Dynamic Research on the Collaborative Governance in Urban and Rural Black-Odorous Water: A Tripartite Stochastic Evolutionary Game Perspective

Abstract

The issue of black-odorous water (BOW) represents a formidable challenge to the current aquatic ecosystems, and its governance exhibits characteristics of low efficiency, susceptibility to relapse, and fragmented management under the Central Environmental Protection Inspection, thereby emerging as a dynamically complex issue in the ecological governance of urban and rural settings. This study introduces Gaussian white noise to simulate environmental uncertainty and design a stochastic evolutionary game model encompassing the central government, local governments, and societal forces based on evolutionary game theory and classical governance theories and concepts. Numerical simulations are conducted to explore trajectories of the strategic evolution of various subjects influenced by numerous factors. Results indicate that under the environment of random disturbances, the strategies of the game subjects show significant fluctuations, but actively cultivating the subject’s initial willingness facilitates collaboration governance in inspection. Moreover, joint construction of a “belief system” by multi-subjects, the intensity of inspection interventions, the integration of heterogeneous resources, and effective punitive measures all influence the governance of BOW, but the efficiency of resource allocation should be considered throughout the governance process. Recommendations are made finally for collaborative governance of urban and rural BOW, promoting the sustainable development of the ecological environment.

1. Introduction

The integrity and robustness of the aquatic ecosystem are pivotal to the prosperity of a nation.They not only underpin the sustainable development of society by providing material foundations and ecological services but also play an irreplaceable role in preserving the intrinsic balance of the entire complex ecosystem [1]. While rapid urbanization and industrialization pose a series of ecological challenges to the aquatic ecosystem, BOW represents a prominent and intricate issue. This environmental problem has occurred both in developed and developing countries around the world [2]. Practically, as a direct manifestation of water quality deterioration, BOW reveals the deficiencies in water management and protection across nations. It directly stands in contrast to the 2030 Sustainable Development Goal (SDG 6), which aims to guarantee the availability and sustainable stewardship of water and sanitation. As of 2023, although 56% of water bodies assessed in more than 120 countries were found to have good water quality, a downward trend in water quality since 2017 has been observed in countries with the most extensive monitoring programs, and currently, only 44% of countries have made good progress in improving water quality [3]. This trend underscores the pressing and severe challenges in water management and protection. The rectification of BOW emerges as a paramount strategic imperative in addressing these formidable ecological challenges. Effective remediation of BOW would not only have a profound impact on the amelioration of aquatic environmental quality in various countries but also constitute an indispensable element within the intricate mosaic of realizing the Sustainable Development Goals.

The issue of BOW refers to conditions where industrial effluents, agricultural non-point sources, and domestic sewage, laden with organic and inorganic pollutants, are discharged into urban and rural aquatic systems. This leads to a rate of oxygen depletion that exceeds the rate of reoxygenation, culminating in hypoxia [4]. The anaerobic degradation process then generates substances such as hydrogen sulfide, ammonia, and thiols, which contribute to the discoloration and odor of the water [5]. The repercussions of BOW often extend beyond the source of pollution, further inducing regional air pollution, affecting the living environment of urban and rural areas, and imposing substantial health hazards on local inhabitants, along with sensory discomfort [6]. However, due to the inherent fragility of water bodies, once contaminated, their self-purification capabilities are compromised, making it difficult for them to revert to a healthy state through natural processes [7]. Moreover, the issue of BOW often spans a broad geographical area and involves numerous stakeholders. Its management presents not only technical hurdles but also necessitates overcoming policy, economic, and social constraints [8]. Consequently, the remediation of BOW has become a dynamic and complex conundrum within the realm of urban and rural environmental stewardship.

Globally, as early as the 1930s, industrialized Western countries had already embarked on the governance of the BOW. The River Thames in London, burdened by the aggregation of urban populations and the direct discharge of untreated industrial and domestic wastewater, became a quintessential example of such water bodies. The local government established the Thames Water Authority and introduced stringent standards and policies for the discharge of domestic and industrial effluents, thereby transitioning from a fragmented to an integrated management system [9]. In recent years, through scientific planning, the pollution control of the River Thames has been divided into rural and urban areas for differentiated management, and a comprehensive water quality monitoring system has been established [10]. The Rhine River, which flows through five countries in Europe, saw the establishment of the International Commission for the Protection of the Rhine in 1950, which fostered a transnational river basin management system and an efficient cooperation mechanism, adopting a unified decision-making and regional management model, breaking the division between departments and regions [11]. The Duwamish River in the United States had been continuously polluted by industry, and later the Environmental Protection Agency, in conjunction with the Washington State Department of Ecology, Boeing, and the public, jointly carried out comprehensive river management and long-term monitoring [12]. Singapore’s Public Utilities Board, in cooperation with the German company, undertook the ecological restoration project of the Kallang River [13]. The phenomenon of BOW seems to be more common in underdeveloped areas. Many developing countries, where wastewater treatment is considered a public utility and is not subject to market price mechanisms, often grapple with inefficiencies in wastewater treatment enterprises [14]. However, they are also making strenuous efforts and investing substantial funds in the development, maintenance, and operation of a national wastewater network and sewage treatment plants, in order to prevent the discharge of untreated sewage into rivers, which can lead to the water bodies becoming black and odorous [15].

In China, the governance of urban and rural BOW has increasingly captured the attention of the government in recent years. On the one hand, the central government has rolled out a series of policy directives to address this escalating environmental challenge since 2015. Notably, the 2018 Implementation Plan for the Battle against Urban Black- Odorous Water Bodies highlights the remediation of urban BOW as one of the seven landmarks in the war against pollution [16]. As the “14th Five-Year Plan” commences, China’s developmental vision has shifted from the pursuit of rapid growth to high-quality development, with a particular emphasis on the concept of green development, which aligns with the Sustainable Development Goals. Hence, strategic documents such as the Opinions on Further Promoting the Nationwide Battle to Prevent and Control Pollution, the 14th Five-Year Plan Action Program for Urban Black- Odorous Waters Treatment and Environmental Protection and the Guidelines for the Management of Black-Odorous Water Bodies in Rural Areas are spearheading efforts to expand the scope of intervention, delve into more profound areas of action, and establish a more enduring and effective management framework for water quality. On the other hand, on a practical level, China has initiated a series of specialized environmental protection initiatives focused on the remediation of BOW since 2018. Concurrently, the Central Environmental Protection Inspection (CEPI) is being orchestrated by the central government to solidify local environmental protection responsibilities. It pays much attention to the issue of BOW in many areas in the inspection process and proposes corresponding rectification demands [17]. Nonetheless, the reality is that the governance of urban and rural BOW still exhibits fleeting improvements and retaliatory rebound intertwined situations [18]. From this perspective, the governance of BOW is not a task that can be accomplished in one fell swoop; rather, it should be regarded as an ongoing process aimed at mitigating the associated phenomena [19]. Confronting this process requires the application of systematic and comprehensive thinking coupled with strategic and intelligent action.

Research across various disciplines has spanned a broad spectrum of topics in BOW governance, with the focus progressively transitioning from ecological implications to political connotations. While these studies predominantly offer linear and static analyses [7,20,21], they have yet to fully capture and dissect the dynamics and complexities of BOW governance within specific institutional settings, nor have they adequately delineate the interplay between the functions and rules of the governance system and its performance. Secondly, the governance of urban and rural BOW is fundamentally a systemic endeavor, emphasizing a holistic understanding. To put it another way, it cannot be effectively addressed by the actions of a single entity or department; rather, it depends on the intricate interplay and behavioral dynamics among the subjects within a territorial governance network, and it is necessary to judiciously consider the feedback mechanisms of the system structure, to foster effective collaborative governance. Moreover, the governance of urban and rural BOW is a process that unfolds over time, characterized by its phased nature and adaptivity. Consequently, the quest for long-term sustainable governance must adhere to the inherent principles of public resource management, heed the outcomes of systemic feedback, and engage in profound deliberation against the intricate backdrop of socioeconomic conditions. This study thus poses the following questions naturally:

(1)

How can the conflicts and coordination relationships between various subjects in the governance of urban and rural BOW under the CEPI be systematically portrayed?

(2)

How do the subjects interact strategically to promote the effective collaborative governance of urban and rural BOW?

