M . Slikker et al. Mathematical Social Sciences 39 2000 55 –70
61
s
w s
a
R
]
Q s 5 o
for all s [ S, where a
are the unanimity coordinates of
R7N,R±5 R R7N,s [S
w
R
s w
7
hN, v j [
for which p s 5 F v for all i [ N and all s [ S.
s [S N,S
i i
s
3. Networks
In this section we will first introduce hypergraphs. After that we will discuss hypergraph communication situations and characterize a class of allocation rules for
these situations. A hypergraph is a pair N, with N the player set and a family of subsets of N.
An element H [ is called a conference. The interpretation of a hypergraph is as follows: communication between players in a hypergraph can only take place within a
conference. Furthermore, communication via this conference cannot take place between a proper subset of this conference, i.e. all players of the conference have to participate in
the communication. Note that a hypergraph is a generalization of a graph, which consists only of conferences with exactly two players.
Next, we consider hypergraph communication situations, first introduced in Myerson 1980. Formally, a hypergraph communication situation is a triple N, v, , where
N, v is a cooperative game and N, a hypergraph. By assuming that every player can communicate with himself we can restrict our attention to hypergraphs N, with
N
7 hH [ 2 uuHu 2j. We will denote the class of all these hypergraph communication
N
situations with player set N by HCS . In a hypergraph communication situation a coalition S 7 N can effect communication
in conferences in S : 5 hH [ uH 7 Sj. Further we define interaction sets of
S, S : 1. every
hij 7 S is an interaction set. 2. every H [ S is an interaction set.
3. if T and T are interaction sets with T T ± 5, then T T is an interaction set.
1 2
1 2
1 2
A set T 7 S is a maximal interaction set of S if T is an interaction set of S, S and there exists no interaction set T 9 of S, S with T , T 9. Following the tradition set in
earlier papers we will also refer to maximal interaction sets as components. We will denote the resulting partition of S in maximal interaction sets by S .
Conform this partition we define the value of coalition S 7 N in N, v, by v
S : 5
O
v C .
C [S
We call N, v the hypergraph-restricted game. An allocation rule g is a function that
N N
assigns to every N, v, [ HCS an element of R . If there is no ambiguity about the
game N, v we will write g instead of gN, v, . For a positive weight-vector
N w
w 5 w [
R the weighted
extended Myerson value, m , is the allocation rule
i i [N 11
7 w
w w
It holds that Q s 5 P N, v , where P denotes the weighted potential for cooperative games as used in the
s
characterization of the weighted Shapley values in Hart and Mas-Colell 1989.
62 M
. Slikker et al. Mathematical Social Sciences 39 2000 55 –70
which assigns to every N, v, the w-weighted Shapley value of the hypergraph- restricted game N, v ,
w w
m N, v, : 5 F N, v . We will characterize the w-weighted extended Myerson value by two properties,
component efficiency and w-fairness. Consider for an allocation rule g these two
properties:
Component efficiency: For all hypergraph communication situations N, v, [
N
HCS it holds for all C [ N :
O
g 5 vC.
i i [C
N
w-Fairness: For all N, v, [ HCS , all H 7 N and all i, j [ H
1 1
] ]
g 2 g \
hHj 5
g 2 g \
hHj.
i i
j j
w w
i j
Component efficiency states that the players in a maximal interaction set divide the value vC amongst themselves. The property w-fairness is an extension of the fairness
property of Myerson 1980. In Myerson 1980 the extended Myerson value is characterized by the properties component efficiency and fairness. The weights represent
the strength of the players: the changes in payoffs for two players as a consequence of forming an additional conference in which they are both involved are proportional to
their weights.
The following lemma shows that the w-weighted extended Myerson value satisfies the two properties component efficiency and w-fairness. In the proof we use some results
of Kalai and Samet 1988. It is shown that the w-weighted Shapley value satisfies the dummy property, additivity, and partnership consistency. The dummy property states
w
that F N,v 5 v
hij for all N,v with vS hij 5 vS 1 vhij for all S 7 N\hij.
i w
w w
Additivity states that F N, v 1 z 5 F N, v 1 F N, z for all cooperative games N, v
and N, z. To describe partnership consistency we need the notion of partnership. A
coalition S 7 N is a partnership in N, v if for all T , S and all R 7 N\S, vR T 5
w
v R. Partnership consistency of F states that for every partnership S in N, v it holds
that
w w
w
F v 5 F F v u , for every i [ S,
s d
i i
S S
w w
where F v 5 o
F v.
