# Cooperative Game-Based Energy Efficiency Management over Ultra-Dense Wireless Cellular Networks
## 4 Bargaining Problem Solutions for Ultradense Wireless Cellular Networks
### Solution of Cooperative Bargaining Game  
According to the aforementioned analysis, we can reformulate the problem in Equations (30) and (31). For deriving the closed-form solutions of this problem, we assume that a wired backhaul exists to connect the MeNBs to the SeNBs; that is, the interference information can be exchanged between them for cooperative power coordination. This assumption was made in [31]. The general EE-coordination problem in the presence of inter-tier and intra-tier interference is difficult to solve even under the precondition of ideal information exchange. Hence, a simplified scheme is proposed to reduce the information-exchange overhead, which solves the aforementioned problem by updating the transmission power. **Corollary 3**: _The closed-form power solution of P-MeNB in Equation (30) is given by:_  
\[p_{m}^{*}=\frac{1-\alpha_{m}-\beta_{m}}{\sum\limits_{n=1}^{N}\sum\limits_{j=1 }^{N}\frac{g_{n,(n,j)}}{g_{nj}}\frac{p_{nj}^{MAX}}{h_{n}}+\lambda_{m}} \tag{32}\]  
_where \(\lambda_{m}\) is the Lagrange multiplier corresponding to the MeNB\({}_{m}\) power constraint._  
**Proof:** For the primal problem P3 expanded in Equation (30), introducing the Lagrangian multipliers \(\lambda_{m}\) corresponding to the MeNB\({}_{m}\) power constraint \(0\leq p_{m}\leq p_{m}^{MAX}\), which yields:  
\[L_{m}=\sum\limits_{m=1}^{M}\left\{\text{ln}p_{m}-(\alpha_{m}+\beta_{m})\text {ln}(a_{sistem}*p_{m}+b_{sistem})\right\}+\sum\limits_{n=1}^{N}\text{ln}h_{n }-\sum\limits_{m=1}^{M}\lambda_{m}(p_{m}-p_{m}^{MAX}) \tag{33}\]  
On the basis of standard optimization techniques and the Karush-Kuhn-Tucker (KKT) conditions [26], the power allocation for MeNB\({}_{m}\) is determined by obtaining the first-order derivative of Equation (33) with respect to \(p_{m}\), which can be given as:  
\[\frac{\partial L_{m}}{\partial p_{m}}=\frac{1}{p_{m}}-\frac{(\alpha_{m}+\beta _{m})\cdot a_{sistem}}{a_{sistem}*p_{m}+b_{sistem}}+\sum\limits_{n=1}^{N}\frac{1 }{h_{n}}\frac{\partial h_{n}}{\partial p_{m}}-\lambda_{m} \tag{34}\]According to the definition of \(h_{n}=\sum\limits_{j=1}^{I}h_{n,j}\) and the definition of \(h_{n,j}\) in Equation (5), we obtain:  
\[ \begin{array}{l}\frac{\partial h_{n}}{\partial p_{m}}=\frac{\partial}{\partial p _{m}}\left\{\sum\limits_{j=1}^{I}h_{n,j}\right\}=\sum\limits_{j=1}^{J}\frac{ \partial h_{n,j}}{\partial p_{m}}\\ =\sum\limits_{j=1}^{I}\frac{\partial}{\partial p_{m}}\frac{\sum\limits_{n^{ \prime}\
eq n,j}p_{n^{\prime}}s_{n^{\prime},(n,j)}+\sum\limits_{m=1}^{M}p_{m} s_{m,(n,j)}+\sigma_{n,j}^{2}}{\sum\limits_{n^{\prime}\
eq n,j}^{2}p_{m}s_{m,(n,j)}+ \sigma_{n,j}^{2}}\\ =-\sum\limits_{j=1}^{I}\frac{S_{m,(n,j)}}{S_{n,j}}h_{n,j}^{2}\end{array}  \tag{35}\]  
By substituting Equation (35) into Equation (34), we obtain:  
\[ \begin{array}{l}\frac{\partial L_{m}}{\partial p_{m}}=\frac{1}{p_{m}}-\frac{ (\alpha_{m}+\beta_{m})\cdot a_{istem}}{a_{istem}*p_{m}+b_{istem}}-\sum\limits_{n=1 }^{N}\frac{1}{h_{n}}\sum\limits_{j=1}^{J}\frac{S_{m,(n,j)}}{S_{n,j}}h_{n,j}^{2 }-\lambda_{m}\\ =\frac{1}{p_{m}}-\frac{(\alpha_{m}+\beta_{m})\cdot a_{istem}}{a_{istem}*p_{m}+b_{istem}}-\sum\limits_{n=1}^{N}\sum\limits_{j=1}^{J}\frac{h_{n,j}^{2 }}{h_{n}}\frac{S_{m,(n,j)}}{S_{n,j}}-\lambda_{m}\end{array}  \tag{36}\]  
According to the analysis in [15], the circuit power consumption \(b_{istem}\) is constant. Here, to obtain the closed-form solution of P-MeNB in Equation (30), we assume that \(b_{istem}=0\). Then,  
\[ \begin{array}{l}\frac{\partial L_{m}}{\partial p_{m}}=\frac{1}{p_{m}}-\frac {(\alpha_{m}+\beta_{m})\cdot a_{istem}}{a_{istem}*p_{m}}-\sum\limits_{n=1}^{N} \sum\limits_{j=1}^{J}\frac{h_{n,j}^{2}}{h_{n}}\frac{S_{m,(n,j)}}{S_{n,j}}- \lambda_{m}\\ =\frac{1}{p_{m}}-\frac{(\alpha_{m}+\beta_{m})}{p_{m}}-\sum\limits_{n=1}^{N} \sum\limits_{j=1}^{J}\frac{h_{n,j}^{2}}{h_{n}}\frac{S_{m,(n,j)}}{S_{n,j}}- \lambda_{m}\end{array}