(3)

What factors affect the collaborative behavioral strategies of the subjects in the governance of urban and rural BOW, and how do these factors exert their influence?

This study utilized evolutionary game theory (EGT) to solve the above questions and to further simulate realistic and uncertain conditions; Gaussian white noise was incorporated to refine the evolutionary game model. It makes the following contributions. Firstly, it introduces an integrated theoretical framework that facilitates a profound analysis of the intricacies and dynamics inherent in urban and rural BOW governance under the CEPI. This not only augments the discourse on the impact mechanisms of CEPI but also broadens the scope of research on governance networks and collaborative governance of urban and rural BOW. Secondly, by considering environmental uncertainties, the study examines the evolution of subjects’ behavioral strategies and their interplay under diverse factors, particularly the inclination towards proactive and efficacious collaborative governance under specific conditions. Such insights lay the theoretical groundwork for enduring mechanisms and sustainable governance trajectories for urban and rural BOW. Thirdly, the theoretical analytical model established in this study, enhanced by stochastic theory and EGT, provides some reference for the transference and emulation of similar ecological systems and governance networks across varied institutional settings. This endeavor marks both a methodological advance in the exploration of network and systemic dynamics and an innovative foray into theoretical scholarship.

The subsequent discourse of this study unfolds as follows. Section 2 presents a related literature review, culminating in the formulation of a theoretical analytical framework. Section 3 and Section 4 delineate the foundational aspects of the game model and integrate Gaussian white noise to optimize it. Section 5 employs simulation tools to analyze the strategic manoeuvers and evolutionary patterns of the principal subjects within the urban and rural BOW governance process, considering the impact of various factors. The paper concludes with the simulation findings, accompanied by policy recommendations and an acknowledgement of drawbacks.

2. Literature Review

2.1. Multidimensional Research Advances of Black-Odorous Water and Its Governance

The issue of BOW stems from the complex interaction of multiple factors and its governance is a multifaceted endeavor that encompasses political, economic, technological, and social systems within the realm of public administration, garnering sustained interest from both practitioners and scholars. The extant literature predominantly concentrates on exploring the causes of BOW and the interplay of pollution factors [22,23,24], the ecological diversity and restoration status within such water bodies [25,26], the pollutant profiles and ecological risks present in the water bodies [27,28], the engineering and management techniques for addressing BOW [29,30,31], and the evaluation [32,33,34] or monitoring systems for BOW [35].

As the thought of the relationship between humans and nature continues to evolve, an increasing number of scholars are beginning to value the protection of aquatic systems. Against this backdrop, studies on environmental behavior and policy are increasingly highlighting the characteristics of “governance”. In the sphere of governance research, certain scholars have examined policies pertinent to the management of BOW [36,37] and have undertaken assessments of the multifaceted effectiveness of governance within regional contexts [38]. Some studies have focused on gauging the level of citizen satisfaction throughout the governance process of BOW [39] or the general public’s perception of government performance in governing these water bodies and its engagement in environmental protection [40]. It has been posited that the public’s willingness is one of the most important bases for decision-makers in the governance process [41].

Some studies have also examined the behavior and attitude of local governments in the management process of BOW, suggesting that underlying disparities in the institutionalized distribution of power across multiple levels of urban governance play a pivotal role [42,43]. Indeed, the development of sewage treatment systems is subject to varying interpretations at the municipal and higher administrative levels which, in turn, has led to both constraints and innovations in local approaches to treatment challenges [44]. Furthermore, decision-makers exhibit a preference for straightforward constraints (precautionary principle) and flexible negotiation tactics (authorization) over intricate assessment and decision-making support methodologies [45] during the construction process.

Some scholars have acknowledged a paradigm shift is essential to transcend the traditional siloed governance models of BOW [46], advocating for the establishment of a cooperative framework among various functional areas or administrative jurisdictions [47] and of inclusive governance networks that encompass individuals, the scientific community, and governmental entities [48]. Moreover, a considerable amount of research recognizes the importance of institutions [49] and overcoming technological bureaucratic culture and institutional structural rigidity [50].

2.2. Environmental Power and Responsibilities Dynamics under Central Environmental Protection Inspection

In a multi-level government system, the distribution of authority and accountability in environmental stewardship has emerged as a critical structural concern in the system analysis [51,52]. In China, the expansive geography and varied regional characteristics contribute to the escalating costs associated with hierarchical administration. The onus of environmental governance is delegated to local governments, underscoring their pivotal role in this domain. Nonetheless, under the Chinese hierarchical management system, the “non-zero-sum” dynamic within environmental governance involving both central and local authorities is strikingly intricate.

As new challenges and dilemmas in environmental governance surface incessantly, the phenomenon of local governments deviating from central policies and environmental governance objectives in their execution has become a fundamental contradiction that cannot be ignored in the national governance system. Therefore, it is imperative for the central government to augment the provision of institutional supply, bolster institutional enforcement, and foster collaborative multilateral coordination and cooperation to effectively enhance governance efficiency.

Among the many environmental governance innovations, the advent of the CEPI has progressively emerged as a pivotal governance mechanism, adept at resolving the entrenched dilemmas within central–local interactions and mitigating the principal–agent risk [53]. Meanwhile, it has effectively catalyzed the emergence of a novel multilateral game within the context of central–local, vertical intergovernmental relations in environmental governance.

The CEPI, through a combination of regular inspections or ad hoc inspections and follow-up inspections, has exerted a normative and deterrent influence on the environmental governance endeavors of various stakeholders, but the “campaign-style governance” approach has drawn skepticism regarding the sustainability of their impact [54]. Local governments, on the one hand, driven by the political pressure exerted by inspections, are compelled to adopt more vigorous measures to address the issue of BOW. On the other hand, they often prioritize economic development tasks in the pursuit of political advancement and increased fiscal revenues [55], engaging in collusive practices with businesses to achieve win–win outcomes [56] while neglecting the potential adverse consequences of economic growth and paying insufficient heed to environmental protection and other governance objectives [57]. Consequently, pollution governance efforts exhibit a pattern of temporary responses to local water pollution issues [18]. The issues of inadequate remediation and management, water bodies reverting to their previous odorous and discolored state, and the failure to address the root causes of pollution are still prominent in many regions [58,59]. The latest data revealed that, during the second round of CEPI, 106 typical cases were identified in the inspection process, with 25 pertaining to issues of BOW, accounting for 23.58%. The ongoing third round of CEPI places special emphasis on the holistic watershed issues and common environmental concerns of the Yangtze River Economic Belt. It has been observed that the issue of BOW reoccurs in six provinces within the Yangtze River Economic Belt in the inspection process, with the situation continuously worsening.

Within the inspection process, there has been a discernible increase in the influence of societal oversight forces, represented by the media and the public. Their exposure of BOW in residential areas not only aids the central government in gathering environmental information and strengthening inspections on local environmental governance but also helps to sustain long-term environmental governance behaviors by local governments, creating a form of social pressure [60]. However, it is noteworthy that there is still a lingering lack of trust from social forces when participating in the defense of or advocacy for environmental health rights and interests.

2.3. Theoretical Framework

The governance of BOW, as a complex engineering issue, requires dynamic processes, interactive subjects, and the continuous optimization of strategies. Faced with such environmental issues, only through collaborative efforts among diverse subjects can a harmonious coexistence be realized amid high levels of uncertainty [61].

In the course of environmental governance, individuals and organizations often enter the governance field as proactive, enthusiastic, and conscious subjects [62]. Some subjects unite by sharing core beliefs, which encompass the recognition of the severity of environmental issues, the valuation of different policies and governance actions, the prioritization of welfare for various groups, and the assessment of the relative authority and roles of different actors [63], thereby consolidating collective action and influencing specific environmental governance actions.

Resources, as the prerequisites and foundations affecting the governance process, affect the choice of behavioral governance methods and the level of governance capacity. Resources generally include policy authority resources, financial resources, leadership resources, public opinions, information, and other resources [64]. Some studies also classify resources into institutional resources, organizational resources, and grassroots resources [65]. These resources within governance subsystems mainly motivate the adjustment of behaviors of various subjects through a particular mechanism, such as redistribution [66].

