The Owen and Banzhaf-Owen values revisited

The Owen and Banzhaf-Owen values revisited J.M. Alonso–Meijide * , B. Casas–M´endez  , A.M. Gonz´alez–Rueda  , and S. Lorenzo–Freire § August 28, 2015 Abstract In this work, we consider games with coalitional structure. We afford two new parallel axiomatic characterizations for the well-known Owen and Banzhaf-Owen coalitional values. Two properties are common to both characterizations: a property of balanced contributions and a property of neutrality. The results prove that the main difference between these two coalitional values is that the former is efficient while the latter verifies a property of 2-efficiency. Keywords: cooperative game, Shapley value, Banzhaf value, coalition structure, coalitional value, balanced contributions. MSC (2000) classification: 91A12. JEL classification: C71. Published in Optimization (2015) Published version available at: https://www.tandfonline.com/journals/gopt20 DOI: 10.1080/02331934.2015.1091823 1 Introduction The main objective of the cooperative game theory is the investigation of solutions (or values) for games with transferable utility (TU games). A value decides the payoffs allocated to each player in a cooperative game. It can be applied to share costs in * Jos´e Mar´ıa Alonso–Meijide. MODESTYA Research Group, Department of Statistics and Operations Research and Faculty of Sciences, University of Santiago de Compostela, Spain, jose- [email protected]  Balbina Casas–M´endez. Corresponding author. Postal address: Faculty of Mathematics, Campus Vida s/n 15782 Santiago de Compostela, Spain. Telephone number: +34 981563100. Fax number: +34 881813197. MODESTYA Research Group, Department of Statistics and Operations Research and Faculty of Mathematics, University of Santiago de Compostela, Spain, bal- [email protected]  A.M. Gonz´alez–Rueda. Department of Statistics and Operations Research and Faculty of Mathematics, University of Santiago de Compostela, Spain, angelman[email protected] § Silvia Lorenzo–Freire. MODES Research Group, Department of Mathematics and Faculty of Computer Science, University of A Coru˜na, Spain, [email protected] economic problems or to measure the power of each agent in a collective decisionmaking system. The Shapley value and the Banzhaf value are two of the best known concepts in this respect. The relevant difference between these two values is that the Shapley value is efficient while the Banzhaf value satisfies a property of 2-efficiency. When enlightenment on affinity among players is disposable, coalitional values are the most appropriate tools to settle the payoffs. These agreements among players are modeled by a set of a priori unions, i.e., a partition of the set of players. The appraisal of the impact that derives from the action of forming unions is of great interest to game theorists. Games with a coalition structure were first considered by Aumann and Dr`eze [10]. A different approach was used by Owen [24] when introducing and axiomatically characterizing the Owen value. In this case, the unions play a game among themselves (the quotient game), and after that the players of a union play an internal game. Both payoffs, in the quotient game for unions and in the internal games for players of the same union, are given by the Shapley value. If both payoffs are given by the Banzhaf value, the Banzhaf-Owen value (cf. Owen [25]) is obtained. In the characterization of the Owen value (Owen [24]), properties of efficiency, additivity, symmetry and null player are used. Later, in a new characterization, V´azquez-Brage et