G . Hamiache Mathematical Social Sciences 37 1999 281 –305
285
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 , . . . , B
, S, B , . . . ,
1 k 21
k 11
B j. The Owen value of the original game coincides with the Shapley values of the
r
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 ,
2 vK ] if T ± [,
K uT K
T K K
v T 5
1
f K [
N •T
5
if T 5 [. By v
u we denote the restriction of the function v to the set T K, and f T K,
T K K
v ,
5 o
f T K, v ,
.
uT K T K
i [K i
uT K T K
An interpretation of the associated game is as follows. Let f 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 structure
kN\T, l. The rationale of such
N \T
an assumption is clear, since T cannot break structural coalitions of
kN\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 and kN, 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, [c T K, v ,
K uT K
2vK ] of Eq. 1, on which T may have designs. Notice that the relevant
T K
coalition structures in those sub-games are kT K,
l. That reflects the special
T K
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. Hamiache Mathematical Social Sciences 37 1999 281 –305
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 would
f
achieve if it could catch and add to its own worth in the original game, vT , all the quantities [f T K, v
, 2vK ], the surpluses of the structural coalitions K,
K uT K
T 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 have
o
i [S
f N,v,5vS.
i
Axiom A2 Additivity Property. For all games, N,v, and N,w, in G, fN,v 1 w,5fN,v,1fN,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 RN, all i [R, and all positive parameters c, f N, cu ,
i R
is strictly positive.
Axiom A5 Associated Consistency. For all games with coalition structure N,v,,
fN,v,5fN,v ,.
f
Axiom A6 Symmetry. For all permutations p of N, p [PN , f N, pv,
hNj5f
p i i
N,v, hNj where for all SN, pvS5vpS.
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,
vR5vN , 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 vR, 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, are
R
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
f is such that the players neither lose nor gain when they perform a manipulation of the kind related to the construction of the associated game. As a consequence, given the
socially recognized sharing rule f, there is not any more need for this kind of manipulation. This type of axiom is generally encountered in the context of reduced
games and consistency. Since Axiom A5 requires the existence of a functional equation
G . Hamiache Mathematical Social Sciences 37 1999 281 –305
287
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
hN,v,[N]uN,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 is
w
such that: v
S 5 vS 1
O
[w S h jj, v
2 v h jj].
w j
S h j j
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
v 5
O O
21 v
T u . 2
R R
T RN T R
We shall show that the associated game v can be expressed as the same linear
w
combination of associated unanimity games, u , [±RN.
R w
Let us compute, for all S N,
[R2[T
O O
21 v
T u S 5
R w R
T RN T R
Following the definition of the associated game,
[R2[T
O O
21 v
T u S 1
O
[w S h jj, u 2 u h jj 5
R j
R R
R T
j
F G
RN T R j [N •S
Using Eq. 2,
[R2[T
5 vS 1
O O
21 v
T
O
[w S h jj,u 2 u h jj] 5
j R
R R
T j
RN T R j [N •S
Changing the order of summations,
[R2[T
5 vS 1
O O O
21 v
T [w S h jj, u 2 u h jj] 5
j R
R j
R T
F G
RN T R j [N •S
288 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
For all R such that R ⁄ S
h jj, the restriction of the function u to the set Sh jj is the
R
null function and the term w S h jj, u 50. Then we can rewrite the last expression as,
j R
5 vS 1
O O O
j R
T
F
RS h j j T R
j [N •S [R2[T
[R2[T
21 v
T w S h jj,u 2
O O
21 v
T u h jj
j R
R R
T
G
RN T R
5 vS 1
O
w S h jj, vu
2 v h jj 5 v S,
f g
j S
h j j w
j j [N •S
then,
[R2[T
v 5
O O
21 v
T u .
w R w
R T
RN T R
Using the linearity property of the Shapley value, and changing the order of summations,
[R2[T
wN,v 5
O O
21 wN,u vT .
w R w
T R
T N T RN
Using Eq. 2 and the linearity property of the Shapley value,
[R2[T
wN,v 5
O O
21 wN,u vT .
