# Hierarchical Resource Management for Mega-LEO Satellite Constellation
## 6 Approaches and Algorithms
### Game-Based Spectrum Planning
This state is known as the Nash equilibrium, which can be expressed as follows:  
\[G_{i}(p^{*},q_{i}^{*},q_{-i}^{*})\geq G_{i}(p^{*},q_{i},q_{-i}^{*}) \tag{2}\]  
where \(G_{i}\) and \(q_{i}^{*}\) are the revenue and optimal demand strategy of the \(i\)-th SBS, respectively, and \(q_{-i}^{*}\) is the optimal demand strategy of SBSs other than the \(i\)-th SBS. The first-order partial derivative of any \(i\)-th SBS's revenue function can be achieved as follows:  
\[\frac{\partial G_{i}(p,q_{i},q_{-i})}{\partial q_{i}}=\frac{a}{1+q_{i}}-\frac {b}{Q}-\frac{Pr^{*}(SBS_{i})}{(C-q_{i})^{2}} \tag{3}\]  
where \(a\) is a numerical value between [1; 2], which is related to the user's QoS requirement (the larger the \(a\) value, the higher the user's QoS requirement, the more urgent its demand for physical resources, and the more it pays to the RRSC), and \(b\) is a constant greater than zero. \(\mathcal{C}\) represents the maximum amount that the RRSC can provide of a certain resource. \(Pr^{*}(SBS_{i})\) is the normalization importance quantification of the \(i\)-th SBS that is carrying out a service, which can be achieved as in [45]. The second-order partial derivative of any \(i\)-th SBS's revenue function can be achieved as follows:  
\[\frac{\partial^{2}G_{i}(p,q_{i},q_{-i})}{\partial q_{i}^{2}}=\frac{a}{1+q_{i} }-\frac{Pr^{*}(SBS_{i})}{(C-q_{i})^{3}}<0 \tag{4}\]  
The second order partial derivative is less than 0 so that the revenue function of an arbitrary SBS is strictly concave, which ensures the existence of the Nash equilibrium. Therefore, the revenue function \(G_{i}(p,q_{i},q_{-i})\) of the SBS exists at the non-cooperative game equilibrium point \(q^{*}\) for the price strategy \(p\) given by the RRSC. Reverse induction is a commonly used method to solve for the optimal strategy in Stackelberg games. However, this method typically requires participants to have complete information about the strategies and payoffs of all other participants. In the context of the above problem, where such complete information may not be available, reverse induction is not suitable. Therefore, we use a distributed iterative method to find the Nash equilibrium in Stackelberg games. Assuming that the RRSC announces a pricing strategy at time \(t\), SBSs need to adjust their resource demand strategies to achieve a Nash equilibrium solution. In this context, a Nash equilibrium refers to a set of strategies where no SBS can improve its utility by unilaterally changing its strategy. The resource demand strategy of each SBS directly determines its revenue function, which is modeled using a simple dynamic approach. Specifically, the change rate of an SBS's resource demand is proportional to the first-order deviation of its revenue function from its optimal level. This relationship is used to model the dynamic adjustment of resource demand strategies by SBSs in response to the pricing strategy announced by the RRSC. \[\Delta q_{i}=\frac{\partial G_{i}(p,q_{i},q_{-i})}{\partial q_{i}} \tag{5}\]  
Furthermore, to model the dynamic adjustment of resource demand strategies by SBSs over time, we introduce a resource demand strategy iteration equation. This equation describes the change in the resource demand strategy of the \(i\)-th SBS between the current moment \(t\) and the next moment \(t+1\), which can be expressed as follows:  
\[q_{i}(t+1)=q_{i}(t)+\theta_{i}\Delta q_{i} \tag{6}\]  
where \(\theta_{i}\) represents the resource demand's adjustment step (or learning rate) for the \(i\)-th SBS. This step size will affect the speed of convergence towards the equilibrium state. An important property of the system is that the revenue function is concave, which ensures that the iterative algorithm used to update resource demands can converge to the Nash equilibrium of the game. The Nash equilibrium is a stable state where no SBS has an incentive to unilaterally change its resource demand given the resource demands of the other SBSs.