al. [27] introduced a property of balanced contributions in the unions and characterized the Owen value as the only coalitional value for singletons satisfying this property and the quotient game property. Other characterizations of the Owen value can be found in Hart and Kurz [17], in Winter [29], and in Hamiache [16], which make use of properties of consistency, or in Albizuri [2]. The principle of balanced contributions has also been used to characterize the Owen value in Amer and Carreras [8], Calvo et al. [13], and G´omez-R´ua and Vidal-Puga ([14] and [15]). The Banzhaf-Owen value was first characterized in Albizuri [1], for the particular case of simple games, employing, among others, a property called delegation, which was already used by Lehrer [19] for a characterization of the Banzhaf value without using the additivity axiom. The first characterization of the Banzhaf-Owen value for the family of TU games was provided by Amer et al. [9]. In the characterization, they introduced the properties of delegation neutrality and delegation transfer. Laruelle and Valenciano [18] showed that the Banzhaf-Owen value of a particular player iof a union Pkcan be identified as the Banzhaf value of a modified TU game. This game is played by the unions other than Pkand by the players in Pk. Alonso-Meijide et al. [3] provide an axiomatic characterization of the Banzhaf-Owen value. They use a property stronger than the balanced contributions property and a property weaker than the quotient game property. The axiomatic system used there is also compared with parallel axiomatizations of the Owen value. The parallel axiomatizations are useful to decide which value should be chosen, taking into account the context where they will be applied. It is also appropriate to make a reference to the paper of van den Brink and van der Laan [12]. In this paper, a class of share values for games with coalitional structure is axiomatized by using a multiplication property. This class contains both the Owen coalitional share function and the Banzhaf coalitional share function. The first one corresponds with the Owen coalitional value, but with sum of payoffs (shares) normalized to one, but the second one is different from the BanzhafOwen coalitional (share) value, that does not satisfy the multiplication property. 2 The aim of the current paper is to provide a new comparison of these two mentioned coalitional values from the point of view of their properties. To this aim, we present two parallel axiomatic characterizations for the Owen and Banzhaf-Owen values. The property of balanced contributions, introduced in V´azquez-Brage et al. [27], and the property of neutrality for reduced games (it is a similar property to delegation neutrality, that was defined by Amer et al. [9]), appear in both characterizations. 2 Preliminaries We recollect here some basic concepts. 2.1 Games and values A cooperative TU game with a finite number of players N={1,2, . . . , n}(or simply agame) is a pair (N, w), where w: 2N→Ris a function that allocates to each coalition of players S⊆Na real number w(S) which represents the utility that every coalition can obtain and verifies w(∅) = 0. For a finite set A, we denote a=|A|. From now on, we will denote by GNthe family of all games with a given set of player Nand by Gthe family of all games. We will denote by f:G → RN, the value that allocates to every game (N, w), a vector f(N, w)=(f1(N, w), . . . , fn(N, w)) ∈RNwhose components represent the payoffs assigned to every player. Two of the more studied values that appear in this context are the Shapley value (Shapley [26]), which is defined as φi(N, w) = PS⊆N\{i} s!