Research on Transboundary Air Pollution Control and Cooperative Strategies Based on Differential Game

Abstract

This paper examines control and cooperation mechanisms for trans-regional air pollution using differential game theory. This study focuses on analyzing pollution control pathways in regions characterized by asymmetric economic development. Three models are constructed: the Nash non-cooperative game, the pollution control cost compensation mechanism, and the collaborative cooperation mechanism. These models are used to investigate optimal pollution control strategies for various regions. The findings indicate that the collaborative cooperation model substantially reduces pollution emissions and enhances overall benefits. Additionally, the pollution control cost compensation mechanism alleviates the burden of pollution control on less developed regions. Numerical analysis confirms the effectiveness of the proposed models and offers theoretical foundations and policy recommendations for regional cooperation in pollution prevention.

1. Introduction and Background

With rapid economic development, increasingly severe environmental pollution has become a significant threat to public life and the sustainable growth of the economy. The extraction of oil, industrial activities, and everyday emissions have contributed to an increasing quantity of pollutant emissions. Notably, regional compound pollution issues—such as ozone, sulfur dioxide, fine particulate matter (PM2.5), and coarse particulate matter (PM10)—have become more prominent. These pollutants not only impede sustainable economic development, but also pose serious risks to public health. The Intergovernmental Panel on Climate Change (IPCC) has identified greenhouse gasses produced by human activities as the primary driver of global climate warming and the increasing frequency of extreme weather events [1]. In densely populated and industrialized areas, pollution transmission between cities and regions is particularly significant, with external sources accounting for 30–40% of pollutants such as sulfur dioxide in some cities. The traditional approach to pollution prevention and management, which focuses on individual administrative regions, is insufficient to address the growing pollution challenges. Furthermore, trans-regional pollution often results in “free-rider” behavior, diminishing the motivation of local governments to tackle pollution [2]. Currently, the main approaches to managing regional pollution are administrative measures imposed by higher-level authorities, such as pollution control subsidies and penalty mechanisms [3]. However, these efforts have yielded unsatisfactory results. The primary reason is that decision-makers in environmental governance tend to prioritize the interests of their own jurisdictions, often overlooking the spillover effects of pollution on neighboring areas. This often leads to opportunistic avoidance of their responsibilities in pollution control. Moreover, differences in economic conditions and pollution control measures across regions further hinder effective trans-regional pollution management. Consequently, addressing trans-regional pollution has become a crucial factor in achieving ecological civilization.

Against this backdrop, this paper examines pollution control between two regions with asymmetric economic development. Due to differences in economic strength, less developed regions often engage in “free-rider” behavior, investing inadequately in pollution control, while more developed regions suffer greater environmental damage [4]. To address this, this study considers three scenarios for regional pollution control: the Nash non-cooperative mechanism, the pollution control cost compensation mechanism, and the collaborative cooperation mechanism. Differential game models are established to conduct a comparative analysis of these scenarios. The models are validated through numerical analysis, and based on the results, this paper proposes relevant management strategies. These findings offer valuable insights for strategy selection in regional cooperative pollution governance.

Environmental pollution and emission reduction are critical challenges for global and regional development and have become focal points in academic research. Numerous scholars have investigated solutions to transboundary air pollution, primarily focusing on optimal control theory. Over recent decades, mathematical and simulation models, including energy models and Integrated Assessment Models (IAMs), have been widely used to develop “optimal” management strategies across regional boundaries. These applications span areas such as air quality management [5], water resource allocation [6], and energy policy planning [7]. Optimization-based models, such as IAMs, generally aim to minimize net present value, identify the least-cost strategy [8], or balance supply and demand through market simulations [9]. However, these approaches often fail to consider the diverse priorities and interests of real-world decision-makers, who may favor their own welfare over a globally optimal solution. Game theory addresses this gap by modeling strategic interactions among actors with potentially conflicting objectives. This study applies game-theoretical approaches to pollution management, capturing complexities that traditional optimization models do not fully address. Studies relevant to this paper primarily address differential game modeling and optimization [10,11], the feasibility of international pollution control cooperation [12,13,14], and pollution control issues across different regions under various environmental policies [15,16,17]. For instance, Kaitala et al. [12] examined the transboundary pollution problem between the Soviet Union and France, concluding that while cooperation benefited France, the Soviet Union required compensation to participate. Yanase’s [13] international pollution control model demonstrated that environmental policies in non-cooperative games deviate from socially optimal levels. Nkuiya [14] analyzed pollution games between countries in a complex environmental system, showing that nations respond differently to pollution threats. Shoude [16] found that emissions trading makes cooperation in asymmetric regions more advantageous than non-cooperation. In transboundary pollution control, administrative intervention is the most common approach, with governments acting directly as players in the decision-making process. List and Mason [15] explored whether environmental regulation in asymmetric games should be implemented by central or local governments. Biancardi [18] and Breton [19] analyzed the formation and stability of cooperative alliances under international environmental agreements (IEAs) between asymmetric countries. Their model showed that non-signatory countries set their emission levels to maximize their own welfare, while signatory countries aim to maximize the welfare of all members. The study found that the number of signatory countries and the severity of environmental damage significantly influence the stability of cooperation. Specifically, a higher initial pollution level requires fewer signatories to reach a stable agreement. This suggests that when global environmental damage is severe, it becomes easier to encourage other countries to join international cooperation efforts. Shoude [16] also examined emissions trading in transboundary pollution.

Game theory has become an effective tool for addressing trans-regional air pollution control by analyzing social conflict and cooperation. Commonly used game models include evolutionary games, cooperative games, and dynamic games. Evolutionary games have been applied to study collaborative air pollution control in the Beijing–Tianjin–Hebei region [20] and pollution control issues in inland waterway shipping [21]. Cooperative games, particularly dynamic models, have been used to examine air pollution reduction under both cooperative and non-cooperative scenarios [22]. Additionally, dynamic games, such as those involving differential game models, have been utilized to construct atmospheric pollution ecological compensation mechanisms [23]. These studies demonstrate the application of traditional game theory, evolutionary games, and the emerging field of differential game theory.

A differential game is a time-continuous system in which multiple participants engage in continuous-time games, optimizing their respective strategies to achieve the Nash equilibrium—a state in which no player can improve their outcome by unilaterally changing their strategy—under conflicting objectives. This theory originated from the U.S. The Air Force’s military confrontation research in the 1950s integrates optimal control theory with game theory. Currently, differential games are primarily applied in areas such as cooperative emission reduction [23], disaster relief cooperation [24], inventory management [25], product pricing [26], and cooperative governance [27]. In the context of dynamic pollution control, Jianyue Liu et al. [28] proposed a strategy to mitigate transboundary air pollution using differential games, while Li et al. [29] investigated cross-border pollution issues through a differential game model.

Early research into applying game theory to pollution control laid a theoretical foundation for the field. Dockner and Long [30], along with Long [31], studied the Cournot-Nash model in the context of international pollution control using differential game methods. Their findings highlighted a significant drawback of non-cooperative strategies: countries prioritizing economic interests over environmental protection tend to increase global emissions when acting independently. This limitation led to further exploration of coordinated international pollution control strategies to improve emission reduction outcomes.

Building on these insights, Wirl [32] examined the effectiveness of tax and permit systems within the Cournot-Nash model framework, providing additional perspectives on the limitations of non-cooperative approaches.

Later, Insley and Forsyth [33] applied the Stackelberg model to pollution control, introducing a leader–follower structure to achieve partial coordination in non-cooperative games. Compared to the Cournot-Nash model, the Stackelberg framework somewhat reduces the tension between individual optimal decisions and social welfare. However, it still falls short of achieving globally optimal emission reduction targets, particularly in economically diverse regions.

Recent studies have advanced these theories by analyzing different game structures and their outcomes under various environmental policies. For example, Nkuiya and Plantinga [34] studied strategic pollution control in a free-trade context, finding that non-cooperative countries struggle to maximize global welfare through emission reductions. In contrast, the Stackelberg strategy, with its leader–follower hierarchy, partially mitigates adverse effects, though its effectiveness remains limited by non-cooperative behavior.

In cross-regional pollution control, balancing cooperative and non-cooperative strategies presents significant challenges. While the Stackelberg framework enables partial coordination through leader–follower dynamics, achieving optimal outcomes requires overcoming the inherent limitations of non-cooperative models like Cournot-Nash. This is especially challenging in economically diverse regions, where less affluent economies may “free-ride”, underinvesting in pollution control and thereby placing additional environmental burdens on neighboring regions.

The existing literature primarily focuses on regional pollution control, benefit distribution mechanisms, and various game theory methods. In contrast, this paper addresses pollution issues in regions with different levels of economic development, considering factors such as pollution stock, pollution control costs, and the environmental impact on different areas. It explores three models: the Nash non-cooperative mechanism, the pollution control cost-sharing mechanism, and the collaborative cooperation model. This paper examines the dynamic trajectories of pollution levels and regional benefits, and investigates the sensitivity of optimal strategies to factors such as short-sighted behavior and the natural environmental decomposition rate. Using numerical analysis, this study explores dynamic changes in total pollution and total benefits, as well as the impact of the discount rate and natural decomposition rate on benefit dynamics. These findings provide valuable insights for developing regional cooperative pollution control strategies.

2. Problem Description and Model Construction

In various regions, industrial enterprises release pollutants, such as sulfur dioxide, particulate matter (e.g., PM2.5), and volatile organic compounds, into the atmosphere during production processes. These emissions contribute to environmental problems locally and in surrounding areas. To mitigate the regional impact of such pollutants, the governments of two neighboring regions have decided to jointly address cross-boundary pollution.

Consider two local governments, Region A and Region B, operating under three potential governance models. The first model, non-cooperative governance, allows each government to maximize its own interests and manage pollution independently. The second model, a compensation framework, involves the more economically developed region sharing the pollution control costs of the other region. In this approach, the developed region first optimizes its decisions (e.g., pollution emissions, control investments, and compensation rate) to maximize its own value, and then the other region maximizes the total benefits based on this framework. The third model, collaborative governance, sees both governments working together to maximize joint interests and manage regional pollution collectively.

This article develops an optimal control game model for cross-boundary pollution, highlighting the benefits of collaborative governance. Nevertheless, cross-boundary pollution often involves multiple regions, each prioritizing self-interest, which can lead to excessive emissions and adverse externalities for others. Therefore, effective environmental governance requires addressing the collective action dilemma and fostering cross-regional collaborative efforts.

For analysis purposes, assume that Region A is more developed overall compared to Region B, and that each region has only one industrial enterprise involved in the production of a specific product. Each unit of product produced emits a certain amount of air pollutants, resulting in environmental damage to both the local region and the neighboring region, represented as ${\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})$, ${\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})$, ${\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})$ and ${\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})$, respectively. Here, ${\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}>0$ and ${\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}>0$ denote the coefficients representing the environmental damage caused by emissions within each respective region. Conversely, ${\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}>0$ and ${\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}>0$ represent the coefficients of environmental damage caused to the neighboring regions.

According to Breton [19], each region’s profit is determined by the production function of its respective enterprise.

(1)${\mathrm{R}}_{\mathrm{i}}\left(\mathrm{t}\right)={\mathsf{\alpha }}_{\mathrm{i}}{\mathrm{g}}_{\mathrm{i}}\left(\mathrm{t}\right)-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathrm{i}}^2\left(\mathrm{t}\right),\mathrm{i}=\mathrm{A},\mathrm{B}$

Here, the parameter $\alpha >0$ represents the utility coefficient of the industrial enterprise, which reflects, to some extent, the production efficiency of the region. g denotes the level of pollution emissions. ${\mathrm{R}}_{\mathrm{i}}\left(\mathrm{t}\right)$ represents the revenue of each region and will not appear in subsequent models; instead, its expanded equation will be used as a substitute.

