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On new characterizations of the Owen value
Operations Research Letters 44 (2016) 491–494
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
On new characterizations of the Owen value Silvia Lorenzo-Freire ∗ MODES Research Group, Department of Mathematics, Faculty of Computer Science, University of A Coruña, Postal code: 15071, A Coruña, Spain
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info
Article history: Received 15 April 2016 Received in revised form 11 May 2016 Accepted 17 May 2016 Available online 26 May 2016
abstract The aim of this paper is to provide new characterizations for the Owen value that generalize different characterizations of the Shapley value. To obtain these characterizations we consider axioms of efficiency, balanced contributions, symmetry, strong monotonicity, additivity and null player in the context of cooperative games with coalition structure. © 2016 Elsevier B.V. All rights reserved.
Keywords: Cooperative game Shapley value Coalition structure Owen value
1. Introduction The Shapley value is one of the most famous solutions in cooperative games. This solution was defined and first characterized by means of axioms of carrier, additivity and anonymity in Shapley [14]. A close characterization of this solution with efficiency, additivity, symmetry and null player can be found in Shubik [15]. Although there are many other interesting characterizations of the Shapley value, two characterizations that also deserve special attention in this paper are the ones by Young [18] by means of efficiency, symmetry and strong monotonicity and Myerson [12] with the axioms of efficiency and balanced contributions. In some cases, the players are grouped forming a coalition structure, which is just a partition of the set of players. With the aim to study the cooperation of the players in these situations, Aumann and Drèze [4] introduced the class of cooperative games with coalition structure. They also defined a solution for this class of games which extends the Shapley value. According to this solution, the payoff of a player is given by the Shapley value of this player in the game restricted to the union he belongs to. But, although this solution still preserves interesting axioms, it does not satisfy efficiency. Later on, Owen [13] defined an efficient solution for cooperative games with coalition structure that also extends the Shapley value. This efficient solution has been successfully applied in the Operations Research field. Some of these applications can be found in Carreras and Owen [7], where the process of coalition
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formation in the Catalonian Parliament is considered, in VázquezBrage et al. [16], by using the Owen value to determine the aircraft landing fees, or in Alonso-Meijide and Bowles [1], where the distribution of power in the International Monetary Fund is studied. Since the Owen value is a very well-known coalitional value, there are many characterizations of this value in the literature. The first one was made by Owen. The axioms involved in the characterization are efficiency, additivity, coalitional symmetry, intracoalitional symmetry and null player. Other interesting characterizations are the one provided by Calvo et al. [6] to characterize the level structure value, and in particular the Owen value, according to efficiency and the principle of balanced contributions applied in the different levels or the characterization that appears in Khmelnitskaya and Yanovskaya [10], where the Owen value is characterized by means of efficiency, strong monotonicity, coalitional symmetry and intracoalitional symmetry. In this paper new characterizations for the Owen value are obtained. All the characterizations make use of axioms that also follow the spirit of the axioms used to characterize the Shapley value in [14,18,12]. In particular, the axiom of intracoalitional balanced contributions, introduced in [6], is a key element. It says that, given two players in the same union, the amounts that both players gain or lose when the other leaves the game should be equal. Other papers where this axiom has been used to characterize the Owen value are Gómez-Rúa and Vidal-Puga [8], comparing the Owen value with other coalitional values, and Lorenzo–Freire [11], where the Owen value is compared with the Banzhaf–Owen value. In Vázquez-Brage et al. [16] and AlonsoMeijide et al. [2] the principle of balanced contributions is also applied to get characterizations of the Owen value, but with a
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S. Lorenzo-Freire / Operations Research Letters 44 (2016) 491–494
slightly different meaning since in this case the players leave the union but not the game. The paper is organized as follows. In Section 2, the concepts used along the paper are introduced and, in Section 3, the results of the paper are collected.
A coalitional value is a map g that assigns to every TU-game with coalition structure (N , v, C ) a vector g (N , v, C ) = (gi (N , v, C ))i∈N . The Owen value (Owen [13]) is the coalitional value defined for all (N , v, C ) ∈ C G and all i ∈ Ch , with Ch ∈ C , by Owi (N , v, C ) = r !(m−r −1)! s!(ch −s−1)! [v( r ∈R Cr ∪ S ∪ {i}) − R⊆M \{h} S ⊆Ch \{i} m! c !
v(
2. Notation and definitions
r ∈R
h
Cr ∪ S )].
3. The characterizations 2.1. TU-games A cooperative game with transferable utility (or TU-game) is a pair (N , v) defined by a finite set of players N ⊂ N (usually, N = {1, 2, . . . , n}) and a function v : 2N → R, that assigns to each coalition S ⊆ N a real number v(S ), called the worth of S, and such that v(∅) = 0. For any coalition S ⊆ N, we assume that s = |S |. GN will denote the family of all TU-games on a given N and G the family of all TU-games. Given S ⊆ N, we denote the restriction of a TU-game (N , v) ∈ GN to S as (S , v). Given two TU-games (N , v), (N , v ′ ) ∈ GN , the TU-game (N , v + ′ v ) is defined by (v + v ′ )(S ) = v(S ) + v ′ (S ) for all S ⊆ N. Given ∅ ̸= T ⊆ N , the unanimity game N , uT is a TUT T game such that u (S ) = 1 if T ⊆ S and u (S ) = 0 otherwise. Since the family (N , uT ) T ∈2N \∅ is a basis for GN , for any TU-game
(N , v), there are unique coefficients {cT }T ∈2N \∅ such that v = T T ∈2N \∅ cT u . A player i ∈ N is a null player in the TU-game (N , v) if v(S ∪ {i}) = v(S ) for all S ⊆ N \ {i}. The set of null players in (N , v) will be denoted by N P (N , v). Two players i, j ∈ N are symmetric in the TU-game (N , v) if v(S ∪ {i}) = v(S ∪ {j}) for all S ⊆ N \ {i, j}. A value is a map f that assigns to every TU-game (N , v) ∈ G a vector f (N , v) = (fi (N , v))i∈N , where each component fi (N , v)
represents the payoff of player i when he participates in the game. The Shapley value (Shapley [14]) is the value defined for all (N , v) ∈ G and all i ∈ N by Shi (N , v) = S ⊆N \{i} s!(n−ns!−1)! [v(S ∪ {i}) − v(S )]. Given an unanimity game N , uT for ∅ ̸= T ⊆ N, the expression of the TU-game is given by particular Shapley value in this Shi N , uT = 1t if i ∈ T and Shi N , uT = 0 otherwise. 2.2. TU-games with coalition structure Let us consider a finite set of players N. A coalition structure over N is a partition of N, i.e., C= {C1 , . . . , Cm } is a coalition structure over N if it satisfies that h∈M Ch = N, where M = {1, . . . , m}, and Ch ∩ Cr = ∅ when h ̸= r. A cooperative game with coalition structure (or TU-game with coalition structure) is a triple (N , v, C ) where (N , v) is a TU-game and C is a coalition structure over N. The set of all TU-games with coalition structure will be denoted by C G , and by C G N the subset where N is the player set. Given S ⊆ N, such that S = h∈M Sh , with ∅ ̸= Sh ⊆ Ch for all h ∈ M, we will denote the restriction of (N , v, C ) ∈ C G N to S as the TU-game with coalition structure (S , v, CS ), where CS = {S1 , . . . , Sm }. If (CN , v, C ) ∈ C G and C = {C1 , . . . , Cm }, the quotient game M , v is the TU-game played by the unions where the player set is given by M = {1, . . . , m}. It is defined as v (R) = v r ∈R Cr for all R ⊆ M. Let us consider (N , v, C ) ∈ C G. A union Ch ∈ C is a null union in (N , v, C ) if h is a null player in M , v C . Two unions, Ch , Cr ∈ C , are symmetric unions in (N , v, C ) if h and r are symmetric players in M , v C . C
Below, we introduce the axioms used to characterize the Owen value.
