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Value of games with two-layered hypergraphs
Mathematical Social Sciences 62 (2011) 114–119
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Value of games with two-layered hypergraphs Takumi Kongo Faculty of Political Science & Economics, Waseda University. 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo, Japan
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Article history: Received 29 October 2009 Received in revised form 19 June 2011 Accepted 23 June 2011 Available online 7 July 2011
abstract This paper studies cooperative games with restricted cooperation among players. We define situations in which a priori unions and hypergraphs coexist simultaneously and mutually depend on each other. We call such structures two-layered hypergraphs. Using a two-step approach, we define a value of the games with two-layered hypergraphs. The value is characterized by Owen’s coalitional value of hypergraph-restricted games and in terms of weighted Myerson value. Further, our value is axiomatically characterized by component efficiency and a coalition size normalized balanced contributions property. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In many social and economic situations, agents form a group and act as if they are a single agent. For example, neighboring countries form a regional economic bloc and stand together against other countries, laborers form a labor union and negotiate jointly with a manager, firms form a cartel and jointly set a uniform price, and politicians form a political party and coordinate their policy. The reason for such an act, in such situations, is that agents forming a group and coordinating their actions in the group leads to better outcomes for each of them if they distribute the profit generated by their cooperation appropriately. Such group formation and surplus distributions are analyzed mainly through cooperative games. Since cooperation needs coordination and coordination requires communication, in real situations, agents who are unable to communicate with each other cannot cooperate even if they wish to do so. By introducing a priori unions on the player set, Aumann and Drèze (1974) considered situations in which only players in the same element of a priori unions can cooperate. Myerson (1977, 1980) studied the restriction of cooperation by (hyper)graphs on the player set. In this setting, only players who are connected with each other can cooperate. By integrating the above two models, Vázquez-Brage et al. (1996) and Alonso-Meijide et al. (2009) investigated the situations in which both a priori unions and graphs exist simultaneously. In these two papers, a priori unions and graphs were independent. The a priori unions affected the definition of solution concepts, as in Owen (1977, 1981) and Alonso-Meijide and Fiestras-Janeiro (2002), while the graphs affected the worth of each coalition in order to represent communication restrictions among agents, as in Myerson (1977) and Owen (1986).
E-mail address: [email protected]. 0165-4896/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2011.06.006
In this paper, we consider the situation in which both a priori unions and hypergraphs coexist simultaneously and mutually depend on each other. That is, given an a priori union, there exists a hypergraph on each element of the a priori union and on the a priori union. We call such structures two-layered hypergraphs and study cooperative games with such structures.1 Any twolayered hypergraph H can be seen as an ordinary hypergraph with a restriction; each element of H must either be a subset of a single member of the given a priori union or be a union of several such members. However, if we take into consideration the layered structures on the set of agents represented by a given a priori union, two-layered hypergraphs and their corresponding usual hypergraphs are not the same. As mentioned in Hart and Kurz (1983), under the existence of an a priori union, interactions among agents will be conducted on two levels: among agents and among groups of agents that constitute the a priori union. Our two-layered hypergraphs well capture communication structures among agents in such situations. Next, we define a one-point solution (value) of the games with two-layered hypergraphs. In the definition of our value, we focus on the layered structure of agents. We decompose a game with a two-layered hypergraph into two problems. One is a problem within each element of the a priori union and the other is the problem between elements of the a priori union. Applying the Myerson value in each problem, we define the value of the game with the two-layered hypergraph.2 Our value is a generalization
1 Khmelnitskaya (2007) also studied a similar situation in which a priori unions and graphs exist simultaneously and mutually depend on each other. However, the interpretation of cooperation (connectedness) among players is different; thus, the value studied in Khmelnitskaya (2007) is different from ours. 2 This kind of ‘‘two-step approach’’ was also taken by van der Laan and van den Brink (2002); van den Brink and van der Laan (2005) and Kamijo (2009) in the class of games with a priori unions.
T. Kongo / Mathematical Social Sciences 62 (2011) 114–119
of the Myerson value of games with hypergraphs and also is a generalization of the value in Aumann and Drèze (1974) and the two-step Shapley value (Kamijo, 2009) of games with a priori unions. In conjunction with the hypergraph restriction of characteristic functions, our value is represented by Owen’s (1977) coalitional value for games with a priori unions. Thus, our value is a natural extension of existing solution concepts and does not depend heavily on the two-step approach. Further, our value is represented by weighted Myerson value. We also axiomatically characterize our value by two axioms. One axiom is an efficiency property called component efficiency and the other is a normalized version of the balanced contributions property. Here, we normalized each player’s contributions to others with respect to the number of players in the element of the a priori union that contains the player. As a generalization of an a priori union, Winter (1989) introduced a levels structure of players, that is a sequence of a priori unions, each a priori union is obtained by unification of elements in the previous a priori union, and each representing the various cooperations between elements of the previous a priori union. By considering situations in which both levels structures and hypergraphs coexist simultaneously and dependently, we can define more general structures called multi-layered hypergraphs. However, the essence of our discussion, coexistence of layered structures of players and communication channels among players, is described enough in our two-layered hypergraphs. Thus, we do not mention this further generalizations in this paper. The paper is organized as follows. Section 2 gives the notation and definition of two-layered hypergraphs. Section 3 presents a value of the games with two-layered hypergraphs. Section 4 characterizes our value via Owen’s coalitional value and the weighted Myerson value. Section 5 provides an axiomatic characterization of our value.. 2. Preliminaries Let N ⊆ N be a finite set of players and |N | = n where | · | is the cardinality of the set. A cooperative game with transferable utility or, simply, a game is a pair (N , v) where v : 2N → R satisfying v(∅) = 0. The Shapley value (Shapley, 1953) of a game (N , v) is an n-dimensional vector (Shi (N , v))i∈N satisfying Shi (N , v) = ∑ s!(n−1−s)! (v(S ∪ {i}) − v(S )) , where s = |S |. S ⊆N \{i} n! Given N, let H ⊆ {S |S ⊆ N , |S | ≥ 2} be a hypergraph. We call each element of hypergraph a conference. Given a hypergraph H and two players i and j, i is connected to j in H if there exists a finite sequence of players i0 , . . . , iK and a finite sequence of conferences S1 , . . . , SK such that (i) i0 = i, (ii) iK = j and (iii) for any k = 1, . . . , K , {ik−1 , ik } ⊆ Sk . The notion of connectedness induces a partition of the player set into maximal connected components. Let N /H = {{i} ∪ {j|j is connected to i in H }|i ∈ N } be a partition of N induced by the connectedness relation in H. Given hypergraph H and T ⊆ N, let H (T ) = {S ∈ H |S ⊆ T } be a restricted hypergraph on T . Connectedness in H (T ) induces a partition T /H of T into components in a similar manner as H induces a partition of N. The Myerson value (Myerson, 1980) of a game with a hypergraph (N , v, H ) is an n-dimensional vector (µi (∑ N , v, H ))i∈N satisfying µi (N , v, H ) = Shi (N , v H ), where v H (S ) = C ∈S /H v(C ) for any S ⊆ N. Let B = {B1 , B2 , . . . , Bm } be a partition on N or an a priori union on N. Let M = {1, 2, . . . , m} be a set of all indices of elements in B and let bk = |Bk | for any k ∈ M. Given N and B, hypergraph H on N is two-layered via B if for each S ∈ H either S ⊆ Bℓ for some ℓ ∈ M, or S is a
115
union of two or more sets Bℓ . Every hypergraph on N is twolayered via at least the coarsest possible partition, i.e., {N } or the finest one, i.e., {{i}}i∈N , and may well be two-layered via a broad variety of other B as well. The two levels of interaction (alluded to earlier) are then as follows: conferences H (Bℓ ) among several players from a single element Bℓ of B, and conferences HM among groups of players that constitute the a priori union: HM = {Q ⊆ M | there exist S ∈ H such that ℓ∈Q Bℓ = S , |Q | ≥ 2}. By definition, H = ℓ∈M H (Bℓ ) ∪ { ℓ∈Q Bℓ |Q ∈ HM }. Hence, our two-layered hypergraphs well capture communication structures among players in a priori unions. A four-tuple (N , v, B, H ) such that H is two-layered via B is a game with a two-layered hypergraph. Let Γ be a set of all games with two-layered hypergraphs. The following is an example of a game with a two-layered hypergraph: Example 1. Let N = {1, 2, 3, 4, 5} and v(S ) = 1 if S = {1, 3}, {1, 5}, {3, 5}, {1, 2, 3}, {1, 2, 5}, {1, 3, 4}, {1, 4, 5}, {2, 3, 5}, {3, 4, 5}, {1, 2, 3, 4}, {1, 2, 4, 5}, and {2, 3, 4, 5}, v(S ) = 2 if S = {1, 3, 5}, {1, 2, 3, 5}, {1, 3, 4, 5}, and N, and v(S ) = 0 otherwise. Let B = {B1 , B2 , B3 } = {{1, 2, 3}, {4}, {5}} and H = {{1, 2}, {2, 3}, {1, 2, 3, 4}, {4, 5}}. Now, H is two-layered via B since {1, 2}, {2, 3} ⊆ B1 , {1, 2, 3, 4} = B1 ∪ B2 , and {4, 5} = B2 ∪ B3 . The first two conferences represent players’ communications within an element of the a priori union and the last two conferences represent players communications among elements of the a priori union. 3. A value of games with two-layered hypergraphs A value of games with two-layered hypergraphs is a mapping that assigns an n-dimensional vector to each game (N , v, B, H ) in Γ . Because there are two levels of interaction, we will define such a value by applying the value of games with hypergraphs two times— once within individual elements of the a priori union, and once among groups of such elements. Mathematically, for players in Bk ∈ B, the game considered in the first step is a triple (Bk , v|Bk , H (Bk )), where v|Bk (S ) = v(S ) for any S ⊆ Bk . The game considered in the second step is a triple (M , vM ,H , HM ) where vM ,H : 2M → R with vM ,H (∅) = 0 is defined as follows: for any Q ⊆ M,
H v (Bℓ ) vM ,H (Q ) = Bℓ v
if Q = {ℓ} otherwise.
