Later, Maschler and Owen. (1992) extend the consistent value to NTU games. Moreover, Hart and Mas"Colell. (1996) develop a bargaining mechanism whose ...
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The consistent coalitional value for hyperplane games Gustavo Bergantiños and Juan Vidal-Puga 10-04
______________________________________________________________________ Facultade de Ciencias Económicas e Empresariais, Campus As Lagoas-Marcosende, 36310 Vigo.
The consistent coalitional value for hyperplane games Gustavo Bergantiños Research Group in Economic Analysis. Universidade de Vigo. Spain. Juan Vidal-Puga Facultade de Ciencias Sociais. Universidade de Vigo. Spain.
Abstract We introduce a value for hyperplane games with coalition structure. This value coincides with the consistent value for trivial coalition structures, and with the Owen value for transferable utility games with coalition structure. Furthermore, we present two characterizations: the …rst one using a consistency property and the second one using properties of balanced contributions.
1. Introduction Some of the most important issues of cooperative game theory are to de…ne “good” values, to study which interesting properties are satis…ed by these values, and to obtain axiomatic characterizations using some of these properties. In cooperative games with transferable utility (T U games), Shapley (1953) introduces the Shapley value. Later, several authors obtain new characterizations of the Shapley value using other properties. For instance, Hart and Mas-Colell We thank David Perez-Castrillo for helpful comments. Financial support from the Ministerio de Ciencia y Tecnologia and FEDER through grant BEC2002-04102-C02-01 and Xunta de Galicia through grant PGIDIT03PXIC30002PN is gratefully aknowledged.
(1989) characterize it with consistency and Myerson (1980) with balanced contributions. There are several extensions of T U games. The most natural one is the extension to games without transferable utility (N T U games). Another extension applies to T U games with coalition structure, which study situations where players are partitioned into several groups. This model was considered by Owen (1977). Of course, a third extension is to N T U games with coalition structure. The Shapley value has many interesting properties in T U games. Then, several authors propose values which are generalizations of the Shapley value in these extended models. In N T U games, the Harsanyi value (Harsanyi (1963)) and the Shapley N T U value (Shapley (1969)) are generalizations of the Shapley value. Maschler and Owen (1989) de…ne the consistent value for hyperplane games, which are a subclass of N T U games. The main idea behind this generalization is to maintain (as far as possible) the consistency property of the Shapley value. Later, Maschler and Owen (1992) extend the consistent value to N T U games. Moreover, Hart and Mas-Colell (1996) develop a bargaining mechanism whose equilibrium payo¤ coincides with the consistent value. Owen (1977) introduces a generalization of the Shapley value, called the Owen value, for T U games with coalition structure. He characterizes his value using axioms similar to those used by Shapley (1953). Later, Winter (1992) characterizes the Owen value with a consistency property and Calvo, Lasaga, and Winter (1996) with balanced contributions. In the volumes 2 and 3 of the handbook of game theory with economic applications, chapters 37 (“Coalition structures” by J: Greenberg), 53 (“The Shapley value”by E. Winter), 54 (“Variations of the Shapley value”by D. Monderer and D: Samet), and 55 (“Values of non-transferable utility games”by R: P: McLean), it is possible to …nd surveys of this literature. It is of our interest to know whether the consistent value and the Owen value can be generalized the same way to hyperplane games. Therefore, we introduce a new value, called the consistent coalitional value. This value is de…ned through a two-stage procedure. In the …rst stage, we divide the utility between the coalitions. In the second stage, agents of every coalition divide the utility obtained in the …rst stage. This value can be characterized with a consistency property. Our characterization generalizes the results about consistency obtained by Maschler and Owen (1989) for the consistent value and Winter (1992) for the Owen value. 2
Moreover, we characterize it with properties of balanced contributions. This characterization generalizes the results about balanced contributions obtained by Hart and Mas-Colell (1996) for the consistent value and Calvo, Lasaga, and Winter (1996) for the Owen value. We believe that these characterizations together with the de…nition make the consistent coalitional value a proper generalization of the consistent and the Owen value. Bergantiños and Vidal-Puga (2004) extend the consistent coalitional value to N T U games with coalition structure following the same procedure of Maschler and Owen (1992). They also extend the characterization based on balanced contributions. Furthermore, Vidal-Puga (2004) proposes a non-cooperative game for which the consistent coalitional value arises as equilibrium payo¤. His results are similar to those presented by Hart and Mas-Colell (1996) for the consistent value. Chae and Heidhues (2004) introduce a value for bargaining problems with coalitional structure. Bergantiños, Casas-Mendez, Fiestras-Janeiro, and VidalPuga (2004) prove that the consistent coalitional value coincides with the value proposed by Chae and Heidhues in the subclass of bargaining problems. The paper is organized as follows. In Section 2 we introduce the notation and some previous results. In Section 3 we de…ne the consistent coalitional value. In Section 4 we study which properties are satis…ed by this value. In Section 5 we present the axiomatic characterizations. Section 6 is devoted to some concluding remarks. Finally, in the Appendix we present the proofs of the results obtained in the paper.
2. De…nitions and Previous Results Given a set N , jN j denotes the cardinal of N: Given x; y 2 P RN ; we say y x when yi xi for each i 2 N and x y is the scalar product xi yi : We denote i2N
N N 0; 8i and RN RN + = x 2 R : xi ++ = x 2 R : xi > 0; 8i . A game without transferable utility, or simply an NTU game, is a pair (N; V ) where N = f1; 2; :::; ng is the set of players and V is a correspondence (characteristic function) which assigns to each coalition S N a subset V (S) RS . This subset represents all the possible payo¤s that members of S can obtain for themselves when playing cooperatively. For S N , if there is no ambiguity, we maintain the notation V when referring to the application V restricted to S as
3
player set. We also denote S = N nS. We say that (N; V ) is a game with transferable utility (or TU game) if there exists a function v : 2N ! R, called the characteristic function, satisfying v (;) = P 0 and for each S N; V (S) = x 2 RS : xi v(S) : Usually we represent a i2S
T U game as the pair (N; v) : We denote the set of all T U games as T U .