• Additivity (ADD). For all (N , v, C ), (N , v ′ , C ) ∈ C G , g (N , v + v ′ , C ) = g (N , v, C ) + g (N , v ′ , C ). • Coalitional marginality (CMA). For all (N , v, C ), (N , v ′ , C ) ∈ C G , ′ if v (S ∪ Ch ) − v(S ) = v ′ (S ) for all S ⊆ N \ Ch then v (S ∪ Ch ) − ′ i∈Ch gi (N , v, C ) = i∈Ch gi (N , v , C ). • Coalitional strong monotonicity (CSM). For all (N , v, C ), (N , v ′ , C ) ∈ C G , if v (S ∪ Ch )−v(S ) ≥ v ′ (S ∪ Ch )−v ′ (S ) for all S ⊆ N \ Ch ′ then i∈Ch gi (N , v, C ) ≥ i∈Ch gi (N , v , C ). • Coalitional symmetry (CSY). For all(N , v, C ) ∈ C G andfor all symmetric coalitions Ch , Cr ∈ C , i∈Ch gi (N , v, C ) = i∈Cr gi (N , v, C ). • Efficiency (EFF). For all (N , v, C ) ∈ C G , i∈N gi (N , v, C ) = v(N ). • Intracoalitional balanced contributions (IBC). For all (N , v, C ) ∈ C G and all i, j ∈ Ch ∈ C , i ̸= j, gi (N , v, C )−gi (N \{j}, v, CN \{j} ) = gj (N , v, C ) − gj (N \ {i}, v, CN \{i} ). • Individual marginality (IMA). For all (N , v, C ), (N , v ′ , C ) ∈ C G , if v (S ∪ {i}) − v(S ) = v ′ (S ∪ {i}) − v ′ (S ) for all S ⊆ N \ {i} then gi (N , v, C ) = gi (N , v ′ , C ). • Individual strong monotonicity (ISM). For all (N , v, C ), (N , v ′ , C ) ∈ C G , if v (S ∪ {i})−v(S ) ≥ v ′ (S ∪ {i})−v ′ (S ) for all S ⊆ N \{i} then gi (N , v, C ) ≥ gi (N , v ′ , C ). • Null player (NP). For all (N , v, C ) ∈ C G and for all i ∈ N, if i is a null player then gi (N , v, C ) = 0. • Null union (NU). For all (N , v, C ) ∈ C G and for all Ch ∈ P, if Ch is a null union then i∈Ch gi (N , v, C ) = 0. The axioms of ADD, EFF, CSY and NP are quite standard in the literature and they have been used in several characterizations of different coalitional values. The null union axiom is used in Calvo and Gutiérrez [5] to characterize the two-step Shapley value. They also show that there is no relation between NP and NU. CMA and IMA are adaptations of the axiom of marginality defined by Young [18] to the context of cooperative games with coalition structure, whereas CSM and ISM follow the spirit of strong monotonicity. IBC was defined in [6] and states that, given two players in the same union, the amounts that both players gain or lose when the other leaves the game should be equal. In LorenzoFreire [11], a new expression based on the Shapley value is obtained for all the coalitional values satisfying intracoalitional balanced contributions. Proposition 3.1 (Lorenzo-Freire [11]). A coalitional value g satisfies IBC if and only if, for all (N , v, C ) ∈ C G and all i ∈ Ch with Ch ∈ C , gi (N , v, C ) = Shi Ch , v g ,C ,
where v g ,C (S ) =
i∈S
gi (N \ Ch ) ∪ S , v, C(N \Ch )∪S
for all S ⊆ Ch .
In Young [18], the Shapley value is characterized by means of efficiency, symmetry and marginality. In the next proposition, EFF and CSY are combined with the two adaptations of marginality to the context of cooperative games with coalition structure. In both cases, we obtain that the total payoff of the players in each union is given by the Shapley value of the union in the quotient game.
S. Lorenzo-Freire / Operations Research Letters 44 (2016) 491–494
493
Table 1 Independence of the axioms in Theorem 3.3. Coalitional value g1 g2 g3 g4
IBC
EFF ✓
×
CSY ✓ ✓
×
✓ ✓ ✓
✓ ✓
×
CSM ✓ ✓ ✓
✓
×
Proposition 3.2. Let us fix (N , v, C ) ∈ C G . (a) If g is a coalitional value thatC satisfies EFF, CSY and CMA then i∈Ch gi (N , v, C ) = Shh (M , v ) for all Ch ∈ C . (b) If g is a coalitional value thatC satisfies EFF, CSY and IMA then i∈Ch gi (N , v, C ) = Shh (M , v ) for all Ch ∈ C . T Proof. Given (N , v, C ) ∈ C G , we know that v = T ∈2N \∅ cT u . Then, for all T ⊆ N, with T ̸= ∅, let us denote Th = Ch ∩ T and C MT = {h ∈ M : Th ̸= ∅}. The quotient game (M , v ) can also be C MT expressed in terms of unanimity games as v = T ∈2N \∅ cT u
since, for all R ⊆ M , v C (R) = v
h∈R
Ch
=
T ∈2N \∅
cT uT
(R). h∈R Ch = T ∈2N \∅ cT u Thus, since the Shapley value satisfies the axiom of linearity, the proof is finished if we prove that, for all Ch ∈
C,
gi (N , v, C ) = Shh M ,
cT
| MT |
i∈Ch
MT
T ∈2N \∅
cT uMT
=
T ∈2N \∅:h∈M
.