R T
R T N T RN
Then the equality wN,v 5wN,v is true for all game N,v if and only if for all T N,
w [R2[T
[R2[T
O
21 wN,u 5
O
21 wN,u .
R R w
R R
T RN T RN
By the induction hypothesis these equalities are true if and only if wN,u 5 wN,u .
N N w
So let us compute w N, u for all i [N,
i N w
[R2[T
w N,u 5
O O
21 u T w N,u .
i N w
N w i
R R
T RN T R
i [R
Since u T 5 u T 1
O
[w T h jj, u
2 u h jj],
N w N
j N
u N
T h j j
j j [N •T
after substitution and re-order of the summations,
G . Hamiache Mathematical Social Sciences 37 1999 281 –305
289
[R2[T
w N,u 2 w N,u 5
O O O
21 w T
h jj, u w N,u .
i N w
i N
j N
u i
R
T h j j
j R
T RN T R
j [N i [R j [
⁄ T
If N ±T h jj the function u
is the null function and the term w T h jj, u
N u
j N
u
T h j j
T h j j
cancels. Then, the only relevant value of the variable T is T 5N\
h jj and the two
corresponding values of the variable R are R5N\
h jj and R5N. Let us substitute these values in the last expression.
w N,u 2 w N,u 5
O
w N,u w N,u 2
O
w N,u w N,u
i N w
i N
j N
i N •
h j j j
N i
N j
j j [N •
hi j j [N
[N 2 1 1
1 ]]] ]]]
] 5
2 5 0.
[N [N 2 1
[N The last expression cancels, then wN,v5wN,v for all N,v[G, completing the proof
w
of Lemma 1. h
Theorem 2. The Owen value satisfies axioms A
1 – A6.
Proof. Given a game with coalition structure N,v,, let us consider its intermediate
game IN ,v ,[IN ] and the intermediate game of the associated game IN ,v
c
,[IN]. We shall show that v 5v . Let us compute for all T IN the
c w
expression v T .
w
v T 5 v T 1
O
[w T h jj, v
2 v h jj].
w j
u
T h j j
j j [IN •T
Let M denote the set of players represented by T, M 5 B N. Using the
i [T i
intermediate game property of the Owen value, v T 5 vM 1
O
[c M K, v ,
2 vK ] 5 v M
w K
u M K
c
M K
K K [
N •M
5 v T ,
c
thus, wIN,v 5 wIN, v .
c w
Since, by Lemma 1, the Shapley value satisfies the associated consistency property, wIN, v 5 wIN, v ,
w
and then wIN, v 5 wIN, v .
c
Using once more the intermediate game property of the Owen value,
290 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
c N,v, 5 c 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 A 1 – A6.
Proof. Let f be a solution that satisfies axioms A1–A6 and is different from the Owen value. From the efficiency axiom, f
hij, v, hhijj5vhij5chij, 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
fN,v,5cN,v,. By use of Eq. 2 and Axiom A2,
[R2[T
fN,v, 5
O O
21 fN,vT u , .
R R
T RN T R
Changing the order of summations,
[R2[T
fN,v, 5
O O
21 fN,vT 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 i [R,
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,vT 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 and f 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,vT 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
subsets R of N, and all i [R. Then, combining Eqs. 3 and 4,
G . Hamiache Mathematical Social Sciences 37 1999 281 –305
291
[N2[T
f N,v, 2 f N,v , 5
O
21 [f N,vT u ,
i i
f i
N T
T N
2 f N,vT u , ].
i N f
From the last equation, we deduce that f 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 v 5cu and using Eq. 1 we get:
N f
fN,cu , 5 fN,cu , 1
O O
N f N
T R
[,T ,N T R [R2[T
21 f
N,
O
[f T K,cu ,
K N
u T K
T K
K
S D
K [
N •T
2 cu K]u , .
