A new axiomatization of the Owen value for games with
coalition structures
* ´
Gerard Hamiache
´ ´
G.R.E.Q.A.M., Centre de la Vieille Charite, 2 rue de la Charite, 13002 Marseille, France Received 9 September 1997; received in revised form 28 February 1998; accepted 31 March 1998
Abstract
This paper presents a new axiomatization of the Owen value for games with coalition structures. The driving force of this new axiomatization is a consistency axiom based on an associated game which is not a reduced game. We show that the Owen value is the only one satisfying, in addition with other rather ‘standard’ properties, a consistency requirement which makes it unnecessary to perform a particular kind of manipulation. 1999 Elsevier Science B.V. All rights reserved.
Keywords: Owen value; Coalition structure; Associated game; Consistency; Iterative functional equation
JEL classification: C71
1. Introduction
The Owen (1977) value is a generalization of the Shapley (1953) value that, to some extent, answers the question: how the Shapley value of a game should be modified in order to reflect membership of players to a priori unions. These unions are the expression of affinities existing between players. Each player is a member of one and only one such union. Formally, these affinities induce a partition of the set of players called a coalition structure.
Another answer to that question was presented in the pioneering work of Aumann and `
Dreze (1974). These two authors consider that there are no lateral payments between coalitions of the coalition structure. Their @-value coincides, in each coalition separately, with the regular Shapley value of the corresponding restricted game.
*Tel.: 133-4-9114-0770; fax:133-4-9190-0227.
E-mail address: [email protected] (G. Hamiache)
In this paper a new axiomatization of the Owen value for games with coalition structure is offered. Besides Owen (1977), more axiomatizations can be found in the literature (Hart and Kurz, 1983; Peleg, 1989; Winter, 1992). It is worth mentioning the interesting survey on games with coalition structures presented in Greenberg (1994).
An axiomatization unveils hidden properties of solution concepts, and reveals relations with other concepts which would not be suspected otherwise. For example, the present work shows that the Owen value is the only one satisfying the property that players neither lose nor gain when performing a manipulation of a particular kind.
The driving force of the present axiomatization is an (associated) consistency axiom based on an associated game, that is not a reduced game. The main difference between the two approaches is that the associated consistency requires the definition of a single game while the usual consistency requires the definition of a new game for each of the coalitions. In this sense the present consistency axiom is less demanding.
In Section 2, we present the formal framework and review the Owen value. In Section 3, the new axiomatization is presented.
2. The Owen value
Let U be a non-empty and finite set of players. A coalition is a non-empty subset of
U.
Definition. A coalitional game with transferable utility (a TU game) is a pair (N,v) where
N
N is a coalition and v is a function satisfying v:2 →Rand v([)50. We denote byG
the set of all these games.
Definition. Let (N,v) be a game and T a subset of N. We say that T is a carrier of (N,v) if
for all Q#N, v(Q )5v(T>Q ).
Definition. Let (N,v) be a game and i a player of N. We say that i is a null player of the
game (N,v) if for all S#N, v(S )5v(S<hij).
Definition. A game (N,v) is called a unanimity game if there exist a non-void subset R of
N, and a real number c such that, v5cu where u is defined for all SR R #N by u (S )R 51 if R#S and u (S )R 50 if R#u S.
Definition. A coalition structure for N is a pair kN, @l, in which @ is a collection of disjoint and non-empty subsets of N, the union of which is the grand coalition.
We shall denote a typical partition of N by @5hB , B , . . . , B1 2 rj with 1#r#[N. In
the following we shall also consider the coalition structure kN, [N]l in which all the members of the partition [N] are singletons, [N]5hhijui[Nj.
Definition. Let kN, @l be a coalition structure and S a subset of N. We denote by
Ik (S ) the index set of S where I (S )5hiuB>S±[, B[@j.
N,@l kN,@l i i
With this notation the number of structural coalitions in @ is [Ik (N ). When no N,@l
confusion may occur, we shall denote the index set of S by I(S ).
]
Given a coalition structurekN,Bl and a subset S of N, we denote by S the set,
]
S5
<
B .i i i[I(S )]
In words, S is the minimal union of structural coalitions which covers S. In particular,
]
we denote BI(hij), the structural coalition which contains player i, by hij.
Definition. LetkN,@lbe a coalition structure and S a subset of N. We definekS,@ l, the S induced coalition structure for S, to be a coalition structure for S in which @S5hS>
Bui[I(S ), B[@j.
i i
Definition. A permutationp on N is a one-to-one mapping on N,p:N→N. We denote
by P(N ) the set of all the permutations on N.
Definition. We say that a permutationp is consistent with the coalition structurekN,@l if the following condition is fulfilled. For all B[@, all i and j[B and all l[N if
p(i ),p(l ),p( j ), then also l[B.
In words, a permutation is consistent with a coalition structure if it is the composition of permutations of the players within structural coalitions and a permutation of the structural coalitions. We denote by S(N,@) the set of all the permutations inP(N ) which are consistent with the coalition structure kN,@l. As a consequence,
[S(N,@)5([I(N )!)?([B !)? ? ? ? ?([B !)
1 [I(N )
Definition. A game with coalition structure is a triplet (N,v,@) where N is a coalition, (N,v) is a game andkN,@lis a coalition structure. Let us denote by G the set of all these games.
Definition. Given a game with coalition structure (N,v,@)[G, we say that two players i
] ]
and j in the same structural coalition (hij5hjj) are substitute if for all coalitions
S#N\hi, jj, v(S<hij)5v(S<hjj).
Definition. Given a game with coalition structure (N,v,@)[G, the intermediate game is
the game with coalition structure (I(N ), v /@, [I(N )]) where v /@ is defined so that for every set T, T#I(N ),
v/@(T )5v
<
B .i
S D
ii[T
Remark. We defined I(B) as a set. But when B[@, I(B) can be considered as a single
Definition. A solution on G is a function s which associates each game with coalition
[N structure (N,v,@)[G, a vector s(N,v,@) of R .
