## 4 Methodology
### Cooperative Resource Allocation Method Based on Nash Bargaining Game and Lagrangian Optimization
#### 4.1.2 Assumptions  
In constructing the model, we make the following assumptions to simplify the analysis and ensure the feasibility of the resource allocation framework:  
**Assumption 1** (Information sharing).: _All nodes can share sufficient information, including utility functions, resource allocations, and environmental states. Specifically, each node's utility function \(U_{n}(P_{n},W_{n})\) is known to all, and each node can observe the network's current state, including interference levels \(I_{n}\) and channel conditions \(\gamma_{ij}\)._  
This assumption simplifies the modeling process and ensures that the Nash bargaining solution can be computed globally. However, in large-scale satellite networks, full information sharing may not be realistic, due to communication delays and overhead. This implies potential scalability challenges when the network grows.  
**Assumption 2** (Interference model).: _The interference \(I_{n,m}\) experienced by node \(n\) from node \(m\) is given by_  
\[I_{n,m}=\sum_{m\
eq n}\frac{P_{n}\gamma_{n,m}}{d_{n,m}^{n}}, \tag{16}\]  
_where \(P_{n}\) is the power allocated to node \(m\), \(\gamma_{n,m}\) is the channel gain between nodes \(n\) and \(m\), and \(d_{n,m}\) is the distance between them._  
This assumption models interference as distance-based attenuation, which is practical and widely used. It enables analytical tractability, but neglects complex factors like atmospheric conditions, multi-path fading, or directional antennas. This may lead to inaccuracies in environments with strong physical dynamics or non-line-of-sight scenarios.  
**Assumption 3**.: _The utility function of each node \(n\) incorporates the following interference:_  
\[U_{n}(P_{n},W_{n})=f_{n}(P_{n},W_{n},I_{n}), \tag{17}\]  
_where \(f_{n}(P_{n},W_{n},I_{n})\) is a function depending on the power, bandwidth, and interference._  
By explicitly modeling interference in the utility function, this assumption ensures that the resource allocation considers the mutual impact between nodes. This reflects realistic network competition but also increases model complexity. Any deviation from the assumed interference model could directly affect the fairness and efficiency of the Nash bargaining solution.  
**Assumption 4** (Bounded Rationality and Limited Knowledge).: _Satellite nodes and user terminals operate under bounded rationality, possessing limited knowledge about global channel states and external interference conditions. Each node makes decisions based on estimated local channel gains \(\gamma_{ij}\) and interference levels \(I_{n}\), where estimation errors are bounded within known margins \(e_{\gamma}\) and \(e_{1}\), respectively. Nodes periodically communicate these estimated values to the SDN controller, which aggregates and utilizes this information in Nash bargaining equilibrium computations._  
Based on Assumptions 3 and 4, we incorporate bounded rationality and limited knowledge, enhancing the realism and applicability of the proposed model, and enabling the Nash bargaining equilibrium to better reflect network behaviors under realistic operational conditions. Consequently, this refinement provides critical insights into equilibrium stability and robustness against channel variability and external interference, addressing key considerations for practical deployment scenarios.