Additionally, confronted with the complexity and uncertainty issues, subjects often struggle to overcome their inherent limitations, resulting in dilemmas in collective action, thereby being unable to effectively address public issues. Feiock’s Institutional Collective Action Framework Theory suggests that the maintenance of specific collaborative behaviors by governance subjects primarily relies on a comprehensive consideration of their benefits and transaction costs [67]. Among these, the collective and selective benefits obtained from collaborative actions are crucial driving factors for achieving multi-subject cooperation. In the governance process of BOW, subjects actively collaborate to improve the local riverine environment, obtaining collective benefits such as green ecological environment gains and long-term collaborative experience for reference. Selective benefits mainly refer to enhancements in reputation, social status, and capital gains [68]. Moreover, for various subjects to achieve concerted cooperation in the governance of urban and rural BOW, they also need to deal with the influence of transaction costs, consisting of execution, organizational, bargaining, and autonomy loss costs. Additionally, they need to address risks such as unfavorable coordination, unequal profit distribution, and cooperation defection, aiming to avoid inefficient and fragmented collective actions in BOW governance. Based on the above analysis, a comprehensive “Belief-Resource-Action” analytical framework can be established to depict the core elements presented in the process of BOW governance (see Figure 1). This framework serves as a foundational analytical framework for examining the behavioral interplay and strategic game among the multiple parties within urban and rural BOW governance, particularly under the purview of CEPI.

2.4. Stochastic Evolutionary Game Model

EGT, which amalgamates game theory and dynamic evolutionary processes, can be utilized to delineate the dynamic interrelationships among agents within governance systems [19], thereby deepening our comprehension of complex systems. It posits that many agents make decisions that are not entirely rational. The attainment of equilibrium in the process of strategy selection is not immediate but rather requires a series of continuous trial-and-error, adjustment, and optimization processes. Even if a certain equilibrium state is momentarily attained, it remains susceptible to disruption as strategies evolve, leading to new adjustments in strategies. EGT has been used in some research areas, such as safety management [69], public services [70], remanufacturing strategies [71], and environmental regulation policies [72,73].

While these studies predominantly initiate from the vantage point of static decisions with complete information and the assumption of rational choice, they merely probe into the behavioral traits and motivational factors of agents within the system or concentrate on the strategic manoeuvers for systemic evolution and equilibrium. There is a gap in addressing the scenario expansions influenced by stochastic elements and the dynamic adjustments occurring within the system [74]. Indeed, considering the intricate and unpredictable nature of social reality, it is challenging to capture the nuances of environmental governance through classical EGT. Thus, the introduction of stochastic evolutionary game models (SEGMs) becomes imperative to align more closely with the real conditions and attributes of intricate environmental governance systems [75]. Such models have been used in regional collaborative governance under enclave economics [76], electronic waste recycling [77], and data sharing in digital government [78], offering references for the research questions explored in this study.

In summary, existing research utilizing multidisciplinary and multi-perspective approaches has made some achievements in understanding BOW and its governance. However, many studies adopt a linear and static approach, focusing on specific characteristics, which results in a diversity and dispersion of studies lacking systematic quantitative empirical research. Research on the issues of BOW governance remains predominantly qualitative, with limited quantitative investigations. These qualitative studies tend to consider the governance of BOW primarily from the sole perspective of government or public behavior, overlooking the network issues involved in the ecosystem governance process and the complexity revealed by the governance network, especially the complex behavioral strategies and interactive relationships among different subjects. Furthermore, urban and rural BOW governance not only pertains to ecological concerns but also encapsulates political actions and implications regarding management and governance. Hence, scholarly discourse on environmental action and institutional or policy frameworks progressively accentuates multidimensional attributes of governance, such as its systemic and adaptive nature. Currently, the CEPI has profound implications for local environmental regulation and governance. Under its influence, the governance of BOW exhibits an overall improving trend, albeit marred by recurring challenges. This indirectly reflects the complexity and bargaining nature of local environmental governance. Urban and rural BOW governance should not be viewed as a static process but rather analyzed and interpreted as a complex and dynamic game field from the system thinking and new public governance perspective.

Therefore, considering the disruptive effects of stochastic factors, this study utilizes the “Belief-Resource-Action” framework as a theoretical basis to design a tripartite stochastic evolutionary game model for BOW governance processes under CEPI. Through simulation analysis, it explores the strategic determinations and developmental trajectories of multi-subject behavior under the influence of different key variables, aiming to provide policy recommendations for the pathways and mechanisms of urban and rural BOW governance.

3. Problem Definition and Model Framing

3.1. Problem Description and Assumptions

This study draws on the systems thinking paradigm to view the governance of urban and rural BOW under the CEPI as a dynamic process with many influencing factors and complex interrelationships among subjects. Among them, the central government primarily assumes the function of the inspector, while local governments encompassing both urban and rural administrations, as principal government entities, are responsible for addressing BOW pollution issues. The media and the public serve as external supervisory forces that are concerned with and vocal about BOW pollution issues. In the governance process, different subjects, guided by their belief systems, continuously assess their respective costs and benefits based on available resources and formulate corresponding action strategies for environmental pollution control. Such governance typically results in two scenarios: successful governance, marked by the amelioration of BOW, or governance failure, leading to collective action dilemmas, where the phenomenon of BOW re-emerges, necessitating the initiation of another round of behavioral games and cooperation.

Grounded in the “Belief-Resource-Action” framework, this study delineates the pertinent influencing factors shaping diverse subjects’ behaviors and formulates a tripartite evolutionary game model. The formulation of a rational evolutionary game model requires adherence to the following assumptions:

Assumption 1. 

The bounded rationality of diverse subjects. Constrained by factors such as differences in demands and limited information exchange, subjects present bounded rationality and cannot fully comprehend the action beliefs and strategic orientations of others in the BOW governance process. It is necessary to gradually adjust strategies to achieve the steady-state conditions of the system. Hence, the strategy set for the inspectors is {Thorough inspection, Not-thorough inspection}, with probability (x, 1 − x). The strategies set by government entities are {Active cooperation with governance, Passive cooperation with governance}, with probability (y, 1 − y). External supervisory forces set strategies as {Active advocacy, Passive advocacy}, with probability (z, 1 − z). Additionally, x, y, z ∈ [0, 1].

Assumption 2. 

The assumption of action parameters in the collaborative governance of urban and rural BOW. Given that environmental governance unfolds as a sequence of behavioral actions initiated by subjects within specific governance scenarios [56], this study directs its attention to the transaction costs, cooperative benefits, and risks associated with the actions of three key subjects: inspectors, government entities, and external supervisory forces [67]. In terms of costs, this study assumes that inspectors incur organization costs C^d for conducting inspections. Government entities engaged in the remediation of BOW generate bargaining costs and execution costs $C_1^z$, as well as autonomy loss costs $C_2^z$. However, when passively cooperating with governance, they are not concerned about the loss of autonomy. External supervisory forces involved in environmental advocacy regarding BOW incur information costs C^w. Regarding benefits, collaborative efforts in ecological environmental governance generate collective benefits and selective benefits [68]. Collaborative governance in the CEPI of BOW by the three key subjects accumulates collaborative relationship capital, forming collective action benefits, as a hedge against negotiation costs, and the green ecological benefits obtained through environmental governance are also the core connotation of collective benefits. Thus, this study assumes that inspectors, government entities, and external supervisory forces participating in the collaborative governance of urban and rural BOW generate collective action benefits and develop common interests [79], thereby resulting in greater collective action benefits [68]; namely, the construction of collective benefits E. When one subject withdraws from collective action, they will lose the collective action benefits augmented by their belief system. If all subjects withdraw from collective action, then there will be no collective benefits. As for selective benefits, inspectors gain their selective benefits R^d from CEPIs, which consist of the reinforcement of departmental authority and public acclaim. The intensity of inspection intervention λ in environmental governance also affects the generation of organization costs C^d and selective benefits R^d. The development rights of government entities gained through neglecting environmental protection in the BOW governance process constitute their selective benefits R^z, which may lead to their reluctance to cooperate with inspections and later governance. The selective benefits R^w of external supervisory forces mainly come from actively participating in environmental governance rewards, and are also influenced by the intensity of inspection intervention λ. In addition, inspectors can provide targeted accountability and penalties P based on governance entities’ cooperation with inspection and remediation actions.

Assumption 3. 