(n−s−1)! n![w(S∪{i})−w(S)], and the Banzhaf value (Banzhaf [11]), that is βi(N, w) = PS⊆N\{i}1 2n−1[w(S∪ {i})−w(S)], for all (N, w)∈ GNand all i∈N. 2.2 Games with a coalition structure Given a finite set of players N={1,2, . . . , n}, we will express by P(N) the set of all partitions of N. Every P={P1, P2, . . . , Pm} ∈ P(N), is called a system of a priori unions or a coalition structure on N. Thus, immediately arise two trivial coalition structures, the one formed only by the grand coalition, PN={N}, and the system where every union is composed by a unique singleton player, Pn= {{1},{2},...,{n}}. Let i∈Nbe a player, we will denote by P(i) the family of a priori unions over Nwhere {i}is a singleton union, that is, there exists a union in the partition which is formed only by the player i. Formally, P∈P(i) if and only if {i} ∈ P. Suppose that we have a player i∈Pk∈P. If the player idecides to leave the union he/she belongs to and forms a union by himself, we will denote this new partition by P−i. Formally, P−i={Ph∈P:h=k}∪{Pk\{i},{i}}.Notice that P−i∈P(i). Given a game (N, w)∈ GNand a system of unions P∈P(N), we define a cooperative game with a coalition structure as the 3-tuple (N, w, P). We will denote 3 by Gcs the set of all cooperative games with a system of unions, and by Gcs Nthe subset when the player set is N. If (N, w, P)∈ Gcs and P={P1, P2, . . . , Pm}, the quotient game is the cooperative game (M, wP) where the set M={1,2, . . . , m}is composed by the representatives of every union and wP(R) = w(∪r∈RPr) for all R⊆M. In other words, the quotient game is the game played by the unions (considering that every union acts as a representative player of the whole union). Note that whenever P=Pn, the quotient game (M, wP) coincides with (N, w). Let (N, w, P)∈ Gcs be a cooperative game with P={P1, P2, . . . , Pm}the coalition structure. Suppose that two players of the same union, i, j ∈Pk∈Pwith i=j, decide to merge together and form a new player p∈ N. Let us denote the new set of players N{i,j}= (N\ {i, j})∪ {p}and the new system of a priori unions P{i,j}=nP{i,j} 1, P{i,j} 2, . . . , P{i,j} mowhere P{i,j} k= (Pk\{i, j})∪{p}and P{i,j} h=Ph for h=k. Thus, the {i, j}-reduced game N{i,j}, w{i,j}, P{i,j}of (N, w, P) is defined by w{i,j}(S) = w((S\{p})∪{i, j}) if p∈Sand w{i,j}(S) = w(S) otherwise, for every S⊆N{i,j}. We will denote by g:Gcs →RNthe coalitional value that allocates to every game (N, w, P) a vector g(N, w, P)=(g1(N, w, P), . . . , gn(N, w, P)) ∈RN, whose components represent the payoffs assigned to every player. We consider here two possibilities of combining the Banzhaf and the Shapley value in two steps, in order to obtain a coalitional value. Definition 2.1 (Owen [24]) The Owen value Φ is the coalitional value defined by: Φi(N, w, P) = X R⊆M\{k}X T⊆Pk\{i} r! (m−r−1)! m! t! (pk−t−1)! pk!hw(Q∪T∪{i})−w(Q∪T)i, for all (N, w, P)∈ Gcs N,i∈Pk, and Pk∈P, where Q=S r∈R Pr. Definition 2.2 (Owen [25]) The Banzhaf-Owen value Ψ is the coalitional value defined by: Ψi(N, w, P) = X R⊆M\{k}X T⊆Pk\{i} 1 2m−1 1 2pk−1hw(Q∪T∪ {i})−w(Q∪T)i, for all (N, w, P)∈ Gcs N,i∈Pk, and Pk∈P, where Q=S r∈R Pr. 2.3 Comparison between the two coalitional values in games with a trivial systems of unions The coalitional values introduced in the previous section can be considered extensions of the Shapley and Banzhaf values in the sense that they coincide with them when the system of unions is the trivial one, that is, each coalition is formed by a unique player. To formalize it, we introduce the concept of coalitional values for singletons. 