In this section, the problem of bandwidth allocation is formulated as a non-cooperative Stackelberg game, in which FU is considered as a leader or primary user, and MUs act as followers or secondary users. In this section, we analytically proved that the aforementioned problem has unique Nash Equilibrium by which the utility of both FU and MUs are maximized.

Since both FU and MUs are rational as well as selfish, and they intend to maximize their utility, game theory is one of the best tools for solving this problem. In this paper, it is assumed that both FU and MUs prefer to work in hybrid access mode where both can maximize their utility. Therefore, it is reasonable to formulate the problem based on Stackelberg game theory.

The Stackelberg game is a strategic game that consists of two types of player: leaders and follower. Leaders select a strategy and act, while the followers will choose their strategy based on leaders’ action. Since the leaders know the follower reaction, they select the best strategy that maximizes their benefits. In this paper the Stackelberg game proceeds in two steps. First, an FBS specifies the amount of bandwidth that intends to share with MUs while it is aware that its move will not only be seen by MUs, but also it will influence the decision of MUs. Thus, FBS determines optimum $\alpha $ to maximize its utility. In the next step, followers should obtain the leader’s strategy, therefore, MUs specify their revenue to maximize their utility while considering FBS determination. This decision affects the other side of the game. So, the solution procedure can be described as follows:

#### 4.1 Macro Users’ Payment

To solve the resource allocation problem, according to Stackelberg game theory method, a backward induction method is used [

35]. First, the problem of having one unique Nash Equilibrium point is indicated where MUs have maximum utility. The problem described by (7) is a convex optimization problem. After solving this problem, FBS is able to specify the optimum value of

$\alpha $ to maximize its utility. In order to solve this problem, we first prove that (7) is a convex.

Afterward, we find the optimal solution by solving first-order optimality conditions of underlying problem [

33]. To this end, the Lagrangian of MUs utility can be derived as:

where

${\lambda}_{0}$ and

${\lambda}_{j}$ are the Lagrange multiplier. Based on KKT condition,

${\lambda}_{j}$ should be equal to 0. To prove the convexity of macro users’ utility in Lagrangian form, the second derivative on mi is calculated as:

The second order derivative of

$L$ with respect to

${m}_{i}$ is always negative, so the problem in (10) is convex; hence, this function has a maximum. In order to solve the convex optimization problem,

${m}_{i}^{\ast}$ should be obtained under Karush–Kuhn–Tucker (KKT) condition. With regard to this, each MU tries to achieve Nash Equilibrium because the best answer can maximize its utility, consequently, MUs compete with each other. Therefore,

${m}_{i}^{\ast}$ is given by:

Because

B is a dependent variable with respect to

${\lambda}_{0}$,

$X$ and

${m}_{i}^{\ast}$ also depend on

${\lambda}_{0}$,

${\lambda}_{0}$ can be obtained from the above condition as:

Appendix A provides detailed analytical discussion for derivation of (21).

By putting

${\lambda}_{0}$ in

$B$,

$X$ is obtained as follows:

where

$X$ demonstrates overall revenue that FBS can get from MUs. Also, the value of parameter

$\mu $ has a vital role in the allocation and price. This parameter both affects the sign of

${m}_{i}^{\ast}$ and determines the strategy of allocation. At first, the condition that

$\mu $ is acceptable should be obtained, then, how

$\mu $ determines a strategy should be specified as well as each bandwidth assigning strategy. The condition that leads to turning the sign of

${m}_{i}^{\ast}$ to a positive state is as follows:

Based on the above condition, two proposed values for parameter

$\mu $ are placed in this range. If

$\mu =k$ is selected, the system can establish fairness among MUs. In this case, the ratio of assigned bandwidth to desired bandwidth is directly dependent on

${Z}_{i}$, furthermore, all macro users pay the same revenue if the priority factor

${Z}_{i}$ is assumed to be one, i.e.,:

On the other hand, if

$\mu =\sqrt{D{w}_{x}}$, not only is the whole revenue, that FBS can get, maximized but also the utility of FBS reaches its maximum. Two other strategies are defined to compete with these proposed methods. The

