58 M
. Slikker et al. Mathematical Social Sciences 39 2000 55 –70
potential game are the weighted extended Myerson values. In Section 5 we show that the argmax set of the weighted potential corresponds to the full cooperation structure
and payoff-equivalent structures. We conclude in Section 6.
2. Potential games
In Ui 1996 a representation theorem for potential games is given in terms of the Shapley value. In this section we will extend the result of Ui 1996 and provide a
representation theorem for weighted potential games in terms of weighted Shapley values. We will first give some definitions.
A game in strategic form will be denoted by G 5 N; S ;
p , where
i i [N i i [N
N 5 h1, . . . , nj denotes the player set, S the strategy space of player i [ N, and
i
p 5 p the payoff function which assigns to every strategy-tuple s 5 s
[
i i [N i i [N
N
P S 5 S a vector in
R . For notational convenience we write s 5 s and
i [N i
2i j j [N \
hi j
s 5 s .
R i i [R
The class of weighted potential games is formally defined in Monderer and Shapley
N
1996. Let w 5 w [
R be a vector of positive weights. A function
i i [N 11
w
Q : P S
→ R is called a w-potential for G if for every i [ N, every s [ S, and every
i [N i
t [ S it holds that
i i
w w
p s , s 2 p t ,s 5 w Q s ,s 2 Q t ,s . 1
i i
2i i
i 2i
i i
2i i
2i
The game G is called a w-potential game if it admits a w-potential. G is called a
N
weighted potential game if G is a w-potential game for some weights w [ R
.
11
In Monderer and Shapley 1996 it is pointed out that the argmax set of a weighted potential game does not depend on a particular choice of a weighted potential, and hence
can be used as an equilibrium refinement. It is also remarked that this refinement is
3
supported by some experimental results. The representation theorem in this section is in terms of cooperative games and
weighted Shapley values. A cooperative game is an ordered pair N, v, where
N
N 5 h1, . . . , nj is the set of players, and v is a real-valued function on the family 2 of
all subsets of N with v5 5 0. Denote the set of all cooperative games with player set N
N
by TU . Weighted Shapley values can easily be defined using unanimity games. For every
4
R 7 N the unanimity game N, u is defined by
R
1, if R 7 T
u T 5 .
2
H
R
0, otherwise
Unanimity games were introduced in Shapley 1953. It is shown that every cooperative
3
In Monderer and Shapley 1996 it is pointed out that this may be a mere coincidence. See also van Huyck et al. 1990 and Crawford 1991.
4
R 7 T denotes that R is a subset of T, R , T denotes that R is a strict subset of T.
M . Slikker et al. Mathematical Social Sciences 39 2000 55 –70
59
game can be written as a linear combination of unanimity games in a unique way, v
5 o
a u , where a are called the unanimity coordinates of N, v.
R7N R
R R R7N
N
Let w 5 w [
R be a vector of positive weights. For all R 7 N define
i i [N 11
w N
w : 5 o
w . The weighted Shapley value F of a cooperative game N, v [ TU
R i [R
i
with unanimity coordinates a
is then defined by
R R7N
w
i w
] F N, v 5
O
a . 3
i R
w
R R7N,i [R
To represent weighted potential games in terms of weighted Shapley values we need the
5
following interaction between cooperative and non-cooperative games. Consider a player set N 5
h1, . . . ,nj and strategy space S 5 P S . Assume that once
i [N i
the players have chosen a strategy profile s [ S they face the cooperative game N, v .
