Authentic or spurious: an evolutionary game analysis of manufacturing enterprises’ greenwashing behavior under dynamic government regulation

Abstract

The phenomenon of greenwashing, wherein companies exaggerate or falsify environmental claims to project a “green” image, has become increasingly prevalent among manufacturing enterprises. This behavior not only highlights a lack of corporate social responsibility, inadequate regulatory supervision, and insufficient public awareness but also underscores the urgent need to mitigate such practices. To explore effective governance mechanisms, this study constructs a tripartite evolutionary game model involving manufacturing enterprises, the government, and the public, incorporating dynamic government regulatory strategies for simulation. The evolutionary stability of strategy choices among these stakeholders is analyzed, and the impact of various factors on their decision-making processes is investigated. The results reveal that among four government regulatory strategies—static reward and static punishment, static reward and dynamic punishment, dynamic reward and static punishment, and dynamic reward and dynamic punishment—only the strategy combining static reward and dynamic punishment leads to an evolutionarily stable strategy (ESS). In this stable state, manufacturing enterprises engage in green innovation, the government enforces strict regulation, and the public actively participates in supervision. Parameter analysis further demonstrates that greenwashing behavior is most strongly influenced by the benefits of green innovation (more than its costs), the issuance of green credit (more than certification revocation), and public supportive incentives (more than resistive pressures), alongside the degree of public environmental concern. Specifically, the benefits of green innovation exert a greater impact on greenwashing behavior than the costs, the issuance of green credit has a more significant effect than the revocation of green certification, and public supportive incentives play a more substantial role than resistive pressures. These results highlight the critical role of dynamic regulation and public engagement in mitigating greenwashing, offering policymakers actionable insights for designing incentive-aligned governance mechanisms while providing enterprises with guidance to align environmental claims with substantive green practices.

Introduction

Amidst the rapid advancement of the global economy, environmental pollution and global warming have emerged as critical concerns within the international community (Fang et al., 2022). Notably, the phenomenon of greenwashing among enterprises, which represents a significant decoupling from corporate social responsibility, has garnered extensive societal attention (Yu et al., 2024a). Certain manufacturing enterprises, driven by the pursuit of short-term profits, engage in the dissemination of exaggerated or false environmental claims, thereby misleading both the public and regulatory bodies. Such conduct not only violates consumer rights but also threatens market expansion (Nygaard and Silkoset, 2023), ultimately impeding genuine green innovation. Consequently, the exploration of effective strategies to mitigate greenwashing practices in manufacturing enterprises and to foster genuine green innovation is of substantial practical significance and urgency.

Evolutionary game theory has been recognized as an effective tool for analyzing complex dynamic systems to optimize practices, as it can simulate the strategic choices and evolutionary trajectories of participants in long-term interactions. In recent years, evolutionary game theory has made significant strides in analyzing the dynamic shifts in individual and collective behaviors, which has led to its broad application across both social and natural sciences. Notably, Ogbo et al. (2022) have shed light on the role of pre-commitment in fostering cooperation by applying evolutionary game theory. Mondal et al. (2024) have conducted an examination of chaotic dynamics within the context of resource management. Additionally, Kastelic et al. (2024) have explored cooperative behavior within the flow of charitable resources. Furthermore, Wang et al. (2025b) have studied the impact of state feedback in public goods games, thereby enriching the understanding of this domain. These studies not only demonstrate the broad applicability of evolutionary game theory but also highlight its unique advantages in addressing dynamic and adaptive strategies. Therefore, this paper employs evolutionary game theory to construct a tripartite evolutionary game model incorporating dynamic regulatory strategies, with the aim of elucidating the governance mechanisms of corporate greenwashing behavior and its relationship with green innovation.

Participants in the mechanisms influencing corporate behavior have been classified differently in the academic community. For instance, some scholars have identified manufacturers, consumers, and government as key stakeholders (Hu and Wang, 2022), while others have emphasized the tripartite evolutionary relationship among manufacturers, retailers, and government (Yu et al., 2024b). From a comprehensive perspective, both governmental and public entities have been identified as crucial actors in the governance of greenwashing behaviors. Governmental intervention, through the establishment and enforcement of stringent environmental regulations, has been demonstrated to provide normative guidance for corporate production and management policies (Chen and Hu, 2018), thereby effectively curbing greenwashing behavior. In addition, the government can also promote external auditing and verification of ESG information (Liao et al., 2023), continuously improving the accuracy of corporate disclosure data and the quality of disclosure, thus reducing the occurrence of greenwashing behavior. Concurrently, public supervision has been evidenced to play a significant role through various channels, including media scrutiny (Yue and Li, 2023), consumer purchasing intentions (Akturan, 2018), and social credit systems (Huang and Sun, 2024), creating strong social pressure that encourages companies to abandon greenwashing and pursue genuine green innovation. Therefore, this study ultimately selects manufacturing enterprises, the government, and the public as the primary participants in the governance mechanism of greenwashing behavior and constructs a tripartite evolutionary game model to analyze how to effectively avoid greenwashing and achieve genuine green innovation. Moreover, it should be noted that traditional game models have predominantly treated government regulation as static and fixed parameters, exemplified by approaches such as green innovation subsidies and carbon taxes (Yu et al, 2022), or conventional tax and subsidy mechanisms (Li et al., 2019). In fact, however, the reward and punishment mechanisms employed by the government to regulate enterprises are not static but rather dynamic processes that can be flexibly adjusted according to market responses and policy effectiveness. Therefore, this study takes into account the dynamic nature of government regulation and explores how the government can suppress corporate greenwashing behavior by adjusting its reward and punishment mechanisms, as well as how this dynamic regulation affects corporate green innovation decisions.

This paper makes the following key contributions. First, it has been observed that previous investigations into greenwashing phenomena have predominantly concentrated on dyadic actor relationships, thereby overlooking the intricate interplay among governmental entities, enterprises, and the public sector (Sun and Zhang, 2019; Wu et al., 2020). This limitation has been identified as a substantial barrier to the comprehensive understanding of greenwashing governance systems. In contrast to these conventional approaches, the current research introduces a novel dynamic governmental regulatory framework and develops a tripartite evolutionary game model incorporating manufacturing enterprises, governmental bodies, and the public. This model not only facilitates a thorough examination of the interactive dynamics among these stakeholders in greenwashing governance and their influence on strategic evolution but also specifically investigates system performance under varying governmental reward-punishment mechanisms. Second, traditional evolutionary game studies have primarily focused on strategy choices under static government regulation (Zhao et al., 2016; Peng et al., 2025), emphasizing technology adoption or innovation incentives (Zheng and Wu, 2024), while overlooking the dynamic regulatory mechanisms under problem governance and the potential dynamic strategy changes among the stakeholders. To address this gap, the current study incorporates dynamic governmental regulation, thereby emphasizing the realistic, context-dependent nature of regulatory interventions. Finally, existing research has been found to provide limited exploration of the public’s role in greenwashing governance, particularly with regard to the inhibitory effects of public supervision on greenwashing behaviors, which have not been sufficiently acknowledged. The present study addresses this oversight by explicitly recognizing the critical importance of public participation and incorporating the public as a principal actor within the evolutionary game model. This theoretical advancement underscores the necessity and novelty of the current research at the conceptual level.

Addressing greenwashing behavior is essential for multiple stakeholders from a practical standpoint. For manufacturing enterprises, genuine green innovation enhances long-term competitiveness and sustainability by reducing operational costs, improving resource efficiency, and meeting consumer demand for eco-friendly products. This also builds a positive corporate image and aligns with global sustainable development trends. For the government, effective regulation to curb greenwashing is crucial for maintaining market integrity and protecting consumer interests through strict environmental standards and monitoring mechanisms. For the public, vigilance and active participation in supervising corporate environmental claims drive enterprises to adopt genuine green practices through informed choices and social pressure. Thus, understanding and addressing greenwashing is both an academic pursuit and a practical necessity for all stakeholders.

The structure of this paper is organized as follows. “Literature review” presents a comprehensive review of the relevant literature. In Section 3, the research problem is delineated, followed by the construction of the model through the specification of parameter settings. Section 4 is devoted to the analysis of the stability of the main actors and the equilibrium of the system under the traditional reward and punishment model. Section 5 introduces the establishment of three distinct dynamic reward and punishment models: (1) Dynamic issuance of green credit coupled with static revocation of green certification, (2) Static issuance of green credit combined with dynamic revocation of green certification, (3) Dynamic issuance of green credit integrated with dynamic revocation of green certification. This section further examines the parameter effects under stable equilibrium conditions. Finally, Section 6 provides a detailed discussion of the models, numerical simulation results, theoretical and practical implications, and limitations of the study.

Literature review

This section reviews literature on greenwashing, green innovation, and evolutionary game theory applications. It aims to establish a theoretical foundation by synthesizing key findings and critically examining influencing factors. Limitations of current research are highlighted, and gaps are identified for our study to address. The role of dynamic regulatory mechanisms in shaping corporate environmental behavior is also explored, setting the stage for our model analysis.

Greenwashing behavior

This study presents a systematic examination of existing scholarship on corporate greenwashing behavior, which has encompassed both influential determinants and their consequential impacts. From an internal governance perspective, it has been demonstrated that corporate digital transformation serves as an effective mechanism for curbing greenwashing practices through the mitigation of financing constraints, enhancement of information transparency, and reinforcement of internal controls. These organizational improvements have been shown to optimize production efficiency and economic returns while simultaneously mitigating risks associated with R&D activities. (Wang et al., 2024b; Lu et al., 2023). From an external governance standpoint, empirical evidence suggests that institutional supervision mechanisms, particularly media exposure and governmental regulatory frameworks (Yue and Li, 2023; Sun and Zhang, 2019), can effectively compel enterprises to adopt authentic green innovation strategies by imposing accountability measures. It has been established that both internal and external supervision mechanisms can effectively prevent companies from adopting greenwashing strategies. Regarding the consequences, the focus is primarily on the impact on consumers. Greenwashing behavior has negative effects on green brand associations and brand credibility, which in turn affects consumers’ emotions and identification with the company, and indirectly influences their purchasing intentions (Akturan, 2018; Volschenk et al., 2022). In fact, the public, as a critical factor influencing corporate behavior, comprises not only media and consumers but also investors, non-governmental organizations, community members, and other stakeholders (Li et al., 2024). These diverse groups can exert influence on corporate greenwashing practices through various channels and mechanisms. However, existing studies on greenwashing behavior often focus on isolated factors without considering the interplay between multiple governance mechanisms. For instance, while some studies highlight the role of digital transformation in reducing greenwashing (Wang et al., 2023), they often overlook the combined effect of external regulatory pressures and public scrutiny. This gap leaves a significant portion of the greenwashing phenomenon unexplained, particularly in contexts where internal governance mechanisms are weak. Additionally, the majority of existing studies adopt a descriptive approach, summarizing the prevalence and types of greenwashing without critically analyzing the underlying drivers and their interdependencies. Our study addresses these limitations by integrating both internal and external governance mechanisms into a comprehensive framework. By employing a tripartite evolutionary game model, we explore the dynamic interactions between manufacturing enterprises, the government, and the public. This approach not only provides a more holistic understanding of greenwashing behavior but also offers insights into how different governance mechanisms can be optimized to effectively curb greenwashing.

