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Axiomatization for the center-of-gravity of imputation set value
Linear Algebra and its Applications 439 (2013) 2205–2215
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Axiomatization for the center-of-gravity of imputation set value Genjiu Xu ∗ , Wenna Wang, Hua Dong Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China
a r t i c l e
i n f o
Article history: Received 3 August 2012 Accepted 23 June 2013 Available online 12 July 2013 Submitted by H. Schneider MSC: 91A12 15A04 15A18 Keywords: TU-game CIS-value Almost inessential game property Associated consistency Matrix approach
a b s t r a c t In this paper, we introduce the almost inessential game (property) to the solution part of cooperative game theory, which generalizes the inessential game (property). Following the framework of Hamiache to characterize the Shapley value, we then define a new associated game to characterize the center-of-gravity of imputation set value (CIS-value) by means of the almost inessential game property, associated consistency, continuity and efficiency. It provides an interpretation to the CIS-value as the essentially unique fixed point of an endogenous transformation by selfevaluation of TU-games. In addition, symmetry and translation covariance are used to axiomatize the CIS-value instead of the almost inessential property. © 2013 Elsevier Inc. All rights reserved.
1. Introduction A cooperative game with transferable utility (or TU-game) describes a situation where players can earn certain values by cooperating. An inessential game says that, by cooperating, the total amount of any coalition is the sum of the amounts that each player in the coalition feels he can obtain when playing alone, which means there are no efficient coalition forms. In practice, we often meet some situations that the efficient cooperation can only work with all participants, to form the grand coalition. On the other hand, the grand coalition forms sometimes directly from individuals, without the process of in-between coalitions forming. We introduce the
*
Corresponding author. E-mail addresses: [email protected] (G. Xu), [email protected] (W. Wang), [email protected] (H. Dong).
0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.06.026
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model of almost inessential game to describe the situation that the game is only determined by the worths of singleton and grand coalitions. Given a game, we are usually interested to know how the “fruits” of cooperation are shared among the players. A solution for TU-games is a function that assigns to every TU-game a payoff vector, where each component of the vector is the payoff assigned to the corresponding individual player. Usually the payoffs assigned to the individual players are based on their contributions to the different coalitions they are or can be member of. In the axiomatic formulation of cooperative games, consistency is an important characteristic of viable and stable solutions. Hamiache [4] presented a consistency axiom to characterize the Shapley value. His axiom of consistency is called associated consistency which is defined in terms of an associated game. Hamiache introduced firstly an associated game for the Shapley value. And the sequence of repeated associated games is also constructed. It is showed that the sequence of repeated associated games converged to an inessential game. Then the Shapley value is obtained from inessential game property, associated consistency and continuity. Motivated from that the center-of-gravity of imputation set value (CIS-value) [2] is a more straightforward value and allocates to each player, besides her stand-alone worth generated by her singleton coalition, an equal share of the surplus (in excess of the sum of all players’ stand-alone worths) generated by the grand coalition, we aim to characterize the CIS-value in this paper, in terms of the almost inessential game property in the framework of Hamiache’s axiomatization for the Shapley value. We introduce the model of almost inessential game, and then a type of associated game. The corresponding sequence of associated game is shown by matrix approach, to converge to an almost inessential game. Then the CIS-value is fully axiomatized by means of the almost inessential game property, associated consistency, continuity and efficiency. 2. Definitions and notations A cooperative game with transferable utility (TU) is a pair N , v , where N is the finite set of n players with n 2 and v : 2 N → R is the characteristic function assigning to each coalition S ∈ 2 N \ {∅} the worth v ( S ), with the convention that v (∅) = 0. We denote by G N the game space of all TU-games N , v on player set N. We define an almost inessential game to describe a game that the efficient (positive or negative) cooperation only works on the grand coalition N. Definition 1. A game N , v is called an almost inessential game, if v ( S ) = tion S N.
i∈ S
v ({i }), for any coali-
The almost inessential game generalizes the concept of inessential game for excluding the case of grand coalition v ( N ) = i ∈ N v ({i }). So, the game is finally decided with the outcome of singleton and grand coalitions. A value on G N is a function φ that assigns a payoff vector φ( v ) = (φi ( v ))i ∈ N ∈ R N to every game N , v . The component φi ( v ) of player i represents an assessment by i of his or her gains for participating in the game N , v . A value φ is efficient if i ∈ N φi ( v ) = v ( N ). A value φ is called to satisfy inessential game property, if φi ( v ) = v ({i }) for any inessential game N , v and each i ∈ N. With the observation of almost inessential game, each (n − 1)-person game (with players in N) is an inessential game, so each player i deserves his individual value. When we consider the n-person game by dividing v ( N ), all players should have equal rights on what is a surplus (when v ( N ) j ∈ N v ({ j })) or a deficit (when v ( N ) < j ∈ N v ({ j })) induced by cooperation. The property of value is defined as follows1 :
Definition 2. A value φ is said to satisfy almost inessential game property, if φi ( v ) = v ({i }) + α [ v ( N ) − j ∈ N v ({ j })] for any almost inessential game N , v and each i ∈ N, where α ∈ [0, 1]. 1
We are grateful to Rodica Branzei for an insightful discussion and comments regarding this issue.
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It is trivial that this property of φ generalizes the inessential game property for any α , too. Recalling the center-of-gravity of imputation set value (CIS-value), introduced by Driessen and Funaki [2], is a solution on G N , which associates with each game N , v and all players i ∈ N,
CISi ( v ) = v {i } +
1 n
v (N ) −
.
v { j}
j∈N
Using a reduced game consistency, Brink and Funaki [1] provided characterizations for a class of equal surplus sharing solutions including the CIS-value. Obviously, the CIS-value is a more straightforward value only related to the worths generated by singleton coalitions and by the grand coalition. From this, we aim to characterize the CIS-value in terms of the almost inessential game property, which follows up on Hamiache’s characterization of the Shapley value as the unique fixed point, namely an inessential game of an endogenous transformation of TU-games (see [4]). A value φ is said to be associated consistent if φ( v ) = φ( v ∗λ ) for any game N , v and its associated game N , v ∗λ (see [4]). According to associated consistency, a value behaves invariant under the adaptation of the game into some kind of associated game. Driessen [3] extended Hamiache’s axiomatization to the enlarged class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. For this enlarged class of values, explicit relationships to the Shapley value are exploited in order to present a uniform approach to obtain axiomatizations of such values with reference to a slightly adapted inessential game property, continuity, and a similar associated consistency. Naumova [9] treated a cooperative game as a bargaining problem with claim point, and studied the weighted entropy solution in terms of associated consistency with respect to associated games by overestimation of characteristic function values for partition members. Given any game N , v and λ ∈ R, Hamiache [4] define the associated game N , v λSh for characterizing the Shapley value as
v λSh ( S ) = v ( S ) + λ
v S ∪ { j} − v ( S ) − v { j} ,
for all S ⊆ N .
j∈N \ S
Here the definition of the associated game is modified as follows: Definition 3. Given any game N , v and λ ∈ R, define its associated game N , v C λ as follows:
vC λ (S) =
v(S) − λ v ( N ),
j∈ S [ v ( S ) −
v ( S \{ j }) − v ({ j })], if S N ; if S = N .
