SGUM-based Database Assisted Spectrum Access - Social Group Utility Maximization - page 23

We then formulate the database assisted spectrum access problem as a SGUM
game ʓ = ( N , { M n } n ∈ N , { f n } n ∈ N ), where the set of white-space users N is the set

of players, the set of vacant channels M n is the set of strategies for each player n , and

the social group utility function f n of each user n is the payoff function of player n .

4.3 Existence of Social-Aware Nash Equilibrium

We next study the existence of SNE of the SGUM game for database assisted spectrum
access. Here we resort to a useful tool of potential game [ 5 ].

Definition 4.1 A game is called a potential game if it admits a potential function
ʦ ( a ) such that for every n and a − n ∈ i = n M i , for any a n , a n ∈ M n , ∈ N

f n ( a n , a − n ) ʦ ( a n , a − n )

− f n ( a n , a − n ) = − ʦ ( a n , a − n ) . (4.4)

An appealing property of the potential game is that it always admits a Nash equilib-
rium, and any strategy profile that maximizes the potential function ʦ ( a ) is a Nash
equilibrium [ 5 ].
For the SGUM game ʓ for database assisted spectrum access, we can show
that it is a potential game. For ease of exposition, we first introduce the physical-
social graph

sp sp

to capture both physical coupling and social coupling
simultaneously. Here the vertex set is the same as the user set

G ={ N , E }

N and the edge set

( n , m ): e s nm e nm where e s nm

sp

e nm · is given as E ={ ≡ = 1, ∀ n , m ∈ N } = 1

if and only if users n and m have social tie between each other (i.e., e nm = 1)

and can also generate interference to each other (i.e., e nm

1). We denote the
set of users that have social ties and can also generate interference to user n as
N

=

sp
n

m : e s nm =

.
Based on the physical-social graph

={ 1, ∀ m ∈ N }

sp , we show in Theorem 4.1 that the SGUM
game ʓ is a potential game with the following potential function

G

N N

1
2

P m d − ʱ ˉ a n ʦ ( a ) =−

mn I { a n = a m } −

p
n

n = 1 n = 1

m ∈ N

ʦ 1 ( a ): due to physical coupling

N

1
2

s nm P m d − ʱ

−

mn I { a n = a m }

. (4.5)

sp
n

n = 1

m ∈ N

ʦ 2 ( a ): due to social coupling

The potential function in ( 4.5 ) can be decomposed into two parts: ʦ 1 ( a ) and ʦ 2 ( a ).
The first part ʦ 1 ( a ) reflects the weighted system-wide interference level (including
background noise) due to physical coupling in the physical domain and the second
part ʦ 2 ( a ) captures the interdependence of user utilities due to social coupling in the
social domain.