The assumption of belief parameters in the collaborative governance of urban and rural BOW. The collaboration of diverse subjects in environmental governance collective action is strongly shaped by their belief systems, stimulating the action process [80]. In the implementation process of the CEPI, the intensity of inspectors’ beliefs α in environmental governance not only moderates their own selective benefits [81] but also regulates the collective benefits generated from the collaborative governance of urban and rural BOW. The willingness β of government entities to cooperate with governance influences bargaining and execution costs, as well as the collective benefits generated from governance. The stronger β is, the lower the costs and the greater the collective benefits. The belief parameter of external supervisory forces is reflected in the subjective intensity of their engagement in environmental affairs, which is denoted by γ, positively influencing collective benefits.

Assumption 4. 

The assumption of resource parameters in the collaborative governance of urban and rural BOW. The policy resources invested by inspectors to urge local governments to address the issue of BOW can influence the dependency of actual organization costs within specific inspection conduction through the authority of policies [64,65], which is denoted by π. Additionally, central governments, acting as inspectors, provide dedicated financial resources J to local governments acting as government entities in treating BOW, regulated by the intensity of the inspector’s beliefs α. If government entities do not actively cooperate with governance, they will not receive specialized financial support. Moreover, a not-thorough inspection by inspectors will also affect the use of specialized funds [82]. During the process of the CEPI, government entities, under thorough inspection by inspectors and active advocacy by external supervisory forces, can mobilize effectively. With the support of manpower mobilization capability ρ, they can reduce bargaining and execution costs, as well as concerns about autonomy loss during governance [83]. External supervisory forces, leveraging technical support resources π, can reduce the cost of opinion expression during their active advocacy process by utilizing the diversity of opinion expression platforms and ensuring smooth platform channels [84].

The parameter configuration and symbolic expressions during urban and rural BOW governance can be seen in

Table 1. Parameter configurations and symbols.

Dimension	Parameters	Symbols	Description
Belief	The intensity of the inspector’s beliefs in environmental governance	α	0 < α < 1
	The willingness of government entities to cooperate with environmental governance	β	0 < β < 1
	The subjective intensity of the engagement of external supervisory forces in environmental governance	γ	0 < γ < 1
Resource	The authority of policies from the inspector to urge environmental governance	π	π > 0
	Financial resources given to government entities to address environmental issues	J	J > 0
	Manpower mobilization capability of government entities in environmental governance	ρ	ρ > 0
	Technical support resources for the opinion expression platforms of external supervisory forces in environmental governance	ε	ε > 0
Action	The organization costs incurred by inspectors in inspection	Cd	Cd > 0
	Accountability and penalties for the government entities’ passive management in inspection	P	P > 0
	Selective benefits for inspectors such as reputation and status in inspection	Rd	Rd > 0;
	The intensity of intervention in inspection	λ	0 < λ < 1
	The bargaining and execution costs of government entities in environmental governance	$C_1^z$	$C_1^z$ > 0
	Autonomy loss costs of government entities in environmental governance	$C_2^z$	$C_2^z$ > 0
	Selective benefits of government entities by neglecting environmental protection	Rz	Rz > 0
	The information cost of environmental advocacy by external supervisory forces	Cw	Cw > 0
	Selective benefits of external supervisory forces for engagement in BOW governance	Rw	Rw > 0
	Construction of collective benefits in BOW governance	E	E > 0

.

3.2. Payoff Matrix and Game Model Construction

In this part, we construct a payoff matrix and game model, building upon the aforementioned assumptions and parameter configurations. The resulting tripartite payoff matrix, depicted in

Table 2. The payoff matrix for diverse subjects in the evolutionary game system.

The Game Parties and Strategy				Strategy of Inspectors
				Thorough Inspection (X)	Not-Thorough Inspection (1 − X)
Strategy of urban-rural government entities	Active cooperation with governance (y)	Strategy of external supervisory forces	Active advocacy (z)	$-\left(1-\pi \right)C^d-\alpha J+\left(1+\alpha \right)R^d+\left(\alpha +\beta +\gamma \right)E$, $-\left(1-\rho \right)\left(C_1^z+C_2^z\right)+\alpha J+R^z+\left(\alpha +\beta +\gamma \right)E$, $-\left(1-\epsilon \right)C^w+\left(1+\lambda \right)R^w+\left(\alpha +\beta +\gamma \right)E$	$-\lambda C^d+R^d+\left(\beta +\gamma \right)E$, $-C_1^z-C_2^z+R^z+\left(\beta +\gamma \right)E$, $-\left(1-\epsilon \right)C^w+\left(1+\lambda \right)R^w+\left(\beta +\gamma \right)E$
			Passive advocacy (1 − z)	$-\left(1-\pi \right)C^d-\alpha J+\left(1+\alpha \right)R^d+\left(\alpha +\beta \right)E$, $-C_1^z-C_2^z+\alpha J+R^z+\left(\alpha +\beta \right)E$, $-C^w+R^w+\left(\alpha +\beta \right)E$	$-\lambda C^d+R^d+\beta E$, $-C_1^z-C_2^z+R^z+\beta E$, $-C^w+R^w+\beta E$
	Passive cooperation with governance (1 − y)		Active advocacy (z)	$-\left(1-\pi \right)C^d+\left(1+\alpha \right)R^d+\left(\alpha +\gamma \right)E+P$, $-\left(1-\rho \right)C_1^z+\left(1+\beta \right)R^z+\left(\alpha +\gamma \right)E-P$, $-\left(1-\epsilon \right)C^w+\left(1+\lambda \right)R^w+\left(\alpha +\gamma \right)E$	$-\lambda C^d+R^d+\left(\beta +\gamma \right)E+\lambda P$, $-C_1^z+R^z+\gamma E-\lambda P$, $-\left(1-\epsilon \right)C^w+\left(1+\lambda \right)R^w+\left(\beta +\gamma \right)E$
			Passive advocacy (1 − z)	$-\left(1-\pi \right)C^d+\left(1+\alpha \right)R^d+\alpha E+P$, $-C_1^z+\left(1+\beta \right)R^z+\alpha E-P$, $-C^w+R^w+\alpha E$	$-\lambda C^d+R^d$, $-C_1^z+R^z$, $-C^w+R^w$

, facilitates the analysis of the benefits and deficits from interactions among the inspector, government entities, and external supervisory forces.

Agents in EGT refine their tactics by emulating others, following a dynamic update protocol. To represent this process, we derived replication dynamics formulas for the three subjects, utilizing the established benefits matrix.

The anticipated benefits for inspectors selecting the strategy of thorough inspection are:

$$\begin{array}{l}{U}_1^d=yz\left[-\left(1-\pi \right)C^d-\alpha J+\left(1+\alpha \right)R^d+\left(\alpha +\beta +\gamma \right)E\right]+y\left(1-z\right)\left[-\left(1-\pi \right)C^d-\alpha J+\left(1+\alpha \right)R^d+\left(\alpha +\beta \right)E\right]\\ +z\left(1-y\right)\left[-\left(1-\pi \right)C^d+\left(1+\alpha \right)R^d+\left(\alpha +\gamma \right)E+P\right]+\left(1-y\right)\left(1-z\right)\left[-\left(1-\pi \right)C^d+\left(1+\alpha \right)R^d+\alpha E+P\right]\end{array}$$

The anticipated benefits for inspectors choosing the strategy of not-thorough inspection are:

$$\begin{array}{l}{U}_2^d=yz\left[-\lambda C^d+R^d+\left(\beta +\gamma \right)E\right]+y\left(1-z\right)\left[-\lambda C^d+R^d+\beta E\right]\\ +z\left(1-y\right)\left[-\lambda C^d+R^d+\left(\beta +\gamma \right)E+\lambda P\right]+\left(1-y\right)\left(1-z\right)\left[-\lambda C^d+R^d\right]\end{array}$$

The average anticipated benefits of the inspectors’ strategy choices are $\overline{U^d}=x{U}_1^d+\left(1-x\right)U_2^d$.