ℓ∈Q
The above definition of characteristic function vM ,H is the same as that found for the intermediate game, or quotient game, in the literature on games with a priori unions, except for the value of singleton coalitions. This different value for singletons is exactly what is needed to prove the following Lemma: Lemma 1. For any Q ⊆ M, v H (
ℓ∈Q
Bℓ ) = (vM ,H )HM (Q ).
Proof. By definition,
Bℓ
H
ℓ∈Q
=
ℓ∈C
Bℓ |C ∈ Q /HM and |C | ≥ 2
∪
{ℓ}∈Q /HM
Bℓ /H .
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T. Kongo / Mathematical Social Sciences 62 (2011) 114–119 Table 1 Values in the game in Example 1.
Thus,
vH
Bℓ
=
ℓ∈Q
−
v(C )
Bℓ )/H C ∈( ℓ∈Q
= =
−
v
Bℓ
C ∈Q /HM ,|C |≥2
ℓ∈C
−
vM ,H (C ) +
C ∈Q /HM ,|C |≥2
=
−
+
−
−
v(C )
{ℓ}∈Q /HM C ∈Bℓ /H
−
v H (Bℓ )
1
2
3
4
5
4 9 2 3 1 2 3 4 2 3 8 15
4 9
4 9 2 3 1 2 3 4 2 3 8 15
1 3
0
1 3 2 3
0
0
0
1 2 1 2 1 5
0 0 0 1 6 8 15
0 1 5
{ℓ}∈Q /HM
vM ,H (C ) = (vM ,H )HM (Q ).
Owen’s coalitional value Ow of a game with a priori union
C ∈Q /HM
By the above two-step approach, we define the value ϕ M on Γ as follows: Definition 1. For each (N , v, B, H ) ∈ Γ such that H is two-layered via B and each i ∈ N with i ∈ Bk ∈ B,
ϕiM (N , v, B, H ) = µi (Bk , v, H ) +
Values/players
ϕ M (N , v, B, H ) Sh(N , v) AD(N , v, B) Ow(N , v, B) SS (N , v, B) µ(N , v, H )
µk (M , vM ,H , HM ) − v (Bk ) H
bk
,
where (Bk , v, H ) = (Bk , v|Bk , H (Bk )). It is worth mentioning the relationships between the above value and values in related studies.
• If there are no conferences in HM , i.e., HM = ∅ (one typical example is the case that an a priori union is the coarsest one,) ϕ M coincides with the Myerson value of (N , v, H ), since the second term equals zero. • If an a priori union is the finest one, ϕ M coincides with the Myerson value of (M , vM ,H , HM ), since the other terms are canceled. • If H contains all two-player coalitions in the same element of an a priori union, i.e., H ⊇ {S |S ⊆ Bℓ ∈ B, |S | = 2, ℓ ∈ M } and if there are no conferences in HM , ϕ M coincides with the Aumann and Dreze value (Aumann and Drèze, 1974), ADi (N , v, B) = Shi (Bk , v|Bk ) for each i ∈ Bk ∈ B. This is because the Myerson value coincides with the Shapley value if the hypergraph contains all coalitions of size two and the Myerson value coincides with the value of the singleton coalition if the hypergraph is empty. Further, if an a priori union is the coarsest one, ϕ M coincides with the Shapley value. • If H contains all two-player coalitions in the same element of an a priori union, and HM contains all coalitions of size two, i.e. HM ⊇ {Q |Q ⊆ M , |Q | = 2}, ϕ M coincides with the twostep Shapley value (Kamijo, 2009), SSi (N , v, B) = Shi (Bk , v) + Shk (M ,vM )−v(Bk ) for i ∈ Bk ∈ B, where vM (Q ) = v( ℓ∈Q Bℓ ) for bk any Q ⊆ M, for the same reason as mentioned above. Therefore, our value is a generalization of all of the aforementioned values. Table 1 is a list of our value and values mentioned in the above list in the game given in Example 1. Note that in the game in Example 1, players 1, 3, and 5 are symmetric with each other in (N , v). Player 2 is a null player in game (N , v), but plays an important role in hypergraph H (B1 ). Player 4 is a null player in game (N , v) and B2 = {4} is also a null player in the intermediate game, but B2 plays an important role in hypergraph HM . 4. Characterization via Owen’s coalitional value By definition, our value seems to depend heavily on the twostep approach. In the following, we characterize our value by applying the coalitional value introduced by Owen (1977).
(N , v, B) is an n-dimensional vector (Owi (N , v, B))i∈N satisfying − q!(m − 1 − q)! − s!(bk − 1 − s)! Owi (N , v, B) = m! bk ! Q ⊆M \{k} S ⊆Bk \{i} × v Bℓ ∪ S ∪ {i} − v Bℓ ∪ S , ℓ∈Q
ℓ∈Q
where i ∈ Bk ∈ B and q = |Q |. It is well known that the value is also represented by the average of each player’s marginal contributions with respect to all consistent orderings with the a priori union.3 Then, the following theorem holds. Theorem 1. For any (N , v, B, H ) ∈ Γ such that H is two-layered via B, ϕ M (N , v, B, H ) = Ow(N , v H , B). Proof. In the case of S = Bk \ {i},
s!(bk −1−s)! bk !
=
1 . bk
Hence,
− q!(m − 1 − q)! 1 m! bk Q ⊆M \{k} H H × v Bℓ ∪ Bk − v Bℓ ∪ Bk \ {i}
Owi (N , v H , B) =
ℓ∈Q
ℓ∈Q
− q!(m − 1 − q)! − s!(bk − 1 − s)! + m! bk ! S (Bk \{i} Q ⊆M \{k} H H × v Bℓ ∪ S ∪ {i} − v Bℓ ∪ S . ℓ∈Q
(1)
ℓ∈Q
Next, by definition, for any Q ⊆ M \ k and any S ( Bk , it holds that
Bℓ ∪ S
H =
Bℓ
H ∪S
H.
