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1
1
Fig. 3.2
For a two-user SGUM game for random access control, as the social tie level
s
s
12
=
s
21
increases from 0 to 1, each user's SNE strategy
q
SNE
migrates from its NE strategy
q
NC
,
NE
for a
standard NCG to its social optimal strategy
q
SO
for NUM, and the social welfare
v
SNE
of the SNE
also migrates correspondingly
Corollary 3.1
Each user's access probability at the SNE is decreasing as its social
tie levels with others increase.
Remark 3.1
. We observe that each user's SNE strategy does not depend on other
users' strategies (also known as a
dominant strategy
), but depends on the user's
social ties with others. Clearly, when a user increases its access probability, it also
increases the collision probabilities of the users within its interference range, and
thus reduces their individual utilities. Therefore, a user would decrease its access
probability when its social ties with those within its interferencerange get stronger
(as illustrated in Fig.
3.2
).
Let
V
(
q
) denote the social welfare of all users, i.e., the total individual utility of
all users:
⊡
⊛
⊝
ʸ
i
q
i
j
∈
I
i
⊞
⊤
N
⊣
log
⊠
−
⊦
.
V
(
q
)
(1
−
q
j
)
c
i
q
i
(3.8)
i
=
1
Proposition 3.1
The social welfare of the SNE is increasing as social tie levels
increase, and reaches the social optimal point when all socialtie levels are equal to 1.
Proof
. Using (
3.8
), setting the first-order derivative of
V
(
q
)to0,wehave
|
I
i
c
i
q
i
−
|+
+
c
i
)
q
i
+
∂V
(
q
)
∂q
i
(
1
1
=
=
0
.
(3.9)
q
i
(1
−
q
i
)
Similar to the proof of Theorem
3.1
, we obtain the social optimal strategy
q
SO
i
that
maximizes
V
(
q
) as the smaller root of Eq. (
3.9
), which is
|
I
i
c
i
2
|
I
i
|+
1
+
c
i
−
|+
1
+
−
4
c
i
q
SO
i
=
.
2
c
i
Since the larger root of Eq. (
3.9
)is
(
|
I
i
|
I
i
|+
+
c
i
+
|+
+
c
i
)
2
−
1
1
4
c
i
2
c
i