In this paper we focus on a subclass of N T U games. We say that (N; V ) is a hyperplane game if for each S N there exists S 2 RS++ satisfying V (S) = x 2 RS :
S
x
v(S)
(2:1)
for some v : 2N ! R with v (;) = 0. We denote by H the set of all hyperplane games. Notice that each T U game is a hyperplane game (just take Si = 1 for each S N and i 2 S). A coalition structure C = fC1 ; C2 ; :::; Cm g over N is a partition of N . We denote a hyperplane game (N; V ) with coalition structure C over N as (N; V; C). We denote by CH the set of all hyperplane games with coalition structure (CT U for TU games). Given S N; we denote the coalition structure C restricted to the players in S as CS , i:e: CS = fCq \ S : Cq 2 C and Cq \ S 6= ;g. For simplicity, we use C i instead of CN nfig . Given G a subset of CH (or H), a value in G is a mapping which assigns to each (N; V; C) 2 G a point (N; V; C) 2 V (N ). Usually we write N ; N (V ) or N (V; C) instead of (N; V; C) : We denote the Shapley value (Shapley (1953)) of the T U game (N; v) as N (or N (v)): We now present the consistent value for hyperplane games following Maschler and Owen (1989). Let be the set of all permutations over N . Given 2 ; we de…ne the set of predecessors of i under as P ( ; i) = fj 2 N : (j) < (i)g: We de…ne the marginal contribution of player i 2 N to the game V in the permutation as n o di ( ) = max xi : (dj ( ))j2P ( ;i) ; xi 2 V (P ( ; i) [ fig) : 4
So, di ( ) is the maximum that player i can obtain in V (P ( ; i) [ fig) after his predecessors obtain their respective dj ( )’s. We denote d( ) = (di ( ))i2N . Given a hyperplane game (N; V ), the consistent value N is the vector of expected marginal contributions, where each 2 is equally likely, i.e. N
=
1 P d( ). j j 2
Maschler and Owen (1989) prove that, given i 2 N; N i
N i
=
1 jN j
P
N i
N nfjg i
+ v (N )
j2N nfig
P
j2N nfig
N j
N nfig j
!
:
(2:2)
For T U games with coalition structure, N (or N (v; C)) denotes the Owen value (Owen (1977)). In trivial coalition structures (C = fN g or C = ffiggi2N ) the Owen value coincides with the Shapley value. Then, the Owen value is a generalization of the Shapley value. Owen (1977) de…nes his value in a two-stage procedure. Let (N; v; C) 2 CT U . Stage 1. The total amount is divided among the coalitions. Owen ! de…nes S the T U game M; v [C] where M = f1; :::; mg and v [C] (Q) = v Cq for each q2Q
Q M: This game (called the game between coalitions) is obtained from (N; v; C) assuming that the players are the coalitions acting as single players. For any coalition!Cq 2 C; Owen says that the total amount obtained by players P N in Cq i:e: must be equal to the Shapley value of coalition Cq in the j j2Cq [C]
game M; v ; i:e: M v [C] : q Stage 2. The players inside each coalition Cq 2 C divide among themselves the amount obtained in Stage 1. For each Cq 2 C; Owen de…nes the T U game (Cq ; v ) ; where v (S) =
M q
v
h i CS[Cq
for each S
Cq : Notice that v (S) represents the
amount that players of S would obtain in Stage 1 if players in Cq n S were not in the game. The Owen value of player i 2 Cq is de…ned as N i
=
Cq i
5
(v ) :
It is well-known that the Owen value (like the Shapley value and the consistent value) can be obtained as an average of marginal contributions depending on equally likely permutations. We say that a permutation 2 is admissible with respect to C if players in the same coalition come together in the order, i: e: given i; j 2 Cq 2 C and k 2 N such that (i) < (k) < (j); then k 2 Cq : We denote the set of all permutations over N admissible with respect to C as C . Then, N i
=
1 j
Cj
X 2
d( ):
C
3. The Consistent Coalitional Value In this section we introduce the consistent coalitional value for hyperplane games. We de…ne it following the two-stage procedure used by Owen (1977) in the de…nition of the Owen value. We end this section by proving that the consistent coalitional value cannot be obtained as an average of marginal contributions depending on equally likely permutations. The value obtained in this way is called the random order value. We de…ne the consistent coalitional value fig then we take i = v(fig) fig .
in a recursive way. If jN j = 1;
i
Assume that we have de…ned N for each hyperplane game (N; V; C) where jN j t: When jN j = t + 1 we de…ne N by extending the two-stage procedure of Owen (1977) to hyperplane games. Let (N; V; C) be a hyperplane game and let S S N and v be associated with V as in (2:1) : Stage 1. ThePtotal amount is divided among the coalitions. Each coalition N N Cq 2 C receives j j = j2Cq
2
1 4 jCj
X
Cr 2CnfCq g
0 @
X
j2Cq
N j
1
N nCr A j
+ v (N )
X
Cr 2CnfCq g
Notice that (3:1) gives a recursive way to compute
X
j2Cr
P
i2Cq
6
N j
N j
N j :
N nCq j
!3
5:
(3:1)
Stage 2. The players inside each coalition Cq 2 C divide among themselves the amount obtained in Stage 1. For each Cq 2 C; we consider the hyperplane game (Cq ; V ) such that, for each S Cq , V (S) = x 2 RS : where
S i
=
S[Cq i
S
for each i 2 S and v (S) =
x P
v (S) ; S[Cq j
S[Cq j
is the amount ob-
j2S
tained by coalition S in Stage 1 applied to the hyperplane game S [ Cq ; V; CS[Cq : The consistent coalitional value N i
N i
=
of player i 2 Cq is de…ned as Cq i
(V ) :
We now explain this de…nition carefully. In Stage 1 we must decide the P total N amount received by each coalition Cq 2 C: In T U games this amount is i . j2Cq P N N In hyperplane games it should be j j : j2Cq
In T U games the total amount received by coalition Cq is the Shapley value of the game M; v [C] played by the coalitions. Nevertheless, in hyperplane games the game between coalitions cannot, in general, be de…ned in a meaningful way. Since we are trying to generalize Owen’s procedure, we will use another property of the Shapley value, which can be generalized to our setting. This property is (2:2), which provides a recursive way for the computation of the Shapley value. Notice that even though (2:2) is written in terms of the consistent value, in T U games the consistent value coincides with the Shapley value. In Stage 2 the total amount received by coalition Cq in the …rst stage must be divided among players in Cq : In this stage, we proceed as in the second stage of Owen’s (1977) using the consistent value instead of the Shapley value. The next proposition shows that is well-de…ned. P N N Proposition 1. is well-de…ned and j j = v (N ) : j2N
Proof. See the Appendix. S i
Remark 1. Let (N; V; C) be a hyperplane game such that for each i 2 N; is constant across coalitions S containing player i, i:e: there exists = ( i )i2N 7
such that Si = i for each S N , i 2 S. Let Cq and Cr be two coalitions of C: Then, v (Cq ) and v (Cr ) represent the maximum utility that players in Cq and Cr can obtain when they compare their respective levels of utility using : If both coalitions join they compare their respective levels of utility with the same vector. Then, it makes sense to compare v (Cq [ Cr ) with v (Cq ) and v (Cr ) : Under this assumption we can de…ne in a meaningful way the game played ! by S the coalitions. We de…ne the T U game M; v [C] where v [C] (S) = v Cq : q2S
For each coalition Cq 2 C; the amount obtained by coalition with P CqNcoincides M the Shapley value of the game played by the coalitions, i:e: v [C] : j j = q j2Cq
The proof of this statement is located in the Appendix. This remark also applies to T U games because in T U games Si = 1 for all S N and i 2 S: Thus, our procedure coincides with Owen’s procedure in T U games.
