# Behavioral Game Theory Model in Pollution Control with Additional Supervision
## 5 Simulation Analysis  
With the help of MATLAB software, simulation experiments were carried out on the conclusions of the above theories, intuitively showing the evolutionary trend of strategies on both sides of the game under different conditions.  
In order to meet the conditions of Case 1 in Table 4, we set \(W_{1}=6\), \(W_{2}=4\), \(C_{2}=2\), \(C_{3}=4\), \(H=3\), and \(P=5\); the two groups of initial values of \(x\) and \(y\) at the beginning of the game were (0.3, 0.7) and (0.8, 0.2). After calculation, \(\mu\in\) (0,1/3) could be obtained; then, we selected \(\mu=0.2\) from the value range for simulation experiment, and the simulation results are shown in Figure 6. The simulation results show that, when the power of third-party supervision is weak, no matter the initial state of the game between the enterprises and the government, it will eventually evolve into the enterprises choosing no pollution control strategy and the government regulators choosing no supervision strategy.  
In order to meet the conditions of Case 2 in Table 4, we set \(W_{1}=6\), \(W_{2}=5\), \(C_{2}=3\), \(C_{3}=2\), \(H=2\), and \(P=5\); the two groups of initial values of \(x\) and \(y\) at the beginning of the game were (0.1, 0.9) and (0.7, 0.2). After calculation, \(\mu\in\) (2/7,1/2) could be obtained, and we selected \(\mu=0.4\) from the value range for simulation experiment. The simulation results are shown in Figure 7. The simulation results show that, no matter the initial state of the game between the pollutant discharge enterprises and government regulators, an appropriate increase in third-party supervision can change the behavior choices of enterprises and the government in the short term, but it cannot make the enterprises and the government tend to a certain stable point. The strategy selection of both sides are correlated in the evolutionary process, the whole system is in a cyclical state of shock, and there is no evolutionary stable strategy.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Case No.** & **Value Ranges of \(\mu\)** & **ESS** \\ \hline
1 & \((0,\min(\frac{C_{2}}{W_{1}},\frac{C_{3}}{H+W_{2}}))\) & \(E_{1}(0,0)\) \\
2 & \((\frac{C_{1}}{H+W_{2}},\frac{C_{3}}{H+W_{2}})\) & Nonexistent \\
3 & \((\frac{C_{1}}{W_{1}},\frac{C_{2}}{H+W_{2}})\) or \((\max(\frac{C_{3}}{W_{1}},\frac{C_{3}}{H+W_{2}}),1)\) & \(E_{3}(1,0)\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Evolutionary stable strategies in three cases.  
In order to meet the conditions of Case 3 in Table 4, we set \(W_{1}\) = 6, \(W_{2}\) = 4, \(C_{2}\) = 3, \(C_{3}\) = 5, \(H\) = 3, and \(P\) = 5; the two groups of initial values of \(x\) and \(y\) at the beginning of the game were (0.2, 0.8) and (0.7, 0.3). After calculation, \(\mu\) (1/2,5/7) or \(\mu\) (5/7,1) could be obtained, and we selected \(\mu\) = 0.6 and \(\mu\) = 0.8 from the two value ranges for comparative simulation experiments. The simulation results are shown in Figure 8. The simulation results show that, no matter the initial state of the game between enterprises and government regulators, affected by the strong power of third-party supervision, enterprises evolve into the pollution control strategy and the government evolves into the no supervision strategy. By comparing Figure 8a,b, it can also be found that, under the same institutional system, increasing the power of third-party supervision can accelerate the evolutionary process of the enterprises choosing the pollution control strategy, but it has little impact on the evolutionary strategy path of the government.  
Figure 6: Simulation results of government–enterprise strategies under the weak power of third-party supervision. In this case, the corresponding value range of \(\mu\) is \((0,\min(\frac{C_{2}}{W_{1}},\frac{C_{2}}{H+W_{2}}))\).  
Figure 7: Simulation results of government–enterprise strategies after increasing the power of third-party supervision. In this case, the corresponding value range of \(\mu\) is \((\frac{C_{2}}{H+W_{2}},\frac{C_{2}}{W_{1}})\).