4 We say that a coalitional value gon Gcs is a coalitional f–value for singletons, where fis a value on G, if g(N, w, Pn) = f(N, w) for all (N, w)∈ G. It is known that the Owen value Φ is a coalitional Shapley value for singletons (Φ(N, w, Pn) = φ(N, w)) and the Banzhaf-Owen value Ψ is a coalitional Banzhaf value for singletons (Ψ(N, w, Pn) = β(N, w)). 3 Characterization results This section contains the main results of the work and several remarks. 3.1 Axioms Below we introduce several properties for a coalitional value g. A1. (2-Efficiency within unions). For all (N, w, P)∈ Gcs, any Pk∈P, and all i, j ∈Pkwith i=j, gi(N, w, P) + gj(N, w, P) = gpN{i,j}, w{i,j}, P{i,j}. A2. (Efficiency). For all (N, w, P)∈ Gcs, X i∈N gi(N, w, P) = w(N). A3. (Balanced contributions within unions). For all (N, w, P)∈ Gcs, any Pk∈P, and all i, j ∈Pk∈Pwith i=j, gi(N, w, P)−gi(N, w, P−j) = gj(N, w, P)−gj(N, w, P−i). A4. (Neutrality for the reduced game). For all (N, w, P)∈ Gcs, any Pk∈P, all i, j ∈Pkwith i=j, and all l∈N\Pk, gl(N, w, P) = glN{i,j}, w{i,j}, P{i,j}. A property with the same savour than A1 was used by Nowak [23] to give an axiomatization of the Banzhaf value without the so-called additivity axiom. According to Nowak, A1 assumes a “reduction” property with an easy interpretation. It was originally discussed in Lehrer [19]. A similar property was also applied in AlonsoMeijide et al. ([3] and [5]) to characterize the Banzhaf-Owen coalitional value. The property of efficiency is standard in the literature and it is usual to find it in the characterizations of the Shapley and Owen value. One important axiom in the literature is the balanced contributions axiom. This property is based on a rule of reciprocity, as introduced by Myerson [22], and it is often used in the literature on the Shapley value. Myerson’s property of balanced 5 contributions asserts that for any two players the gain or loss to each player when the other “leaves” the game should be equal. Property A3 follows this principle but taking into account the role of the unions. Thus, A3 states that the loss (or gain) of a player i∈Pkwhen a player j∈Pkdecides to leave the union and remain alone is the same as the loss (or gain) of player jwhen player idecides to leave the union. This property was introduced in V´azquez-Brage et al. [27] and it was used by Alonso–Meijide et al. [4] in the axiomatic characterization of the symmetric coalitional binomial semivalues. On the other hand, adaptations of this property were used by ´ Alvarez-Mozos and Tejada [7] to characterize extensions of coalitional values to the model of games with levels structure of cooperation. Moreover, the principle of balanced contributions has also been used in other contexts, e.g., Lorenzo-Freire et al. [21] in generalized bankruptcy situations. A similar property to A4, called delegation neutrality, was defined by Amer et al. [9]1. It says that the merger of two players of the same union does not affect the payoffs of the players outside this union. 3.2 Main results Now, we present a new axiomatic characterization for the Owen value. Theorem 3.1 The Owen value is the only coalitional Shapley value for singletons that satisfies A2, A3, and A4. Proof. (a)Existence. In V´azquez-Brage et al. [27] it is proved that the Owen value is a coalitional Shapley value and it satisfies the property of balanced contributions within unions (A3). Besides, in Owen [24] it is shown that the Owen value satisfies efficiency (A2). Since it is straightforward to prove that the Owen value satisfies the property of neutrality for the reduced game (A4), we omit the proof. (b)Uniqueness. To prove uniqueness, let us suppose that there exists other coalitional value gin the conditions established in this theorem. We should then prove that g= Φ. To this aim, we distinguish two cases:  If P=Pn={{1},{2},...,{n}}, since both values are coalitional Shapley values for singletons, we know that for all TU game (N, w) : g(N, w, P) = φ(N, w) = Φ(N, w, P).  