ℓ∈Q
ℓ∈Q
Hence, together with Lemma 1,
v
H
Bℓ ∪ S
= (vM ,H )HM (Q ) + v H (S ).
ℓ∈Q
Thus, in conjunction with Lemma 1, Eq. (1) =
− q!(m − 1 − q)! 1 (vM ,H )HM (Q ∪ {k}) m! bk Q ⊆M \{k} − (vM ,H )HM (Q ) − v H (Bk \ {i}) − q!(m − 1 − q)! − s!(bk − 1 − s)! + m! bk ! Q ⊆M \{k} S (B \{i} k
3 An ordering (on the player set) is consistent with an a priori union if all players in the same element of the a priori union appear successively.
T. Kongo / Mathematical Social Sciences 62 (2011) 114–119
× (vM ,H )HM (Q ) + v H (S ∪ {i}) − (vM ,H )HM (Q ) − v H (S ) − q!(m − 1 − q)! 1 = m! bk Q ⊆M \{k} HM × (vM ,H ) (Q ∪ {k}) − (vM ,H )HM (Q ) − q!(m − 1 − q)! 1 v H (Bk \ {i}) − m ! b k Q ⊆M \{k} − q!(m − 1 − q)! − s!(bk − 1 − s)! + m! bk ! Q ⊆M \{k} S (Bk \{i} H × v (S ∪ {i}) − v H (S ) . ∑ q!(m−1−q)! Since Q ⊆M \{k} = 1, m!
Myerson value4 with respect to weight vector w = (wi )i∈N such that wi = b1 for any i ∈ Bk ∈ B. k
Proof. From Theorem 1, the coincidence between the marginal contributions vectors and the modified marginal contributions vectors mentioned in the above, and Theorems 2 and 3 in Kamijo (2009), it follows that ϕ M (N , v, B, H ) = Ow(N , v H , B) = Shw (N , (v H )B ), where Shw is the weighted Shapley value with respect to weight vector Theorem 2, and v B is w defined as ∑ B defined as v (S ) = v( Bℓ ⊆S ,ℓ∈M Bℓ ) + Bℓ ∩S ̸=Bℓ ,ℓ∈M v(Bℓ ∩ S ) for any S ⊆ N. Since H is two-layered via B, we can show that (v H )B = v H . Therefore, we obtain, Shw (N , (v H )B ) = Shw (N , v H ) = µw (N , v, H ). (2)
− q!(m − 1 − q)! Eq. (2) = bk Q ⊆M \{k} m! × (vM ,H )HM (Q ∪ {k}) − (vM ,H )HM (Q ) 1 H 1 H v (Bk \ {i}) + v (Bk ) − v H (Bk ) − 1
bk
bk
− s!(bk − 1 − s)! + v H (S ∪ {i}) − v H (S ) b ! k S (B \{i} k
=
1 bk
+ +
Shk (M , (vM ,H )HM ) −
1 bk
1
v H (Bk )
v H (Bk ) − v H (Bk \ {i})
bk
− s!(bk − 1 − s)! v H (S ∪ {i}) − v H (S ) b ! k S (B \{i} k
=
1 bk
+
Shk (M , (vM ,H )HM ) −
1 bk
v H (Bk )
− s!(bk − 1 − s)! v H (S ∪ {i}) − v H (S ) b ! k S ⊆B \{i} k
=
1 bk
Shk (M , (vM ,H )HM ) −
= µi (Bk , v, H ) +
1 bk
117
v H (Bk ) + Shi (Bk , v H )
µk (M , vM ,H , HM ) − v H (Bk ) bk
= ϕ (N , v, B, H ). M
The above result might appear to suggest that the two-step Shapley value and the Owen value were the same. However, they are different. Kamijo (2009, Theorem 2) gave the Owen value-like representation of the two-step Shapley value in the class of all games with a priori unions. In this representation, the modified marginal contributions vectors are used instead of the marginal contributions vectors. In the class of games with two-layered hypergraphs, the connectedness obtained by the two-layered hypergraphs renders the two vectors the same. Since two-layered hypergraphs can be seen as generalizations of a priori unions, our result justifies using the modified marginal contributions vectors instead of the original vectors in the class of games with a priori unions. In addition, Theorem 1 can be seen as a generalization of Theorem 2 in Kamijo (2009). Further, the coincidence between the two vectors mentioned above implies the following generalization of Theorem 3 in Kamijo (2009). Theorem 2. For any (N , v, B, H ) ∈ Γ such that H is two-layered via B, ϕ M (N , v, B, H ) = µw (N , v, H ), where µw is the weighted
5. Axiomatic characterization Next, we give an axiomatic characterization of our value via two axioms. The first axiom is a fundamental one in the literature of games with restricted communications among players. Component efficiency: For any C ∈ N /H, i∈C ϕi (N , v, B, H ) = v(C ). The above property requires that, given communicationrestricted structures (now two-layered hypergraphs), the sum of the value among players in a maximal connected component is equal to the worth of the component. This property is related to efficiency. To define the second axiom, we have to consider the effect of a player’s isolation from the two-layered hypergraphs. Since we consider a priori unions and hypergraphs simultaneously, a player’s isolation affects both structures. First, consider the effect on a priori unions. Let i ∈ N, B = {B1 , B2 , . . . , Bm }, and i ∈ Bk ∈ B. Then, if i is isolated from the structure, the structure becomes B−i = (B \ Bk ) ∪ {Bk \ {i}} ∪ {{i}}. A player’s isolation affects only the element to which the player belongs. Next, consider the effect on hypergraphs. We assume that for forming a conference by a set of players, unanimous approval of the players is needed. Then, a player’s isolation induces the deletion of all conferences that contain the player and all conferences that contain the element of the a priori union that contains the player. Let H be a two-layered hypergraph via B and let i ∈ N. Then, player i’s isolation induces hypergraph H−i = H (N \ {i}). Note that H−i is two-layered via B−i . Therefore, given (N , v, B, H ) ∈ Γ , for any {i, j} ⊆ N, value ϕ M for player i ∈ Bk of a game where player j ∈ Bh is isolated is as follows:
∑
ϕiM (N , v, B−j , H−j ) µ (B \ {j}, v, H ) if Bk = Bh µi (Bk , v, H ) −j i k = H + µk (M , vM ,H , HM −h ) − v (Bk ) if Bk ̸= Bh , 5 bk
where HM −h = HM (M \ {h}). In the case where Bk = Bh , j’s isolation affects i’s value both in the game within Bk and in the game among elements of the a priori union. In the game within Bk , we isolate j from the restricted hypergraph on Bk . In the game
4 See e.g., Slikker and van den Nouweland (2000) for details. 5 More precisely, we have to consider in a different way the case where B = h {j}. In this case, Bh becomes the empty set after j’s deletion. Thus, we should write, µk (M \ {h}, vM ,H |M \{h} , HM −h ) instead of µk (M , vM ,H , HM −h ). However, by component decomposability of the Myerson value (see van den Nouweland, 1993), it holds that µk (M \ {h}, vM ,H |M \{h} , HM −h ) = µk (M , vM ,H , HM −h ) and hence, it is not significant. Therefore, in this case we also write in the above way.
