As is known to us, subcarrier and power allocation impose great influence on system performance. In uplink cooperative OFDMA system, subcarrier is public resource which is in competition among different users. However, power of each user can be controlled independently according to their own demand and power capacity. So, we develop a semidistributed allocation framework, in which we are pursuing for the combination of merits of centralized and distributed schemes, and can effectively achieve the subcarrier and power allocation. In the proposed scheme, relay selection can be achieved with subcarrier allocation under the control of a centre, i.e. the base station, while the next step is achieved by each user.
On the other hand, resource allocation, which contains relay selection, subcarrier and power allocation, is an NPhard problem which has great complexity. For simplicity, the division of the original problem into two progressive suboptimal ones, which is at the cost of a loss of optimality, and is convenient for us to solve the problem.
3.1. Centralized joint relay selection and subcarrier allocation
The relay selection and power allocation framework can be represented by binary assignment variables {x}_{km}^{n}, which can alternatively equals to 0 or 1. While {x}_{km}^{n}=1, the nth subcarrier is matched for the communications between the mth user and the kth relay, vice versa. If k = 0, the value of {x}_{km}^{n} demonstrates whether the n th subcarrier is matched directly for the communications between the m th user and the base station. The (K + 1) × M × N dimension 01 matrix X=\left({x}_{km}^{n}\right) can be defined as the relayusersubcarrier matching matrix, i.e. the relay selection and subcarrier allocation matrix.
Therefore, we can optimize the framework aiming at maximizing the system capacity. However, waterfilling algorithm cannot ensure the QoS of farther users whose CSI is not as good as the nearer ones, i.e. the nearfar effect is more obvious [4]. Hence, we can introduce QoS requirement to ensure that each user can reach a minimum rate. The optimization problem for joint relay selection and subcarrier allocation in cooperative OFDMA networks can be formulated as following [2].
\underset{{x}_{km}^{n}}{argmax}{C}_{\mathsf{\text{sum}}}=\sum _{k=0}^{K}\sum _{m=1}^{M}\sum _{n=1}^{N}{c}_{km}^{n}{x}_{km}^{n}
(3)
s.t.\phantom{\rule{0.3em}{0ex}}\sum _{k=0}^{K}\sum _{m=1}^{M}{x}_{km}^{n}\le 1
{x}_{km}^{n}=\left\{0,1\right\}
\sum _{k=0}^{K}\sum _{m=1}^{M}{c}_{km}^{n}\ge {\stackrel{\u0304}{c}}_{min}
where the 1 × m vector {\stackrel{\u0304}{c}}_{min} denotes the minimum rate of each user.
The optimizing framework is therefore formulated as an integer programming problem. By introducing a 1 × m dual vector λ_{
m
} and introducing subgradient method, we can make use of the iterative algorithm proposed in [2] with regardless of power allocation as following.
{x}_{km}^{n}\left(iter\right)=\{\begin{array}{l}1,(k,m)=({k}^{*},{m}^{*})=\underset{k,m}{\mathrm{arg}\mathrm{max}}\left[1+{\lambda}_{m}\left(iter\right)\right]{c}_{km}^{n}\hfill \\ 0,o.w.\hfill \end{array}
(4)
{\lambda}_{m}\left(iter+1\right)={\left[{\lambda}_{m}\left(iter\right)+\alpha \left(iter\right)\left({\overline{c}}_{\mathrm{min}}{c}_{km}^{n}{x}_{km}^{n}\left(iter\right)\right)\right]}^{+}
(5)
where iter represents the iterative times, and α(iter) means a proper iterative step length which is related to the iterative times.
Theorem 1 [12] If \sum _{iter}\alpha \left(iter\right)\to \infty, and α(iter)→0 as iter → ∞, then the optimizing goal can converge upon the optimal value.
According to Theorem 1, we can entitle a nonconvergent infinite series to α(iter), whose items incline towards zero while the iterative time goes to infinite. Therefore, we can evaluate \alpha \left(iter\right)=\frac{1}{iter}. By iteration of Equations 4 and 5, the optimal solution can be obtained accordingly. It is achieved by the base station in the centre.
3.2. Distributed power allocation scheme
When subcarriers' allocation has been finished, we can focus on power allocation with the goal of maximizing each user's rate. We can pay more attention to the even time slot, in which we can distinguish cooperative and noncooperative modes. By introducing game theory, the power allocation scenario can be corresponded with the three basic components of game theory as following.