The associated game may be considered as an adaptation of a given game such that it reflects a pessimistic self-evaluation of worths of coalitions. From the point of view that any coalition is formed as its members joining one by one, a coalition S may evaluate its own net-worth during the last step of its formation from all possible members i ∈ S joining. The worth v C λ ( S ) of coalition S in the associated game differs from the initial worth v ( S ), by taking into account the possible (weighted) loss of benefits v ( S ) − v ( S \{ j }) − v ({ j }) derived from any member i ∈ S being the last player joining the coalition. C Obviously, for the associated game N , v C λ , we have v λ ( S ) = v ( S ) for all singleton coalitions S = {i }, i ∈ N and the grand coalition N. Therefore, Proposition 2.1. The CIS-value verifies the associated consistency with respect to N , v C λ . 3. Axiomatization of the CIS-value with the almost inessential game property For characterizing the CIS-value, we introduce the sequence of repeated associated games
∗C }∞ C { N , v m λ m=0 as, the term of order m, in this sequence, is the associated game N , v λ of the term of order (m − 1). The convergence of the sequence plays a pivotal role in Hamiache’s axiomatization
framework for a value in terms of associated consistency. Now we present the result of convergence ∗C }∞ . It will be proved in next section by matrix approach. of the sequence { N , v m λ m=0
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2 Lemma 3.1. For any game N , v ∈ G N and 0 < λ < n− , the sequence of repeated associated games 1 m ∗C ∞ { N , v λ }m=0 converges and, that the limit game is almost inessential.
A value φ satisfies continuity, if for every convergent sequence of games { N , v k }k∞=0 , the limit of which is the game N , v˜ , the sequence of values converge too, that is limk→∞ φ( v k ) = φ( v˜ ). Following Hamiache’s framework, for the convergent sequence of repeated associated games, the corresponding sequence of values verifies by associated consistency together with continuity, that
C ∗C = · · · = φ( v˜ ). φ( v ) = φ v Cλ = φ v Cλ λ = · · · = φ v m λ Therefore, the value φ( v ) is identified by the value φ( v˜ ) of the limit game N , v˜ . From the almost inessential game property, φ( v ) is identified with the worths of all singleton coalitions and the grand coalition of the limit game. 2 Theorem 3.2. For 0 < λ < n− , the CIS-value is the unique value verifying the almost inessential game prop1 erty, associated consistency, continuity and efficiency.
Proof. The almost inessential game property of the CIS-value is trivial, and by Proposition 2.1 the CIS-value verifies the associated consistency. For a convergent sequence of games { N , v k }k∞=0 , denote the limit of which by the game N , v˜ . By the matrix representation in Section 4, the corresponding sequence of CIS-value satisfies
lim CIS( v k ) = lim M C v k = M C lim v k = M C v˜ = CIS( v˜ ).
k→∞
k→∞
k→∞
Therefore the CIS-value verifies the continuity axiom. Consider a value φ satisfying these four axioms. For any game N , v , we show that φ( v ) = CIS( v ). ∗C }∞ From Lemma 3.1, the sequence of repeated associated games { N , v m λ m=0 converges for 0 < λ < 2 . n−1
Denote the limit game by N , v˜ , and N , v˜ is almost inessential.
By associated consistency and continuity of φ , it holds that φ( v ) = φ( v˜ ). And the almost inessen˜ ({ j })] for all i ∈ N and α ∈ [0, 1]. By tial game property of φ yields φi ( v˜ ) = v˜ ({i }) + α [ v˜ ( N ) − j∈N v
α = n1 . On the other hand, v˜ ({i }) = v ({i }), i ∈ N, and v˜ ( N ) = v ( N ). In summary, 1 φ( v ) = v {i } + v (N ) − v { j} .
the efficiency,
n
i∈N
j∈N
Similarly, since the CIS-value also verifies these four axioms, it follows that
CIS( v ) =
v {i } +
1 n
[v (N ) −
v { j} j∈N
From this, we conclude φ( v ) = CIS( v ).
. i∈N
2
Remark 1. Since every inessential game is almost inessential, a value verifying the inessential game property implies that it satisfies the almost inessential game property. With Hamiache’s associated game, the Shapley value is also characterized with the present axiom system. However the CIS-value cannot be axiomatized in terms of the inessential game property unless a new definition of associated game such that the corresponding sequence converges to an inessential game. 4. Proof of Lemma 3.1 by matrix approach The set G N of all n-person games with player set N is identified with the vector space R2 −1 , for always v (∅) = 0. Any linear operator on G N have a matrix interpretation, such as a linear value and any type of associated game. From this, matrix approach was applied to study associated consistency n
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of the Shapley value by Xu et al. [10] and Hamiache [5], respectively. Xu et al. [11] then derived another type of associated game for the Shapley value in terms of similarity of matrices. Recently, this approach has also been applied by Hamiache [6] to study the efficient Aumann–Drèze value and the efficient Myerson value for TU-games with coalition and communication structures. The type of coalitional matrices was introduced in [10] for the presentation of linear operators on game space. A matrix M is called row (respectively, column)-coalitional if its rows (respectively, columns) are indexed by all coalitions S ⊆ N. M is called square-coalitional if it is both row-coalitional and column-coalitional. Here we define a row-coalitionalmatrix M to be almost inessential, if the row-vector m S of M indexed by coalition S verifies m S = i ∈ S m{i } for all S N. Now we represent the CIS-value and the corresponding associated game in matrix version. For any game N , v , the CIS-value can be represented by the (column-coalitional) CIS standard matrix M C as CIS( v ) = M C · v, where the matrix M C = [ M C ]i ∈ N , S ⊆ N is given by
M
C
i, S
⎧1 , ⎪ n ⎪ ⎪ ⎨ 1 − n1 , = ⎪ −1, ⎪ ⎪ ⎩ n 0,
if S = N ; if S = {i }; if S = { j }, j ∈ N \{i }; otherwise.
Note that for two-person coalitions {i , j }, the associated game can be v C λ ({i , j }) = (1 − 2λ) v ({i , j })+ 2λ[ v ({i }) + v ({ j })]. The square-coalition matrix M λC = [ M λC ] S , T is given as listed below. The matrix M λC is a lower triangular matrix with diagonal entries 1 (n + 1 times) and 1 − sλ, 2 s n − 1. size
[ M λC ] S , T
T
s=1
1
T=S
s=2
2λ
T = { j }, j ∈ S
3sn−1
λ
T = { j }, j ∈ S
3sn−1
λ
T = S \{ j }, j ∈ S
2sn−1
1 − sλ
T=S
s=n
1
T=S
else
0
Then, the associated consistency of CIS-value with respect to N , v C λ presented in Proposition 2.1, turns out that the CIS standard matrix M C is invariant under multiplication with the associated transformation matrix M λC . That is M C = M C M λC . The sequence of repeated associated games ∗C }∞ C { N , v m λ m=0 can be represented in terms of M λ , as (m−1)∗C
∗C vm = M λC · v λ λ
m = M λC · v ,
for all m 0.
∗C }∞ So, convergence of the sequence { N , v m λ m=0 is identified with the convergence of the sequence
∞ . To prove Lemma 3.1, similar to the technique having been applied of power matrices {( M λC )m }m =0 in [10], we analyze the diagonal decomposition of M λC , by studying its eigenvalues and eigenvectors.2
Proposition 4.1. Eigenvalues of the associated transformation matrix M λC are of the form μk = 1 − kλ, k ∈ {2, 3, . . . , n − 1}, or k = 0 (eigenvalue 1). In the setting of eigenvalue 1, the corresponding eigenspace has dimension n + 1 (the only free variables are x N and x j , j ∈ N and every eigenvector is almost-inessential). nConcerning any eigenvalue of the form 1 − kλ, 2 k n − 1, the corresponding eigenspace has dimension k (the only free variables are x S , s = k). 2 The presentation of Proposition and the corresponding proof have been substantially improved following the referees’ very helpful comments.