Green innovation

Green innovation plays a pivotal role in generating value for both enterprises and society, with manufacturing enterprises serving as key contributors to this process. Existing research has predominantly focused on identifying and analyzing the factors influencing green innovation, which can be broadly categorized into macro- and micro-level factors. At the macro level, financial environments and government measures, such as subsidies and green credit policies, have been shown to significantly influence corporate green innovation. For example, Han et al. (2025), utilizing panel data, demonstrated that data elements, through their integration with technology, labor, and capital, significantly improve both the quality and quantity of low-carbon innovation, providing a governance mechanism distinct from traditional financial incentives. Similarly, Du and Guo (2023) and Xu et al. (2024) provided evidence that green credit policies effectively enhance corporate green innovation behavior. Furthermore, An et al. (2025) concluded that government green subsidies exert a significantly positive impact on corporate green innovation. At the micro level, Wang et al. (2024a), through a meta-analysis, established a significant correlation between corporate social responsibility and green innovation. Shi et al. (2024) further revealed that investors’ green preferences amplify the positive effects of green finance on green innovation. Additionally, stakeholder pressure has been identified as a positive driver of green innovation. Despite these contributions, existing studies on green innovation often neglect the role of dynamic regulatory mechanisms in influencing corporate behavior. Most research assumes static regulatory environments, which may not reflect the adaptive nature of real-world policies. Moreover, the interplay between macro-level financial incentives and micro-level stakeholder pressures is rarely explored in a dynamic context. This gap limits the applicability of current findings to evolving regulatory landscapes and market conditions. Our study bridges this gap by incorporating dynamic regulatory strategies into the analysis of green innovation. By examining how dynamic green credit policies and certification mechanisms influence corporate decision-making, we provide a more realistic and context-dependent understanding of green innovation drivers. This approach not only enriches the existing literature but also offers practical insights for policymakers aiming to design effective green innovation incentives.

Evolutionary game of greenwashing behavior

Evolutionary game theory focuses on the interactions among different participants, with the evolutionary equilibrium of the system being achieved through continuous strategy adjustments. Regarding influencing factors, Sun and Zhang (2019) employed the method of evolutionary game theory to construct two evolutionary models, analyzing the impacts of government punishment mechanisms and tax-subsidy mechanisms on greenwashing behavior and green innovation strategies, respectively. Similarly, Wang et al. (2022) developed a three-stage game-theoretic model to examine the influence of the False-Claims Ban (FCB) regulatory policy on corporate greenwashing behavior. Regarding the consequences, a game-theoretic model was employed by Wu et al. (2020) to investigate greenwashing behavior in corporate social responsibility (CSR) and the effects of information transparency on corporate strategies and social welfare. Similarly, Huang et al. (2020), utilizing a two-stage game-theoretic model, analyzed the competitive pricing strategies of green enterprises and brown (greenwashing) enterprises, offering an in-depth examination of the impact of greenwashing behavior in emerging markets. Furthermore, Dong et al. (2023) applied a game-theoretic approach to develop a model involving manufacturers, e-commerce platforms, and third-party logistics companies, exploring the influence of logistics companies’ greenwashing behavior on manufacturers’ decision-making processes.

However, existing studies using evolutionary game theory to analyze greenwashing behavior often focus on dyadic relationships, such as those between enterprises and the government or between enterprises and consumers. This approach overlooks the complex interplay among multiple stakeholders, including the public, which plays a crucial role in shaping corporate behavior through social pressure and supervision. Moreover, these studies typically assume static regulatory environments, which may not capture the dynamic nature of real-world regulatory strategies. In fact, to establish an effective governance system for greenwashing behavior and further achieve genuine green innovation, the roles of the government and the public are recognized as crucial and indispensable. On the one hand, the government can focus on improving legislative and law enforcement systems, with enforcement efforts against greenwashing behavior being intensified to strengthen constraints on corporate actions. On the other hand, the development of evaluation standards and management systems for green innovation intermediary institutions can be prioritized, thereby creating a comprehensive support system and a conducive policy environment for green innovation. This ensures that green innovation is safeguarded at the institutional level. Meanwhile, the public can leverage market mechanisms to penalize corporate greenwashing behavior. For instance, consumers may refuse to purchase products associated with greenwashing, and investors may withdraw financial support from companies involved in such practices, thereby constraining improper corporate behavior. Additionally, public participation in supervision can effectively compensate for the limitations of government regulation (Liu et al., 2023), forming a robust constraint on the behavioral choices of both enterprises and the government. This collaborative approach can lead to a win-win situation among the government, enterprises, and the public. Given these considerations, the construction of an evolutionary game model involving three parties — enterprises, the government, and the public — is deemed particularly necessary. Such a model can provide an in-depth analysis of the dynamic interactions among these key actors and offer a solid theoretical foundation for precise policy-making aimed at governing greenwashing behavior and promoting green innovation.

Dynamic reward and punishment mechanisms in evolutionary games

Under dynamic reward and punishment mechanisms, the behavior of participants in an evolutionary game system can be more effectively analyzed. It has been demonstrated by existing research that manufacturers’ free-riding behavior can be effectively curbed, and their transition toward low-carbon practices can be incentivized, through a combination of differentiated carbon tax policies and subsidy policies (Yu et al., 2024b). Moreover, among the three types of dynamic carbon tax and subsidy mechanisms, greater effectiveness has been observed in the bilateral dynamic carbon tax and subsidy mechanism, which provides stronger incentives for low-carbon manufacturing processes to be adopted by manufacturers (Chen and Hu, 2018). Additionally, greater advantages for manufacturers’ decision-making have been identified in strategies based on dynamic carbon taxes and subsidies when compared to static strategies (Hu and Wang, 2022). Despite these contributions, existing studies on dynamic reward and punishment mechanisms often focus on specific contexts, such as carbon taxes and subsidies, without considering the broader implications for greenwashing behavior. Moreover, these studies typically assume that dynamic mechanisms are universally applicable, without critically examining the conditions under which they are most effective. This gap limits the generalizability of current findings and leaves questions about the applicability of dynamic mechanisms in different regulatory environments. Since dynamic reward and punishment mechanisms can be adapted to demonstrate their characteristics according to specific strategic goals in different contexts (Chen and Hu, 2018), dynamic issuance of green credit and dynamic revocation of green certification are introduced in this study to explore the evolutionary game behavior within the system of greenwashing behavior among manufacturing enterprises.

Model analysis and construction

In this section, we construct a tripartite evolutionary game model involving manufacturing enterprises, the government, and the public. The goal is to develop a theoretical framework capturing dynamic interactions among stakeholders under dynamic regulatory strategies. We define strategies, specify parameters, and outline payoff matrices, laying the groundwork for subsequent stability and equilibrium analysis under different regulatory scenarios.

Strategy analysis of participants

In the research on the evolutionary game model involving manufacturing enterprises, the government, and the public, the formation of a governance system for greenwashing behavior is explored. Among these participants, manufacturing enterprises are the main actors in greenwashing behavior and green innovation, while the government plays a significant role in regulating greenwashing behavior. Additionally, due to limitations in human resources, material resources, and timeliness, the government finds it challenging to comprehensively and effectively regulate the behavior of manufacturing enterprises. In contrast, the public can monitor whether the production processes and product quality of manufacturing enterprises are genuine, as well as whether the government is strictly regulating these enterprises, thereby further compensating for the deficiencies in government supervision.

The behavioral strategies of manufacturing enterprises are categorized as {Green Innovation, Greenwashing}. The probability of the green innovation strategy being chosen is denoted as x, and the probability of the greenwashing strategy being chosen is denoted as 1 − x, where $x\in (0,1)$. Here, green innovation is defined as the innovative approaches adopted by enterprises to reduce the negative environmental impacts associated with the design and development of products, services, or processes, thereby enhancing efficiency and sustainability (Mao and Lin, 2024). In contrast, greenwashing refers to the dissemination of misleading information, exaggerated claims, or inaccurate labeling by enterprises or organizations in marketing and public relations communications, which is used to create a false green image for the public (Nygaard and Silkoset, 2023).

The government’s behavioral strategies are defined as {Stringent Regulation, Lenient Regulation}. The probability of the stringent regulation strategy being chosen is denoted as y, and the probability of the lenient regulation strategy being chosen is denoted as $1-y$, where $y\in (0,1)$. Stringent regulation refers to the imposition of mandatory regulations and high standards on enterprises by the government to promote green innovation, with enterprises engaged in green innovation being rewarded and those involved in greenwashing being penalized (Liao and Xiao, 2025). This approach is aimed at fostering a positive government image, enhancing government credibility, and reducing the loss of public interests. In contrast, lenient regulation involves a reduction in direct government intervention in the behavioral choices of manufacturing enterprises (Cheng et al., 2024), with enterprises being encouraged to independently manage their environmental responsibilities. Under this strategy, if greenwashing behavior is detected by the public, a certain degree of public pressure is generated, and government credibility may be undermined.

The public’s behavioral strategies are defined as {Participate in Supervision, Not Participate in Supervision}. The probability of the strategy to participate in supervision being chosen is denoted as z, and the probability of not participating in supervision being chosen is denoted as $1-z$, where $z\in (0,1)$. Participating in supervision refers to the active involvement of the general public in the supervision of corporate environmental behavior and government regulatory activities (Mao and Lin, 2024). This includes monitoring and evaluating both corporate and government actions, such as initiating and engaging in discussions on social media platforms to increase attention towards corporate behavior and government performance. This process generates social pressure that compels corporations to enhance their environmental responsibility and the government to strengthen its regulatory control over corporate actions. In contrast, non-participation in oversight means that the public takes no measures to intervene in corporate greenwashing behavior or government regulatory activities.

Model construction

In the context of the green economy, this study constructs a tripartite evolutionary game model involving manufacturing enterprises, the government, and the public, as depicted in Fig. 1. The parameters selected aim to capture key dynamics in green innovation and greenwashing behaviors. For manufacturing enterprises, parameters reflect the economic trade-offs between green innovation and greenwashing, balancing potential market advantages against costs and risks. For the government, regulatory costs and benefits highlight the balance between enforcement efforts and societal gains. Public participation is characterized by the costs and benefits of engaging in supervision, emphasizing the role of societal pressure in promoting genuine green practices. Additionally, parameters for green credit and certification revocation capture the government’s role in incentivizing and penalizing corporate behavior. Public concern for greenness underscores market pressures driving sustainable innovation. These parameters provide a comprehensive framework that balances theoretical rigor with practical relevance in environmental governance. The specific descriptions of the parameters are shown in Table 1.