Assume that reducing each unit of pollutant requires a unit of investment in the region. Consequently, when the pollution control investment is ${\mathrm{C}}_{\mathrm{i}}\left({\mathrm{h}}_{\mathrm{i}}(\mathrm{t}\right)),\mathrm{i}=\mathrm{A},\mathrm{B}$, the amount of pollutant emissions reduced is ${\mathsf{\omega }}_{\mathrm{i}}{\mathrm{h}}_{\mathrm{i}}\left(\mathrm{t}\right),\mathrm{i}=\mathrm{A},\mathrm{B},{\mathsf{\omega }}_{\mathrm{i}}>0$.

(2)${\mathrm{C}}_{\mathrm{i}}\left({\mathrm{h}}_{\mathrm{i}}(\mathrm{t}\right))={\displaystyle \frac{1}{2}}{\mathrm{c}}_{\mathrm{i}}{\mathrm{h}}_{\mathrm{i}}^2(\mathrm{t}),\mathrm{i}=\mathrm{A},\mathrm{B}$

Here, ${\mathrm{c}}_{\mathrm{i}}$ represents the pollution control cost coefficient, measuring the differences in pollution control technology between the two regions. The magnitude of this coefficient reflects the cost of pollution treatment in the region. ${\mathrm{C}}_{\mathrm{i}}\left({\mathrm{h}}_{\mathrm{i}}(\mathrm{t}\right))$ denotes the pollution control costs for Region A or B, which will not appear in later models; instead, it will be replaced by its expanded form.

The squared term ${\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathrm{i}}^2\left(\mathrm{t}\right)$ represents a cost factor reflecting the increasing negative impact of rising pollution emissions, with marginal losses escalating in a nonlinear manner. In other words, as emissions increase, the rate at which losses intensify also rises. Thus, this profit function illustrates the balance between production benefits and pollution costs. Similarly, the squared term ${\displaystyle \frac{1}{2}}{\mathrm{c}}_{\mathrm{i}}{\mathrm{h}}_{\mathrm{i}}^2(\mathrm{t})$ in the abatement cost function suggests that the marginal cost of abatement also increases nonlinearly with additional investment in abatement. The coefficient “${\displaystyle \frac{1}{2}}$” adjusts the growth rates of these marginal costs and losses, enhancing the model’s realism, drawing on forms from classical economics’ utility or loss functions [19].

Assume that a higher environmental management authority imposes an additional environmental tax $\mathsf{\tau }$ on the government of the pollutant-emitting region. This tax serves as an extra levy aimed at protecting and improving the natural environment.

Assume that the pollutant stock $\mathrm{x}(\mathrm{t})$ is represented by the following dynamic differential equation.

(3)$\stackrel{·}{\mathrm{x}}(\mathrm{t})={\displaystyle \sum _{\mathrm{i}=\mathrm{A},\mathrm{B}}{\mathrm{g}}_{\mathrm{i}}(\mathrm{t})}-{\displaystyle \sum _{\mathrm{i}=\mathrm{A},\mathrm{B}}{\mathsf{\omega }}_{\mathrm{i}}{\mathrm{h}}_{\mathrm{i}}(\mathrm{t})}-\mathsf{\zeta }\mathrm{x}(\mathrm{t})$

And $\mathrm{x}\left(0\right)={\mathrm{x}}_0,\mathrm{x}\left(\mathrm{t}\right)\ge 0$.

Here, $\mathsf{\zeta }>0$ represents the natural decomposition rate of the pollutant.

Assuming that pollutant emissions cause pollution in both regions and lead to social losses, both regions must consider the adverse effects of environmental damage in their decision-making. Suppose the negative effect is represented by a linear function ${\mathrm{d}}_{\mathrm{i}}\left(\mathrm{x}\right),\mathrm{i}=\mathrm{A},\mathrm{B}$, with ${\mathrm{d}}_{\mathrm{A}}=\mathrm{d}$, ${\mathrm{d}}_{\mathrm{B}}=\mathsf{\beta }\mathrm{d},(0<\mathsf{\beta }<1)$, where the parameter $\mathsf{\beta }$ represents the amplified negative effect in the less economically developed region. This parameter indicates that the less developed region has a lower capacity to bear environmental damage compared to the other region.

This study explores cross-border air pollution from a broad perspective, emphasizing the necessity of regional collaboration rather than focusing on specific pollution sources or meteorological factors. The model operates under the following simplified assumptions:

This study examines transboundary air pollution from a macro perspective, aiming to highlight the importance of regional cooperation rather than focusing on detailed pollution sources or meteorological influences. Our model operates with the following simplified assumptions:

Binary State Model: We analyze two hypothetical regions with different levels of economic development and environmental impact factors. Although real-world scenarios involve complex interactions among multiple regions, this two-region model provides a foundational approach to exploring cooperation dynamics.

Neutral Spatial Orientation: The model does not incorporate directional environmental factors, such as wind patterns, which could influence pollutant movement between regions. This neutral spatial assumption reflects our broad focus on cooperation principles rather than on meteorological specifics.

Generalized Pollutant Representation: Emissions are generalized rather than classified by source or type, and control measures are uniformly applied. This approach allows us to concentrate on overarching pollution control strategies rather than sector-specific solutions.

Homogeneous Cost and Control Assumptions: Cost and control parameters are generalized for each region. While acknowledging that real-world variations in production, technology, and industry structure introduce heterogeneity, this assumption simplifies our analysis.

Use of a Degradation Function: This study employs a linear degradation function to represent the natural decay of pollutants, simplifying the model to create a more accessible analytical framework. While this linear assumption does not capture the nonlinear processes present in actual atmospheric chemistry, it effectively represents the overall trend of pollutant concentrations on a macro scale. This approach offers valuable preliminary insights into transboundary pollution control by focusing on the impact of regional cooperation in decision-making, rather than the complexities of atmospheric transport and chemical reactions.

We acknowledge that real-world atmospheric processes involve numerous nonlinear mechanisms, including interactions among pollutants and variable weather conditions, which can affect pollutant degradation and transformation. Future research could incorporate nonlinear degradation models or more sophisticated chemical kinetics equations to address these complexities. However, within our current framework, the linear degradation function still provides a theoretical basis for evaluating cooperative control strategies and highlights the potential benefits of collaboration in pollution reduction.

These assumptions clarify that the model’s findings are intended to support high-level policy strategies for regional cooperation, rather than provide direct solutions to specific local issues. With these assumptions, the conclusions drawn from our model retain a degree of robustness under simplified conditions, laying a foundation for future studies in more complex environmental contexts.

3. Differential Game Model

3.1. Nash Non-Cooperative Game

In this scenario, each local government focuses on maximizing its own interests and independently manages environmental pollution within its region. The objective utility functions and constraints for both parties are as follows, with each local government aiming to maximize its utility. The utility functions for Region A and Region B are, respectively, as follows:

(4)${\mathrm{U}}_{\mathrm{A}}=\underset{{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})}{\max }{\displaystyle {\int }_0^{\infty }{\mathrm{e}}^{-\mathrm{p}\mathrm{t}}}\left\{\begin{array}{l}{\mathsf{\alpha }}_{\mathrm{A}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathrm{A}}^2(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{c}}_{\mathrm{A}}{\mathrm{h}}_{\mathrm{A}}^2(\mathrm{t})-\mathsf{\tau }[{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-{\mathsf{\omega }}_{\mathrm{A}}{\mathrm{h}}_{\mathrm{A}}(\mathrm{t})]\\ -{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-\mathrm{d}\mathrm{x}(\mathrm{t})\end{array}\right\}\mathrm{d}\mathrm{t}$

(5)${\mathrm{U}}_{\mathrm{B}}=\underset{{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})}{\max }{\displaystyle {\int }_0^{\infty }{\mathrm{e}}^{-\mathrm{p}\mathrm{t}}}\left\{\begin{array}{l}{\mathsf{\alpha }}_{\mathrm{B}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathrm{B}}^2(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{c}}_{\mathrm{B}}{\mathrm{h}}_{\mathrm{B}}^2(\mathrm{t})-\mathsf{\tau }[{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-{\mathsf{\omega }}_{\mathrm{B}}{\mathrm{h}}_{\mathrm{B}}(\mathrm{t})]\\ -{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-\mathsf{\beta }\mathrm{d}\mathrm{x}(\mathrm{t})\end{array}\right\}\mathrm{d}\mathrm{t}$

$\mathrm{s}.\mathrm{t}. \stackrel{·}{\mathrm{x}}(\mathrm{t})={\displaystyle \sum _{\mathrm{i}=\mathrm{A},\mathrm{B}}{\mathrm{g}}_{\mathrm{i}}(\mathrm{t})}-{\displaystyle \sum _{\mathrm{i}=\mathrm{A},\mathrm{B}}{\mathsf{\omega }}_{\mathrm{i}}{\mathrm{h}}_{\mathrm{i}}(\mathrm{t})}-\mathsf{\zeta }\mathrm{x}(\mathrm{t})$

Here, $\mathrm{p}>0$ denotes the discount rate.

Solving the optimal control problem for Equations (4) and (5) yields the optimal solution, with the equilibrium results indicated by the superscript “N*” to distinguish them.

Proposition 1.

Under the Nash non-cooperative game model, the equilibrium results for each region are as follows:

(1) The optimal pollution emission amount for Region $\mathrm{i}(\mathrm{i}=\mathrm{A},\mathrm{B})$ is

(6)$\left\{\begin{array}{l}{\mathrm{g}}_{\mathrm{A}}^{{\mathrm{N}}^{\ast }}={\displaystyle \frac{\left({\mathsf{\alpha }}_{\mathrm{A}}-\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)-\mathrm{d}}{\mathsf{\rho }+\mathsf{\zeta }}}\\ {\mathrm{g}}_{\mathrm{B}}^{{\mathrm{N}}^{\ast }}={\displaystyle \frac{\left({\mathsf{\alpha }}_{\mathrm{B}}-\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)-\mathsf{\beta }\mathrm{d}}{\mathsf{\rho }+\mathsf{\zeta }}}\end{array}\right.$

(2) The optimal pollution control investment amount for Region A (B) is

(7) $\left\{\begin{array}{l}{\mathrm{h}}_{\mathrm{A}}^{{\mathrm{N}}^{\ast }}={\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{A}}\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\mathrm{d}\right]}{{\mathrm{c}}_{\mathrm{A}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\\ {\mathrm{h}}_{\mathrm{B}}^{{\mathrm{N}}^{\ast }}={\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{B}}\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\mathsf{\beta }\mathrm{d}\right]}{{\mathrm{c}}_{\mathrm{B}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\end{array}\right.$

(3) The profit function for Region A (B) is

(8a) $\begin{array}{l}{\mathrm{V}}_{\mathrm{A}}^{{\mathrm{N}}^{\ast }}(\mathrm{x})=-{\displaystyle \frac{\mathrm{d}\mathrm{x}}{\mathsf{\rho }+\mathsf{\zeta }}}+{\displaystyle \frac{{\mathrm{c}}_{\mathrm{B}}{\left(\mathrm{d}+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right)}^2{{\mathsf{\omega }}_{\mathrm{A}}}^2+2{\mathrm{c}}_{\mathrm{A}}\mathrm{d}\left(\mathrm{d}\mathsf{\beta }+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right){{\mathsf{\omega }}_{\mathrm{B}}}^2}{2\mathsf{\rho }{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{\mathrm{d}}^2\left(1+2\mathsf{\beta }\right)-2\mathrm{d}\left(\mathsf{\rho }+\mathsf{\zeta }\right)\left({\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\alpha }}_{\mathrm{B}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}-\mathsf{\beta }{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}-2\mathsf{\tau }\right)}{2\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{\left({\mathsf{\alpha }}_{\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right)}^2+2{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}-{\mathsf{\alpha }}_{\mathrm{B}}\right)+2\mathsf{\tau }\left(-{\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\right)+{\mathsf{\tau }}^2}{2\mathsf{\rho }}}\end{array}$