Condition (3:1) is a generalization of condition (2:2) for the game played by the coalitions. We can also generalize condition (2:2) for the game played by the agents of each coalition Cq 2 C. For all i 2 Cq 2 C; N i 1 jCq j
P
j2Cq nfig
N i
N nfjg i
+
P
N j
N i
N j
j2Cq
= P
j2Cq nfig
N j
N nfig j
!
:
(3:2)
The next proposition will be very useful in the proofs of our results. Proposition 2. The consistent coalitional value is the only value in CH satisfying conditions (3:1) and (3:2) : Proof. See the Appendix. When C = fN g, (3:2) coincides with (2:2) and when C = ffiggi2N , (3:1) coincides with (2:2) : Since is the only value in hyperplane games satisfying (2:2) we conclude that = in trivial coalition structures. Moreover, by Remark 1, for T U games with coalition structure the consistent coalitional value coincides with the Owen value. Thus, the consistent coalitional value is a generalization of the consistent value and the Owen value for hyperplane games with coalition structure.
8
We know that the Shapley value, the consistent value, and the Owen value are obtained as an average of marginal contributions depending on equally likely permutations. Thus, it seems reasonable to generalize these values in the same way. Given a hyperplane game (N; V; C), the random order coalitional value zN is de…ned as the vector of expected marginal contributions when all the admissible permutations with respect to C are equally likely, i.e. 1 X zN = C d( ): j j C 2
In T U games, McLean (1991) de…nes the random order coalitional structure values. z is the natural generalization to hyperplane games of McLean’s values when all the admissible permutations are equally likely and the rest of permutations have probability 0: It is remarkable that Maschler and Owen (1992) even suggest the name random order value instead of consistent value. We now compute and z in the following example. Example 1. Let (N; V; C) be the hyperplane game such that N = f1; 2; 3g and V (fig) = fxi 2 Rfig : xi 0g; 8i 2 N; V (f1; 2g) = f(x1 ; x2 ) 2 Rf1;2g : 4x1 + x2 1g; V (f1; 3g) = f(x1 ; x3 ) 2 Rf1;3g : x1 + x3 1g; 1 V (f2; 3g) = f(x2 ; x3 ) 2 Rf2;3g : x2 + x3 g; 4 and V (N ) = fx 2 RN : If C = ff1; 2g; f3gg, we obtain that N
=
19 7 6 ; ; 32 32 32
P
xi
1g:
i2N
and zN =
16 10 6 ; ; 32 32 32
:
This example shows that and z are di¤erent. Even though and z are natural generalizations of the Owen value, we believe that is a more suitable value: We will prove that satis…es more interesting properties and can be characterized generalizing axiomatic characterizations of the Owen value and the consistent value. 9
4. Properties In this section we present several desirable properties and study which of them are satis…ed by the consistent coalitional value. Some of these properties are well-known in the literature of N T U games. Others are introduced in this paper generalizing properties of T U games. P N N A value satis…es e¢ ciency (EF) if for each (N; V; C) 2 CH; j j = j2N
v (N ) :
Given (N; V; C) 2 CH we say that two players i; j 2 N are symmetric if two properties hold: For each S N nfi; jg; if x 2 V (S [ fig), yj = xi ; and yk = xk for each k 2 S then, y 2 V (S [ fjg). For each S fi; jg ; if x 2 V (S); yi = xj , yj = xi ; and yk = xk for each k 2 Sn fi; jg then, y 2 V (S). A value satis…es individual symmetry (IS) if for each pair of symmetric N players i; j 2 Cq 2 C; N i = j : We now present the property of covariance in hyperplane games following Maschler and Owen (1989). Let (N; V; C) and (N; Ve ; C) be two hyperplane games such that for each S N; n o S V (S) = x 2 RS : S x v(S) and Ve (S) = x 2 RS : e x ve (S) :
We say that (N; V; C) and (N; Ve ; C) are equivalent under a linear transformation of player i’s utility if there exist two constants a 2 R++ and b 2 R such that S S S b S for all S N; ei = i , ej = Sj if j 6= i; ve(S) = v (S) + i if i 2 S; and a a e ve(S) = v (S) if i 2 = S: Notice that if (N; V; C) and N; V ; C are equivalent under a linear transformation of player i’s utility, then x e 2 V~ (S) if and only if there exists x 2 V (S) satisfying x ei = axi + b and x ej = xj if j 2 S n fig. We say that a value satis…es covariance ( COV) if, given two hyperplane games (N; V; C) and (N; Ve ; C) equivalent under a linear transformation of some player i’s utility, N j
Ve ; C =
a
N i (V; C) N j (V; C)
10
+b
if j = i if j = 6 i:
Hart and Mas-Colell (1989) characterize the Shapley value as the only value in T U games satisfying consistency and other properties. A value in T U satis…es consistency (CON S) if and only if for each (N; v) 2 T U; S N; and i 2 S; N i
P
where vS (T ) = v T [ S
S i
(v) =
T [S j
(vS )
(v) for each T
S:
j2S
Winter (1992) extends the de…nition of consistency to T U games with coalition structure. A value in CT U satis…es consistency (CON S) if and only if for each (N; v; C) 2 CT U; Cq 2 C; S Cq ; and i 2 S; N i
(v; C) =
S i
(vS ; fSg) :
Maschler and Owen (1989) show that, if we de…ne the property of consistency of Hart and Mas-Colell (1989) in hyperplane games as in the T U case, there is no value satisfying consistency and other “basic”properties (for instance, e¢ ciency). Thus, they provide a weaker de…nition of consistency for hyperplane games. A value in H satis…es l-consistency if for each hyperplane game (N; V; C), l jN j; and i 2 N; X
S i
jN j 1 l 1
(VS ) =
S N;i2S;jSj=l
n where VS (T ) = x 2 RT : x;
T [S i
It is straightforward to prove 8 < X VS (T ) = x 2 RT : :
i2S that VS
2V T [S
N i
(V )
o
for all T
S:
is a hyperplane game and
T [S xi i
i2T
v T [S
X
T [S i
i2S
T [S i
9 = ;
:
We now present a generalization of l-consistency to hyperplane games with coalition structure. A value in CH satis…es l-consistency if for each hyperplane game (N; V; C); Cq 2 C, l jCq j ; and i 2 Cq ; X
S Cq ;i2S;jSj=l
S i
(VS ; fSg) = 11
jCq j 1 l 1
N i
(V; C) :
We call 2-consistency bilateral consistency ( BCONS). Notice that our bilateral consistency generalizes in the natural way the consistency of Hart and Mas-Colell (1989), the consistency of Winter (1992), and the bilateral consistency of Maschler and Owen (1989). Myerson (1980) characterizes the Shapley value using e¢ ciency and balanced contributions. He says that a value in T U satis…es balanced contributions (BC) if for each (N; v) 2 T U and i; j 2 N; N i
N nfjg i
N j
=
N nfig : j
Calvo et al: (1996) generalize the property of balanced contributions for T U games with coalition structure obtaining two properties: BCAC and BCAP 1 . A value in CT U satis…es balanced contributions among coalitions (BCAC) if for each (N; v; C) 2 CT U and Cq ; Cr 2 C with q 6= r, X X N nC X X N nC q r N N = : j j j j j2Cq
j2Cq
j2Cr
j2Cr
A value in CT U satis…es balanced contributions among players in the same coalition (BCAP) if for each i; j 2 Cq 2 C with i 6= j, N i
N nfjg i
N j
=
N nfig : j
Hart and Mas-Colell (1996) introduce, in Proposition 4, a generalization of balanced contributions for N T U games. We present it in the case of hyperplane games. A value in H satis…es average balanced contributions (ABC) if for each (N; V ) 2 H and i 2 N; X X N nfjg N nfig N N N N = : i i j j i j j2N nfig
j2N nfig
We now introduce the extension of these properties to CH. A value in CH satis…es average balanced contributions among coalitions ( ABCAC) if for each (N; V; C) 2 CH and Cq 2 C; 2 3 " # X X X X N nCq N nCr 5 N N N N 4 = : j j j j j j Cr 2CnfCq g
j2Cq
Cr 2CnfCq g
1
j2Cr
Calvo et al: (1996) present these two balanced properties as only one. In this paper, we think that is more natural the formulation as two properties.