Fig. 1: The game-theoretic relationship among manufacturing enterprises, the government, and the public.

Each stakeholder has two different strategies to choose from, and the different combinations of their strategies will affect their respective payoffs.

Table 1 Description of Parameters in the Evolutionary Game Model.

In the modeling process, several assumptions and limitations were made to simplify the analysis and focus on the core dynamics of greenwashing and green innovation. The model assumes homogeneous participants with binary strategy choices (e.g., green innovation vs. greenwashing for enterprises, stringent vs. lenient regulation for the government, and participation vs. non-participation in supervision for the public). This simplification overlooks the heterogeneity of real-world stakeholders and the potential for mixed strategies. Additionally, the model assumes that enterprises, the government, and the public are boundedly rational participants. Over time, their strategic choices will gradually converge to optimal strategies. The regulatory mechanisms, such as green credit allocation and certification revocation, are modeled with linear functions and static parameters, which may not fully capture the complexity and dynamic nature of real-world policies. These assumptions and limitations should be considered when interpreting the findings, as they may affect the model’s applicability and the robustness of the conclusions. Future research could address these limitations by incorporating more complex, dynamic, and heterogeneous elements to enhance the model’s realism and accuracy.

Manufacturing enterprises, recognized as the primary agents of green innovation, are regulated by the government and monitored by the public. The various strategy combinations among manufacturing enterprises, the government, and the public are systematically outlined in Table 2. For example, when the strategy combination is (1, 1, 1), the production cost ${C}_{1}$ associated with adopting a green innovation strategy is incurred by the manufacturing enterprise. However, the enterprise is able to obtain the production revenue ${R}_{1}$ generated from green innovation, the additional benefit $\alpha \ast {R}_{1}$ derived from public green concern due to its green innovation behavior (Wang et al., 2025a), the benefit P resulting from green credit issued under strict government regulation (Chen et al., 2024), and the supportive incentive L provided by public supervision for its green innovation efforts (Wang et al., 2025a). The costs ${C}_{4}$ related to human resources, materials, and finances required for strict regulation, as well as the green credit P issued to the enterprise, are borne by the government. In return, the government gains the governance benefit ${R}_{3}$ arising from the enterprise’s green innovation and the reputational reward F received from the public for its regulatory actions (Liu et al., 2024; Wang et al., 2025a). The public incurs the cost ${C}_{5}$ of participating in supervision and the supportive consumption and promotional expenses L associated with the enterprise’s green innovation. At the same time, the public obtains the utility ${U}_{1}$ derived from using green innovation products or services and the income H resulting from safeguarding its own interests under strict government regulation (Mao and Lin, 2024).

Table 2 Strategy Combinations and Payoff Matrices of the Participants.

To achieve strategic stability and system evolution, the replicator dynamics equations for the behavioral strategies of manufacturing enterprises, the government, and the public are constructed within the tripartite evolutionary game model in this study. In evolutionary game theory, the replicator dynamics and the evolutionarily stable strategy (ESS) are recognized as two core processes.

Model analysis of static green credit issuance and revocation of green certification

This section analyzes the traditional static regulatory mechanism of green credit issuance and green certification revocation. We examine the strategic behavior of enterprises, the government, and the public under this framework. Stability analysis of main actors and equilibrium points reveals limitations of static strategies in addressing greenwashing and promoting green innovation. Parameter assignments are explained based on real data or literature.

Stability analysis of participants

Manufacturing enterprises

It is assumed that the expected payoff for a manufacturing enterprise adopting a green innovation strategy is denoted as ${E}_{11}$ while the expected payoff for adopting a greenwashing strategy is represented as ${E}_{12}$. The average expected payoff for the enterprise is $\overline{{E}_{1}}$.

$${E}_{11}=Lz-Py+(1+\alpha ){R}_{1}-{C}_{1}-2y(\alpha {R}_{1}+{R}_{1}z-{C}_{1}z)$$

(1)

$${E}_{12}=2y[(1-\alpha ){R}_{2}(z-1)-{C}_{2}z]+Qy(1-2z)+(1-\alpha ){R}_{2}-M-C2$$

(2)

$$\overline{{E}_{1}}=x{E}_{11}+(1-x){E}_{12}$$

(3)

The replicator dynamics equation for the enterprise’s behavioral strategy is:

$$F(x)=\frac{dx}{dt}=x({E}_{11}-\overline{{E}_{1}})=-x(x-1)[{C}_{2}-{C}_{1}+{R}_{1}-{R}_{2}+({R}_{1}+{R}_{2})\alpha +(P+Q)y+(L+M)z]$$

(4)

The first-order derivative of x is:

$$\frac{d[F(x)]}{dx}=(2x-1)[{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-({R}_{1}+{R}_{2})\alpha -(P+Q)y-(L+M)z]$$

For ease of expression, let:

$$G(y,z)={C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-({R}_{1}+{R}_{2})\alpha -(P+Q)y-(L+M)z$$

(5)

According to the stability theorem of differential equations, the following condition must be satisfied for the probability of an enterprise choosing the green innovation strategy to be in a stable state:

$$\left\{\begin{array}{c}F(x)=0\\ \frac{d[F(x)]}{dx} < 0\end{array}\right.$$

(6)

Since $\frac{\partial G(y,z)}{\partial y}=-(P+Q) < 0$, $G(y,z)$ is a decreasing function of y, it follows that:

• (1) When ${\rm{y}}=\frac{{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-({R}_{1}+{R}_{2})\alpha -(L+M)z}{P+Q}={y}^{\ast }$, $G(y,z)=0$, and $\frac{d[F(x)]}{dx}=0$, manufacturing enterprises are unable to determine their stable strategy points.

• (2) When ${\rm{y}} < {y}^{\ast }$, $G(y,z) > 0$, $x=0$ satisfy $\frac{d[F(x)]}{dx} < 0$, $x=0$ is the ESS for the enterprise. Enterprises tend to adopt greenwashing behavior and are unwilling to engage in green innovation.

• (3) When ${\rm{y}} > {y}^{\ast }$, $G(y,z) < 0$, $x=1$ satisfy $\frac{d[F(x)]}{dx} < 0$, $x=1$ is the ESS for the enterprise. Enterprises tend to engage in green innovation and are unwilling to adopt greenwashing behavior.

The evolutionary stages of the enterprise’s strategy are illustrated in Fig. 2:

Fig. 2: The evolutionary phase diagram of enterprise strategies.

Evolutionary phase diagram of enterprise strategies showing: a strategy selection at critical threshold y = y*; b dominance of greenwashing when yy*. Among them, the probability that an enterprise stably adopts a greenwashing strategy is represented by the volume V11, and the probability that it stably adopts a green innovation strategy is represented by the volume V12.

In Fig. 2, the cross-section ${\rm{y}}0z$ is shown to pass through the point $(0,0,\frac{{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-({R}_{1}+{R}_{2})\alpha }{L+M})$, where the probability of an enterprise stably adopting greenwashing behavior strategies is represented by volume ${V}_{11}$, and the probability of stably adopting green innovation strategies is represented by volume ${V}_{12}$. The calculation formula is provided as follows:

$$\begin{array}{c}{V}_{11}={\int }_{0}^{1}{\int }_{0}^{\frac{{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-({R}_{1}+{R}_{2})\alpha }{L+M}}\displaystyle\frac{{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-({R}_{1}+{R}_{2})\alpha -z(L+M)}{P+Q}dzdx(ESS:x\to 0)\\ \,=\,\displaystyle\frac{{[{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}-({R}_{1}+{R}_{2})\alpha ]}^{2}}{2(P+Q)(L+M)}\end{array}$$

(7)

$${V}_{12}=1-{V}_{11}$$

(8)

Corollary 1: A negative correlation is observed between the green innovation of enterprises and ${C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}$. In contrast to greenwashing behavior, the benefits derived from green innovation are positively correlated with the following factors: the public’s concern regarding the environmental performance of manufacturing enterprises α, the government’s issuance of green credit and revocation of green certifications $P+Q$, as well as public support and resistance $L+M$.

Proof 1: In the model, we define ${R}_{2}-{C}_{2}$ as the net profit from greenwashing behavior, and ${R}_{1}-{C}_{1}$ as the net profit from green innovation. Consequently, the expression ${C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}$ represents the disparity in net profits between greenwashing and green innovation behaviors for manufacturing enterprises. The step of taking partial derivatives is aimed at analyzing how this net profit gap changes with variations in different parameters within the model. Specifically, we calculate the first-order partial derivatives of ${V}_{12}$ with respect to each element to understand how the probability of enterprises choosing green innovation strategies is affected when factors such as green concern, supportive motivation, resistive pressure, or others change. These analytical results help us comprehend which factors may promote or inhibit corporate green innovation under different conditions. Based on the probability expression ${V}_{12}$ of enterprises adopting green innovation strategies, the first-order partial derivatives of each element are calculated as $\frac{\partial {V}_{12}}{\partial [{C}_{1}-{C}_{2}+{R}_{2}-{R}_{1}]} < 0$, $\frac{\partial {V}_{12}}{\partial \alpha } > 0$, $\frac{\partial {V}_{12}}{\partial P} > 0$, $\frac{\partial {V}_{12}}{\partial Q} > 0$, $\frac{\partial {V}_{12}}{\partial L}$, $\frac{\partial {V}_{12}}{\partial M} > 0$. Consequently, the probability of enterprises choosing green innovation strategies can be enhanced through the following approaches: (1) The returns on corporate green innovation are increased; (2) Elevating public concern regarding the environmental performance of manufacturing enterprises; (3) Government issuance of green credit and revocation of green certifications; (4) Fostering public supportive motivation and resistive pressure toward green innovation in manufacturing enterprises.

Conclusion 1: (1) When deciding whether to pursue green innovation, enterprises primarily consider whether such innovation can yield sufficient economic returns. Therefore, the enhancement of financial benefits associated with green innovation is identified as a critical factor in incentivizing enterprises to adopt green innovation strategies. (2) Public concern regarding the environmental performance of enterprises can be transformed into market pressure, which compels enterprises to prioritize green innovation. Through the promotion of public environmental awareness via education and campaigns, consumers can be encouraged to favor green and eco-friendly products. This shift in consumer preference drives enterprises to align with market demands by more actively pursuing green innovation strategies. (3) Additionally, corporate behavior can be influenced by governments through financial mechanisms. Green credit, such as low-interest loans provided to enterprises implementing green innovation projects, can be utilized to reduce financing costs and incentivize green investments. Simultaneously, for enterprises that fail to comply with environmental regulations or engage in greenwashing, green certifications can be revoked by the government, thereby increasing their market credibility losses. This approach is effective in discouraging unsustainable practices and encouraging genuine green innovation. (4) At the same time, public supportive motivation can be demonstrated through the purchase of green products and participation in green initiatives. Such positive market feedback can serve as an encouragement for enterprises to continue their green innovation efforts. Conversely, resistive pressure can be applied through consumer boycotts of non-eco-friendly products and public condemnation, forcing enterprises to bear the costs of complaints and compensation. This further compels enterprises, under the risks of market and reputational losses, to reconsider their environmental strategies and lean toward adopting green innovation pathways.