Noncooperative players: the M_{
d
} noncooperative users and the K relays which conflict with each other shown in Equations 1 and 2.
The action profile: the transmitting strategy of each player.
The set of utility functions: an income which is related to the rate of each player.
Therefore, the game can be expressed as
\underset{{x}_{km}^{n}}{argmax}\phantom{\rule{1em}{0ex}}{u}_{s}\left({p}_{s}^{n}\left(2\right)\right)
(6)
s.t.\sum _{n}{p}_{s}^{n}\left(2\right)\le {p}_{max},\phantom{\rule{1em}{0ex}}s=\left\{1,2,\dots ,K,K+1,\dots ,K+{M}_{d}\right\}
where {p}_{s}^{n}\left(2\right) is the transmitting power of node s in the even time slot, and the elements of p_{max} are the peak values of the K relays and the M_{
d
} noncooperative users.
The design of the utility function exerts great influence on the system performance. For noncooperative users, with the purpose of adjusting the transmitting power in the even time slot, we can design the utility function of the m th noncooperative user as
{u}_{m}\left({p}_{0m}^{n}(2)\right)=\frac{B}{N}{\displaystyle \sum _{n\in {\psi}_{d,m}}{\mathrm{log}}_{2}\left(1+SIN{R}_{d,m}^{n}(2)\right)}{\alpha}_{d}{\displaystyle \sum _{n\in {\psi}_{d,m}}{\left{h}_{d,m}^{n}\right}^{2}}{p}_{0m}^{n}(2)
(7)
s.t.\sum _{n\in {\psi}_{d,m}}{p}_{0m}^{n}\left(2\right)\le {p}_{m,max}
where ψ_{
d, m
} is the subcarrier set of the m th noncooperative user obtained in Section 3.1, {\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)=\frac{{p}_{0m}^{n}\left(2\right){\left{h}_{d,m}^{n}\right}^{2}}{{\sigma}^{2}+{I}_{0}} is the signaltonoiseandinterferenceratio (SINR) of the m th noncooperative user upon the n th subcarrier in the even time slot, α_{
d
} is a nonnegative cost factor whose unit is bps/W, in which the notation W denotes the power unit Watt, and p_{m, max}denotes the power constraint of the m th noncooperative user. In the utility function above, the first term represents the goal to maximize the m th noncooperative users' rate, and the second term is the cost function which aims at reducing the transmitting power. Particularly, path attenuation is an important factor to determine the cost originating from power consumption. In light of this, the 2norm of channel coefficients is introduced into the cost function to regulate the power consumption in different channel conditions.
Similarly, for relays which are applied to forward cooperative users' information in the odd slot, the utility function can be designed as following.
{u}_{k}\left({p}_{km}^{n}\left(2\right)\right)=\frac{B}{N}\sum _{n\in {\psi}_{r,k}}{log}_{2}\left(1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\right){\alpha}_{r}\sum _{n\in {\psi}_{r,k}}{p}_{km}^{n}\left(2\right){\left{h}_{b,k}^{n}\right}^{2}\beta \sum _{n\in {\psi}_{r,k}}{\left(\frac{{p}_{km}^{n}\left(2\right)}{{p}_{km}^{n}\left(1\right)}1\right)}^{2}
(8)
s.t.\sum _{n\in {\psi}_{r,k}}{p}_{km}^{n}\left(2\right)\le {p}_{k,max}
where ψ_{
r, k
} is the subcarrier set of the k th relay for the communication between the k th relay and its corresponding cooperative users, {\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)=\frac{{p}_{km}^{n}\left(2\right){\left{h}_{b,k}^{n}\right}^{2}}{{\sigma}^{2}+{I}_{b}} is the SINR of the m th direct user upon the n th subcarrier in the even time slot and p_{k, max}denotes the power constraint of the k th relay. α_{
r
} is a nonnegative cost factor whose unit is bps/W, while β is a nonnegative weight factor for the pricing item whose unit is bps. Employing a utility function in the form of (7) will lead to a unilateral SINR increase in even time slot, and the capacity of a relay user is confined by the minimum SINR of the two kinds of time slots. To get the balance in the two slots, the introduction of the third item is necessary. The squared term in the expression grows fast when the SINR in the even slot increases and exceeds that in the odd slot. Thus, the price rises to punish the relay from allocating too much power on that subcarrier [9].
3.3. The existence of NE
When it comes to a game model, NE is always our expectation. NE is an important concept in a noncooperative game, in which each actor is content to keep the present strategy rather than change it. In other words, NE is the profile of each actor's optimal strategy.