N R
Using the consistency axiom for cu , fN,cu , 5fN,cu , ,
N N f
N [R2[T
O O
21 f
N,
O
[f T K,cu ,
2 cu K]u ,
K N
u T K
N R
T K
T R
K
S D
[,T ,N T R K [
N •T
5 0. Notice that for all T such that T K ±N, cu
is the null function on the set T K j
N u
T K
and f T K, cu ,
50. Then, we are only concerned with T satisfying
K N
u T K
T K
T K 5N, i.e.
[IN\T 51 and K5N\T. Note that in this case, for all T ±[, u N\T 5
N
0. By Axioms A1 and A3, f R, cu , 50 for all i [ ⁄ R,
i R
R [R2[T
O O
21 f R,f
N,cu , u , 5 0, ;i [ N.
5
i N •T
N R
R T
R [,T ,N
T R [IN •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 , 5 0.
i j
N R
R T
R j
S D
[,T ,N T R
j [N •T [IN •T 51 i[R
Changing the order of the summations,
[R2[T
O O O
21 f R,f N,cu , u , 5 0.
i j
N R
R j
T R
[,T ,N T R
j [N [IN •T 51 i[R
j [ ⁄ T
So,
292 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
[R2[T
O O O
21 f R,f N,cu , u , 5 0.
6
i j
N R
R j
R T
i [R T R
j [N [IN •R 1 [IN•T 51
j [ ⁄ T
[,T ,N
It is convenient, at this point, to split the computations into the following four cases: 1 [IN51; 2 [IN5[N; 3 r2 and there is no singleton structural coalition; and
4 r 2 and we have both singleton and non-singleton structural coalitions.
Case 1.
[IN51. Note that we assume that N has at least two elements. In this case Eq. 6 becomes,
[R2[T
O O O
21 f R,f N,cu ,B u ,B 5 0,
;i [ N. 7
i j
N R
R j
R T
RN 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 j 5i.
[N2[T
0 5
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 • hi j
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 if L 5 [,
[T t
O
21 5
O
21 5
.
S D H
t 2 1
otherwise
T t 51
[,T L
We perform the three summations over T in Eqs. 8a–8c, when L 5N\
h jj in the term
of Eq. 8a, L 5R\ h jj in the term of Eq. 8b, and L5R\hij in the term of Eq. 8c,
[N
0 5 2
O
21 f N,f N,cu , u ,B 2
9a
i j
N N
j j [N
G . Hamiache Mathematical Social Sciences 37 1999 281 –305
293
[R
2
O O
21 f R,f N,cu , u , 2
9b
i j
N R
R j
R R,N
j [N • hi j
i [R [R
2
O
21 f R,f N,cu , u , .
9c
i i
N R
R R
R,N i [R
R± hi j
Using Axioms A1 and A2, we simplify Eq. 9a. Dissociating the case R5 hij in the term
of Eq. 9c, that is f hij, f N, cu , u , and using the efficiency axiom for the
i i
N hi j
hi j
game hij, u , ,
hi j hi j
[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 • hi j
i [R [R
2
O
21 f R,f N,cu , u ,
i i
N R
R R
3
R,N i [R
2 21 f N,cu , .
i N
4
Gathering the relevant terms,
[N [R
[1 1 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
[1 1 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 value f coincides with the Owen value for all t-player games, t ,
[N,
[N22
c
[N t 11
[N 2 1 ]]
[1 1 21 ]f N,cu ,B 5 2
O
21
S D
i N
t t 1 1
t 50 [N21
[N21
c 21
c
t
[N 2 1 ]]
]]] 5
O
21
S D
2 .
t t 1 1
[N
t 50
Lemma 2. For all integers n and x it holds that,
n
1 1
n
t
]] ]]]]]
S D
O
21 5
. t
n 1 x x 1 t
t 50
n 1 1
S D
n 1 1
294 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
Proof. See Appendix A.
Then for all i in N,
[N
c 21
c c
[N [N
] ]]]
] [1 1 21
]f N,cu , 5 1
5 [1 1 21 ]
.
i N
[N [N
[N When
[N is an even number f N, cu , 5c[N for all i in N.
i N
But, when the number of players is odd and [IN51, 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 same
N
payment c [N. And the value f coincides in this case with the Owen value.