(8b) $\begin{array}{l}{\mathrm{V}}_{\mathrm{B}}^{{\mathrm{N}}^{\ast }}(\mathrm{x})=-{\displaystyle \frac{\mathsf{\beta }\mathrm{dx}}{\mathsf{\rho }+\mathsf{\zeta }}}+{\displaystyle \frac{\mathrm{d}\mathsf{\beta }\left(\mathrm{d}+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right){{\mathsf{\omega }}_{\mathrm{A}}}^2}{\mathsf{\rho }{\mathrm{c}}_{\mathrm{A}}{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}+{\displaystyle \frac{+{\left(\mathrm{d}\mathsf{\beta }+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right)}^2{{\mathsf{\omega }}_{\mathrm{B}}}^2}{2\mathsf{\rho }{\mathrm{c}}_{\mathrm{B}}{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{\mathrm{d}}^2\mathsf{\beta }\left(2+\mathsf{\beta }\right)+2\mathrm{d}\left(\mathsf{\rho }+\mathsf{\zeta }\right)\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+\mathsf{\beta }\left(-{\mathsf{\alpha }}_{\mathrm{A}}-{\mathsf{\alpha }}_{\mathrm{B}}+{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+2\mathsf{\tau }\right)\right)}{2\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{{\mathsf{\alpha }}_{\mathrm{B}}}^2+2\left(-{\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right){\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+2\mathsf{\tau }{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}-2{\mathsf{\alpha }}_{\mathrm{B}}{\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+\mathsf{\tau }\right)}^2}{2\mathsf{\rho }}}\end{array}$

The optimal profit function for local governments under this model of environmental collaborative governance is as follows:

${\mathrm{V}}^{{\mathrm{N}}^{\mathrm{\ast }}}\left(\mathrm{x}\right)={\mathrm{V}}_{\mathrm{A}}^{{\mathrm{N}}^{\mathrm{\ast }}}\left(\mathrm{x}\right)+{\mathrm{V}}_{\mathrm{B}}^{{\mathrm{N}}^{\mathrm{\ast }}}\left(\mathrm{x}\right) = \left(8\mathrm{a}\right) + \left(8\mathrm{b}\right)$

(4) The dynamic change process of the pollutant stock over time t is

(9)$\begin{array}{c}{\mathrm{x}}^{\mathrm{N}}(\mathrm{t})={\displaystyle \frac{{\mathrm{A}}_{\mathrm{N}}}{\mathsf{\zeta }}}+\left({\mathrm{x}}_0-{\displaystyle \frac{{\mathrm{A}}_{\mathrm{N}}}{\mathsf{\zeta }}}\right){\mathrm{e}}^{-\mathsf{\zeta }\mathrm{t}}\hfill \\ \left\{\begin{array}{l}\stackrel{·}{\mathrm{x}}(\mathrm{t})={\mathrm{A}}_{\mathrm{N}}-\mathsf{\zeta }\mathrm{x}(\mathrm{t})\\ {\mathrm{x}}^{\mathrm{N}}(0)={\mathrm{x}}_0\\ \\ {\mathrm{A}}_{\mathrm{N}}=\left\{\left\{\begin{array}{l}{\displaystyle \frac{\left({\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\alpha }}_{\mathrm{B}}-\mathsf{\zeta }-2\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)-\mathrm{d}\left(1+\mathsf{\beta }\right)}{\mathsf{\rho }+\mathsf{\zeta }}}\\ -{\displaystyle \frac{{{\mathsf{\omega }}_{\mathrm{A}}}^2\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\mathrm{d}\right]}{{\mathrm{c}}_{\mathrm{A}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}-{\displaystyle \frac{{{\mathsf{\omega }}_{\mathrm{B}}}^2\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\mathsf{\beta }\mathrm{d}\right]}{{\mathrm{c}}_{\mathrm{B}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\end{array}\right.\right\}\end{array}\right.\hfill \end{array}$

Proof of Proposition 1.

A detailed proof of Proposition 1 is provided in Appendix A. The proof of Proposition 1 is complete. □

Theorem 1.

In a non-cooperative game, the dynamic path of pollutant stock $x^N\left(t\right)$ is influenced by the pollutant stock level and can be divided into the following three cases: When ${\mathrm{x}}_0<{\displaystyle \frac{{\mathrm{A}}_{\mathrm{N}}}{\mathsf{\zeta }}}$, the pollutant stock shows an increasing trend over time. When ${\mathrm{x}}_0={\displaystyle \frac{{\mathrm{A}}_{\mathrm{N}}}{\mathsf{\zeta }}}$, the pollutant stock remains constant over time. When ${\mathrm{x}}_0>{\displaystyle \frac{{\mathrm{A}}_{\mathrm{N}}}{\mathsf{\zeta }}}$, the pollutant stock shows a decreasing trend over time.

Proof of Theorem 1.

Taking the first derivative of ${\mathrm{x}}^{\mathrm{N}}\left(\mathrm{t}\right)$ with respect to time t, we obtain $({\mathrm{x}}^{\mathrm{N}}{)}^{\prime }(\mathrm{t})=-\mathsf{\zeta }({\mathrm{x}}_0-{\displaystyle \frac{{\mathrm{A}}_{\mathrm{N}}}{\mathsf{\zeta }}}){\mathrm{e}}^{-\mathsf{\zeta }\mathrm{t}}$. Clearly, when ${\mathrm{x}}_0<{\displaystyle \frac{{\mathrm{A}}_{\mathrm{N}}}{\mathsf{\zeta }}}$, $({\mathrm{x}}^{\mathrm{N}}{)}^{\prime }(\mathrm{t})>0$, meaning ${\mathrm{x}}^{\mathrm{N}}\left(\mathrm{t}\right)$ is monotonically increasing over time. When ${\mathrm{x}}_0={\displaystyle \frac{{\mathrm{A}}_{\mathrm{N}}}{\mathsf{\zeta }}}$, $({\mathrm{x}}^{\mathrm{N}}{)}^{\prime }(\mathrm{t})=0$, meaning ${\mathrm{x}}^{\mathrm{N}}\left(\mathrm{t}\right)$ remains constant. When ${\mathrm{x}}_0>{\displaystyle \frac{{\mathrm{A}}_{\mathrm{N}}}{\mathsf{\zeta }}}$, $({\mathrm{x}}^{\mathrm{N}}{)}^{\prime }(\mathrm{t})<0$, meaning ${\mathrm{x}}^{\mathrm{N}}\left(\mathrm{t}\right)$ is monotonically decreasing over time. The proof of Theorem 1 is complete. □

3.2. Stackelberg Game

In this case, a pollution control cost compensation mechanism is introduced within the “non-cooperative game” model to reduce the total pollution stock. Under this mechanism, the more economically developed Region A shares pollution control costs with Region B. Region A first selects a decision that maximizes its own utility function, and based on this, Region B subsequently chooses a decision that maximizes its total benefit. Let $\mathsf{\theta }$ denote the pollution control cost compensation rate. The utility functions for Region A and Region B are defined as follows:

(10)${\mathrm{U}}_{\mathrm{A}}=\underset{{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})}{\max }{\displaystyle {\int }_0^{\infty }{\mathrm{e}}^{-\mathrm{p}\mathrm{t}}}\left\{\begin{array}{l}{\mathsf{\alpha }}_{\mathrm{A}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathrm{A}}^2(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{c}}_{\mathrm{A}}{\mathrm{h}}_{\mathrm{A}}^2(\mathrm{t})-\mathsf{\tau }[{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-{\mathsf{\omega }}_{\mathrm{A}}{\mathrm{h}}_{\mathrm{A}}(\mathrm{t})]\\ -{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-\mathrm{d}\mathrm{x}(\mathrm{t})\end{array}\right\}\mathrm{d}\mathrm{t}$

(11)${\mathrm{U}}_{\mathrm{B}}=\underset{{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})}{\max }{\displaystyle {\int }_0^{\infty }{\mathrm{e}}^{-\mathrm{p}\mathrm{t}}}\left\{\begin{array}{l}{\mathsf{\alpha }}_{\mathrm{B}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathrm{B}}^2(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{c}}_{\mathrm{B}}{\mathrm{h}}_{\mathrm{B}}^2(\mathrm{t})-\mathsf{\tau }[{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-{\mathsf{\omega }}_{\mathrm{B}}{\mathrm{h}}_{\mathrm{B}}(\mathrm{t})]\\ -{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-\mathsf{\beta }\mathrm{d}\mathrm{x}(\mathrm{t})\end{array}\right\}\mathrm{d}\mathrm{t}$

Solving the optimal control problem for Equations (10) and (11) yields the optimal solution, with the equilibrium results denoted by the superscript “S*” for distinction.

Proposition 2.

Under the Stackelberg game model (pollution control cost compensation mechanism), the equilibrium results for each region are as follows:

(1) The optimal pollution emission amount for Region $\mathrm{i}(\mathrm{i}=\mathrm{A},\mathrm{B})$ is

(12)$\left\{\begin{array}{l}{\mathrm{g}}_{\mathrm{A}}^{{\mathrm{N}}^{\ast }}={\displaystyle \frac{\left({\mathsf{\alpha }}_{\mathrm{A}}-\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)-\mathrm{d}}{\mathsf{\rho }+\mathsf{\zeta }}}\\ {\mathrm{g}}_{\mathrm{B}}^{{\mathrm{N}}^{\ast }}={\displaystyle \frac{\left({\mathsf{\alpha }}_{\mathrm{B}}-\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)-\mathsf{\beta }\mathrm{d}}{\mathsf{\rho }+\mathsf{\zeta }}}\end{array}\right.$

(2) The optimal pollution control investment amount for Region A (B) is

(13) $\left\{\begin{array}{l}{\mathrm{h}}_{\mathrm{A}}^{{\mathrm{S}}^{\ast }}={\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{A}}\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\mathrm{d}\right]}{{\mathrm{c}}_{\mathrm{A}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\\ {\mathrm{h}}_{\mathrm{B}}^{{\mathrm{S}}^{\ast }}={\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{B}}\left[\mathrm{d}\left(2+\mathsf{\beta }\right)+\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)\right]}{{2\mathrm{c}}_{\mathrm{B}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\end{array}\right.$

(3)The optimal pollution control cost compensation rate for Region A (B) is

(14) ${\mathsf{\theta }}^{*}={\displaystyle \frac{\mathrm{d}\left(2-\mathsf{\beta }\right)-\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)}{\mathrm{d}\left(2+\mathsf{\beta }\right)+\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)}},\mathrm{d}>{\displaystyle \frac{\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)}{2{\mathrm{c}}_{\mathrm{B}}-\mathsf{\beta }}}$

