## 4 Analysis of the Stability Strategy of Each Game Subject
### Stability Analysis of Equilibrium Point of Tripartite Evolutionary Game System  
According to [PERSON]'s research conclusion, in an asymmetric game, if the condition of information asymmetry holds, the evolutionary stable strategy is a pure strategy. Therefore, it is only necessary to explore the eight partial equilibrium points in \(O_{1}(0,0,0),O_{2}(1,0,0),O_{3}(0,1,0),O_{4}(0,0,1),O_{5}(1,1,0),O_{6}(1,0,1),O_{7 }(0,1,1),O_{8}(1,1,1)\). The Jacobian matrix of the three-way evolutionary game system can be obtained by taking partial derivatives of \(F(x),F(y),F(z)\) with respect to \(x\), \(y\) and \(z\) and letting \(F(x)=0,F(y)=0\), \(F(z)=0\):  
\[J= \begin{pmatrix}\frac{\partial F(x)}{\partial x}&\frac{\partial F(x)}{\partial y }&\frac{\partial F(x)}{\partial z}\\ \frac{\partial F(y)}{\partial x}&\frac{\partial F(y)}{\partial y}&\frac{ \partial F(y)}{\partial z}\\ \frac{\partial F(z)}{\partial x}&\frac{\partial F(z)}{\partial y}&\frac{ \partial F(z)}{\partial z}\\ \end{pmatrix} = \begin{pmatrix}(1-2x)[z(R_{S}^{G}+&x(1-x)(-R_{SS})&x(1-x)(R_{S}^{ G}+C_{S}^{G})\\ C_{S}^{G})-yR_{SS}-C_{S}]&\\ y(1-y)(1-\omega_{12})&(1-2y)[x(1-\omega_{12}z)&\\ (V_{AB}-V_{PB})&(V_{AB}-V_{PB})+&y(1-y)[-(1-x)\omega_{1}V_{PB}\\ (V_{AB}-V_{PB})&(1-\omega_{12})V_{PB}-C_{P}]&-x\omega_{1}V_{AB}]\\ &\begin{array}{c}(1-2z)[y(C_{G}^{G}+\\ -z(1-z)x(R_{S}^{G}+C_{S}^{G})&z(1-z)(C_{G}^{G}+&R_{-U}-R_{G}-R_{U})\\ R_{-U}-R_{G}-R_{U})&-x(R_{S}^{G}+C_{S}^{G})+R_{G}\\ &+R_{U}+C_{S}^{G}-C_{G}]\end{array}\end{pmatrix} \]  
The Jacobian matrix of the system at point \(O_{1}(0,0,0)\) is  
\[J= \begin{pmatrix}-C_{S}&0&0\\ 0&(1-\omega_{2})V_{PB}-C_{P}&0\\ 0&0&R_{G}+R_{U}+C_{S}^{G}-C_{G}\end{pmatrix} ,\]  
the eigenvalues of the Jacobian matrix \(\lambda= \begin{pmatrix}-C_{S}&\\ (1-\omega_{2})V_{PB}-C_{P}\\ R_{G}+R_{U}+C_{S}^{G}-C_{G}\end{pmatrix} \). Similarly, the eigenvalues of the other seven points can be obtained, as shown in Table 4.  
The stability conditions for the equilibrium points in Table 4 are as follows:  
\[ \begin{array}{l}\text{Condition }1:-C_{G}^{G}-R_{-U}>C_{S}^{G}-C_{G}^{G}\\ \text{Condition }2:C_{G}<R_{G}+R_{U}+C_{S}^{G},R_{S}^{G}+C_{S}^{G}<C_{S},(1- \omega_{1})V_{PB}<C_{P},\\ \text{Condition }3:(1-\omega_{1})V_{AB}<C_{P},C_{S}<R_{S}^{G}+C_{S}^{G},R_{S}^{G}+C_{G} <R_{G}+R_{U},\\ \text{Condition }4:-C_{G}^{G}-R_{-U}<C_{S}^{G}-C_{G}^{G},R_{S}^{G}-C_{S}\text{ }<R_{SS}-C_{S}^{G},C_{P}<(1-\omega_{1})V_{PB},\\ \text{Condition }5:-R_{S}^{G}-C_{S}^{G}-C_{G}^{G}-R_{-U},C_{P}<(1-\omega_{1})V_{AB},R_{S}^ {G}-C_{S}>R_{SS}-C_{S}^{G}\end{array} .\]  
When the eigenvalues of the Jacobian matrix are all negative, the equilibrium point is the evolutionary stability strategy (ESS) of the system. The analysis of the system stability point can be discussed by Scen \begin{table}
\begin{tabular}{l c c c} \hline
**Equilibrium** & \multicolumn{3}{c}{**Eigenvalues**} & **Stable Situation** \\ \cline{2-4}
**Point O** & \(\mathbf{\lambda_{1}}\) & \(\mathbf{\lambda_{2}}\) & \(\mathbf{\lambda_{3}}\) & **Stable Situation** \\ \hline \(O_{1}(0,0,0)\) & \(-C_{S}\) & \(V_{PB}-C_{P}\) & \(R_{G}+R_{U}+C_{S}^{G}-C_{G}\) & Unstable Point \\ \(O_{2}(1,0,0)\) & \(C_{S}\) & \(V_{AB}-C_{P}\) & \(R_{G}+R_{U}-R_{S}^{G}-C_{G}\) & Unstable Point \\ \(O_{3}(0,1,0)\) & \(-R_{SS}-C_{S}\) & \(C_{P}-V_{PB}\) & \(C_{S}^{G}-C_{G}+C_{S}^{G}+R_{-U}\) & (Condition 1) \\ \(O_{4}(0,0,1)\) & \(R_{S}^{G}+C_{S}^{G}-C_{S}\) & \((1-\omega_{1})V_{PB}-C_{P}\) & \(C_{G}-R_{G}-R_{U}-C_{S}^{G}\) & ESS \\ \(O_{5}(1,1,0)\) & \(R_{SS}+C_{S}\) & \(C_{P}-V_{AB}\) & \(C_{G}^{G}+R_{-U}-R_{S}^{G}-C_{G}\) & Unstable Point \\ \(O_{6}(1,0,1)\) & \(C_{S}-R_{S}^{G}-C_{S}^{G}\) & \((1-\omega_{1})V_{AB}-C_{P}\) & \(R_{S}^{G}+C_{G}-R_{G}-R_{U}\) & (Condition 3) \\ \(O_{7}(0,1,1)\) & \(R_{S}^{G}+C_{S}^{G}-R_{SS}-C_{S}\) & \(C_{P}-(1-\omega_{1})V_{PB}\) & \(-\big{(}C_{S}^{G}-C_{G}\big{)}-C_{G}^{G}-R_{-U}\) & ESS \\ \(O_{8}(1,1,1)\) & \(R_{SS}+C_{S}-R_{S}^{G}-C_{S}^{G}\) & \(C_{P}-(1-\omega_{1})V_{AB}\) & \(R_{S}^{G}+C_{G}-C_{G}^{G}-R_{-U}\) & (Condition 5) \\ \hline \end{tabular}
\end{table}