# Behavioral Game Theory Model in Pollution Control with Additional Supervision
## 1 Introduction
Thus, the main reasons for the difficulty of environmental pollution control can be summarized into two points: first, because pollution control requires cost investment, enterprises choose to discharge pollutants directly to the environment in pursuit of high profits; second, local governments may relax or even not regulate pollutant emissions because of their own interests, failing to give full play to their regulatory responsibilities. When the government regulatory mechanism fails, an external force is needed to restrict their behaviors. With the deepening of the research on the interaction between government and enterprise environmental strategies, the authors of [11] drew the following conclusion: as an independent regulatory subject other than the government and enterprises, a third party could get rid of the interest temptation and external punishment constraints, as well as play a regulatory role more fairly. With the gradual enhancement of public awareness of rights, more and more incidents of environmental pollution rights protection were reported [12], and the regulatory role of the rapidly developing media and the internet is becoming increasingly prominent in environmental issues [13], which shows that third-party supervision is playing a more and more important role in environmental pollution regulation. Existing research [14; 15; 16; 17] provides a theoretical basis for third-party supervision of environmental pollution, but there are also some limitations, which is mainly reflected in the following two aspects: in most game analyses, third-party supervision is set as the game subject, whereas, in fact, it does not have a direct game relationship with the government or enterprises, but acts as an external factor affecting the strategic choice of government and enterprises; thus, the conclusions fail in explanation; furthermore, the benefits of the game subject in the existing models often only focus on the economic benefits, ignoring the nonmaterial return of the government such as reputation income from active supervision and the public blame due to dereliction of duty, as well as the nonmaterial losses of enterprises such as the negative comments from consumers caused by environmental pollution. Evolutionary game theory combines the analysis of game theory with the analysis of dynamic evolutionary process; it no longer models the game players as completely rational game subjects, but takes the bounded rationality of the game players as the analysis framework. This theory holds that the members of the game group constantly adjust and modify their strategies through learning, imitation, and trial and error, finally reaching a certain equilibrium state. At this time, the game players tend toward a stable strategy, which is called the evolutionary stable strategy (ESS) [18]. The whole evolutionary process is consistent with the actual economic life situation; hence, it is widely used in all fields of social studies [19; 20; 21; 22; 23; 24; 25]. Through in-depth analysis of the environmental pollution issue, it is easy to find that government regulators and pollutant discharge enterprises also follow a similar law: enterprises constantly adjust their environmental behaviors according to the regulation policies of the government, while government regulators constantly update and improve environmental regulation policies according to the environmental behaviors of enterprises. The strategy selection of the two follows the basic characteristics of evolutionary game theory. Therefore, from the perspective of evolutionary game theory, it is very consistent with the actual situation to study the behavior law of the government and enterprises in the pollution control issue. In view of the literature review and comprehensive analysis, this paper proposes an evolutionary game model of government and enterprises subject to the participation of third-party supervision, and makes the following contributions:  
1.