S j [S
j w
Lemma 3.1. The w-weighted extended Myerson value, m , satisfies component
efficiency and w-fairness .
w N
Proof. First we will show that m satisfies component efficiency. Let N, v, [ HCS
C N \C
and C a maximal interaction set of N, . We define two games N, v and N, v .
For all T 7 N let
M . Slikker et al. Mathematical Social Sciences 39 2000 55 –70
63
C
v T : 5 v T C ,
N \C
v
T : 5 v T \C .
C N \C
Since C is a maximal interaction set of N, it holds that v 5 v 1 v . Since all
N \C
i [ C are dummy players in the game N, v , we conclude from the dummy player
w N \C
property of the w-weighted Shapley value, that F v 5 0 for all i [ C. In the same
i w
C
way we find for all i [ N\C that F v 5 0. Using this and the additivity of the
i
w-weighted Shapley values we find
w w
C w
N \C
O
F v 5
O
F v 1
O
F v
i i
i i [C
i [C i [C
w C
w C
C
5
O
F v 5
O
F v 5 v N 5 v C 5 vC ,
i i
i [C i [N
where the fourth equality follows from the efficiency of the w-weighted Shapley value. Secondly, we will show that the w-weighted extended Myerson value satisfies
N 9
w-fairness. Let N, v, [ HCS and H [ . Define 9: 5 \
hHj and v9: 5 v 2 v . For all T 7 N with H≠T we then have
v 9T 5
O
v R 2
O
v R 5 0
R[T R[T 9
since T 5 T 9. This means that H is a partnership in v9. From partnership
w
consistency of F , it follows for all i [ H that w
i w
w w
w
]] F v9 5 F
O
F v9 u 5
O
F v9
i i
j H
j
SS D D
S D
j [H j [H
O
w
j j [H
So, for all i, j [ H
w w
F v9 F v9
j i
]] ]]
5 .
w w
i j
From this we find
w w
w w
w w
F v9 m 2 m 9
m 2 m 9 F v9
j j
j i
i i
]]]]] ]]
]] ]]]]]
5 5
5 ,
w w
w w
i i
j j
where the first and third equalities follow from the definition of the game N, v9 and the
w
additivity of the w-weighted Shapley values. Hence, m satisfies w-fairness.
h The following theorem shows that the w-weighted extended Myerson value is the
unique rule that is component efficient and w-fair.
w
Theorem 3.1. The w-weighted extended Myerson value m
is the unique rule that satisfies component efficiency and w-fairness
.
w
Proof. From Lemma 3.1 we know that m satisfies component efficiency and w-fairness.
64 M
. Slikker et al. Mathematical Social Sciences 39 2000 55 –70
w
We only need to show here that m is the unique solution concept which satisfies these
properties.
1 2
Suppose there are two rules g
and g
which satisfy component efficiency and w-fairness. Let N, v, be a communication situation with a minimum number of
1 2
conferences such that g ± g . By component efficiency it follows that ± 5.
1
Let H [ and hi, jj 7 H. From w-fairness of g we then find
1 1
1 1
1 1
] ]
g 2 g \
hHj 5
g 2 g \
hHj .
s d
s d
i i
j j
w w
i j
2
Using this, the minimality of , and the w-fairness of g respectively, we find
1 1
1 1
w
g 2 w g 5 w g \ hHj 2 w g \hHj
j i
i j
j i
i j
2 2
5 w g \
hHj 2 w g \hHj
j i
i j
2 2
5 w g 2 w g .
j i
i j
So
1 2
1 2
g 2 g g 2 g
j j
i i
]]]]] ]]]]]
5 .
w w
i j
This expression is valid for all pairs hi, jj for which there exists an H [ with
hi, jj 7 H. Hence, it is also valid for all pairs hs, tj that are in the same maximal interaction set.
Let C [ N and i [ C. For all j [ C we now have 1
1
1 2
1 2
] ]
g 2 g 5 g 2 g .
s d
s d
j j
i i
w w
j i
1 2
1 2
1 ]
Let d: 5 g 2 g . Then for all j [ C : g 2 g 5 w d. Component
s d
i i
j j
j w
i
1 2
efficiency of g and g gives us
1 2
O
g 5
O
g 5 vC.
j j
j [C j [C
Thus,
1 2
0 5
O
g 2 g 5
O
w d.
s d
j j
j j [C
j [C N
Since w [ R
it follows that d 5 0. Since C was chosen arbitrarily, we conclude that
11 1
2
g 5 g . h
4. Network formation