One normally requires that the solutions satisfy a set of properties, usually called axioms. In the following we present a set of axioms proposed by Peleg (1989). Since this reference may be difficult to find, we reproduce here his main result.
Axiom P1 (Pareto Optimality). For every game (N,v,@)[G,
O
c(N,v,@)5v(N ).i i i[N
Axiom P2 (Restricted Equal Treatment Property). For every game with coalition
structure (N,v,@), and any two substitute players i and j, c(N,v,@)5c(N,v,@).
i j
Axiom P3 (Null Player Property). If player i is a null player in the game (N,v), then
ci(N,v,@)50.
Axiom P4 (Additivity Property). For any two games (N,v ,@) and (N,v ,@) in G,
1 2
c(N,v 1v ,@)5c(N,v ,@)1c(N,v ,@),
1 2 1 2
where for all subsets S of N, (v11v )(S )2 5v (S )1 1v (S ).2
Axiom P5 (Intermediate Game Property). For every game with coalitional structure
(N,v,@) and every structural coalition B[@,
O
c(N,v,@)5c (I(N ),v/@, [I(N )]).j I(B )
j
j[B
Peleg (1989) used the axioms above to characterize the Owen value.
Theorem 1. (Peleg (1989)) There is a unique solution c on the set of games with coalition structures which satisfies P1 –P5. This solution coincides with the Owen value,
1 i i
]]]
c(N,v,@)5
O
[v(P <hij)2v(P )], ;i[N.i [S(N,@) p p
p p[S(N,@)
i
Where Pp5hj[Nup( j),p(i)jis the set of all the players of N preceding player i in a permutation p, p[P(N).
Remark. In the two cases [I(N)51 and [I(N)5[N in which the only structural coalition is N and all the singletons of players are structural coalitions, respectively, the
Owen value coincides with the Shapley (1953) value since in these two cases S(N, @)5P(N).
value. In the second stage, a claims game is constructed. This is used to determine the distribution of the value received by the intermediates at the first stage, among the members of the respective structural coalitions. Within each structural coalition we define the claims game so that the worth of a coalition S, S#B [@, is the value of S in
k
the intermediate game in which the structural coalition is hB , . . . , B1 k21, S, Bk11, . . . ,
Brj. The Owen value of the original game coincides with the Shapley values of the claims games defined for each of the different structural coalitions.
3. The new axiomatization
In the following we present our alternative characterization of the Owen value based on an associated game. The basic construction of this game is based on a new approach which, to the best of my knowledge, has been used only in Hamiache (in press), in the context of cooperative games with communication structures. Given f a solution for games with coalition structures and (N,v,@) a game, we define the associated game
*
(N,v ,@) so that,
f
v(T )1
O
[f (T<K,v ,@ )2v(K )] if T±[,K uT<K T<K K
*
v (T )5 (1)
f K[@
N•T
5
0 if T5 [.By vuT<K we denote the restriction of the function v to the set T<K, and fK(T<K, vu ,@ )5o f(T<K, v ,@ ).
T<K T<K i[K i uT<K T<K
An interpretation of the associated game is as follows. Letf be a solution on the set of games with coalition structures. Such a solution may be understood as a recipe how to share some worth among a population of players when a game with coalition structure is given. But even within this socially recognized rule of distribution, players may try to reach extra payments by use of manipulations on the type of coalition structures they accept to form. Let us suppose that players try to improve their position by adoption of the so called ‘divide and rule’ behavior; that is: the coalition considers its opponent coalitions as isolated avoiding to accord them the credit of their cooperation. Let T be a coalition. According to the divide and rule behavior of coalition T, a partition of the set
N\T is determined. To be fully consistent with the given coalition structure the natural
partition is imposed by the induced coalition structurekN\T,@ l. The rationale of such N \T
an assumption is clear, since T cannot break structural coalitions ofkN\T, @ l, these N \T structural coalitions are the basic units players in coalition T are able to discern. In other words, the divide and rule behavior of coalition T creates a refinement@ <@ of the
T N \T
original partition, which corresponds to the superimposition of the two coalition structures kN, @l andkN, hT, N\Tjl.
According to its divide and rule behavior, coalition T invites each structural coalition,
K, of kN\T, @ l to play the partial game (T<K, v , @ ). This cooperation
N \T uT<K T<K
between T and K produces a surplus for the structural coalition K, [cK(T<K, vuT<K, @T<K)2v(K )] of Eq. (1), on which T may have designs. Notice that the relevant
attachment of players in T to people in the same structural coalition of@that were not enrolled in coalition T. That is, when K meets players of T that used to be in the same structural coalition of @, they renew their structural coalition.
*
The worth of coalition T in the associated game, v (T ), is the worth coalition T wouldf
achieve if it could catch and add to its own worth in the original game, v(T ), all the quantities [fK(T<K, vuT<K,@T<K)2v(K )], the surpluses of the structural coalitions K, K[@ , when they are invited by T to play the game (T<K, v , @ ).
N \T uT<K T<K
We now formulate our system of axioms.
Axiom A1 (Efficiency). For all games (N,v,@) and all carriers S of (N,v), we haveo i[S
fi(N,v,@)5v(S ).
Axiom A2 (Additivity Property). For all games, (N,v,@) and (N,w,@) in G, f(N,v1
w,@)5f(N,v,@)1f(N,w,@).
Axiom A3 (Independence of Irrelevant Players). For all games (N,v,@), all carriers S of (N,v) and all players i in S, f(N,v,@)5f(S,v ,@ ). Where v is the restriction of
i i uS S uS
function v to the domain S#N.
Axiom A4 (Positivity). For all R#N, all i[R, and all positive parameters c,fi(N, cu ,R @) is strictly positive.
Axiom A5 (Associated Consistency). For all games with coalition structure (N,v,@),
*
f(N,v,@)5f(N,v ,f@).
Axiom A6 (Symmetry). For all permutations p of N, p[P(N ), fpi(N, pv, hNj)5fi (N,v,hNj) where for all S#N,pv(S )5v(pS ).
The first (efficiency) and the second axiom (additivity) are standard axioms in the context of Shapley and Owen value.