Furthermore, replication dynamic equations for the inspectors’ strategy in urban and rural BOW governance can be deduced:

$$\begin{array}{l}{E}_{\left(x\right)}^d={\displaystyle \frac{dx}{dt}}=x\left(U_1^d-\overline{U^d}\right)=x\left(1-x\right)\left(U_1^d-U_2^d\right)\\ =x\left(1-x\right)\left\{yz\left[\left(2\beta -\alpha \right)E+\left(\lambda -1\right)P\right]+y\left(P-\alpha J\right)-z\left(\beta E+\lambda P\right)+\left(\pi +\lambda -1\right)C^d+\alpha R^d+\alpha E+P\right\}\end{array}$$

(1)

The anticipated benefits for government entities choosing the strategy of active cooperation with governance are:

$$\begin{array}{l}{U}_1^z=xz\left[-\left(1-\rho \right)\left(C_1^z+C_2^z\right)+\alpha J+R^z+\left(\alpha +\beta +\gamma \right)E\right]+x\left(1-z\right)\left[-C_1^z-C_2^z+\alpha J+R^z+\left(\alpha +\beta \right)E\right]\\ +z\left(1-x\right)\left[-C_1^z-C_2^z+R^z+\left(\beta +\gamma \right)E\right]+\left(1-x\right)\left(1-z\right)\left[-C_1^z-C_2^z+R^z+\beta E\right]\end{array}$$

The anticipated benefits for government entities choosing the strategy of passive cooperation with governance are:

$$\begin{array}{l}{U}_2^z=xz\left[-\left(1-\rho \right)C_1^z+\left(1+\beta \right)R^z+\left(\alpha +\gamma \right)E-P\right]+x\left(1-z\right)\left[-C_1^z+\left(1+\beta \right)R^z+\alpha E-P\right]\\ +z\left(1-x\right)\left[-C_1^z+R^z+\gamma E-\lambda P\right]+\left(1-x\right)\left(1-z\right)\left[-C_1^z+R^z\right]\end{array}$$

The average anticipated benefits of the government entities’ strategy choices are $\overline{U_2}=y{U}_{21}+\left(1-y\right)U_{22}$.

Furthermore, replication dynamic equations for the government entities’ strategy in urban and rural BOW governance can be deduced:

$$\begin{array}{l}{E}_{\left(y\right)}^z={\displaystyle \frac{dy}{dt}}=y\left(U_1^z-\overline{U^z}\right)=y\left(1-y\right)\left(U_1^z-U_2^z\right)\\ =y\left(1-y\right)\left[xz\left(\rho C_2^z-\lambda P\right)+x\left(\alpha J-\beta R^z+P\right)+z\left(R^z+\lambda P\right)-C_2^z+\beta E\right]\end{array}$$

(2)

The anticipated benefits for external supervisory forces choosing the strategy of active advocacy are:

$$\begin{array}{l}{U}_1^w=xy\left[-\left(1-\epsilon \right)C^w+\left(1+\lambda \right)R^w+\left(\alpha +\beta +\gamma \right)E\right]+x\left(1-y\right)\left[-\left(1-\epsilon \right)C^w+\left(1+\lambda \right)R^w+\left(\alpha +\gamma \right)E\right]\\ +y\left(1-x\right)\left[-\left(1-\epsilon \right)C^w+\left(1+\lambda \right)R^w+\left(\beta +\gamma \right)E\right]+\left(1-x\right)\left(1-y\right)\left[-\left(1-\epsilon \right)C^w+\left(1+\lambda \right)R^w+\left(\beta +\gamma \right)E\right]\end{array}$$

The anticipated benefits for external supervisory forces choosing the strategy of passive advocacy are:

$$\begin{array}{l}{U}_2^w=xy\left[-C^w+R^w+\left(\alpha +\beta \right)E\right]+x\left(1-y\right)\left[-C^w+R^w+\alpha E\right]\\ +y\left(1-x\right)\left[-C^w+R^w+\beta E\right]+\left(1-x\right)\left(1-y\right)\left[-C^w+R^w\right]\end{array}$$

The average anticipated benefits of the external supervisory forces’ strategy choices are $\overline{U_3}=z{U}_{31}+\left(1-z\right)U_{32}$.

Furthermore, replication dynamic equations for the external supervisory forces’ strategy in urban and rural BOW governance can be obtained:

$$\begin{array}{l}{E}_{\left(z\right)}^w={\displaystyle \frac{dz}{dt}}=z\left(U_1^w-\overline{U^w}\right)=z\left(1-z\right)\left(U_1^w-U_2^w\right)\\ =z\left(1-z\right)\left[xy\beta E-x\beta E+y\beta E+\epsilon C^w+\lambda R^w+\left(\beta +\gamma \right)E\right]\end{array}$$

(3)

Based on the replication dynamic equations in Equations (1)~(3), a replication dynamic equation set is formed for the evolutionary game involving inspectors, government entities, and external supervisory forces in urban and rural BOW governance within the context of the CEPI, as depicted in (4):

$$\left\{\begin{array}{l}{E}_{\left(x\right)}^d=x\left(1-x\right)\left\{\begin{array}{l}yz\left[\left(2\beta -\alpha \right)E+\left(\lambda -1\right)P\right]+y\left(P-\alpha J\right)-z\left(\beta E+\lambda P\right)\\ +\left(\pi +\lambda -1\right)C^d+\alpha R^d+\alpha E+P\end{array}\right\}\hfill \\ E_{\left(y\right)}^z=y\left(1-y\right)\left[xz\left(\rho C_2^z-\lambda P\right)+x\left(\alpha J-\beta R^z+P\right)+z\left(R^z+\lambda P\right)-C_2^z+\beta E\right]\hfill \\ E_{\left(z\right)}^w=z\left(1-z\right)\left[xy\beta E-x\beta E+y\beta E+\epsilon C^w+\lambda R^w+\left(\beta +\gamma \right)E\right]\hfill \end{array}\right.$$

(4)

4. Stochastic Evolutionary Game Model Construction

4.1. Integration of Gaussian White Noise for Stochastic Evolutionary Model Construction

The episodic and uncertain nature of environmental contamination issues and the complexity of environmental governance collaboration indicate that the deterministic simulation environment of traditional EGT may not be suitable for the multi-agent collaborative game process of urban and rural BOW governance examined in this study. Thus, it is necessary to introduce Gaussian white noise as a random perturbation within the replicated dynamic equations portraying an evolutionary game involving multiple parties, depicted as follows (5):

$$\left\{\begin{array}{l}dx\left(t\right)=\left\{\begin{array}{l}yz\left[\left(2\beta -\alpha \right)E+\left(\lambda -1\right)P\right]+y\left(P-\alpha J\right)-z\left(\beta E+\lambda P\right)\\ +\left(\pi +\lambda -1\right)C^d+\alpha R^d+\alpha E+P\end{array}\right\}x\left(t\right)dt+\sigma x\left(t\right)d\omega \left(t\right)\hfill \\ \begin{array}{l}dy\left(t\right)=\left[xz\left(\rho C_2^z-\lambda P\right)+x\left(\alpha J-\beta R^z+P\right)+z\left(R^z+\lambda P\right)-C_2^z+\beta E\right]y\left(t\right)dt+\sigma y\left(t\right)d\omega \left(t\right)\\ dz\left(t\right)=\left[xy\beta E-x\beta E+y\beta E+\epsilon C^w+\lambda R^w+\left(\beta +\gamma \right)E\right]z\left(t\right)dt+\sigma z\left(t\right)d\omega \left(t\right)\end{array}\hfill \end{array}\right.$$

(5)

In these equations, $\omega \left(t\right)$ is a standard (one-dimensional) Brown motion, an irregular arbitrary motion process that can be used to portray the interference of random factors. $\Delta \omega \left(t\right)=\left[\omega \right(t+h)-\omega (t\left)\right]\sim N(0,\sqrt{h})$, where step size $h>0$, and $d\omega \left(t\right)$ represents white Gaussian noise. $\sigma x\left(t\right)d\omega \left(t\right),\sigma y\left(t\right)d\omega \left(t\right),\mathrm{and}\sigma z\left(t\right)d\omega \left(t\right)$ represent the stochastic perturbation terms affecting each subsystem (i.e., inspectors, government entities, and external supervisory forces) within the game model, where the disturbance intensity is denoted by $\sigma$.