C Let us now assume that for i∈Ch gi (N , v, C ) = Shh M , v any TU-game with coalition structure (N , v, C ) with I (N , v, C ) ≤ q, where q is a non-negative integer. Suppose that there exists
(N , v, C ) ∈ C G with I (N , v, C ) = q + 1, i.e., v = q+1 Let us define T = l=1 T l .
q +1 l =1
l
cT l uT .
(a) For each Ch ∈ C , we distinguish two possibilities: (1) h ̸∈ MT . The TU-game with coalition structure (N , v h , C ), where v h
=
l
l∈{1,...,q+1}:h∈M l T
cT l uT , satisfies that
I (N , v h , C ) ≤ q. By induction hypothesis
=
c l T l∈{1,...,q+1}:h∈M l |M l | . T T
i∈Ch
gi (N , v h , C )
l
cT l uT (S ∪ Ch )
l∈{1,...,q+1}:h∈M l T
= v (S ∪ Ch ) − v(S ),
for all S ⊆ N \ {Ch }.
Thus, by CMA
gi (N , v, C ) =
i∈Ch
i∈Ch
gi (N , v h , C ) =
l∈{1,...,q+1}:h∈M l T
(2) h ∈M T . By EFF gi (N , v, C ) = v(N ) − gi (N , v, C ) r ∈MT i∈Cr
=
q +1
r ̸∈MT i∈Cr
cT l −
l =1
= |MT |
q +1 l =1
cT l |MT l | − |MT | |MT l |
q +1 cT l . |MT l | l =1
EFF ✓
CSY ✓ ✓
×
✓ ✓ ✓
✓ ✓
×
ISM ✓ ✓ ✓
✓
×
In case |MT | ≥ 2, we know that v(S∪ Ch ) = v(S ∪ Cr ) for all r ∈ MT and CSY implies that i∈Ch gi (N , v, C ) =
q+1
c l T
l =1 | M l | T
.
(b) For each Ch ∈ C we have two cases: h 1. h ̸∈ MT . By induction hypothesis i∈Ch gi (N , v , C ) = c l T l∈{1,...,q+1}:h∈M l |M | . Moreover, for all i ∈ Ch , Tl
T
l
v h (S ∪ {i}) − v h (S ) =
cT l uT (S ∪ {i})
l∈{1,...,q+1}:i∈T l
= v (S ∪ {i}) − v(S ), for all S ⊆ N \ {i}. By IMA gi (N , v, C ) = gi (N , v h , C ) for all i ∈ Ch , what implies that gi (N , v, C ) = gi (N , v h , C ) i∈Ch
i∈Ch
=
l∈{1,...,q+1}:h∈M l T
cT l
|MT l |
.
2. h ∈ MT . The proof of this case follows the same arguments to the proof in part (a). Young [18] also characterizes the Shapley value with efficiency, symmetry and strong monotonicity. In the next theorem we show that it is possible to characterize the Owen value just by adding IBC to EFF, CSY and CSM or ISM. Theorem 3.3. (a) The Owen value is the only coalitional value that satisfies IBC, EFF, CSY and CSM. (b) The Owen value is the only coalitional value that satisfies IBC, EFF, CSY and ISM. Proof. It is straightforward to prove that the Owen value satisfies IBC, EFF, CSY, CSM and ISM. Then, it only remains to prove the uniqueness. (a) CSM is stronger than CMA. Then, by Proposition 3.2 part (a) we know that any coalitional value g satisfying CSM, CSY and EFF verifies that i∈Ch gi (N , v, C ) = Shh (M , v C ) for all Ch ∈ C and all (N , v, C ) ∈ C G . Since g satisfies IBC, by Proposition 3.1 we have that for all i ∈ Ch with Ch ∈ C , gi (N , v, C ) = Shi Ch , v g ,C , where v g ,C (S ) =
gi (N \ Ch ) ∪ S , v, C(N \Ch )∪S = Shh (M ,
i∈S
v C(N \Ch )∪S ) for all S ⊆ Ch .
Moreover,
v h ( S ∪ Ch ) − v h ( S ) =
IBC
×
T
For any TU-game with coalition structure (N , v, C ), let us denote I (N , v, C ) as the minimum number of unanimity games necessary to represent v in terms of these unanimity games. Also in both cases, the proof will be done by induction on the cardinality of I (N , v, C ). If I (N , v, C ) = v(S ) = 0 for all S ⊆ N. 0, it means that According to EFF, i∈N gi (N , v, C ) = h∈M i∈Ch gi (N , v, C ) = 0. Since any pair of unions in this game are symmetric, by CSY we deduce that i∈Ch gi (N , v, C ) = 0 for all Ch ∈ C .
Coalitional value g5 g2 g6 g4
cT l
|MT l |
.
It means that g is uniquely determined and leads us to deduce that for all (N , v, C ) ∈ C G and all i ∈ Ch with Ch ∈ C , gi (N , v, C ) = Owi (N , v, C ). (b) Since ISM is stronger than IMA, by Proposition 3.2 part (b) we have that any coalitional value g which satisfies CSY, EFF and ISM verifies that i∈Ch gi (N , v, C ) = Shh (M , v C ) for all Ch ∈ C and all (N , v, C ) ∈ C G . Since g satisfies IBC, we can follow the same procedure as in part (a) to obtain that g (N , v, C ) = Ow(N , v, C ). Remark 3.4. The axioms in Theorem 3.3 are independent. The independence is justified in Table 1 and the coalitional values considered in the Table are defined below:
• g 1 corresponds to the coalitional value defined for all (N , v, C ) ∈ C G andall i ∈ Ch , with Ch ∈ C , by g 1 (N , v, C ) = Shi (Ch , v) + 1 Shh M , v C − v(Ch ) . This coalitional value is known as the ch two-step Shapley value and was defined by Kamijo [9].