Suppose now that |P|< n. In this case, we define q= maxh∈M|Ph|(note that 2≤q≤n) and Mp={h∈M:|Ph|=p}, with 1 ≤p≤q. Then, the proof 1In the section of Final Remarks, we discuss in more detail the properties of delegation neutrality and neutrality for the reduced game, as well as 2-efficiency within unions and the so-called delegation transfer. 6 goes by induction on the number p. We use the recursive procedure indicated below: –We start with p= 1. If M1=∅, then go to the next stage. On the contrary, if M1=∅, we can consider k∈M1. Suppose then that Pk={i}and let us choose a union Phwith h=k, such that |Ph|>1. Then, we consider the steps described below: * Step 1. Let us take two different players j1, j2∈Phand consider (N{j1,j2}, w{j1,j2}, P{j1,j2}) as the {j1, j2}-reduced game of (N, w, P ). Then, by A4, we have that gi(N, w, P) = gi(N{j1,j2}, w{j1,j2}, P{j1,j2}) and Φi(N, w, P)=Φi(N{j1,j2}, w{j1,j2}, P{j1,j2}). We define (N1, w1, P1)=(N{j1,j2}, w{j1,j2}, P {j1,j2}). If  P1 h = 1, go to Step 2. Otherwise, since  P1 h >1, we choose other two different players j3, j4∈P1 hand repeat the procedure given above, obtaining the {j3, j4}-reduced game of (N1, w1, P1).We denote this coalitional game by (N2, w2, P2). Finally, after |Ph|−1 iterations, we get a new coalitional game denoted by (N|Ph|−1, w|Ph|−1, P|Ph|−1), where |P|Ph|−1 h|= 1 and such that, by A4, gi(N, w, P) = gi(N|Ph|−1, w|Ph|−1, P|Ph|−1) and Φi(N, w, P) = Φi(N|Ph|−1, w|Ph|−1, P|Ph|−1). * Step 2. Let us take the game (N|Ph|−1, w|Ph|−1, P|Ph|−1).If we carry out the same procedure to that described in Step 1 for all the unions Ph′with h′∈M\(M1∪{k, h}), we obtain a coalitional game denoted2 by (NPh′∈M\{k}(|Ph′|−1), wPh′∈M\{k}(|Ph′|−1), P Ph′∈M\{k}(|Ph′|−1)). This coalitional game satisfies that i∈NPh′∈M\{k}(|Ph′|−1),PPh′∈M\{k}(|Ph′|−1) is a coalitional structure with only one player in each union and, moreover, by A4, gi(N, w, P) = gi(NPh′∈M\{k}(|Ph′|−1), wPh′∈M\{k}(|Ph′|−1), PPh′∈M\{k}(|Ph′|−1)) and Φi(N, w, P) = Φi(NPh′∈M\{k}(|Ph′|−1), wPh′∈M\{k}(|Ph′|−1), PPh′∈M\{k}(|Ph′|−1)). Then, taking into account that gand Φ are coalitional Shapley values for singletons, gi(NPh′∈M\{k}(|Ph′|−1), wPh′∈M\{k}(|Ph′|−1), PPh′∈M\{k}(|Ph′|−1)) =φi(NPh′∈M\{k}(|Ph′|−1), wPh′∈M\{k}(|Ph′|−1)) = Φi(NPh′∈M\{k}(|Ph′|−1), wPh′∈M\{k}(|Ph′|−1), PPh′∈M\{k}(|Ph′|−1)) and we conclude that gi(N, w, P)=Φi(N, w, P). 2This notation corresponds to the number of iterations used to obtain a coalitional game with all the unions formed by isolated players. 7 –For 2 ≤p≤q, suppose that the payoffs are determined for every iin a union Ph∈Psuch that |Ph|< p. We will determine the payoffs for players in a union Phwith |Ph|=p. Remember that Mp={k∈M:|Pk|=p}. If Mp=∅, then go to the next stage. Otherwise, fix k∈Mp. If we choose i∈Pk, by A3 we know that for all j∈Pk\ {i}, gi(N, w, P)−gj(N, w, P) = gi(N, w, P−j)−gj(N, w, P−i), Φi(N, w, P)−Φj(N, w, P)=Φi(N, w, P−j)−Φj(N, w, P−i), and, since |(P−j)k|=|(P−i)k|=p−1, by applying stage p−1 we deduce that gi(N, w, P−j) = Φi(N, w, P−j) and gj(N, w, P−i) = Φj(N, w, P−i). It means that, for all j∈Pk\ {i}, gi(N, w, P)−gj(N, w, P)=Φi(N, w, P)−Φj(N, w, P).