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T. Kongo / Mathematical Social Sciences 62 (2011) 114–119
among elements of the a priori union, the element Bk \{j} is isolated, since j’s absence renders impossible the unanimous approval of forming a conference including Bk . By the component efficiency of the Myerson value, the element obtains its value as a singleton coalition, and thus the second term of the value is canceled. In the case where Bk ̸= Bh , j’s isolation affects i’s value only in the game among elements of the a priori union. Following the above argument, the next property is well defined. Normalized balanced contributions property: For any i, j ∈ N where i ∈ Bk ∈ B and j ∈ Bh ∈ B, bk ϕi (N , v, B, H ) − ϕi (N , v, B−j , H−j )
= bh ϕj (N , v, B, H ) − ϕj (N , v, B−i , H−i ) . The above two axioms characterize our value ϕ M . Theorem 3. ϕ M is the unique value satisfying both component efficiency and the normalized balanced contributions property. Proof. First, we show that ϕ M satisfies component efficiency. Take C ∈ N /H and consider two cases: (i) C is a subset of a single element of a priori union B, or (ii) C is a union of several such elements. In case (i), there exists Bk ∈ B such that Bk ⊇ C and there exists no conference Q ∈ HM such that Q ∋ k, i.e., {k} ∈ Q /HM . Hence, by component efficiency of the Myerson value in the game between elements of the a priori union, µk (M , vM ,H , HM ) = vM ,H ({k}). By any i ∈ C , the second definition, vM ,H ({k}) = v H (Bk ). Thus, for ∑ M M term of the value ϕ is canceled. Therefore, i∈C ϕi (N , v, B, H ) = ∑ i∈C µi (Bk , v, H ) = v(C ), where the equality follows by the component efficiency of the Myerson value of the game within the element of the a priori union. In case (ii), there exists D ∈ M /HM such that C = ℓ∈D Bℓ . ∑ ∑ ∑ M In this case, ϕ ( N , v, B , H ) = (µ ( B , v, H) + i ℓ i∈C i ℓ∈D i∈Bℓ µℓ (M ,vM ,H ,HM )−v H (Bℓ )
). For each ℓ ∈ D, component efficiency of ∑ H the Myerson value implies that i∈Bℓ µi (Bℓ , v, H ) = v (Bℓ ) and ∑ HM HM ℓ∈D µℓ (M , vM ,H , HM ) = vM ,H (D). By definition, vM ,H (D) = v( ℓ∈D Bℓ ) = v(C ). Hence, we obtain the desired result. Next, we show that ϕ M satisfies the normalized balanced contributions property. Take any {i, j} ⊆ N. If i, j ∈ Bk ∈ B, the second term of ϕiM (N , v, B, H ) is the same as the second term of ϕjM (N , v, B, H ). Hence, bℓ
bk ϕiM (N , v, B, H ) − bk ϕjM (N , v, B, H )
= bk µi (Bk , v, H ) − bk µj (Bk , v, H ).
(3)
By definition, in both H−i and H−j , there is no conference that contains Bk . Thus, the second terms of both ϕiM (N , v, B−j , H−j ) and ϕjM (N , v, B−i , H−i ) are equal to zero, and it holds that bk ϕiM (N , v, B−j , H−j ) − bk ϕjM (N , v, B−i , H−i )
= bk µi (Bk \ {j}, v, H−j ) − bk µj (Bk \ {i}, v, H−i ).
(4)
By component decomposability (see van den Nouweland, 1993) and the balanced contributions property (see Myerson, 1980) of the Myerson value in the game within the element of the a priori union, the right hand sides of Eqs. (3) and (4) are equal. Thus, we obtain the desired result. If i ∈ Bk , j ∈ Bh , and k ̸= h then bk ϕiM (N , v, B, H ) − bh ϕjM (N , v, B, H )
= bk µi (Bk , v, H ) + µk (M , vM ,H , HM ) − v (Bk ) H
− bh µj (Bh , v, H ) − µh (M , vM ,H , HM ) + v H (Bh ),
(5)
and, bk ϕiM (N , v, B−j , H−j ) − bh ϕjM (N , v, B−i , H−i )
= bk µi (Bk , v, H−j ) + µk (M , vM ,H , HM −h ) − v H−j (Bk ) − bh µj (Bh , v, H−i ) − µh (M , vM ,H , HM −k ) + v H−i (Bh ).
(6)
Given that i ∈ Bk , j ∈ Bh and k ̸= h and by the fact that there is no conference that contains players in different elements of the a priori union in H, it holds that Bk /H = Bk /H−j and Bh /H = Bh /H−i . Thus, µi (Bk , v, H ) = µi (Bk , v, H−j ), µj (Bh , v, H ) = µj (Bh , v, H−i ), v H (Bk ) = v H−j (Bk ), and v H (Bh ) = v H−i (Bh ). In conjunction with the balanced contributions property of the Myerson value in the game among elements of the a priori union, the right hand sides of Eqs. (5) and (6) are equal. Therefore, we obtain the desired result. Finally, we show the uniqueness of the value that satisfies the two axioms. Let ϕ be a value that satisfies the two axioms. We use induction with respect to the number of conferences in hypergraph H. If |H | = 0, N /H = {{i}}i∈N . By component efficiency, ϕi (N , v, B, H ) = v({i}) and the value is uniquely determined. Let a be a natural number. Assume that for any hypergraph that contains fewer than a conferences, ϕ is uniquely determined and consider the case where hypergraph H contains a conferences. Take any component C ∈ N /H. If |C | = 1, component efficiency implies that ϕi (N , v, B, H ) = v({i}). If |C | > 1, take i ∈ C . For any j ∈ C \ {i} with j ∈ Bh ∈ B, the normalized balanced contributions property and supposition above together imply bk ϕi (N , v, B, H ) − bh ϕj (N , v, B, H )
= bk ϕiM (N , v, B−j , H−j ) − bh ϕjM (N , v, B−i , H−i ). Also, by component efficiency,
−
ϕi (N , v, B, H ) = v(C ).
i∈C
These |C | equalities form a regular system of linear equations in |C | variables and that system has a unique solution. Vázquez-Brage et al. (1996, see their Theorem 3) characterized their value in a slightly different model from ours, using three axioms: component efficiency, balanced contributions for the graph, and fairness in the quotient. Our normalized balanced contributions property becomes equivalent to balanced contributions for the graph if two players i and j are in the same element of the a priori union. Thus, comparing our result to their Theorem 3, our normalized balanced contributions property (for players in different elements in the a priori union) is a counterpart of fairness in the quotient. Thus differences between these two properties constitute the key distinction between their approach and ours. Acknowledgments The author would like to thank Rene van den Brink, Yukihiko Funaki, Yoshio Kamijo, Gerard van der Laan, and Koichi Suga for helpful comments and discussions. The author is grateful to an associate editor and anonymous referees for useful comments and suggestions. The author would like to acknowledge the financial support provided by the Japan Society for the Promotion of Science (JSPS). References Alonso-Meijide, J., Alvarez-Mozos, M., Fiestras-Janeiro, M., 2009. Values of games with restricted communication and a priori unions. Mathematical Social Sciences 58, 202–213. Alonso-Meijide, J., Fiestras-Janeiro, M., 2002. Modification of the Banzhaf value for games with a coalition structure. Annals of Operations Research 109, 213–227.
T. Kongo / Mathematical Social Sciences 62 (2011) 114–119 Aumann, R.J., Drèze, J.H., 1974. Cooperative games with coalition structures. International Journal of Game Theory 3, 217–237. Hart, S., Kurz, M., 1983. Endogenous formation of coalitions. Econometrica 51, 1047–1064. Kamijo, Y., 2009. A two-step Shapley value in a cooperative game with a coalition structure. International Game Theory Review 11, 207–214. Khmelnitskaya, A., 2007. Values for graph-restricted games with coalition structure. Memorandum No. 1848, Department of Applied Mathematics, Faculty EEMCS, University of Twente, The Netherlands. Myerson, R.B., 1977. Graphs and cooperation in games. Mathematics of Operations Research 2, 225–229. Myerson, R.B., 1980. Conference structures and fair allocation rules. International Journal of Game Theory 9, 169–182. Owen, G., 1977. Value of games with a priori unions. In: Henn, R., Moeschlin, O. (Eds.), Mathematical Economics and Game Theory. Springer-Verlag. Owen, G., 1981. Modification of the Banzhaf–Coleman index for games with a priori unions. In: Holler, M. (Ed.), Power, Voting, and Voting Power. Physica-Verlag, pp. 223–238.
