## 4 Optimization and Analysis
Meanwhile, constraints on the average transmission and number of concurrent EH-STs need to be taken into consideration. For a fixed network size \(U\), the Lagrange operator decreases with the increase of \(M\); however, due to the concavity of \(g(\eta)\), larger \(M\) will increase the packet loss rate, which is negative to network throughput. Moreover, the larger the throughput or transmission probability of other EH-STs, i.e., \(G(\eta^{*})\) or \(P(\eta^{*})\), the larger the Lagrange operator \(\Gamma(\eta^{*})\); thus, a stringent transmission probability is needed to reduce the packet collisions. The Lagrange operator optimally controls the transmission probability of each EH-ST. Therefore, in order to achieve the maximum throughput of the network, the average transmission probability needs to be properly designed based on the Lagrange operator. To solve Equation (25), we simplified it to a more general optimization problem, which is represented as follows. For \(\gamma\geq 0\):  
\[\eta^{(\gamma)}=\operatorname*{arg\,max}_{\eta\in\Omega}\left[G(\eta)-\gamma P (\eta)\right], \tag{27}\]  
and \(G(\eta)-\gamma P(\eta)=\sum_{e=1}^{e_{max}}\pi_{\eta}(e)\left[g(\eta(e))-\gamma \eta(e)\right]\). Based on this simplified optimization problem, we can further prove the existence and uniqueness of the symmetric NE in Equation (25), thus finding the solution of Equation (25), which is locally optimal for the original optimization problem (Equation (22)). Due to the concavity of \(g(\eta)\)[10], \(\eta^{(\gamma)}\) has the following properties. **Proposition 1**. : _(1) \(\eta^{(\gamma)}\) is uniquely defined, i.e.,_  
\[G(\eta^{(\gamma)})-\gamma P(\eta^{(\gamma)})>G(\eta)-\gamma P(\eta),\forall \eta\
eq\eta^{(\gamma)}; \tag{28}\]  
_(2) \(\eta^{(\gamma)}\) is continuous in \(\gamma\); (3) \(0<P(\eta)\leq\beta(B\geq I)\), \(G(\eta)\geq G(\eta_{PLR_{th}})\)._  
**Remark 2**. : _The first property can be proved by the concavity of \(g(x)\); thus, \(x_{\gamma}^{*}=\operatorname*{arg\,max}_{x\in[0,1]}g(x)-\gamma x\) has a unique solution. Due to the continuity of \(g(x)\), \(\eta^{(\gamma)}\) is continuous in \(\gamma\). For the third property, the transmission probability \(\eta\in\Omega\) cannot be larger than the energy harvesting rate \(\beta(B\geq I)\), where \(B\) is number of acquired energy units and \(I\) is the number of transmitted packet replicas. Moreover, \(g(\eta)\) is a concave function which is related to \(\mathit{PLR}\), and the relationship between \(g(\eta)\) and \(\mathit{PLR}\) is dictated in Equation (9). Since \(\mathit{PLR}\) increases with the traffic load increasing, \(\mathit{PLR}\) should be below a \(\mathit{PLR}\) threshold. Because a large \(\mathit{PLR}\) means more packets cannot be detected successfully, which will cause a large delay penalty for the satellite networks,is the maximum packet transmission probability that ensures the \(\mathit{PLR}\) of the network the being below the \(\mathit{PLR}\) threshold._  
From Equations (25) and (27), we can obtain that \(\eta^{*}\) is optimal for Equation (25) if and only if \(\eta^{*}=\eta^{(\gamma^{*})}\), for \(\gamma^{*}>0\), and \(\Gamma(\eta^{(\gamma^{*})})=\gamma^{*}\). In order to prove the existence of the unique solution of Equation (25), we need to prove the following propositions first. Proposition 2: \(P(\eta^{(\gamma)})\) _is a non-increasing function of \(\gamma\), and \(P(\eta^{(0)})>0\), \(P(\eta^{(\infty)})=0\)._  
Proof of Proposition 2: See Appendix A. Proposition 3: \(\Gamma(\eta^{(\gamma)})\) _is a non-decreasing, continuous function of \(\gamma\), and \(\Gamma(\eta^{(0)})>0\), \(\Gamma(\eta^{(\infty)})=g^{\prime}(0)\)._  
Proof of Proposition 3: See Appendix B. Since the proposed policy is a symmetric policy, we further analyze the characteristics of the symmetric NE condition for the game. According to the above propositions, the following theorem proves the existence and uniqueness of the symmetric NE, i.e., the solution of Equation (25).