Let R be a carrier of the game (N,v). By Axiom 3, the worth of this coalition,
v(R)5v(N ), is distributed independently of the existing affinities between players of N\R
and is even independent of the affinities players in R may have with players in N\R. In other words the distribution of the worth v(R), depends only on the coalition structure
kR, @ l. Axiom A3 permits us to develop a recursive approach to the value. R
By Axiom A4, the payments to players of R in the game cu , with c positive, areR strictly positive. This axiom can be understood as an individual rationality assumption that ensures the participation of all the players in R to the unanimity game.
The fifth axiom (associated consistency) means that the solution f is such that the original game and its associated game have the same value. In other words, the function
based on an associated game and not on a reduced game, we adopt a different terminology, namely the associated consistency axiom, to differentiate the two concepts. Axiom A6 (symmetry) ensures that if two players are substituted in the game (N,v,[N]), they have the same value. In fact we shall need this axiom only when the number of players in N is odd.
Lemma 1. The Owen value is consistent on the set h(N,v,[N])u(N,v)[Gj.
Proof. As already seen above the Owen value c coincides with the Shapley value w
*
when the structural coalitions are singletons. In this case the associated game (N,v ) isw
such that:
*
v (S )5v(S )1
O
[w(S<hjj,v )2v(hjj)].w j S<hjj
j j[N•S
The lemma is true for one-player game since in these cases the game and its associated game coincide. We shall prove Lemma 1 by induction on the number of players. So let us assume that the lemma is true for all t-player games when t,[N.
It is a well known fact that for all games (N,v) the function v can be expressed as the following linear combination,
[R2[T
v5
O O
(21) v(T ) u . (2)R
R T
R#N T#R
*
We shall show that the associated game vw can be expressed as the same linear
*
combination of associated unanimity games, (u ) ,R w [±R#N.
Let us compute, for all S#N,
[R2[T
*
O O
(21) v(T )(u ) (S )5R w
R T
R#N T#R
Following the definition of the associated game,
[R2[T
O O
(21) v(T ) u (S )1O
[w(S<hjj, u )2u (hjj) 5R j R R
R T
F
jG
R#N T#R j[N•S
Using Eq. (2),
[R2[T
5v(S )1
O O
(21) v(T )O
[w(S<hjj,u )2u (hjj)]5j R R
R T j
R#N T#R j[N•S
Changing the order of summations,
[R2[T
5v(S )1
O O O
(21) v(T )[w(S<hjj, u )2u (hjj)] 5j R R
j
F
R TG
For all R such that R#⁄ S<hjj, the restriction of the function u to the set S<R hjjis the null function and the termwj(S<hjj, u )R 50. Then we can rewrite the last expression as,
5v(S )1
O O O
j
F
R TR#S<hjjT#R j[N•S
[R2[T [R2[T
(21) v(T )w(S<hjj,u )2
O O
(21) v(T )u (hjj)j R R
R T
G
R#N T#R
*
5v(S )1
O
f
w(S<hjj,vu )2v(hjj)g
5v (S ),j S<hjj w
j j[N•S then,
[R2[T
* *
vw5
O O
(21) v(T )(u ) .R w
R T
R#N T#R
Using the linearity property of the Shapley value, and changing the order of summations,
[R2[T
* *
w(N,vw)5
O O
(21) w(N,(u ) )v(T ).R w
T R
T#N T#R#N
Using Eq. (2) and the linearity property of the Shapley value,
[R2[T
w(N,v)5
O O
(21) w(N,u )v(T ).R
T R
T#N T#R#N
*
Then the equalityw(N,v )w 5w(N,v) is true for all game (N,v) if and only if for all T#N,
[R2[T [R2[T
*
O
(21) w(N,u )R 5O
(21) w(N,(u ) )).R wR R
T#R#N T#R#N
By the induction hypothesis these equalities are true if and only if
*
w(N,u )N 5w(N,(u ) ).N w
*
So let us compute wi(N, (u ) )) for all i[N w N,
[R2[T
* *
wi(N,(u ) )N w 5
O O
(21) (u ) (T )N w wi(N,u ).RR T
R#N T#R i[R
Since
*
(u ) (T )N w 5(u )(T )N 1
O
[wj(T<hjj, uNuT<hjj)2u (N hjj)], jj[N•T
[R2[T
*
wi(N,(u ) )N w 2wi(N,u )N 5
O O O
(21) wj(T<hjj, uNuT<hjj)wi(N,u ).Rj R T
R#N T#R j[N
i[R j[⁄ T
If N±T<hjj the function uNuT<hjj is the null function and the termwj(T<hjj, uNuT<hjj) cancels. Then, the only relevant value of the variable T is T5N\hjj and the two corresponding values of the variable R are R5N\hjjand R5N. Let us substitute these
values in the last expression.
*
wi(N,(u ) )N w 2wi(N,u )N 5
O
wj(N,u )N wi(N,uN•hjj)2O
wj(N,u )N wi(N,u )Nj j
j[N•hij j[N
[N21 1 1
]]] ]]] ]
5 [ 2 50.
N [N21 [N
*
The last expression cancels, thenw(N,v)5w(N,v ) for all (N,v)w [G, completing the proof of Lemma 1. h
Theorem 2. The Owen value satisfies axioms A1 – A6.
Proof. Given a game with coalition structure (N,v,@), let us consider its intermediate
*
game (I(N ),v /@,[I(N )]) and the intermediate game of the associated game (I(N ),v /c
* *
@,[I(N )]). We shall show that v /c @5(v /@) . Let us compute for all Tw #I(N ) the
*
expression (v /@) (T ).w
*
(v/@) (T )w 5(v/@)(T )1
O
[w(T<hjj,v/@ )2v/@(hjj)].j uT<hjj
j j[I(N )•T
Let M denote the set of players represented by T, M5<i[T Bi#N. Using the
intermediate game property of the Owen value,
* *
(v/@) (T )5v(M )1
O
[c (M<K,v ,@ )2v(K )]5v (M )w K uM<K M<K c
K K[@N•M
*
5(vc/@)(T ),
thus,
* *
w(I(N ),v @5w(I(N ), (v/@) ).
c w
Since, by Lemma 1, the Shapley value satisfies the associated consistency property,
*
w(I(N ),v/@)5w(I(N ), (v/@) ), w
and then
*
w(I(N ),v/@)5w(I(N ),v /@). c
*
c (N,v,@)5c (N,v ,@) for all K[@.