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Owen, G., 1986. Values of the graph-restricted games. SIAM Journal on Algorithm and Discrete Method 7, 210–220. Shapley, L.S., 1953. A value for n-person games. In: Roth, A.E. (Ed.), The Shapley Value. Cambridge University Press, pp. 41–48. Slikker, M., van den Nouweland, A., 2000. Communication situations with asymmetric players. Mathematical Methods of Operations Research 52, 39–56. van den Brink, R., van der Laan, G., 2005. A class of consistent share functions for games in coalition structure. Games and Economic Behavior 51, 193–212. van den Nouweland, A., 1993. Games and Graphs in Economic Situations. Ph.D. Thesis, Tilburg University, Netherland. van der Laan, G., van den Brink, R., 2002. A Banzhaf share function for cooperative games in coalition structure. Theory and Decision 53, 61–86. Vázquez-Brage, M., García-Jurado, I., Carreras, F., 1996. The Owen value applied to games with graph-restricted communication. Games and Economic Behavior 12, 42–53. Winter, E., 1989. A value for cooperative games with levels structure of cooperation. International Journal of Game Theory 18, 227–240.
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Value of games with two-layered hypergraphs Takumi Kongo Faculty of Political Science & Economics, Waseda University. 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo, Japan
article
info
Article history: Received 29 October 2009 Received in revised form 19 June 2011 Accepted 23 June 2011 Available online 7 July 2011
abstract This paper studies cooperative games with restricted cooperation among players. We define situations in which a priori unions and hypergraphs coexist simultaneously and mutually depend on each other. We call such structures two-layered hypergraphs. Using a two-step approach, we define a value of the games with two-layered hypergraphs. The value is characterized by Owen’s coalitional value of hypergraph-restricted games and in terms of weighted Myerson value. Further, our value is axiomatically characterized by component efficiency and a coalition size normalized balanced contributions property. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In many social and economic situations, agents form a group and act as if they are a single agent. For example, neighboring countries form a regional economic bloc and stand together against other countries, laborers form a labor union and negotiate jointly with a manager, firms form a cartel and jointly set a uniform price, and politicians form a political party and coordinate their policy. The reason for such an act, in such situations, is that agents forming a group and coordinating their actions in the group leads to better outcomes for each of them if they distribute the profit generated by their cooperation appropriately. Such group formation and surplus distributions are analyzed mainly through cooperative games. Since cooperation needs coordination and coordination requires communication, in real situations, agents who are unable to communicate with each other cannot cooperate even if they wish to do so. By introducing a priori unions on the player set, Aumann and Drèze (1974) considered situations in which only players in the same element of a priori unions can cooperate. Myerson (1977, 1980) studied the restriction of cooperation by (hyper)graphs on the player set. In this setting, only players who are connected with each other can cooperate. By integrating the above two models, Vázquez-Brage et al. (1996) and Alonso-Meijide et al. (2009) investigated the situations in which both a priori unions and graphs exist simultaneously. In these two papers, a priori unions and graphs were independent. The a priori unions affected the definition of solution concepts, as in Owen (1977, 1981) and Alonso-Meijide and Fiestras-Janeiro (2002), while the graphs affected the worth of each coalition in order to represent communication restrictions among agents, as in Myerson (1977) and Owen (1986).
E-mail address: [email protected]. 0165-4896/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2011.06.006
In this paper, we consider the situation in which both a priori unions and hypergraphs coexist simultaneously and mutually depend on each other. That is, given an a priori union, there exists a hypergraph on each element of the a priori union and on the a priori union. We call such structures two-layered hypergraphs and study cooperative games with such structures.1 Any twolayered hypergraph H can be seen as an ordinary hypergraph with a restriction; each element of H must either be a subset of a single member of the given a priori union or be a union of several such members. However, if we take into consideration the layered structures on the set of agents represented by a given a priori union, two-layered hypergraphs and their corresponding usual hypergraphs are not the same. As mentioned in Hart and Kurz (1983), under the existence of an a priori union, interactions among agents will be conducted on two levels: among agents and among groups of agents that constitute the a priori union. Our two-layered hypergraphs well capture communication structures among agents in such situations. Next, we define a one-point solution (value) of the games with two-layered hypergraphs. In the definition of our value, we focus on the layered structure of agents. We decompose a game with a two-layered hypergraph into two problems. One is a problem within each element of the a priori union and the other is the problem between elements of the a priori union. Applying the Myerson value in each problem, we define the value of the game with the two-layered hypergraph.2 Our value is a generalization
1 Khmelnitskaya (2007) also studied a similar situation in which a priori unions and graphs exist simultaneously and mutually depend on each other. However, the interpretation of cooperation (connectedness) among players is different; thus, the value studied in Khmelnitskaya (2007) is different from ours. 2 This kind of ‘‘two-step approach’’ was also taken by van der Laan and van den Brink (2002); van den Brink and van der Laan (2005) and Kamijo (2009) in the class of games with a priori unions.
T. Kongo / Mathematical Social Sciences 62 (2011) 114–119
of the Myerson value of games with hypergraphs and also is a generalization of the value in Aumann and Drèze (1974) and the two-step Shapley value (Kamijo, 2009) of games with a priori unions. In conjunction with the hypergraph restriction of characteristic functions, our value is represented by Owen’s (1977) coalitional value for games with a priori unions. Thus, our value is a natural extension of existing solution concepts and does not depend heavily on the two-step approach. Further, our value is represented by weighted Myerson value. We also axiomatically characterize our value by two axioms. One axiom is an efficiency property called component efficiency and the other is a normalized version of the balanced contributions property. Here, we normalized each player’s contributions to others with respect to the number of players in the element of the a priori union that contains the player. As a generalization of an a priori union, Winter (1989) introduced a levels structure of players, that is a sequence of a priori unions, each a priori union is obtained by unification of elements in the previous a priori union, and each representing the various cooperations between elements of the previous a priori union. By considering situations in which both levels structures and hypergraphs coexist simultaneously and dependently, we can define more general structures called multi-layered hypergraphs. However, the essence of our discussion, coexistence of layered structures of players and communication channels among players, is described enough in our two-layered hypergraphs. Thus, we do not mention this further generalizations in this paper. The paper is organized as follows. Section 2 gives the notation and definition of two-layered hypergraphs. Section 3 presents a value of the games with two-layered hypergraphs. Section 4 characterizes our value via Owen’s coalitional value and the weighted Myerson value. Section 5 provides an axiomatic characterization of our value.. 