## 3 Bimatrix Stackelberg Game
### Optimal Mixed Strategy for UAV  
The Stackelberg game articulated in Problem (11) can be reformulated as a binary integer programming problem. Such problems are generally NP-hard, posing computational challenges. According to the minimax theorem [36], a set of mixed strategies in zero-sum games is a Nash equilibrium if and only if both strategies are minimax strategies. It is well-established that linear programming can efficiently find such minimax strategies within polynomial time [37].  
After spectrum sensing, the UAV and the jammer construct their respective payoff matrices, denoted as \(\mathbf{A}\) and \(\mathbf{B}\). These matrices are defined as  
\[\mathbf{A}=\begin{bmatrix}u_{U}(f_{U1}^{c},f_{J\,1}^{c})&\cdots&u_{U}(f_{U1}^{c},f _{J\,M}^{c})\\ \vdots&\ddots&\vdots\\ u_{U}(f_{UM^{\prime}}^{c},f_{J\,1}^{c})&\cdots&u_{U}(f_{UM^{\prime}}^{c},f_{J \,M}^{c})\end{bmatrix}, \tag{12}\]\[B=\begin{bmatrix}u_{f}(f^{c}_{U\Pi^{\prime}},f^{c}_{I\,1})&\cdots&u_{f}(f^{c}_{U \Pi^{\prime}},f^{c}_{I\,M})\\ \vdots&\ddots&\vdots\\ u_{f}(f^{c}_{U\Pi^{\prime}},f^{c}_{I\,1})&\cdots&u_{f}(f^{c}_{U\Pi^{\prime}},f^{ c}_{I\,M})\end{bmatrix}. \tag{13}\]  
Leveraging the principles of the minimax theorem, the complex interaction that transpires between the UAV and the jammer at the onset of each discrete time slot is elegantly reformulated into a bimatrix Stackelberg game framework. Within this mathematical setting, the pursuit of identifying an optimal mixed strategy for the UAV effectively becomes synonymous with discovering a minimax strategy, a strategy that seeks to minimize the maximum expected utility, symbolically represented as \(u_{I}\), which the adversarial jammer can potentially attain.  
In the game of the UAV and the jammer, an optimal mixed strategy to which to commit can be found in polynomial time using linear programming.  
Proof of Theorem.: For the jammer's every pure strategy, i.e.,  
\[\pi_{I}\in\Pi_{I}=\underbrace{[1,0,\cdots,0]^{T},[0,1,0,\cdots,0]^{T},\cdots,[0,\cdots,0,1]^{T}}_{M}, \tag{14}\]  
where \(M\) is the total number of strategies and \(1\) indicates that the jammer may choose the corresponding center frequency \(f^{c}_{J}\), we compute a mixed strategy for the UAV:  
\[\pi_{U}=[p_{f^{c}_{U\Pi^{\prime}}},p_{f^{c}_{U\Pi^{\prime}}2},\cdots,p_{f^{c}_ {U\Pi^{\prime}}m},\cdots,p_{f^{c}_{U\Pi^{\prime}}M}]^{T}, \tag{15}\]  
where \(p_{f^{c}_{U\Pi^{\prime}}m}\) represents the probability that the UAV would choose the center frequency \(f^{c}_{U\Pi^{\prime}}\), with \(\sum_{m=1}^{M}p_{f^{c}_{U\Pi^{\prime}}m}=1\). Two prerequisites must be satisfied: (1) playing \(\pi_{I}\) is the best response for the jammer, and (2) the mixed strategy maximizes the UAV's utility under constraint (1). Such a mixed strategy can be produced by the following linear program:  
\[\begin{split}&\max\limits_{\pi_{U}\in\Pi_{U}}\quad\sum\limits _{m=1}^{M}p_{f^{c}_{U\Pi^{\prime}}m}u_{U}(f^{c}_{U\Pi^{\prime}},f^{c}_{\pi_{I}} ),\\ &\text{s.t.}\left\{\begin{array}{l}\sum\limits_{m=1}^{M}p_{f^{c} _{U\Pi^{\prime}}m}u_{f}(f^{c}_{U\Pi^{\prime\prime}},f^{c}_{J})\geq\sum\limits _{m=1}^{M}p_{f^{c}_{U\Pi^{\prime}}m}u_{f}(f^{c}_{U\Pi^{\prime\prime}},f^{c^{ \prime}}_{J}),\\ \sum\limits_{m=1}^{M}p_{f^{c}_{U\Pi}m}=1,\\ p_{f^{c}_{U\Pi}m}\geq 0.\end{array}\right.\end{split} \tag{16}\]  
For each possible strategy \(\pi_{I}\in\Pi_{I}\) of the jammer, the UAV considers it as the jammer's optimal response, meaning that the jammer's payoff from selecting the central frequency \(f^{c}_{J}\) is greater than payoffs of any other central frequency \(f^{c^{\prime}}_{J}\). Then, the UAV computes the corresponding optimal mixed strategy. Among all the optimal objective function values obtained, the UAV selects the one that maximizes its own payoff as the final optimal payoff. The corresponding mixed strategy represents the optimal strategy for the UAV, while the pure strategy represents the optimal response strategy for the jammer.