Theorem 2 [10] The function must be quasiconcave (or quasiconvex) if it is concave (or convex).
Theorem 3 [10] There is an NE at least in game G = [N, {P_{
i
}}, u_{
i
}(·)], if

(1)
P_{
i
} are the subsets of R^{n} and are nonempty, compact and convex;

(2)
u_{
i
} are continuous in P_{
i
};

(3)
u_{
i
} are quasiconcave in P_{
i
}.
where N is the set of players, P_{
i
} and u_{
i
} are the action profile and the utility function of player i, i = 1,2,..., N.
According to these theorems, we can give the proof of the existence of NE in the game scenario in this article as follow.
Proof: It is obviously that the first and the second conditions are satisfied according to the expressions of P_{
i
} and u_{
i
}(·). We can focus on the rest one.
In our game model, for the mth noncooperative user, we can get the firstorder and secondorder derivative functions of its utility of Equation 7 as
\frac{\partial {u}_{m}\left({p}_{0m}^{n}\left(2\right)\right)}{\partial {p}_{0m}^{n}\left(2\right)}=\frac{B}{Nln2}\cdot \frac{{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)}{1+{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)}\cdot \frac{1}{{p}_{0m}^{n}\left(2\right)}{\alpha}_{d}{\left{h}_{d,m}^{n}\right}^{2}
(9)
\frac{\partial {{u}_{m}}^{2}\left({p}_{0m}^{n}\left(2\right)\right)}{\partial {\left({p}_{0m}^{n}\left(2\right)\right)}^{2}}=\frac{B}{Nln2}\cdot \frac{1}{{\left(1+{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\right)}^{2}}\cdot {\left(\frac{1+{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)}{{p}_{0m}^{n}\left(2\right)}\right)}^{2}<0
(10)
Similarly, for the k th relay, we can get the firstorder and secondorder derivative functions of its utility of Equation 9 as
\frac{\partial {u}_{k}\left({p}_{km}^{n}\left(2\right)\right)}{\partial {p}_{km}^{n}\left(2\right)}=\frac{B}{Nln2}\cdot \frac{{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)}{1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)}\cdot \frac{1}{{p}_{km}^{n}\left(2\right)}{\alpha}_{r}{\left{h}_{b,k}^{n}\right}^{2}\frac{2\beta}{{\left({p}_{km}^{n}\left(2\right)\right)}^{2}}\left({p}_{km}^{n}\left(2\right){p}_{km}^{n}\left(1\right)\right)
(11)
\frac{\partial {{u}_{k}}^{2}\left({p}_{km}^{n}\left(2\right)\right)}{\partial {\left({p}_{km}^{n}\left(2\right)\right)}^{2}}=\frac{B}{Nln2}\cdot \frac{1}{{\left(1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\right)}^{2}}\cdot {\left(\frac{{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)}{{p}_{km}^{n}\left(2\right)}\right)}^{2}\frac{2\beta}{{\left({p}_{km}^{n}\left(2\right)\right)}^{2}}<0
(12)
Therefore, for both relays and direct users, their utility functions are quasiconcave in {p}_{km}^{n}\left(2\right) and {p}_{0m}^{n}\left(2\right), respectively. Consequently, the NE exists in the proposed framework.
3.4. The uniqueness of NE
The uniqueness of NE demonstrates that each node can reach a converging strategy profile. The authors of [11] proposed an approach to prove it. As a matter of fact, the uniqueness can also be obtained from its original meaning. Considering the power strategies profile \stackrel{\u0304}{P} can be obtained by iteration, whether the positive term series \left\stackrel{\u0304}{P}\left(t+1\right)\stackrel{\u0304}{P}\left(t\right)\right can reach its convergence determines the uniqueness of NE.