494
S. Lorenzo-Freire / Operations Research Letters 44 (2016) 491–494 Table 2 Independence of the axioms in Theorem 3.5. Coalitional value g1 g2 g7 g3 g4
IBC
× ✓ ✓ ✓ ✓
EFF ✓
× ✓ ✓ ✓
ADD ✓ ✓
× ✓ ✓
CSY ✓ ✓ ✓
×
NU ✓ ✓ ✓ ✓
✓
×
• g 2 is the symmetric coalitional Banzhaf value defined in [3]. • g 3 is the coalitional value defined by Vidal-Puga in [17]. • g 4 is the coalitional value defined for all (N , v, C ) ∈ C G and all i ∈ Ch , with Ch ∈ C , by gi4 (N , v, C ) = Shi (Ch , v1 ), where v1 (S ) = v((N \mCh )∪S ) for all S ⊆ Ch , S ̸= ∅. • Let us consider (N, v, C ) ∈ C G . Then, given Ch ∈ C , let us assume that Ch = i1 , . . . , ich and i1 < · · · < ich . For all 1 ≤ j ≤ ch , the coalitional value g 5 is defined as gi5j {i } , where (Ch , v2 ) is (N , v, C ) = v2 l≤j {il } − v2 l
h if S ̸∈ N P M , v Ch , S ̸= ∅, v3 (S ) = C M \N P M ,v (N \Ch )∪S and v3 (S ) = 0 otherwise.
C(N \C )∪S h
Remark 3.7. • The first characterization of the Owen value appears in [13] and it is characterized by means of EFF, ADD, CSY, NP and the axiom of intracoalitional symmetry. Another characterization of this coalitional value is obtained in [10] by means of EFF, CSY, ISM and intracoalitional symmetry. Intracoalitional symmetry (ISY) says that two symmetric players belonging to the same union should obtain the same payoff. Formally, a coalitional value satisfies this axiom if, for
Coalitional value g5 g2 g7 g6 g4
IBC
× ✓ ✓ ✓ ✓
EFF ✓
× ✓ ✓ ✓
ADD ✓ ✓
× ✓ ✓
CSY ✓ ✓ ✓
×
NP ✓ ✓ ✓ ✓
✓
×
all (N , v, C ) ∈ C G and all symmetric players i, j ∈ Ch , with Ch ∈ C , gi (N , v, C ) = gj (N , v, C ). The characterizations in Theorems 3.3 and 3.5 parts (b) only differ from the characterizations in [10,13] in one axiom. In both cases IBC is replaced with ISY. However, ISY and IBC are not connected: – The two-step Shapley value defined by Kamijo satisfies ISY but not IBC. – Let us consider (N , v, C ) ∈ C G and i ∈ Ch , with Ch ∈ C . The coalitional value given by gi8 (N , v, C ) = Shi (Ch , v4 ), where for all S ⊆ Ch , S ̸= ∅, v4 (S ) = v({j ∈ (N \ Ch ) ∪ S : j ≤ iS }) − v({j ∈ N \ Ch : j ≤ iS }), with iS = max{j : j ∈ S }, satisfies IBC but it does not satisfy ISY. • It is not possible to obtain new characterizations for the Owen value by replacing IBC with ISY in Theorems 3.3 and 3.5 parts (a). In fact, the two-step Shapley value also satisfies ISY, EFF, ADD, CSY, CSM and NU. Acknowledgments Financial support from Ministerio de Ciencia y Tecnología through grant ECO2011-23460 and Ministerio de Economía y Competitividad through grant MTM2014-53395-C3-1-P is gratefully acknowledged. The author is also very grateful for the interesting suggestions made by the associate editor. References [1] J.M. Alonso-Meijide, C. Bowles, Generating functions for coalitional power indices: An application to the IMF, Ann. Oper. Res. 137 (2005) 21–44. [2] J.M. Alonso-Meijide, B. Casas-Méndez, A.M. González-Rueda, S. Lorenzo-Freire, The Owen and the Banzhaf–Owen values revisited, Optimization 65 (6) (2016) 1277–1291. [3] J.M. Alonso-Meijide, M.G. Fiestras-Janeiro, Modification of the Banzhaf value for games with a coalition structure, Ann. Oper. Res. 109 (2002) 213–227. [4] R.J. Aumann, J. Drèze, Cooperative games with coalition structures, Internat. J. Game Theory 3 (1974) 217–237. [5] E. Calvo, E. Gutiérrez, Solidarity in games with a coalition structure, Math. Social Sci. 60 (2010) 196–203. [6] E. Calvo, J. Lasaga, E. Winter, The principle of balanced contributions and hierarchies of cooperation, Math. Social Sci. 31 (1996) 171–182. [7] F. Carreras, G. Owen, Evaluation of the Catalonian parliament, 1980–1984, Math. Social Sci. 15 (1988) 87–92. [8] M. Gómez-Rúa, J. Vidal-Puga, The axiomatic approach to three values in games with coalition structure, European J. Oper. Res. 207 (2010) 795–806. [9] Y. Kamijo, A two-step Shapley value in a cooperative game with a coalition structure, Int. Game Theory Rev. 11 (2) (2009) 207–214. [10] A. Khmelnitskaya, E. Yanovskaya, Owen coalitional value without additivity axiom, Math. Methods Oper. Res. 66 (2007) 255–261. [11] S. Lorenzo-Freire, New characterizations of the Owen and Banzhaf–Owen values using the intracoalitional balanced contributions property. Working paper, MODES Research Group, 2016. Available from: http://dm.udc.es/modes/sites/default/files/owen_banzhaf1.pdf. [12] R.B. Myerson, Conference structures and fair allocation rules, Internat. J. Game Theory 9 (1980) 169–182. [13] G. Owen, Values of games with a priori unions, in: R. Henn, O. Moeschlin (Eds.), Essays in Mathematical Economics and Game Theory, Springer, 1977, pp. 76–88. [14] L.S. Shapley, A value for n-person games, in: H.W. Kuhn, A.W. Tucker (Eds.), Contributions to the Theory of Games II, Princeton University Press, 1953, pp. 307–317. [15] M. Shubik, Incentives, decentralized control, the assignment of joint costs and internal princing, Manage. Sci. 8 (1962) 325–343. [16] M. Vázquez-Brage, A. van den Nouweland, I. García-Jurado, Owen’s coalitional value and aircraft landing fees, Math. Social Sci. 34 (1992) 273–286. [17] J. Vidal-Puga, The Harsanyi paradox and the right to talk in bargaining among coalitions, Math. Social Sci. 64 (2012) 214–224. [18] H.P. Young, Monotonic solutions of cooperative games, Internat. J. Game Theory 14 (2) (1983) 65–72.
