## 3 Model and Analysis
### Construction and Analysis of Stochastic Evolutionary Game Model
#### 3.3.2 Game Model Analysis  
Model stability analysis is based on the existence and uniqueness of solutions. The existence and uniqueness of solutions to Equations (8)-(11) are proved using the local Lipschitz condition and the linear growth condition [68], and the proof procedure is shown in Appendix A. Thus, stability analysis can be performed. For Equations (8)-(11), when \(t=0\), it results in \(x(t)=y(t)=z(t)=u(t)=0\). This shows that the zero solution is the equilibrium solution of the equation when there is no perturbation in the gaming system [67]. In practice, Gaussian white noise is bound to disturb the gaming system, which is therefore judged based on the stability theorem for stochastic differential equations [69; 70; 71]. Firstly, given the stochastic differential equation as shown in Equation (12)  
\[dx(t)=f(t,x(t))dt+g(t,x(t))dw(t),x(t_{0})=x_{0} \tag{12}\]  
Let there exist a function \(V(t,x)\) and positive constants \(v_{1},v_{2}\) such that \(v_{1}|x|^{p}\leq V(t,x)\leq v_{2}|x|^{p},t\geq 0\). **Theorem 1. **_If there exists a positive constant \(\mu\), such that \(LV(t,x)\leq-\mu V(t,x),t\geq 0\), then the zero solution of Equation (12) is a P-order moment exponentially stable, and \(E|x(t,x_{0})|^{p}<(\frac{v_{1}}{v_{2}})|x_{0}|^{p}e^{\mu t}\), \(t\geq 0\) holds._  
**Theorem 2. **_If there exists a positive constant \(\mu\), such that \(LV(t,x)\geq-\mu V(t,x),t\geq 0\), then the zero solution of Equation (12) is a P-order moment exponentially unstable, and \(E|x(t,x_{0})|^{p}\geq(\frac{v_{1}}{v_{2}})|x_{0}|^{p}e^{\mu t}\), \(t\geq 0\) holds._  
According to the above theorem, taking \(V_{t}(t,x)=x,V_{t}(t,y)=y,V_{t}(t,z)=z,V_{t}(t,u)=u\); \(x,y,z,u\in[0,1],v_{1}=v_{2}=1,p=1\) for Equations (8)-(11) yields Equations (13)-(16):  
\[LV(t,x)=[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y} +(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x \tag{13}\]  
\[LV(t,y)=[(1+x\beta_{YX}+z\beta_{YZ}+u\beta_{YU}-x\alpha_{YX})E_{Y}-C_{Y}+\Pi_{ Y}+G_{Y}+xN_{Y}]y \tag{14}\]  
\[LV(t,z)=[(1+x\beta_{ZX}+y\beta_{ZY}+u\beta_{ZU}-x\alpha_{ZX}-y\alpha_{ZY})E_{ Z}-C_{Z}+\Pi_{Z}+G_{Z}+xN_{Z}]z \tag{15}\]  
\[LV(t,u)=[(1+x\beta_{UX}+y\beta_{UY}+z\beta_{UZ}-x\alpha_{UX}-y\alpha_{UY}-z \alpha_{UZ})E_{U}-C_{U}+\Pi_{U}+G_{U}+xN_{U}]u \tag{16}\]  
If the P-order moment exponents of the zero solutions of Equations (13)-(16) are stable, then Equations (8) and (9) are satisfied. A more detailed demonstration process can be found in Appendix A.  
\[x[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y}+(1-z)N _{Z}+(1-u)N_{U}+G_{X}]\leq-x \tag{17}\]  
\[y[(1+x\beta_{YX}+z\beta_{YZ}+u\beta_{YU}-x\alpha_{YX})E_{Y}-C_{Y}+\Pi_{Y}+G_{Y }+xN_{Y}]\leq-y \tag{18}\]  
\[z[(1+x\beta_{ZX}+y\beta_{ZY}+u\beta_{ZU}-x\alpha_{ZX}-y\alpha_{ZY})E_{Z}-C_{ Z}+\Pi_{Z}+G_{Z}+xN_{Z}]\leq-z \tag{19}\]  
\[u[(1+x\beta_{UX}+y\beta_{UY}+z\beta_{UZ}-x\alpha_{UX}-y\alpha_{UY}-z\alpha_{ UZ})E_{U}-C_{U}+\Pi_{U}+G_{U}+xN_{U}]\leq-u \tag{20}\]  
The condition satisfying Equation (17) is derived as X1 or X2 by performing the reduction computation. The derivation is shown in Appendix A. **X1. **_When \(x\in(0,1]\) and \(\beta_{XY}E_{X}-N_{Y}>0\), \(y\leq[-1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{Y}-(1-z)N_{Z}-(1-u )N_{U}-G_{X}]/(\beta_{XY}E_{X}-N_{Y})\) and \(0\leq-1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{Y}-(1-z)N_{Z}-(1-u)N _{U}-G_{X}\) should be satisfied._  
**X2. **_When \(x\in(0,1]\)_and \(\beta_{XY}E_{X}-N_{Y}<0\), \(y\geq[-1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{Y}-(1-z)N_{Z}-(1-u )N_{U}-G_{X}]/(\beta_{XY}E_{X}-N_{Y})\)and \(-1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{Y}-(1-z)N_{Z}-(1-u)N_{U}-G _{X}\leq 0\) should be satisfied._  
Similarly, the conditions for satisfying Equations (18)-(20) are Y1, Z1, and U1, respectively:  
**Y1. **: _It should satisfy \(z\leq[-1-(1+x\beta_{YX}+u\beta_{YU}-xa_{YX})E_{Y}+C_{Y}-\Pi_{Y}-G_{Y}-xN_{Y}]/\beta_ {YZ}E_{Y}\) and \(0\leq-1-(1+x\beta_{YX}+u\beta_{YU}-xa_{YX})E_{Y}+C_{Y}-\Pi_{Y}-G_{Y}-xN_{Y}\)._  
**Z1.