K K c
Using the Owen’s two-stages interpretation of his value, we see that at the first stage, in the original game and in its associated game, the same value is distributed among the members of a given structural coalition. At the second stage the claim of a coalition which is a subset of a structural coalition is equal to its value in the intermediate game when this coalition substitutes itself to the structural coalition it is a subset of. Again by use of Lemma 1, the claims of each relevant coalition is the same in the original game as in its associated game. As a consequence the Owen value of the two games are equal and we have proved that the Owen value satisfies the associated consistency property. Axioms A1–A4 and A6 are clearly satisfied by the Owen value. h
Theorem 3. The Owen value is the unique solution satisfying axioms A1 – A6.
Proof. Letf be a solution that satisfies axioms A1–A6 and is different from the Owen value. From the efficiency axiom, f(hij, v, hhijj)5v(hij)5c(hij, v, hhijj). Then the theorem is true for 1-player games. We prove the theorem by induction on the number of players, so, let us assume that the theorem is true for all t-player games with t,[N. We
prove in the following that the theorem is also true for [N-player games, that is
f(N,v,@)5c(N,v,@).
By use of Eq. (2) and Axiom A2,
[R2[T
f(N,v,@)5
O O
(21) f(N,v(T ) u ,@).R
R T
R#N T#R
Changing the order of summations,
[R2[T
f(N,v,@)5
O O
(21) f(N,v(T ) u ,@).R
T R
T#N T#R
Since R is a carrier of the game (N, cu ), we deduce from Axiom A3 that for all iR [R,
f(N, cu , @)5f(R, cu , @ ), and then we can rewrite the last expression as
i R i R R
[R2[T
f(N,v,@)5
O O
(21) f(R,v(T ) u ,@ ), ;i[N. (3)i i R R
T R
T#N T#R
i[R
* *
By Axiom A5,f(N,v,@)5f(N,v ,@) for all i[N andf(R, cu , @ )5f(R, (cu ) ,
i i f i R R i R f
@ ) for all i[R, R
[R2[T
* *
f(N,v ,@)5
O O
(21) f(R,(v(T ) u ) ,@ ), ;i[N. (4)i f i R f R
T R
T#N T#R i[R
By the induction hypothesis, we have f(R, (cu ), @ )5c(R, (cu ), @ ) for all strict
i R R i R R
[N2[T
*
f(N,v,@)2f(N,v ,@)5
O
(21) [f(N,v(T ) u ,@)i i f i N
T T#N
*
2f(N,(v(T ) u ) ,@)].
i N f
*
From the last equation, we deduce thatf(N,v,@)5f(N,v ,@) for all games (N,v,@) in
i i f
*
G if and only if f(N, cu , @)5f(N, (cu ) ,@) for all real numbers c.
i N i N f
*
Applying the additivity axiom to Eq. (3) for v5(cu )N f and using Eq. (1) we get:
*
f(N,(cu ) ,N f@)5f(N,cu ,N@)1
O O
T R
[,T,N T#R
[R2[T
(21) f N,
O
[f (T<K,cu ,@ K NuT<K T<K KS
K[@N•TD
2cu (K )]u ,@).
N R
*
Using the consistency axiom for cu , f(N,(cu ) ,@)5f(N,cu ,@),
N N f N
[R2[T
O O
(21) f N,O
[f (T<K,cu ,@ 2cu (K )]u ,@)K NuT<K T<K N R
T R
S
KD
[,T,N T#R K[@ N•T
50.
Notice that for all T such that T<K±N, cuNuT<K is the null function on the set T<Kj
and f (T<K, cu , @ )50. Then, we are only concerned with T satisfying K NuT<K T<K
T<K5N, i.e.[I(N\T )51 and K5N\T. Note that in this case, for all T±[, u (N\T )N 5
0. By Axioms A1 and A3, fi(R, cu ,R @R)50 for all i[⁄ R,
[R2[T
O O
(21) fi(R,fN•T(N,cu ,N@) u ,R @R)50, ;i[N. (5)T R
[,T,N T#R
[I(N•T )51 i[R
Let us now fix i and rewrite the above expression,
[R2[T
O O
(21) f R,O
f(N,cu ,@) u ,@ 50.i j N R R
T R
S
jD
[,T,N T#R j[N•T
[I(N•T )51 i[R
Changing the order of the summations,
[R2[T
O O O
(21) f(R,f(N,cu ,@) u ,@ )50.i j N R R
j T R
[,T,N T#R j[N
[I(N•T )51 i[R j[⁄ T
[R2[T
O O
O
(21) f(R,f(N,cu ,@) u ,@ )50. (6)i j N R R
j R T
i[R T#R j[N
[I(N•R )#1[I(N•T )51
j[⁄ T
[,T,N
It is convenient, at this point, to split the computations into the following four cases: (1)
[I(N )51; (2)[I(N )5[N; (3) r$2 and there is no singleton structural coalition; and (4) r$2 and we have both singleton and non-singleton structural coalitions.
Case 1. [I(N )51.
Note that we assume that N has at least two elements. In this case Eq. (6) becomes,
[R2[T
O O O
(21) fi(R,fj(N,cu ,B ) u ,B )N R R 50, ;i[N. (7)j R T
R#N T#R j[N
i[R j[⁄ T
[,T,N
We distinguish also between the two cases R5N and R±N and in the second case between j±i and j5i.
