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log
h k p k
n k + j = k g jk p j
.
s ik
c k p k
(3.10)
k = i
3.3.2
Game Analysis
We first have the following result.
Theorem 3.2 The SGUM-based power control game is a supermodular game, and
thus there exists at least one SNE.
Proof . Using ( 3.10 ), we have
∂f i ( p i , p i )
∂p i
1
p i
s ik g ik
n k + j = k g jk p j
=
c i .
k = i
Since each term in theabove summation term is decreasing in p j ,
j
N \
i ,it
follows that
2 f i ( p i , p i )
∂p i ∂p j
> 0,
j
N \
i
which implies that the social group utility function f i ( p i , p i ) is supermodular. It
follows from [ 5 ] that there exists at least one NE.
Since the SGUM-based power control game is a supermodular game, it follows
from [ 6 ] that users can start from any strategies (e.g., p
, 0)) and use
asynchronous best response updates such that their strategies will monotonically
converge to a SNE.
For ease of exposition, in the rest of this section we will focus on the SGUM-based
power control game with two users, because the two-user case can shed light on the
impact of social ties on users' strategies and social welfare. Furthermore, in general,
the game with more than two users does not yield closed-form SNE strategies, and
hence is much more difficult to quantify the impact.
=
(0,
···
Theorem 3.3. For the two-user SGUM-based power control game, there exists a
unique SNE, which is
ʱ 1 +
ʱ 2 +
p SNE
1
ʱ 1 , p SNE
=
ʲ 1
=
ʲ 2
ʱ 2
2
where
s 12 g 12 +
c 1 n 2
g 12
n 2
c 1 g 12
ʱ 1
, ʲ 1
2 c 1 g 12
and
s 21 g 21 + c 2 n 1 g 21
2 c 2 g 21
n 1
c 2 g 21 .
ʱ 2
, ʲ 2
 
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