## 3 Transmission Scenarios
### UR-MCJST Scenario  
The \(S\) and CJs play the game known as the leader-follower Stackelberg game. In this game, the source acts as the leader, and CJs are assigned the role of the followers, as shown in Figure 3. The \(S\), which is the leader, will first allocate the time fraction \(\alpha\) to be used during the first and second phases, then the chosen set of CJs \(\mathcal{J}\subseteq\{1,\ldots,N\}\), where \(N\) is the total number of available CJs, as well as the power ratio \(\beta\). Here, \(\alpha\) is normalized to one. The power ratio \(\beta\) will be defined as in [48]:  
\[\beta=\frac{\mathbb{E}[Y^{2}_{\{CJ_{i_{1}},\frac{\alpha}{2}\}}]}{\mathbb{E}[Y^ {2}_{\{CJ_{i_{3}},1-\alpha\}}]}, \tag{1}\]  
where \(\mathbb{E}[Y^{2}_{\{CJ_{i_{1}},\frac{\alpha}{2}\}}]\) is the average power for the \(i\)th CJ denoted as \(CJ_{i}\) for transmitting a jamming signal during the first phase, while \(\mathbb{E}[Y^{2}_{\{CJ_{i_{3}},1-\alpha\}}]\) will be \(CJ_{i}\)'s average power for its own transmission during the third phase. \(K\) is a factor that represents the number of transmitted symbols by the PU denoted as \(S\) for the complete normalized interval. \(S\) transmits the symbols \(X_{\{t_{1},\frac{\alpha}{2}\}}\) during the first time slot only, i.e., we have \(X_{\{t_{2},\frac{\alpha}{2}\}}=0\) and \(X_{\{t_{3},1-\alpha\}}=0\). Consequently, the total power that is represented as \(P_{\text{s}}\) can be calculated by employing the following equation, which is the revised form of Equation (1) in [48]:  
\[P_{\text{s}} = \frac{1}{K}\sum_{t_{1}=1}^{\left\lfloor\frac{\alpha}{2}K\right \rfloor}\mathbb{E}[X^{2}_{\{t_{1},\frac{\alpha}{2}\}}]+\frac{1}{K}\sum_{t_{2} =\left\lfloor\frac{\alpha}{2}K+1\right\rfloor}^{\left\lfloor\frac{\alpha}{2} K+1\right\rfloor}\mathbb{E}[X^{2}_{\{t_{2},\frac{\alpha}{2}\}}] \tag{2}\] \[+\frac{1}{K}\sum_{t_{3}=\left\lfloor\alpha K+1\right\rfloor}^{K }\mathbb{E}[X^{2}_{\{t_{3},1-\alpha\}}]\] \[= \frac{1}{K}\sum_{t_{1}=1}^{\left\lfloor\frac{\alpha}{2}K\right\rfloor }\mathbb{E}[X^{2}_{\{t_{1},\frac{\alpha}{2}\}}]=\frac{\alpha}{2}\,\mathbb{E}[X ^{2}_{\{t_{1},\frac{\alpha}{2}\}}],\]  
and thus, we have:  
\[\mathbb{E}[X^{2}_{\{t_{1},\frac{\alpha}{2}\}}]=\frac{2P_{\text{s}}}{\alpha}\;. \tag{3}\]  
Figure 3: Stackelberg interaction between the PU (leader) and the cooperative jamming node (follower). Furthermore, the CJs will not be transmitting during the second transmission phase, i.e., \(\mathbb{E}[Y^{2}_{\{CJ_{i_{1},2},\frac{6}{2}\}}]=0\). Therefore, the \(i\)th CJ's transmission power will be given by:  
\[P_{CJ_{i}} = \frac{1}{K}\sum_{t_{1}=1}^{\left\lfloor\frac{\pi}{K}\right\rfloor} \mathbb{E}[Y^{2}_{\{CJ_{i_{1},2},\frac{6}{2}\}}]+ \tag{4}\] \[\frac{1}{K}\sum_{t_{2}=\left\lfloor\frac{\pi}{2}K+1\right\rfloor} \mathbb{E}[Y^{2}_{\{CJ_{i_{2},2},\frac{6}{2}\}}]+\] \[\frac{1}{K}\sum_{t_{3}=\left\lfloor\pi K+1\right\rfloor}^{\left \lfloor K\right\rfloor}\mathbb{E}[Y^{2}_{\{CJ_{i_{3},1}-\alpha\}}]\] \[\beta=\frac{\mathbb{E}[Y^{2}_{\{CJ_{i_{1}},\frac{\alpha}{2}\}}]}{\mathbb{E}[Y^ {2}_{\{CJ_{i_{3}},1-\alpha\}}]}, \tag{1}\]0  
By integrating the power ratio \(\beta\) from Equation (1) into Equation (4), we obtain the following expression:  
\[\beta=\frac{\mathbb{E}[Y^{2}_{\{CJ_{i_{1}},\frac{\alpha}{2}\}}]}{\mathbb{E}[Y^ {2}_{\{CJ_{i_{3}},1-\alpha\}}]}, \tag{1}\]1  
where \(\kappa_{1}=\frac{\beta}{\frac{\alpha\beta}{2}+1-\alpha}\) and  
\[\beta=\frac{\mathbb{E}[Y^{2}_{\{CJ_{i_{1}},\frac{\alpha}{2}\}}]}{\mathbb{E}[Y^ {2}_{\{CJ_{i_{3}},1-\alpha\}}]}, \tag{1}\]2  
During the first phase, \(S\) sends the signal \(X_{\{t_{1},\frac{6}{2}\}}\) as its message to the receiver denoted by \(D\) and an untrusted relay \(R\).