282 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
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 →
R and v[ 5 0. We denote by G 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, vQ 5 vT 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, vS 5 vS
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, v 5 cu where u is defined for all S N by u S 5 1
R R
R
if R S and u S 5 0 if R u S.
R
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 5 hB , B , . . . , B j with 1r[N. In
1 2
r
the following we shall also consider the coalition structure kN, [N]l in which all the
members of the partition [N] are singletons, [N]5 hhijui[Nj.
In the following, we shall have to deal with two types of coalitions, those that are members of a coalition structure and those that are subsets of players but are not
necessarily members of . To avoid confusion, where it may occur, we shall use the term structural coalition for members of a coalition structure.
G . Hamiache Mathematical Social Sciences 37 1999 281 –305
283
Definition. Let
kN, l be a coalition structure and S a subset of N. We denote by I
S the index set of S where I S 5
hiuB S±[, B [j.
kN,l kN,l
i i
With this notation the number of structural coalitions in is [I
N . When no
kN,l
confusion may occur, we shall denote the index set of S by IS . ]
Given a coalition structure kN,Bl and a subset S of N, we denote by S the set,
] S 5
B .
i i
i [IS
] In words, S is the minimal union of structural coalitions which covers S. In particular,
] we denote B
, the structural coalition which contains player i, by hij.
I hi j
Definition. Let
kN,l be a coalition structure and S a subset of N. We define kS, l, the
S
induced coalition structure for S, to be a coalition structure for S in which 5 hS
S
B ui[IS, B [j.
i i
Definition. A permutation p on N is a one-to-one mapping on N, p :N
→ N. We denote
by PN the set of all the permutations on N.
Definition. We say that a permutation p is consistent with the coalition structure
kN,l if the following condition is fulfilled. For all B [, all i and j [B and all l [N if
pi,pl,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 SN, the set of all the permutations in PN which
are consistent with the coalition structure kN,l. As a consequence,
[SN, 5 [IN ? [B ? ? ? ? ? [B
1 [IN
Definition. A game with coalition structure is a triplet N,v, where N is a coalition, N,v is a game and
kN,l is 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 hij 5h jj are substitute if for all coalitions
S N\
hi, jj, vShij5vSh jj.
Definition. Given a game with coalition structure N,v,[G, the intermediate game is the game with coalition structure IN , v , [IN ] where v is defined so that for
every set T, T IN ,
v T 5 v
B .
i
S D
i i [T
Remark. We defined I B as a set. But when B[, IB can be considered as a single
player in the intermediate game.
284 G
. Hamiache Mathematical Social Sciences 37 1999 281 –305
Definition. A solution on G is a function s which associates each game with coalition
[N
structure N,v,[G, a vector sN,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, 5 vN.
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 c N,v,50.
i
Axiom P4 Additivity Property. For any two games N,v , and N,v , in G,
1 2
cN,v 1 v , 5 cN,v , 1 cN,v , ,
1 2
1 2
where for all subsets S of N, v 1v S 5v S 1v S .
1 2
1 2
Axiom P5 Intermediate Game Property. For every game with coalitional structure N,v, and every structural coalition B [,
O
c N,v, 5 c IN ,v , [IN ].
j IB
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 P
1 –P5. This solution coincides with the Owen value, 1
i i
]]] c N,v, 5
O
[vP hij 2 vP ], ;i [ N.
i p
p
[SN,
p p [SN,
i
Where P 5 hj[Nup j,pij is the set of all the players of N preceding player i in a
p
permutation p , p[PN.
Remark. In the two cases
[IN51 and [IN5[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, 5PN.
Owen 1977 proposed a heuristic two-stage construction of his value. In the first stage, the intermediate game is played and gives to each structural coalition its Shapley
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