12
A value in CH satis…es average balanced contributions among players in the same coalition ( ABCAP) if for each (N; V; C) 2 CH and i 2 Cq ; X
j2Cq nfig
N i
N i
N nfjg i
=
X
N j
N j
N nfig j
:
j2Cq nfig
Notice that ABCAC generalizes, in the natural way, BC, BCAC; and ABC whereas ABCAP generalizes BC, BCAP; and ABC: The following result appears in Maschler and Owen (1989). Lemma 1. Given a hyperplane game (N; V; C) and i 2 S Cq 2 C; (Sn fig ; VS ; fSn figg) = Sn fig ; VSnfig ; fSn figg : Maschler and Owen (1989) prove that satis…es l-consistency for all l = 1; :::; n: In the next proposition we obtain a similar result for : Proposition 3. The consistent coalitional value satis…es l-consistency for each l = 1; :::; n. Proof. See the Appendix. In the next theorem we study which of these properties are satis…ed by the consistent coalitional value. Theorem 1. The consistent coalitional value satis…es EF , IS, COV , BCON S; ABCAP; and ABCAC. Proof. See the Appendix. Then, satis…es the same interesting properties that the Owen value satis…es in T U games and the consistent value satis…es in hyperplane games.
5. Axiomatic Characterizations In this section we present two axiomatic characterizations of the consistent coalitional value. Hart and Mas-Colell (1989) characterize the Shapley value in the class of T U games as the only value satisfying EF; SY M (the usual symmetry property), COV; and CON S: Later, Winter (1992) and Maschler and Owen (1989) extend this result in two di¤erent ways. 13
Winter (1992) extends it to the class of T U games with coalition structure. He proves that the Owen value is the only value satisfying EF; IS; COV; CON S; and Game Between Coalitions Property (GBCP ). A value satis…es GBCP if for each T U game (N; v; C) and Cq 2 C; X [C] ; fM g : i (N; v; C) = Cq M; v i2Cq
This property says that the amount received by a coalition in the game played by the coalitions coincides with the sum of the amounts received by the members of this coalition in the original game. This property cannot be exported to hyperplane games. Nevertheless, it is possible to obtain the following result: Proposition 4. The Owen value is the only value in CT U satisfying EF; IS; COV; CON S; and BCAC: Proof. It is similar to the proof of Winter’s result about the characterization of the Owen value. We omit it. Maschler and Owen (1989) extend the result of Hart and Mas-Colell (1989) to the class of hyperplane games. They prove that the consistent value is the only value satisfying EF; SY M; COV; and BCON S: In Theorem 2 below we generalize the results of Hart and Mas-Colell (1989), Winter (1992), and Maschler and Owen (1989) to hyperplane games with coalition structure. Theorem 2: The consistent coalitional value is the only value in CH satisfying EF; IS, COV; BCON S; and ABCAC: Proof. See the Appendix. Remark 2. The properties used in this theorem are independent (see the Appendix). Myerson (1980) characterizes the Shapley value in T U games as the only value satisfying EF and BC: Calvo et al: (1996) extend this result to CT U proving that the Owen value is the only single value satisfying EF; BCAP; and BCAC: Hart and Mas-Colell (1996) extend Myerson’s result to N T U games proving that the consistent value is the only value satisfying EF and ABC 2 : 2
This result appears in Proposition 4 of Hart and Mas-Colell (1996) in a di¤erent way. It is easy to check that, in CH; it could be rewritten as we did.
14
In Theorem 3 below we generalize the results of Myerson (1980), Calvo et al. (1996), and Hart and Mas-Colell (1996) to hyperplane games with coalition structure. Theorem 3. The consistent coalitional value is the only value in CH satisfying EF; ABCAC, and ABCAP: Proof. See the Appendix. Remark 3. The properties used in this theorem are independent. Bergantiños and Vidal-Puga (2004) extend this result to N T U games and prove that these properties are independent. We can obtain the independence of the properties in CH using the same rules as in Bergantiños and Vidal-Puga (2004). The results obtained in this section about the consistent coalitional value and the relation with other values can be summarized in the following table:
TU games
Without coalition structure Shapley EF EF SY M BC COV CON S
Consistent EF EF Hyperplane SY M ABC games COV BCON S
With coalition structure Owen EF EF IS BCAP COV BCAC CON S BCAC Coalitional consistent EF EF IS ABCAP COV ABCAC BCON S ABCAC
Hence, the consistent coalitional value is the right generalization of the Owen value and the consistent value to hyperplane games with coalition structure if we focus on the properties of consistency and balanced contributions.