2. Preliminaries Let N ⊆ N be a finite set of players and |N | = n where | · | is the cardinality of the set. A cooperative game with transferable utility or, simply, a game is a pair (N , v) where v : 2N → R satisfying v(∅) = 0. The Shapley value (Shapley, 1953) of a game (N , v) is an n-dimensional vector (Shi (N , v))i∈N satisfying Shi (N , v) = ∑ s!(n−1−s)! (v(S ∪ {i}) − v(S )) , where s = |S |. S ⊆N \{i} n! Given N, let H ⊆ {S |S ⊆ N , |S | ≥ 2} be a hypergraph. We call each element of hypergraph a conference. Given a hypergraph H and two players i and j, i is connected to j in H if there exists a finite sequence of players i0 , . . . , iK and a finite sequence of conferences S1 , . . . , SK such that (i) i0 = i, (ii) iK = j and (iii) for any k = 1, . . . , K , {ik−1 , ik } ⊆ Sk . The notion of connectedness induces a partition of the player set into maximal connected components. Let N /H = {{i} ∪ {j|j is connected to i in H }|i ∈ N } be a partition of N induced by the connectedness relation in H. Given hypergraph H and T ⊆ N, let H (T ) = {S ∈ H |S ⊆ T } be a restricted hypergraph on T . Connectedness in H (T ) induces a partition T /H of T into components in a similar manner as H induces a partition of N. The Myerson value (Myerson, 1980) of a game with a hypergraph (N , v, H ) is an n-dimensional vector (µi (∑ N , v, H ))i∈N satisfying µi (N , v, H ) = Shi (N , v H ), where v H (S ) = C ∈S /H v(C ) for any S ⊆ N. Let B = {B1 , B2 , . . . , Bm } be a partition on N or an a priori union on N. Let M = {1, 2, . . . , m} be a set of all indices of elements in B and let bk = |Bk | for any k ∈ M. Given N and B, hypergraph H on N is two-layered via B if for each S ∈ H either S ⊆ Bℓ for some ℓ ∈ M, or S is a
115
union of two or more sets Bℓ . Every hypergraph on N is twolayered via at least the coarsest possible partition, i.e., {N } or the finest one, i.e., {{i}}i∈N , and may well be two-layered via a broad variety of other B as well. The two levels of interaction (alluded to earlier) are then as follows: conferences H (Bℓ ) among several players from a single element Bℓ of B, and conferences HM among groups of players that constitute the a priori union: HM = {Q ⊆ M | there exist S ∈ H such that ℓ∈Q Bℓ = S , |Q | ≥ 2}. By definition, H = ℓ∈M H (Bℓ ) ∪ { ℓ∈Q Bℓ |Q ∈ HM }. Hence, our two-layered hypergraphs well capture communication structures among players in a priori unions. A four-tuple (N , v, B, H ) such that H is two-layered via B is a game with a two-layered hypergraph. Let Γ be a set of all games with two-layered hypergraphs. The following is an example of a game with a two-layered hypergraph: Example 1. Let N = {1, 2, 3, 4, 5} and v(S ) = 1 if S = {1, 3}, {1, 5}, {3, 5}, {1, 2, 3}, {1, 2, 5}, {1, 3, 4}, {1, 4, 5}, {2, 3, 5}, {3, 4, 5}, {1, 2, 3, 4}, {1, 2, 4, 5}, and {2, 3, 4, 5}, v(S ) = 2 if S = {1, 3, 5}, {1, 2, 3, 5}, {1, 3, 4, 5}, and N, and v(S ) = 0 otherwise. Let B = {B1 , B2 , B3 } = {{1, 2, 3}, {4}, {5}} and H = {{1, 2}, {2, 3}, {1, 2, 3, 4}, {4, 5}}. Now, H is two-layered via B since {1, 2}, {2, 3} ⊆ B1 , {1, 2, 3, 4} = B1 ∪ B2 , and {4, 5} = B2 ∪ B3 . The first two conferences represent players’ communications within an element of the a priori union and the last two conferences represent players communications among elements of the a priori union. 3. A value of games with two-layered hypergraphs A value of games with two-layered hypergraphs is a mapping that assigns an n-dimensional vector to each game (N , v, B, H ) in Γ . Because there are two levels of interaction, we will define such a value by applying the value of games with hypergraphs two times— once within individual elements of the a priori union, and once among groups of such elements. Mathematically, for players in Bk ∈ B, the game considered in the first step is a triple (Bk , v|Bk , H (Bk )), where v|Bk (S ) = v(S ) for any S ⊆ Bk . The game considered in the second step is a triple (M , vM ,H , HM ) where vM ,H : 2M → R with vM ,H (∅) = 0 is defined as follows: for any Q ⊆ M,
H v (Bℓ ) vM ,H (Q ) = Bℓ v
if Q = {ℓ} otherwise.
ℓ∈Q
The above definition of characteristic function vM ,H is the same as that found for the intermediate game, or quotient game, in the literature on games with a priori unions, except for the value of singleton coalitions. This different value for singletons is exactly what is needed to prove the following Lemma: Lemma 1. For any Q ⊆ M, v H (
ℓ∈Q
Bℓ ) = (vM ,H )HM (Q ).
Proof. By definition,
Bℓ
H
ℓ∈Q
=
ℓ∈C
Bℓ |C ∈ Q /HM and |C | ≥ 2
∪
{ℓ}∈Q /HM
Bℓ /H .
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T. Kongo / Mathematical Social Sciences 62 (2011) 114–119 Table 1 Values in the game in Example 1.
Thus,
vH
Bℓ
=
ℓ∈Q
−
v(C )
Bℓ )/H C ∈( ℓ∈Q
= =
−
v
Bℓ
C ∈Q /HM ,|C |≥2
ℓ∈C
−
vM ,H (C ) +
C ∈Q /HM ,|C |≥2
=
−
+
−
−
v(C )
{ℓ}∈Q /HM C ∈Bℓ /H
−
v H (Bℓ )
1
2
3
4
5
4 9 2 3 1 2 3 4 2 3 8 15
4 9
4 9 2 3 1 2 3 4 2 3 8 15
1 3
0
1 3 2 3
0
0
0
1 2 1 2 1 5
0 0 0 1 6 8 15
0 1 5
{ℓ}∈Q /HM
vM ,H (C ) = (vM ,H )HM (Q ).
Owen’s coalitional value Ow of a game with a priori union
C ∈Q /HM
By the above two-step approach, we define the value ϕ M on Γ as follows: Definition 1. For each (N , v, B, H ) ∈ Γ such that H is two-layered via B and each i ∈ N with i ∈ Bk ∈ B,
ϕiM (N , v, B, H ) = µi (Bk , v, H ) +
Values/players
ϕ M (N , v, B, H ) Sh(N , v) AD(N , v, B) Ow(N , v, B) SS (N , v, B) µ(N , v, H )
µk (M , vM ,H , HM ) − v (Bk ) H
bk
,
where (Bk , v, H ) = (Bk , v|Bk , H (Bk )). It is worth mentioning the relationships between the above value and values in related studies.
• If there are no conferences in HM , i.e., HM = ∅ (one typical example is the case that an a priori union is the coarsest one,) ϕ M coincides with the Myerson value of (N , v, H ), since the second term equals zero. • If an a priori union is the finest one, ϕ M coincides with the Myerson value of (M , vM ,H , HM ), since the other terms are canceled. • If H contains all two-player coalitions in the same element of an a priori union, i.e., H ⊇ {S |S ⊆ Bℓ ∈ B, |S | = 2, ℓ ∈ M } and if there are no conferences in HM , ϕ M coincides with the Aumann and Dreze value (Aumann and Drèze, 1974), ADi (N , v, B) = Shi (Bk , v|Bk ) for each i ∈ Bk ∈ B. This is because the Myerson value coincides with the Shapley value if the hypergraph contains all coalitions of size two and the Myerson value coincides with the value of the singleton coalition if the hypergraph is empty. Further, if an a priori union is the coarsest one, ϕ M coincides with the Shapley value. • If H contains all two-player coalitions in the same element of an a priori union, and HM contains all coalitions of size two, i.e. HM ⊇ {Q |Q ⊆ M , |Q | = 2}, ϕ M coincides with the twostep Shapley value (Kamijo, 2009), SSi (N , v, B) = Shi (Bk , v) + Shk (M ,vM )−v(Bk ) for i ∈ Bk ∈ B, where vM (Q ) = v( ℓ∈Q Bℓ ) for bk any Q ⊆ M, for the same reason as mentioned above. Therefore, our value is a generalization of all of the aforementioned values. Table 1 is a list of our value and values mentioned in the above list in the game given in Example 1. Note that in the game in Example 1, players 1, 3, and 5 are symmetric with each other in (N , v). Player 2 is a null player in game (N , v), but plays an important role in hypergraph H (B1 ). Player 4 is a null player in game (N , v) and B2 = {4} is also a null player in the intermediate game, but B2 plays an important role in hypergraph HM . 4. Characterization via Owen’s coalitional value By definition, our value seems to depend heavily on the twostep approach. In the following, we characterize our value by applying the coalitional value introduced by Owen (1977).