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
On new characterizations of the Owen value Silvia Lorenzo-Freire ∗ MODES Research Group, Department of Mathematics, Faculty of Computer Science, University of A Coruña, Postal code: 15071, A Coruña, Spain
article
info
Article history: Received 15 April 2016 Received in revised form 11 May 2016 Accepted 17 May 2016 Available online 26 May 2016
abstract The aim of this paper is to provide new characterizations for the Owen value that generalize different characterizations of the Shapley value. To obtain these characterizations we consider axioms of efficiency, balanced contributions, symmetry, strong monotonicity, additivity and null player in the context of cooperative games with coalition structure. © 2016 Elsevier B.V. All rights reserved.
Keywords: Cooperative game Shapley value Coalition structure Owen value
1. Introduction The Shapley value is one of the most famous solutions in cooperative games. This solution was defined and first characterized by means of axioms of carrier, additivity and anonymity in Shapley [14]. A close characterization of this solution with efficiency, additivity, symmetry and null player can be found in Shubik [15]. Although there are many other interesting characterizations of the Shapley value, two characterizations that also deserve special attention in this paper are the ones by Young [18] by means of efficiency, symmetry and strong monotonicity and Myerson [12] with the axioms of efficiency and balanced contributions. In some cases, the players are grouped forming a coalition structure, which is just a partition of the set of players. With the aim to study the cooperation of the players in these situations, Aumann and Drèze [4] introduced the class of cooperative games with coalition structure. They also defined a solution for this class of games which extends the Shapley value. According to this solution, the payoff of a player is given by the Shapley value of this player in the game restricted to the union he belongs to. But, although this solution still preserves interesting axioms, it does not satisfy efficiency. Later on, Owen [13] defined an efficient solution for cooperative games with coalition structure that also extends the Shapley value. This efficient solution has been successfully applied in the Operations Research field. Some of these applications can be found in Carreras and Owen [7], where the process of coalition
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http://dx.doi.org/10.1016/j.orl.2016.05.003 0167-6377/© 2016 Elsevier B.V. All rights reserved.
formation in the Catalonian Parliament is considered, in VázquezBrage et al. [16], by using the Owen value to determine the aircraft landing fees, or in Alonso-Meijide and Bowles [1], where the distribution of power in the International Monetary Fund is studied. Since the Owen value is a very well-known coalitional value, there are many characterizations of this value in the literature. The first one was made by Owen. The axioms involved in the characterization are efficiency, additivity, coalitional symmetry, intracoalitional symmetry and null player. Other interesting characterizations are the one provided by Calvo et al. [6] to characterize the level structure value, and in particular the Owen value, according to efficiency and the principle of balanced contributions applied in the different levels or the characterization that appears in Khmelnitskaya and Yanovskaya [10], where the Owen value is characterized by means of efficiency, strong monotonicity, coalitional symmetry and intracoalitional symmetry. In this paper new characterizations for the Owen value are obtained. All the characterizations make use of axioms that also follow the spirit of the axioms used to characterize the Shapley value in [14,18,12]. In particular, the axiom of intracoalitional balanced contributions, introduced in [6], is a key element. It says that, given two players in the same union, the amounts that both players gain or lose when the other leaves the game should be equal. Other papers where this axiom has been used to characterize the Owen value are Gómez-Rúa and Vidal-Puga [8], comparing the Owen value with other coalitional values, and Lorenzo–Freire [11], where the Owen value is compared with the Banzhaf–Owen value. In Vázquez-Brage et al. [16] and AlonsoMeijide et al. [2] the principle of balanced contributions is also applied to get characterizations of the Owen value, but with a
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S. Lorenzo-Freire / Operations Research Letters 44 (2016) 491–494
slightly different meaning since in this case the players leave the union but not the game. The paper is organized as follows. In Section 2, the concepts used along the paper are introduced and, in Section 3, the results of the paper are collected.
A coalitional value is a map g that assigns to every TU-game with coalition structure (N , v, C ) a vector g (N , v, C ) = (gi (N , v, C ))i∈N . The Owen value (Owen [13]) is the coalitional value defined for all (N , v, C ) ∈ C G and all i ∈ Ch , with Ch ∈ C , by Owi (N , v, C ) = r !(m−r −1)! s!(ch −s−1)! [v( r ∈R Cr ∪ S ∪ {i}) − R⊆M \{h} S ⊆Ch \{i} m! c !
v(
2. Notation and definitions
r ∈R
h
Cr ∪ S )].
3. The characterizations 2.1. TU-games A cooperative game with transferable utility (or TU-game) is a pair (N , v) defined by a finite set of players N ⊂ N (usually, N = {1, 2, . . . , n}) and a function v : 2N → R, that assigns to each coalition S ⊆ N a real number v(S ), called the worth of S, and such that v(∅) = 0. For any coalition S ⊆ N, we assume that s = |S |. GN will denote the family of all TU-games on a given N and G the family of all TU-games. Given S ⊆ N, we denote the restriction of a TU-game (N , v) ∈ GN to S as (S , v). Given two TU-games (N , v), (N , v ′ ) ∈ GN , the TU-game (N , v + ′ v ) is defined by (v + v ′ )(S ) = v(S ) + v ′ (S ) for all S ⊆ N. Given ∅ ̸= T ⊆ N , the unanimity game N , uT is a TUT T game such that u (S ) = 1 if T ⊆ S and u (S ) = 0 otherwise. Since the family (N , uT ) T ∈2N \∅ is a basis for GN , for any TU-game
(N , v), there are unique coefficients {cT }T ∈2N \∅ such that v = T T ∈2N \∅ cT u . A player i ∈ N is a null player in the TU-game (N , v) if v(S ∪ {i}) = v(S ) for all S ⊆ N \ {i}. The set of null players in (N , v) will be denoted by N P (N , v). Two players i, j ∈ N are symmetric in the TU-game (N , v) if v(S ∪ {i}) = v(S ∪ {j}) for all S ⊆ N \ {i, j}. A value is a map f that assigns to every TU-game (N , v) ∈ G a vector f (N , v) = (fi (N , v))i∈N , where each component fi (N , v)
represents the payoff of player i when he participates in the game. The Shapley value (Shapley [14]) is the value defined for all (N , v) ∈ G and all i ∈ N by Shi (N , v) = S ⊆N \{i} s!(n−ns!−1)! [v(S ∪ {i}) − v(S )]. Given an unanimity game N , uT for ∅ ̸= T ⊆ N, the expression of the TU-game is given by particular Shapley value in this Shi N , uT = 1t if i ∈ T and Shi N , uT = 0 otherwise. 2.2. TU-games with coalition structure Let us consider a finite set of players N. A coalition structure over N is a partition of N, i.e., C= {C1 , . . . , Cm } is a coalition structure over N if it satisfies that h∈M Ch = N, where M = {1, . . . , m}, and Ch ∩ Cr = ∅ when h ̸= r. A cooperative game with coalition structure (or TU-game with coalition structure) is a triple (N , v, C ) where (N , v) is a TU-game and C is a coalition structure over N. The set of all TU-games with coalition structure will be denoted by C G , and by C G N the subset where N is the player set. Given S ⊆ N, such that S = h∈M Sh , with ∅ ̸= Sh ⊆ Ch for all h ∈ M, we will denote the restriction of (N , v, C ) ∈ C G N to S as the TU-game with coalition structure (S , v, CS ), where CS = {S1 , . . . , Sm }. If (CN , v, C ) ∈ C G and C = {C1 , . . . , Cm }, the quotient game M , v is the TU-game played by the unions where the player set is given by M = {1, . . . , m}. It is defined as v (R) = v r ∈R Cr for all R ⊆ M. Let us consider (N , v, C ) ∈ C G. A union Ch ∈ C is a null union in (N , v, C ) if h is a null player in M , v C . Two unions, Ch , Cr ∈ C , are symmetric unions in (N , v, C ) if h and r are symmetric players in M , v C . C
Below, we introduce the axioms used to characterize the Owen value.
