## 4 Methodology
### Cooperative Resource Allocation Method Based on Nash Bargaining Game and Lagrangian Optimization
#### 4.1.3 Dynamic Adjustment Mechanism Based on Game Model
Similarly, if \(U_{n}\) is also concave with respect to \(W_{n}\) (which is typically the case for resource allocation problems involving Shannon capacity-based utility functions), then each node's utility function \(U_{n}\) is concave. Now, considering the overall utility maximization problem,  
\[ \begin{split}&\max_{\mathbf{P},\mathbf{W}}U=\sum_{n=1}^{K}U_{n}(P_{n},W _{n}),\\ &\text{s.t.}\quad\sum_{n=1}^{K}P_{n}\leq P_{\text{max}},\quad \sum_{n=1}^{K}W_{n}\leq W_{\text{max}}.\end{split}  \tag{21}\]  
Since the sum of concave functions remains concave, and as the constraints are linear, the optimization problem remains a convex optimization problem. Thus, it can be efficiently solved using standard convex optimization methods. Theorem 2: _Given the interference constraints and power limitations, the system's utility can be optimized using the Nash bargaining solution in the game, and the utility of each node will not be lower than its minimum utility._  
Proof: First, we assume that the total utility of the system is the sum of the utilities of all nodes. Each node \(n\)'s utility function \(U_{n}\) is given by  
\[U_{n}=\sum_{f=1}^{F}\sum_{t=1}^{T}\frac{\delta_{n,f,t}\gamma_{n,f,t}}{(1+P_{n,f,t}\gamma_{n,f,t})\ln 2} \tag{22}\]  
According to Nash bargaining game theory, the total utility needs to be maximized, i.e., solving the following maximization problem to obtain the optimal solution:  
\[U^{*}=\arg\max_{P_{n},W_{n}}\prod_{n=1}^{K}(U_{n}-U_{n}^{\text{min}}) \tag{23}\]This problem has constraints on power and bandwidth maximums (i.e., \(P_{n}\leq p_{n}^{\max}\) and \(W_{n}\leq W_{n}^{\max}\)). By introducing Lagrange multipliers, we can transform this problem into a dual problem involving power and bandwidth constraints, as well as maximizing the utility function. Consider the following optimization problem:  
\[ \begin{array}{rl}\max_{P_{n},W_{n}}&\prod_{n=1}^{K}\left(U_{n}-U_{n}^{\min} \right)\\ \text{s.t.}&\sum_{n=1}^{K}P_{n}\leq P_{\text{total}}\\ &\sum_{n=1}^{K}W_{n}\leq W_{\text{total}}\\ &P_{n}\leq P_{n}^{\max},\quad\forall n\in\{1,...,K\}\\ &W_{n}\leq W_{n}^{\max},\quad\forall n\in\{1,...,K\}\end{array}  \tag{24}\]  
The Lagrangian function (25) converts the constrained optimization problem (24) into an unconstrained form by introducing dual variables \(\lambda\) and \(\mu\). Here, \(\lambda\) and \(\mu\) penalize violations of the total power and bandwidth constraints. \[L=\arg\max_{P_{n},W_{n}}\prod_{n=1}^{K}(U_{n}-U_{n}^{\min})+\lambda\left(\sum _{n=1}^{K}P_{n}-P_{\text{total}}\right)+\mu\left(\sum_{n=1}^{K}W_{n}-W_{\text{ total}}\right) \tag{25}\]  
By optimizing the Lagrange multipliers, we can find the optimal power and bandwidth allocation, proving that the game process can maximize the system's total utility, while satisfying the minimum utility constraints of each node. To achieve a fair and efficient resource distribution among satellite nodes, the proposed CMPT framework integrates a Nash Bargaining Game model that simulates the resource competition between satellites and users. The detailed implementation of this game-theoretic resource allocation process is presented in Algorithm 1. ```
1:Input: satellite nodes \(N\), ground station nodes \(M\), channel gains \(\gamma_{ij}\), power limits \(P_{n}^{\max}\), bandwidth limits \(W_{n}^{\max}\). 2:Output: Initial resource allocation \((P_{n}^{*},W_{n}^{*})\) for each satellite node \(n\). 3:for each node \(n\in\{1,2,\ldots,N+M\}\)do
4:\(U_{n}(P_{n},W_{n})=f_{n}(P_{n},W_{n},I_{n})\);
5:for each time slot \(t\)do
6:for each frequency \(f\)do
7: Calculate utility as Equation (22);
8:endfor
9:endfor
10: Compute Nash bargaining solution defined in Equation (23);
11:endfor
12:Return\((P_{n}^{*},W_{n}^{*})\).