(N , v, B) is an n-dimensional vector (Owi (N , v, B))i∈N satisfying − q!(m − 1 − q)! − s!(bk − 1 − s)! Owi (N , v, B) = m! bk ! Q ⊆M \{k} S ⊆Bk \{i} × v Bℓ ∪ S ∪ {i} − v Bℓ ∪ S , ℓ∈Q
ℓ∈Q
where i ∈ Bk ∈ B and q = |Q |. It is well known that the value is also represented by the average of each player’s marginal contributions with respect to all consistent orderings with the a priori union.3 Then, the following theorem holds. Theorem 1. For any (N , v, B, H ) ∈ Γ such that H is two-layered via B, ϕ M (N , v, B, H ) = Ow(N , v H , B). Proof. In the case of S = Bk \ {i},
s!(bk −1−s)! bk !
=
1 . bk
Hence,
− q!(m − 1 − q)! 1 m! bk Q ⊆M \{k} H H × v Bℓ ∪ Bk − v Bℓ ∪ Bk \ {i}
Owi (N , v H , B) =
ℓ∈Q
ℓ∈Q
− q!(m − 1 − q)! − s!(bk − 1 − s)! + m! bk ! S (Bk \{i} Q ⊆M \{k} H H × v Bℓ ∪ S ∪ {i} − v Bℓ ∪ S . ℓ∈Q
(1)
ℓ∈Q
Next, by definition, for any Q ⊆ M \ k and any S ( Bk , it holds that
Bℓ ∪ S
H =
Bℓ
H ∪S
H.
ℓ∈Q
ℓ∈Q
Hence, together with Lemma 1,
v
H
Bℓ ∪ S
= (vM ,H )HM (Q ) + v H (S ).
ℓ∈Q
Thus, in conjunction with Lemma 1, Eq. (1) =
− q!(m − 1 − q)! 1 (vM ,H )HM (Q ∪ {k}) m! bk Q ⊆M \{k} − (vM ,H )HM (Q ) − v H (Bk \ {i}) − q!(m − 1 − q)! − s!(bk − 1 − s)! + m! bk ! Q ⊆M \{k} S (B \{i} k
3 An ordering (on the player set) is consistent with an a priori union if all players in the same element of the a priori union appear successively.
T. Kongo / Mathematical Social Sciences 62 (2011) 114–119
× (vM ,H )HM (Q ) + v H (S ∪ {i}) − (vM ,H )HM (Q ) − v H (S ) − q!(m − 1 − q)! 1 = m! bk Q ⊆M \{k} HM × (vM ,H ) (Q ∪ {k}) − (vM ,H )HM (Q ) − q!(m − 1 − q)! 1 v H (Bk \ {i}) − m ! b k Q ⊆M \{k} − q!(m − 1 − q)! − s!(bk − 1 − s)! + m! bk ! Q ⊆M \{k} S (Bk \{i} H × v (S ∪ {i}) − v H (S ) . ∑ q!(m−1−q)! Since Q ⊆M \{k} = 1, m!
Myerson value4 with respect to weight vector w = (wi )i∈N such that wi = b1 for any i ∈ Bk ∈ B. k
Proof. From Theorem 1, the coincidence between the marginal contributions vectors and the modified marginal contributions vectors mentioned in the above, and Theorems 2 and 3 in Kamijo (2009), it follows that ϕ M (N , v, B, H ) = Ow(N , v H , B) = Shw (N , (v H )B ), where Shw is the weighted Shapley value with respect to weight vector Theorem 2, and v B is w defined as ∑ B defined as v (S ) = v( Bℓ ⊆S ,ℓ∈M Bℓ ) + Bℓ ∩S ̸=Bℓ ,ℓ∈M v(Bℓ ∩ S ) for any S ⊆ N. Since H is two-layered via B, we can show that (v H )B = v H . Therefore, we obtain, Shw (N , (v H )B ) = Shw (N , v H ) = µw (N , v, H ). (2)
− q!(m − 1 − q)! Eq. (2) = bk Q ⊆M \{k} m! × (vM ,H )HM (Q ∪ {k}) − (vM ,H )HM (Q ) 1 H 1 H v (Bk \ {i}) + v (Bk ) − v H (Bk ) − 1
bk
bk
− s!(bk − 1 − s)! + v H (S ∪ {i}) − v H (S ) b ! k S (B \{i} k
=
1 bk
+ +
Shk (M , (vM ,H )HM ) −
1 bk
1
v H (Bk )
v H (Bk ) − v H (Bk \ {i})
bk
− s!(bk − 1 − s)! v H (S ∪ {i}) − v H (S ) b ! k S (B \{i} k
=
1 bk
+
Shk (M , (vM ,H )HM ) −
1 bk
v H (Bk )
− s!(bk − 1 − s)! v H (S ∪ {i}) − v H (S ) b ! k S ⊆B \{i} k
=
1 bk
Shk (M , (vM ,H )HM ) −
= µi (Bk , v, H ) +
1 bk
117
v H (Bk ) + Shi (Bk , v H )
µk (M , vM ,H , HM ) − v H (Bk ) bk
= ϕ (N , v, B, H ). M
The above result might appear to suggest that the two-step Shapley value and the Owen value were the same. However, they are different. Kamijo (2009, Theorem 2) gave the Owen value-like representation of the two-step Shapley value in the class of all games with a priori unions. In this representation, the modified marginal contributions vectors are used instead of the marginal contributions vectors. In the class of games with two-layered hypergraphs, the connectedness obtained by the two-layered hypergraphs renders the two vectors the same. Since two-layered hypergraphs can be seen as generalizations of a priori unions, our result justifies using the modified marginal contributions vectors instead of the original vectors in the class of games with a priori unions. In addition, Theorem 1 can be seen as a generalization of Theorem 2 in Kamijo (2009). Further, the coincidence between the two vectors mentioned above implies the following generalization of Theorem 3 in Kamijo (2009). Theorem 2. For any (N , v, B, H ) ∈ Γ such that H is two-layered via B, ϕ M (N , v, B, H ) = µw (N , v, H ), where µw is the weighted
5. Axiomatic characterization Next, we give an axiomatic characterization of our value via two axioms. The first axiom is a fundamental one in the literature of games with restricted communications among players. Component efficiency: For any C ∈ N /H, i∈C ϕi (N , v, B, H ) = v(C ). The above property requires that, given communicationrestricted structures (now two-layered hypergraphs), the sum of the value among players in a maximal connected component is equal to the worth of the component. This property is related to efficiency. To define the second axiom, we have to consider the effect of a player’s isolation from the two-layered hypergraphs. Since we consider a priori unions and hypergraphs simultaneously, a player’s isolation affects both structures. First, consider the effect on a priori unions. Let i ∈ N, B = {B1 , B2 , . . . , Bm }, and i ∈ Bk ∈ B. Then, if i is isolated from the structure, the structure becomes B−i = (B \ Bk ) ∪ {Bk \ {i}} ∪ {{i}}. A player’s isolation affects only the element to which the player belongs. Next, consider the effect on hypergraphs. We assume that for forming a conference by a set of players, unanimous approval of the players is needed. Then, a player’s isolation induces the deletion of all conferences that contain the player and all conferences that contain the element of the a priori union that contains the player. Let H be a two-layered hypergraph via B and let i ∈ N. Then, player i’s isolation induces hypergraph H−i = H (N \ {i}). Note that H−i is two-layered via B−i . Therefore, given (N , v, B, H ) ∈ Γ , for any {i, j} ⊆ N, value ϕ M for player i ∈ Bk of a game where player j ∈ Bh is isolated is as follows:
∑
ϕiM (N , v, B−j , H−j ) µ (B \ {j}, v, H ) if Bk = Bh µi (Bk , v, H ) −j i k = H + µk (M , vM ,H , HM −h ) − v (Bk ) if Bk ̸= Bh , 5 bk
where HM −h = HM (M \ {h}). In the case where Bk = Bh , j’s isolation affects i’s value both in the game within Bk and in the game among elements of the a priori union. In the game within Bk , we isolate j from the restricted hypergraph on Bk . In the game
4 See e.g., Slikker and van den Nouweland (2000) for details. 5 More precisely, we have to consider in a different way the case where B = h {j}. In this case, Bh becomes the empty set after j’s deletion. Thus, we should write, µk (M \ {h}, vM ,H |M \{h} , HM −h ) instead of µk (M , vM ,H , HM −h ). However, by component decomposability of the Myerson value (see van den Nouweland, 1993), it holds that µk (M \ {h}, vM ,H |M \{h} , HM −h ) = µk (M , vM ,H , HM −h ) and hence, it is not significant. Therefore, in this case we also write in the above way.