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Proof. Due to the lower triangular shape of M λC , all its diagonal entries are eigenvalues of the form μk = 1 − kλ, k ∈ {2, 3, . . . , n − 1}, or k = 0 (eigenvalue 1). The corresponding eigenvector equation That is, M λC · x = μk · x reduces to [μk · I − M λC ] · x = 0. size
coalition
s=1
S = { j}
(simplified) eigenvector equation
[(1 − kλ) − 1] · x j = 0 −kλ · x j
s=2
S = {i , j }
k = 0 or x j = 0
[(1 − kλ) − (1 − 2λ)] · x S − 2λ · (xi + x j ) = 0 (2 − k)x S − 2(xi + x j ) = 0 [(1 − kλ) − (1 − sλ)] · x S − λ · j ∈ S x j − λ · j ∈ S x S \{ j } = 0 (s − k) · x S − j ∈ S x j − j ∈ S x S \{ j } = 0
3sn−1
s=n
S=N
[(1 − kλ) − 1] · xN = 0 −kλ · xN = 0
k = 0 or x N = 0
Concerning the solution part, we distinguish two cases. 1). Suppose k = 0. Then in particular, x N = 0 as well as x j = 0 for all j ∈ N, (2 − k)x S = 0 if s = 2, and (s − k)x S = j ∈ S x S \{ j } if 3 s n − 1. Thus, x S is free if s = k, x S = 0 if s < k, and
1 x S = s− j ∈ S x S \{ j } for all k < s n − 1. Hence, the dimension of the eigenspace corresponding to k k any eigenvalue 1 − kλ is n (where 2 k n − 1). 2). Suppose k = 0 (eigenvalue 1). Then in particular x j is free, x N is free too, x S = xi + x j if [x S \{ j } + x j ] if 3 s n − 1. By induction, it is easy to prove that S = {i , j }, as well as sx S = j ∈ S xS = j ∈ S x j for all 2 s n − 1, that is any eigenvector corresponding to eigenvalue 1 is almostinessential and the dimension of the eigenspace is n + 1. 2
By Proposition 4.1, for the associated transformation matrix M λC , the dimension of the eigenspace for each eigenvalue equals the multiplicity of the eigenvalue. From the theory of linear algebra, we conclude that matrix M λC is diagonalizable. −1 , where D C = diag (1, . . . , Lemma 4.2. Let M λC be the associated transformation matrix. Then M λC = PDC λP λ 1, 1 − 2λ, . . . , 1 − 2λ, . . . , 1 − (n − 1)λ, . . . , 1 − (n − 1)λ) and P consists of eigenvectors of M λC corresponding to eigenvalues 1, 1 − kλ (k = 2, 3, . . . , n − 1).
We make use of the following properties of row-coalitional matrices. It is slightly modified from Lemma 2.5 in [10] and can be proved similarly. Lemma 4.3. Let M be a row-coalitional matrix and A be a matrix. (1) If M is almost inessential, then the row-coalitional matrix M A is almost inessential. (2) If A is invertible, then M A is almost inessential if and only if M is almost inessential. (3) For every game N , v ∈ G N , if M is almost inessential, then the game N , M · v is almost inessential. Using the previous results, we now present the proof of the convergence of the sequence of repeated associated games. 2 Proof of Lemma 3.1. If 0 < λ < n− , then −1 < 1 − kλ < 1 for all k = 2, 3, . . . , n − 1. By Lemma 4.2, 1 we have
lim M λC m→∞
m
m m = lim P D Cλ P −1 = P lim D Cλ P −1 = PDP−1 , m→∞
m→∞
where D = diag(1, . . . , 1, 0, . . . , 0) and the element 1 repeats n + 1 times.
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Consider the row-coalitional matrix PD, we obtain
, . . . , 0 PD = x1 , x2 , . . . , xn+1 , 0
(0 denotes a zero column vector)
where column vectors xi (i = 1, 2, . . . , n + 1) are different eigenvectors of M λC corresponding to eigenvalue 1, there are almost inessential from Proposition 4.1. Therefore PD is almost inessential. By Lemma 4.3 (2), we know that PDP −1 is also almost inessential. ∗C }∞ , we have By the matrix representation of { N , v m λ m=0
∗C lim v m = lim M λC λ
m→∞
m
m→∞
· v = PDP−1 · v .
Due to Lemma 4.3 (3), the game N , PDP −1 · v is an almost inessential game. It is concluded that the sequence of repeated associated games converges to an almost inessential game. 2 5. The second axiomatization of the CIS-value The Equal Allocation of Non-Separable Costs (EANS) value, see Moulin [8], also known as the Egalitarian Non-Separable Contribution value in [2], assigns to each N , v ∈ G N the payoffs
EANSi ( v ) = SC i ( v ) +
1
n
v (N ) −
SC j ( v ) ,
for all i ∈ N ,
j∈N
where SC j ( v ) = v ( N ) − v ( N \{ j }). Notice that the EANS-value and the CIS-value are each other’s dual, that is EANS( v ) = CIS( v D ) for any game N , v and its dual game N , v D given by v D ( S ) = v ( N ) − v ( N \ S ) for all S ⊆ N. Hwang [7] modified the definition of the associated game of Hamiache to characterize the EANSvalue as
vE λ (S) = v(S) + λ
v S ∪ { j } − v ( S ) − SC j ( v ) ,
for all S ⊆ N .
j∈N \ S
Although the corresponding sequence of repeated associated games converges. The limit game N , v˜ is not an inessential game, but the sum of an inessential game and a constant game. He characterized the EANS-value, by means of associated consistency, continuity, efficiency (Pareto optimality), symmetry and translation covariance. A value φ on G N is said to satisfy symmetry, if φπ i (π v ) = φi ( v ) for any game N , v and any i ∈ N, where π is a permutation on N and N , π v is defined by π v (π S ) = v ( S ) for all S ⊆ N; translation ∈ R N , where N , v + α ) = for any game N , v and any α is defined covariance, if φ( v + α φ( v ) + α )( S ) = v ( S ) + i ∈ S αi for all S ⊆ N. With reference to our associated game N , v Cλ , the by ( v + α CIS-value can also be axiomatized with Hwang’s axiom system. 2 Theorem 5.1. For 0 < λ < n− , the CIS-value is the unique value verifying associated consistency, continuity, 1 efficiency, symmetry, and translation covariance.
Proof. We have shown that the CIS-value verifies associated consistency, continuity and efficiency. It is straightforward to verify that the CIS-value satisfies symmetry and translation covariance. Now we treat the uniqueness proof. Let φ be a value on G N satisfies the five axioms. For any game ∗C }∞ N , v and 0 < λ < n−2 1 , the sequence of repeated associated games { N , v m λ m=0 converges to an almost inessential game. Denote the limit game by N , v˜ . It can be rewritten as
v˜ ( S ) = w ( S ) +
xi ,
for all S ⊆ N ,
i∈ S
where xi = v˜ ({i }) for all i ∈ N and
w(S) =
0, v˜ ( N ) −
if S N ; j∈ S
v˜ ({ j }), if S = N .
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For all i ∈ N, because of translation covariance, we have φi ( v˜ ) = φi ( w ) + v˜ ({i }). By associated consistency and continuity, φi ( v ) = φi ( v˜ ). Using the properties of symmetry to the game N , w , we have φi ( w ) = α [ v˜ ( N ) − j ∈ N v˜ ({ j })], α ∈ [0, 1]. In addition, α = n1 from efficiency. Therefore, for all i ∈ N,
1 φi ( v ) = φi ( v˜ ) = φi ( w ) + v˜ {i } = v˜ {i } + v˜ ( N ) − v˜ { j } .
n
j∈N
Since v˜ ({i }) = v ({i }) for all i ∈ N and v˜ ( N ) = v ( N ). It follows that φ( v ) = CIS( v ).