Proof: For the m th noncooperative user, let Equation 9 be zero, we can get
{p}_{0m}^{n}\left(2\right)=\frac{B}{{\alpha}_{d}N{\left{h}_{d,m}^{n}\right}^{2}ln2}\frac{{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)}{1+{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)}
(13)
Therefore, we can get the value of {p}_{0m}^{n}\left(2\right) by iteration of Equation 13 and {\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)=\frac{{p}_{0m}^{n}\left(2\right){\left{h}_{d,m}^{n}\right}^{2}}{{\sigma}^{2}+{I}_{0}}. When it comes to the iterative expression, as the iterative time t → ∞, we can get the positive term series
\begin{array}{c}\left{p}_{0m}^{n}\left(t+1\right){p}_{0m}^{n}\left(t\right)\right\\ =\left\frac{B}{{\alpha}_{d}{\left{h}_{d,m}^{n}\right}^{2}Nln2}\frac{{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t\right)}{1+{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t\right)}\frac{B}{{\alpha}_{d}{\left{h}_{d,m}^{n}\right}^{2}Nln2}\frac{{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t1\right)}{1+{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t1\right)}\right\\ =\frac{B}{{\alpha}_{d}{\left{h}_{d,m}^{n}\right}^{2}Nln2}\cdot \frac{\left{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t\right){\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t1\right)\right}{\left(1+{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t\right)\right)\left(1+{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t1\right)\right)}\end{array}
Let \delta \left(t\right)=\frac{1}{\left(1+{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t\right)\right)\left(1+{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t1\right)\right)}, the equation above can be expressed as
\begin{array}{c}\left{p}_{0m}^{n}\left(t+1\right){p}_{0m}^{n}\left(t\right)\right\\ =\frac{B}{{\alpha}_{d}{\left{h}_{d,m}^{n}\right}^{2}Nln2}\cdot \delta \left(t\right)\cdot \left{\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t\right){\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t1\right)\right\\ =\frac{B}{{\alpha}_{d}{\left{h}_{d,m}^{n}\right}^{2}Nln2}\cdot \delta \left(t\right)\cdot \left\frac{{\left{h}_{d,m}^{n}\right}^{2}{p}_{0m}^{n}\left(2\right)\left(t\right)}{{\sigma}^{2}+{I}_{0}}\frac{{\left{h}_{d,m}^{n}\right}^{2}{p}_{0m}^{n}\left(2\right)\left(t1\right)}{{\sigma}^{2}+{I}_{0}}\right\\ =\frac{B}{{\alpha}_{d}Nln2}\cdot \frac{\delta \left(t\right)}{{\sigma}^{2}+{I}_{0}}\cdot \left{p}_{0m}^{n}\left(2\right)\left(t1\right){p}_{0m}^{n}\left(2\right)\left(t2\right)\right\\ ={\left(\frac{B}{{\alpha}_{d}Nln2}\right)}^{2}\cdot \frac{\delta \left(t\right)\delta \left(t2\right)}{{\left({\sigma}^{2}+{I}_{0}\right)}^{2}}\cdot \left{p}_{0m}^{n}\left(2\right)\left(t3\right){p}_{0m}^{n}\left(2\right)\left(t4\right)\right\\ ={\left(\frac{B}{{\alpha}_{d}Nln2\left({\sigma}^{2}+{I}_{0}\right)}\right)}^{\frac{t1}{2}}\cdot \prod _{i=3}^{t}\delta \left(i\right)\cdot \left{p}_{0m}^{n}\left(2\right)\left(2\right){p}_{0m}^{n}\left(2\right)\left(1\right)\right\end{array}
(14)
\forall {\mathsf{\text{SINR}}}_{d,m}^{n}\left(2\right)\left(t\right)\ne 0, δ(t)<1. So, the second term of the equation above inclines to zero as t → ∞. If we adjust the factor α_{
d
} to an adequate value to ensure \frac{B}{{\alpha}_{d}Nln2\left({\sigma}^{2}+{I}_{0}\right)}<1, i.e. {\alpha}_{d}>\frac{B}{Nln2\left({\sigma}^{2}+{I}_{0}\right)}, \underset{t\to \infty}{lim}{\left(\frac{B}{{\alpha}_{d}Nln2\left({\sigma}^{2}+{I}_{0}\right)}\right)}^{\frac{t1}{2}}=0. And \left{p}_{0m}^{n}\left(2\right)\left(2\right){p}_{0m}^{n}\left(2\right)\left(1\right)\right is a definite value. Thus, the righthand side of Equation 14 inclines towards zero. So, does it if \frac{B}{{\alpha}_{d}Nln2{\left({\sigma}^{2}+{I}_{0}\right)}^{2}}=1. In a word, while {\alpha}_{d}\ge \frac{B}{Nln2\left({\sigma}^{2}+{I}_{0}\right)}, the righthand side of Equation 14 is inclining to zero as t → ∞, i.e. the series {p}_{0m}^{n}\left(t+1\right){p}_{0m}^{n}\left(t\right) is absolutely convergent. It means that by iteration, the solution of NE {p}_{0m}^{n}\left(2\right) can reach a certain value. NE is therefore unique.