(4) The profit function for Region A (B) is

(15a) $\begin{array}{l}{\mathrm{V}}_{\mathrm{A}}^{{\mathrm{S}}^{\ast }}(\mathrm{x})=-{\displaystyle \frac{\mathrm{d}\mathrm{x}}{\mathsf{\rho }+\mathsf{\zeta }}}+{\displaystyle \frac{2\left({\mathrm{d}}^2\left(1+2\mathsf{\beta }\right)-2\mathrm{d}\left(\mathsf{\rho }+\mathsf{\zeta }\right)\left({\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\alpha }}_{\mathrm{B}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}-\mathsf{\beta }{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}-2\mathsf{\tau }\right)\right)}{\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{2\left({\left({\mathsf{\alpha }}_{\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right)}^2+2{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}-{\mathsf{\alpha }}_{\mathrm{B}}\right)+2\mathsf{\tau }\left(-{\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\right)+{\mathsf{\tau }}^2\right)}{\mathsf{\rho }}}\\ +{\displaystyle \frac{4{\mathrm{c}}_{\mathrm{B}}{\left(\mathrm{d}+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right)}^2{{\mathsf{\omega }}_{\mathrm{A}}}^2+{\mathrm{c}}_{\mathrm{A}}{\left(\mathrm{d}\left(2+\mathsf{\beta }\right)+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right)}^2{{\mathsf{\omega }}_{\mathrm{B}}}^2}{2{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\end{array}$

(15b) $\begin{array}{l}{\mathrm{V}}_{\mathrm{B}}^{{\mathrm{S}}^{\ast }}(\mathrm{x})=-{\displaystyle \frac{\mathsf{\beta }\mathrm{dx}}{\mathsf{\rho }+\mathsf{\zeta }}}+{\displaystyle \frac{\mathrm{d}\mathsf{\beta }\left(\mathrm{d}+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right){{\mathsf{\omega }}_{\mathrm{A}}}^2}{{\mathrm{c}}_{\mathrm{A}}\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}+{\displaystyle \frac{\mathsf{\tau }{{\mathsf{\omega }}_{\mathrm{B}}}^2\left(\mathrm{d}\mathsf{\beta }+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right)\left(\mathrm{d}\left(2+\mathsf{\beta }\right)+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\right)}{4{\mathrm{c}}_{\mathrm{B}}\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{{\mathsf{\omega }}_{\mathrm{B}}}^2\left({\mathrm{d}}^2\mathsf{\beta }\left(2+\mathsf{\beta }\right)+2\mathrm{d}\left(\mathsf{\rho }+\mathsf{\zeta }\right)\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+\mathsf{\beta }\left(-{\mathsf{\alpha }}_{\mathrm{A}}-{\mathsf{\alpha }}_{\mathrm{B}}+{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+2\mathsf{\tau }\right)\right)\right)}{2\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{{\mathsf{\omega }}_{\mathrm{B}}}^2\left({{\mathsf{\alpha }}_{\mathrm{B}}}^2+2\left(-{\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right){\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+2{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}\mathsf{\tau }-2{\mathsf{\alpha }}_{\mathrm{B}}\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+\mathsf{\tau }\right)+{\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+\mathsf{\tau }\right)}^2\right)}{2\mathsf{\rho }}}\end{array}$

The optimal profit function for local governments under this model of environmental collaborative governance is as follows:

${\mathrm{V}}^{{\mathrm{S}}^{\ast }}(\mathrm{x})={\mathrm{V}}_{\mathrm{A}}^{{\mathrm{S}}^{\ast }}(\mathrm{x})+{\mathrm{V}}_{\mathrm{B}}^{{\mathrm{S}}^{\ast }}(\mathrm{x})= \left(15\mathrm{a}\right) + \left(15\mathrm{b}\right)$

(5) The dynamic change process of the pollutant stock over time t is

(16) $\begin{array}{c}{\mathrm{x}}^{\mathrm{S}}(\mathrm{t})={\displaystyle \frac{{\mathrm{A}}_{\mathrm{S}}}{\mathsf{\zeta }}}+\left({\mathrm{x}}_0-{\displaystyle \frac{{\mathrm{A}}_{\mathrm{S}}}{\mathsf{\zeta }}}\right){\mathrm{e}}^{-\mathsf{\zeta }\mathrm{t}}\hfill \\ \left\{\begin{array}{l}\stackrel{·}{\mathrm{x}}(\mathrm{t})={\mathrm{A}}_{\mathrm{S}}-\mathsf{\zeta }\mathrm{x}(\mathrm{t})\\ {\mathrm{x}}^{\mathrm{S}}(0)={\mathrm{x}}_0\\ \\ {\mathrm{A}}_{\mathrm{S}}=\left\{\begin{array}{l}{\displaystyle \frac{\left({\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\alpha }}_{\mathrm{B}}-\mathsf{\zeta }-2\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)-\mathrm{d}\left(1+\mathsf{\beta }\right)}{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\\ -{\displaystyle \frac{{{\mathsf{\omega }}_{\mathrm{A}}}^2\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\mathrm{d}\right]}{{\mathrm{c}}_{\mathrm{A}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}-{\displaystyle \frac{{{\mathsf{\omega }}_{\mathrm{B}}}^2\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\left(2+\mathsf{\beta }\right)\mathrm{d}\right]}{{2\mathrm{c}}_{\mathrm{B}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\end{array}\right\}\end{array}\right.\hfill \end{array}$

Proof of Proposition 2.

The detailed proof of Proposition 2 can be found in Appendix A. The proof of Proposition 2 is complete. □

Theorem 2.

In Stackelberg game, the dynamic path of pollutant stock ${\mathrm{x}}^{\mathrm{S}}(\mathrm{t})$ is influenced by the pollutant stock level and can be divided into the following three cases: When ${\mathrm{x}}_0<{\displaystyle \frac{{\mathrm{A}}_{\mathrm{S}}}{\mathsf{\zeta }}}$, the pollutant stock shows an increasing trend over time. When ${\mathrm{x}}_0={\displaystyle \frac{{\mathrm{A}}_{\mathrm{S}}}{\mathsf{\zeta }}}$, the pollutant stock remains constant over time. When ${\mathrm{x}}_0>{\displaystyle \frac{{\mathrm{A}}_{\mathrm{S}}}{\mathsf{\zeta }}}$, the pollutant stock shows a decreasing trend over time.

Proof of Theorem 2.

The detailed proof of Theorem 2 is similar to that of Theorem 1 and is therefore omitted for brevity. The proof of Theorem 2 is complete. □

3.3. Cooperative Game

In this case, collaborative governance is applied, meaning that the two regions cooperate to manage regional environmental pollution based on maximizing their joint interests. Therefore, the joint utility objective function for the two neighboring regions can be described as follows:

(17)${\mathrm{U}}_{\mathrm{A}\mathrm{B}}=\underset{{\mathrm{g}}_{\mathrm{A}}(\mathrm{t}),{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})}{\max }{\displaystyle {\int }_0^{\infty }{\mathrm{e}}^{-\mathrm{p}\mathrm{t}}}\left\{\begin{array}{l}{\mathsf{\alpha }}_{\mathrm{A}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})+{\mathsf{\alpha }}_{\mathrm{B}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathrm{A}}^2(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathrm{B}}^2(\mathrm{t})-{\displaystyle \frac{1}{2}}{\mathrm{c}}_{\mathrm{A}}{\mathrm{h}}_{\mathrm{A}}^2(\mathrm{t})\\ -{\displaystyle \frac{1}{2}}{\mathrm{c}}_{\mathrm{B}}{\mathrm{h}}_{\mathrm{B}}^2(\mathrm{t})-\mathsf{\tau }[{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-{\mathsf{\omega }}_{\mathrm{A}}{\mathrm{h}}_{\mathrm{A}}(\mathrm{t})]-\mathsf{\tau }[{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-{\mathsf{\omega }}_{\mathrm{B}}{\mathrm{h}}_{\mathrm{B}}(\mathrm{t})]\\ -{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}{\mathrm{g}}_{\mathrm{B}}(\mathrm{t})-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}{\mathrm{g}}_{\mathrm{A}}(\mathrm{t})-(1+\mathsf{\beta })\mathrm{d}\mathrm{x}(\mathrm{t})\end{array}\right\}\mathrm{d}\mathrm{t}$

Solving the optimal control problem for Equation (17) yields the optimal solutions for the benefit of the two regions, with the equilibrium results denoted by the superscript “C*” for distinction.

Proposition 3.

Under the Cooperative game model, the equilibrium results for each region are as follows:

(1) The optimal pollution emission amount for Region $\mathrm{i}(\mathrm{i}=\mathrm{A},\mathrm{B})$ is

(18)$\left\{\begin{array}{l}{\mathrm{g}}_{\mathrm{A}}^{{\mathrm{C}}^{\ast }}={\displaystyle \frac{\left({\mathsf{\alpha }}_{\mathrm{A}}-\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)-\left(1+\mathsf{\beta }\right)\mathrm{d}}{\mathsf{\rho }+\mathsf{\zeta }}}\\ {\mathrm{g}}_{\mathrm{B}}^{{\mathrm{C}}^{\ast }}={\displaystyle \frac{\left({\mathsf{\alpha }}_{\mathrm{B}}-\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)-\left(1+\mathsf{\beta }\right)\mathrm{d}}{\mathsf{\rho }+\mathsf{\zeta }}}\end{array}\right.$

(2) The optimal pollution control investment amount for Region A (B) is

(19) $\left\{\begin{array}{l}{\mathrm{h}}_{\mathrm{A}}^{{\mathrm{C}}^{\ast }}={\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{A}}\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\left(1+\mathsf{\beta }\right)\mathrm{d}\right]}{{\mathrm{c}}_{\mathrm{A}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\\ {\mathrm{h}}_{\mathrm{B}}^{{\mathrm{C}}^{\ast }}={\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{B}}\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\left(1+\mathsf{\beta }\right)\mathrm{d}\right]}{{\mathrm{c}}_{\mathrm{B}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\end{array}\right.$

(3) The total benefit function for the region A (B) is

(20) $\begin{array}{l}{\mathrm{V}}_{\mathrm{A}\mathrm{B}}^{{\mathrm{C}}^{\ast }}(\mathrm{x})=-{\displaystyle \frac{\left(1+\mathsf{\beta }\right)\mathrm{d}\mathrm{x}}{\mathsf{\rho }+\mathsf{\zeta }}}+{\displaystyle \frac{{\mathrm{d}}^2{\left(1+\mathsf{\beta }\right)}^2}{\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{{\mathsf{\alpha }}_{\mathrm{A}}}^2+{{\mathsf{\alpha }}_{\mathrm{B}}}^2-2{\mathsf{\alpha }}_{\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+\mathsf{\tau }\right)-2{\mathsf{\alpha }}_{\mathrm{B}}\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+\mathsf{\tau }\right)}{2\mathsf{\rho }}}\\ +{\displaystyle \frac{{{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}}^2+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}\left(2{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}\right)+2{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+\mathsf{\tau }\right)+2\mathsf{\tau }\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+\mathsf{\tau }\right)}{2\mathsf{\rho }}}\\ -{\displaystyle \frac{{\mathsf{\alpha }}_{\mathrm{A}}\mathrm{d}\left(1+\mathsf{\beta }\right)+\mathrm{d}\left(1+\mathsf{\beta }\right)\left({\mathsf{\alpha }}_{\mathrm{B}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}-2\mathsf{\tau }\right)}{\mathsf{\rho }\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\\ +{\displaystyle \frac{{\mathrm{c}}_{\mathrm{B}}{\left(\mathrm{d}\left(1+\mathsf{\beta }\right)+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right)}^2{{\mathsf{\omega }}_{\mathrm{A}}}^2+{\mathrm{c}}_{\mathrm{A}}{\left(\mathrm{d}\left(1+\mathsf{\beta }\right)+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right)}^2{{\mathsf{\omega }}_{\mathrm{B}}}^2}{2{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\end{array}$

(4) The dynamic change process of the pollutant stock over time t is

(21)$\begin{array}{c}{\mathrm{x}}^{\mathrm{C}}(\mathrm{t})={\displaystyle \frac{{\mathrm{A}}_{\mathrm{C}}}{\mathsf{\zeta }}}+\left({\mathrm{x}}_0-{\displaystyle \frac{{\mathrm{A}}_{\mathrm{C}}}{\mathsf{\zeta }}}\right){\mathrm{e}}^{-\mathsf{\zeta }\mathrm{t}}\hfill \\ \left\{\begin{array}{l}\stackrel{·}{\mathrm{x}}(\mathrm{t})={\mathrm{A}}_{\mathrm{C}}-\mathsf{\zeta }\mathrm{x}(\mathrm{t})\\ {\mathrm{x}}^{\mathrm{C}}(0)={\mathrm{x}}_0\\ \\ {\mathrm{A}}_{\mathrm{C}}=\left\{\begin{array}{l}{\displaystyle \frac{\left({\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\alpha }}_{\mathrm{B}}-\mathsf{\zeta }-2\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)-2\mathrm{d}\left(1+\mathsf{\beta }\right)}{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\\ -{\displaystyle \frac{\left({\mathrm{c}}_{\mathrm{A}}{{\mathsf{\omega }}_{\mathrm{B}}}^2+{\mathrm{c}}_{\mathrm{B}}{{\mathsf{\omega }}_{\mathrm{A}}}^2\right)\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\left(1+\mathsf{\beta }\right)\mathrm{d}\right]}{{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\end{array}\right\}\end{array}\right.\hfill \end{array}$

Proof of Theorem 3.

