## 4 Optimization and Analysis  
The optimization problem (Equation (22)) can be separated into two parts. The term \(M\cdot G(\eta)\) represents the expected network total throughput when there are \(M\) EH-STs co-transmitting packets in the same frame, and the term \(\binom{U}{M}P(\eta)^{M-1}\left(1-P(\eta)\right)^{U-M}\) is the probability of \(M\) EH-STs transmitting packets concurrently in the same frame, which can be regarded as a binomial operator. Moreover, \(g(\eta)\) is concave; thus, Equation (22) is a nonconvex optimization problem. In order to determine the approximated solutions of Equation (22), this paper exploits a game theoretic method which is mentioned in Reference [36] to formulate the random access procedure. Specifically, we use a game theoretic method to model the optimization problem, where each EH-ST \(u\) acts as a player which optimizes its own transmission policy \(\eta_{u}\) to maximize the network throughput (Equation (19)). This section first analyzes the characteristics of the general Nash equilibrium (NE) of this game. The transmission policy profile is defined as \(\eta^{\star}=(\eta_{1}^{\star},\eta_{2}^{\star},\ldots,\eta_{U}^{\star})\), which is the joint policy. According to the definition of NE, if any EH-ST \(u\) adopts the policy \(\eta_{u}\
eq\eta_{u}^{\star}\), while all the other EH-STs use the policy \(\eta_{i}^{\star}\) (\(i\
eq u\)) to achieve the NE, then a smaller network throughput is obtained. The NE condition must satisfy all the EH-STs, and no player can improve the throughput by deviating unilaterally, i.e.,  
\[T_{\eta_{u}^{\star},\eta_{-u}^{\star}}\geq T_{\eta_{u},\eta_{-u}^{\star}}, \forall u\in U,\forall\eta_{u}\in\Omega^{U}. \tag{23}\]  
By the definition of NE, if an NE (not necessarily symmetric) exists for the game, it must solve \(\forall u\):  
\[\eta_{u}^{\star}=\underset{\eta_{u}\in\Omega}{\arg\max}\,T_{\eta_ {u},\eta_{-u}^{\star}} \tag{24}\] \[=\underset{\eta_{u}\in\Omega}{\arg\max}\,\mathbb{E}\Bigg{[}\binom{ U}{M}\Bigg{]}\Bigg{[}G(\eta_{u})\prod_{i\
eq u,j}P(\eta_{i}^{\star})\prod_{j\
eq u,i} \left(1-P(\eta_{j}^{\star})\right)\] \[+(1-P(\eta_{u}))\Bigg{(}\sum_{n\
eq u,i}G(\eta_{n}^{\star})\prod_ {i\
eq u,j}P(\eta_{i}^{\star})\prod_{j\
eq u,i,n}\left(1-P(\eta_{j}^{\star}) \right)\] \[-\sum_{m\
eq u,j}G(\eta_{m}^{\star})\prod_{i\
eq u,j,m}P(\eta_{i} ^{\star})\prod_{j\
eq u,j}\left(1-P(\eta_{j}^{\star})\right)\Bigg{)}\Bigg{]}\] \[=\underset{\eta_{u}\in\Omega}{\arg\max}\,G(\eta_{u})-P(\eta_{u}) \Bigg{(}\sum_{n\
eq u,i}\frac{G(\eta_{u}^{\star})}{1-P(\eta_{j}^{\star})}- \sum_{m\
eq u,j}\frac{G(\eta_{m}^{\star})}{P(\eta_{i}^{\star})}\Bigg{)}\,.\]In the last step, the optimization formula is divided by the term \((\prescript{M}{}{U})\prod_{j\
eq u,j}P(\eta_{i}^{*})\prod_{j\
eq u,j}\left(1{-}P( \eta_{j}^{*})\right)\), and we remove the additive term, which is independent of \(\eta_{u}\). Furthermore, we impose the symmetric policy \(\eta_{u}^{*}=\eta^{*},\forall u\), and obtain the symmetric NE as follows:  
\[\eta^{*}=\operatorname*{arg\,max}_{\eta\in\Omega}G(\eta)-\Gamma(\eta^{*})P( \eta), \tag{25}\]  
where \(\Gamma(\eta)\) is defined as:  
\[\Gamma(\eta)=G(\eta)\left(\frac{U-M}{1-P(\eta)}-\frac{M-1}{P(\eta)}\right). \tag{26}\]  
\(\eta^{*}\) in Equation (25) is the policy that is simultaneously optimal for all the EH-STs, and any single EH-ST unilaterally deviates from the equilibrium condition \(\eta^{*}\) yielding a smaller network throughput. \(G(\eta)\) in Equation (25) is the throughput contributed by EH-ST \(u\) when there are \(M\) EH-STs accessing the channel simultaneously. The term \(\Gamma(\eta^{*})\) can be regarded as a Lagrange operator, which is associated to control the transmission probability of EH-ST \(u\) and further controls the number of concurrent EH-STs and packet loss rate due to packet collisions. Therefore, the overall objective of Equation (25) is to maximize the individual throughput, so as to maximize the network throughput.