Similarly, when it comes to the k th relay, let Equation 11 be zero, we can get the iterative expression of {p}_{km}^{n}\left(2\right) as following.
{p}_{km}^{n}\left(2\right)\left(t+1\right)=\frac{b+\sqrt{c+d\frac{{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right)}{1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right)}}}{2a}
(15)
where a=\frac{2\beta}{{\left({p}_{km}^{n}\left(1\right)\right)}^{2}}, b=\frac{2\beta}{{p}_{km}^{n}\left(1\right)}{\alpha}_{r}{\left{h}_{b,k}^{n}\right}^{2}, c=\frac{4{\beta}^{2}}{{\left({p}_{km}^{n}\left(1\right)\right)}^{2}}\frac{4\beta {\alpha}_{r}{\left{h}_{b,k}^{n}\right}^{2}}{{p}_{km}^{n}\left(1\right)}+{\alpha}_{r}{\left{h}_{b,k}^{n}\right}^{4} and d=\frac{2\beta B}{{\left({p}_{km}^{n}\left(1\right)\right)}^{2}N}.
Therefore,
\begin{array}{c}\left{p}_{km}^{n}\left(2\right)\left(t+1\right){p}_{km}^{n}\left(2\right)\left(t\right)\right\\ =\frac{1}{2a}\left\sqrt{c+d\frac{{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right)}{1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right)}}\sqrt{c+d\frac{{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t1\right)}{1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t1\right)}}\right\\ =\frac{1}{2a}\cdot \frac{1}{\sqrt{c+d\frac{{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right)}{1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right)}}+\sqrt{c+d\frac{{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t1\right)}{1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t1\right)}}}\cdot \frac{\left{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right){\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t1\right)\right}{\left(1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right)\right)\left(1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t1\right)\right)}\\ =\frac{1}{2a}\sqrt{\frac{d}{c}}\frac{{\left{h}_{b,k}^{n}\right}^{2}}{{\sigma}^{2}+{I}_{0}}\cdot \frac{1}{\sqrt{c+d\frac{{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right)}{1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right)}}+\sqrt{c+d\frac{{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t1\right)}{1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t1\right)}}}\cdot \frac{\left{p}_{km}^{n}\left(2\right)\left(t1\right){p}_{km}^{n}\left(2\right)\left(t2\right)\right}{\left(1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t\right)\right)\left(1+{\mathsf{\text{SINR}}}_{k,m}^{n}\left(2\right)\left(t1\right)\right)}\\ ={\left(\frac{1}{2a}\sqrt{\frac{d}{c}}\frac{{\left{h}_{b,k}^{n}\right}^{2}}{{\sigma}^{2}+{I}_{0}}\right)}^{\frac{t}{2}}\cdot \left{p}_{km}^{n}\left(2\right)\left(2\right){p}_{km}^{n}\left(2\right)\left(1\right)\right\\ \cdot \prod _{i=2}^{t}\frac{1}{\sqrt{c+d\frac{SIN{R}_{k,m}^{n}\left(2\right)\left(i\right)}{1+SIN{R}_{k,m}^{n}\left(2\right)\left(i\right)}}+\sqrt{c+d\frac{SIN{R}_{k,m}^{n}\left(2\right)\left(i1\right)}{1+SIN{R}_{k,m}^{n}\left(2\right)\left(i1\right)}}}\cdot \prod _{i=2}^{t}\frac{1}{\left(1+SIN{R}_{k,m}^{n}\left(2\right)\left(i1\right)\right)\left(1+SIN{R}_{k,m}^{n}\left(2\right)\left(i1\right)\right)}\end{array}
Imitating the explanation of the counterpart of direct users, the second term is a finite value, and the last two terms are smaller than 1. By adjusting the relationship among a, c and d to make sure \frac{1}{2a}\sqrt{\frac{d}{c}}\frac{{\left{h}_{b,k}^{n}\right}^{2}}{{\sigma}^{2}+{I}_{0}}\le 1, the series {p}_{km}^{n}\left(t+1\right){p}_{km}^{n}\left(t\right) is absolutely convergent. NE is therefore unique.
Conclusively, NE of our game approach is unique. In other words, all of the relays and noncooperative users can automatically choose a strategy of transmitting power, and no one will unilaterally change the power value. But, as far as we are concerned, the Pareto efficiency cannot be ensured because this is a suboptimal scheme.
3.5. Relay selection, subcarrier and power allocation algorithm
According to the discussion above, we can get the progressive optimization of relay selection, subcarrier and power allocation mechanism as Figure 2.