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among elements of the a priori union, the element Bk \{j} is isolated, since j’s absence renders impossible the unanimous approval of forming a conference including Bk . By the component efficiency of the Myerson value, the element obtains its value as a singleton coalition, and thus the second term of the value is canceled. In the case where Bk ̸= Bh , j’s isolation affects i’s value only in the game among elements of the a priori union. Following the above argument, the next property is well defined. Normalized balanced contributions property: For any i, j ∈ N where i ∈ Bk ∈ B and j ∈ Bh ∈ B, bk ϕi (N , v, B, H ) − ϕi (N , v, B−j , H−j )
= bh ϕj (N , v, B, H ) − ϕj (N , v, B−i , H−i ) . The above two axioms characterize our value ϕ M . Theorem 3. ϕ M is the unique value satisfying both component efficiency and the normalized balanced contributions property. Proof. First, we show that ϕ M satisfies component efficiency. Take C ∈ N /H and consider two cases: (i) C is a subset of a single element of a priori union B, or (ii) C is a union of several such elements. In case (i), there exists Bk ∈ B such that Bk ⊇ C and there exists no conference Q ∈ HM such that Q ∋ k, i.e., {k} ∈ Q /HM . Hence, by component efficiency of the Myerson value in the game between elements of the a priori union, µk (M , vM ,H , HM ) = vM ,H ({k}). By any i ∈ C , the second definition, vM ,H ({k}) = v H (Bk ). Thus, for ∑ M M term of the value ϕ is canceled. Therefore, i∈C ϕi (N , v, B, H ) = ∑ i∈C µi (Bk , v, H ) = v(C ), where the equality follows by the component efficiency of the Myerson value of the game within the element of the a priori union. In case (ii), there exists D ∈ M /HM such that C = ℓ∈D Bℓ . ∑ ∑ ∑ M In this case, ϕ ( N , v, B , H ) = (µ ( B , v, H) + i ℓ i∈C i ℓ∈D i∈Bℓ µℓ (M ,vM ,H ,HM )−v H (Bℓ )
). For each ℓ ∈ D, component efficiency of ∑ H the Myerson value implies that i∈Bℓ µi (Bℓ , v, H ) = v (Bℓ ) and ∑ HM HM ℓ∈D µℓ (M , vM ,H , HM ) = vM ,H (D). By definition, vM ,H (D) = v( ℓ∈D Bℓ ) = v(C ). Hence, we obtain the desired result. Next, we show that ϕ M satisfies the normalized balanced contributions property. Take any {i, j} ⊆ N. If i, j ∈ Bk ∈ B, the second term of ϕiM (N , v, B, H ) is the same as the second term of ϕjM (N , v, B, H ). Hence, bℓ
bk ϕiM (N , v, B, H ) − bk ϕjM (N , v, B, H )
= bk µi (Bk , v, H ) − bk µj (Bk , v, H ).
(3)
By definition, in both H−i and H−j , there is no conference that contains Bk . Thus, the second terms of both ϕiM (N , v, B−j , H−j ) and ϕjM (N , v, B−i , H−i ) are equal to zero, and it holds that bk ϕiM (N , v, B−j , H−j ) − bk ϕjM (N , v, B−i , H−i )
= bk µi (Bk \ {j}, v, H−j ) − bk µj (Bk \ {i}, v, H−i ).
(4)
By component decomposability (see van den Nouweland, 1993) and the balanced contributions property (see Myerson, 1980) of the Myerson value in the game within the element of the a priori union, the right hand sides of Eqs. (3) and (4) are equal. Thus, we obtain the desired result. If i ∈ Bk , j ∈ Bh , and k ̸= h then bk ϕiM (N , v, B, H ) − bh ϕjM (N , v, B, H )
= bk µi (Bk , v, H ) + µk (M , vM ,H , HM ) − v (Bk ) H
− bh µj (Bh , v, H ) − µh (M , vM ,H , HM ) + v H (Bh ),
(5)
and, bk ϕiM (N , v, B−j , H−j ) − bh ϕjM (N , v, B−i , H−i )
= bk µi (Bk , v, H−j ) + µk (M , vM ,H , HM −h ) − v H−j (Bk ) − bh µj (Bh , v, H−i ) − µh (M , vM ,H , HM −k ) + v H−i (Bh ).
(6)
Given that i ∈ Bk , j ∈ Bh and k ̸= h and by the fact that there is no conference that contains players in different elements of the a priori union in H, it holds that Bk /H = Bk /H−j and Bh /H = Bh /H−i . Thus, µi (Bk , v, H ) = µi (Bk , v, H−j ), µj (Bh , v, H ) = µj (Bh , v, H−i ), v H (Bk ) = v H−j (Bk ), and v H (Bh ) = v H−i (Bh ). In conjunction with the balanced contributions property of the Myerson value in the game among elements of the a priori union, the right hand sides of Eqs. (5) and (6) are equal. Therefore, we obtain the desired result. Finally, we show the uniqueness of the value that satisfies the two axioms. Let ϕ be a value that satisfies the two axioms. We use induction with respect to the number of conferences in hypergraph H. If |H | = 0, N /H = {{i}}i∈N . By component efficiency, ϕi (N , v, B, H ) = v({i}) and the value is uniquely determined. Let a be a natural number. Assume that for any hypergraph that contains fewer than a conferences, ϕ is uniquely determined and consider the case where hypergraph H contains a conferences. Take any component C ∈ N /H. If |C | = 1, component efficiency implies that ϕi (N , v, B, H ) = v({i}). If |C | > 1, take i ∈ C . For any j ∈ C \ {i} with j ∈ Bh ∈ B, the normalized balanced contributions property and supposition above together imply bk ϕi (N , v, B, H ) − bh ϕj (N , v, B, H )
= bk ϕiM (N , v, B−j , H−j ) − bh ϕjM (N , v, B−i , H−i ). Also, by component efficiency,
−
ϕi (N , v, B, H ) = v(C ).
i∈C
These |C | equalities form a regular system of linear equations in |C | variables and that system has a unique solution. Vázquez-Brage et al. (1996, see their Theorem 3) characterized their value in a slightly different model from ours, using three axioms: component efficiency, balanced contributions for the graph, and fairness in the quotient. Our normalized balanced contributions property becomes equivalent to balanced contributions for the graph if two players i and j are in the same element of the a priori union. Thus, comparing our result to their Theorem 3, our normalized balanced contributions property (for players in different elements in the a priori union) is a counterpart of fairness in the quotient. Thus differences between these two properties constitute the key distinction between their approach and ours. Acknowledgments The author would like to thank Rene van den Brink, Yukihiko Funaki, Yoshio Kamijo, Gerard van der Laan, and Koichi Suga for helpful comments and discussions. The author is grateful to an associate editor and anonymous referees for useful comments and suggestions. The author would like to acknowledge the financial support provided by the Japan Society for the Promotion of Science (JSPS). References Alonso-Meijide, J., Alvarez-Mozos, M., Fiestras-Janeiro, M., 2009. Values of games with restricted communication and a priori unions. Mathematical Social Sciences 58, 202–213. Alonso-Meijide, J., Fiestras-Janeiro, M., 2002. Modification of the Banzhaf value for games with a coalition structure. Annals of Operations Research 109, 213–227.
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