# 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
Figure 2 shows the dynamic evolutionary trend of government-enterprise strategies in this case. **Corollary 1**. : _The weak power of third-party supervision is not enough to promote the evolution of the behavioral decisions of government regulators and pollutant discharge enterprises in an environment-friendly direction._  
**Proposition 2**. : _When \(\mu\in(\frac{C_{3}}{H+W_{2}},\frac{C_{2}}{W_{1}})\), there is no evolutionary steady state for the dynamic evolutionary system._  
Proof of Proposition 2.: According to the stability analysis results of each equilibrium point in Proposition 2 of Table 3, there is no evolutionary stable strategy for the dynamic evolutionary government-enterprise system. From the value of \(\mu\) in Proposition 2, we can get \(R-C_{1}-C_{2}<R-C_{1}-\mu W_{1}\), \(H-C_{3}>(1-\mu)H-\mu W_{2}\), and \(0<x_{e},y_{e}<1\). Compared with the strategy selection of the two actors in Proposition 1, the behavior choice of the pollutant discharge enterprises has not changed; however, when the enterprises choose no pollution control strategy, the government will change its choice to the supervision strategy. At this time, the supervision and no supervision behaviors of the government coexist, which shows that, under the influence of third-party supervision, government decision  
Figure 2: Dynamic evolutionary phase diagram of government-enterprise strategies under the weak power of third-party supervision. making will change with the change in the strategy of the pollutant discharge enterprises, evolving from a stable strategy selection state to an unstable state. The evolutionary path of the system exhibits dynamic change with certain uncertainty. Figure 3 describes the evolutionary process of the government-enterprise strategies in this case. \(\square\)  
**Corollary 2**. : _Although an appropriate increase in the power of third-party supervision changes the choice of the government and enterprise strategies in the short term, due to the mutual influence of the strategies on both sides of the game, the evolutionary system does not pursue a stable state in this situation._  
**Proposition 3**. : _When \(\mu\in(\frac{C_{2}}{W_{1}},\frac{C_{3}}{H+W_{2}})\), the dynamic evolutionary system is stable at the equilibrium point \(E_{3}(1,0)\) (pollution control, no supervision)._  
Proof of Proposition 3.: According to the stability analysis results of each equilibrium point in Proposition 3 of Table 3, the evolutionary stable strategy of pollutant discharge enterprises and government regulators is (pollution control, no supervision). According to the value of \(\mu\), we can get \(R-C_{1}-C_{2}>R-C_{1}-\mu W_{1}\) and \(H-C_{3}<(1-\mu)H-\mu W_{2}\). Compared with Proposition 1, the behavior choice of the government has not changed, i.e., the no supervision strategy of the government is still stable; however, when the government chooses the no supervision strategy, the enterprises will change to the pollution control strategy, which shows that the pollutant discharge enterprises will tend to choose the pollution control strategy under the influence of third-party supervision. \(\square\)  
In Proposition 3, the benefits of the no supervision strategy of government regulators are always greater than those of the supervision strategy, and the benefits of the pollution control strategy of pollutant discharge enterprises are always higher than those of the no pollution control strategy. After long-term evolution, government regulators finally tend to choose the no supervision strategy, while enterprises are stable in the pollution control strategy. According to the value of \(\mu W_{1}-P\) in \(x_{e}\), the value of \(\mu\) in this proposition can be divided into two cases. Firstly, when \(\mu W_{1}-P<0\), we can get \(\mu\in(\frac{C_{2}}{W_{1}},\frac{C_{3}}{H+W_{2}},\frac{P}{W_{1}}))\); at this time, the critical values of \(x_{e}\) and \(y_{e}\) are both \(0\), and point \(E_{5}\) can be regarded as coinciding with point \(E_{3}(0,0)\). The dynamic evolutionary phase diagram is shown in Figure 4a.