15
6. Concluding Remarks The Shapley value and the consistent value have two important properties. Firstly, they have an intuitive de…nition (they can be computed through the vector of marginal contributions). Secondly, these values can be characterized with nice properties (consistency and balanced contributions). The Owen value can be de…ned using either the two-stage procedure or the vector of marginal contributions over admissible permutations. Moreover, it can also be characterized with consistency and balanced contributions. Hyperplane games with coalition structure generalize the three classes of games mentioned before. In this setting, whereas generalizes the de…nition based in the two-stage procedure, z generalizes the de…nition of the Owen value based in marginal contributions. This fact is not surprising. There are results from T U games that cannot be generalized in the same way to N T U games. For instance, the consistent value and the Shapley N T U value are generalizations of the Shapley value. The consistent value generalizes the characterizations of the Shapley value based in the properties of consistency and balanced contributions, but the Shapley N T U value (Shapley (1969)) generalizes the classical axiomatization of the Shapley value (Aumann (1985)). Bergantiños and Vidal-Puga (2004) extend both values to N T U games. They characterize with properties of EF; ABCAC; and ABCAP: Moreover, they prove that z satis…es, in N T U games, properties of EF and ABCAC: In hyperplane games it is straightforward to prove that z also satis…es COV: Nevertheless, it satis…es neither BCON S (in hyperplane games) nor ABCAP . Then z can be characterized neither with consistency nor balanced contributions. N T U games with coalition structure are also studied by Winter (1991), who characterizes the Game Coalition Structure Value. This value is a generalization of the Harsanyi value for N T U games and the Owen value for T U games with coalition structure. Winter characterizes this value with six axioms: EF; COV; conditional additivity, independence of irrelevant alternatives, inessential games, and unanimity games. We believe that is a more natural extension of the Owen value. Our main arguments are the de…nition (which generalizes the original definition of the Owen value) and theorems 2 and 3 (which extend characterizations of the Owen value).
16
7. Appendix 7.1. Proof of Proposition 1 We …rst prove that
satis…es (3:1) : Since X X N N N j j = j
is an e¢ cient value, for each Cq 2 C; Cq j
(V ) = v (Cq ) ;
j2Cq
j2Cq
which satis…es (3:1) by de…nition. Using an induction argument over the number of agents jN j ; it is easy to prove N that there exists a unique obtained through the two stage procedure. P point N N We now prove that = v (N ) : i i i2N
X
N i
X
=
Cq 2C
i2N
Since
N i
X
0 @
X
N j
j2Cq
1
NA : j
satis…es (3:1) ; last expression coincides with 0
2
0
X
X
1
X
X
!31
N N nCr A N N nCq 5A @ 1 4 @ + v (N ) j j j j jCj Cq 2C j2Cq j2Cr Cr 2CnCq Cr 2CnCq 0 0 1 !1 X 1 @ X X @ X N N nCr A X X N N nCq A = + v (N ) j j j j jCj C 2C j2Cq Cq 2C Cr 2CnCq j2Cr Cr 2CnCq q 0 0 1 !1 X X X X 1 @ N N nCr A N N nCq A @ = + v (N ) j j jCj C ;C 2C;C 6=C j2C j j j2C C ;C 2C;C 6=C q
r
q
r
q
q
= v (N ) :
17
r
q
r
r
7.2. Proof of Remark 1 Since M; v [C] is a T U game and = in T U games, applying (2:2) we obtain that, given q 2 M 0 1 X X 1 @ M M nfrg M nfqg A = + v [C] (M ) q q r jCj r2M nfqg r2M nfqg 0 1 X 1 @ X M nfrg M nfqg A + v (N ) : = q r jCj r2M nfqg
M q
We now prove that for all Cq 2 C; M q
on jM j : If jM j = 1; then
=
P
r2M nfqg
j
N j :
We proceed by induction
j2Cq
= v (N ) and
P
j
N j
= v (N ) : Assume that the
j2Cq
result holds when jM j t: We prove it when jM j = t + 1: Since satis…es (3:1) and Si = i for all S N such that i 2 S; 2 0 1 X X X 1 4 X @X N nCr A N + v (N ) j j = j j j jCj j2C j2C j2C Cr 2CnCq
q
Cr 2CnCq
q
By induction hypothesis, 2 X 1 4 X N j j = jCj j2C
M nfrg q
+ v (N )
r2M nfqg
q
X
r2M nfqg
N nCq j
r
3
M nfqg 5 r
=
!3
5:
M q .
7.3. Proof of Proposition 2 Using an induction argument over the number of agents jN j ; it is easy to prove that there exists a unique value satisfying conditions (3:1) and (3:2) : We only need to prove that satis…es (3:1) and (3:2) : We know that satis…es (3:1) : We now prove that satis…es (3:2) : If we apply (2:2) to the hyperplane game (Cq ; V ) we obtain that for all i 2 Cq ; N i
= =
Cq i
(V ) 1
jCq j
Cq i
0 @
X
Cq i
Cq nfjg i
(V ) + v (Cq )
j2Cq nfig
X
j2Cq nfig
18
Cq j
Cq nfig j
1
(V )A :
C nfjg
In this expression, i q (V ) represents the consistent value of player i in the hyperplane game (Cq n fjg ; V ) : V is obtained from (N; V; C) as in Stage 2 C nfjg N nfjg of the de…nition of : By de…nition, i = iq V j where V j is obtained from (N n fjg ; V; C j ) as in Stage 2 of the de…nition of : It is trivial to see that V coincides with V j in Cq n fjg : Thus, N i
Since
=
1 jCq j
N i
0 @
X
N nfjg i
N i
X
+ v (Cq )
j2Cq nfig
N j
j2Cq nfig
satis…es (3:1) ; v (Cq ) =
P
N j
N j :
Then,
1
N nfig A : j
satis…es (3:2) :
j2Cq
7.4. Proof of Proposition 3 We proceed by induction on l. The result is trivially true for l = 1. Assume it is true when l t: We prove it when l = t + 1: If we apply the induction hypothesis to the game (N n fjg ; V; C j ) with j 2 Cq n fig (if Cq = fig, the result is trivially true for Cq ) then, P
