# 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
Secondly, when \(\mu W_{1}-P>0\), we can get \(\mu\in(\frac{P}{W_{1}},\frac{C_{3}}{H+W_{2}})\); at this time, the critical value of \(x_{e}\) is \(0\), the critical value of \(y_{e}\) is \(1\), and point \(E_{5}\) can be regarded as coinciding with the point \(E_{3}(0,1)\). The dynamic evolutionary phase diagram is shown in Figure 4b. Figure 3: Dynamic evolutionary phase diagram of government–enterprise strategies after increasing the power of third-party supervision. **Corollary 3**. : _When the power of third-party supervision increases to an extent, it can promote enterprises to tend to choose the pollution control strategy, but it is not enough to promote the government to evolve into the supervision strategy._  
**Proposition 4**. : _When \(\mu\in(\max(\frac{C_{2}}{W_{1}},\frac{C_{3}}{H+W_{2}}),1)\), the dynamic evolutionary system is stable at the equilibrium point \(E_{3}(1,0)\) (pollution control, no supervision)._  
Proof of Proposition 4.: According to the stability analysis results of each equilibrium point in Proposition 4 of Table 3, the evolutionary stable strategy of the 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 \(H-C_{3}>(1-\mu)H-\mu W_{2}\). At this time, the evolutionary stable strategy of the system is the same as that in Proposition 3; however, when the pollutant discharge enterprises choose no control pollution strategy, the government will change its choice to the supervision strategy, which shows that the current power of third-party supervision can keep enterprises in a stable pollution control strategy, as well as cause a change in the intensity of government supervision. When the government does not care about the loss of reputation, they will still choose the no supervision strategy because of the strong supervision efficiency of third-party supervision. \(\square\)  
In Proposition 4, 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, enterprises finally tend to choose the pollution control strategy. The government regulators have no stable strategy choice, but constantly adjust and modify their own strategies along with enterprises strategies, finally stabilizing to no supervision strategy. According to the value of \(\mu W_{1}-P\) in \(x_{e}\), the value of \(\mu\) in this proportion can be divided into two cases. Firstly, when \(\mu W_{1}-P<0\), we can get \(\mu\in(\max(\frac{C_{2}}{W_{1}},\frac{C_{3}}{H+W_{2}}),\frac{P}{W_{1}})\); at this time, \(0<x_{e}<1\), the critical value of \(y_{e}\) is 0, and point \(E_{5}\) can be regarded as located on the \(x\)-axis. The dynamic evolutionary phase diagram of government-enterprise strategies is shown in Figure 5a. Secondly, when \(\mu W_{1}-P>0\), we can get \(\mu\in(\max(\frac{C_{3}}{H+W_{2}},\frac{P}{W_{1}}),1)\); in this situation, \(0<x_{e}<1\), the critical value of \(y_{e}\) is 1, and point \(E_{5}\) can be regarded as located on the straight line \(y=1\). The dynamic evolutionary phase diagram is shown in Figure 5b. Figure 4: Dynamic evolutionary phase diagram of government–enterprise strategies after increasing the power of third-party supervision. The increase is higher than that in Figure 3. (**a**) The case of \(\mu\in(\frac{C_{2}}{W_{1}},\min(\frac{C_{3}}{H+W_{2}},\frac{P}{W_{1}}))\); (**b**) the case of \(\mu\in(\frac{P}{W_{1}},\frac{C_{3}}{H+W_{2}})\). **Corollary 4**. : _The strong power of third-party supervision coordinating with the government supervision pushes enterprise stability in the pollution control strategy, changes the intensity of government supervision, and replaces the government supervision to a certain extent in this situation._  
By summarizing the values of \(\mu\) and evolutionary results in the above four propositions, three cases can be obtained, as shown in Table 4.