pk−r−t−2)! (m+pk−1)! [v(Q′∪T∪ {i})−v(Q′∪T)] =X R⊆M\{k}X T⊆Pk\{i,j} (r+t+ 1)!(m+pk−r−t−3)! (m+pk−1)! [v(Q∪T∪ {j}∪{i})−v(Q∪T∪ {j})] . Thus, Γi(N, v, P−j) =X R⊆M\{k}X T⊆Pk\{i,j}(r+t)!(m+pk−r−t−2)! (m+pk−1)! [v(Q∪T∪ {i})−v(Q∪T)] +(r+t+ 1)!(m+pk−r−t−3)! (m+pk−1)! [v(Q∪T∪ {j}∪{i})−v(Q∪T∪ {j})] =X R⊆M\{k}X T⊆Pk\{i,j} (r+t)!(m+pk−r−t−2)! (m+pk−1)! [v(Q∪T∪ {i})−v(Q∪T)] +X R⊆M\{k}X T⊆Pk\{i},j∈T (r+t)!(m+pk−r−t−2)! (m+pk−1)! [v(Q∪T∪ {i})−v(Q∪T)] = Γi(N, v, P). 3. Γ satises the property of neutrality for the reduced game (A6). Let (N, v, P)∈Gcs and i, j ∈Pk∈P with i=j , l∈Pa with a=k , and pa=|Pa|. Let us consider Nij, vij, Pij as the {ij} - reduced game of (N, v, P) , such that for all R⊆M, Q =Sh∈{1,...,m}:h∈RPh , Qij =Sh∈{1,...,m}:h∈RPij h=Q if k /∈R , Qij =Sh∈{1,...,m}:h∈RPij h= (Q\{i, j})∪ {p} if k∈R , and pij a=|Pij a| . 8 ΓlNij, vij, Pij =X R⊆M\{a}X T⊆Pij a\{l} (r+t)!(m+pij a−r−t−2)! (m+pij a−1)! vij(Qij ∪T∪ {l})−vij(Qij ∪T) =X R⊆M\{a},k/∈RX T⊆Pij a\{l} (r+t)!(m+pij a−r−t−2)! (m+pij a−1)! vij(Qij ∪T∪ {l})−vij(Qij ∪T) +X R⊆M\{a},k∈RX T⊆Pij a\{l} (r+t)!(m+pij a−r−t−2)! (m+pij a−1)! vij(Qij ∪T∪ {l})−vij(Qij ∪T). And, taking into account the denitions of Qij and vij , we obtain ΓlNij, vij, Pij =X R⊆M\{a},k/∈RX T⊆Pa\{l} (r+t)!(m+pa−r−t−2)! (m+pa−1)! vij(Q∪T∪ {l})−vij(Q∪T) +X R⊆M\{a},k∈RX T⊆Pa\{l} (r+t)!(m+pa−r−t−2)! (m+pa−1)! vij ((Q\{i, j})∪ {p} ∪ T∪ {l}) −X R⊆M\{a},k∈RX T⊆Pa\{l} (r+t)!(m+pa−r−t−2)! (m+pa−1)! vij ((Q\{i, j})∪ {p} ∪ T) =X R⊆M\{a},k/∈RX T⊆Pa\{l} (r+t)!(m+pa−r−t−2)! (m+pa−1)! [v(Q∪T∪ {l})−v(Q∪T)] +X R⊆M\{a},k∈RX T⊆Pa\{l} (r+t)!(m+pa−r−t−2)! (m+pa−1)! [v(Q∪T∪ {l})−v(Q∪T)] =X R⊆M\{a}X T⊆Pa\{l} (r+t)!(m+pa−r−t−2)! (m+pa−1)! [v(Q∪T∪ {l})−v(Q∪T)] = Γl(N, v, P). 4. Γ satises the 1-quotient game property (A5). We omit the proof because it is straightforward to deduce that A6 implies A5. 5. Since Γ satises A4 and A4 implies A7, we deduce that Γ satises indierence for PS unions (A7). □ Alonso-Meijide et al. [2] proved that the Banzhaf-Owen coalitional value is the unique coalitional Banzhaf value that satises A4 and A5. We obtain a similar result for the coalitional value Γ. In this case, the proposed value is a coalitional Shapley value instead of a coalitional Banzhaf value. Theorem 3.3 Γ is the only coalitional value that satises A1-A5. Equivalently, Γ is the only coalitional Shapley value that satises A4 and A5. Proof . a) Existence . We know, by Proposition 3.2, that Γ satises A1-A5. b) Uniqueness . We only prove that the rst statement in the theorem is true. The second statement follows from the rst statement and Proposition 3.1. 9 Γ1(N, v, P) = Z1 0 ∂g ∂q1 (t, t, t)dt =Z1 0 (t−t2)dt =1 6 and Γ2(N, v, P) = Z1 0 ∂g ∂q2 (t, t, t)dt =Z1 0 (t−t2)dt =1 6. To compute Γ3(N, v, P) and Γ4(N, v, P) , we replace both q1 and q2 by p1 in (1). Thus, we deduce that f(p1, q3, q4) = p1·q3+p1·q3·q4−p1·p1·q3·q4=p1·q3+p1·q3·q4−p2 1·q3·q4. Reducing the higher exponents we have that g(p1, q3, q4) = p1·q3+p1·q3·q4−p1·q3·q4=p1·q3, and so, ∂g ∂q3 (p1, q3, q4) = p1 and ∂g ∂q4 (p1, q3, q4)=0. Hence, Γ3(N, v, P) = Z1 0 ∂g ∂q3 (t, t, t)dt =Z1 0 tdt =1 2 and Γ4(N, v, P) = Z1 0 ∂g ∂q4 (t, t, t)dt =Z1 0 0dt = 0. Acknowledgements Financial support from Ministerio de Ciencia y Tecnología and FEDER through grants MTM2011- 27731-C03-02 and ECO2011-23460 is gratefully acknowledged. The authors are also very grateful for the interesting suggestions given by three anonymous referees. References References [1] J.M. Alonso-Meijide, M. Álvarez-Mozos, M.G. 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