## 4 Analysis of the Stability Strategy of Each Game Subject
### Stability Analysis of Local Governments  
The expected payoffs of the local governments when they actively respond (\(U_{2}\)) and negatively respond (\(U_{1-z}\)) are as follows:  
\[ \begin{array}{l}U_{2}=xy(-R_{S}^{G}-C_{G})+x(1-y)(R_{G}+R_{U}-R_{S}^{G}-C_{G}) +(1-x)y(C_{S}^{G}-C_{G})+(1-x)(1-y)(R_{G}+R_{U}+C_{S}^{G}-C_{G})\\ U_{1-z}=xy(-C_{G}^{G}-R_{-U})+(1-x)y(-C_{G}^{G}-R_{-U})\end{array}  \tag{11}\]  
Thus, the average payoff of the local government (\(\overline{U}_{3}\)) is as follows:  
\[\overline{U}_{3}=zU_{2}+(1-z)U_{1-z} \tag{12}\]  
According to Equation (12), the replicator dynamics equation of the local government can be calculated as:  
\[F(z)=dz/dt=z(U_{2}-\overline{U}_{3})=z(1-z)[y(C_{G}^{G}+R_{-U}-R_{G}-R_{U})-x( R_{S}^{G}+C_{S}^{G})+R_{G}+R_{U}+C_{S}^{G}-C_{G}] \tag{13}\]  
Furthermore, the first partial derivative of \(F(z)\) and the set \(G(x,y)\) are, respectively:  
\[dF(z)/dz=(1-2z)[y(C_{G}^{G}+R_{-U}-R_{G}-R_{U})-x(R_{S}^{G}+C_{S}^{G})+R_{G}+R_ {U}+C_{S}^{G}-C_{G}] \tag{14}\]  
\[G(y,z)=y(C_{G}^{G}+R_{-U}-R_{G}-R_{U})-x(R_{S}^{G}+C_{S}^{G})+R_{G}+R_{U}+C_{S }^{G}-C_{G} \tag{15}\]  
When \(F(z)=0\) and \(dF(z)/dz\)\(<\)0, \(z\) is the evolution equilibrium point. The following analysis can be obtained:  
(1) When \(x=[y(C_{G}^{G}+R_{-U}-R_{G}-R_{U})+R_{G}+R_{U}+C_{S}^{G}-C_{G}]/(R_{S}^{G}+C_{S }^{G})\)  
(Let \(x**=[y(C_{G}^{G}+R_{-U}-R_{G}-R_{U})+R_{G}+R_{U}+C_{S}^{G}-C_{G}]/(R_{S}^{G}+C_{ S}^{G})\)), there is \(dF(z)/dz\equiv 0\); no matter what the value of z is, it is in a stable state. At this time, the strategy selection ratio of the local government will not change with time. (2) When \(x\
eq x**\), there are two cases:  
1 Because of \(\partial G(x,y)/\partial x\)\(<\)0, we know that \(G(x,y)\) is decreasing with respect to x. When \(x\)\(<\)\(x**\), there are \(dF(z)/dz|_{z=0}\)\(>\)0 and \(dF(z)/dz|_{z=1}\)\(<\)0, so \(z=1\) is the evolution equilibrium point. At this point, local governments tend to choose actively respond. 2 When \(x\)\(>\)\(x**\), there are \(dF(z)/dz|_{z=0}\)\(<\)0 and \(dF(z)/dz|_{z=1}\)\(>\)0, so \(z=0\) is the evolution equilibrium point. At this point, local governments tend to choose to negatively respond. The local governments replication dynamic phase diagram is shown in Figure 4.  
It can be seen from Figure 4 that the volume \(V_{C_{1}}\) of \(C_{1}\) is the probability that local governments choose to actively respond, and the volume \(V_{C_{2}}\) of \(C_{2}\) is the probability that local governments choose to negatively respond, according to the calculation:  
\[ \begin{array}{l}V_{C_{1}}=\mathop{\int\!\!\!\!\!\!\int}\limits_{C_{1}}[y(C_{G }^{G}+R_{-U}-R_{G}-R_{U})+R_{G}+R_{U}+C_{S}^{G}-C_{G}]/(R_{S}^{G}+C_{S}^{G})dydz =\\ (R_{G}+R_{U}+C_{S}^{G}+R_{-U}+2C_{S}^{G}-2C_{G})/2(R_{S}^{G}+C_{S}^{G})\end{array} \]  
Figure 4: Phase diagram of the local government strategy evolution. **Proposition 5**. : _The probability that the local government chooses to actively respond is positively correlated with\(R_{G}\), \(C_{G}^{G}\)and \(R_{-U}\), and it is negatively correlated with \(C_{G}\)and \(R_{S}^{G}\)._  
Proof. : According to the expression of actively respond probability \(C_{1}\), the first partial derivative of each element can be obtained:\(dV_{C_{1}}/dR_{G}\)>0, \(dV_{C_{1}}/dC_{G}^{G}\)>0, \(dV_{C_{1}}/dR_{-U}\)>0, \(dV_{C_{1}}/dC_{G}\)<0, and \(dV_{C_{1}}/dR_{S}^{G}\)<0. Therefore, the increase in \(R_{G}\), \(C_{G}^{G}\) and \(R_{-U}\) or the decrease in \(C_{G}\) and \(R_{S}^{G}\) can improve the probability that the local government will actively respond. Proposition 5 shows that although high response costs and high subsidies will reduce the probability that local governments will respond positively, once the situation worsens, local governments will be punished by the central government and lose their reputation. For the local government, the loss of credibility and the punishment of the central government are irreversible; in reality, they often do not choose to negatively respond.