• Additivity (ADD). For all (N , v, C ), (N , v ′ , C ) ∈ C G , g (N , v + v ′ , C ) = g (N , v, C ) + g (N , v ′ , C ). • Coalitional marginality (CMA). For all (N , v, C ), (N , v ′ , C ) ∈ C G , ′ if v (S ∪ Ch ) − v(S ) = v ′ (S ) for all S ⊆ N \ Ch then v (S ∪ Ch ) − ′ i∈Ch gi (N , v, C ) = i∈Ch gi (N , v , C ). • Coalitional strong monotonicity (CSM). For all (N , v, C ), (N , v ′ , C ) ∈ C G , if v (S ∪ Ch )−v(S ) ≥ v ′ (S ∪ Ch )−v ′ (S ) for all S ⊆ N \ Ch ′ then i∈Ch gi (N , v, C ) ≥ i∈Ch gi (N , v , C ). • Coalitional symmetry (CSY). For all(N , v, C ) ∈ C G andfor all symmetric coalitions Ch , Cr ∈ C , i∈Ch gi (N , v, C ) = i∈Cr gi (N , v, C ). • Efficiency (EFF). For all (N , v, C ) ∈ C G , i∈N gi (N , v, C ) = v(N ). • Intracoalitional balanced contributions (IBC). For all (N , v, C ) ∈ C G and all i, j ∈ Ch ∈ C , i ̸= j, gi (N , v, C )−gi (N \{j}, v, CN \{j} ) = gj (N , v, C ) − gj (N \ {i}, v, CN \{i} ). • Individual marginality (IMA). For all (N , v, C ), (N , v ′ , C ) ∈ C G , if v (S ∪ {i}) − v(S ) = v ′ (S ∪ {i}) − v ′ (S ) for all S ⊆ N \ {i} then gi (N , v, C ) = gi (N , v ′ , C ). • Individual strong monotonicity (ISM). For all (N , v, C ), (N , v ′ , C ) ∈ C G , if v (S ∪ {i})−v(S ) ≥ v ′ (S ∪ {i})−v ′ (S ) for all S ⊆ N \{i} then gi (N , v, C ) ≥ gi (N , v ′ , C ). • Null player (NP). For all (N , v, C ) ∈ C G and for all i ∈ N, if i is a null player then gi (N , v, C ) = 0. • Null union (NU). For all (N , v, C ) ∈ C G and for all Ch ∈ P, if Ch is a null union then i∈Ch gi (N , v, C ) = 0. The axioms of ADD, EFF, CSY and NP are quite standard in the literature and they have been used in several characterizations of different coalitional values. The null union axiom is used in Calvo and Gutiérrez [5] to characterize the two-step Shapley value. They also show that there is no relation between NP and NU. CMA and IMA are adaptations of the axiom of marginality defined by Young [18] to the context of cooperative games with coalition structure, whereas CSM and ISM follow the spirit of strong monotonicity. IBC was defined in [6] and states that, given two players in the same union, the amounts that both players gain or lose when the other leaves the game should be equal. In LorenzoFreire [11], a new expression based on the Shapley value is obtained for all the coalitional values satisfying intracoalitional balanced contributions. Proposition 3.1 (Lorenzo-Freire [11]). A coalitional value g satisfies IBC if and only if, for all (N , v, C ) ∈ C G and all i ∈ Ch with Ch ∈ C , gi (N , v, C ) = Shi Ch , v g ,C ,
where v g ,C (S ) =
i∈S
gi (N \ Ch ) ∪ S , v, C(N \Ch )∪S
for all S ⊆ Ch .
In Young [18], the Shapley value is characterized by means of efficiency, symmetry and marginality. In the next proposition, EFF and CSY are combined with the two adaptations of marginality to the context of cooperative games with coalition structure. In both cases, we obtain that the total payoff of the players in each union is given by the Shapley value of the union in the quotient game.
S. Lorenzo-Freire / Operations Research Letters 44 (2016) 491–494
493
Table 1 Independence of the axioms in Theorem 3.3. Coalitional value g1 g2 g3 g4
IBC
EFF ✓
×
CSY ✓ ✓
×
✓ ✓ ✓
✓ ✓
×
CSM ✓ ✓ ✓
✓
×
Proposition 3.2. Let us fix (N , v, C ) ∈ C G . (a) If g is a coalitional value thatC satisfies EFF, CSY and CMA then i∈Ch gi (N , v, C ) = Shh (M , v ) for all Ch ∈ C . (b) If g is a coalitional value thatC satisfies EFF, CSY and IMA then i∈Ch gi (N , v, C ) = Shh (M , v ) for all Ch ∈ C . T Proof. Given (N , v, C ) ∈ C G , we know that v = T ∈2N \∅ cT u . Then, for all T ⊆ N, with T ̸= ∅, let us denote Th = Ch ∩ T and C MT = {h ∈ M : Th ̸= ∅}. The quotient game (M , v ) can also be C MT expressed in terms of unanimity games as v = T ∈2N \∅ cT u
since, for all R ⊆ M , v C (R) = v
h∈R
Ch
=
T ∈2N \∅
cT uT
(R). h∈R Ch = T ∈2N \∅ cT u Thus, since the Shapley value satisfies the axiom of linearity, the proof is finished if we prove that, for all Ch ∈
C,
gi (N , v, C ) = Shh M ,
cT
| MT |
i∈Ch
MT
T ∈2N \∅
cT uMT
=
T ∈2N \∅:h∈M
.