2
Remark 2. For the class of almost inessential games, we can show that the almost inessential game property is equivalent to the two axioms: symmetry, and translation covariance. So Theorem 5.1 can be thought as a corollary derived from Theorem 3.2. Acknowledgements This research has been supported by the National Natural Science Foundation of China (Grant Nos. 70901063, 71171163, 71271171 and 71311120091), the NPU Foundation for Fundamental Research (Grant No. JC20110276), the graduate starting seed fund of NPU (Grant No. Z2013159). Appendix A. The game theoretical proof of Lemma 3.1 The associated game is equivalent to
vC λ (S) =
(1 − sλ) v ( S ) + λ
j∈ S
v ( S \{ j }) + λ
j∈ S
v ({ j }), if S N ; if S = N .
v ( N ),
∗ ∞ For each game N , v in G N , we define the sequence of associated game { N , v m λ }m=0 , where (m+1)∗
0∗
v λ = v, v 1λ∗ = v ∗λ , and v λ (m+1)∗
vλ
∗ ∗ ∗ = (vm λ )λ . Since v λ ({ j }) = v ({ j }) for all j ∈ N, we have that
∗ ( S ) = vm λ (S) − λ
∗ m∗ vm λ ( S ) − v λ S \{ j } − v { j } ,
for all ∅ = S N .
j∈ S
∗ Hence the term v m λ ( S ) can be expressed as a linear combination of v ( T ) and v ({ j }) for all T ⊆ S and j ∈ S, that is
m∗
vλ (S) =
T ⊆S
γmS ( T ) v ( T ) +
j∈ S
βmS v ({ j }), if S N ; if S = N ;
v ( N ),
(A.1)
S where γmS ( T ) ∈ R, βm ∈ R and γmS (∅) = 0. The following two lemmas concern the determination of the (m + 1)-repeated associated game (m+1)∗ N , v λ . We simplify the notation of the coefficients as well (to be justified later) in the sense that they do not depend on coalitions nor players, but merely on the coalition sizes.
∗ Lemma A.1. Concerning the representation (1) of the m-repeated associated game N , v m λ , the coefficients
γms (t ) and βm satisfy the following recursive relationships: βm+1 = (1 − λ)βm + λ,
where β0 = 0;
s s where 0s (s) = 1; m+1 (s) = (1 − sλ) m (s), s s s −1 m+1 (t ) = (1 − sλ) m (t ) + (s − t )λ m (t ),
γ
γ
γ
γ
γ
γ
for all t s < n.
Proof. Let S N. On the one hand, by applying formula (1) to m + 1, we have (m+1)∗
vλ
(S) =
T ⊆S
γms +1 (t ) v ( T ) +
j∈ S
βm+1 v { j } .
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On the other hand, the inductive approach (through the induction hypothesis) and a few combinatorial calculations yield the following chain of equalities: (m+1)∗
vλ
∗ ∗ ( S ) = vm λ λ(S ) ∗ = (1 − sλ) v m λ (S) + λ
∗ vm λ S \{ j } + λ
j∈ S
∗ vm λ { j}
j∈ S
s = (1 − sλ) γm (t ) v ( T ) + βm v { j } T ⊆S
+λ
j∈ S
j∈ S
γms−1 (t ) v ( T ) +
T ⊆ S \{ j }
= (1 − sλ)γ
j∈ S
+λ
βm v {i } + λ v { j}
i ∈ S \{ j }
s m (s) v ( S ) + (1 −
+ (1 − sλ)
sλ)
j∈ S
s m (t ) v ( T )
γ
T S
βm v { j } + λ γms−1 (t ) v ( T ) j ∈ S T ⊆ S \{ j }
βm v {i } + λ v { j}
j ∈ S i ∈ S \{ j }
= (1 − sλ)γms (s) v ( S ) +
j∈ S
(1 − sλ)γms (t ) + (s − t )λγms−1 (t ) v ( T )
T S
+ (1 − sλ)βm + (s − 1)λβm + λ v { j } . j∈ S
This completes the proof
2
Lemma A.2. 1. The solution of the recursive formula βm+1 = (1 − λ)βm + λ is given by βm = 1 − (1 − λ)m for all m 1, where β0 = 0. 2. The solution of the recursive formula γms +1 (t ) = (1 − sλ)γms (t ) + (s − t )λγms−1 (t ) for all t s < n is given by γ0s (s) = 1 and
γms (t ) =
s−t m s−t (−1)s−t −d 1 − λ(s − d) . d =0
d
(A.2)
Proof. 1. For m = 1, the above expression of βm reduces to β1 = λ as it should be. Let m 1 be arbitrary, suppose βm = 1 − (1 − λ)m , then
βm+1 = (1 − λ)βm + λ = (1 − λ) 1 − (1 − λ)m + λ = 1 − (1 − λ)m+1 . 2. For m = 1, the above expression (2) reduces to γ1s (s) = 1 − sλ, as well as γ1s (s − 1) = λ as it should be. Let m 1 be arbitrary. In case t = s, the above expression (2) reduces to γms (t ) = (1 − sλ)m which solves the first equality γms +1 = (1 − sλ)γms (s). Suppose t < s, the above expression (2) applied to m + 1 can be rewritten as follows:
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G. Xu et al. / Linear Algebra and its Applications 439 (2013) 2205–2215
γms +1 (t ) =
s−t m+1 s−t (−1)s−t −d 1 − λ(s − d) d
d =0
s−t m s−t −d s − t = (−1) 1 − λ(s − d) (1 − sλ + dλ) d
d =0
= (1 − sλ)γms (t ) + λ
s−t m s−t (−1)s−t −d d 1 − λ(s − d) d
d =0
s m (t ) + λ
= (1 − sλ)γ
s−t m s−t s−t −d (−1) (s − t ) 1 − λ(s − d) d−1 d =1
= (1 − sλ)γms (t ) + (s − t )λ
s− 1−t
(−1)s−1−t −d
d =0 s m (t ) + (s
= (1 − sλ)γ This completes the proof.
m s−1−t 1 − λ(s − 1 − d) d
− t )λγms−1 (t ).
2
∗ Corollary A.3. Concerning the representation (1) of the m-repeated associated game N , v m λ , the coefficients γms (t ) and βm satisfy the following convergence results: ∞ = {1 − (1 − λ)m }∞ converges to 1. 1. For all 0 < λ < 2, the sequence {βm }m =0 m=0
2 ∞ converges to 0. 2. Let 0 < λ < n− , it holds for all t s < n, that the sequence {γms (t )}m =0 1 2 ∗ ∞ 3. Let 0 < λ < n− , then the sequence of m-repeated games { N , v m λ }m=0 converges to the limit game 1 N , v˜ given by
v˜ ( S ) =
v { j} ,
for all ∅ = S N ;
j∈ S
v˜ ( N ) = v ( N ). Proof. Let λ > 0 and fix the coalition size s n − 1. ∞ 1. When |1 − λ| < 1, i.e. 0 < λ < 2, the sequence {βm }m =0 converges to 1. 2. In view of the formula (2) concerning γms (t ), t s, it suffices to study the behavior of the fundamental expression 1 − λ(s − d) for all 0 d s − t. We have 1 − λ(s − d) 1 − λ(n − 1) > −1, provided 2 0 < λ < n− . 1 3. Let S ⊆ N , S = ∅. For sufficiently large m, it follows from the representation (1) of the ∗ m-repeated associated game N , v m λ and the convergency results of parts 1–2. 2 From Corollary A.3, we can get that the limit of the repeated associated game sequence is an almost inessential game for any game N , v in G N . References [1] R. van den Brink, Y. Funaki, Axiomatizations of a class of equal surplus sharing solutions for TU-games, Theory and Decision 67 (2009) 303–340. [2] T.S.H. Driessen, Y. Funaki, Coincidence of and collinearity between game theoretic solutions, OR Spektrum 13 (1991) 15–30. [3] T.S.H. Driessen, Associated consistency and values for TU-games, Internat. J. Game Theory 30 (2010) 467–482. [4] G. Hamiache, Associated consistency and Shapley value, Internat. J. Game Theory 30 (2001) 279–289. [5] G. Hamiache, A matrix approach to the associated consistency with an application to the Shapley value, Int. Game Theory Rev. 12 (2010) 175–187. [6] G. Hamiache, A matrix approach to TU-games with coalition and communication structures, Soc. Choice Welf. 38 (2012) 85–100. [7] Y. Hwang, Associated consistency and equal allocation of nonseparable costs, Econom. Theory 28 (2006) 709–719.
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[8] H. Moulin, The separability axiom and equal sharing method, J. Econom. Theory 36 (1985) 120–148. [9] N.I. Naumova, Associated consistency based on utility functions of coalitions, UBS 26 (2009) 79–99. [10] G. Xu, T.S.H. Driessen, H. Sun, Matrix analysis for associated consistency in cooperative game theory, Linear Algebra Appl. 428 (2008) 1571–1586. [11] G. Xu, T.S.H. Driessen, H. Sun, Matrix approach to dual similar associated consistency for the Shapley value, Linear Algebra Appl. 430 (2009) 2896–2897.
