## 4 Analysis of the Stability Strategy of Each Game Subject
### Stability Analysis of Green Smart Supply Chain Enterprises
 } & \(R_{S}^{G}-C_{S},(1-\omega_{1})V_{PB}-C_{P},C_{S}^{G}-C_{G}\) & \(-C_{S},V_{AB}-C_{P},-C_{G}^{G}-R_{-U}\) \\  & \(R_{SS}-C_{S}^{G},0,R_{G}+R_{U}+C_{S}^{G}-C_{G}\) & \(-C_{S},0,0\) \\ \cline{1-1}  & \(R_{SS}-C_{S}^{G},(1-\omega_{1})V_{PB}-C_{P},C_{S}^{G}-C_{G}\) & \(R_{SS},V_{PB}-C_{P},-C_{G}^{G}-R_{-U}\) \\ \cline{1-1}  & \(R_{SS},V_{PB}-C_{P},-C_{G}^{G}-R_{-U}\) \\ \cline{1-1}  & \(R_{SS},V_{PB}-C_{P}^{G}-C_{G}^{G}-R_{-U}\) \\ \cline{1-1}  & \(R_{SS},V_{PBAccording to Equation (2), the replicator dynamics equation of green smart supply chain enterprises can be calculated as:  
\[F(x)=dx/dt=x(U_{x}-\overline{U}_{1})=x(1-x)[z(R_{S}^{G}+C_{S}^{G})-yR_{SS}-C_{S}] \tag{3}\]  
Furthermore, the first partial derivative of \(F(x)\) and the set \(G(y:z)\) are, respectively:  
\[dF(x)/dx=(1-2x)[z(R_{S}^{G}+C_{S}^{G})-yR_{SS}-C_{S}] \tag{4}\]  
\[G(y,z)=z\Big{(}R_{S}^{G}+C_{S}^{G}\Big{)}-yR_{SS}-C_{S} \tag{5}\]  
When \(F(x)=0\) and \(dF(x)/dx\)<0, \(x\) is evolutionarily stable, the following analysis can be obtained:  
(1) When \(z=yR_{SS}+C_{S}/R_{S}^{G}+C_{S}^{G}\)(Let \(z*=yR_{SS}+C_{S}/R_{S}^{G}+C_{S}^{G}\)), there is \(dF(x)/dx\equiv 0\), no matter what the value of \(x\) is, it is in a stable state. At this time, the strategy selection ratio of green smart supply chain enterprises will not change with time. (2) When \(z\
eq z*\), there are two cases:  
1 1 Because of \(\partial G(y,z)/\partial z\)>0, we know that \(G(y,z)\) is increasing with respect to \(z\). When \(z\)<\(z*\), there are \(dF(x)/dx|_{x=0}\)<0 and \(dF(x)/dx|_{x=1}\)>0, so \(x=0\) is the evolution equilibrium point. At this point, green smart supply chain enterprises tend to choose to negatively respond. 2 2 When \(z\)>\(z*\), there are \(dF(x)/dx|_{x=0}\)>0, \(dF(x)/dx|_{x=1}\)<0, so \(x=1\) is the evolution equilibrium point. At this point, green smart supply chain enterprises tend to choose to actively respond. The green smart supply chain enterprises replication dynamic phase diagram is shown in Figure 2.  
It can be seen from Figure 2 that the volume \(V_{A_{1}}\) of \(A_{1}\) is the probability that green smart supply chain enterprises choose to actively respond, and the volume \(V_{A_{2}}\) of \(A_{2}\) is the probability that green smart supply chain enterprises choose to negatively respond, according to the calculation: \(V_{A_{1}}=\iint\limits_{A_{1}}{(yR_{SS}+C_{S})}/{(R_{S}^{G}+C_{S}^{G})}ydx=1- (R_{SS}+2C_{S})/2(R_{S}^{G}+C_{S}^{G})\), \(V_{A_{2}}=1-V_{A_{1}}\). **Proposition 1**. : _The probability that green smart supply chain enterprises choose to actively respond is positively related to \(R_{S}^{G}\)and \(C_{S}^{G}\), and it is negatively related to \(C_{S}\)and\(R_{SS}\)._  
Proof. : According to the probability \(V_{A_{1}}\) that green smart supply chain enterprises actively respond, the first partial derivative of each element can be obtained: \(dV_{A_{1}}/dR_{S}^{G}\)>0, \(dV_{A_{1}}/dC_{S}^{G}\)>0, \(dV_{A_{1}}/dC_{S}\)<0 and \(dV_{A_{1}}/dR_{SS}\)<0. Therefore, the increase in \(R_{S}^{G}\) and \(C_{S}^{G}\) or the decrease in \(C_{S}\) and \(R_{SS}\) can improve the probability that green smart supply chain enterprises actively respond. \(\square\)  
Figure 2: Phase diagram of green smart supply chain enterprises strategy evolution. Proposition 1 shows that local governments can not only reduce the illegal gains of green smart supply chain enterprises and reduce the environmental pollution of the production process by strict supervision but also increase illegal penalties to avoid green smart supply chain enterprises choosing to negatively respond. It is also possible to increase the probability that green smart supply chain enterprises to actively respond by reducing costs and increasing financial subsidies. **Proposition 2.