Since the detailed proof of Theorem 3 closely resembles that of Theorem 1, it is omitted here for conciseness. The proof of Theorem 3 is complete. □

Theorem 3.

In Stackelberg game, the dynamic path of pollutant stock ${\mathrm{x}}^{\mathrm{C}}(\mathrm{t})$ is influenced by the pollutant stock level and can be divided into the following three cases: When ${\mathrm{x}}_0<{\displaystyle \frac{{\mathrm{A}}_{\mathrm{C}}}{\mathsf{\zeta }}}$, the pollutant stock shows an increasing trend over time. When ${\mathrm{x}}_0={\displaystyle \frac{{\mathrm{A}}_{\mathrm{C}}}{\mathsf{\zeta }}}$, the pollutant stock remains constant over time. When ${\mathrm{x}}_0>{\displaystyle \frac{{\mathrm{A}}_{\mathrm{C}}}{\mathsf{\zeta }}}$, the pollutant stock shows a decreasing trend over time.

Proof of Proposition 3.

The detailed proof of Proposition 3 can be found in Appendix A. The proof of Proposition 3 is complete. □

4. Comparison of Equilibrium Results

By performing a comparative analysis of the optimal solutions under the Nash non-cooperative game, the Stackelberg game, and the Cooperative game (i.e., Propositions 1 through 3), the following results are obtained:

${\mathrm{g}}_{\mathrm{A}}^{{\mathrm{N}}^{\ast }}={\mathrm{g}}_{\mathrm{A}}^{{\mathrm{S}}^{\ast }}>{\mathrm{g}}_{\mathrm{A}}^{{\mathrm{C}}^{\ast }}, {\mathrm{g}}_{\mathrm{B}}^{{\mathrm{N}}^{\ast }}={\mathrm{g}}_{\mathrm{B}}^{{\mathrm{S}}^{\ast }}>{\mathrm{g}}_{\mathrm{B}}^{{\mathrm{C}}^{\ast }}, {\mathrm{h}}_{\mathrm{A}}^{{\mathrm{N}}^{\ast }}={\mathrm{h}}_{\mathrm{A}}^{{\mathrm{S}}^{\ast }}<{\mathrm{h}}_{\mathrm{A}}^{{\mathrm{C}}^{\ast }}, {\mathrm{h}}_{\mathrm{B}}^{{\mathrm{C}}^{\ast }}>{\mathrm{h}}_{\mathrm{B}}^{{\mathrm{S}}^{\ast }}>{\mathrm{h}}_{\mathrm{B}}^{{\mathrm{N}}^{\ast }}$

The above results indicate the following:

(1) Regardless of whether the Stackelberg game or the Nash non-cooperative game is used, the total optimal pollution emission levels in both regions remain unchanged. However, with the compensation mechanism, pollution control investment in the less developed region may increase.

(2) Under the cooperative game, total air pollution emissions in both regions decrease, and both regions also increase their pollution control investments.

Based on Equations (9), (16), and (21), the differences in total air pollutant stock across the three game models are as follows:

$\left\{\begin{array}{l}{\mathrm{x}}^{\mathrm{N}}(\mathrm{t})-{\mathrm{x}}^{\mathrm{C}}(\mathrm{t})={\displaystyle \frac{\mathrm{d}\left[{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}\left(\mathsf{\beta }+1\right)+{\mathrm{c}}_{\mathrm{B}}{{\mathsf{\omega }}_{\mathrm{A}}}^2\mathsf{\beta }+{\mathrm{c}}_{\mathrm{A}}{{\mathsf{\omega }}_{\mathrm{B}}}^2\right]}{{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}>0\\ {\mathrm{x}}^{\mathrm{S}}(\mathrm{t})-{\mathrm{x}}^{\mathrm{C}}(\mathrm{t})={\displaystyle \frac{{2\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}\mathrm{d}\left(\mathsf{\beta }+1\right)+{2\mathrm{c}}_{\mathrm{B}}{{\mathsf{\omega }}_{\mathrm{A}}}^2\mathsf{\beta }\mathrm{d}+{\mathrm{c}}_{\mathrm{A}}{{\mathsf{\omega }}_{\mathrm{B}}}^2\left[\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\mathrm{d}\mathsf{\beta }\right]}{{2\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}>0\\ {\mathrm{x}}^{\mathrm{N}}(\mathrm{t})-{\mathrm{x}}^{\mathrm{S}}(\mathrm{t})={\displaystyle \frac{{{\mathsf{\omega }}_{\mathrm{B}}}^2\left[2\mathrm{d}-\mathrm{d}\mathsf{\beta }-\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)\right]}{{2\mathrm{c}}_{\mathrm{B}}\left(\mathsf{\rho }+\mathsf{\zeta }\right)}}\end{array}\right.$

From this, we derive Proposition 4:

Proposition 4.

To reduce the total pollutant stock in both regions, the cooperative game method is superior to both the Nash non-cooperative game and Stackelberg game methods. Additionally, when comparing the effectiveness of the Stackelberg game and Nash non-cooperative game methods, we find the following: When $\mathrm{d}>{\displaystyle \frac{\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)}{2-\mathsf{\beta }}}$, the Stackelberg game method is more effective in reducing pollutant stock than the Nash non-cooperative game method. When $\mathrm{d}<{\displaystyle \frac{\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)}{2-\mathsf{\beta }}}$, the Nash non-cooperative game method is more effective in reducing pollutant stock than the Stackelberg game method.

Based on Equations (8a), (8b), (15a), (15b) and (20), the differences in total benefits between the two regions under the three game models are as follows:

$\left\{\begin{array}{l}{\mathrm{V}}_{\mathrm{B}}^{{\mathrm{S}}^{\ast }}(\mathrm{x})-{\mathrm{V}}_{\mathrm{B}}^{{\mathrm{N}}^{\ast }}(\mathrm{x})={\displaystyle \frac{\left(2\mathrm{d}-\mathrm{d}\mathsf{\beta }-\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)\right)\left(\mathrm{d}\mathsf{\beta }+\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)\right){{\mathsf{\omega }}_{\mathrm{B}}}^2}{4{\mathrm{c}}_{\mathrm{B}}\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}>0\\ \\ {\mathrm{V}}_{\mathrm{A}\mathrm{B}}^{{\mathrm{C}}^{\ast }}(\mathrm{x})-{\mathrm{V}}_{\mathrm{A}}^{{\mathrm{N}}^{\ast }}(\mathrm{x})-{\mathrm{V}}_{\mathrm{B}}^{{\mathrm{N}}^{\ast }}(\mathrm{x})=\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}\left(\mathrm{d}\left(1+{\mathsf{\beta }}^2\right)+2\left(\mathsf{\beta }{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)\right)}{2\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{\mathrm{d}\left({\mathrm{c}}_{\mathrm{B}}\mathrm{d}{\mathsf{\beta }}^2{{\mathsf{\omega }}_{\mathrm{A}}}^2+{\mathrm{c}}_{\mathrm{A}}\mathrm{d}{{\mathsf{\omega }}_{\mathrm{B}}}^2\right)}{2{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\end{array}\right\}>0\\ {\mathrm{V}}_{\mathrm{A}\mathrm{B}}^{{\mathrm{C}}^{\ast }}(\mathrm{x})-{\mathrm{V}}_{\mathrm{A}}^{{\mathrm{S}}^{\ast }}(\mathrm{x})-{\mathrm{V}}_{\mathrm{B}}^{{\mathrm{S}}^{\ast }}(\mathrm{x})=\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}\left({\mathrm{c}}_{\mathrm{A}}\mathrm{d}\left(1+{\mathsf{\beta }}^2\right)+2{\mathrm{c}}_{\mathrm{A}}\left(\mathsf{\beta }{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\right)\left(\mathsf{\rho }+\mathsf{\zeta }\right)+\mathrm{d}{\mathsf{\beta }}^2{{\mathsf{\omega }}_{\mathrm{A}}}^2\right)}{{2\mathrm{c}}_{\mathrm{A}}\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{\left(\mathrm{d}\mathsf{\beta }+\mathsf{\tau }\left(\mathsf{\rho }+\mathsf{\zeta }\right)\right)}^2{{\mathsf{\omega }}_{\mathrm{B}}}^2}{8{\mathrm{c}}_{\mathrm{B}}\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\end{array}\right\}>0\end{array}\right.$

From this, we derive Proposition 5:

Proposition 5.

The instantaneous benefits of both regions are higher under the Stackelberg game method than under the Nash non-cooperative game method. Furthermore, the instantaneous benefits of both regions are higher under the Cooperative game method than under either the Nash non-cooperative game or the Stackelberg game methods.

Propositions 4 and 5 highlight the substantial advantages of collaborative pollution management approaches. From a management perspective, cooperative strategies not only reduce pollutant stock more effectively, but also significantly enhance instantaneous benefits for both regions compared to non-cooperative or sequential approaches. Specifically, the Cooperative game model maximizes both environmental and economic outcomes by aligning regional interests, enabling shared responsibility, and reducing cumulative environmental damage.

Furthermore, in scenarios where full cooperation may be challenging, implementing a compensation mechanism—as illustrated in the Stackelberg game—still leads to improved environmental and economic outcomes relative to non-cooperative methods. These findings underscore the critical role of regional coordination in tackling complex, cross-boundary environmental issues. Cooperation and appropriate compensation mechanisms can drive Pareto improvements by both reducing pollution and increasing economic benefits. This study thus reinforces the importance of joint governance models in promoting sustainable environmental management and enhancing mutual benefits across regions.

5. Numerical Analysis

In this section, we expand on the conclusions drawn in the previous section through numerical modeling using Mathematica 13.2. The baseline parameter settings follow the research findings of Li and Shoude [16], as detailed below: ${\mathsf{\alpha }}_{\mathrm{A}}=150, {\mathsf{\alpha }}_{\mathrm{B}}=130, \mathrm{d}=5,$ $\mathsf{\beta }=0.85, {\mathrm{c}}_{\mathrm{A}}=1, {\mathrm{c}}_{\mathrm{B}}=1.2, \mathsf{\rho }=0.1, \mathsf{\zeta }=0.2, \mathsf{\tau }=5, {\mathrm{x}}_0=400, {\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}=0.1, {\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}=0.1, {\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}=0.1,$ ${\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}=0.1, {\mathsf{\omega }}_{\mathrm{A}}=1, {\mathsf{\omega }}_{\mathrm{B}}=2$. Based on these parameter settings, we use Equations (6), (7), (12), (13), (18), and (19) to calculate the optimal pollution emission levels and pollution control investment amounts for both regions across the three game models. A comparative analysis of the results is then provided.

