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1
1
Fig. 3.5
For the two-user SGUM-based power control game, as the social tie level
s
s
12
=
s
21
increases from 0 to 1, each user's SNE strategy
p
SNE
migrates from its NE strategy
p
NC
,
NE
for a
standard NCG to its social optimal strategy
p
SO
for NUM, and the social welfare
v
SNE
of the SNE
also migrates correspondingly
we have
∂
ʱ
1
+
ʱ
1
ʲ
1
−
∂p
SNE
1
∂s
12
∂ʱ
1
∂s
12
=
∂ʱ
1
⊛
⊞
ʱ
1
ʱ
1
+
1
2
c
1
⊝
⊠
=
ʲ
1
−
1
<
0
.
So
p
SNE
1
is decreasing in
s
12
. Similarly, we can show that
p
SNE
2
is decreasing in
s
21
.
Proposition 3.2
For the two-user SGUM-based power control game, the social
welfare of the SNE is increasing as social tie levels increase, and reaches the social
optimal point when all social tie levels are equal to 1.
Proof
. Since
log
h
1
p
1
n
1
+
g
21
p
2
V
(
p
1
,
p
2
)
=
−
c
1
p
1
log
h
2
p
2
n
2
+
+
−
c
2
p
2
g
12
p
1
we have
∂V
(
p
1
,
p
2
)
∂p
1
1
p
1
−
g
12
n
2
+
=
g
12
p
1
−
c
1
.
Using the same argument as in the proof of Theorem
3.3.
, the optimal value
p
SO
1
of
p
1
for
V
(
p
1
,
p
2
) is the unique solution of
1
p
1
−
g
12
n
2
+
g
12
p
1
−
c
1
=
0
.