Contents lists available at SciVerse ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Axiomatization for the center-of-gravity of imputation set value Genjiu Xu ∗ , Wenna Wang, Hua Dong Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China
a r t i c l e
i n f o
Article history: Received 3 August 2012 Accepted 23 June 2013 Available online 12 July 2013 Submitted by H. Schneider MSC: 91A12 15A04 15A18 Keywords: TU-game CIS-value Almost inessential game property Associated consistency Matrix approach
a b s t r a c t In this paper, we introduce the almost inessential game (property) to the solution part of cooperative game theory, which generalizes the inessential game (property). Following the framework of Hamiache to characterize the Shapley value, we then define a new associated game to characterize the center-of-gravity of imputation set value (CIS-value) by means of the almost inessential game property, associated consistency, continuity and efficiency. It provides an interpretation to the CIS-value as the essentially unique fixed point of an endogenous transformation by selfevaluation of TU-games. In addition, symmetry and translation covariance are used to axiomatize the CIS-value instead of the almost inessential property. © 2013 Elsevier Inc. All rights reserved.
1. Introduction A cooperative game with transferable utility (or TU-game) describes a situation where players can earn certain values by cooperating. An inessential game says that, by cooperating, the total amount of any coalition is the sum of the amounts that each player in the coalition feels he can obtain when playing alone, which means there are no efficient coalition forms. In practice, we often meet some situations that the efficient cooperation can only work with all participants, to form the grand coalition. On the other hand, the grand coalition forms sometimes directly from individuals, without the process of in-between coalitions forming. We introduce the
*
Corresponding author. E-mail addresses: [email protected] (G. Xu), [email protected] (W. Wang), [email protected] (H. Dong).
0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.06.026
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model of almost inessential game to describe the situation that the game is only determined by the worths of singleton and grand coalitions. Given a game, we are usually interested to know how the “fruits” of cooperation are shared among the players. A solution for TU-games is a function that assigns to every TU-game a payoff vector, where each component of the vector is the payoff assigned to the corresponding individual player. Usually the payoffs assigned to the individual players are based on their contributions to the different coalitions they are or can be member of. In the axiomatic formulation of cooperative games, consistency is an important characteristic of viable and stable solutions. Hamiache [4] presented a consistency axiom to characterize the Shapley value. His axiom of consistency is called associated consistency which is defined in terms of an associated game. Hamiache introduced firstly an associated game for the Shapley value. And the sequence of repeated associated games is also constructed. It is showed that the sequence of repeated associated games converged to an inessential game. Then the Shapley value is obtained from inessential game property, associated consistency and continuity. Motivated from that the center-of-gravity of imputation set value (CIS-value) [2] is a more straightforward value and allocates to each player, besides her stand-alone worth generated by her singleton coalition, an equal share of the surplus (in excess of the sum of all players’ stand-alone worths) generated by the grand coalition, we aim to characterize the CIS-value in this paper, in terms of the almost inessential game property in the framework of Hamiache’s axiomatization for the Shapley value. We introduce the model of almost inessential game, and then a type of associated game. The corresponding sequence of associated game is shown by matrix approach, to converge to an almost inessential game. Then the CIS-value is fully axiomatized by means of the almost inessential game property, associated consistency, continuity and efficiency. 2. Definitions and notations A cooperative game with transferable utility (TU) is a pair N , v , where N is the finite set of n players with n 2 and v : 2 N → R is the characteristic function assigning to each coalition S ∈ 2 N \ {∅} the worth v ( S ), with the convention that v (∅) = 0. We denote by G N the game space of all TU-games N , v on player set N. We define an almost inessential game to describe a game that the efficient (positive or negative) cooperation only works on the grand coalition N. Definition 1. A game N , v is called an almost inessential game, if v ( S ) = tion S N.
i∈ S
v ({i }), for any coali-
The almost inessential game generalizes the concept of inessential game for excluding the case of grand coalition v ( N ) = i ∈ N v ({i }). So, the game is finally decided with the outcome of singleton and grand coalitions. A value on G N is a function φ that assigns a payoff vector φ( v ) = (φi ( v ))i ∈ N ∈ R N to every game N , v . The component φi ( v ) of player i represents an assessment by i of his or her gains for participating in the game N , v . A value φ is efficient if i ∈ N φi ( v ) = v ( N ). A value φ is called to satisfy inessential game property, if φi ( v ) = v ({i }) for any inessential game N , v and each i ∈ N. With the observation of almost inessential game, each (n − 1)-person game (with players in N) is an inessential game, so each player i deserves his individual value. When we consider the n-person game by dividing v ( N ), all players should have equal rights on what is a surplus (when v ( N ) j ∈ N v ({ j })) or a deficit (when v ( N ) < j ∈ N v ({ j })) induced by cooperation. The property of value is defined as follows1 :
Definition 2. A value φ is said to satisfy almost inessential game property, if φi ( v ) = v ({i }) + α [ v ( N ) − j ∈ N v ({ j })] for any almost inessential game N , v and each i ∈ N, where α ∈ [0, 1]. 1
We are grateful to Rodica Branzei for an insightful discussion and comments regarding this issue.
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It is trivial that this property of φ generalizes the inessential game property for any α , too. Recalling the center-of-gravity of imputation set value (CIS-value), introduced by Driessen and Funaki [2], is a solution on G N , which associates with each game N , v and all players i ∈ N,
CISi ( v ) = v {i } +
1 n
v (N ) −
.
v { j}
j∈N
Using a reduced game consistency, Brink and Funaki [1] provided characterizations for a class of equal surplus sharing solutions including the CIS-value. Obviously, the CIS-value is a more straightforward value only related to the worths generated by singleton coalitions and by the grand coalition. From this, we aim to characterize the CIS-value in terms of the almost inessential game property, which follows up on Hamiache’s characterization of the Shapley value as the unique fixed point, namely an inessential game of an endogenous transformation of TU-games (see [4]). A value φ is said to be associated consistent if φ( v ) = φ( v ∗λ ) for any game N , v and its associated game N , v ∗λ (see [4]). According to associated consistency, a value behaves invariant under the adaptation of the game into some kind of associated game. Driessen [3] extended Hamiache’s axiomatization to the enlarged class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. For this enlarged class of values, explicit relationships to the Shapley value are exploited in order to present a uniform approach to obtain axiomatizations of such values with reference to a slightly adapted inessential game property, continuity, and a similar associated consistency. Naumova [9] treated a cooperative game as a bargaining problem with claim point, and studied the weighted entropy solution in terms of associated consistency with respect to associated games by overestimation of characteristic function values for partition members. Given any game N , v and λ ∈ R, Hamiache [4] define the associated game N , v λSh for characterizing the Shapley value as
v λSh ( S ) = v ( S ) + λ
v S ∪ { j} − v ( S ) − v { j} ,
for all S ⊆ N .
j∈N \ S
Here the definition of the associated game is modified as follows: Definition 3. Given any game N , v and λ ∈ R, define its associated game N , v C λ as follows:
vC λ (S) =
v(S) − λ v ( N ),
j∈ S [ v ( S ) −
v ( S \{ j }) − v ({ j })], if S N ; if S = N .