The comparison shows that the results in Table 1 align with the equilibrium conclusions for each model.

5.1. Optimal Pollution Stock and Instantaneous Benefit Trajectory

Figure 1a,c illustrate the evolution of total pollutant stock and total revenue for both regions over time under the three game models, showing gradual convergence to a steady state. Figure 1b depicts the evolution of each region’s revenue over time under the Nash non-cooperative game and Stackelberg game models, which also converge to stability over time.

From Figure 1a,c, it is evident that collaborative governance in air pollution control has a distinct advantage in reducing pollutant stock and increasing regional revenues.

5.2. Numerical Analysis in Steady State

In this section, we selected two representative factors—the discount rate and the natural decomposition rate of pollutants—to analyze their impact on steady-state Nash equilibrium outcomes. These factors often differ significantly across regions with varying levels of economic development.

Figure 2a–c illustrate the effect of different discount rates on steady-state Nash equilibrium outcomes. The slope of the graphs shows a positive correlation between the discount rate and pollutant stock (i.e., a higher discount rate corresponds to a larger pollutant stock, and vice versa). Conversely, the discount rate is negatively correlated with each region’s revenue and the total revenue of both regions (i.e., a higher discount rate results in lower revenue or total revenue, and vice versa).

The discount rate reflects a region’s time value of money, economic environment, and market risk level. A higher discount rate typically indicates greater uncertainty about future returns, leading investors to seek higher returns to offset risk. In contrast, a lower discount rate suggests reduced capital costs, a stable economic environment, and more predictable returns. The graphs indicate that as the discount rate increases, the slopes of the pollutant stock and revenue curves tend to decrease. Our analysis suggests that as the discount rate rises, regions may exhibit “short-sighted” behavior, prioritizing short-term gains and consequently reducing investment in pollution control technology.

When the discount rate remains excessively high, the benefits of the “cost compensation mechanism” over the “non-cooperative mechanism” in the Nash non-cooperative game and Stackelberg game models—such as reduced pollutant stock and increased revenue—may become less pronounced.

Figure 3a–c illustrate the impact of varying natural decomposition rates of pollutants on the steady-state Nash equilibrium outcomes. The natural decomposition rate of pollutants refers to the rate at which pollutants are removed from the environment through natural processes, such as microbial degradation or chemical decomposition. A higher decomposition rate generally indicates a stronger self-purification capacity of the ecosystem, effectively reducing the long-term accumulation of pollutants. Conversely, a lower decomposition rate suggests that pollutants persist longer in the environment, resulting in more severe ecological impacts.

Figure 3a,c show that the natural decomposition rate of pollutants significantly affects the total pollutant stock across all three game models. A higher decomposition rate helps mitigate environmental damage in both regions.

Figure 3b indicates that as the natural decomposition rate increases in both regions, the curve slope is steeper for the economically developed Region A. This suggests that the more developed region is more sensitive to environmental pollution and thus needs to invest more in pollution control. In contrast, the less developed region may exhibit “free-riding” behavior. As the natural decomposition rate rises, the curve initially shows concave growth before declining. Correspondingly, the revenue curves for both regions first dip, then rise, eventually stabilizing.

An interesting phenomenon emerges: in mathematical simulations, as the natural decomposition rate steadily increases, the revenue curves for the non-cooperative and cost-compensation mechanisms converge. From a “mathematical limit” perspective, as the natural decomposition rate rises substantially, the distinction among the three game models may become less relevant, with game revenue curves converging. This observation suggests that, regardless of regional economic development levels, attention should be given to the local natural decomposition rate of pollutants. Excessive environmental damage could weaken the ecosystem’s self-purification capacity, leading to permanent ecological issues.

6. Conclusions and Recommendations

This paper explores three governance models for air pollution management, focusing on the dynamic changes in pollution stock: the regional non-cooperative governance mechanism, the pollution control cost-sharing mechanism, and the collaborative cooperation mechanism. Using a differential game model, the feedback Nash equilibrium was obtained, leading to the following conclusions:

(1) When pollution causes significant environmental damage, the pollution control cost-sharing mechanism can achieve Pareto improvements in both environmental and economic outcomes. However, the collaborative cooperation mechanism is optimal in terms of regional benefits and environmental results, as shown by the equilibrium trajectories in Propositions 1–4 and Figure 1.

(2) Due to the negative externalities of environmental pollution and the absence of effective cooperation mechanisms between regions, free-rider behavior often emerges during collaborative governance. Through numerical analysis, this paper derives the optimal dynamic benefits of cooperation over time, ensuring that individual rationality aligns with collective rationality.

(3) This paper also analyzes the effects of steady-state equilibrium factors, such as the discount rate and the natural decomposition rate of pollutants, on pollution stock and decision-making. The analysis shows that when both regions exhibit “short-sighted” behavior, maximizing short-term benefits leads to reduced investment in pollution control technologies. As the natural decomposition rate increases, pollution stock decreases significantly. At the same time, Region A is more sensitive to environmental damage and thus gains more from improvements in the environment’s self-purification rate compared to Region B.

In air pollution governance, regional coordination and cooperation are crucial. By establishing cross-regional environmental governance mechanisms, local governments can share the responsibility for pollution control, effectively reducing pollution while enhancing economic benefits, creating a win-win situation for both environmental protection and economic development. A fair and reasonable pollution control cost-sharing mechanism is essential, as it can prevent free-rider behavior, ensure equitable distribution of responsibilities, and encourage less developed regions to actively participate in pollution control. To support long-term sustainable environmental governance, policymakers should design appropriate incentive mechanisms. Policy and legal measures can guide local governments to focus on long-term environmental goals, reducing the tendency to underinvest in pollution control in favor of short-term economic gains. Additionally, increasing funding and resources for long-term environmental projects will significantly improve the effectiveness of pollution control efforts. In ecologically fragile regions, greater investments in ecological restoration are necessary to prevent further degradation. Cross-regional cooperation and fair cost-sharing for pollution control can enhance the ecological recovery stock of the entire region. Without clear definitions of responsibilities by the national government, local governments may face challenges between cooperation and non-cooperation. Therefore, governments at all levels should strengthen regional joint prevention and control mechanisms, clearly define roles in environmental protection and governance, and ensure active participation from all parties to form a united front in tackling air pollution.

Building on the preceding conclusions, this study introduces a set of management insights and actionable strategies to strengthen institutional frameworks for transboundary pollution control. Drawing on real-world examples, these insights illustrate practical strategies and essential elements of effective institutional design. Together, they offer policymakers a comprehensive approach to address the complex challenges of transboundary air pollution, providing pathways to develop adaptable, equitable, and resilient cooperative mechanisms across regions with diverse economic and environmental conditions.

6.1. Management Insights

(1) Dynamic Adaptation to Economic Asymmetries: This analysis emphasizes the need for institutions capable of adapting to economic disparities among regions. In cases where economically disadvantaged regions engage in “free-rider” behavior due to limited resources, institutions like the Mekong River Commission (MRC) [35] in Southeast Asia exemplify frameworks that offer flexible support mechanisms. Management strategies should encourage adaptive cost-sharing structures, such as those in the CLRTAP model [36], to promote fair participation in pollution control across regions with economic inequalities.

(2) Integration of Cost Compensation for Effective Cooperation: This study shows that when wealthier regions contribute to pollution control costs for less economically developed regions, it aligns regional interests with collective environmental goals. The International Commission for the Protection of the Danube River (ICPDR) serves as a model where cost compensation bridges economic disparities among member states [37]. This finding highlights the importance of compensation mechanisms in reducing the economic burden of pollution control, thus promoting effective cooperation among regions with diverse economies.

(3) Role of Short-Term and Long-Term Incentives: The findings reveal that institutions perform better when they balance long-term goals with attainable short-term incentives. For instance, the Great Lakes Water Quality Agreement (GLWQA) between the U.S. and Canada demonstrates sustained cooperation through regular goal assessments [38]. Similar approaches could benefit transboundary institutions in Asia, like EANET [39], by enabling economically diverse regions to commit to long-term pollution reduction while achieving visible short-term progress.

(4) Sensitivity to Environmental Resilience and Natural Decomposition Rates: This research highlights that natural decomposition rates significantly impact cooperation dynamics. Institutions operating in ecologically sensitive areas, such as the Amazon Cooperation Treaty Organization (ACTO) in South America [40], could prioritize pollution control in regions most vulnerable to environmental damage. HELCOM (Baltic Sea) is another example, effectively focusing on pollution in fragile marine ecosystems [41]. Institutions in Asia, like the ASEAN Haze Agreement, could apply this approach to direct resources toward regions with limited natural resilience, maximizing the effectiveness of transboundary pollution management.

(5) Encouraging Collaborative Investments in Technology and Stock Building: Given the economic and technological disparities identified, institutions like the Nile Basin Initiative (NBI) [42], which supports stock building in economically disadvantaged regions, demonstrate the potential of shared investments. Management strategies that promote cooperative investments in pollution control technology can enable under-resourced regions to access effective solutions, strengthen cooperation, and achieve balanced environmental outcomes across regions with varying economic capabilities.

By implementing these management insights, policymakers can create more adaptive and resilient institutional frameworks that promote equitable regional cooperation. Such frameworks can address the economic and environmental complexities involved in transboundary air pollution control, ultimately achieving a win-win outcome for both environmental protection and economic development.

6.2. Policy Recommendations

To enhance the effectiveness of institutional efforts in managing transboundary air pollution, the following policy recommendations are proposed, with examples from real-world institutions illustrating practical applications:

(1) Implement Adaptive Frameworks Based on Regional Conditions: Flexible frameworks that adapt to local economic and environmental conditions improve institutional effectiveness. For instance, the CLRTAP model in Europe demonstrates how structures that provide economic and technical support for less developed regions can foster cooperation. A similar approach could strengthen the ASEAN Haze Agreement in addressing financial and technological disparities in Southeast Asia.

(2) Enhance Governance and Accountability through Transparent Mechanisms: Transparent monitoring, reporting, and verification mechanisms build trust and reduce the risk of free-riding. The Great Lakes Water Quality Agreement (GLWQA) provides a strong example of bilateral accountability, while the Nile Basin Initiative (NBI) illustrates how transparency inconsistencies can undermine cooperation. EANET could benefit from adopting more rigorous transparency practices.

(3) Strengthen Public Engagement to Foster Cooperation: Engaging the public can increase institutional support and encourage government action on transboundary pollution. HELCOM (Baltic Sea) has successfully utilized public advocacy, and similar strategies could benefit Asia. For example, the Nile Basin Initiative has involved local communities to raise awareness—a model that could inspire the ASEAN Haze Agreement and SAARC Environmental Cooperation Program.

(4) Facilitate Cross-Regional Resource Sharing: Multilateral resource sharing, as seen in the Amazon Cooperation Treaty Organization (ACTO), can aid regions with limited resources in addressing transboundary pollution. Establishing similar mechanisms in Asia could enhance the EANET and ASEAN Haze Agreement, enabling regions to share technical resources and expertise.

By implementing these recommendations, policymakers can build more resilient and cooperative institutional frameworks better suited to address the complexities of transboundary air pollution.