Corollary 2: During the evolutionary process, the probability of enterprises adopting green innovation increases with the enhancement of government regulation, public supervision, and public green awareness.

Proof 2: Upon conducting a stability analysis of corporate strategies, it is found that when $z < \frac{C1-C2+R1-R2-(R1+R2)\alpha -(P+Q)y}{L+M}$ and $y < y\ast$, the condition ${\frac{dF(x)}{dx}|}_{x=0} < 0$ is fulfilled, leading to the ESS of the enterprise being $x=0$. Consequently, as $\alpha$, $y$, and $z$ increase, the stable strategy of the enterprise transitions from $x=0$ (greenwashing behavior) to $x=1$ (green innovation).

Conclusion 2: It is concluded that an increase in the rate of public concern for environmental performance, along with the probability of government regulation and public supervision, facilitates enterprises in adopting green innovation as a stable production strategy. The heightened attention to corporate environmental performance by the public can serve as an incentive for enterprises to place a greater emphasis on green innovation. It is observed that green credit policies exert a positive effect on enhancing the productivity of corporate green innovation, while the revocation of green certifications leads to increased market and operational costs for enterprises, thereby encouraging the adoption of green innovation. strategies Both public support and resistance are identified as significant driving forces behind corporate green innovation. Furthermore, the role of media and non-governmental organizations in amplifying public scrutiny through the exposure of unsustainable corporate practices is recognized, which in turn increases the reputational pressure on enterprises.

Government

It is assumed that the expected return associated with the government’s strict regulatory strategy is denoted as ${E}_{21}$, while the expected return for the lenient regulatory strategy is represented as ${E}_{22}$. The average expected return for the government is calculated as $\overline{{E}_{2}}$.

$${E}_{21}=x({R}_{3}-P+{C}_{3})+zF-{C}_{3}-{C}_{4}$$

(9)

$${E}_{22}=({R}_{3}+{C}_{3}-J)x-Gz-{C}_{3}$$

(10)

$$\overline{{E}_{2}}=y{E}_{21}+(1-y){E}_{22}$$

(11)

The replicator dynamic equation for the government’s behavioral strategy is as follows:

$$F(y)=\frac{dy}{dt}=y({E}_{21}-\overline{{E}_{2}})=y(y-1)[{C}_{4}-(F+G)z+(P-J)x]$$

(12)

The first derivative of $y$ is as follows:

$$\frac{d[F(y)]}{dy}=(2y-1)[{C}_{4}-(F+G)z+(P-J)x]$$

For ease of expression, let:

$$H(x,z)={C}_{4}-(F+G)z+(P-J)x$$

(13)

According to the stability theorem of differential equations, the following condition must be satisfied for the probability of the government’s choice of a strict regulatory strategy to remain in a stable state:

$$\left\{\begin{array}{c}F(y)=0\\ \frac{d[F(y)]}{dy} < 0\end{array}\right.$$

(14)

Since $\frac{\partial H(x,z)}{\partial z}=-(F+G) < 0$, $H(x,z)$ is a decreasing function of z, it follows that:

• (1) When $z=\frac{{C}_{4}+(P-J)x}{F+G}={z}^{\ast }$ and $H(x,z)=0$, the condition $\frac{d[F(y)]}{dy}=0$ is satisfied, and the government cannot determine its stable strategy point.

• (2) When $z < {z}^{\ast }$, $H(x,z) > 0$, and $y=0$ satisfies $\frac{d[F(y)]}{dy} < 0$, $y=0$ represents the government’s ESS. The government tends to prefer lenient regulation over strict regulation.

• (3) When $z > {z}^{\ast }$, $H(x,z) < 0$, and $y=1$ satisfies $\frac{d[F(y)]}{dy} < 0$, $y=1$ represents the government’s ESS. The government tends to prefer strict regulation over lenient regulation.

The evolutionary phases of the government’s strategy are illustrated in Fig. 3:

Fig. 3: The evolutionary phase diagram of government strategies.

Evolutionary phase diagram of government regulatory strategies showing: a policy equilibrium at critical threshold y = y*; b Lenient regulation dominance when yy*. Volumes V₂₁ and V₂₂ represent the probabilities of stable adoption of lenient and strict regulatory strategies, respectively.

In Fig. 3, the probability of the government stably adopting a lenient regulatory strategy is represented by volume ${V}_{21}$, while the probability of stably adopting a strict regulatory strategy is represented by the volume ${V}_{22}$. The calculation formula is provided as follows:

$${V}_{21}={\int }_{0}^{1}{\int }_{0}^{\frac{{C}_{4}}{J-P}}\frac{{C}_{4}+(P-J)x}{F+G}dxdy(ESS:y\to 0)=\frac{{{C}_{4}}^{2}}{2(F+G)(J-P)}$$

(15)

$${V}_{22}=1-{V}_{21}$$

(16)

Corollary 3: A negative correlation is observed between strict regulation by the government and regulatory costs ${C}_{4}$, as well as the issuance of green credit $P$. In contrast to lenient regulation, the benefits of strict regulation are positively correlated with the reputational rewards $F$ received from the public, the losses $G$ incurred due to public reporting, and the investment in corporate green innovation, along with the losses $J$ resulting from the suppression of innovation under lenient regulation.

Proof 3: By calculating the first-order partial derivatives of each element in the equation, namely $\frac{\partial {V}_{22}}{{C}_{4}} < 0$, $\frac{\partial {V}_{22}}{\partial P} < 0$, $\frac{\partial {V}_{22}}{\partial F} > 0$, $\frac{\partial {V}_{22}}{\partial G} > 0$, and $\frac{\partial {V}_{22}}{\partial J} > 0$, the following measures can be identified to increase the probability of strict regulation being adopted by the government: (1) Reducing the costs associated with strict regulation; (2) Decreasing the allocation of green credit; (3) Enhancing reputational rewards from the public for government actions; (4) Increasing the losses incurred by the government due to public reporting; (5) Raising the investment in corporate green innovation and the losses from innovation suppression under lenient regulation.

Conclusion 3: (1) A strict regulatory strategy can be adopted by the government through the simplification of regulatory procedures, improvement of regulatory efficiency, or utilization of technological means to reduce resource consumption in regulation (2). Additionally, the allocation of green credit can be reduced, which may diminish the financial incentives for corporate green innovation, thereby increasing the necessity for strict regulatory measures to be implemented by the government to ensure environmentally responsible corporate behavior (3). At the same time, enhancing positive public evaluation of the government’s strict regulatory actions, such as social recognition, favorable media coverage, or increasing the significant losses incurred by the government due to public reporting of non-compliant behavior, will prompt the government to strengthen regulation to mitigate losses arising from inadequate supervision (4). Finally, when weighing the stringency of regulation, the government considers its impact on corporate innovation potential. If lenient regulation leads to suppressed investment and innovation in green initiatives, stricter regulation may be opted for by the government to avoid such losses.

Corollary 4: During the evolutionary process, when the benefits derived from issuing green credit to green innovation enterprises under strict government regulation exceed the losses caused by investment and innovation suppression under lenient regulation, the probability of strict government regulation is increased as the proportion of manufacturing enterprises choosing greenwashing behavior and public participation in supervision rises. Conversely, the probability of strict government regulation is also increased as the proportion of manufacturing enterprises choosing green innovation and public participation in supervision grows.

Proof 4: Based on the stability strategy analysis of the government, when $P > J$, $x > \frac{C4-(F+G)z}{J-P}$, $z < {z}^{\ast }$ and the conditions $H(x,z) > 0$ and ${\frac{dF(y)}{dy}|}_{y=0} < 0$ are satisfied, the ESS of the enterprise is $y=0$. Therefore, as $x$ decreases and $z$ increases, the government’s stable strategy shifts from $y=0$ (lenient regulation) to $y=1$ (strict regulation). When $P < J$, $x < \frac{C4-(F+G)z}{J-P}$, $z < {z}^{\ast }$, and the conditions $H(x,z) > 0$ and ${\frac{dF(y)}{dy}|}_{y=0} < 0$ are satisfied, the ESS of the enterprise is determined to be $y=0$. Consequently, as $x$ and $z$ increase, the government’s stable strategy transitions from $y=0$ (lenient regulation) to $y=1$ (strict regulation).

Conclusion 4: To encourage the implementation of strict regulation by the government on corporate green production practices, public intervention is identified as an effective approach.

When $P > J$, the probability of enterprises choosing green innovation strategies is reduced, and the probability of public participation in supervision is increased, which is conducive to the adoption of strict regulation as a stable regulatory strategy by the government. This is because a lower probability of enterprises choosing green innovation strategies may indicate insufficient incentives or high costs, making enterprises less willing to proactively adopt green innovation. Meanwhile, increased public participation in supervision can provide more information and feedback, enabling the government to implement regulation more effectively. Both factors suggest a need for stricter regulation, and the government may perceive this trend as necessary, thus opting for strict regulation as a stable strategy.

When $P < J$, an increase in the probability of enterprises choosing green innovation strategies and public participation in supervision is conducive to the adoption of strict regulation as a stable regulatory strategy by the government. When both corporate and public behaviors are inclined toward green innovation, this trend is perceived by the government as sustainable. Therefore, strict regulation is more likely to be chosen by the government as a stable strategy to maintain this positive trend and prevent enterprises from resorting to greenwashing practices.