The associated game may be considered as an adaptation of a given game such that it reflects a pessimistic self-evaluation of worths of coalitions. From the point of view that any coalition is formed as its members joining one by one, a coalition S may evaluate its own net-worth during the last step of its formation from all possible members i ∈ S joining. The worth v C λ ( S ) of coalition S in the associated game differs from the initial worth v ( S ), by taking into account the possible (weighted) loss of benefits v ( S ) − v ( S \{ j }) − v ({ j }) derived from any member i ∈ S being the last player joining the coalition. C Obviously, for the associated game N , v C λ , we have v λ ( S ) = v ( S ) for all singleton coalitions S = {i }, i ∈ N and the grand coalition N. Therefore, Proposition 2.1. The CIS-value verifies the associated consistency with respect to N , v C λ . 3. Axiomatization of the CIS-value with the almost inessential game property For characterizing the CIS-value, we introduce the sequence of repeated associated games
∗C }∞ C { N , v m λ m=0 as, the term of order m, in this sequence, is the associated game N , v λ of the term of order (m − 1). The convergence of the sequence plays a pivotal role in Hamiache’s axiomatization
framework for a value in terms of associated consistency. Now we present the result of convergence ∗C }∞ . It will be proved in next section by matrix approach. of the sequence { N , v m λ m=0
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2 Lemma 3.1. For any game N , v ∈ G N and 0 < λ < n− , the sequence of repeated associated games 1 m ∗C ∞ { N , v λ }m=0 converges and, that the limit game is almost inessential.
A value φ satisfies continuity, if for every convergent sequence of games { N , v k }k∞=0 , the limit of which is the game N , v˜ , the sequence of values converge too, that is limk→∞ φ( v k ) = φ( v˜ ). Following Hamiache’s framework, for the convergent sequence of repeated associated games, the corresponding sequence of values verifies by associated consistency together with continuity, that
C ∗C = · · · = φ( v˜ ). φ( v ) = φ v Cλ = φ v Cλ λ = · · · = φ v m λ Therefore, the value φ( v ) is identified by the value φ( v˜ ) of the limit game N , v˜ . From the almost inessential game property, φ( v ) is identified with the worths of all singleton coalitions and the grand coalition of the limit game. 2 Theorem 3.2. For 0 < λ < n− , the CIS-value is the unique value verifying the almost inessential game prop1 erty, associated consistency, continuity and efficiency.
Proof. The almost inessential game property of the CIS-value is trivial, and by Proposition 2.1 the CIS-value verifies the associated consistency. For a convergent sequence of games { N , v k }k∞=0 , denote the limit of which by the game N , v˜ . By the matrix representation in Section 4, the corresponding sequence of CIS-value satisfies
lim CIS( v k ) = lim M C v k = M C lim v k = M C v˜ = CIS( v˜ ).
k→∞
k→∞
k→∞
Therefore the CIS-value verifies the continuity axiom. Consider a value φ satisfying these four axioms. For any game N , v , we show that φ( v ) = CIS( v ). ∗C }∞ From Lemma 3.1, the sequence of repeated associated games { N , v m λ m=0 converges for 0 < λ < 2 . n−1
Denote the limit game by N , v˜ , and N , v˜ is almost inessential.
By associated consistency and continuity of φ , it holds that φ( v ) = φ( v˜ ). And the almost inessen˜ ({ j })] for all i ∈ N and α ∈ [0, 1]. By tial game property of φ yields φi ( v˜ ) = v˜ ({i }) + α [ v˜ ( N ) − j∈N v
α = n1 . On the other hand, v˜ ({i }) = v ({i }), i ∈ N, and v˜ ( N ) = v ( N ). In summary, 1 φ( v ) = v {i } + v (N ) − v { j} .
the efficiency,
n
i∈N
j∈N
Similarly, since the CIS-value also verifies these four axioms, it follows that
CIS( v ) =
v {i } +
1 n
[v (N ) −
v { j} j∈N
From this, we conclude φ( v ) = CIS( v ).
. i∈N
2
Remark 1. Since every inessential game is almost inessential, a value verifying the inessential game property implies that it satisfies the almost inessential game property. With Hamiache’s associated game, the Shapley value is also characterized with the present axiom system. However the CIS-value cannot be axiomatized in terms of the inessential game property unless a new definition of associated game such that the corresponding sequence converges to an inessential game. 4. Proof of Lemma 3.1 by matrix approach The set G N of all n-person games with player set N is identified with the vector space R2 −1 , for always v (∅) = 0. Any linear operator on G N have a matrix interpretation, such as a linear value and any type of associated game. From this, matrix approach was applied to study associated consistency n
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of the Shapley value by Xu et al. [10] and Hamiache [5], respectively. Xu et al. [11] then derived another type of associated game for the Shapley value in terms of similarity of matrices. Recently, this approach has also been applied by Hamiache [6] to study the efficient Aumann–Drèze value and the efficient Myerson value for TU-games with coalition and communication structures. The type of coalitional matrices was introduced in [10] for the presentation of linear operators on game space. A matrix M is called row (respectively, column)-coalitional if its rows (respectively, columns) are indexed by all coalitions S ⊆ N. M is called square-coalitional if it is both row-coalitional and column-coalitional. Here we define a row-coalitionalmatrix M to be almost inessential, if the row-vector m S of M indexed by coalition S verifies m S = i ∈ S m{i } for all S N. Now we represent the CIS-value and the corresponding associated game in matrix version. For any game N , v , the CIS-value can be represented by the (column-coalitional) CIS standard matrix M C as CIS( v ) = M C · v, where the matrix M C = [ M C ]i ∈ N , S ⊆ N is given by
M
C
i, S
⎧1 , ⎪ n ⎪ ⎪ ⎨ 1 − n1 , = ⎪ −1, ⎪ ⎪ ⎩ n 0,
if S = N ; if S = {i }; if S = { j }, j ∈ N \{i }; otherwise.
Note that for two-person coalitions {i , j }, the associated game can be v C λ ({i , j }) = (1 − 2λ) v ({i , j })+ 2λ[ v ({i }) + v ({ j })]. The square-coalition matrix M λC = [ M λC ] S , T is given as listed below. The matrix M λC is a lower triangular matrix with diagonal entries 1 (n + 1 times) and 1 − sλ, 2 s n − 1. size
[ M λC ] S , T
T
s=1
1
T=S
s=2
2λ
T = { j }, j ∈ S
3sn−1
λ
T = { j }, j ∈ S
3sn−1
λ
T = S \{ j }, j ∈ S
2sn−1
1 − sλ
T=S
s=n
1
T=S
else
0
Then, the associated consistency of CIS-value with respect to N , v C λ presented in Proposition 2.1, turns out that the CIS standard matrix M C is invariant under multiplication with the associated transformation matrix M λC . That is M C = M C M λC . The sequence of repeated associated games ∗C }∞ C { N , v m λ m=0 can be represented in terms of M λ , as (m−1)∗C
∗C vm = M λC · v λ λ
m = M λC · v ,
for all m 0.
∗C }∞ So, convergence of the sequence { N , v m λ m=0 is identified with the convergence of the sequence
∞ . To prove Lemma 3.1, similar to the technique having been applied of power matrices {( M λC )m }m =0 in [10], we analyze the diagonal decomposition of M λC , by studying its eigenvalues and eigenvectors.2
Proposition 4.1. Eigenvalues of the associated transformation matrix M λC are of the form μk = 1 − kλ, k ∈ {2, 3, . . . , n − 1}, or k = 0 (eigenvalue 1). In the setting of eigenvalue 1, the corresponding eigenspace has dimension n + 1 (the only free variables are x N and x j , j ∈ N and every eigenvector is almost-inessential). nConcerning any eigenvalue of the form 1 − kλ, 2 k n − 1, the corresponding eigenspace has dimension k (the only free variables are x S , s = k). 2 The presentation of Proposition and the corresponding proof have been substantially improved following the referees’ very helpful comments.