7. Future Research Directions

To enhance the model’s applicability to real-world scenarios, future research could explore several key areas. First, integrating meteorological factors, such as wind direction, into pollution control strategies by incorporating meteorological models could improve the accuracy of cooperative impact assessments, especially in pollutant transmission between regions. Second, adopting a multi-pollutant framework that includes industry-specific control costs and strategies would better capture the diverse characteristics of real-world pollution. Third, examining nonlinear degradation functions, particularly those involving complex atmospheric reactions, could more accurately represent pollutant dynamics. Fourth, expanding the model to account for interactions among multiple regions and diverse economic and environmental states would increase its relevance for governance in multi-stakeholder contexts. Fifth, incorporating dynamic policy adjustments and economic feedback mechanisms would allow cooperative strategies to be more adaptable and viable in response to evolving environmental and regulatory conditions. Lastly, this study treats cooperative and non-cooperative decisions as exogenous; future research could improve the model by making cooperative decision-making endogenous within various game-theoretic structures. For instance, using frameworks like the Stackelberg model (e.g., Barrett, 1994 [43]; Nkuiya, 2020 [44]; Diamantoudi and Sartzetakis, 2006 [45]) or the Cournot-Nash model (e.g., Nkuiya, 2012 [46]; Schopf, 2024 [47]), regions could treat cooperation as an endogenous choice, enabling analysis of how parties optimally decide between competition and cooperation under different incentive mechanisms. Making cooperation endogenous would provide deeper insight into the motivations behind regional collaboration and how these dynamics evolve under various game-theoretic conditions.

Author Contributions

Conceptualization, G.T. and C.Y.; Methodology, G.T. and C.Y.; Software, C.Y.; Validation, C.Y.; Formal analysis, G.T. and F.Y.; Investigation, F.Y.; Data curation, C.Y. and F.Y.; Writing—original draft, C.Y.; Writing—review & editing, G.T.; Visualization, C.Y.; Project administration, G.T.; Funding acquisition, F.Y. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Table

Figure 1. (a): Pollutant stock trajectories under three scenarios; (b): benefit trajectories of the two regions under non-cooperative and compensation mechanisms; (c): the total benefit trajectories of the two regions under three scenarios.

Figure 2. (a): Impact of discount rate on pollutant stock; (b): impact of discount rate on the benefits of the two regions; (c): impact of discount rate on the total benefits of the two regions under three scenarios.

Figure 3. (a): Impact trajectory of pollutants natural decomposition rate on pollutant stock; (b): impact trajectory of pollutants natural decomposition rate on the benefits of the two regions; (c): impact trajectory of pollutants natural decomposition rate on the total benefits of the two regions under three scenarios.

Table 1

Comparison of Indicators under different game models.

Game Model	${\mathbf{g}}_{\mathbf{A}}^{\ast }$	${\mathbf{g}}_{\mathbf{B}}^{\ast }$	${\mathbf{h}}_{\mathbf{A}}^{{\mathbf{S}}^{\ast }}$	${\mathbf{h}}_{\mathbf{B}}^{{\mathbf{S}}^{\ast }}$
Nash Non-Cooperative Game	130.2	112.7	19.7	28.6
Stackelberg Game	130.2	112.7	19.7	42.1
Cooperative game	116.1	96.1	33.8	56.4
Comparison	${\mathrm{g}}_{\mathrm{A}}^{{\mathrm{N}}^{\ast }}={\mathrm{g}}_{\mathrm{A}}^{{\mathrm{S}}^{\ast }}>{\mathrm{g}}_{\mathrm{A}}^{{\mathrm{C}}^{\ast }}$	${\mathrm{g}}_{\mathrm{B}}^{{\mathrm{N}}^{\ast }}={\mathrm{g}}_{\mathrm{B}}^{{\mathrm{S}}^{\ast }}>{\mathrm{g}}_{\mathrm{B}}^{{\mathrm{C}}^{\ast }}$	${\mathrm{h}}_{\mathrm{A}}^{{\mathrm{N}}^{\ast }}={\mathrm{h}}_{\mathrm{A}}^{{\mathrm{S}}^{\ast }}>{\mathrm{h}}_{\mathrm{A}}^{{\mathrm{C}}^{\ast }}$	${\mathrm{h}}_{\mathrm{B}}^{\mathrm{C}*}>{\mathrm{h}}_{\mathrm{B}}^{{\mathrm{S}}^{\ast }}>{\mathrm{h}}_{\mathrm{B}}^{{\mathrm{N}}^{\ast }}$

Appendix A

The proof process for Proposition 1 is as follows:

To determine the optimal conditions for the optimal control problem, this paper applies the Pontryagin Maximum Principle. The Hamiltonian functions satisfying Equations (4) and (5) are as follows:(A1)$$\mathsf{\rho }{\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}(\mathrm{x})=\underset{{\mathrm{g}}_{\mathrm{A}},{\mathrm{h}}_{\mathrm{A}}}{\max }\left\{{\mathsf{\alpha }}_{\mathrm{A}}{\mathrm{g}}_{\mathrm{A}}-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathrm{A}}^2-{\displaystyle \frac{1}{2}}{\mathrm{c}}_{\mathrm{A}}{\mathrm{h}}_{\mathrm{A}}^2-\mathsf{\tau }\left({\mathrm{g}}_{\mathrm{A}}-{\mathsf{\omega }}_{\mathrm{A}}{\mathrm{h}}_{\mathrm{A}}\right)-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}{\mathrm{g}}_{\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}{\mathrm{g}}_{\mathrm{B}}-\mathrm{d}\mathrm{x}\left.+{\displaystyle \frac{\partial {\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}}{\partial \mathrm{x}}}({\displaystyle \sum _{\mathrm{i}=\mathrm{A},\mathrm{B}}{\mathrm{g}}_{\mathrm{i}}}-{\displaystyle \sum _{\mathrm{i}=\mathrm{A},\mathrm{B}}{\mathsf{\omega }}_{\mathrm{i}}{\mathrm{h}}_{\mathrm{i}}}-\mathsf{\zeta }\mathrm{x})\right\}\right.$$ (A2)$$\mathsf{\rho }{\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}(\mathrm{x})=\underset{{\mathrm{g}}_{\mathrm{B}},{\mathrm{h}}_{\mathrm{B}}}{\max }\left\{{\mathsf{\alpha }}_{\mathrm{B}}{\mathrm{g}}_{\mathrm{B}}-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathrm{B}}^2-{\displaystyle \frac{1}{2}}{\mathrm{c}}_{\mathrm{B}}{\mathrm{h}}_{\mathrm{B}}^2-\mathsf{\tau }\left({\mathrm{g}}_{\mathrm{B}}-{\mathsf{\omega }}_{\mathrm{B}}{\mathrm{h}}_{\mathrm{B}}\right)-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}{\mathrm{g}}_{\mathrm{B}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}{\mathrm{g}}_{\mathrm{A}}-\mathsf{\beta }\mathrm{d}\mathrm{x}\right.\left.+{\displaystyle \frac{\partial {\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}}{\partial \mathrm{x}}}({\displaystyle \sum _{\mathrm{i}=\mathrm{A},\mathrm{B}}{\mathrm{g}}_{\mathrm{i}}}-{\displaystyle \sum _{\mathrm{i}=\mathrm{A},\mathrm{B}}{\mathsf{\omega }}_{\mathrm{i}}{\mathrm{h}}_{\mathrm{i}}}-\mathsf{\zeta }\mathrm{x})\right\}$$

According to the Pontryagin Maximum Principle, the first-order necessary conditions are derived as follows:(A3)$${\mathrm{g}}_{\mathrm{A}}^{\mathrm{N}}={\mathsf{\alpha }}_{\mathrm{A}}-\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\displaystyle \frac{\partial {\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}}{\partial \mathrm{x}}}\hspace{1em}{\mathrm{g}}_{\mathrm{B}}^{\mathrm{N}}={\mathsf{\alpha }}_{\mathrm{B}}-\mathsf{\tau }-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+{\displaystyle \frac{\partial {\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}}{\partial \mathrm{x}}}\hspace{1em}{\mathrm{h}}_{\mathrm{A}}^{\mathrm{N}}={\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{A}}\left(\mathsf{\tau }-\frac{\partial {\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}}{\partial \mathrm{x}}\right)}{{\mathrm{c}}_{\mathrm{A}}}}\hspace{1em}{\mathrm{h}}_{\mathrm{B}}^{\mathrm{N}}={\displaystyle \frac{{\mathsf{\omega }}_{\mathrm{B}}\left(\mathsf{\tau }-\frac{\partial {\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}}{\partial \mathrm{x}}\right)}{{\mathrm{c}}_{\mathrm{B}}}}$$

By substituting Equation (A3) into Equations (A1) and (A2) and simplifying, we obtain (A4a)$$\begin{array}{l}\mathsf{\rho }{\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}(\mathrm{x})=-\left(\mathrm{d}+\mathsf{\zeta }{\displaystyle \frac{\partial {\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}}{\partial \mathrm{x}}}\right)\mathrm{x}+{\displaystyle \frac{\left({\mathrm{c}}_{\mathrm{A}}+{{\mathsf{\omega }}_{\mathrm{A}}}^2\right){\left(\frac{\partial {\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}}{\partial \mathrm{x}}\right)}^2}{2{\mathrm{c}}_{\mathrm{A}}}}\\ +{\displaystyle \frac{\frac{\partial {\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}}{\partial \mathrm{x}}\left({\mathrm{c}}_{\mathrm{B}}\left({\mathsf{\alpha }}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{A}}+{\mathsf{\alpha }}_{\mathrm{B}}{\mathrm{c}}_{\mathrm{A}}-{\mathrm{c}}_{\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+2\mathsf{\tau }\right)-\mathsf{\tau }{{\mathsf{\omega }}_{\mathrm{A}}}^2\right)-{\mathrm{c}}_{\mathrm{A}}\mathsf{\tau }{{\mathsf{\omega }}_{\mathrm{B}}}^2+{\mathrm{c}}_{\mathrm{A}}\left({\mathrm{c}}_{\mathrm{B}}+{{\mathsf{\omega }}_{\mathrm{B}}}^2\right){\left(\frac{\partial {\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}}{\partial \mathrm{x}}\right)}^2\right)}{{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}}}\\ +{\displaystyle \frac{{\mathrm{c}}_{\mathrm{A}}\left({{\mathsf{\alpha }}_{\mathrm{A}}}^2+{{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}}^2-2{\mathsf{\alpha }}_{\mathrm{B}}{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}+2{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+2\mathsf{\tau }\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\right)+{\mathsf{\tau }}^2-2{\mathsf{\alpha }}_{\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+\mathsf{\tau }\right)\right)+{\mathsf{\tau }}^2{{\mathsf{\omega }}_{\mathrm{A}}}^2-2{\mathrm{c}}_{\mathrm{A}}{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\frac{\partial {\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}}{\partial \mathrm{x}}}{2{\mathrm{c}}_{\mathrm{A}}}}\end{array}$$ (A4b)$$\begin{array}{l}\mathsf{\rho }{\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}(\mathrm{x})=-\left(\mathsf{\beta }\mathrm{d}+\mathsf{\zeta }{\displaystyle \frac{\partial {\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}}{\partial \mathrm{x}}}\right)\mathrm{x}+{\displaystyle \frac{\left({\mathrm{c}}_{\mathrm{B}}+{{\mathsf{\omega }}_{\mathrm{B}}}^2\right){\left(\frac{\partial {\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}}{\partial \mathrm{x}}\right)}^2}{2{\mathrm{c}}_{\mathrm{B}}}}\\ +{\displaystyle \frac{{\mathrm{c}}_{\mathrm{B}}\left({{\mathsf{\alpha }}_{\mathrm{B}}}^2-2\left({\mathsf{\alpha }}_{\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right){\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+2\mathsf{\tau }{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}-2{\mathsf{\alpha }}_{\mathrm{B}}\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+\mathsf{\tau }\right)+{\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+\mathsf{\tau }\right)}^2+{\mathsf{\tau }}^2{{\mathsf{\omega }}_{\mathrm{B}}}^2-2{\mathrm{c}}_{\mathrm{B}}{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}\frac{\partial {\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}}{\partial \mathrm{x}}\right)}{2{\mathrm{c}}_{\mathrm{B}}}}\\ +{\displaystyle \frac{\frac{\partial {\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}}{\partial \mathrm{x}}\left({\mathrm{c}}_{\mathrm{B}}\left({\mathsf{\alpha }}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{A}}+{\mathsf{\alpha }}_{\mathrm{B}}{\mathrm{c}}_{\mathrm{A}}-{\mathrm{c}}_{\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+2\mathsf{\tau }\right)-\mathsf{\tau }{{\mathsf{\omega }}_{\mathrm{A}}}^2\right)-{\mathrm{c}}_{\mathrm{A}}\mathsf{\tau }{{\mathsf{\omega }}_{\mathrm{B}}}^2+{\mathrm{c}}_{\mathrm{B}}\left({\mathrm{c}}_{\mathrm{A}}+{{\mathsf{\omega }}_{\mathrm{A}}}^2\right)\frac{\partial {\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}}{\partial \mathrm{x}}\right)}{{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}}}\end{array}$$