[N2[T
05
O O
(21) f(N,f(N,cu ,@)u ,@)1 (8a)i j N N
j T
[,T,N j[N
j[⁄ T
[R2[T
1
O O O
(21) f(R,f(N,cu ,@) u ,@ )1 (8b)i j N R R
j R T
R,N T#R j[N•hij
i[R j[⁄ T
[,T,N
[R2[T
1
O O
(21) f(R,f(N,cu ,@) u ,@ ). (8c)i i N R R
R T
R,N T#R i[R i[⁄ T
[,T,N
Using the fact that
[L
[L 0 if L5 [,
[T t
O
(21) 5O
(21)S D H
5 .t 21 otherwise
T t51
[,T#L
We perform the three summations over T in Eqs. (8a)–(8c), when L5N\hjjin the term of Eq. (8a), L5R\hjj in the term of Eq. (8b), and L5R\hij in the term of Eq. (8c),
[N
05 2
O
(21) f(N,f(N,cu ,@)u ,B )2 (9a)i j N N
j
[R
2
O O
(21) f(R,f(N,cu ,@) u ,@ )2 (9b)i j N R R
j R
R,N j[N•hij
i[R
[R
2
O
(21) f(R,f(N,cu ,@) u ,@ ). (9c)i i N R R
R R,N
i[R R±hij
Using Axioms A1 and A2, we simplify Eq. (9a). Dissociating the case R5hijin the term of Eq. (9c), that isf(hij,f(N, cu ,@) u ,@ ) and using the efficiency axiom for the
i i N hij hij
game (hij, u , @ ),
hij hij
[N [R
(21) f(N,cu ,B )5 2
O O
(21) f(R,f(N,cu ,@) u ,@ )i N i j N R R
j R
R,N j[N•hij
i[R
[R
2
O
(21) f(R,f(N,cu ,@) u ,@ )i i N R R
R
3
R,N i[R2(21)f(N,cu ,@) .
i N
4
Gathering the relevant terms,
[N [R
[11(21) ]f(N,cu ,@)5 2
O O
(21) f(R,f(N,cu ,@) u ,@ ).i N i j N R R
j R
R,N j[N
i[R
Using the additivity axiom and the efficiency axiom we get:
[N [R
[11(21) ]f(N,cu ,@)5 2
O
(21) f(R,cu ,@ ). (10)i N i R R
R R,N
i[R
Using the induction assumption, that the valuef coincides with the Owen value for all
t-player games, t,[N,
[N22
c
[N t11 [N21
]]
[11(21) ]fi(N,cu ,B )N 5 2
O
(21) t11S
tD
t50[N21 [N21 c (21) c
t [N21
]] ]]]
5
O
(21)S
D
2 .t
t11 [N
t50
Lemma 2. For all integers n and x it holds that,
n
1 1
n
t
]] ]]]]]
S D
O
(21) t 5 .n1x x1t
t50 (n11)
S D
Proof. See Appendix A.
Then for all i in N,
[N
c (21) c c
[N [N
] ]]] ]
[11(21) ]f(N,cu ,@)5 1 5[11(21) ] .
i N [N [N [N
When [N is an even number fi(N, cu ,N @)5c /[N for all i in N.
But, when the number of players is odd and[I(N )51, the equations do not determine the value of the unanimity game on the coalition structure @. We then use Axiom 6 (symmetry) to ensure that in the unanimity game u , all the players have the sameN payment c /[N. And the value f coincides in this case with the Owen value.
Case 2. [I(N )5[N.
In this case all the structural coalitions of@ are singletons. We rewrite Eq. (6). The set R, may receive only two values, R5N and R5N\hjjwhenever i±j. In both cases the only available T is T5N\hjj.
O
f(N•hjj,f(N,cu ,@) u ,@ )5O
f(N,f(N,cu ,@) u ,@ ).i j N N•hjj N•hjj i j N N N
j j
j[N j[N
i±j
Using the induction hypothesis for the left-hand side term, and using the axioms of additivity and efficiency for the right-hand side term,
1
]]]
O
f(N,cu ,@)5f(N,cu ,@ ).j N i N N
[N21 j
j[N
i±j
By use of the efficiency axiom,
1
]]][c2f(N,cu ,@)]5f(N,cu ,@ ),
i N i N N
[N21 then,
c
]
f(N,cu ,@)5 5c(N,cu ,@),
i N [N i N
which proves the theorem in Case 2.
Case 3. r$2 and ;t[h1, 2, . . . , rj,[Bt.1.
In words, there are at least two structural coalitions and they all have at least two elements.
We change in Eq. (6) the names of the variables T and R so that they become N\T and
[T2[R
O O O
(21) f(N•R,f(N,cu ,@) u ,@ )50i j N N•R N•R
j R T
i[⁄ R R#T j[N
[I(R )#1[I(T )51
j[T
] ] ] ]
.We distinguish between three cases, [ j5i], [ j±i and hjj5hij], and [hjj±hij].
[T2[R
O O
(21) f(N•R,f(N,cu ,@) u ,@ )1 (11a)i i N N•R N•R
R T
i[⁄ R R#T
[I(R )#1[I(T )51
i[T
[T2[R
1
O O O
(21) f(N•R,f(N,cu ,@)u ,@ )1 (11b)i j N N•R N•R
j R T
] i[⁄ R R#T j[hij•hij
[I(R )#1[I(T )51
j[T
[T2[R
1
O O O
(21) f(N•R,f(N,cu ,@) u ,@ )50.i j N N•R N•R
j R T
] i[⁄ R R#T j[N•hij
[I(R )#1[I(T )51
j[T
(11c)
If, in the term of Eq. (11a),[I(R)50, that is R5[, then the summation over T cancels.
