The symbols used in the modeling and problem formulation are summarized in Table 1.
Symbol Meaning
The game of the system
The space of external states
The set of CRBSs
M The maximum channel number which can be demanded
i( ) A The parameter in the cost function B The parameter in the cost function
V i The value which CRBS i can earn from obtaining a channel V The vector of values for all the CRBSs
D i The set of channel numbers which CRBS i can demand
, ( )
pi j t Probability for CRBS i to demand j channels at time t
i( )t
p Probability vector for CRBS i at time t
Table 1 Symbols used in the modeling and system formultaion.
6
The CR network can be implemented for different scenarios [2]. In our work, we considered one of the CR network architectures, CR network access architecture, where the primary network gives out the residual channels to the spectrum broker, and the spectrum broker distributes the residual channels to the CRBSs according to their demands. Finally the CR users can utilize the spectrum resources from the CRBSs. The architecture is shown in Fig. 3. In our model, there are N CRBSs which serve different numbers of CR users. CRBS i demands d t channels from the spectrum i( ) broker at time t. Each CRBS can demand at most M channels. Upon successfully obtaining a channel, the CRBS-connected CR users can utilize the resources and benefit from sharing the residual channels.
On the other hand, the primary network gives out a number of residual channels. Note that the number of residual channels is time-varying with a fixed statistic character, and the CR network is not able to interfere with how many channels the primary network gives out. So when the primary network is fully occupied by the primary users, the
Fig. 3 The overview of CR network access architecture.
7
spectrum broker will detect no residual channels. The CR network then can never get any spectrum resources from the spectrum broker. In this situation, no matter how many channels the CRBSs demand, they are not allowed to share any resources with primary network. We will formulate this mechanism by an economical model in the next section.
To make the system more practical, we imposed the following assumptions.
The number of residual channels given out from the primary network is time varying. Its statistics are fixed but unknown to the CRBSs.
The system is decentralized which means the CRBSs have no information about how many channels other CRBSs demand. They act individually.
Notably, the only information available for decision making is the action-reward history of individual players.
2.1 Game-theoretic model
In this section, we present the game-theoretic formulation of the system. We considered the system as a Cournot Game (Cournot Competition) with external state. The players are the CRBSs. The game can be represented as a 4-tuple:
( , ,M u,{ }i i∈ )
=
where is the space of external states (number of residual channels), is the set of players, M is the maximum number of channels that a CRBS can demand, and { }ui i∈ is the utility function of player i that depends on his own decision as well as the decisions of other players. The description of the utility function u is given below. i
At time t , CRBS i demand d channels from the spectrum broker. The cost i
8
( , )
C D t for demanding a channel depends on the total number of channels demanded
by all the CRBSs. In a typical Cournot Game model, the cost for demanding a channel is given by
( , ) ( ) ( ), ( ) iN1 i( )
C D t = A t + ×B D t D t =
∑
= d t (1)where D t represents the total number of channels demanded by all the CRBSs at ( ) time t, B is a constant and A t is a parameter indicating the availability of the ( ) residual channels at time t. In fact, A t increases while the number of the residual ( ) channels decreases in a linear fashion, which causes the cost to get higher. Note that
( )
A t changes with time since the number of the residual channels is time-varying with a fixed statistic character which is not known by the CRBSs.
On the other hand, each CRBS benefits from successfully obtaining a channel from the spectrum broker. CRBS i earns a value V when a channel is obtained. The value i V i depends on how many CR users are under the CRBS's service. We could then formulate the utility function u for CRBS i i as
The objective for each CRBS is to decide how many channels d to demand in order i
to maximize its utility u . Each CRBS desires to demand more channels to obtain a i higher utility value for itself. However the cost C increases when the total demand
9
increases, which will decrease the value of Vi− and thus decrease the utility. Since C the CRBSs have no information of each other's decision, they need to compete for their own benefit. Formally,
( ) : max ( , ),
i i
d∈ u d di i −i ∀ ∈i
(4)
2.3 Analysis of Nash equilibrium
With the utility function defined in (2), we show the existence of the NE point for the proposed Cournot game model in the following proposition.
Proposition 1. The game is an exact potential game (EPG) which possesses at least one pure strategy Nash equilibrium (NE).
Proof: From the cost equation (1) and the utility function (2), we considered the function
: 1 N +
When CRBS i decides to demand another number of channels and changes its strategy from d to i di′, the utility function will also change from u d di( ,i −i) to
( , )
i i i
u d d′ − .
It can be readily verified that
( , ) ( , ) ( , ) ( , )
i i i i i i i i i i
u d d′ − −u d d− = Φ d d′ − − Φ d d− (6)
10
As shown in (6), if the unilateral change of the strategy d can increase the utility i u , i the increased utility will contribute the exact amount of improvement to the function Φ. Therefore, as the players compete for better unilateral utility, they also benefit the total system. pure strategy NE is always guaranteed. This pure strategy NE is also a local maximum of the potential function Φ. Thus, Proposition 1 is proved. ■
However, we should note two things about NE.
A NE point doesn't guarantee the global maximum of the potential function Φ.
There could be more than just one NE point, and the number of NE points of is difficult to be confirmed.
11