T i
(VT ) =
T Cq nfjg:i2T;jT j=l 1
jCq j 2 l 2
N nfjg i
(V ) :
(7:1)
We want to prove that, for each Cq 2 C and i 2 Cq ; l
N i
P
S i
S Cq :i2S;jSj=l
(VS ) = l
N jCq j 1 i l 1
N i
(V ) :
(7:2)
To do so, we analyze the left side of this expression. Assume that i 2 S Cq and jSj = l: Applying (3:2) to (S; VS ; fSg), which is also a hyperplane game, we obtain X X X N Snfjg N S N Snfig S l N (V ) = (V ) + (V ) (VS ) : S S S i i i j j j i j j2Snfig
If we compute
j2S
j2Snfig
in the game VS we obtain that X X N N S (V ) = v (N ) S j j j j2S
j2S
19
N j
(V ) :
Hence, l
N i
S i
X
(VS ) =
N i
Snfjg i
(VS )+v (N )
j2Snfig
N j
N j
(V )
possible sets S X l N i
X
N j
Snfig j
(VS ) :
j2Snfig
j2S
jCq j 1 l 1
Since there are
X
Cq with i 2 S and jSj = l, S i
(VS ) =
S Cq :i2S;jSj=l
X
S Cq :i2S;jSj=l
X
0 @
S Cq :i2S;jSj=l
X
N i
j2Snfig
0 X @
N j
1
jCq j 1 (VS )A + v (N ) l 1 1 0 X X N A @ j (V ) Snfjg i
S Cq :i2S;jSj=l
j2S
N j
j2Snfig
Rearranging the order of summation, we have 0 1 X X jCq j 1 N Snfjg @ (VS )A + v (N ) i i l 1 j2Cq nfig S Cq :i;j2S;jSj=l 0 1 0 X X X X N N @ A @ (V ) j j j2N nfig
S Cq :i2S;j 2S;jSj=l =
j2Cq nfig
S Cq :i;j2S;jSj=l
We now analyze the four terms separately: 1. By Lemma 1, the …rst term is equal to 0 X X N @ i j2Cq nfig
T i
T Cq nfjg:i2T;jT j=l 1
which coincides, by (7:1); with jCq j 2 l 2
X
j2Cq nfig
20
N i
Snfig j
N nfjg i
1
(VT )A
(V ) :
N j
1
(VS )A :
Snfig j
1
(VS )A :
2. Since v(N ) =
P
N j
N j
(V ), the second term is equal to
j2N
jCq j 1 l 1
+
N i
N j
X
jCq j 1 l 1
(V ) +
X
jCq j 1 l 1
N j
N j
N j
(V )
j2Cq nfig
(V ) :
j2N nCq
3. The third term is equal to 0 X X @ j2Cq nfig
N i
N j
N j
S Cq :i2S;j 2S;jSj=l =
1
(V )A
X
j2N nCq
0 @
X
N j
N j
S Cq :i2S;j 2S;jSj=l =
1
(V )A :
For each j 2 Cq n fig, there are jClq j 1 2 possible sets S; such that S Cq ; jCq j 1 i 2 S; j 2 = S; and jSj = l. For each j 2 N nCq , there are l 1 possible sets S; such that S Cq ; i 2 S; j 2 = S; and jSj = l. Then, this term coincides with jCq j 2 l 1
X
N j
N j
(V )
j2Cq nfig
4. By Lemma 1, the fourth term is equal to 0 X X N @ j j2Cq nfig
X
jCq j 1 l 1
T j
which coincides, by (7:1), with
Since
jCq j 1 l 1
=
jCq j 2 l 1
jCq j 1 l 1
+ N i
jCq j 2 l 2 N i
X
N j
N j
j2N nCq
T Cq nfig:j2T;jT j=l 1
jCq j 2 l 2
N j
N nfig j
1
(VT )A ;
(V ) :
j2Cq nfig
; adding terms 2 and 3 we obtain
(V ) +
jCq j 2 l 2 21
X
j2Cq nfig
N j
N j
(V ) :
(V ) :
Then, l
N i
X
S i
jCq j 2 l 2
(VS ) =
S Cq :i2S;jSj=l
+
jCq j 1 l 1
N i
N i
N nfjg i
N i
(V ) +
j2Cq nfig
X
jCq j 2 l 2
(V ) +
X N j
N j
N nfig j
(V )
(V ) :
j2Cq nfig
In Theorem 1 we will prove, without using this proposition, that satis…es ABCAP . Hence, X X jCq j 2 jCq j 1 N N N N nfjg S l N (V ) = (V ) + S i i i i i (V ) i l 2 l 1 j2Cq nfig
S Cq :i2S;jSj=l
jCq j 1 l 1
+
jCq j 2 l 2
(jCq j l
N i
N i
jCq j 1 l 1
1) = l
N i
N i
N nfjg i
(V )
j2Cq nfig
jCq j 1 l 1
= Since
X
jCq j 2 l 2
+
jCq j 1 l 1
N i
X
jCq j 2 l 2
(V ) +
(V ) N i
N i
(V ) :
j2Cq nfig
; the last expression coincides with
N i
(V ) ;
which is the right side of (7:2). 7.5. Proof of Theorem 1 By Proposition 1, satis…es EF: By Proposition 3, straightforward to check that satis…es IS: We now prove that satis…es ABCAP: Since satis…es (3:2), X X X N N N N nfjg N jCq j N = + j j i i i i j2Cq
j2Cq nfig
=
X
N i
N nfjg i
+
N i
=
N j
N i
+
X
N j
N nfjg i
N i
j2Cq nfig
22
+ jCq j
N i
N i
N nfig j
N j
j2Cq nfig N i
N nfig j
j2Cq nfig
j2Cq nfig
X
satis…es BCON S: It is
+
X
j2Cq nfig
N j
N j
N nfig j
:
Then, X
N j
N nfig j
N j
X
=
j2Cq nfig
N i
N nfjg i
N i
;
j2Cq nfig
which means that
satis…es ABCAP:
We now prove that
P
satis…es ABCAC: By EF , v(N ) =
Cr 2C
Applying this to (3:1) we obtain that, for all Cq 2 C; X N N jCj j j =
P
N j
N j
j2Cr
!
.
j2Cq
=
X
Cr 2CnfCq g
=
X
Cr 2CnfCq g
0 @
X
N j
j2Cq
0 @
X
N j
j2Cq
If we subtract (jCj
1
N nCr A j
+
N nCr A j
+
1
1)
P
X
X
Cr 2C
X
N j
N j
N j
X
=
j2Cq
Cr 2CnfCq g
+
X
N j
N j
P
=
@
Cr 2CnfCq g
X
Cr 2CnfCq g
X
N nCr j
N j
P
N j
N j
N j
N j
j2Cq
N j
j2Cq
X
N nCq j
j2Cr
X
N j
N nCq j
j2Cr
N j
N nCq j
N j
j2Cr
Cr 2CnfCq g
Cr 2CnfCq g
0
X
X
+
j2Cq
N j
N j
j2Cr
j2Cq
X
N j
!
!
1
A+ !
in both sides,
X
N j
N j
j2Cq
:
Hence, X
Cr 2CnfCq g
0 @
X
j2Cq
which means that
N j
N j
N nCr j
1
A=
satis…es ABCAC:
23
X
Cr 2CnfCq g
X
j2Cr
N j
N j
N nCq j
!
;
! !
:
We now prove that satis…es COV: Given i 2 Cq 2 C; let N; Ve ; C be obtained from (N; V; C) by a change in player i’s utility. Let a and b be the corresponding constants. We proceed by induction on the number of coalitions in C: If C has only one coalition (C = fN g) then, = : Since satis…es COV , N i N j
Ve =
Ve = a
N i
Ve =
Ve =
N j
N i N j
(V ) + b = a N j
(V ) =
N i
(V ) + b and
(V ) for each j 2 N n fig.
Assume the result holds when jCj = t. We prove it when jCj = t + 1. P ~N N e By (3:1) ; jCj V = j j j2Cq
X
Cr 2CnfCq g
0 @
X
~N j
1
Ve A + v~ (N )
N nCr j
j2Cq
(V ) when j 6= i:
N Since ~ i =
jCj
X
j2Cq
~N j
N j
N i
a Ve
N , ~j =
=
N j
X
0 @
+v (N ) + X
Cr 2CnfCq g
X
b
0 @
Cr 2CnfCq g
= jCj
N nCr i
~N
X
j2Cq
N j
X
N j
N nCr j
j2Cr
(V ) +
X
a
X
N i
b
a N nCr i
N i
X
N nCr j
N j
j2Cq
X
N j
(V )A + v (N )
N nCq j
(V ) + jCj 24
N nCq j
!