4.2. Investigation of the Existence and Stability of the Equilibrium Solution in the Model

Firstly, we examine the stochastic evolutionary game system (5) at the initial moment $t=0$, and let $x\left(0\right)=y\left(0\right)=z\left(0\right)=0$; therefore, it follows that

$$\left\{\begin{array}{l}\left\{\begin{array}{l}yz\left[\left(2\beta -\alpha \right)E+\left(\lambda -1\right)P\right]+y\left(P-\alpha J\right)-z\left(\beta E+\lambda P\right)\\ +\left(\pi +\lambda -1\right)C^d+\alpha R^d+\alpha E+P\end{array}\right\}\times 0+0\times d\omega \left(t\right)=0\hfill \\ \begin{array}{l}\left[xz\left(\rho C_2^z-\lambda P\right)+x\left(\alpha J-\beta R^z+P\right)+z\left(R^z+\lambda P\right)-C_2^z+\beta E\right]\times 0+0\times d\omega \left(t\right)=0\\ \left[xy\beta E-x\beta E+y\beta E+\epsilon C^w+\lambda R^w+\left(\beta +\gamma \right)E\right]\times 0+0\times d\omega \left(t\right)=0\end{array}\hfill \end{array}\right.$$

Thus, $d\omega \left(t\right){|}_{t=0}={\omega }^{\prime }\left(t\right)dt{|}_{t=0}=0$, indicating that, in the absence of white noise interference, the dynamic system remains stable in the zero-solution state, and the equilibrium solution of the equations is a zero solution. Furthermore, the impact of stochastic perturbation disturbances on the persistence of the equilibrium of the stability of the evolutionary game system is considered by utilizing the stochastic differential equation stability discriminant theorem. For a given stochastic differential equation,

$$dx(t)=f(t,x(t))dt+g(t,x(t))d\omega (t),x(t_0)=x_0$$

we suppose the presence of function $V(t,x)$, which is continuously differentiable, and positive constants $c_1$ and $c_2$, satisfying $c_1{\left|x\right|}^p\le V(t,x)\le c_2{\left|x\right|}^p$, where $t\ge 0$.

Provided that positive constant $\gamma$ such that $LV(t,x)\le -\gamma V(t,x)$ is identified, then the zero solution of this p-order stochastic differential equation exhibits finite moment exponential stability and holds $E|x(t,x_0){|}^p<(c_2/c_1)|x_0{|}^p{e}^{\gamma t}$ $t\ge 0$.

Provided that positive constant $\gamma$ such that $LV(t,x)\ge \gamma V(t,x)$ is identified, then the zero solution of this p-order stochastic differential equation exhibits non-exponential stability and holds $E|x(t,x_0){|}^p\ge (c_2/c_1)|x_0{|}^p{e}^{\gamma t}$, $t\ge 0$.

Among these, $LV(t,x)=V_t(t,x)+V_x(t,x)f(t,x)+{\displaystyle \frac{1}{2}}{g}^2(t,x)V_{xx}(t,x)$.

For the stochastic evolutionary system, let $V_t(t,x)=x$, $V_t(t,y)=y$, $V_t(t,z)=z$, $x,y,z\in [0,1]$, $c_1=c_2=1$, $p=1$, and $\gamma =1$; thus,

$$LV(t,x)=f(t,x)=x\left\{\begin{array}{l}yz\left[\left(2\beta -\alpha \right)E+\left(\lambda -1\right)P\right]+y\left(P-\alpha J\right)-z\left(\beta E+\lambda P\right)\\ +\left(\pi +\lambda -1\right)C^d+\alpha R^d+\alpha E+P\end{array}\right\}$$

(6)

$$LV(t,y)=f(t,y)=y\left[xz\left(\rho C_2^z-\lambda P\right)+x\left(\alpha J-\beta R^z+P\right)+z\left(R^z+\lambda P\right)-C_2^z+\beta E\right]$$

(7)

$$LV(t,z)=f(t,z)=z\left[xy\beta E-x\beta E+y\beta E+\epsilon C^w+\lambda R^w+\left(\beta +\gamma \right)E\right]$$

(8)

According to the aforementioned stability criteria, in order for the zero-solution moment of (6)~(8) to be exponentially stable, it is necessary to satisfy the following:

$$x\left\{yz\left[\left(2\beta -\alpha \right)E+\left(\lambda -1\right)P\right]+y\left(P-\alpha J\right)-z\left(\beta E+\lambda P\right)+\left(\pi +\lambda -1\right)C^d+\alpha R^d+\alpha E+P\right\}\le -x$$

(9)

$$y\left[xz\left(\rho C_2^z-\lambda P\right)+x\left(\alpha J-\beta R^z+P\right)+z\left(R^z+\lambda P\right)-C_2^z+\beta E\right]\le -y$$

(10)

$$z\left[xy\beta E-x\beta E+y\beta E+\epsilon C^w+\lambda R^w+\left(\beta +\gamma \right)E\right]\le -z$$

(11)

As $x,y,z\in [0,1]$, upon the simplification of the aforementioned equations, it becomes evident that, for the zero moment of the solution to achieve exponential stability, condition (12) must be fulfilled:

$$\left\{\begin{array}{c}\beta E-P-\alpha J+\left(\pi +\lambda -1\right)C^d+\alpha R^d+P+1\le 0\\ \left(\rho -1\right)C_2^z+\left(R^z-E\right)\beta +\alpha J+P+R^z+1\le 0\\ \beta E+\epsilon C^w+\lambda R^w+\left(\beta +\gamma \right)E+1\le 0\end{array}\right.$$

(12)

4.3. Taylor Expansion of the Evolution Equation

The above equation represents the nonlinear Itô system of the stochastic differential equations, which do not require analytical solutions. Instead, stochastic Taylor expansion can be employed to address them. Considering that $t_0=0$ and $t\in \left[0,T\right]$, the interval $t\in \left[0,T\right]$ is segmented into $0=t_0<t_1<t_2<\dots <t_N=T$, with an average step length of $t_N=nh$, where $n=1,2,3,\dots ,N$. Here, $x\left(t_0\right)=x_0$, and $y\left(t_0\right)=y_0$ are indicated, with $x_0,y_0\in R$.

We apply the forward Euler method to expand the equations of the stochastic game system (5), obtaining Equation set (13):

$$\left\{\begin{array}{l}{x}_{n+1}=x_n+x\left\{\begin{array}{l}yz\left[\left(2\beta -\alpha \right)E+\left(\lambda -1\right)P\right]+y\left(P-\alpha J\right)-z\left(\beta E+\lambda P\right)\\ +\left(\pi +\lambda -1\right)C^d+\alpha R^d+\alpha E+P\end{array}\right\}h+\Delta {\omega }_n\sigma x(n)\hfill \\ y_{n+1}=y_n+y\left[xz\left(\rho C_2^z-\lambda P\right)+x\left(\alpha J-\beta R^z+P\right)+z\left(R^z+\lambda P\right)-C_2^z+\beta E\right]h+\Delta {\omega }_n\sigma y\left(n\right)\hfill \\ z_{n+1}=z_n+z\left[xy\beta E-x\beta E+y\beta E+\epsilon C^w+\lambda R^w+\left(\beta +\gamma \right)E\right]h+\Delta {\omega }_n\sigma z\left(n\right)\hfill \end{array}\right.$$

(13)

5. Numerical Simulation Analysis

This study developed a tripartite stochastic evolutionary game model for the process of urban and rural BOW governance in the context of CEPI based on the “Belief-Resource-Action” framework, aiming to explore how inspectors, government entities, and external supervisory forces can achieve a consensus state of cooperative governance through a multidimensional parameter analysis. For this purpose, numerical simulations were conducted in line with stochastic expansion.

Considering the difficulty in obtaining and the quantitative obstacles of certain parameters, such as collective benefits, belief intensity, and other theoretical parameters, we conducted online and telephone interviews with multiple stakeholders over a time frame of 25 September 2023 to 20 November 2023. Specifically, we surveyed government officials from public institutions at different levels and types, including 14 interviewees from county-level and township-level environmental departments and 8 interviewees from their superior inspection units, 5 interviewees from media professionals, and 10 interviewees from residents of townships involved in the inspection and remediation of BOW. Two rounds of expert ratings were conducted to set the parameters.

Ultimately, the parameter values for this study were as follows: C^d = 8, $C_1^z$ = 10, $C_2^z$ = 15, C^w = 4, R^d = 5, R^z = 6, R^w = 3, P = 7, E = 4, and J = 5. All coefficient parameters were set to a median value of 0.5 to examine the intrinsic cooperation relationships among the three parties. In further simulation analysis, this study focused on the effects of initial intentions and certain key variables on the evolutionary trajectories of governance cooperation processes.

