In this paper we model the conflict between the group of polluting firms in a country and any social planner in the same country who attempts to control the volume of emissions generated during the production process. Both players of the game have their own control policies, i.e., the rate of emissions on behalf of the polluting firms and the rate of pollution control (e.g., pollution abatement or environmental taxation) on behalf of the home country. The common state variable of the model is the number of polluting firms, which aims to be minimized via the country’s control policy, but on the polluters’ side it is beneficial to be maximized. Regarding the game model, its setup belongs to the special class of differential games, which are called ‘state separable differential games’. An important property of these games is that the open-loop Nash equilibrium coincides with the Markovian (closed-loop) equilibrium and, in the case of hierarchical moves, analytical solutions are easily obtained. The game proposed here is analyzed for both types of equilibrium, i.e., Nash and Stackelberg. In the simultaneous move game (i.e., the Nash game) we find the equilibrium’s analytical expressions of the controls for both players, as well as the stationary value of the stock of polluting firms. A sensitivity analysis of the model’s crucial variables takes place. In the hierarchical move game (i.e., the Stackelberg game) we find the equilibrium values of the controls, as well as of the state variable. As a result, a comparison between the two types of equilibrium for the game takes place. The analysis of the comparison reveals that the conflict is more intensive (since both controls have greater values) for the case in which the polluting firms act as the leader in the hierarchical move game.

中文翻译：

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动态纳什和斯塔克伯格模式对污染企业控制的一些结果

在本文中，我们对一个国家的污染企业集团与该国家中试图控制生产过程中产生的排放量的任何社会计划者之间的冲突进行建模。游戏的两个参与者都有自己的控制策略，即代表污染公司的排放率和代表本国的污染控制率（例如，污染消除或环境税）。该模型的常见状态变量是污染企业的数量，其目的是通过国家的控​​制政策来使其最小化，但从污染者的角度来看，最大化是有利的。关于游戏模型，其设置属于差分游戏的特殊类别，称为“状态可分离差分游戏”。这些博弈的一个重要特性是，开环纳什均衡与马尔可夫（闭环）均衡重合，并且在分层移动的情况下，很容易获得解析解。针对两种均衡类型，即纳什和斯塔克尔伯格，对这里提出的博弈进行了分析。在同时移动博弈（即纳什博弈）中，我们找到两个参与者的控制权的均衡分析表达式，以及污染企业股票的固定价值。对模型的关键变量进行敏感性分析。在分层移动游戏（即Stackelberg游戏）中，我们找到控件以及状态变量的平衡值。结果，在游戏的两种平衡类型之间进行了比较。