Public

It is assumed that the expected return associated with the public’s participation in the supervision strategy is denoted as ${E}_{31}$, while the expected return for the non-participation supervision strategy is represented as ${E}_{32}$. The average expected return for the public is calculated as $\overline{{E}_{3}}$.

$${E}_{31}=({U}_{1}-{U}_{2}-L-M)x+(H+I)y+M-{U}_{2}-{C}_{5}-I$$

(17)

$${E}_{32}=({U}_{1}-{U}_{2})x+{U}_{2}$$

(18)

$$\overline{{E}_{3}}=z{E}_{31}+(1-z){E}_{32}$$

(19)

The replicator dynamics equation of the public behavior strategy is as follows:

$$F(z)=\frac{dz}{dt}=x({E}_{31}-\overline{{E}_{3}})=z(z-1)[{C}_{5}+I-M-(H+I)y+(L+M)x]$$

(20)

The first derivative of $z$ is as follows:

$$\frac{d[F(z)]}{dz}=(2z-1)[{C}_{5}+I-M-(H+I)y+(L+M)x]$$

For ease of expression, let:

$$I(y,z)={C}_{5}+I-M-(H+I)y+(L+M)x$$

(21)

In accordance with the stability theorem of differential equations, the probability that the public’s choice to participate in the supervision strategy remains stable must fulfill the following condition:

$$\left\{\begin{array}{c}F(z)=0\\ \frac{d[F(z)]}{dz} < 0\end{array}\right.$$

(22)

Since $\frac{\partial I(y,x)}{\partial y}=-(H+I) < 0$, $I(y,x)$ is a decreasing function of $y$, it follows that:

• (1) When $y=\frac{{C}_{5}+I-M+(L+M)x}{H+I}={y}^{\ast \ast }$, $I(y,x)=0$ and the condition $\frac{d[F(z)]}{dz}=0$ is satisfied, the public cannot determine its stable strategy point.

• (2) When $y < {y}^{\ast \ast }$, $I(y,x) > 0$ and $z=0$ satisfies $\frac{d[F(z)]}{dz} < 0$, $z=0$ represents the public’s ESS. That is, the public tends to refrain from participating in the supervision of enterprises and the government.

• (3) When $y > {y}^{\ast \ast }$, $I(y,x) < 0$ and $z=1$ satisfies $\frac{d[F(z)]}{dz} < 0$, $z=1$ represents the public’s ESS. That is, the public tends to participate in the supervision of enterprises and the government.

The evolutionary phases of the public’s strategy are illustrated in Fig. 4.

Fig. 4: The evolutionary phase diagram of public strategies.

Evolutionary phase diagram of public supervision strategies showing: a participation equilibrium at critical threshold z = z*; b non-supervision dominance when zz*. Volumes V₃₁ and V₃₂ represent the probabilities of stable adoption of non-supervision and active supervision strategies, respectively.

In Fig. 4, the probability of the public stably adopting a non-supervision strategy is represented by volume ${V}_{31}$, while the probability of stably adopting a supervision strategy is represented by the volume ${V}_{32}$. The calculation formula is as follows:

$${V}_{31}={\int }_{0}^{1}{\int }_{0}^{1}\frac{{C}_{5}+I-M+(L+M)x}{H+I}dxdz(ESS:z\to 0)=\frac{2{C}_{5}+3I-M}{2(H+I)}$$

(23)

$${V}_{32}=1-{V}_{31}$$

(24)

Corollary 5: A negative correlation is identified between public participation in supervision and the supervision costs denoted as ${C}_{5}$. Compared to non-participation, the benefits of public participation in supervision are positively correlated with the resistive pressure $M$ exerted against the greenwashing behavior of manufacturing enterprises and with the protection of public rights $H$ within the framework of strict government regulation.

Proof 5: By calculating the first-order partial derivatives of each element in ${V}_{32}$, namely $\frac{\partial {V}_{32}}{\partial {C}_{5}} < 0$, $\frac{\partial {V}_{32}}{\partial I} < 0$, $\frac{\partial {V}_{32}}{\partial M} > 0$, and $\frac{\partial {V}_{32}}{\partial H} > 0$, the following measures can be identified to increase the probability of public participation in supervision: (1) Reducing the costs associated with public participation in supervision; (2) Decreasing the loss of public rights caused by lenient government regulation; (3) Increasing the resistive pressure against greenwashing behavior by manufacturing enterprises, for instance, through compensation amounts paid by enterprises in response to public complaints; (4) Enhancing the protection of public rights under strict government regulation.

Corollary 6: Throughout the evolutionary process, the probability of public participation in supervision increases as the probability of corporate green innovation decreases and the probability of strict government regulation increases.

Proof 6: Based on the stability strategy analysis conducted for the public sector, it is demonstrated that when conditions $x > \frac{M-C5-I+(H+I)y}{L+M}$, $y < {y}^{\ast \ast }$, and $I(y,x) > 0$ are met, condition ${\frac{dF(z)}{dz}|}_{z=0} < 0$ is satisfied, leading to the establishment of $z=0$ as the ESS for the enterprise. Consequently, a decrease in $x$ coupled with an increase in $y$ is observed to result in a transition of the public’s stable strategy from $z=0$ (non-participation in supervision) to $z=1$ (participation in supervision).

Conclusion 6: Reducing the probability of enterprises choosing green innovation strategies and increasing the probability of strict government regulation are conducive to the public opting for participation in supervision as a stable strategy. Due to insufficient efforts by enterprises in environmental protection and sustainable development, the government adopts strict regulation to constrain corporate behavior, compelling enterprises to take necessary environmental measures. At this point, the public may recognize that their role has become more critical and, therefore, may choose to participate in supervision to supplement the shortcomings of government regulation, ensuring transparency and accountability in corporate behavior.

Stability analysis of system equilibrium points

For the achievement of an ESS, continuous strategy adjustments are made by participants in response to existing interest considerations. In order to identify the equilibrium points within the evolutionary game framework, the joint replicator dynamic equations are established, encompassing the tripartite system consisting of enterprises, government entities, and the public, as presented below:

$$\left\{\begin{array}{l}F(x)=\frac{dx}{dt}=-x(x-1)[{C}_{2}-{C}_{1}+{R}_{1}-{R}_{2}+({R}_{1}+{R}_{2})\alpha +(P+Q)y+(L+M)z]\\ F(y)=\frac{dy}{dt}=y(y-1)[{C}_{4}-(F+G)z+(P-J)x]\\ F(z)=\frac{dz}{dt}=z(z-1)[{C}_{5}+I-M-(H+I)y+(L+M)x]\end{array}\right.$$

(25)

Given the assumption in mixed strategy game models that players make strategic choices only with knowledge of the strategies of others, which does not align with the real-world scenario of information asymmetry, it follows that mixed strategy equilibria are not evolutionarily stable in asymmetric dynamic games (Selten, 1980; Shan et al., 2021; He et al., 2023). Consequently, the analysis is confined to the pure strategy equilibrium points within the evolutionary game system. Let $F(x)=0$, $F(y)=0$ and $F(z)=0$. Consequently, eight pure strategy equilibrium points are identified in the system, which are formally denoted as ${E}_{1}(0,0,0)$, ${E}_{2}(0,1,0)$, ${E}_{3}(0,0,1)$, ${E}_{4}(0,1,1)$, ${E}_{5}(1,0,0)$, ${E}_{6}(1,1,0)$, ${E}_{7}(1,0,1)$, and ${E}_{8}(1,1,1)$.

To analyze the stability of these equilibrium points, the Jacobian matrix of the tripartite evolutionary game system is first constructed as follows:

$$\begin{array}{ll}J=\left[\begin{array}{ll}\partial F(x)/\partial x \qquad \partial F(x)/\partial y \qquad \partial F(x)/\partial z\\ \partial F(y)/\partial x \qquad \partial F(y)/\partial y \qquad \partial F(y)/\partial z\\ \partial F(z)/\partial x \qquad \partial F(z)/\partial y \qquad \partial F(z)/\partial z\end{array}\right]\\ =\left[\begin{array}{ll}(1-2x)(\begin{array}{ll}{C}_{2}-{C}_{1}+{R}_{1}-{R}_{2}+\alpha {R}_{1}+\\ \alpha {R}_{2}+Lz+Mz+Py+Qy\end{array}) \qquad (x-{x}^{2})(P+Q) \qquad (x-{x}^{2})(L+M)\\ ({y}^{2}-y)(P-J) \qquad (2y-1)(\begin{array}{ll}{C}_{4}-Fz+Gz\\ +Pz-Jx\end{array}) \qquad (y-{y}^{2})(F+G)\\ ({z}^{2}-z)(L+M) \qquad (z-{z}^{2})(H+I) \qquad (2z-1)(\begin{array}{ll}{C}_{5}+I-M-Hy\\ -Iy+Lx+Mx\end{array})\end{array}\right]\end{array}$$

According to Lyapunov’s first method (indirect method), the stability characteristics of equilibrium points can be determined as follows: an equilibrium point is classified as evolutionarily stable when all three eigenvalues are found to be negative; it is identified as unstable when all three eigenvalues are observed to be positive; and it is characterized as a saddle point when one or two eigenvalues are determined to be positive (Lyapunov, 1992). The stability of each equilibrium point is shown in Table 3.

Table 3 Stability Analysis of the Equilibrium Point.

This paper selects He Steel Group Co., Ltd. as a research sample for investigation. Established in 1997, He Steel Group Co., Ltd.‘s main business includes the processing of ferrous metals, with products spanning various series such as bars, wires, profiles, coated sheets, and wide and heavy plates. The company is involved in the production of materials for automotive, petroleum, railway, and other construction industries, making it a typical heavy polluting enterprise. Its subsidiary, Tangshan Medium and Thick Plate Company, has been involved in numerous environmental violations, and in 2021, it was found to have falsified its online monitoring data in an attempt to evade regulatory emissions of air pollutants. Following reports from concerned individuals and media exposure, the government-imposed penalties including business suspension for rectification and a downgrade of its ecological environment credit rating. Subsequently, the company actively responded to environmental protection policies, continuously implemented the concept of green development, and advanced in ultra-low emission technology innovations, significantly improving its performance in green governance and environmental investment. Therefore, this study integrates the case of He Steel Group Co., Ltd. and its actual environmental performance, and based on the parameter assignment methods from relevant literature (Chen and Hu, 2018; He and Sun, 2022; Liu et al., 2024), discussions were held with experts in related fields. It was concluded that the following parameter assignments are consistent with the model settings and are representative: ${R}_{1}=200$, ${R}_{2}=300$, ${C}_{1}=100$, ${C}_{2}=40$, ${R}_{3}=140$, ${C}_{3}=100$, ${U}_{1}=100$, ${U}_{2}=80$, ${C}_{4}=100$, $P=20$, $Q=20$, $F=40$, $G=40$, $J=120$, ${C}_{5}=30$, $L=50$, $M=50$, $H=60$, $I=10$, and $\alpha =0.2$.

The evolutionary trajectory is illustrated in Fig. 5a, where oscillatory patterns are observed in variables $x$ and $z$, suggesting the non-existence of a stable strategy within the evolutionary game framework. In this context, dynamic mixed strategies are adopted by entities $x$ and $z$. For enhanced visualization of the evolutionary strategies employed by each entity, a three-dimensional dynamic evolutionary trend simulation was implemented, with the corresponding results being presented in Fig. 5b. Through this simulation, it is demonstrated that the system fails to achieve a stable equilibrium state, instead exhibiting infinite cyclical behavior around the central point $({x}_{0},0,{z}_{0})$. These simulation outcomes are found to be consistent with the previously established theoretical inferences. From a managerial standpoint, the strategy associated with the equilibrium point $(1,1,1)$ is regarded as optimal. This strategic configuration represents a collaborative governance paradigm wherein green innovation is pursued by enterprises, stringent regulations are enforced by governmental bodies, and active monitoring of greenwashing behaviors is undertaken by the public. Accordingly, particular attention is directed toward the regulatory mechanisms employed by the government, specifically focusing on the dynamic allocation of green credit and the potential revocation of green certifications, with the ultimate objective of establishing an optimal collaborative governance framework.