(V )
b
N i
a
;
+ jCj
b
!
:
Ve =
;
(V ) + !
(V )
1
j2Cr N j
N j
j2Cr
Cr 2CnfCq g
N nCr j
(V ) + b and
j2Cq nfig N i
Ve
N nCq j
j
when j 6= i; and v~ (N ) = v (N ) +
Cr 2CnfCq g
=
X
Cr 2CnfCq g
Ve = a
N nCr i
By the induction hypothesis, N nCr j
X
N i
a ( )
b
N i
a
1 A
where the last equality comes because Given k 2 Cq , by (3:2) ; N
jCq j ~ k
N k
Ve =
X
~N
Ve +
N nfjg k
k
satis…es (3:1) : X
~N j
j2Cq
j2Cq nfkg
By the induction hypothesis and ( ), N
jCq j ~ i
N i
Ve
X
=
N i
N nfjg i
N j
(V ) +
(V ) +
j2Cq nfig
+
X
N j
j2Cq
X
=
N nfjg i
N i
j
N nfkg j
j2Cq nfkg
Ve :
a
a
X
N nfig j
N j
(V )
j2Cq nfig
(V ) +
X
N j
N j
(V )
j2Cq
j2Cq nfig
X
N nfig j
N j
j2Cq nfig
= jCq j
~N
N i
b
N i
b
X
Ve
N j
N i
N i
(V ) + jCq j
(V ) + jCq j
N i
b
a
N i
b
a
;
where the last equality comes because satis…es (3:2) : Therefore, N Ve = a N i i (V ) + b: By the induction hypothesis and ( ), if k 2 Cq n fig then, jCq j
N k
N k
Ve
=
X
N k
N nfjg k
N j
N nfkg j
X
j2Cq nfkg
= jCq j
N k
N k
N nfjg k
N i
(V )
j2Cq nfk;ig
=
N j
N j
(V ) +
j2Cq
j2Cq nfkg
X
X
(V ) +
(V ) +
X
N j
N nfkg i N j
(V )
j2Cq N k
(V ) ;
where the last equality comes because Therefore, N Ve = N k k (V ) :
satis…es (3:2) :
25
(V )
b
N i
a b
N i
a X
j2Cq nfkg
N j
N nfkg j
(V )
Given Cr 2 C n fCq g ; using arguments similar to those used for Cq we can conclude that X N X N N ~ N Ve = (V ) : j
j
j
j2Cr
N j
j
j2Cr
Now, using (3:2) it is not di¢ cult to conclude that for each j 2 Cr ; (V ) : Thus,
satis…es COV:
N j
Ve =
7.6. Proof of Theorem 2 In Theorem 1 we proved that the consistent coalitional value satis…es these …ve properties. We now prove the uniqueness. Let e be a value satisfying these …ve properties. We will show that e = . We proceed by induction on the number of players. If there is only one player, then, by EF; fi = max fx : x 2 V (fig)g = i : fig When jN j = 2; by COV we can assume without loss of generality that i = fjg = 1: There are two possible coalition structures, C 1 = ffi; jgg or C 2 = j ffig ; fjgg : Given a 2 R; let (N; v a ) be the T U game given by v a (fig) = v a (fjg) = a and a v (N ) = 1: Since e satis…es EF and IS; we conclude that 1 eN va; C 1 = e N va; C 1 = : i j 2
Since e satis…es EF and ABCAC; we conclude that
e N va; C 2 = e N va; C 2 = 1 : i j 2
A similar result can be obtained for : Since any hyperplane game with two players (N; V; C) can be obtained from v a (for some a) by linear transformation of utilities of players, and and e satisfy COV; it is straightforward to prove that, for each i 2 N;
Moreover,
eN = i
N i
=
v (N ) +
N i v
(fig) 2
26
N i
N j v
(fjg)
:
N i
N j
N i
N j
N i
=
eN
N j
i
eN = j
N i v
N j v
(fig)
(fjg) :
Assume that e = when 2 n t. We prove e = We …rst prove that, for each Cq 2 C; P
N j
N j
P
=
j2Cq
N j
j2Cq
(7:3)
when n = t + 1.
eN:
(7:4)
j
eS = By the induction hypothesis P N N we know that, for each S 6= N; Cq 2 C, by (3:1), jCj j j =
S
: Given
j2Cq
X
=
Cr 2CnfCq g
X
=
Cr 2CnfCq g
0 @
0 @
X
1
N nCr A j
N j
j2Cq
X
N j
j2Cq
jCj
N j
N j
e N nCr A + v (N ) j
j2Cq
Cr 2CnfCq g
+
P
X
0 @
Cr 2CnfCq g
X
=
Cr 2CnCq
+
X
P
Cr 2C
X
=
+ v (N )
1
Since e satis…es EF , v(N ) = X
X
0 @
Cr 2CnCq
X
N j
j2Cq
X
X
X
1
eN j
N j
j2Cr
eN j
e N nCq j
!
:
j2Cq
j
!
e N A + jCj j !
e N nCq j
j2Cq
eN
N j
1
P Since e satis…es ABCAC; we conclude that 27
X
e N nCq j
j
j2Cq
N j
!
e N . Then, j
j
N j
N nCq j
N j
j2Cr
!