C Let us now assume that for i∈Ch gi (N , v, C ) = Shh M , v any TU-game with coalition structure (N , v, C ) with I (N , v, C ) ≤ q, where q is a non-negative integer. Suppose that there exists
(N , v, C ) ∈ C G with I (N , v, C ) = q + 1, i.e., v = q+1 Let us define T = l=1 T l .
q +1 l =1
l
cT l uT .
(a) For each Ch ∈ C , we distinguish two possibilities: (1) h ̸∈ MT . The TU-game with coalition structure (N , v h , C ), where v h
=
l
l∈{1,...,q+1}:h∈M l T
cT l uT , satisfies that
I (N , v h , C ) ≤ q. By induction hypothesis
=
c l T l∈{1,...,q+1}:h∈M l |M l | . T T
i∈Ch
gi (N , v h , C )
l
cT l uT (S ∪ Ch )
l∈{1,...,q+1}:h∈M l T
= v (S ∪ Ch ) − v(S ),
for all S ⊆ N \ {Ch }.
Thus, by CMA
gi (N , v, C ) =
i∈Ch
i∈Ch
gi (N , v h , C ) =
l∈{1,...,q+1}:h∈M l T
(2) h ∈M T . By EFF gi (N , v, C ) = v(N ) − gi (N , v, C ) r ∈MT i∈Cr
=
q +1
r ̸∈MT i∈Cr
cT l −
l =1
= |MT |
q +1 l =1
cT l |MT l | − |MT | |MT l |
q +1 cT l . |MT l | l =1
EFF ✓
CSY ✓ ✓
×
✓ ✓ ✓
✓ ✓
×
ISM ✓ ✓ ✓
✓
×
In case |MT | ≥ 2, we know that v(S∪ Ch ) = v(S ∪ Cr ) for all r ∈ MT and CSY implies that i∈Ch gi (N , v, C ) =
q+1
c l T
l =1 | M l | T
.
(b) For each Ch ∈ C we have two cases: h 1. h ̸∈ MT . By induction hypothesis i∈Ch gi (N , v , C ) = c l T l∈{1,...,q+1}:h∈M l |M | . Moreover, for all i ∈ Ch , Tl
T
l
v h (S ∪ {i}) − v h (S ) =
cT l uT (S ∪ {i})
l∈{1,...,q+1}:i∈T l
= v (S ∪ {i}) − v(S ), for all S ⊆ N \ {i}. By IMA gi (N , v, C ) = gi (N , v h , C ) for all i ∈ Ch , what implies that gi (N , v, C ) = gi (N , v h , C ) i∈Ch
i∈Ch
=
l∈{1,...,q+1}:h∈M l T
cT l
|MT l |
.
2. h ∈ MT . The proof of this case follows the same arguments to the proof in part (a). Young [18] also characterizes the Shapley value with efficiency, symmetry and strong monotonicity. In the next theorem we show that it is possible to characterize the Owen value just by adding IBC to EFF, CSY and CSM or ISM. Theorem 3.3. (a) The Owen value is the only coalitional value that satisfies IBC, EFF, CSY and CSM. (b) The Owen value is the only coalitional value that satisfies IBC, EFF, CSY and ISM. Proof. It is straightforward to prove that the Owen value satisfies IBC, EFF, CSY, CSM and ISM. Then, it only remains to prove the uniqueness. (a) CSM is stronger than CMA. Then, by Proposition 3.2 part (a) we know that any coalitional value g satisfying CSM, CSY and EFF verifies that i∈Ch gi (N , v, C ) = Shh (M , v C ) for all Ch ∈ C and all (N , v, C ) ∈ C G . Since g satisfies IBC, by Proposition 3.1 we have that for all i ∈ Ch with Ch ∈ C , gi (N , v, C ) = Shi Ch , v g ,C , where v g ,C (S ) =
gi (N \ Ch ) ∪ S , v, C(N \Ch )∪S = Shh (M ,
i∈S
v C(N \Ch )∪S ) for all S ⊆ Ch .
Moreover,
v h ( S ∪ Ch ) − v h ( S ) =
IBC
×
T
For any TU-game with coalition structure (N , v, C ), let us denote I (N , v, C ) as the minimum number of unanimity games necessary to represent v in terms of these unanimity games. Also in both cases, the proof will be done by induction on the cardinality of I (N , v, C ). If I (N , v, C ) = v(S ) = 0 for all S ⊆ N. 0, it means that According to EFF, i∈N gi (N , v, C ) = h∈M i∈Ch gi (N , v, C ) = 0. Since any pair of unions in this game are symmetric, by CSY we deduce that i∈Ch gi (N , v, C ) = 0 for all Ch ∈ C .
Coalitional value g5 g2 g6 g4
cT l
|MT l |
.
It means that g is uniquely determined and leads us to deduce that for all (N , v, C ) ∈ C G and all i ∈ Ch with Ch ∈ C , gi (N , v, C ) = Owi (N , v, C ). (b) Since ISM is stronger than IMA, by Proposition 3.2 part (b) we have that any coalitional value g which satisfies CSY, EFF and ISM verifies that i∈Ch gi (N , v, C ) = Shh (M , v C ) for all Ch ∈ C and all (N , v, C ) ∈ C G . Since g satisfies IBC, we can follow the same procedure as in part (a) to obtain that g (N , v, C ) = Ow(N , v, C ). Remark 3.4. The axioms in Theorem 3.3 are independent. The independence is justified in Table 1 and the coalitional values considered in the Table are defined below:
• g 1 corresponds to the coalitional value defined for all (N , v, C ) ∈ C G andall i ∈ Ch , with Ch ∈ C , by g 1 (N , v, C ) = Shi (Ch , v) + 1 Shh M , v C − v(Ch ) . This coalitional value is known as the ch two-step Shapley value and was defined by Kamijo [9].