s
Furthermore, assume that the players have made a pre-play agreement on the allocation rule that determines their payoffs for any chosen cooperative game. If the players have
agreed on allocation rule g this implies that player i obtains g N, v if strategy profile
i s
s [ S is played. In Ui 1996 it is shown that in analyzing potential games it suffices to consider
collections of cooperative games where the value of a coalition does not depend on the strategies of the players outside this coalition: v R only depends on s . We will show a
s R
similar result for weighted potential games. First we define the following set of collections of cooperative games cf. Ui, 1996:
N S
: 5 hhN, v j
[ TU
uv R 5 v R if s 5 t for all s, t [ S, R 7 Nj. 4
N,S s
s [S s
t R
R s
Denote the unanimity coordinates of the game v by a
. It can be shown that the
s R R7N
condition in definition 4 can be rewritten in terms of these unanimity coordinates,
N S s
t
5 hhN, v j
[ TU
ua 5 a if s 5 t for all s, t [ S, R 7 Nj. 5
N,S s
s [S R
R R
R
We can now state the main result of this section, which states that the class of weighted potential games can be represented in terms of weighted Shapley values. In Ui
6
1996 this theorem is shown for unweighted potential games and Shapley-values.
N
Theorem 2.1. Let G 5 N; S ;
p be a game in strategic form and w [
R .
i i [N i i [N
11
G is a w-potential game if and only if there exists hN, v j
[ such that
s s [S
N,S w
p s 5 F v , for all i [ N and all s [ S. 6
i i
s
Proof. First we will prove the if-part of the theorem. Assume there exists
w
hN, v j [
with p s 5 F v , for all i [ N and s [ S. Define
s s [S
N,S i
i s
5
Note that we consider weighted potentials for non-cooperative games, as opposed to Hart and Mas–Colell 1989 where weighted Shapley values are characterized using weighted potentials for cooperative games.
6 w
w
If there is no ambiguity about the underlying player set we will simply write F v instead of F N, v.
60 M
. Slikker et al. Mathematical Social Sciences 39 2000 55 –70
s
a
R w
] Q s: 5
O
. 7
w
R R7N,R±5
w
We will show that Q is a w-potential of G. Let i [ N, s [ S, and t [ S , then
i i
w w
p s 2 p t , s 5 F v 2 F v
i i
i 2i
i s
i t ,s
i 2i
s t ,s
i 2i
a a
R R
] ]]
5 w
O
2 w
O
i i
w w
R R
R7N,i [R R7N,i [R
s t ,s
i 2i
a a
R R
] ]]
5 w
O
2 w
O
i i
w w
R R
R7N,R±5 R7N,R±5
w w
5 w Q s 2 Q t , s ,
i i
2i
where the third equality follows from 5.
w
To prove the only-if-part assume G is a w-potential game, with potential Q . Define for all s [ S and all R 7 N
p s
i w
]] w
O
2 n 2 1Q s , if R 5 N
S D
H J
R
w
i i [N
s
p s a 5
8
i w
R
]] w
2 1 Q s ,
if R 5 N\
hij, i [ N
H J
R
w
5
i
0, otherwise
s
which determine v 5 o
a u for all s [ S.
s R7N
R R
We will show that hN,v j
[ . Let R 7 N, s, t [ S with s 5 t . For R 5 N or R
s s [S
N,S R
R s
t
with uRu n 2 2 we immediately find that a 5 a . It remains to consider R with
R R
w w
uRu 5 n 2 1. Let i [ N and R 5 N\hij then p s 2 p t 5 w Q s 2 Q t so
i i
i
p s p t
i i
s w
w t
]] ]]
a 5 w 2
1 Q s 5 w
2 1 Q t
5 a .
H J H
J
R R
R R
w w
i i
So, hN, v j
[ .
s s [S
N,S w
Finally, we will show that for all i [ N and s [ S it holds that F v 5 p s.
i s
i
Therefore, let i [ N and s [ S. Then
s
a
R w
] F v 5 w
O
i s
i
w
R R7N,i [R
p s p s
j j
w w
]] ]]
5 w
O
2 n 2 1Q s 1
O
2 1 Q s
H S D S
DJ
i
w w
j j
j [N j [N, j ±i
p s
i
]] 5 w
5 p s.
H J
i i
w
i
This completes the proof. h
Note that if G is a w-potential game then an associated potential is given by
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
