M . Slikker et al. Mathematical Social Sciences 39 2000 55 –70
67
5. Potential maximizing strategies
In this section we will consider potential maximizing strategies in the conference
w
formation game G m . Throughout this section we will assume that the underlying
cooperative game N, v is superadditive, i.e. vR T vR 1 vT , for all disjoint R, T 7 N. We will show that the strategy resulting in the complete conference structure, the
structure with all subsets of players in the set of conferences, is a potential maximizing strategy. Furthermore, we will show that all potential maximizing strategies result in the
same payoffs as the strategy corresponding to the full cooperation structure.
First we need some notation to denote the structures that will result according to the conference formation game with a weighted extended Myerson value used as
N \
hi j
] ]
] ]
allocation rule. Let s 5 s , . . . ,s be the strategy tuple with s 5 2 for all i [ N.
1 n
i
This strategy implies that player i is willing to cooperate with all subsets of the other ]
] players. The corresponding set of conferences will be denoted by : 5 s 5
hT [
N
2 uuTu 2j. A set of conferences is called essentially complete with regard to solution
] ]
concept g iff and are payoff-equivalent, i.e. g 5 g . To facilitate the proof
of the main theorem in this section we will first prove two lemmas. The first lemma states that a player is never worse off forming an additional conference, extending a
result of Myerson 1977.
N N
Lemma 5.1. Let N, v, [ HCS , H 7 N and w [
R . For all i [ H it holds that
11 w
w
m hHj m .
i i
hH j
Proof. Let v9: 5 v 2 v . The superadditivity of v implies that v9R 0 for all
R 7 N since every maximal interaction set in N, hHjR is the union of one or
more maximal interaction sets in N, R. For all R with H≠R it follows that
hH j
v R 5 v R and hence v9R 5 0.
Let i [ H. Since v9R 5 0 for all R 7 N\
hij we have v
9R
hij v9R, for all R 7 N\hij.
10
w N \
hi j
From Weber 1988 it follows that there exists a probability distribution p on 2
i
such that
w w
F v9 5
O
p Rv9R hij 2 v9R.
11
i i
R7N \
hi j H
w
Combining equations 10 and 11 completes the proof since v9 5 v 2 v
and F satisfies additivity.
h The following lemma considers a specific deviation of a player, say i. If player i
deviates to a strategy which is a superset of his original strategy and this deviation does not influence his payoff, then the payoffs of all the other players remain unchanged as
well.
68 M
. Slikker et al. Mathematical Social Sciences 39 2000 55 –70
N
9
Lemma 5.2. Let w [ R
. Then for all i [ N, all s [
S , and all s , s [ S with
11 2i
2i i
i i
w w
w w
m m
m m
9 9
9
s 7 s , it holds that if f s , s
5 f s , s
then f s , s
5 f s , s
.
i i
i i
2i i
i 2i
i 2i
i 2i
9
Proof. If s , s 5 s , s
, then the statement in the theorem is obviously true.
i 2i
i 2i
N
9
Otherwise, since s 7 s there exist k [ N and H , . . . ,H [ 2
with i [ H for all
i i
1 k
j
9 9
j [ h1, . . . kj such that s , s 5 s , s hH , . . . ,H j. Define : 5 s , s
i 2i
i 2i
1 k
i 2i
and for all j [ h1, . . . ,kj
9
: 5 s , s hH , . . . ,H j.
j i
2i 1
j w
w w
w
Since m 5 m , it follows from lemma 5.1 that m
5 m for all
i i
k i
j 21 i
j
j [ h1, . . . ,kj.
w w
j j 21
For j [ h1, . . . , kj, define v9: 5 v 2 v
. Since m 5 m
it follows
i j
i j 21
from 10 and 11 that v
9R
hij 5 v9R, for all R 7 N\hij,
12
w 8
since p R . 0 for all R 7 N\ hij. Consider an arbitrary l [ N\hij and S 7 N\hlj. Using
i
equation 12 and the fact that v9T 5 0 for every T with H ÆT we have
j
v 9R
hlj 5 v9R hlj\hij 5 0
13 and
v
9R 5 v9R\
hij 5 0. 14
w
It follows that F v9 5 0 and hence, by the additivity of the weighted Shapley values
l
that
w w
w w
j j 21
m 5 F v 5 F v
5 m
. 15
l j
l l
l j 21
We conclude that
w w
m w
w w
m
9
f s , s
5 m 5 m 5 . . . 5 m 5 f
s , s .
16
i 2i
1 k
i 2i
This completes the proof. h
We can now state our main theorem. It states that if a weighted Myerson value is used as an allocation then potential maximizing strategies result in essentially complete
hypergraphs.
w
Theorem 5.1. Let w be a vector of positive weights and let P be a weighted potential
w w
] for the conference formation game
m . Then s [ argmax P . Further, if t [
w w
argmax P then t is essentially complete for
m .
1
Proof. Let i [ N, s [ S and s
[ S . Define the following conference sets: : 5
i i
2i 2i
2 2
1
] ]
s , s and : 5 s , s . From s 7s we conclude that 7 . Furthermore,
i 2i
i 2i
i i
8
This follows for example from the expression for weighted Shapley values of Kalai and Samet 1988.
M . Slikker et al. Mathematical Social Sciences 39 2000 55 –70
69
1 2
note that if H [ \ , then i [ H. If we apply Lemma 5.1 repeatedly for all
1 2
H [ \ then
w w
m w
1 w
2 m
] f
s , s 5
m m 5 f s , s
. 17
i i
2i i
i i
i 2i
] We conclude that s is a weakly dominant strategy.
] Consider the n-tuple of weakly dominant strategies s and an arbitrary n-tuple of
i n i
] ]
strategies t. Construct a sequence s with s 5 s , . . . , s , t
, . . . , t . This
i 50 1
i i 11
n n
] ]
construction implies that s 5 t and s 5s. Since s is a weakly dominant strategy it holds
w i 11
w i
w i 11
w i
for all i [ h0, . . . , n 2 1j that m
s m
s , so P s P s . Thus
i 11 i 11
w w
n w
n 21 w
1 w
w
] P s 5 P s P s
. . . P s P s 5 P t. 18
This completes the proof of the first part of the theorem.
w w
] Furthermore, since P s P t for all strategy-tuples t [ S it follows that if t is a
w w
] potential maximizing strategy then P s 5 P t. But then every inequality in 18 has
to hold with equality for this strategy-tuple t. Since 1
w k
w k 21
w k
w k 21
] P s 2 P s
5 m s 2 m s
s d
k k
w
k w
k w
k 21
for all k [ h1, . . . , nj it follows that m s 5 m s
for all k [ h1, . . . ,nj.
k k
w k
w k 21
From Lemma 5.2 we then conclude that m s 5 m s
for all k [ h1, . . . ,nj
and hence,
w w
w n
w
] m t 5 m s 5 . . . 5 m s 5 m s.
w
We conclude that if t [ argmax P then it holds that t is essentially complete for
w
m . h
If we drop the superadditivity assumption in Theorem 5.1 the full cooperation structure may fail to form. This follows easily by considering a non-superadditive
two-person cooperative game. Then the potential maximizing strategy profiles result in the formation two one-player coalitions.
6. Concluding remarks
