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
In this section we will model the process that leads to the formation of a conference structure as a game in strategic form. The game is a generalization of the linking game
as formulated in Myerson 1991 and discussed in Qin 1996 and Dutta et al. 1998.
M . Slikker et al. Mathematical Social Sciences 39 2000 55 –70
65
We will show that the only component efficient allocation rules that result in a weighted potential game are the weighted extended Myerson values.
Let g be an allocation rule and N, v a cooperative game. Define the conference
g
formation game GN, v, g determined by the tuple N; S
; f where for all
i i [N i
i [N
i [ N
N \
hi j
S : 5 hTuT 7 2
j
i
represents the strategy set of player i . A strategy of player i denotes the set of coalitions player i wants to join to form conferences. A strategy profile s 5 s , . . . , s [ P
S ,
1 n
i [N i
induces a set of conferences s given by s: 5
hHu uHu 2; H\hij [ s , i [ Hj.
i
The interpretation is that a conference is formed if and only if all players in this
g g
conference are willing to form it. The payoff function f 5 f is then defined as the
i i [N
allocation rule applied to the conference structure formed,
g
f s 5 gs.
In case there is no ambiguity about the underlying cooperative game we will simply write G
g instead of GN, v, g . In the remainder we will consider an arbitrary game N, v.
In order to prove that weighted extended Myerson values are the only allocation rules that are component efficient and that lead to conference formation games which are
weighted potential games, we need two lemmas.
N
Lemma 4.1. Let g be an allocation rule and w [ R
. If the conference formation
11
game G g is a w-potential game, then for all hypergraphs N, , all H 7 N and all
i, j [ H 1
1 ]
]
g 2 g \
hHj 5
g 2 g \
hHj. 9
i i
j j
w w
i j
w
Proof. Since G g is a w-potential game, Gg has a w-potential P . We will show that
g satisfies equation 9.
N
Let N, be a hypergraph, so N, v, [ HCS . Define for all k [ N, s : 5
hH\hkjuH [ , k [ Hj.
k
Then it holds that s 5 . Let H [ and i [ H, then for all j [ H\
hij we get
w w
w
P s \ hH\hijj, s , s 5 P s \hH\hijj, s \hH\h jjj, s 5 P s , s \hH\h jjj, s ,
i j
2ij i
j 2ij
i j
2ij
since the three strategy tuples all result in the formation of the same conferences, the conferences in \
hHj, and hence, they all result in the same payoffs. Using this we find for all i, j [ H
66 M
. Slikker et al. Mathematical Social Sciences 39 2000 55 –70
1 1
g g
] ]
g 2 g \
hHj 5
f s 2 f s \ hH\hijj, s
i i
i i
i 2i
w w
i i
w w
5 P s 2 P s \ hH\hijj, s
i 2i
w w
5 P s 2 P s \ hH\h jjj, s
j 2j
1
g g
] 5
f s 2 f s \ hH\h jjj, s
j j
j 2j
w
j
1 ]
5 g 2 g \
hHj.
j j
w
j
This completes the proof. h
The following lemma shows that the conference formation game corresponding to an arbitrary cooperative game with a weighted extended Myerson value used as an
allocation rule is a weighted potential game.
w
Lemma 4.2. The conference formation game G
m is a w-potential game.
Proof. Consider the following set of cooperative games, indexed by the set of strategy
w s
s
profiles of G m ,
hN, v j
. Let R 7 N and s 5 s , s [ S. Since v
R 5
s [S R
N \R
s
o v
C and R s does not depend on s it follows that v
R does not
C [R s N \R
w
s m
w
depend on s . This implies that
hN, v j
[ . Since f
s 5 m s 5
N \R s [S
N,S i
i w
s w
F v by definition, it follows by Theorem 2.1 that G
m is a w-potential game. h
i
In Qin 1996 it is shown that there is a unique allocation rule that is component efficient and results in a link formation game with a potential in the restricted case where
only bilateral communication between the players is possible, i.e. uHu 5 2 for all H [ .
If we combine the results of the lemmas above we can extend the result of Qin 1996 in two directions. First, we consider hypergraphs and not only graphs, and second, we
allow for asymmetric players.
Theorem 4.1. Let g be a solution concept that is component efficient. The conference
formation game G g is a weighted potential game if and only if g coincides with a
weighted extended Myerson value for all hypergraphs N, .
Proof. Suppose that the conference formation game G g is a weighted potential game.
From Lemma 4.1 it follows that there exist weights w for which g satisfies equation 9.
Since g is component efficient, it then follows, by the proof of Theorem 3.1, that g
w
coincides with m for all hypergraphs N, .
The reverse statement follows by Lemma 4.2. h
M . Slikker et al. Mathematical Social Sciences 39 2000 55 –70
67
5. Potential maximizing strategies