5.1. Analysis of the Impact of the Initial Probability in Stochastic Game System

The initial probability settings reflect the variations in subjects’ behavioral choices under different initial willingness. Figure 2 illustrates the evolutionary trends of the collaborative governance of urban and rural BOW among the three subjects, with the initial willingness set to 0.2, 0.4, 0.6, and 0.8, in a complex governance environment. The solid line, dashed line, and dotted line represent the trajectories of the actions taken by inspectors, government entities, and external supervisory forces, respectively.

It is evident that, under lower initial willingness, tripartite cooperation exhibits significant divergence and fluctuations in urban and rural BOW governance, especially when government entities lack the motivation to actively cooperate with governance in the inspection. Inspectors and external supervisory forces show a limited willingness to cooperate in the joint governance of BOW under the former context. With a thorough inspection by inspectors and the public voice of external supervisory forces, the willingness of government entities to actively cooperate with governance remains low, though there is a partial turnaround. Comparing the process of increasing the initial willingness of the tripartite cooperation from 0.2 to 0.8, the convergence rate of the inspectors and external supervisory forces choosing proactive governance strategies significantly improves. Meanwhile, government entities gradually shift from a tendency of passive cooperation with governance towards active cooperation with governance, accompanied by enhanced action efficiency, with the strengthening of initial willingness. It is evident that motivational willingness plays a significant role in shaping subject behavioral strategies during the inspection and remediation processes of BOW.

It is important to note that Figure 2 also reveals significant discrepancies and opinion gaps in the governance process, depicted as troughs in the simulation graph. The following sections examine whether various parameters could improve or intervene in the trends of the governance game systems of urban and rural BOW.

5.2. Analysis of the Impact of Diverse Variables in Stochastic Game System

5.2.1. The Impact of Parameter α on the Tripartite Adopted Tactics

The analysis of the impact of the inspector’s belief strength α, ranging from 0.1 to 0.9, on the strategies of multiple subjects is demonstrated in Figure 3. Herein, inspectors and external supervisory forces exhibit a clear inclination towards proactive strategies. Notably, the convergence rate of the inspector’s strategy significantly improves with the intensity of the inspection belief, while the strategies of government entities show a distinct transition from passive to proactive.

Although the inspector’s belief is just one core belief parameter within the “belief system” of BOW governance actions, the trends depicted in Figure 3 reflect the inherent influential role of belief parameters in achieving consensus in environmental governance actions. Specifically, an increase in belief strength not only internally drives inspectors to improve the organization efficiency of the CEPI and enhance the thoroughness of inspections but also positively incentivizes government entities to engage in BOW governance and cooperate with inspection work. This promotes the construction and integration of a common belief system among multiple subjects in the governance process of urban and rural BOW, facilitating the execution and orderly transition of related governance. In reality, different subjects may have divergent belief orientations, and which belief occupies a core dominant position in the construction of the belief system has a significant influence on overall governance behavior [76]. Figure 3 serves as an exploration into enhancing the inspector’s belief as the core concept in the belief system, with the intensity of the government entities’ cooperation and the subjective intensity of the engagement of external supervisory forces serving as important supplements to the belief system, reflecting the high-level promotion of execution philosophy in CEPI.

5.2.2. The Impact of Parameter λ on the Tripartite Adopted Tactics

Figure 4 illustrates the strategy variations of the different subjects under different levels of inspection intervention intensity (λ = 0.1, 0.3, 0.5, 0.7, and 0.9). Inspection intervention intensity refers to the action force exerted by higher authorities as inspectors to advance the work of CEPI for the remediation of BOW. The simulation results indicate that inspection intervention intensity impacts all three subjects. Inspectors, directly representing the authority of the Central Committee of the Communist Party of China and the State Council, often adopt a “high-profile” setup, especially with the participation of the Central Commission for Discipline Inspection and the Organization Department of the CPC Central Committee, effectively elevating environmental tasks to political missions and creating potential accountability pressure. Under different levels of inspection intervention intensity, inspectors tend to vigorously inspect local areas, pushing for thorough inspections and exerting pressure on local party committees, governments, and enterprises. This breaks through the ambiguity in the hierarchy of environmental governance, urging government entities to be more proactive on the ground, coordinate effectively at higher levels, and increase investment in pollution control.

For government entities, especially in the context of urban and rural BOW governance, intervention by inspection forces often triggers a defensive stance from a government-centric perspective, leading to passive attitudes and behavior in pollution control efforts. However, a further escalation of inspection intervention intensity will continue to amplify the political costs of government entities’ inaction, strengthening their understanding, execution, and mobilization capabilities, thereby prompting governments to adjust governance willingness and behaviors. Additionally, as the intervention intensity λ increases, the convergence speed of external supervisory forces towards active advocacy in governance accelerates. This suggests that, when inspectors assign high attention to local governance bodies, it also guides the transformation of environmental governance models, fostering the formation of a multifaceted environmental co-governance structure.

5.2.3. The Impact of Parameter J on the Tripartite Adopted Tactics

Figure 5 shows the impact of different levels of governance-specific financial resource support offered to government entities on the evolution of tripartite behaviors. Resource dimensions include policy resources, financial resources, and human resources, with financial support being the most direct and common way of providing resources in governance processes. In practice, CEPI is typically coupled with central special funding support. That is, the central government provides assistance to environmental governance projects in accordance with the results and recommendations of CEPI, ensuring the smooth and timely completion of environmental rectification tasks during the inspection process. The simulation results present that the provision of governance financial resource support significantly influences the behavioral strategies of both inspectors and government entities. It can alleviate the fiscal budget constraints faced by local governments, as the main actors in urban and rural BOW governance, thereby mobilizing more governance resources towards relevant environmental governance issues and, in turn, enhancing the efficiency and authority of the inspector’s work.

Although government entity behavior quickly shifts from passive cooperation to active cooperation with governance in response to inspection work when financial resources increase, when the supply of specific financial resources reaches a certain threshold, government entity strategies stabilize, indicating redundancy in increasing resource support. This also raises concerns from the perspective of inspectors regarding excessive investment in financial resources in the later stages. Overburdening with downward financial resource allocation can actually limit the financial capacity of higher authorities and restrict their policy support layout in other areas, especially when considering the lack of further governance support incentives after the stabilization of government entity strategies. This may, in turn, suppress the inspectors’ thoroughness and willingness during inspections. Hence, it is imperative to emphasize the avoidance of adverse outcomes resulting from resource redundancy and inefficient allocation during the inspection and rectification processes of BOW.

5.2.4. The Impact of Parameter P on the Tripartite Adopted Tactics

Figure 6 presents typical tools of reward and punishment in inspection work—accountability and penalties—which play a crucial role in urban and rural BOW governance. Figure 6 exhibits the dynamics of the behavior of the three subjects as inspection accountability and penalties (P) range from 5 to 25. It can be observed that increasing accountability and penalties have a significant stimulating effect on all three subjects. Specifically, as P increases, the equilibrium state of government entities shifts from converging to 0 to converging to 1, implying a behavior transition from passive to active cooperation with governance in tackling issues of BOW. Moreover, the probability of government entities adopting active strategies shows a trend of increasing convergence. It is suggested that, under the decentralized governance structure in China, when inspectors or higher-level authorities incorporate economic penalties (such as fines), political penalties (such as party discipline and performance penalties or case investigations), and moral penalties into the utility function of government entities, it serves to deter local governments, reshape their performance outlook, and effectively mobilize their initiative and enthusiasm for governance, thereby achieving better-than-expected pollution control outcomes.

6. Discussion

6.1. Conclusions

The governance of urban and rural BOW, as a major challenge in the environmental domain, has garnered increasing attention from decision-makers and academia. In an era of ongoing transformation in environmental governance, the implementation of the CEPI has had a profound impact on the governance of urban and rural BOW. However, real-world environmental governance practices often exhibit inefficiency, repetition, and fragmentation. The existing literature often adopts a single-entity explanatory perspective when interpreting the governance process of BOW, outlining static governance processes and entity behaviors, while paying insufficient attention to collaborative integrated governance networks, dynamic governance processes, and the interference of complex external environmental factors. Therefore, this study analyses the key theoretical concepts influencing the governance process, forming a “Belief-Resource-Action” analytical framework for the governance of urban and rural BOW under the CEPI. Based on this framework, this study analyzes the collective governance interactions among the central government, local government, and external supervisory forces. Subsequently, it develops a tripartite stochastic evolutionary game model to simulate the strategic intentions and developmental trajectories of different subjects participating in the governance process under the influence of various parameters.