e N nCr
N j
X
j2Cr
e N nCr A +
j2Cr
X
Cr 2CnfCq g
Cr 2CnfCq g
N j
j2Cr
X
N j
N j
X
N j
j2Cq
eN
:
=
P
j2Cq
N j
eN: j
j
N We now prove that e N N: We denote by VS and VeS i = i for each i 2 Cq the reduced games associated with and e ; respectively. N By (7:4) ; if Cq = fig ; we conclude that e N i = i . Assume that Cq 6= fig : For each j 2 Cq nfig ; we consider S = fi; jg. We know that VS and VeS are hyperplane games. We denote by vS and v~S the characteristic functions associated with VS and VeS ; respectively. By the de…nition of reduced game and the induction hypothesis,
VeS (fig) = VS (fig) and VeS (fjg) = VS (fjg) :
Hence, vS (fig) = v~S (fig) and vS (fjg) = v~S (fjg) : P Since e satis…es EF; we conclude that v (N ) =
N k
k2N
n VeS (S) = (xi ; xj ) 2 Rfi;jg :
By the e¢ ciency of e and (7:3), N i
N i
Then, X
N i
j2Cq nfig
X
N i
j2Cq nfig
e S V~S + i
e S V~S i
e S V~S + i
e S V~S i
N j
j2Cq nfig
X
N j
j2Cq nfig
Since e satis…es BCON S, N i
(jCq j
1)
N i
eN i
eN i +
X
j2Cq nfig
j2Cq nfig
X
N i
N ~S i v
e S V~S = j P
N j xj
e S V~S = j
e S V~S = j
j2Cq nfig
N j
+
e S V~S = j
N j
X
N j
N i xi
N i
eN + i
(fig)
N i
(jCq j
X
28
N i
N j
X
X
j2Cq
j2Cq nfig
N j
o eN : j
eN j
N ~S j v
(fjg) :
1) e N i +
j2Cq nfig
e Sj V~S =
e S V~S = j
eN + i
v~S (fig)+
e S V~S = (jCq j i
N j
N i
e N : Then, k
X
N j
j2Cq nfig
X
N ~S j v
eN j
(fjg) :
j2Cq nfig
1) e N i . Hence,
N j
eN j
v~S (fig) +
X
j2Cq nfig
N ~S j v
(fjg) :
A similar analysis for N i
yields X +
N i
N j
S j
X
(VS ) =
1)
N i
N i
X
N j
S j
N j
j2Cq
j2Cq nfig
(jCq j
N j
N i
(VS ) =
X
vS (fig) +
j2Cq nfig
j2Cq nfig
X
N j vS
(fjg) :
j2Cq nfig
By (7:4), N i
eN + i
P
j2Cq nfig
N j
e S V~S = j
N i
N i
eN i
N i
+
P
N j
S j
(VS ) :
(7:5)
j2Cq nfig
Since vS (fig) = v~S (fig) and vS (fjg) = v~S (fjg) for all j 2 Cq n fig ; (jCq j (jCq j
1)
1)
N i
N i
P
N j
e S V~S = j
j2Cq nfig N S j j (VS ) : j2Cq nfig
P
(7:6)
Adding (7:5) and (7:6), jCq j which means that e N i =
N i
N i
= jCq j
N i :
N i
eN i
7.7. Proof of Remark 2
ABCAC is independent of the rest of properties because the consistent value satis…es EF , IS; COV; and BCON S but not ABCAC: Using arguments similar to those used by Winter (1992), we can conclude that the rest of properties are independent. 7.8. Proof of Theorem 3 By Theorem 1, we know that satis…es these properties. We now prove the uniqueness. We proceed by induction on jN j. The result is trivially true for jN j = 1. Assume the result holds when jN j t: We prove it when jN j = t + 1: Let ~ be a value satisfying these properties. 29
Since ~ satis…es EF and ABCAC; using arguments similar to those used in the proof of Theorem 2, we can conclude that, for each Cq 2 C; X X N N N ~N j j = j j : j2Cq
j2Cq
By (3:2), jCq j P
Since
N i
N i
N i
N nfjg i
N i
+
N j
N j
P
=
N j
X
N j
X
N j
j2Cq
j2Cq nfig
j2Cq
jCq j
X
=
N nfig : j
N j
j2Cq nfig
~ N , by the induction hypothesis, j
j2Cq N i
=
X
N i
~ N nfjg + i
X
N j
~N
N i
~ N nfjg +
N i
i
X
N j
~ N nfig j
X
N j
~ N nfig
~N j
j
j2Cq nfig
~ N nfjg
N i
~N + jCq j i
i
j2Cq nfig
+
X
j2Cq nfig
~N + i
j2Cq nfig
=
j
j2Cq
j2Cq nfig
=
X
X
N j
~N j
N i
~N i
~ N nfig : j
j2Cq nfig
Then, jCq j
N i
~N = i
N i
X
N j
~ N nfig
~N j
j
X
N i
~N i
~ N nfjg : i
j2Cq nfig
j2Cq nfig
Since ~ satis…es ABCAP; we conclude that the last expression is equal to 0: Hence, ~ N = N : i
i
8. References Aumann, R., 1985. An axiomatization of the non-transferable utility value. Econometrica 53, 599-612. Bergantiños, G., Casas-Mendez, B., Fiestras-Janeiro, G., Vidal-Puga, J.J., (2004). A focal-point solution for bargaining problems with coalition structure. Mimeo, Universidade de Vigo. 30
Bergantiños, G., Vidal-Puga, J.J., 2004. The consistent coalitional value for N T U games. Available at Calvo, E., Lasaga, J., Winter, E., 1996. The principle of balanced contributions and hierarchies of cooperation. Mathematical Social Sciences 31, 171-182. Chae, S., Heidhues, P., 2004. A group bargaining solution. Mathematical Social Sciences 48, 37-53. Greenberg, J., 1994. Coalition structures, in: Aumann, R., Hart, S. (Eds.), Handbook of game theory with economic applications (Volume 2). Elsevier Science, Amsterdam/ New York, pp. 1305-1337. Harsanyi, J., 1963. A simpli…ed bargaining model for the n-person cooperative game. International Economic Review 4, 194-220. Hart, S., Mas-Colell, A., 1989. Potential, value, and consistency. Econometrica 57, 589-614. Hart, S., Mas-Colell, A., 1996. Bargaining and value. Econometrica 64, 357380. Maschler, M., Owen, G., 1989. The consistent Shapley value for hyperplane games. International Journal of Game Theory 18, 389-407. Maschler, M., Owen, G., 1992. The consistent Shapley value for games without side payments, in: Selten R., (Ed.), Rational Interaction. Springer-Verlag, New York, pp. 5-12. McLean, R.P., 1991. Random order coalition structure values. International Journal of Game Theory 20, 109-127. McLean, R.P. 2002. Values of non-transferable utility games, in: Aumann, R., Hart, S. (Eds.), Handbook of game theory with economic applications (Volume 3). Elsevier Science, Amsterdam/ New York, pp. 2077-2120. Monderer, D., Samet, D., 2002. Variations of the Shapley value, in: Aumann, R., Hart, S. (Eds.), Handbook of game theory with economic applications (Volume 3). Elsevier Science, Amsterdam/ New York, pp. 2055-2076. Myerson, R.B., 1980. Conference structures and fair allocation rules. International Journal of Game Theory 9, 169-182. Owen, G. 1977. Values of games with a priori unions, in: Henn R., O. Moeschlin, O., (Eds.), Essays in Mathematical Economics and Game Theory. Springer-Verlag, New York, pp. 76-88. Shapley, L., 1953. A value for n–person games. Annals of Mathematics Studies 28, 307-317. Shapley, L., 1969. Utility comparison and the theory of games, in La Decision. Editions du CNRS, Paris, pp. 251-263. 31
Vidal-Puga, J.J., 2004. A bargaining approach to the Owen value and the Nash solution with coalition structure. Economic Theory (forthcoming).. Winter, E., 1991. On non-transferable utility games with coalition structure. International Journal of Game Theory 20, 53-63. Winter, E., 1992. The consistency and potential for values of games with coalition structure. Games and Economic Behavior 4, 132-144. Winter, E., 2002. The Shapley value, in: Aumann, R., Hart, S. (Eds.), Handbook of game theory with economic applications (Volume 3). Elsevier Science, Amsterdam/ New York, pp. 2025-2054.
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