]
And if[I(R)51 the summation over T does not cancel only if R5hij•hij, in that case
]
the only available T is T5hij. The term of Eq. (11b) cancels, since for any set R,
]
R#hij•hij, the summation over T cancels. In the term of Eq. (11c), the summation over
] ]
T will not cancel only if R5hjjor if R5hjj•hjj. In both cases the only available T is
] ]] ]
T5hjj. Gathering all those results and using the notation, h2ij5N•hij,
]]
] ]
05(21)f(h2ij<hij,f(N,cu ,@) u ,@ )1 i i N h2ij<hij h2ij<hij
]] ] ]
1
O
f(h2jj,f(N,cu ,@) u ,@ )2i j N h2jj h2jj
j
]
i[h2jj
]] ] ]
2
O
f(h2jj<hjj,f(N,cu ,@) u ,@ ). i j N h2jj<hjj h2jj<hjjj
]
i[h2jj
] ]] ]]
Since for all j[N,[hjj±1, the setsh2jjandh2jj<hjjare strict subsets of the grand coalition N. Then by use of the induction hypothesis for games with less than [N
(21) 1 1
]] ]] ]]
05 f(N,cu ,B )1 ]
O
f(N,cu ,@)i N j N
r r21 [hij j
]
i[h2jj
1 1
] ]]
2 ]
O
f(N,cu ,@).j N
r [hij j
]
i[h2jj
Gathering the relevant terms,
1 1
]] ]]
f(N,cu ,B )5 ]
O
f(N,cu ,@).i N r21 j N
[hij j
]
i[h2jj
We learn from the above expression that for any k[h1, . . . , rjthe valuef (N,u ,@)
Bk N
is distributed equally among the players of B . Thus we need only to findf (N, u ,@)
k Bk N
for all the structural coalitions over @. Summing up the above expression for all the
]
players in hij, and by use of the efficiency axiom we get, 1
] ]] ]
fhij(N,cu ,B )N 5r21[c2fhij(N,cu ,B )],N which leads to,
c
] ]
f (N,cu ,@)5 .
hij N r
]
Since for all i[N the value fh (N, cu , @) is shared equally among the players in
ij N
]
structural coalitions hij, we get
c 1
] ]]
fi(N,cu ,B )N 5r ]5ci(N,cu ,B ),N
[hij
which proves Theorem 2 in Case 3.
Case 4. r$2 and 1#p,r.
In words, there are at least two structural coalitions, when singleton and non-singleton structural coalitions effectively coexist. Remember that p is the number of structural coalitions which are singleton.
Eqs. (11a)–(11c) can be rewritten in the following form,
[T2[R
05
O O
(21) f(N,f(N,cu ,@)u ,@)1 (12a)i i N N•R
R T
] R#T
R#hij
]
i[⁄ R T#hij
[T2[R
1
O O O
(21) f(N,f(N,cu ,@)u ,@)1 (12b)i j N N•R
j R T
] R#T
j[N R#hij
] ]
i[⁄ R T#hij
j[hij
j[T j±i
[T2[R
1
O O O
(21) fi(N,fj(N,cu ,N@)uN•R,@)50. (12c)j R T
] R#T j[N R#hjj
] ] i[⁄ R T#hjj
j[⁄hij
j[T
]
Case 4a. Let i be a player such that [hij51.
The only available R in the term of Eq. (12a) is R5[, and then the only possible
]
value of T is T5hij. In the term of Eq. (12b) there is no j available then this term cancels. Then Eqs. (12a)–(12c) can be rewritten as
[T2[R
f(N,f(N,cu ,@)u ,@)5
O O O
(21) f(N,f(N,cu ,@)u ,@).i i N N i j N N•R
j R T
] R#T
j[N R#hjj
]
j±i T#hjj
j[T
] ]
We distinguish between [hjj51 and [ hjj±1.
fi(N,fi(N,cu ,N@)u ,N@)5 (13a)
[T2[R
5
O O O
(21) f(N•R,f(N,cu ,@)u ,@ )1 (13b)i j N N•R N•R
j R T
] R#T j[N R#hjj
] ] T#hjj
[hjj51
j[T j±i
[T2[R
1
O O O
(21) f(N•R,f(N,cu ,@)u ,@ ). (13c)i j N N•R N•R
j R T
] R#T
j[N R#hjj
] ]
T#hjj [hjj±1
j[T j±i
]
In the term of Eq. (13b) the only two available values of R are R5[and R5hjj, in both
] ] ]
cases T5hjj. The term of Eq. (13c) does not cancel only if R5hjj or R5hjj•hjj, in
] ]
both cases T5hjj. Together with the fact that i±j if [hjj±1 Eqs. (13a)–(13c) can be rewritten as,
f(N,f(N,cu ,B )u ,B )5 2
O
f(N,f(N,cu ,@)u ,@)1 (14a)i i N N i j N N
j
j[N
]
[hjj51
]]
] ]
O
f(h2jj,f(N,cu ,@)u ,@ )2 (14b)i j N h2jj h2jj
j j[N
]
[hjj51
j±i
]]
] ]
2
O
f(h2jj<hjj,f(N,cu ,@)u ,@ )1 (14c)i j N h2jj<hjj h2jj<hjj
j j[N
]
[hjj±1
]] ] ]
1
O
fi(h2jj,fj(N,cu ,B )uN h2jj,Bh2jj). (14d) jj[N
]
[hjj±1
Gathering the two terms of Eq. (14a) gives Eq. (15a) below. Using the induction hypothesis for games with less than [N players and the additivity axiom, Eq. (14b)
gives Eq. (15b), Eqs. (14c) and (14d) give Eq. (15c).
O
f(N,f(N,cu ,@)u ,@)5 (15a)i j N N
j j[N
]
[hjj51
1 1
]] ]]
5
O
f(N,cu ,@)2 f(N,cu ,@)2 (15b)j N i N
r21 j r21 j[N
]
[hjj51
1 1
] ]]
2
O
f(N,cu ,@)1O
f(N,cu ,@). (15c)j N j N
r j r21 j
j[N j[N
] ]
[hjj±1 [hjj±1
By the efficiency axiom the sum of the left-hand side term of Eq. (15b) and of the right-hand side term of Eq. (15c) equals c /(r21). Using once more the efficiency axiom, the right-hand side term of Eq. (15c) gives the right-hand side term of Eq. (16b),
c
]]
O
f(N,f(N,cu ,@)u ,@)5 2 (16a)i j N N r21
j j[N
]
[hjj51
1 1
]] ]
2 f(N,cu ,@)2 c2
O
f(N,cu ,@) . (16b)i N j N
r21 r j j[N
3
]4
[hjj51
]
pc p
]]]
F
]O
f N,O
f(N,cu ,@)u ,@ 5 1i j N N r(r21) r
i j
i[N
1
j[N2
] ]
[hij51 [hjj51
1
]]
G
2
O
f(N,cu ,@).j N
r21 j j[N
]
[hjj51
The last expression is an iterative functional equation of order two,
px p 1
]]]
F
] ]]G
j(j(x))5 1 2 j(x) forj(x)5
O
f(N,xu ,@), (17)j N
r r21
r(r21) j
j[N
]
[hjj51
of the general form,
j(j(x))5ax1bj(x), (18)
where a±0 and b±0 since r and p are integers such that r$2 and 1#p,r. The
properties of the functionj, that are deduced from the properties of the functionf, are:
• j(0)50;
• j(x) has the sign of x;
• j(x1y)5j(x)1j( y);
• j(x),x for all x.0.