Fig. 5: Static green credit allocation and dynamic evolution of static green certification revocation.

a is the strategy evolution path diagram of each stakeholder under static green credit allocation and static green certification revocation, while b is the three-dimensional dynamic evolution trend simulation of the stakeholders’ evolving strategies.

Model analysis of dynamic allocation of green credit and revocation of green certifications

This section explores dynamic regulatory mechanisms and their impact on stakeholders’ behavior. We analyze three scenarios combining dynamic and static elements of green credit allocation and certification revocation. The goal is to demonstrate how dynamic strategies can more effectively influence greenwashing and green innovation compared to static ones.

Dynamic allocation of green credit and static revocation of green certifications

In the context of dynamic green credit allocation, a linear transformation mechanism is implemented for green credit quantification. The green credit, which was previously maintained as a fixed constant $P$, is reformulated as a linear function ${P}^{\ast }=axP$ (Chang et al., 2017), while the value associated with green certification revocation, $Q$, is maintained at its original level. Within this framework, $a$ is defined as the linear dynamic reward coefficient, $x$ is established as the indicator of positive behavioral performance by the entity, and $P$ is characterized as the maximum attainable green credit quota for enterprises, typically satisfying condition $ax > 1$. Based on these parameter definitions, the replicator dynamic equations for the tripartite evolutionary game system are derived as follows:

$$\left\{\begin{array}{l}F(x)=\frac{dx}{dt}=-x(x-1)[{C}_{2}-{C}_{1}+{R}_{1}-{R}_{2}+({R}_{1}+{R}_{2})\alpha +(axP+Q)y+(L+M)z]\\ F(y)=\frac{dy}{dt}=y(y-1)[{C}_{4}-(F+G)z+(axP-J)x]\\ F(z)=\frac{dz}{dt}=z(z-1)[{C}_{5}+I-M-(H+I)y+(L+M)x]\end{array}\right.$$

Within the system, there exist eight pure strategy equilibrium points: ${E}_{1}(0,0,0)$, ${E}_{2}(0,1,0)$, ${E}_{3}(0,0,1)$, ${E}_{4}(0,1,1)$, ${E}_{5}(1,0,0)$, ${E}_{6}(1,1,0)$, ${E}_{7}(1,0,1)$, and ${E}_{8}(1,1,1)$. The local equilibrium points are analyzed using the Jacobian matrix, and it is determined that the system lacks a stable equilibrium point. The simulation results are depicted in Fig. 6a and b. As illustrated in Fig. 6a, compared to the static allocation of green credit and the static revocation of green certifications, fluctuations in variables $x$ and $z$ persist under the mechanism of dynamic allocation of green credit and static revocation of green certifications. However, it is observed that the amplitude of these fluctuations is reduced while their frequency increases. Figure 6b reveals that under the dynamic allocation of green credit, the evolutionary trend diagram exhibits more complex trajectories. This suggests that the dynamic allocation of green credit may destabilize the system, leading to frequent shifts between different strategies.

Fig. 6: Dynamic allocation of green credit and dynamic evolution of static revocation of green certification.

a is the strategy evolution path diagram of each stakeholder under the mechanism of dynamic green credit allocation and static green certification revocation, while b is the three-dimensional dynamic evolution trend simulation of the stakeholders’ evolving strategies under this mechanism.

Static allocation of green credit and dynamic revocation of green certifications

To better mitigate greenwashing behaviors and drive corporate green innovation, a linear dynamic revocation scheme for green certifications is considered. Given that green certifications are positively correlated with corporate green innovation (Chen et al., 2024), and uncertified eco-labels are often associated with greenwashing (Damberg et al., 2024). Consequently, it is hypothesized that the revocation of green certifications may prompt greenwashing behaviors among enterprises. The dynamic revocation of green certifications is defined as ${Q}^{\ast }=b(1-x)Q$, while the value of allocated green credit $P$ remains constant. Here, $b$ represents the linear dynamic penalty coefficient, and $Q$ denotes the maximum loss incurred by an enterprise due to the revocation of green certifications, typically satisfying $b(1-x) > 1$. The replicator dynamic equations for the tripartite evolutionary game system are presented as follows:

$$\left\{\begin{array}{l}F(x)=\frac{dx}{dt}=-x(x-1)[{C}_{2}-{C}_{1}+{R}_{1}-{R}_{2}+({R}_{1}+{R}_{2})\alpha +Py+b(1-x)Qy+(L+M)z]\\ F(y)=\frac{dy}{dt}=y(y-1)[{C}_{4}-(F+G)z+(P-J)x]\\ F(z)=\frac{dz}{dt}=z(z-1)[{C}_{5}+I-M-(H+I)y+(L+M)x]\end{array}\right.$$

Upon analysis of the system’s equilibrium points, it is observed that the system consistently converges to the unique stable point ${E}_{8}(1,1,1)$. This convergence indicates that green innovation is adopted by enterprises, strict regulation is enforced by the government, and active participation in supervision is demonstrated by the public. Consequently, an ideal stable convergence state is achieved by the system, leading to the establishment of a multi-party collaborative governance system aimed at addressing greenwashing behaviors, as depicted in Fig. 7.

Fig. 7: Dynamic evolution of static green credit allocation and dynamic green certification revocation.

a illustrates the strategy evolution paths of all stakeholders under the mechanism of static green credit allocation and dynamic green certification revocation, while b presents a three-dimensional simulation of the dynamic evolution trends of the stakeholders’ evolving strategies under this mechanism.

Dynamic allocation of green credit and dynamic revocation of green certifications

In addressing the dynamic allocation of green credit and the dynamic revocation of green certifications, a strategy that integrates both linear dynamic allocation of green credit and dynamic revocation of green certifications is employed. It is postulated that the government’s allocation of green credit to corporate green innovation initiatives and the revocation of green certifications are influenced by the likelihood of corporate greenwashing behaviors. Within the model, the dynamic allocation of green credit function ${P}^{\ast }=axP$ and the dynamic revocation of green certifications function ${P}^{\ast }=axP$ are introduced. The replicator dynamic equations for the tripartite evolutionary game system are presented as follows:

$$\left\{\begin{array}{l}F(x)=\frac{dx}{dt}=-x(x-1)\left[{C}_{2}-{C}_{1}+{R}_{1}-{R}_{2}+({R}_{1}+{R}_{2})\right.\\\left.\alpha +axPy+b(1-x)Qy+(L+M)z\right]\\ F(y)=\frac{dy}{dt}=y(y-1)[{C}_{4}-(F+G)z+(axP-J)x]\\ F(z)=\frac{dz}{dt}=z(z-1)[{C}_{5}+I-M-(H+I)y+(L+M)x]\end{array}\right.$$

Within the system, there exist eight pure strategy equilibrium points: ${E}_{1}(0,0,0)$, ${E}_{2}(0,1,0)$, ${E}_{3}(0,0,1)$, ${E}_{4}(0,1,1)$, ${E}_{5}(1,0,0)$, ${E}_{6}(1,1,0)$, ${E}_{7}(1,0,1)$, and ${E}_{8}(1,1,1)$. Utilizing the Jacobian matrix to analyze the local equilibrium points, it is established that the system continues to lack a stable equilibrium point. The simulation outcomes are depicted in Fig. 8a and b. Upon comparison of Fig. 8a with 6a, it is observed that, in contrast to the strategy of static green credit disbursement and static green certification revocation, the strategy involving dynamic green credit disbursement and dynamic green certification revocation continues to demonstrate significant periodic fluctuations in $x$ and $z$. Although the intensity of these fluctuations is somewhat diminished while their frequency is increased. Further analysis of Fig. 8b reveals that under the strategy of dynamic green credit disbursement and dynamic green certification revocation, the evolutionary trajectory diagram displays a more complex and intricate form.

Fig. 8: Dynamic evolution of dynamic green credit allocation and dynamic green certification revocation.

a depicts the strategy evolution paths of all stakeholders under the mechanism of dynamic green credit allocation and dynamic green certification revocation, while b presents a three-dimensional simulation of the dynamic evolution trends of the stakeholders’ evolving strategies under this mechanism.

Additionally, to facilitate a comparison of convergence behavior between scenarios involving dynamic allocation of green credit paired with static revocation of green certifications and those involving dynamic allocation of green credit alongside dynamic revocation of green certifications, initial values were assigned to three distinct categories ($x=y=z=0.3$, $x=y=z=0.5$, $x=y=z=0.7$). This setup was designed to simulate the effects of variations in the initial values of $x$, $y$, and $z$ on the evolution of $x$, with the outcomes depicted in Fig. 9a and b. From a comprehensive viewpoint, within the model that incorporates both dynamic allocation of green credit and dynamic revocation of green certifications, the initial conditions are observed to have a more significant impact on the final convergence state. Regarding the rate of convergence, the introduction of dynamic revocation of green certifications adds a layer of complexity, thereby extending the duration required for the system to attain a stable state. As a result, the evolutionary model that integrates dynamic allocation of green credit with dynamic revocation of green certifications demonstrates a slower convergence speed.

Fig. 9: Convergence speed of evolution.

This is used to compare the convergence of strategies of all stakeholders under the mechanisms of dynamic green credit allocation and static green certification revocation a versus dynamic green credit allocation and dynamic green certification revocation b.

Sensitivity analysis under static green credit disbursement and dynamic green certification revocation

A sensitivity analysis of the variables P and Q was conducted to compare their impact on the strategic decisions of manufacturing enterprises under four distinct mechanisms, with the outcomes depicted in the accompanying figures. Upon meticulous examination of Figure a–d, it is observed that fluctuations in the values of P and Q significantly influence the strategic choices of manufacturing enterprises. In Figure a, b, and d, a label value approaching 0 indicates a greater propensity for manufacturing enterprises to engage in greenwashing behaviors, whereas a label value nearing 1 suggests a preference for the adoption of green innovation strategies. However, despite variations in P and Q values, the regions where enterprises opt for green innovation strategies remain exceedingly limited, to the point of being negligible. This suggests that under these combined mechanisms, there is a lack of sufficient incentives for enterprises to undertake green innovation. In marked contrast, Figure c illustrates the impact of changes in P and Q values on the strategic choices of manufacturing enterprises under the combined mechanism of static green credit issuance and dynamic revocation of green certification. The analysis reveals a notable increase in the likelihood of enterprises adopting green innovation strategies as the combined sum of P and Q values exceeds 5. This trend becomes particularly pronounced when the sum of P and Q values surpasses 10, at which point the probability of enterprises choosing green innovation approaches or even reaches near certainty. This finding indicates that the combined strategy of static green credit issuance and dynamic revocation of green certification can effectively incentivize enterprises to engage in green innovation and reduce instances of greenwashing. The significance of investigating the combined strategy of static green credit issuance and dynamic revocation of green certification is thus evident. This combination provides enterprises with immediate economic incentives while increasing compliance pressure through the dynamic revocation of green certification. It not only fosters green innovation among enterprises but also enhances the effectiveness of government regulation, promoting a green transition across society. Through such a strategy, governments can more effectively guide corporate behavior and achieve the objectives of environmental policies Fig. 10.