According to Equations (A4a) and (A4b), let (A5)$${\mathrm{V}}_{\mathrm{A}}^{\mathrm{N}}(\mathrm{x})={\mathrm{k}}_{\mathrm{A}}^{\mathrm{N}}\mathrm{x}+{\mathrm{b}}_{\mathrm{A}}^{\mathrm{N}}, {\mathrm{V}}_{\mathrm{B}}^{\mathrm{N}}(\mathrm{x})={\mathrm{k}}_{\mathrm{B}}^{\mathrm{N}}\mathrm{x}+{\mathrm{b}}_{\mathrm{B}}^{\mathrm{N}}$$

Here, ${\mathrm{k}}_{\mathrm{A}}^{\mathrm{N}}$, ${\mathrm{b}}_{\mathrm{A}}^{\mathrm{N}}$, ${\mathrm{k}}_{\mathrm{B}}^{\mathrm{N}}$, and ${\mathrm{b}}_{\mathrm{B}}^{\mathrm{N}}$ are constants to be determined, and the linear optimal value function of x is the solution to the HJB equation.

Substituting Equation (A5) into Equations (A4a) and (A4b), we obtain (A6a)$$\begin{array}{l}\mathsf{\rho }({\mathrm{k}}_{\mathrm{A}}^{\mathrm{N}}\mathrm{x}+{\mathrm{b}}_{\mathrm{A}}^{\mathrm{N}})=-\left(\mathrm{d}+\mathsf{\zeta }{\mathrm{k}}_{\mathrm{A}}^{\mathrm{N}}\right)\mathrm{x}+{\displaystyle \frac{\left({\mathrm{c}}_{\mathrm{A}}+{{\mathsf{\omega }}_{\mathrm{A}}}^2\right){\left({\mathrm{k}}_{\mathrm{A}}^{\mathrm{N}}\right)}^2}{2{\mathrm{c}}_{\mathrm{A}}}}\\ +{\displaystyle \frac{{\mathrm{k}}_{\mathrm{A}}^{\mathrm{N}}\left({\mathrm{c}}_{\mathrm{B}}\left({\mathsf{\alpha }}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{A}}+{\mathsf{\alpha }}_{\mathrm{B}}{\mathrm{c}}_{\mathrm{A}}-{\mathrm{c}}_{\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+2\mathsf{\tau }\right)-\mathsf{\tau }{{\mathsf{\omega }}_{\mathrm{A}}}^2\right)-{\mathrm{c}}_{\mathrm{A}}\mathsf{\tau }{{\mathsf{\omega }}_{\mathrm{B}}}^2+{\mathrm{c}}_{\mathrm{A}}\left({\mathrm{c}}_{\mathrm{B}}+{{\mathsf{\omega }}_{\mathrm{B}}}^2\right){\left({\mathrm{k}}_{\mathrm{B}}^{\mathrm{N}}\right)}^2\right)}{{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}}}\\ +{\displaystyle \frac{{\mathrm{c}}_{\mathrm{A}}\left({{\mathsf{\alpha }}_{\mathrm{A}}}^2+{{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}}^2-2{\mathsf{\alpha }}_{\mathrm{B}}{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}+2{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+2\mathsf{\tau }\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\right)+{\mathsf{\tau }}^2-2{\mathsf{\alpha }}_{\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+\mathsf{\tau }\right)\right)+{\mathsf{\tau }}^2{{\mathsf{\omega }}_{\mathrm{A}}}^2-2{\mathrm{c}}_{\mathrm{A}}{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}{\mathrm{k}}_{\mathrm{B}}^{\mathrm{N}}}{2{\mathrm{c}}_{\mathrm{A}}}}\end{array}$$ (A6b)$$\begin{array}{l}\mathsf{\rho }({\mathrm{k}}_{\mathrm{B}}^{\mathrm{N}}\mathrm{x}+{\mathrm{b}}_{\mathrm{B}}^{\mathrm{N}})=-\left(\mathsf{\beta }\mathrm{d}+\mathsf{\zeta }{\mathrm{k}}_{\mathrm{B}}^{\mathrm{N}}\right)\mathrm{x}+{\displaystyle \frac{\left({\mathrm{c}}_{\mathrm{B}}+{{\mathsf{\omega }}_{\mathrm{B}}}^2\right){\left({\mathrm{k}}_{\mathrm{B}}^{\mathrm{N}}\right)}^2}{2{\mathrm{c}}_{\mathrm{B}}}}\\ +{\displaystyle \frac{{\mathrm{c}}_{\mathrm{B}}\left({{\mathsf{\alpha }}_{\mathrm{B}}}^2-2\left({\mathsf{\alpha }}_{\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right){\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+2\mathsf{\tau }{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}-2{\mathsf{\alpha }}_{\mathrm{B}}\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+\mathsf{\tau }\right)+{\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+\mathsf{\tau }\right)}^2+{\mathsf{\tau }}^2{{\mathsf{\omega }}_{\mathrm{B}}}^2-2{\mathrm{c}}_{\mathrm{B}}{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}{\mathrm{k}}_{\mathrm{A}}^{\mathrm{N}}\right)}{2{\mathrm{c}}_{\mathrm{B}}}}\\ +{\displaystyle \frac{{\mathrm{k}}_{\mathrm{B}}^{\mathrm{N}}\left({\mathrm{c}}_{\mathrm{B}}\left({\mathsf{\alpha }}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{A}}+{\mathsf{\alpha }}_{\mathrm{B}}{\mathrm{c}}_{\mathrm{A}}-{\mathrm{c}}_{\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+2\mathsf{\tau }\right)-\mathsf{\tau }{{\mathsf{\omega }}_{\mathrm{A}}}^2\right)-{\mathrm{c}}_{\mathrm{A}}\mathsf{\tau }{{\mathsf{\omega }}_{\mathrm{B}}}^2+{\mathrm{c}}_{\mathrm{B}}\left({\mathrm{c}}_{\mathrm{A}}+{{\mathsf{\omega }}_{\mathrm{A}}}^2\right){\mathrm{k}}_{\mathrm{A}}^{\mathrm{N}}\right)}{{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}}}\end{array}$$

According to Equations (A6a) and (A6b), the values of ${\mathrm{k}}_{\mathrm{A}}^{\mathrm{N}}$, ${\mathrm{b}}_{\mathrm{A}}^{\mathrm{N}}$, ${\mathrm{k}}_{\mathrm{B}}^{\mathrm{N}}$, and ${\mathrm{b}}_{\mathrm{B}}^{\mathrm{N}}$ are determined as follows:(A7a)$${\mathrm{k}}_{\mathrm{A}}^{\mathrm{N}}=-{\displaystyle \frac{\mathrm{d}}{\mathsf{\rho }+\mathsf{\zeta }}}$$ (A7b)$${\mathrm{k}}_{\mathrm{B}}^{\mathrm{N}}=-{\displaystyle \frac{\mathsf{\beta }\mathrm{d}}{\mathsf{\rho }+\mathsf{\zeta }}}$$ (A8a)$$\begin{array}{l}{\mathrm{b}}_{\mathrm{A}}^{\mathrm{N}}={\displaystyle \frac{{\mathrm{c}}_{\mathrm{B}}{\left(\mathrm{d}+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right)}^2{{\mathsf{\omega }}_{\mathrm{A}}}^2+2{\mathrm{c}}_{\mathrm{A}}\mathrm{d}\left(\mathrm{d}\mathsf{\beta }+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right){{\mathsf{\omega }}_{\mathrm{B}}}^2}{2\mathsf{\rho }{\mathrm{c}}_{\mathrm{A}}{\mathrm{c}}_{\mathrm{B}}{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{\mathrm{d}}^2\left(1+2\mathsf{\beta }\right)-2\mathrm{d}\left(\mathsf{\rho }+\mathsf{\zeta }\right)\left({\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\alpha }}_{\mathrm{B}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}-\mathsf{\beta }{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}-2\mathsf{\tau }\right)}{2\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{\left({\mathsf{\alpha }}_{\mathrm{A}}-{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right)}^2+2{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}-{\mathsf{\alpha }}_{\mathrm{B}}\right)+2\mathsf{\tau }\left(-{\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{A}}\right)+{\mathsf{\tau }}^2}{2\mathsf{\rho }}}\end{array}$$ (A8b)$$\begin{array}{l}{\mathrm{b}}_{\mathrm{B}}^{\mathrm{N}}={\displaystyle \frac{\mathrm{d}\mathsf{\beta }\left(\mathrm{d}+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right){{\mathsf{\omega }}_{\mathrm{A}}}^2}{\mathsf{\rho }{\mathrm{c}}_{\mathrm{A}}{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}+{\displaystyle \frac{+{\left(\mathrm{d}\mathsf{\beta }+\left(\mathsf{\rho }+\mathsf{\zeta }\right)\mathsf{\tau }\right)}^2{{\mathsf{\omega }}_{\mathrm{B}}}^2}{2\mathsf{\rho }{\mathrm{c}}_{\mathrm{B}}{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{\mathrm{d}}^2\mathsf{\beta }\left(2+\mathsf{\beta }\right)+2\mathrm{d}\left(\mathsf{\rho }+\mathsf{\zeta }\right)\left({\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+\mathsf{\beta }\left(-{\mathsf{\alpha }}_{\mathrm{A}}-{\mathsf{\alpha }}_{\mathrm{B}}+{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+2\mathsf{\tau }\right)\right)}{2\mathsf{\rho }{\left(\mathsf{\rho }+\mathsf{\zeta }\right)}^2}}\\ +{\displaystyle \frac{{{\mathsf{\alpha }}_{\mathrm{B}}}^2+2\left(-{\mathsf{\alpha }}_{\mathrm{A}}+{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{A}}\right){\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}+2\mathsf{\tau }{\mathsf{\epsilon }}_{\mathrm{A}\mathrm{B}}-2{\mathsf{\alpha }}_{\mathrm{B}}{\left({\mathsf{\epsilon }}_{\mathrm{B}\mathrm{B}}+\mathsf{\tau }\right)}^2}{2\mathsf{\rho }}}\end{array}$$

Substituting (A7a) and (A7b), as well as (A8a) and (A8b), into Equations (A3) and (A5) yields Equations (6), (7), (8a), (8b), and (9).

The proof of Proposition 1 complete.

The proofs of Propositions 2 and 3 follow a similar process to that of Theorem 1. For brevity, they are not repeated here.

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