Case 2.
[IN5[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\
h jj whenever i±j. In both cases the
only available T is T 5N\
h jj.
O
f N• h jj,f N,cu , u
, 5
O
f N,f N,cu , u , .
i j
N N •
h j j N •
h j j 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 , 5 f N,cu , .
j N
i N
N
[N 2 1
j j [N
i ±j
By use of the efficiency axiom, 1
]]][c 2 f N,cu , ] 5 f N,cu , ,
i N
i N
N
[N 2 1 then,
c ]
f N,cu , 5 5
c N,cu , ,
i N
i N
[N which proves the theorem in Case 2.
Case 3. r 2 and ;t [
h1, 2, . . . , rj, [B .1.
t
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 N\R, respectively,
G . Hamiache Mathematical Social Sciences 37 1999 281 –305
295
[T2[R
O O O
21 f N•R,f N,cu , u
, 5 0
i j
N N •R
N •R j
R T
i [ ⁄ R
RT j [N
[IR 1 [IT 51 j [T
] ] ]
] .We distinguish between three cases, [ j 5i], [ j ±i and
h jj5hij], and [h jj ±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 RT
[IR 1 [IT 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 RT
j [ hi j•hi j
[IR 1 [IT 51 j [T
[T2[R
1
O O O
21 f N•R,f N,cu , u
, 5 0.
i j
N N •R
N •R j
R T
] i [
⁄ R RT
j [N • hi j
[IR 1 [IT 51 j [T
11c If, in the term of Eq. 11a,
[IR50, that is R5[, then the summation over T cancels. ]
And if [IR51 the summation over T does not cancel only if R 5hij•hij, in that case
] the only available T is T 5
hij. 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 R 5 h jj or if R 5h jj•h jj. In both cases the only available T is
] ]]
] T 5
h jj. Gathering all those results and using the notation, h 2 ij 5 N•hij, ]]
] ]
0 5 21 f h 2 ij hij,f N,cu , u
, 1
i i
N h2i jhi j
h2i jhi j
]]
] ]
1
O
f h 2 jj,f N,cu , u
, 2
i j
N h2j j
h2j j j
] i [
h2j j
]]
] ]
2
O
f h 2 jj h jj,f N,cu , u
, .
i j
N h2j jh j j
h2j jh j j j
] i [
h2j j
] ]]
]] Since for all j [N,
[h jj±1, the sets h 2 jj and h 2 jj h jj are strict subsets of the grand coalition N. Then by use of the induction hypothesis for games with less than
[N players
296 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
21 1
1 ]]
]] ]] 0 5
f N,cu ,B 1
O
f N,cu , ]
i N
j N
r r 2 1
[hij
j ]
i [ h2j j
1 1
] ]] 2
O
f N,cu , . ]
j N
r [hij
j ]
i [ h2j j
Gathering the relevant terms, 1
1 ]] ]]
f N,cu ,B 5
O
f N,cu , . ]
i N
j N
r 2 1 [hij
j ]
i [ h2j j
We learn from the above expression that for any k [ h1, . . . , rj the value f N,u ,
B N
k
is distributed equally among the players of B . Thus we need only to find f N, u ,
k B
N
k
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
] ]
]] f N,cu ,B 5
[c 2 f N,cu ,B ],
hi j N
hi j N
r 2 1 which leads to,
c
]
] f N,cu , 5 .
hi j N
r
]
Since for all i [N the value f N, cu , is shared equally among the players in
hi j N
] structural coalitions
hij, we get c
1 ] ]]
f N,cu ,B 5 5
c N,cu ,B, ]
i N
i N
r [hij
which proves Theorem 2 in Case 3.
