Information Technology Reference
In-Depth Information
Theorem 4.1
The SGUM game ʓ for database assisted spectrum access is a po-
tential game with the potential function
Φ
(
a
)
given in
(
4.5
)
, and hence has a
SNE.
The proof is given in Appendix. Note that when
s
nm
=
0 for any user
n
,
m
∈
N
(i.e., all users are selfish), the potential function
ʦ
(
a
)
ʦ
1
(
a
), which does not
involve the social coupling part
ʦ
2
(
a
). In this case, the SGUM game
ʓ
for database
assisted spectrum access degenerates to the non-cooperative spectrum access game.
When
s
nm
=
=
1 for any user
n
,
m
∈
N
(i.e., all users are fully altruistic), the potential
=
n
=
1
u
n
(
a
), which is the system-wide utility. In this case, the
SGUM becomes the network utility maximization.
We next design a distributed spectrum access algorithm that can achieve the SNE
of the SGUM game
ʓ
for database assisted spectrum access.
function
ʦ
(
a
)
4.4
Distributed Spectrum Access Algorithm
In this section we study the distributed spectrum access algorithm design.
4.4.1
Algorithm Design Principles
According to the property of potential game, any channel selection profile
a
that
maximizes the potential function
ʦ
(
a
) is a Nash equilibrium [
5
]. We hence design
a distributed spectrum access algorithm that achieves the SNE of the SGUM
ʓ
by
maximizing the potential function
ʦ
(
a
).
To proceed, first consider the problem that the users collectively compute the
optimal channel selection profile such that the potential function is maximized, i.e.,
max
ʦ
(
a
)
.
(4.6)
≡
n
=
1
M
n
a
∈
ʩ
The problem (
4.6
) involves a combinatorial optimization over the discrete solution
space
ʩ
. In general, it is very challenging to solve such a problem exactly especially
when the system size is large (i.e., the solution space
ʩ
is large).
With this observation, it is plausible to search for approximate solutions to the
potential function maximization problem. To this end, rewrite
We then consider to approach the potential function maximization solution ap-
proximately. To proceed, we first write the problem (
4.6
) as the following equivalent
randomized problem:
max
(
q
a
≥
0:
a
∈
ʩ
)
q
a
ʦ
(
a
)
a
∈
ʩ
s.t.
a
∈
ʩ
q
a
=
1,
(4.7)