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Proof. Due to the lower triangular shape of M λC , all its diagonal entries are eigenvalues of the form μk = 1 − kλ, k ∈ {2, 3, . . . , n − 1}, or k = 0 (eigenvalue 1). The corresponding eigenvector equation That is, M λC · x = μk · x reduces to [μk · I − M λC ] · x = 0. size
coalition
s=1
S = { j}
(simplified) eigenvector equation
[(1 − kλ) − 1] · x j = 0 −kλ · x j
s=2
S = {i , j }
k = 0 or x j = 0
[(1 − kλ) − (1 − 2λ)] · x S − 2λ · (xi + x j ) = 0 (2 − k)x S − 2(xi + x j ) = 0 [(1 − kλ) − (1 − sλ)] · x S − λ · j ∈ S x j − λ · j ∈ S x S \{ j } = 0 (s − k) · x S − j ∈ S x j − j ∈ S x S \{ j } = 0
3sn−1
s=n
S=N
[(1 − kλ) − 1] · xN = 0 −kλ · xN = 0
k = 0 or x N = 0
Concerning the solution part, we distinguish two cases. 1). Suppose k = 0. Then in particular, x N = 0 as well as x j = 0 for all j ∈ N, (2 − k)x S = 0 if s = 2, and (s − k)x S = j ∈ S x S \{ j } if 3 s n − 1. Thus, x S is free if s = k, x S = 0 if s < k, and
1 x S = s− j ∈ S x S \{ j } for all k < s n − 1. Hence, the dimension of the eigenspace corresponding to k k any eigenvalue 1 − kλ is n (where 2 k n − 1). 2). Suppose k = 0 (eigenvalue 1). Then in particular x j is free, x N is free too, x S = xi + x j if [x S \{ j } + x j ] if 3 s n − 1. By induction, it is easy to prove that S = {i , j }, as well as sx S = j ∈ S xS = j ∈ S x j for all 2 s n − 1, that is any eigenvector corresponding to eigenvalue 1 is almostinessential and the dimension of the eigenspace is n + 1. 2
By Proposition 4.1, for the associated transformation matrix M λC , the dimension of the eigenspace for each eigenvalue equals the multiplicity of the eigenvalue. From the theory of linear algebra, we conclude that matrix M λC is diagonalizable. −1 , where D C = diag (1, . . . , Lemma 4.2. Let M λC be the associated transformation matrix. Then M λC = PDC λP λ 1, 1 − 2λ, . . . , 1 − 2λ, . . . , 1 − (n − 1)λ, . . . , 1 − (n − 1)λ) and P consists of eigenvectors of M λC corresponding to eigenvalues 1, 1 − kλ (k = 2, 3, . . . , n − 1).
We make use of the following properties of row-coalitional matrices. It is slightly modified from Lemma 2.5 in [10] and can be proved similarly. Lemma 4.3. Let M be a row-coalitional matrix and A be a matrix. (1) If M is almost inessential, then the row-coalitional matrix M A is almost inessential. (2) If A is invertible, then M A is almost inessential if and only if M is almost inessential. (3) For every game N , v ∈ G N , if M is almost inessential, then the game N , M · v is almost inessential. Using the previous results, we now present the proof of the convergence of the sequence of repeated associated games. 2 Proof of Lemma 3.1. If 0 < λ < n− , then −1 < 1 − kλ < 1 for all k = 2, 3, . . . , n − 1. By Lemma 4.2, 1 we have
lim M λC m→∞
m
m m = lim P D Cλ P −1 = P lim D Cλ P −1 = PDP−1 , m→∞
m→∞
where D = diag(1, . . . , 1, 0, . . . , 0) and the element 1 repeats n + 1 times.
G. Xu et al. / Linear Algebra and its Applications 439 (2013) 2205–2215
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Consider the row-coalitional matrix PD, we obtain
, . . . , 0 PD = x1 , x2 , . . . , xn+1 , 0
(0 denotes a zero column vector)
where column vectors xi (i = 1, 2, . . . , n + 1) are different eigenvectors of M λC corresponding to eigenvalue 1, there are almost inessential from Proposition 4.1. Therefore PD is almost inessential. By Lemma 4.3 (2), we know that PDP −1 is also almost inessential. ∗C }∞ , we have By the matrix representation of { N , v m λ m=0
∗C lim v m = lim M λC λ
m→∞
m
m→∞
· v = PDP−1 · v .
Due to Lemma 4.3 (3), the game N , PDP −1 · v is an almost inessential game. It is concluded that the sequence of repeated associated games converges to an almost inessential game. 2 5. The second axiomatization of the CIS-value The Equal Allocation of Non-Separable Costs (EANS) value, see Moulin [8], also known as the Egalitarian Non-Separable Contribution value in [2], assigns to each N , v ∈ G N the payoffs
EANSi ( v ) = SC i ( v ) +
1
n
v (N ) −
SC j ( v ) ,
for all i ∈ N ,
j∈N
where SC j ( v ) = v ( N ) − v ( N \{ j }). Notice that the EANS-value and the CIS-value are each other’s dual, that is EANS( v ) = CIS( v D ) for any game N , v and its dual game N , v D given by v D ( S ) = v ( N ) − v ( N \ S ) for all S ⊆ N. Hwang [7] modified the definition of the associated game of Hamiache to characterize the EANSvalue as
vE λ (S) = v(S) + λ
v S ∪ { j } − v ( S ) − SC j ( v ) ,
for all S ⊆ N .
j∈N \ S
Although the corresponding sequence of repeated associated games converges. The limit game N , v˜ is not an inessential game, but the sum of an inessential game and a constant game. He characterized the EANS-value, by means of associated consistency, continuity, efficiency (Pareto optimality), symmetry and translation covariance. A value φ on G N is said to satisfy symmetry, if φπ i (π v ) = φi ( v ) for any game N , v and any i ∈ N, where π is a permutation on N and N , π v is defined by π v (π S ) = v ( S ) for all S ⊆ N; translation ∈ R N , where N , v + α ) = for any game N , v and any α is defined covariance, if φ( v + α φ( v ) + α )( S ) = v ( S ) + i ∈ S αi for all S ⊆ N. With reference to our associated game N , v Cλ , the by ( v + α CIS-value can also be axiomatized with Hwang’s axiom system. 2 Theorem 5.1. For 0 < λ < n− , the CIS-value is the unique value verifying associated consistency, continuity, 1 efficiency, symmetry, and translation covariance.
Proof. We have shown that the CIS-value verifies associated consistency, continuity and efficiency. It is straightforward to verify that the CIS-value satisfies symmetry and translation covariance. Now we treat the uniqueness proof. Let φ be a value on G N satisfies the five axioms. For any game ∗C }∞ N , v and 0 < λ < n−2 1 , the sequence of repeated associated games { N , v m λ m=0 converges to an almost inessential game. Denote the limit game by N , v˜ . It can be rewritten as
v˜ ( S ) = w ( S ) +
xi ,
for all S ⊆ N ,
i∈ S
where xi = v˜ ({i }) for all i ∈ N and
w(S) =
0, v˜ ( N ) −
if S N ; j∈ S
v˜ ({ j }), if S = N .
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For all i ∈ N, because of translation covariance, we have φi ( v˜ ) = φi ( w ) + v˜ ({i }). By associated consistency and continuity, φi ( v ) = φi ( v˜ ). Using the properties of symmetry to the game N , w , we have φi ( w ) = α [ v˜ ( N ) − j ∈ N v˜ ({ j })], α ∈ [0, 1]. In addition, α = n1 from efficiency. Therefore, for all i ∈ N,
1 φi ( v ) = φi ( v˜ ) = φi ( w ) + v˜ {i } = v˜ {i } + v˜ ( N ) − v˜ { j } .
n
j∈N
Since v˜ ({i }) = v ({i }) for all i ∈ N and v˜ ( N ) = v ( N ). It follows that φ( v ) = CIS( v ).