Lemma 3. j(x)5px /r is the only solution to Eq. (17) satisfying the four above
properties.
Proof. A proof adapted from Nabeya (1974) is given in Appendix B.
We are able now to reformulate Eqs. (16a) and (16b) using the resultj(c)5pc /r and
the additivity axiom,
p c 1 1 pc
]f(N,cu ,@)5]]2]]f(N,cu ,@)2]
F
c2]G
,i N i N
r r21 r21 r r
which leads to
c ]
]
fi(N,cu ,B )N 5r5ci(N,cu ,B ) for all iN [N such thathij5hij.
]
Case 4b. Let i be a player such that [hij±1.
]
For all R, in the term of Eq. (12a), which is different fromhij•hijthe summation over
] ]
]
Eq. (12b) the set R<hijis different fromhijthe summation over T always cancels. Then Eqs. (12a)–(12c) can be rewritten as
]] ] ]
f(h2ij<hij,f(N,cu ,@)u ,@ )5 i i N h2ij<hij h2ij<hij
[T2[R
5
O O O
(21) f(N•R,f(N,cu ,@)u ,@ ).i j N N•R N•R
j R T
] ]R#hjj R#T j[h2ij ]
T#hjj
j[T
] ]
We distinguish between [hjj51 and [hjj±1,
]] ] ]
f(h2ij<hij,@ ,f(N,cu ,@)u )5 (19a)
i h2ij<hij i N h2ij<hij [T2[R
5
O O O
(21) fi(N•R,fj(N,cu ,N@)uN•R,@N•R)1 (19b)j R T
] ] R#hjj R#T j[h2ij ]
] T#hjj [hjj51
j[T
[T2[R
O O O
(21) f(N•R,f(N,cu ,@)u ,@ ). (19c)i j N N•R N•R
j R T
] ] R#hjj R#T j[h2ij ]
] T#hjj [hjj±1
j[T
]
In the term of Eq. (19b), the only two available values of R are R5[ and R5hjj, in
] ] ]
both cases T5hjj. The term of Eq. (19c) does not cancel only if R5hjjor R5hjj•hjj,
]
in both cases T5hjj. Using Axiom 3, Eqs. (19a)–(19c) can be rewritten as,
]]
] ]
f(h2ij<hij,f(N,cu ,@)u ,@ )5 i i N h2ij<hij h2ij<hij
1
5
O
(21) f(N,f(N,cu ,@)u ,@)1i j N N
j
]
j[h2ij
]
[hjj51
]] ] ]
1
O
f(h2jj,f(N,cu ,@)u ,@ )1i j N h2jj h2jj
j
]
j[h2ij
]
[hjj51
]]
1 ] ]
1
O
(21) f(h2jj<hjj,f(N,cu ,@)u ,@ )1 i j N h2jj<hjj h2jj<hjjj
]
j[h2ij
]
[hjj±1
]] ] ]
1
O
f(h2jj,f(N,cu ,@)u ,@ ).i j N h2jj h2jj
j
]
j[h2ij
]
Using the induction hypothesis for games with less than[N players and the additivity
axiom,
1
]f(N,cu ,B )5 2f N,
O
f(N,cu ,@) u ,@ 1i N i j N N
r j
j[N
1
]2
[hjj51
1 1 1 1
]] ]] ] ]]
1 ]
O
f(N,cu ,B )2 ]O
f(N,cu ,@)1j N j N
r21 [hij j r [hij j
]
j[N j[h2ij
] ]
[hjj51 [hjj±1
1 1
]] ]]
1 ]
O
f(N,cu ,@).j N
r21 [hij j
]
j[h2ij
]
[hjj±1
Using the fact that j(c)5pc /r and the additivity axiom,
1 p 1 1 pc
]f(N,cu ,@)5 2]f(N,cu ,@)1]] ]] ]]
i N i N
r r r21 [hij r
1 1
]
]]] ]]
1 ]
O
f(N,cu ,@)2f (N,cu ,@) .j N hij N
r(r21) [hij j j[N
3
]4
[hjj±1
Using the efficiency, and once more the result j(c)5pc /r, p11 1 1 pc
]]fi(N,cu ,N@)5]] ]] ]]
r r21 [hij r
1 1 pc
]
]]] ]] ]
1 ]
F
c2 2fh (N,cu ,@)G
ij N
r r(r21) [hij
p11 1 1 c 1 pr1r2p
]
]]f(N,cu ,@)1]]] ]]]f (N,cu ,B )5] ]]]
F
]]]G
.i N hij N
r r(r21) [hij r [hij r(r21)
]
The above equation proves that all the players of the structural coalitionhijare equally
]
treated, that is, the value fh (N, cu , @) is shared equally among the players of the
ij N
structural coalition. Thus, we need only to find the total value of the players of structural
]
coalitions. Summing up the above equation over all the players in hij,
c
] ]
f (N,cu ,@)5 ,
hij N r then,
c 1
] ]]
f(N,cu ,@)5 ]5c(N,cu ,@),
i N r i N
which completes the proof of Theorem 3. h
4. Conclusion
We have seen a new axiomatization of the Owen value in which the motor is a consistency like axiom. The same type of associated game has been used in Hamiache (in press) to define an alternative to the Myerson (1977) value on graphs.