This study reveals that, in the course of the governance of urban and rural BOW, for local governments, the premise of choosing proactive governance behavior strategies lies in the increase in environmental pressure exerted by CEPI and the increase in social pressure exerted by external supervisory forces, such as the media and the public. With the regulatory pressure exerted by the central government through inspection actions, local governments are prompted to change their environmental attitudes. Meanwhile, by responding to public concerns, inspections aim to stimulate media and public participation in governance. This ultimately fosters a willingness for environmental governance among different subjects, achieving an ideal state and pattern of co-governance involving central inspection, proactive execution by local governments, and participation by social forces. This lays the foundation for advancing the governance and rectification of issues of BOW. It can be said that this approach, which relies on top-down political power to stimulate multi-subject participation in environmental remediation, epitomizes a governance model rooted in political mobilization. It can rapidly aggregate resources and strength in a short period, thereby swiftly advancing the rectification process of BOW and achieving short-term governance effects. Conversely, in Western countries that have successfully implemented BOW governance, the regulatory approach is predominantly anchored in legal frameworks and subject to judicial scrutiny. A case in point is the Clean Water Act in the United States, which enshrines enduring and stable regulatory oversight for BOW governance. Nonetheless, such a system, while ensuring consistency and permanence, may concurrently engender a trade-off in the form of diminished agility in policy adaptation and enforcement.

However, the impact of the CEPI on the collective engagement of the central government, local governments, and social forces in the governance of urban and rural BOW is positively correlated with the strength of the belief system. The higher the intensity of the belief system, the greater the probability that the central government and social forces will choose proactive environmental governance behavior strategies. While local governments, under the influence of increasing belief intensity, transition from passive to positive environmental behavior strategies. The prevalence of this phenomenon is notably pronounced in East Asian countries within the Confucian cultural sphere, exemplified by South Korea and Japan, which adhere to the harmonious governance philosophy of humanity and nature throughout the course of river channel rectification initiatives. They champion the principles of social order and collective well-being, effectively transmuting administrative convictions and political resolve into tangible and positive environmental stewardship measures.

The integration of heterogeneous governance resources could guide the central and local governments to adopt proactive behavior strategies, but there is a potential for diminishing the marginal rates of substitution in heterogeneous resource investment, which may affect the willingness and likelihood of the central government to adopt thorough inspection strategies when resources become redundant. Indeed, a heterogeneity characterizes the investment and allocation of resources in the governance processes of BOW across various nations. Consistent with prior research, the integration of heterogeneous resources in the governance of BOW in this study is predominantly led by governmental administrative directives, relying on legitimate hierarchical constraints and unidirectional support from higher authorities. In contrast, many countries are increasingly favoring hybrid management strategies, expanding the governance networks and hierarchies at the practical level to achieve broader resource integration and societal participation. Popular approaches include market mechanisms and collaborative ventures between the public and private sectors for shared risk and profit. However, this trend may encounter legitimacy challenges, requiring a robust and independent regulatory agency to ensure transparency and fairness in governance. This agency should actively contribute to resource allocation governance and ensure its efficiency and sustainability, preventing resource wastage or unfair benefit distribution due to inadequate oversight.

Currently, the adoption of a broader collaborative strategy has become the mainstream among nations striving to address the issue of BOW. The findings of this paper reveal that, by constructing belief systems and mobilizing heterogeneous resources to encourage the participation of different subjects in the governance of BOW, strengthening necessary action will guide the accumulation of cooperation and consensus among them, resulting in collective benefits by addressing environmental issues and continuously influencing their choice of proactive action strategies. The Chinese experience demonstrates that the construction and expansion of collaborative governance networks, coupled with various governance mechanisms and regulations, achieve a tacit understanding of knowledge and experience, which is key to realizing collective action. This approach is conducive to identifying and resolving potential conflicts and obstacles in the BOW governance process, ensuring the coherence and consistency of governance measures. As knowledge and experience continue to accumulate and disseminate, collective wisdom is formed, offering a richer and more diverse perspective for tackling the complex challenges of BOW. It also fosters mutual understanding and trust among parties, creating a sustainable collaborative effect and propelling the governance of BOW towards deeper levels and broader scopes of development.

Additionally, enhancing the intensity of inspection interventions and incentive punishment measures will significantly reduce the passive governance behavior of local governments, increasing the probability of choosing proactive governance behavior strategies. The other two parties, under these intensified action factors, consistently have strong motivation for thorough supervision and active advocacy behavior strategies, thereby enabling the effective governance of urban and rural BOW.

It is noteworthy that the conclusions drawn from this study are rooted in the context of Chinese institutions and politics, delving into the intricacies and dynamics of urban and rural BOW practices in China. It offers what is termed the “Chinese experience”, thereby enhancing the global endeavor to achieve the objectives of availability and sustainable stewardship of water and sanitation. However, when confronted with analogous aquatic environmental issues across various nations worldwide, the efficacy and applicability of these conclusions necessitate thorough and prudent consideration within the framework of each country’s institutional background.

6.2. Recommendations

Building upon the analysis and conclusions of the tripartite stochastic evolutionary game model, the following recommendations can be offered by this study:

(1)

Enhance governance willingness and adjust the consistency of belief systems. On the one hand, the central government could flexibly utilize mechanisms such as spiritual preaching activities, conference mobilization, and administrative interviews to deconstruct old perceptions under the pressure of normative institutions, enabling local governments to construct a new consensus and identity for ecological civilization construction in the new era. On the other hand, it is necessary to guarantee the authentic participation of the people, enhancing their sense of ownership and aligning them with the guiding consensus released by the central authority, facilitating their integration into the governance of urban and rural BOW to form an effective complementary mechanism.

(2)

Integrate diverse resources and adapt the agency of governance behaviors. Tailored configurations of heterogeneous resources within the environmental governance system that influence the expansion of capabilities are required for various subjects. This involves not only relying on policy mobilization through administrative authority but also actively utilizing financial resources such as preferential policies and special subsidies; mobilizing human resources, including the media, experts, and scholars; and employing a variety of technical means to broaden channels for opinion expression and information communication. Such an approach can encourage the proactivity, initiative, and creativity of environmental governance behaviors among central, local, and social actors, thereby achieving effective resource governance outcomes.

(3)

Guide collaborative mechanisms to adjust the tacit understanding of cooperative behaviors. Relying on the political authority of the central government and under the premise of achieving continuous policy and goal alignment, specific mechanisms and rules can assist in clarifying responsibilities and the division of labor among the various entities involved in the management of issues of BOW, preventing the fragmentation of power and responsibility. Simultaneously, strengthening the construction of horizontal integration and vertical linkage mechanisms, as well as comprehensive coordination networks, can enable adaptation to differences and changes through effective communication, thus achieving action coupling and sustainable experience. In the process of pollution governance, enhancing vertical incentive and constraint mechanisms for various subject behaviors, namely, comprehensive assessments and punishments, can overcome the inherent shortcomings of unidimensional hierarchical systems and enhance the effectiveness of collaborative governance.

This study also has certain limitations. Firstly, the assignment of variables was primarily obtained through surveys, which may have involved individual biases or misjudgments. Secondly, although this study employs numerical simulation methods to model real-world problems, it has not yet fully utilized a broad range of empirical data for in-depth empirical verification. Concurrently, we acknowledge that our consideration only extends to the central government, local governments, and societal forces; however, the governance process of urban and rural BOW is a systemic and complex endeavor that encompasses other stakeholders, such as adjacent governments at the same administrative level, related polluting enterprises, and third-party environmental remediation enterprises (also known as environmental service companies). The actual dynamics of strategic evolution within the governance process of urban and rural BOW are likely to be more intricate than the scenarios presented herein. Future iterations of this research could benefit from the complexity theory, namely chaos theory and agent-based modeling, to depict random perturbation thoroughly and improve the simulation systems, while also from the inclusion of these potential stakeholders to develop a more holistic game-theoretic model. Moreover, by using a wider range of variables and data sources, our findings could be supplemented and refined through case study analysis, econometric techniques, and data mining methods, thereby offering a more nuanced and in-depth portrayal of the governance of urban and rural BOW.