Fig. 10: Impact of various reward and punishment mechanisms on the strategic choices of manufacturing enterprises in green innovation.

Among them a represent static reward and static punishment mechanism, b represents the dynamic reward and static punishment mechanism, c represents the static reward and dynamic punishment mechanism, and d represents the dynamic reward and dynamic punishment mechanism. The figures demonstrate the impact of P and Q values on green innovation strategies among manufacturing enterprises under various regulatory frameworks. Notably, when P and Q values exceed 5 under the static green credit and dynamic certification revocation mechanism, as shown in c, the probability of adopting green innovation strategies significantly rises. This underscores the pivotal role of incentive and penalty mechanisms in promoting genuine green innovation and curbing greenwashing.

Based on the insights gained, the current study proceeds to analyze the convergence effects of parameter variations within the evolutionary model under the static green credit issuance and dynamic revocation of green certification mechanism. The parameters analyzed include: the benefits derived from green innovation by manufacturing enterprises (${R}_{1}$), the costs associated with green innovation for these enterprises (${C}_{1}$), the costs incurred by the government for strict regulation (${C}_{4}$), the costs related to public supervision (${C}_{5}$), the benefits received by green innovation enterprises from government green credit disbursement ($P$), the losses suffered by greenwashing enterprises due to government green certification revocation ($Q$), the supportive motivation of the public towards green innovation by manufacturing enterprises ($L$), the resistive pressure exerted by the public against greenwashing by manufacturing enterprises ($M$), the reputational rewards gained by the government for strict regulation from the public ($F$), the losses experienced by the government from public reporting under lax regulation ($G$), the protection of public rights ensured by the government under strict regulation ($H$), the loss of public rights under lax regulation by the government ($I$), and the level of public concern for the degree of greenness of manufacturing enterprises ($\alpha$).

The impact of variations in revenue generated from corporate green innovation was investigated by assigning the values ${R}_{1}$ = 200, 250, and 300, with the simulation results illustrated in Fig. 11. As demonstrated in Fig. 11, an increase in revenue derived from corporate green innovation was observed to accelerate the rate of evolutionary stability. With the rise in ${R}_{1}$, the probability of enterprises adopting green innovation production strategies was found to increase, whereas the probability of the government implementing strict regulatory strategies was noted to decrease. This suggests that when corporate green innovation revenue reaches a sufficiently high level, the government may perceive that enterprises possess greater intrinsic motivation to voluntarily pursue green innovation, thereby leading to a reduction in governmental intervention. Therefore, it is recommended that appropriate incentives, such as tax reductions, low-interest loans, and innovation funds, be provided to small and medium-sized enterprises (SMEs), startups, and technology-intensive enterprises with high innovation potential, to encourage their investment in green technologies and sustainable development practices.

Fig. 11

The impact of the change in the payoff R₁ obtained by manufacturing enterprises from green innovation under the mechanism of static green credit allocation and dynamic green certification revocation on the strategy evolution of the participants.

To analyze the effect of variations in the costs associated with green innovation for manufacturing enterprises, the costs of stringent government regulation, and the costs of public supervision, the values ${C}_{1}$ = 100, 120, 140, ${C}_{4}$ = 100, 120, 140, and ${C}_{5}$ = 30, 60, 90 were assigned. The simulation results are illustrated in Fig. 12a−c. As shown in Fig. 12a, an increase in the cost of green innovation for manufacturing enterprises was found to accelerate the rate of stable evolution. With the rise in ${C}_{1}$, the probability of enterprises adopting green innovation production strategies was observed to decrease, whereas the probability of the government implementing strict regulatory strategies was seen to increase. From Fig. 12b and c, it can be observed that increasing the costs of strict government regulation and public supervision also accelerates the speed of stable evolution. As ${C}_{4}$ and ${C}_{5}$ increase, the probability of enterprises adopting green innovation production strategies rises, while the probabilities of the government adopting strict regulatory strategies and the public choosing to participate in supervision decrease.

Fig. 12: Cost factors influencing strategic evolution under dynamic regulatory mechanism.

Under the mechanism of static green credit allocation and dynamic green certification revocation, the impacts of the green innovation cost C1 of manufacturing enterprises a, the strict regulatory cost C4 of the government b, and the public supervision cost C5 c on the strategy evolution of the participants.

As illustrated in Fig. 12b and c, the influence of variations in the cost of strict government regulation is observed to be more pronounced than that of changes in the cost of public participation in supervision. Furthermore, when compared to the cost of green innovation (Fig. 12a), the effect of changes in the revenue generated from corporate green innovation is found to be more significant (Fig. 11). This suggests that both costs and revenues are critical factors in the decision-making process for enterprises adopting green innovation strategies, with revenues playing a more pivotal role. Consequently, it is essential for policymakers to design policies flexibly rather than relying solely on mandatory regulation. By balancing the costs and incentives for enterprises, an increase in the revenue derived from corporate green innovation can be promoted, thereby enhancing the likelihood of enterprises adopting green innovation production strategies.

To examine the influence of green credit allocation to enterprises engaged in green innovation and the repercussions of green certification revocation for those involved in greenwashing behaviors, values of P = 20, 40, 60 and Q = 20, 70, 120 were designated, with the simulation outcomes presented in Fig. 13. As evidenced in Fig. 13a, the revenue derived from the government’s allocation of green credit to green innovation enterprises is found to expedite the pace of stable evolution. With the increment of P, the probability of manufacturing enterprises adopting green innovation production strategies rises, while the probability of the government choosing strict regulation decreases. This suggests that the government’s capacity to sway enterprises towards green innovation production strategies can be achieved through the modulation of green credit benefits.

Fig. 13: Payoff -loss effects on strategic evolution under dynamic regulation.

Under the mechanism of static green credit allocation and dynamic green certification revocation, the impacts of the change in payoff P for the government from allocating green credit to green innovation enterprises a and the change in loss Q for enterprises engaging in greenwashing due to the revocation of green certification by the government b on the strategy evolution of the participants.

From Fig. 13b, it can be observed that the losses incurred by enterprises due to the government’s revocation of green certifications for greenwashing behaviors also accelerate the speed of stable evolution. As Q increases, there is a rise in the probability of manufacturing enterprises adopting green innovation production strategies, while a decrease in the probability of the government choosing strict regulation is noted. This indicates that the government can exert influence over enterprises’ green innovation production strategies through the revocation of green certifications.

In comparison, it is suggested that the impact of the government’s incentive measures, such as the allocation of green credit, is more pronounced than that of the punitive measures involving the revocation of green certifications.

The impact of public supportive motivation for green innovation in manufacturing enterprises and public resistance pressure against greenwashing behaviors was analyzed with assigned values of L = 50, 100, 150 and M = 50, 100, 150, and the simulation outcomes are presented in Fig. 14. As shown in Fig. 14a, as L increases, the probability of manufacturing enterprises adopting green innovation production strategies rises, while a decrease in the probability of the public choosing to participate in supervision is observed. When L reaches a certain threshold, the public’s strategic evolution no longer trends toward participation in supervision. Figure 14b illustrates that as M increases, the probability of manufacturing enterprises adopting green innovation production strategies also increases, while the probability of the public opting to participate in supervision decreases. In comparison, the impact of public supportive motivation for green innovation is more significant, indicating that bolstering public support for green innovation can act as an incentive mechanism to motivate enterprises to invest in green innovation. Consequently, it is advisable for enterprises to utilize media and social platforms for publicity, as well as offline promotional activities, to enhance public support.

Fig. 14: Dual effects of public engagement on strategy evolution.

Under the mechanism of static green credit allocation and dynamic green certification revocation, the impacts of the change in supportive motivation L from the public towards green innovation of manufacturing enterprises a and the change in resistive pressure M from the public against greenwashing behavior of manufacturing enterprises b on the strategy evolution of the participants.

To analyze the impact of public reputational rewards for strict government regulation and the losses incurred by public reporting of lax government regulation, the values F = 40, 60, 80 and G = 40, 60, 80 were assigned, and the simulation results are illustrated in Fig. 15. As shown in Fig. 15a and b, increasing the public reputational rewards for strict government regulation and the losses resulting from public reporting of lax government regulation can accelerate the speed of stable evolution. As F and G increase, the probability of the public choosing to participate in supervision is reduced, while the probability of the government opting for strict regulation is increased. However, overall, the significance of these two factors is found to be relatively limited, and their effects are nearly identical.

Fig. 15: Symmetric effects of public reputational incentives on strategy evolution.

Under the mechanism of static green credit allocation and dynamic green certification revocation, the impacts of the change in reputational reward F from the public for the government's strict regulation (a) and the change in loss G caused by the public's reporting of the government's lenient regulation (b) on the strategy evolution of the participants.

To assess the impact of the protection of public rights under strict government regulation and the loss of public rights under lax government regulation, values of H were set to 60, 110, and 160, while values of I were assigned to 10, 60, and 110. The simulation results are shown in Fig. 16. As can be seen from Fig. 16a and b, the enhancement of public rights protection through strict government regulation and the mitigation of public rights loss through lax regulation can expedite the rate of stable evolution. With the increase of $H$ and $I$, there is noted a decrease in the probability of the public opting to participate in supervision, alongside an increase in the probability of the government favoring strict regulation. In comparison, the effect of $I$ is more pronounced than that of $H$. Nonetheless, the overall influence of both is deemed not to be substantial.

Fig. 16: Regulatory enforcement effects on strategic evolution.

Under the mechanism of static green credit allocation and dynamic green certification revocation, the impacts of the change H in the protection of public interests due to the government's strict regulation a and the change I in the loss of public interests due to the government's lenient regulation b on the strategy evolution of the participants.