2
Remark 2. For the class of almost inessential games, we can show that the almost inessential game property is equivalent to the two axioms: symmetry, and translation covariance. So Theorem 5.1 can be thought as a corollary derived from Theorem 3.2. Acknowledgements This research has been supported by the National Natural Science Foundation of China (Grant Nos. 70901063, 71171163, 71271171 and 71311120091), the NPU Foundation for Fundamental Research (Grant No. JC20110276), the graduate starting seed fund of NPU (Grant No. Z2013159). Appendix A. The game theoretical proof of Lemma 3.1 The associated game is equivalent to
vC λ (S) =
(1 − sλ) v ( S ) + λ
j∈ S
v ( S \{ j }) + λ
j∈ S
v ({ j }), if S N ; if S = N .
v ( N ),
∗ ∞ For each game N , v in G N , we define the sequence of associated game { N , v m λ }m=0 , where (m+1)∗
0∗
v λ = v, v 1λ∗ = v ∗λ , and v λ (m+1)∗
vλ
∗ ∗ ∗ = (vm λ )λ . Since v λ ({ j }) = v ({ j }) for all j ∈ N, we have that
∗ ( S ) = vm λ (S) − λ
∗ m∗ vm λ ( S ) − v λ S \{ j } − v { j } ,
for all ∅ = S N .
j∈ S
∗ Hence the term v m λ ( S ) can be expressed as a linear combination of v ( T ) and v ({ j }) for all T ⊆ S and j ∈ S, that is
m∗
vλ (S) =
T ⊆S
γmS ( T ) v ( T ) +
j∈ S
βmS v ({ j }), if S N ; if S = N ;
v ( N ),
(A.1)
S where γmS ( T ) ∈ R, βm ∈ R and γmS (∅) = 0. The following two lemmas concern the determination of the (m + 1)-repeated associated game (m+1)∗ N , v λ . We simplify the notation of the coefficients as well (to be justified later) in the sense that they do not depend on coalitions nor players, but merely on the coalition sizes.
∗ Lemma A.1. Concerning the representation (1) of the m-repeated associated game N , v m λ , the coefficients
γms (t ) and βm satisfy the following recursive relationships: βm+1 = (1 − λ)βm + λ,
where β0 = 0;
s s where 0s (s) = 1; m+1 (s) = (1 − sλ) m (s), s s s −1 m+1 (t ) = (1 − sλ) m (t ) + (s − t )λ m (t ),
γ
γ
γ
γ
γ
γ
for all t s < n.
Proof. Let S N. On the one hand, by applying formula (1) to m + 1, we have (m+1)∗
vλ
(S) =
T ⊆S
γms +1 (t ) v ( T ) +
j∈ S
βm+1 v { j } .
G. Xu et al. / Linear Algebra and its Applications 439 (2013) 2205–2215
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On the other hand, the inductive approach (through the induction hypothesis) and a few combinatorial calculations yield the following chain of equalities: (m+1)∗
vλ
∗ ∗ ( S ) = vm λ λ(S ) ∗ = (1 − sλ) v m λ (S) + λ
∗ vm λ S \{ j } + λ
j∈ S
∗ vm λ { j}
j∈ S
s = (1 − sλ) γm (t ) v ( T ) + βm v { j } T ⊆S
+λ
j∈ S
j∈ S
γms−1 (t ) v ( T ) +
T ⊆ S \{ j }
= (1 − sλ)γ
j∈ S
+λ
βm v {i } + λ v { j}
i ∈ S \{ j }
s m (s) v ( S ) + (1 −
+ (1 − sλ)
sλ)
j∈ S
s m (t ) v ( T )
γ
T S
βm v { j } + λ γms−1 (t ) v ( T ) j ∈ S T ⊆ S \{ j }
βm v {i } + λ v { j}
j ∈ S i ∈ S \{ j }
= (1 − sλ)γms (s) v ( S ) +
j∈ S
(1 − sλ)γms (t ) + (s − t )λγms−1 (t ) v ( T )
T S
+ (1 − sλ)βm + (s − 1)λβm + λ v { j } . j∈ S
This completes the proof
2
Lemma A.2. 1. The solution of the recursive formula βm+1 = (1 − λ)βm + λ is given by βm = 1 − (1 − λ)m for all m 1, where β0 = 0. 2. The solution of the recursive formula γms +1 (t ) = (1 − sλ)γms (t ) + (s − t )λγms−1 (t ) for all t s < n is given by γ0s (s) = 1 and
γms (t ) =
s−t m s−t (−1)s−t −d 1 − λ(s − d) . d =0
d
(A.2)
Proof. 1. For m = 1, the above expression of βm reduces to β1 = λ as it should be. Let m 1 be arbitrary, suppose βm = 1 − (1 − λ)m , then
βm+1 = (1 − λ)βm + λ = (1 − λ) 1 − (1 − λ)m + λ = 1 − (1 − λ)m+1 . 2. For m = 1, the above expression (2) reduces to γ1s (s) = 1 − sλ, as well as γ1s (s − 1) = λ as it should be. Let m 1 be arbitrary. In case t = s, the above expression (2) reduces to γms (t ) = (1 − sλ)m which solves the first equality γms +1 = (1 − sλ)γms (s). Suppose t < s, the above expression (2) applied to m + 1 can be rewritten as follows:
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G. Xu et al. / Linear Algebra and its Applications 439 (2013) 2205–2215
γms +1 (t ) =
s−t m+1 s−t (−1)s−t −d 1 − λ(s − d) d
d =0
s−t m s−t −d s − t = (−1) 1 − λ(s − d) (1 − sλ + dλ) d
d =0
= (1 − sλ)γms (t ) + λ
s−t m s−t (−1)s−t −d d 1 − λ(s − d) d
d =0
s m (t ) + λ
= (1 − sλ)γ
s−t m s−t s−t −d (−1) (s − t ) 1 − λ(s − d) d−1 d =1
= (1 − sλ)γms (t ) + (s − t )λ
s− 1−t
(−1)s−1−t −d
d =0 s m (t ) + (s
= (1 − sλ)γ This completes the proof.
m s−1−t 1 − λ(s − 1 − d) d
− t )λγms−1 (t ).
2
∗ Corollary A.3. Concerning the representation (1) of the m-repeated associated game N , v m λ , the coefficients γms (t ) and βm satisfy the following convergence results: ∞ = {1 − (1 − λ)m }∞ converges to 1. 1. For all 0 < λ < 2, the sequence {βm }m =0 m=0
2 ∞ converges to 0. 2. Let 0 < λ < n− , it holds for all t s < n, that the sequence {γms (t )}m =0 1 2 ∗ ∞ 3. Let 0 < λ < n− , then the sequence of m-repeated games { N , v m λ }m=0 converges to the limit game 1 N , v˜ given by
v˜ ( S ) =
v { j} ,
for all ∅ = S N ;
j∈ S
v˜ ( N ) = v ( N ). Proof. Let λ > 0 and fix the coalition size s n − 1. ∞ 1. When |1 − λ| < 1, i.e. 0 < λ < 2, the sequence {βm }m =0 converges to 1. 2. In view of the formula (2) concerning γms (t ), t s, it suffices to study the behavior of the fundamental expression 1 − λ(s − d) for all 0 d s − t. We have 1 − λ(s − d) 1 − λ(n − 1) > −1, provided 2 0 < λ < n− . 1 3. Let S ⊆ N , S = ∅. For sufficiently large m, it follows from the representation (1) of the ∗ m-repeated associated game N , v m λ and the convergency results of parts 1–2. 2 From Corollary A.3, we can get that the limit of the repeated associated game sequence is an almost inessential game for any game N , v in G N . References [1] R. van den Brink, Y. Funaki, Axiomatizations of a class of equal surplus sharing solutions for TU-games, Theory and Decision 67 (2009) 303–340. [2] T.S.H. Driessen, Y. Funaki, Coincidence of and collinearity between game theoretic solutions, OR Spektrum 13 (1991) 15–30. [3] T.S.H. Driessen, Associated consistency and values for TU-games, Internat. J. Game Theory 30 (2010) 467–482. [4] G. Hamiache, Associated consistency and Shapley value, Internat. J. Game Theory 30 (2001) 279–289. [5] G. Hamiache, A matrix approach to the associated consistency with an application to the Shapley value, Int. Game Theory Rev. 12 (2010) 175–187. [6] G. Hamiache, A matrix approach to TU-games with coalition and communication structures, Soc. Choice Welf. 38 (2012) 85–100. [7] Y. Hwang, Associated consistency and equal allocation of nonseparable costs, Econom. Theory 28 (2006) 709–719.
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