Case 4. r 2 and 1p ,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
0 5
O O
21 f N,f N,cu , u
, 1 12a
i i
N N •R
R T
] RT
R hi j
] i [
⁄ R T
hi j i [T
G . Hamiache Mathematical Social Sciences 37 1999 281 –305
297
[T2[R
1
O O O
21 f N,f N,cu , u
, 1 12b
i j
N N •R
j R
T ]
RT j [N R
hi j ]
] i [
⁄ R T
hi j j [
hi j j [T
j ±i [T2[R
1
O O O
21 f N,f N,cu , u
, 5 0. 12c
i j
N N •R
j R
T ]
RT j [N R
h j j ]
] i [
⁄ R T
h j j j [
⁄
hi j 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 T 5
hij. 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 ]
RT j [N R
h j j ]
j ±i T
h j j j [T
] ]
We distinguish between [h jj51 and [ h jj±1.
f N,f N,cu , u , 5 13a
i i
N N
[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
] RT
j [N R
h j j ]
] T
h j j [h j j51
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
] RT
j [N R
h j j ]
] T
h j j [h j j±1
j [T j ±i
] In the term of Eq. 13b the only two available values of R are R5[ and R5
h jj, in both ]
] ]
cases T 5 h jj. The term of Eq. 13c does not cancel only if R 5h jj or R 5h jj•h jj, in
] ]
both cases T 5 h jj. Together with the fact that i±j if [h jj±1 Eqs. 13a–13c can be
rewritten as, f N,f N,cu ,Bu ,B 5 2
O
f N,f N,cu , u , 1 14a
i i
N N
i j
N N
j j [N
] [h j j51
j ±i
298 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
]]
] ]
O
f h 2 jj,f N,cu , u
, 2
14b
i j
N h2j j
h2j j j
j [N ]
[h j j51 j ±i
]]
] ]
2
O
f h 2 jj h jj,f N,cu , u
, 1
14c
i j
N h2j jh j j
h2j jh j j j
j [N ]
[h j j±1
]]
] ]
1
O
f h 2 jj,f N,cu ,Bu
,B .
14d
i j
N h2j j
h2j j j
j [N ]
[h j j±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
] [h j j51
1 1
]] ]]
5
O
f N,cu , 2 f N,cu , 2
15b
j N
i N
r 2 1 r 2 1
j j [N
] [h j j51
1 1
] ]]
2
O
f N,cu , 1
O
f N,cu , . 15c
j N
j N
r r 2 1
j j
j [N j [N
] ]
[h j j±1 [h j j±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 r 21. 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
r 2 1
j j [N
] [h j j51
1 1
]] ]
2 f N,cu , 2
c 2
O
f N,cu , . 16b
i N
j N
r 2 1 r
j j [N
3 4
] [h j j51
] Summing the above expression for all i such that
[hij 5 1 we get:
G . Hamiache Mathematical Social Sciences 37 1999 281 –305
299
pc p
]]] ]
F
O
f N,
O
f N,cu , u , 5 1
i j
N N
r rr 2 1
i j
i [N j [N
1 2
] ]
[hi j51 [h j j51
1 ]]
G
2
O
f N,cu , .
j N
r 2 1
j j [N
] [h j j51
The last expression is an iterative functional equation of order two, px
p 1
]]] ]
]]
F G
jjx 5 1
2 jx
for jx 5
O
f N,xu , , 17
j N
r r 2 1
rr 2 1
j j [N
] [h j j51
of the general form, jjx 5 ax 1 bjx,
18 where a ±0 and b ±0 since r and p are integers such that r 2 and 1p ,r. The
properties of the function j, that are deduced from the properties of the function f, are: •
j050; •
jx has the sign of x; •
jx1y5jx1j y; •
jx,x for all x.0.
