# Behavioral Game Theory Model in Pollution Control with Additional Supervision
## 3 Model Descriptions  
Hypothesis 1: The game model has two players: the pollutant discharge enterprises and the government regulators, and both are bounded rationality. The pollutant discharge enterprises have two strategic choices: the pollutant is discharged after complete treatment and the pollutant is discharged directly after partial treatment or even without treatment, which is abbreviated as (pollution control, no pollution control). We assume the probability of the enterprises choosing the pollution control strategy is \(x\); hence, the probability of no pollution control is \(1-x\). The government regulators can choose a strict supervision strategy, loose supervision, or even no supervision strategy, which is recorded as (supervision, no supervision). We assume that the probability of choosing the supervision strategy is \(y\); hence, the probability of no supervision is \(1-y\), which satisfies \(0\leq x,y\leq 1\). Hypothesis 2: The enterprises can obtain economic benefits \(R\) if they invest in production costs \(C_{1}\) to carry out production. If the pollutants produced in the production process are not treated and discharged directly, they will pollute the environment. At this time, the enterprises have no cost input; however, if they are found by the government regulators, they will be punished by \(P\), such as fines or suspension of business for rectification. If enterprises pay attention to environmental protection, they will adopt a pollution control strategy, which will result in pollution control costs \(C_{2}\), such as manpower and technical costs. Hypothesis 3: Government regulators need to invest in human, financial, and other regulatory costs \(C_{3}\) to supervise pollutant discharge enterprises, but they will be trusted  
Figure 1: Interactive relationships among the government, enterprises, and a third party. \({}^{1}\) Non-governmental organizations. and praised by higher authorities and the public because of their active supervision and gain reputation return \(H\). In particular, when the enterprises actively control pollution and the government chooses no supervision strategy, the government will gain reputation return \(H\) without any cost. Hypothesis 4: When the enterprises directly discharge pollutants and the government chooses no supervision strategy, the pollutant discharge behaviors of enterprises may be discovered by third-party supervision. Suppose that the probability of the third party supervising the pollutant discharge behaviors of enterprises is \(\mu\) (\(0\leq\mu\leq 1\)), the size of \(\mu\) expresses the power of third-party supervision, and a larger value of \(\mu\) represents a stronger power of third-party supervision. Once the pollutant discharge behavior of enterprises is disclosed by the third party, they will bear a series of losses caused by the damage of social image, which can be expressed as \(W_{1}\), and the government will also be criticized by the public and punished by their superiors for their laziness, which can be expressed as \(W_{2}\). At this time, the benefits of enterprises and the government are a function of \(\mu\). When \(\mu=0\), third-party supervision did not play a role, and the government did not implement its supervision responsibilities; thus, the pollution discharge behavior of the enterprises will not be discovered, with the return of the enterprise and the government being \(R-C_{1}\) and \(H\). When \(\mu=1\), the return of the enterprises is \(R-C_{1}-W_{1}\), and the return of the government is \(-W_{2}\). Hypothesis 5: Third-party supervision is restricted by various conditions whereby they cannot directly supervise the pollutant discharge enterprises. When the enterprises have pollutant discharge behavior, if the government implements supervision, it will discover the pollutant discharge behavior before the third party.