494
S. Lorenzo-Freire / Operations Research Letters 44 (2016) 491–494 Table 2 Independence of the axioms in Theorem 3.5. Coalitional value g1 g2 g7 g3 g4
IBC
× ✓ ✓ ✓ ✓
EFF ✓
× ✓ ✓ ✓
ADD ✓ ✓
× ✓ ✓
CSY ✓ ✓ ✓
×
NU ✓ ✓ ✓ ✓
✓
×
• g 2 is the symmetric coalitional Banzhaf value defined in [3]. • g 3 is the coalitional value defined by Vidal-Puga in [17]. • g 4 is the coalitional value defined for all (N , v, C ) ∈ C G and all i ∈ Ch , with Ch ∈ C , by gi4 (N , v, C ) = Shi (Ch , v1 ), where v1 (S ) = v((N \mCh )∪S ) for all S ⊆ Ch , S ̸= ∅. • Let us consider (N, v, C ) ∈ C G . Then, given Ch ∈ C , let us assume that Ch = i1 , . . . , ich and i1 < · · · < ich . For all 1 ≤ j ≤ ch , the coalitional value g 5 is defined as gi5j {i } , where (Ch , v2 ) is (N , v, C ) = v2 l≤j {il } − v2 l
h if S ̸∈ N P M , v Ch , S ̸= ∅, v3 (S ) = C M \N P M ,v (N \Ch )∪S and v3 (S ) = 0 otherwise.
C(N \C )∪S h
Remark 3.7. • The first characterization of the Owen value appears in [13] and it is characterized by means of EFF, ADD, CSY, NP and the axiom of intracoalitional symmetry. Another characterization of this coalitional value is obtained in [10] by means of EFF, CSY, ISM and intracoalitional symmetry. Intracoalitional symmetry (ISY) says that two symmetric players belonging to the same union should obtain the same payoff. Formally, a coalitional value satisfies this axiom if, for
Coalitional value g5 g2 g7 g6 g4
IBC
× ✓ ✓ ✓ ✓
EFF ✓
× ✓ ✓ ✓
ADD ✓ ✓
× ✓ ✓
CSY ✓ ✓ ✓
×
NP ✓ ✓ ✓ ✓
✓
×
all (N , v, C ) ∈ C G and all symmetric players i, j ∈ Ch , with Ch ∈ C , gi (N , v, C ) = gj (N , v, C ). The characterizations in Theorems 3.3 and 3.5 parts (b) only differ from the characterizations in [10,13] in one axiom. In both cases IBC is replaced with ISY. However, ISY and IBC are not connected: – The two-step Shapley value defined by Kamijo satisfies ISY but not IBC. – Let us consider (N , v, C ) ∈ C G and i ∈ Ch , with Ch ∈ C . The coalitional value given by gi8 (N , v, C ) = Shi (Ch , v4 ), where for all S ⊆ Ch , S ̸= ∅, v4 (S ) = v({j ∈ (N \ Ch ) ∪ S : j ≤ iS }) − v({j ∈ N \ Ch : j ≤ iS }), with iS = max{j : j ∈ S }, satisfies IBC but it does not satisfy ISY. • It is not possible to obtain new characterizations for the Owen value by replacing IBC with ISY in Theorems 3.3 and 3.5 parts (a). In fact, the two-step Shapley value also satisfies ISY, EFF, ADD, CSY, CSM and NU. Acknowledgments Financial support from Ministerio de Ciencia y Tecnología through grant ECO2011-23460 and Ministerio de Economía y Competitividad through grant MTM2014-53395-C3-1-P is gratefully acknowledged. The author is also very grateful for the interesting suggestions made by the associate editor. References [1] J.M. Alonso-Meijide, C. Bowles, Generating functions for coalitional power indices: An application to the IMF, Ann. Oper. Res. 137 (2005) 21–44. [2] J.M. Alonso-Meijide, B. Casas-Méndez, A.M. González-Rueda, S. Lorenzo-Freire, The Owen and the Banzhaf–Owen values revisited, Optimization 65 (6) (2016) 1277–1291. [3] J.M. Alonso-Meijide, M.G. Fiestras-Janeiro, Modification of the Banzhaf value for games with a coalition structure, Ann. Oper. Res. 109 (2002) 213–227. [4] R.J. Aumann, J. Drèze, Cooperative games with coalition structures, Internat. J. Game Theory 3 (1974) 217–237. [5] E. Calvo, E. Gutiérrez, Solidarity in games with a coalition structure, Math. Social Sci. 60 (2010) 196–203. [6] E. Calvo, J. Lasaga, E. Winter, The principle of balanced contributions and hierarchies of cooperation, Math. Social Sci. 31 (1996) 171–182. [7] F. Carreras, G. Owen, Evaluation of the Catalonian parliament, 1980–1984, Math. Social Sci. 15 (1988) 87–92. [8] M. Gómez-Rúa, J. Vidal-Puga, The axiomatic approach to three values in games with coalition structure, European J. Oper. Res. 207 (2010) 795–806. [9] Y. Kamijo, A two-step Shapley value in a cooperative game with a coalition structure, Int. Game Theory Rev. 11 (2) (2009) 207–214. [10] A. Khmelnitskaya, E. Yanovskaya, Owen coalitional value without additivity axiom, Math. Methods Oper. Res. 66 (2007) 255–261. [11] S. Lorenzo-Freire, New characterizations of the Owen and Banzhaf–Owen values using the intracoalitional balanced contributions property. Working paper, MODES Research Group, 2016. Available from: http://dm.udc.es/modes/sites/default/files/owen_banzhaf1.pdf. [12] R.B. Myerson, Conference structures and fair allocation rules, Internat. J. Game Theory 9 (1980) 169–182. [13] G. Owen, Values of games with a priori unions, in: R. Henn, O. Moeschlin (Eds.), Essays in Mathematical Economics and Game Theory, Springer, 1977, pp. 76–88. [14] L.S. Shapley, A value for n-person games, in: H.W. Kuhn, A.W. Tucker (Eds.), Contributions to the Theory of Games II, Princeton University Press, 1953, pp. 307–317. [15] M. Shubik, Incentives, decentralized control, the assignment of joint costs and internal princing, Manage. Sci. 8 (1962) 325–343. [16] M. Vázquez-Brage, A. van den Nouweland, I. García-Jurado, Owen’s coalitional value and aircraft landing fees, Math. Social Sci. 34 (1992) 273–286. [17] J. Vidal-Puga, The Harsanyi paradox and the right to talk in bargaining among coalitions, Math. Social Sci. 64 (2012) 214–224. [18] H.P. Young, Monotonic solutions of cooperative games, Internat. J. Game Theory 14 (2) (1983) 65–72.