# Behavioral Game Theory Model in Pollution Control with Additional Supervision
## 4 Evolutionary Stability Analysis of Government-Enterprise Strategies under Third-Party Supervision
### Formatting of Mathematical Components  
The stability of the equilibrium points \(E_{i}\) (\(i=1,2,3,4\)) is analyzed using the judgment method of local stability of the evolutionary system and Jacobian matrix \(J\), according to the different value ranges of the third-party supervision \(\mu\). The specific results are shown in Table 3.  
**Proposition 1**. : _When \(\mu\in(0,\min(\frac{C_{2}}{W_{1}},\frac{C_{3}}{H+W_{2}}))\), the dynamic evolutionary is stable at the equilibrium point \(E_{1}(0,0)\) (no pollution control, no supervision)._  
Proof of Proposition 1.: According to the stability analysis results of each equilibrium point in Proposition 1 of Table 3, the evolutionary stable strategy of pollutant discharge enterprises and government regulators is (no pollution control, no supervision). On the basis of Proposition 1, with \(R-C_{1}-C_{2}<R-C_{1}-\mu W_{1}\) and \(H-C_{3}<(1-\mu)H-\mu W_{2}\) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Proposition** & **Analysis** & \(E_{1}\)**(0,0)** & \(E_{2}\)**(0,1)** & \(E_{3}\)**(1,0)** & \(E_{4}\)**(1,1)** \\ \hline \multirow{3}{*}{1} & _Det(J)_ & _+_ & _+_ & \(-\) & \(-\) \\  & _Tr(J)_ & _\(-\)_ & _+_ & Uncertain & Uncertain \\  & Stability & ESS & Unstable & Saddle point & Saddle point \\ \hline \multirow{3}{*}{2} & _Det(J)_ & _\(-\)_ & _\(-\)_ & \(-\) & \(-\) \\  & _Tr(J)_ & Uncertain & Uncertain & Uncertain & Uncertain \\  & Stability & Saddle point & Saddle point & Saddle point & Saddle point \\ \hline \multirow{3}{*}{3} & _Det(J)_ & _\(-\)_ & _+_ & \(+\) & \(-\) \\  & _Tr(J)_ & Uncertain & _+_ & \(-\) & Uncertain \\  & Stability & Saddle point & Unstable & ESS & Saddle point \\ \hline \multirow{3}{*}{4} & _Det(J)_ & _+_ & \(-\)_ & \(+\) & \(-\) \\  & _Tr(J)_ & _+_ & Uncertain & \(-\) & Uncertain \\ \cline{1-1}  & Stability & Unstable & Saddle point & ESS & Saddle point \\ \hline \hline \end{tabular}
\end{table} 
Table 3: System stability analysis of each equilibrium point with different values of \(\mu\). two conclusions can be drawn. Firstly, the strategy selection of the pollutant discharge enterprises changes with the change in government strategy. When the government regulators choose the supervision strategy, and the benefit of pollutant discharge enterprises choosing a pollution control strategy is greater than that of choosing no pollution control strategy, then bounded rational enterprises will choose pollution control strategy; when government regulators choose no supervision strategy, and the benefit of pollutant discharge enterprises choosing no pollution control strategy is greater than that of choosing a pollution control strategy, then the bounded rational enterprises will choose no pollution control strategy. In this situation, the pollution control behavior and the lack of pollution control behavior of enterprises coexist. Secondly, no matter whether pollutant discharge enterprises choose the pollution control strategy or not, when the benefit of the government choosing no supervision strategy is always greater than that of choosing a supervision strategy, then the government with limited rationality will choose no supervision strategy. After a long-term evolutionary game, the system tends to the steady state that pollutant discharge enterprises choose no pollution control strategy and the government regulators choose no supervision strategy. \(\square\)  
The above analysis procedure of the evolutionary stable strategy validates Proposition 1. At this time, the critical value of \(x_{e}\) is 0, and \(0<y_{e}<1\); hence, the point \(E_{5}\) can be regarded as located on the \(y\)-axis. In Proposition 1, the benefits of no supervision strategy of government regulators are always greater than those of a supervision strategy. After long-term evolution, the government regulators finally tend to choose no supervision strategy. Pollutant discharge enterprises have no stable strategy choices, but constantly adjust and revise their own strategies along with the government strategies, finally stabilizing to no pollution control strategy.