I believe that the present approach can be fruitfully applied to other existing solution concepts and, more importantly, can be used to develop new types of solutions.
Acknowledgements
´ ´
My thanks go to Louis-Andre Gerard-Varet and to Michel Le Breton who read a previous version of this paper. I am also indebted to Issie Sweetko and Marc Teboulle for their help. The author thanks the referee for very detailed comments which led to an improvement of the paper.
Appendix A
Proof of Lemma 2. Let the first and the nth difference of the function f be given by,
n n21 n21
Df(x)5f(x11)2f(x) and Df(x)5D f(x11)2D f(x), respectively. It can easily be
shown that in general,
n
n
n n2t
S D
D f(x)5
O
(21) t f(x1t).t50
n
n n2t
Since we want to compute the expression ot50 (21) ( )(1 /(xt 1t)), the most natural
way is to choose f(x)51 /x, and what is left to be proved is that
1 1
n n ]]]]] ]]]
(21) D f(x)5 5 .
n1x x
S
n1xD
(n11)
S D
nn11
We prove this formula by induction on n. We begin with n51.
1 1 1 1
]] ] ]]] ]]]
F
G
2D(1 /x)5 2 2 5 5 .
x11 x x(x11) x11
2
S D
2
Let the induction hypothesis be true for all t#n21. For t5n21 we have, 1
n21 n21
]]]]]
(21) D (1 /x)5 .
n1x21
S
D
n n
n21 n21 n21
(21) (21) (21) 1 1
n
]]] ]]]]] ]]] ]]] ]]]]
D (1 /x)5 2 5 2 ,
n1x n1x21 n
F
n1x n1x21G
S
D
S
D
n n n
S
nD
nS
nD
arranging the expression between brackets we get,
n21
(21) n!(x21)! n
]]] ]]]
D f(x)5
F
(2n) .G
n (n1x)!After cancellations,
n!(x21)! 1 (n11)!(x21)!
n n ]]] ]] ]]]]]
(21) D f(x)5 5
F
G
,n11
(n1x)! (n1x)!
which proves Lemma 2. h
Appendix B
t
Proof of Lemma 3. We define the successive compositions of the function j, j 5
t21 t
j+j . It can easily be checked thatj (x) can be expressed as a linear combination of x and of j(x),
n
j (x)5a xn 1bnj(x). n11
Let us computej (x),
n11 n
j (x)5j (j(x))5anj(x)1b [axn 1bj(x)], n11
j (x)5ab xn 1[an1bb ]n j(x)5an11x1bn11j(x).
The recursion on the coefficients can be written in a matrix form as follows,
an11 0 a an an
5 5A ,
S D S DS D S D
bn11 1 b bn bnwith a150 and b151 or, alternatively, a051 and b050. The matrix A is diagonalizable since we have different eigenvalues:
p 1
] ]]
l 51 r and l 5 22 r21.
A set of eigenvectors (v ,v ) corresponding to the eigenvalues (1 2 l1,l2), respectively, is
1 2p /r
v 5
S D
and v 5S D
,1 2
r21 1 n11
we can then write j (x) as,
0 1
n11 n 21 n11 21
j (x)5(x,j(x)) PD P
S D
5(x,j(x)) PD PS D
,1 0
1 2p /r P5
S
D
.r21 1
21
Its inverse P is given by,
r p
1
21
]]]
P 5
S
2(r21)r rD
.r1pr2p
So,
n n
1 p 1
n11
]]]
FF
]S
]]D G
j (x)5 p
S D
2p 2 xr1pr2p r r21 n n
p 1
] ]]
F
S
D G G
1 p(r21)
S D
1r 2 j(x) .r r21 Or rewriting the above expression,
n
p(r21) p 1 n11
]]] ]
F S
]]D G
j (x)5
S D
j(x)2 2 x r1pr2p r r21n
r 1 p
]]]
S
]]D
]1 2
F
j(x)2 xG
.r1pr2p r21 r
n11 n
We compute the difference, j (x)2j (x),
n21
p(r21) p r2p 1
n11 n
]]] ] ]]
F
]]G
j (x)2j (x)5 2
S D F
G
j(x)1 x r1pr2p r r r21n21
r 1 r p
]]]
S
]]D
]] ]2 2
F
G F
j(x)2 xG
. (20)r1pr2p r21 r21 r
n11
Since j(x),x for all positive x and since in this casej(x).0 it follows thatj (x)2
n
j (x),0 for all x.0. Now, assume that there exists an x positive such that [j(x)1x /(r2
1)]±0 and [j(x)2px /r]±0. Notice that j(x)5px /r and j(x)5 2x /(r21) are two solutions of the functional Eq. (17).
The first term of the right-hand side expression of Eq. (20) is always negative but the second term has alternate signs, following the parity of n.
n11 n
p 1
] ]
When p51 the expression r2r21 is negative and the signs of j (x)2j (x) alternate for sufficiently large values of n which leads to a contradiction. The consequence is that there exists a unique positive solution to the given functional equation:
1
] j(x)5 x.
r
n
p 1
] ]
When p$2 the expression r 2r21 is positive and we have to extend the seriesj to the negative n.
21 2n
Since matrix A is regular we can define its inverse A and its powers A . We define
`
a2n 2n a0 a0 1
5A for 5 .
S D S D
b2n b0S D S D
b0 0 We definea0
2k 2k
j (x)5a2kx1b2kj(x)5(x,j(x))A
S D
b .0
2k 2k11
We show thatj+j (x)5j(a2kx1b2kj(x))5j (x). Since, for all k, the coefficients
a2k and b2k are rational numbers, by the additivity property of j,
2k
j(j (x))5a2kj(x)1b2kj(j(x))5a2kj(x)1b2k[ax1bj(x)], or in matrix form,
a2k a2k11
2k 2k11
j(j (x))5(x,j(x))A
S D
b 5(x,j(x))S D
b 5j (x).2k 2k11
21
The eigenvalues of A are r /p and 2(r21) with the same set of eigenvectors we computed for A. Then expression Eq. (20) is the same expression for positive and negat