${\mu}_{low}$ is considered as the lower bound and

${\mu}_{up}$ is the upper bound. If

${Z}_{i}$ is assumed equal to one, the system by selecting

${\mu}_{low}$ deprives one of the macro users which needs minimum desired bandwidth from allocation, and the system promotes macro users requiring higher desired bandwidth. Also, choosing

${\mu}_{up}$ removes MU with maximum desired bandwidth and promotes the macro user which needs less; this is not rational and useful for the system. In this case, if the allocated bandwidth is more than the desired bandwidth, system adjusts the assigned bandwidth to the desired one. Considering the analytical results of MUs payment, FBS can maximize its utility function based on revenue which was obtained from (22). A leader knows that its strategy will influence the follower’s decision, so it specifies

${\alpha}^{\ast}$ which maximizes its utility. Substituting (22) with (2), the utility function of FBS is expressed as:

where

$\eta ={\mathsf{\Omega}}_{m}\frac{{w}_{f}({\mu}^{2}-2\mu k+\mu {k}^{2}+D{w}_{x}-\mu D{w}_{x})}{\mu ({k}^{2}-D{w}_{x})}.$In order to find the optimum value of

${\alpha}^{\ast}$, first, the utility function of FBS that has a maximum point should be proved. Next, based on the first derivative, the maximum point will be obtained. The first and second derivatives of UFBS with respect to

α are calculated as:

When $\alpha >\frac{{w}_{0}}{{w}_{f}}$, the second derivative of UFBS with respect to $\alpha $, is always negative, UFBS is concave in terms of $\alpha $, so it has a maximum point. On the other side, when $\alpha <\frac{{w}_{0}}{{w}_{f}}$, the second derivative of UFBS with respect to α, is always positive, UFBS has a minimum. To further clarify the point, it is from the second derivative that the function falls down at first, then, it approaches the minimum point. Next, it goes up and reaches the maximum point of UFBS in ${\alpha}^{\ast}$ and the function decreases again.

When the following conditions are satisfied:

the optimum

${\alpha}^{\ast}$ by assigning the first derivative of UFBS to 0 can be found. In this case, there are a local minimum point

${\alpha}_{1}$ and a maximum point

${\alpha}_{2}$ which are obtained as follows:

The optimal value should have two features: first,

${\alpha}^{\ast}$ should satisfy criteria which are presented in (32) and the other feature is that

${\alpha}^{\ast}$ should be a global maximum point in this criterion, so the optimum value should be on the interior range of domain:

The local maximum

${\alpha}_{2}$ should be a global maximum which is expressed as:

When ${\omega}_{f}\ge \eta \frac{{w}_{f}}{{w}_{f}-1}$, it can be easily proved that ${\alpha}_{2}$ is the global maximum and ${U}_{{}_{FBS}}^{\ast}={U}_{FBS}({\alpha}_{2})$.

When $\eta =\frac{a{\mathsf{\Omega}}_{f}{w}_{f}}{4}$, MUs provide maximum revenue for FBS. In this case, MUs motivate FBS to share more bandwidth and decrease ${\alpha}^{\ast}$. By substituting this value into (28), ${\alpha}^{\ast}$ is obtained as ${\alpha}^{\ast}=\frac{{w}_{0}}{{w}_{f}}$, so this is the minimum value of ${\alpha}^{\ast}$ if and only if ${U}_{FBS}(0)<{U}_{FBS}({\alpha}_{2})$.

If ${\alpha}^{\ast}=0$, it means that FBS prefers to assign the whole bandwidth to MUs only and if ${\alpha}^{\ast}=1$, the system allocates the desired bandwidth that FU desires, so the satisfaction parameter is equal to 1. The system does not assign more than the desired bandwidth to user. This causes the maximum value of ${\alpha}^{\ast}$ to be less than $1$. In other words, the system always works in hybrid access mode because it is not reasonable for the system to allocate greater bandwidth than the desired bandwidth to FU, i.e., closed access.