## 4 Methodology
### Cooperative Resource Allocation Method Based on Nash Bargaining Game and Lagrangian Optimization
#### 4.1.3 Dynamic Adjustment Mechanism Based on Game Model  
In this study, we adopt the Nash Bargaining Game (NBG) model for resource allocation in satellite networks, due to its ability to ensure fairness, adaptability, and efficiency in dynamic environments. Unlike traditional non-cooperative game models, which may lead to suboptimal global outcomes due to selfish decision-making by individual nodes, the NBG model guarantees a Pareto-optimal solution, while ensuring that each participant receives a utility greater than their disagreement point, thereby preventing unfair allocations. Additionally, NBG facilitates dynamic resource allocation by allowing adjustments in response to real-time changes in network topology, ensuring robustness under fluctuating resource constraints. From an optimization perspective, the convexity of the utility function within the NBG framework enables efficient problem-solving using Lagrangian multipliers and subgradient methods, leading to a globally optimal resource distribution. Under the assumptions stated above--including Assumption 4 regarding channel variability, external interference, and limited player knowledge--the Nash bargaining solution is shown to be stable and yields a fair and efficient resource allocation. In scenarios where these factors deviate significantly, adjustments to the disagreement point or utility estimation may be necessary. Given these advantages, the NBG model is particularly well-suited for our proposed collaborative multi-path resource allocation framework, providing a balance between fairness, efficiency, and computational feasibility in large-scale satellite networks. Let there be \(K\) participants, where each node \(n\)'s utility function \(U_{n}\) satisfies the following conditions:  
1. Individual rationality: For each node \(n\), its utility \(U_{n}\) cannot be lower than some minimum utility \(U_{n}^{\min}\), i.e., \(U_{n}^{*}\geq U_{n}^{\min}\). 2. Feasibility: The solution to the utility function \(U^{*}=(U_{1}^{*},U_{2}^{*},\ldots,U_{K}^{*})\) must lie within the utility space \(U\). 3. Pareto optimality: The resource allocation is Pareto optimal, meaning no node can increase its utility without making others worse off. In the Nash bargaining game, the optimal solution \(U^{*}\) can be obtained by maximizing the utility difference:  
\[U^{*}=\arg\max_{U\in U,U_{2}\geq U_{\min}}\prod_{k=1}^{K}(U_{k}-U_{k}^{\min}) \tag{18}\]  
This formula maximizes the utility difference for each node, to ensure optimal resource allocation. To further refine the resource allocation process, we introduce the following lemmas and theorems to support the mathematical derivation of our approach. Lemma 1: _If the utility function \(U_{n}\) for each node is concave with respect to resource allocation constraints, then the overall utility maximization problem is a convex optimization problem._  
Proof: The utility function \(U_{n}\) of each node typically depends on the allocated power \(P_{n}\) and bandwidth \(W_{n}\). To account for differences in satellite capacities, we normalize the utility by the total available resource capacity \(R_{n}\). Specifically, the normalized utility function is given as follows:  
\[U_{n}=\frac{1}{R_{n}}\sum_{f=1}^{F}\sum_{t=1}^{T}\frac{\delta_{n,f,t}\gamma_{n,f,t}}{(1+P_{n,f,t}\gamma_{n,f,t})\ln 2}. \tag{19}\]  
where \(\gamma_{n,f,t}\) represents the channel gain, \(P_{n,f,t}\) is the power allocated to node \(n\) at frequency \(f\) and time slot \(t\), and \(\delta_{n,f,t}\) denotes the resource availability. To determine the concavity of \(U_{n}\), we compute the second-order derivative of \(U_{n}\) with respect to \(P_{n,f,t}\):  
\[\frac{\partial^{2}U_{n}}{\partial P_{n,f,t}^{2}}=-\frac{\delta_{n,f,t}\gamma_ {n,f,t}^{3}}{(1+P_{n,f,t}\gamma_{n,f,t})^{2}\ln 2}. \tag{20}\]  
Since this second-order derivative is always non-positive, \(U_{n}\) is a concave function with respect to \(P_{n}\).