To assess the impact of fluctuations in the public’s concern regarding the environmental performance of manufacturing enterprises, the parameter $\alpha$ was assigned values of 0.2, 0.3, and 0.4. The resulting simulation outcomes are depicted in Fig. 17. According to Fig. 17, as $\alpha$ increases, the probability of manufacturing enterprises choosing green innovation production strategies is enhanced. This indicates that heightened public concern for environmentally friendly production methods elevates the external pressures and expectations placed upon enterprises to partake in green innovation activities. The government can foster the adoption of green consumption concepts by implementing educational and promotional campaigns and by bolstering the disclosure of information, thereby motivating consumers to opt for green products and escalating public concern for the environmental performance of enterprises.

Fig. 17

Under the mechanism of static green credit allocation and dynamic green certification revocation, the impact of the change in the public’s concern α about the greenness of manufacturing enterprises on the strategy evolution of the participants.

Drawing from the sensitivity analysis of the aforementioned parameters, it becomes apparent that the influence of stringent government regulation is more significant than that of public supervision, and that both can shape corporate behavior. Consequently, when addressing greenwashing and promoting green innovation, it is imperative for the government to consider both economic and social incentives, and to accord greater weight to the role and perspectives of the public. Through such an integrated strategy, it is possible to more effectively encourage enterprises to avoid greenwashing and engage in genuine green innovation. Moreover, the impact of stringent government regulation exhibits a certain degree of lag compared to that of public supervision, which may stem from the time gap between the implementation and the effectiveness of government regulatory policies.

Conclusion

In this concluding section, we summarize our study’s key findings. We formulated a tripartite evolutionary game model to examine greenwashing behaviors under four distinct regulatory mechanisms. The analysis, assuming bounded rationality, explores how various factors and incentive-punishment schemes impact the system’s evolution. Conclusions and implications are presented based on this research.

Research findings

Based on the simulation analysis of the tripartite evolutionary game model, the following conclusions are drawn from this study:

Firstly, the issuance of green credit and the revocation of green certification within government regulation both promote enterprises to choose green innovation production strategies. However, under the traditional static mechanisms of green credit allocation and green certification revocation that are implemented by the government, the behavioral strategies of enterprises and the public do not achieve an evolutionary stable point. In this context, there is a tendency for enterprises to still engage in greenwashing behavior, while the government leans towards adopting a lax regulatory strategy.

Secondly, under the mechanism where static green credit disbursement is coupled with dynamic green certification revocation, the system is observed to converge towards a stable strategy. In this strategy, enterprises are inclined towards green innovation, the government maintains strict regulation, and the public actively engages in supervision. Green credit is found to offer enterprises direct economic incentives, which mitigate their financial risks and costs associated with green innovation. This economic incentive is immediate, facilitating enterprises’ swift utilization of credit funds for investment and innovation activities. In contrast, the revocation of green certification is noted to potentially adversely affect a company’s reputation and market position. This punitive measure could elicit resistance from enterprises, thereby dampening their enthusiasm for green innovation. Consequently, it is recommended that the government devise dynamic regulatory measures that integrate static green credit disbursement with dynamic green certification revocation, with the aim of establishing a collaborative governance system to address greenwashing behavior.

Lastly, it is evident that public participation in supervision can serve as an effective counterbalance to the deficiencies of lax government regulation and, to a certain extent, prompt the government to enforce stricter regulatory measures. Additionally, such public engagement can significantly diminish corporate greenwashing behavior through supportive actions, including spontaneous consumption and publicity, as well as by increasing scrutiny of green practices. Overall, the influence of public supervision on governmental actions is considered to be more substantial than its influence on corporate behavior, and the effect of stringent government regulation on corporate behavior is found to be more significant than that of public supervision.

Research significance

Theoretically, this study first constructs a tripartite evolutionary game model, thereby providing a novel theoretical framework for understanding the interactions among multiple stakeholders in the governance of greenwashing and the realization of green innovation. In particular, the incorporation of public concern for greenness further confirms that consumers’ latent psychological and environmental values influence their purchasing decisions when other conditions have similar impacts (Hu and Wang, 2022). This finding not only reveals the importance of consumers’ psychological and environmental values in market mechanisms but also underscores the crucial role of the public in the governance of corporate greenwashing and the realization of green innovation. Consumers’ psychological and environmental values not only affect their purchasing decisions but also influence corporate behavior through market feedback mechanisms, thereby driving enterprises to shift from greenwashing to genuine green innovation. Moreover, public supervision and participation can effectively compensate for the insufficiencies of government regulation, forming a dual constraint mechanism on enterprises and further enhancing the importance of the public in green governance. Secondly, the study differentiates between the dynamic combinations of two policy instruments—green credit disbursement and green certification revocation—thus providing a theoretical basis for evaluating the effectiveness of different policy tool combinations. Specifically, the model exhibits an ESS under the combination of static green credit disbursement and dynamic green certification revocation, which contrasts with the prevailing view in most current studies that validate the superiority of bilateral dynamic reward-punishment mechanisms. This finding highlights the flexibility of the government to adjust in a timely manner according to actual conditions and market responses, thereby emphasizing the importance of dynamic government regulation from another perspective (Chen and Hu, 2018; Liu et al., 2024). When formulating policies, the government needs to take into account market conditions, corporate behavior, and public feedback, and flexibly adjust the combination of policy instruments to achieve more effective governance outcomes. This dynamic adjustment capability is crucial for addressing the complex and changing market environment and corporate behavior, and it also provides policymakers with more flexible policy options.

Additionally, the parameter effect test validates the significant impact of green credit on corporate green innovation, aligning with existing research (Chen et al., 2024; Xu et al., 2023). As an important policy instrument, green credit can directly provide economic incentives to enterprises, reducing the costs and risks associated with green innovation, thereby facilitating the implementation of green innovation behaviors. This finding further underscores the key role of green credit in promoting corporate green transformation and also provides empirical support for the government in formulating green financial policies. Finally, the study expands the scope of the public from the traditional consumer perspective to include a broader range of stakeholders, such as investors and community members (Wang et al., 2023), thereby enriching the connotation of public participation. This expansion not only highlights the collaborative role of multiple stakeholders in green governance but also provides a more comprehensive perspective for the theory of green governance. By incorporating a broader range of stakeholders, this study reveals the importance of multi-party co-governance in achieving sustainable development goals. This co-governance mechanism not only effectively curbs greenwashing behavior but also promotes the realization of green innovation through multi-party collaboration, providing a theoretical foundation for the construction of a more comprehensive green governance system.

In practice, manufacturing enterprises, as the primary agents of green innovation, should actively avoid making exaggerated or false environmental claims that mislead consumers and regulatory authorities. By refraining from greenwashing and adopting genuine green production processes, enterprises not only fulfill their corporate social responsibility but also enhance their performance and achieve sustainable development. Genuine green innovation enables enterprises to reduce operational costs, improve resource efficiency, and meet the growing consumer demand for eco-friendly products. This not only helps in building a positive corporate image but also aligns with the global trend towards sustainable development. Moreover, enterprises that engage in green innovation are more likely to gain a competitive advantage in the market, as consumers increasingly prefer products and services that are environmentally friendly. Therefore, it is in the best interest of manufacturing enterprises to invest in green technologies and sustainable practices. Traditional static regulatory approaches may lack the flexibility needed to adapt to market fluctuations and changing corporate behaviors. Governments should consider implementing dynamic and differentiated regulatory strategies that can be adjusted in response to corporate greenwashing behavior, green innovation performance, and public feedback. This flexibility is crucial for effectively encouraging corporate green innovation. For instance, green credit policies can provide financial incentives for enterprises to engage in green innovation, while dynamic certification mechanisms can penalize those involved in greenwashing. By adopting such strategies, governments can create a more adaptive and responsive regulatory environment that promotes sustainable business practices. Additionally, dynamic regulatory strategies can help governments to better align their policies with the actual needs of the market and enterprises, thereby enhancing the effectiveness of environmental governance.

Moreover, the government should provide technical support to enterprises to facilitate the development of green technologies and reduce the costs associated with green innovation. This can be achieved through initiatives such as research and development grants, technology transfer programs, and collaboration with academic institutions. By lowering the barriers to green innovation, governments can encourage more enterprises to adopt sustainable practices, thereby fostering a greener economy. Technical support can also help enterprises to overcome the challenges associated with green innovation, such as high initial investment costs and technological uncertainties. This, in turn, can lead to a more widespread adoption of green technologies and practices across industries. Public participation and supervision are essential for curbing corporate greenwashing. The public can provide valuable feedback to the government, helping to identify regulatory gaps and prompting policy adjustments when necessary. This feedback mechanism is instrumental in shaping more effective environmental policies. Therefore, when addressing greenwashing and promoting green innovation, governments should consider both economic and social incentives. Establishing whistleblower reward mechanisms and valuing public opinion can significantly enhance the effectiveness of green governance. By involving the public in the regulatory process, governments can create a more transparent and accountable system that encourages genuine green practices among enterprises. Public participation can also exert social pressure on enterprises to engage in sustainable practices, as consumers and other stakeholders increasingly demand transparency and accountability from companies regarding their environmental impact.

In addition, governments should focus on enhancing public awareness and education regarding green issues. By increasing public understanding of environmental challenges and the importance of sustainable practices, governments can foster a more informed and engaged citizenry that is better equipped to hold enterprises accountable for their environmental claims. Public awareness campaigns, educational programs, and community engagement initiatives can all play a role in promoting a culture of sustainability and encouraging consumers to make environmentally responsible choices. This, in turn, can create a positive feedback loop, where increased consumer demand for green products and services drives further investment in green innovation by enterprises. Furthermore, the collaboration between the government, enterprises, and the public is crucial for the successful implementation of green governance policies. Effective communication and cooperation among these stakeholders can help to ensure that policies are well-designed, effectively implemented, and appropriately adjusted based on feedback and changing circumstances. For example, public-private partnerships can facilitate the sharing of resources, knowledge, and expertise, leading to more innovative and effective solutions to environmental challenges. Similarly, stakeholder consultations and collaborative decision-making processes can help to build trust and ensure that policies are aligned with the needs and expectations of all parties involved.

In summary, addressing greenwashing and promoting green innovation require a multi-faceted approach that involves the active participation of manufacturing enterprises, dynamic and adaptive regulatory strategies by the government, and robust public engagement and supervision. By combining economic incentives with social pressures, providing technical support to enterprises, and fostering a culture of sustainability through public education and awareness, governments can create a more effective and sustainable green governance system. This collaborative approach not only helps to curb greenwashing but also drives genuine green innovation, ultimately contributing to a more sustainable and environmentally responsible economy.

Limitations and future prospects

Firstly, in this study, consideration was limited to three parties within the governance system of greenwashing behavior. Future research could expand the analytical framework to include other relevant stakeholders, such as suppliers and downstream enterprises. Secondly, the simulation data employed in this study were predicated on hypothetical conditions. Subsequent research might benefit from conducting empirical analyses with real-world data to further substantiate the findings.