Lemma 3. j x5 px 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 result jc5pc r and the additivity axiom,
p c
1 1
pc ]
]] ]]
] ]
f N,cu , 5 2
f N,cu , 2
F
c 2
G
,
i N
i N
r r 2 1
r 2 1 r
r which leads to
c ]
] f N,cu ,B 5
5 c N,cu ,B for all i [ N such that
hij 5 hij.
i N
i N
r ]
Case 4b. Let i be a player such that
[hij±1. ]
For all R, in the term of Eq. 12a, which is different from hij•hij the summation over
] ]
T cancels. When R 5 hij•hij, the only possible value of T is T 5hij.Since in the term of
300 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
] Eq. 12b the set R
hij is different from hij the summation over T always cancels. Then Eqs. 12a–12c can be rewritten as
]]
] ]
f h 2 ij hij,f N,cu , u
, 5
i i
N h2i jhi j
h2i jhi j [T2[R
5
O O O
21 f N•R,f N,cu , u
, .
i j
N N •R
N •R j
R T
] ]
RT R
h j j j [
h2i j ]
T h j j
j [T
] ]
We distinguish between [h jj 5 1 and [h jj ± 1,
]]
] ]
f h 2 ij hij,
,f N,cu , u 5
19a
i h2i jhi j
i N
h2i jhi j [T2[R
5
O O O
21 f N•R,f N,cu , u
, 1
19b
i j
N N •R
N •R j
R T
] ]
RT R
h j j j [
h2i j ]
] T
h j j [h j j51
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
] ]
RT R
h j j j [
h2i j ]
] T
h j j [h j j±1
j [T
] In the term of Eq. 19b, the only two available values of R are R5[ and R 5
h jj, in ]
] ]
both cases T 5 h jj. The term of Eq. 19c does not cancel only if R 5h jj or R 5h jj•h jj,
] in both cases T 5
h jj. Using Axiom 3, Eqs. 19a–19c can be rewritten as, ]]
] ]
f h 2 ij hij,f N,cu , u
, 5
i i
N h2i jhi j
h2i jhi j 1
5
O
21 f N,f N,cu , u , 1
i j
N N
j ]
j [ h2i j
] [h j j51
]]
] ]
1
O
f h 2 jj, f N,cu , u
, 1
i j
N h2j j
h2j j j
] j [
h2i j ]
[h j j51
]]
1 ]
]
1
O
21 f h 2 jj h jj,f N,cu , u
, 1
i j
N h2j jh j j
h2j jh j j j
] j [
h2i j ]
[h j j±1
]]
] ]
1
O
f h 2 jj,f N,cu , u
, .
i j
N h2j j
h2j j j
] j [
h2i j ]
[h j j±1
G . Hamiache Mathematical Social Sciences 37 1999 281 –305
301
Using the induction hypothesis for games with less than [N players and the additivity
axiom, 1
]f N,cu ,B 5 2 f N,
O
f N,cu , u , 1
i N
i j
N N
r
j j [N
1 2
] [h j j51
1 1
1 1
]] ]] ] ]]
1
O
f N,cu ,B 2
O
f N,cu , 1 ]
]
j N
j N
r 2 1 r
[hij [hij
j j
] j [N
j [ h2i j
] ]
[h j j51 [h j j±1
1 1
]] ]] 1
O
f N,cu , . ]
j N
r 2 1 [hij
j ]
j [ h2i j
] [h j j±1
Using the fact that jc5pc 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
r 2 1 r
[hij 1
1
]
]]] ]] 1
O
f N,cu , 2 f N,cu , . ]
j N
hi j N
rr 2 1 [hij
j j [N
3 4
] [h j j±1
Using the efficiency, and once more the result jc5pc r, p 1 1
1 1
pc ]]
]] ]] ] f N,cu , 5
]
i N
r r 2 1
r [hij
1 1
pc
]
]]] ]] ]
1
F
c 2 2
f N,cu ,
G
]
hi j N
r rr 2 1
[hij p 1 1
1 1
c 1
pr 1 r 2 p
]
]] ]]] ]]
] ]] ]]]
f N,cu , 1 f N,cu ,B 5
. ]
]
F G
i N
hi j N
r r
rr 2 1 rr 2 1
[hij [hij
] The above equation proves that all the players of the structural coalition
hij are equally
]
treated, that is, the value f N, cu , is shared equally among the players of the
hi j 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 ,
hi j N
r then,
c 1
] ]] f N,cu , 5
5 c N,cu , ,
]
i N
i N
r [hij
302 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
which completes the proof of Theorem 3. h
4. Conclusion
