## 4 Relative Jammer Screening Algorithm Based on Mixed-Strategy Game
### Nash Equilibrium Point Solution  
The primary jammer evaluates the system gains using the constructed utility function. All primary jammers play a hybrid game for resource allocation optimisation. Solving the Nash equilibrium under mixed strategies requires assigning a probability to each pure strategy. A pure strategy \(X_{ip}\) for the \(i_{p}\)th primary jammer is selected with the probability \(P^{ip}=\left\{p_{1}^{ip},p_{2}^{ip},\ldots,p_{K}^{ip}\right\}\), where \(p_{k}^{ip}\geq 0,\sum_{k=1}^{K}P_{k}^{ip}=1\) should be satisfied.  
Given the strategy formula \(G=\left\{N,\left\{X_{ip}\right\},\left\{p_{K}^{ip}\right\}\right\}\) and the mixed-strategy outcome \(\left\{p_{1},P_{2},\ldots,p_{K}\right\}\in p^{ip}\), the expected value of the return obtained in the mixed-strategy situation selected by the \(i_{p}\)th primary jammer is shown in (13).  
\[E\left(P_{k}^{ip}\right)=\sum_{\alpha\in X}p^{ip}(x)P_{ip}(x)=\sum_{\alpha=\{x _{1},x_{2},\ldots,x_{K}\}\in X}p_{1}(x_{1})\cdot p_{2}(x_{2})\cdot\ldots\cdot p _{K}(x_{K})P_{ip}(x) \tag{13}\]  
The probabilistic equation for the game decision is established based on the fact that all decisions have the same expected return, along with the equation \(\sum_{k=1}^{K}p_{k}^{ip}=1\). This system of multivariate nonlinear equations is solved with the help of [PERSON]'s iterative method to obtain the corresponding situation probability \(p^{ip}\). The initial value of the [PERSON] iteration needs to satisfy \((0,1)\), the iteration threshold to be set to \(55\), and the convergence condition to be \(\max\left(\left|x_{i}^{m}-x_{i}^{m-1}\right|\right)\leq 10^{-3}\).  
The existence of Nash equilibrium points for the mixed game requires the help of [PERSON]'s immovable point theorem. The mixed-strategy probability space can be expressed as Equation (14).  
\[\sum_{ip}=\left\{p^{ip}\in R^{|S_{ip}|}|p^{ip}(x)\geq 0,\sum_{x\in S_{ip}}p^{ip} (x)=1\right\} \tag{14}\]  
From Equation (14), each \(\sum_{ip}\) is a simplex. The overall strategy probability space is \(\sum=\sum_{1}\times\sum_{2}\times\ldots\times\sum_{n}\), which is tight and convex. Since the utility function is continuous, the optimal response varies continuously with the decisions of the other primary jammers. Thus, optimal responses are upper semicontinuous. An optimal response is the set of strategies that maximise expected utility, and the set of solutions for maximising the linear function on a convex set is convex. It follows from [PERSON]'s theorem that there exists an optimal response of each primary jammer to the other jammers, i.e., a Nash equilibrium point exists.  
Each primary jammer builds a pseudo-random function using the solved strategy probabilities. Each primary jammer independently selects their strategies with the help of a pseudo-random function, and the corresponding combination of strategies serves as the optimal set of strategies under the Nash equilibrium. In order to minimise the effect of randomness, a pseudo-random function will be used several times to obtain the strategies, and the set of strategies with the highest number of occurrences will be used as the set of relative jammers.