(1) Let us now consider the other unions Ph,h=k. There are two cases: * First case. There does not exist any union Ph,h=ksuch that |Ph| = 1. By the property A2, we have that, X j∈Pk gj(N, w, P) = w(N)−X j∈N\Pk gj(N, w, P) and X j∈Pk Φj(N, w, P) = w(N)−X j∈N\Pk Φj(N, w, P). Since |Ph|= 1 for all h=k, by Stage 1 we deduce that for all j∈N\Pk, gj(N, w, P) = Φj(N, w, P ). Taking into account the last two equalities, we obtain that: X j∈Pk gj(N, w, P) = X j∈Pk Φj(N, w, P).(2) Finally, by equations (1) and (2), we conclude that for all i∈Pk, gi(N, w, P) = Φi(N, w, P ). * Second case. Suppose that there exists a union Phwith h=k, such that |Ph|>1. In this case we proceed as in Steps 1-2, obtaining the game (NPh′∈M\{k}(|Ph′|−1), wPh′∈M\{k}(|Ph′|−1), PPh′∈M\{k}(|Ph′|−1)), such that PPh′∈M\{k}(|Ph′|−1) k=Pkand where |PPh′∈M\{k}(|Ph′|−1) h|= 1 for all h=k. Applying A4 and taking into account that this new coalitional game is in the conditions of First case, we conclude that for all i∈Pk, gi(N, w, P) =gi(NPh′∈M\{k}(|Ph′|−1), wPh′∈M\{k}(|Ph′|−1), PPh′∈M\{k}(|Ph′|−1)) = Φi(NPh′∈M\{k}(|Ph′|−1), wPh′∈M\{k}(|Ph′|−1), PPh′∈M\{k}(|Ph′|−1)) = Φi(N, w, P).□ 8 Remark 3.2 Independence of the properties of Theorem 3.1. a. The coalitional value given by gi(N, w, P) = w(N)/m |Pk|for all (N, w, P)∈ Gcs and all i∈N, where Pk∈Pis the union such that i∈Pk, satisfies A2, A3, and A4, but it is not a coalitional Shapley value for singletons. b. The coalitional value Γ given in Alonso-Meijide et al. [5], is a coalitional Shapley value for singletons and satisfies A3 and A4, but not A2. c. The coalitional value given by gi(N, w, P) = φk(M, wP)/|Pk|for all (N, w, P)∈ Gcs and all i∈N, where Pk∈Pis the union such that i∈Pk, is a coalitional Shapley value for singletons and satisfies A2 and A4, but not A3. d. The coalitional value given by g(N, w, P) = φ(N, w) for all (N, w, P)∈ Gcs is a coalitional Shapley value for singletons and satisfies A2 and A3, but not A4. A parallel axiomatic characterization for the Banzhaf-Owen value can be given just replacing the property efficiency with 2-efficiency within unions and taking into account that the Banzhaf-Owen value is a coalitional Banzhaf value. Theorem 3.3 The Banzhaf-Owen value is the only coalitional Banzhaf value for singletons that satisfies A1, A3, and A4. Proof. (a)Existence. In Alonso-Meijide et al. [3] it is shown that the Banzhaf-Owen value is a coalitional Banzhaf value for singletons and it satisfies the property of balanced contributions within unions (A3). It is straightforward to prove that the Banzhaf-Owen value satisfies the property of 2-efficiency within unions (A1) and the property of the neutrality for the reduced game (A4). So, we omit the proof. (b)Uniqueness. The proof of uniqueness follows similar lines to that of the uniqueness in Theorem 3.1. Thus, let us suppose that there exists other coalitional value gin the conditions established in this theorem. We should then prove that g= Ψ. To this aim, we distinguish two cases:  If P=Pn={{1},{2},...,{n}}, since both values are coalitional Banzhaf values for singletons, we know that for all TU game (N, w): g(N, w, P) = β(N, w) = Ψ(N, w, P).  Suppose now that |P|< n. It means that we can find Pk∈Psuch that pk=|Pk| ≥ 2. We define q= maxk∈M|Pk|(note that 2 ≤q≤n) and Mp={k∈M:|Pk|= p}, with 1 ≤p≤q. Then, the proof goes by induction on the number p. We use the recursive procedure indicated below: 9 [3] Alonso–Meijide, J.M., Carreras, F., Fiestras–Janeiro, M.G., and Owen, G. 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