# Cooperative Game-Based Energy Efficiency Management over Ultra-Dense Wireless Cellular Networks
## 3 Cooperative Game in Ultra-Dense Wireless Cellular Networks
### Utility Design and Bargaining Cooperative Game Formulation  
In our case, as energy giants of the network, the MeNBs and SeNBs are the players, which cooperate or bargain to divide the available network energy. Thus, bargaining cooperative game-based interference-aware power coordination is formulated to improve network EE. The following theorem shows the existence and uniqueness of the NBS that satisfies the aforementioned axioms. **Theorem 1:**_The bargaining cooperative game is:_  
\[G=\left\langle\left\{MeNB_{m},SeNB_{n},m\in\mathfrak{M},n\in\right\};\left\{p_{ m},p_{n},m\in\mathfrak{M},n\in\right\};u\right\rangle \tag{15}\]  
_For the players (MeNBs and SeNBs), \(m\in M,\ n\in N\), the available actions are adjusting the transmission power \(p_{m}\leq p_{m}^{max},p_{n}\leq p_{n}^{max}\). Here, we introduce the Nash-product function \(u=\prod_{m=1}^{M}u_{m}\prod_{n=1}^{N}u_{n}\), where \(u_{m}\) and \(u_{n}\) are the utility functions for MeNB\({}_{m}\) and SeNB\({}_{n}\), respectively. Assuming a unique and fair solution, \(F\left(\Gamma,d\right)\in\arg\max\limits_{u\in\Gamma,u\geq d}\prod_{m=1}^{M} \left(u_{m}-u_{m}^{min}\right)\prod_{n=1}^{N}\left(u_{n}-u_{n}^{min}\right)\) satisfies all the axioms in Definition 2._  
**Proof:** The proof of this theorem is omitted because of space limitations. A similar detailed proof can be found in [33], which is proven to be efficient and fair. \(\square\)  
**Remark 1:**_It is important to design the utility function of such a game, which must reflect the influence of the changing strategies of the base stations on the network EE. In previous studies, game theory-based power coordination for SE and EE optimization, which achieves an optimal tradeoff between SE and EE, was widely adopted as the utility function. However, the DE, which is another critical issue for measuring the network EE when multiple SeNBs are densely overlaid on the macro-cell, has been neglected in these game-theoretic formulations._  
In order to jointly address SE, DE, and EE, and obtain an optimal tradeoff between them, \(u_{m}\) and \(u_{n}\) are introduced with two adjustable parameters \(\alpha\) and \(\beta\). **Lemma 1:**_The utility function \(u_{m}\) of a player MeNB\({}_{m},\ m\in M\), is defined as \(u_{m}=\left(EE_{m}-EE_{m}^{0}\right)^{\alpha_{m}}\left(DE_{m}-DE_{m}^{0}\right) ^{\beta_{m}}\left(SE_{m}-SE_{m}^{0}\right)^{1-\alpha_{m}-\beta_{m}}.\) The utility function \(u_{n}\) of a player SeNB\({}_{n}\), \(n\in N\), is defined as \(u_{n}=\left(EE_{n}-EE_{n}^{0}\right)^{\alpha_{m}}\left(DE_{n}-DE_{n}^{0}\right) ^{\beta_{n}}\left(SE_{n}-SE_{n}^{0}\right)^{1-\alpha_{n}-\beta_{n}},\) where \(EE_{m}\), \(DE_{m}\), \(SE_{m}\), \(DE_{n}\), and \(SE_{n}\) are the EE, DE, and SE functions, which can be calculated using Equations (9)-(15). \(EE_{m}^{0}\), \(DE_{m}^{0}\) and \(SE_{m}^{0}\) represent the minimum requirements of the EE, DE, and SE, respectively, toward MeNB\({}_{m}\). \(EE_{m}^{0}\), \(DE_{n}^{0}\) and \(SE_{n}^{0}\) represent the minimum requirements of the EE, DE, and SE, respectively, toward the SeNB\({}_{n}\). The coefficients \(0\leq\alpha_{m}\leq 1\), \(0\leq\beta_{m}\leq 1\), \(0\leq\alpha_{n}\leq 1\), and \(0\leq\beta_{n}\leq 1\) are adopted to maintain a balance between the EE, DE, and SE._  
**Remark 2:**_In contrast to the utility function in [31], our utility function is more comprehensive and considers the cost. It reveals the effect of the power-coordination behaviors between different base stations on the network cost. On the other hand, the coefficients \(\alpha_{m}\), \(\beta_{m}\), \(\alpha_{n}\) and \(\beta_{n}\) are offered for players to balance the three energy-efficiency metrics. They are determined by the distribution of users associated with SeNB\({}_{n}\) and MeNB\({}_{m}\). If a base station is associated with a low number of users, the downlink power is coordinated to focus more on the EE and DE; otherwise, the SE is considered.