Carbon Offsetting-Driven Multi-Actor Low-Carbon Collaborative Evolutionary Game Analysis

Abstract

The proactive strategic choice for low-carbon collaboration among various sectors of society is to promote low-carbon transformation of the industrial chain through carbon offsetting. This study delves into the strategy selection and game process of carbon offset actions with participation from businesses, government, and public, thus revealing the dynamic evolutionary relationship of the behavior of each stakeholder. A multi-agent low-carbon collaboration evolutionary game model is established, driven by carbon compensation. The game process undergoes an evolutionary trend simulation, strategy evolution analysis, and key parameter sensitivity analysis, ultimately identifying the optimal cooperative mode and key influencing factors among various stakeholders. The study found that an evolutionary equilibrium and stable strategy exists in the game process of enterprise, government, and public participation in carbon offsetting. The initial participation willingness of each stakeholder has an impact on the strategy choices of other stakeholders. Behaviors such as leading by example, punishment for violators, reasonable subsidy intensity, and active public supervision have a positive effect in promoting carbon offsetting policies and low-carbon collaboration. The research findings offer theoretical insights into promoting efficient multi-party green cooperation and accomplishing low-carbon transformation of the industrial chain under the ‘dual-carbon’ goal.

1. Introduction

As environmental protection continues to advance, a joint development trend towards new urbanization and industrial structural transformation has resulted in a significant improvement in the green level of the industrial chain. In the early stages of green development, the clean industrial chain involves environmental protection concepts such as pollution control, waste disposal, and the conservation and protection of resources [1]. However, given the severity of the ecological environment, the green transformation of the industrial chain requires refined development. Against this backdrop, a social operating system that combines low-carbon concepts with economic development has emerged. Consequently, the achievement of clean production, scientific carbon emission reduction, and the development of a low-carbon economy have become academic focal points [2,3,4].

Carbon offsetting has become an important policy for reducing carbon emissions and controlling greenhouse effects with development potential. It is a market mechanism that compensates carbon sink agents for their carbon reduction efforts through economic or non-economic means [5]. On the one hand, carbon offsetting increases the carbon emission cost of enterprises, which fundamentally limits carbon emissions [6]. On the other hand, as an emerging green management system, it achieves effective resource coordination, provides financial support, and balances economic and environmental benefits [7]. After the issuance of the Kyoto Protocol, the European Union became the first to establish carbon offsetting as a flexible mechanism and stipulated that enterprises’ carbon offsetting should be based on the reduction amount of greenhouse gases specified in Kyoto Protocol projects [8]. In 2020, the European Climate Law set carbon neutrality targets for EU countries from a legal perspective and established a legal framework guiding the implementation of carbon offsetting policies [9]. At the technological and economic levels, the Clean Development Mechanism (CDM) requires developed countries to provide developing countries with green technology and funding support to offset their own carbon emissions [10]. In 2008, the introduction of the “China Green Carbon Fund Compensation Mark” marked the implementation of China’s carbon offsetting system. As of 2020, China’s forest coverage rate reached 24.1%, with a forest stock of 19 billion cubic meters, and a cumulative reduction of approximately 5.8 billion tons of carbon dioxide emissions [11].

In general, carbon offsetting is in its initial stages of development and has not yet developed standardized and replicable models. The reason for this is the uneven distribution of development, with implementation of carbon offsetting concentrated in energy industries such as coal, steel, and electricity, and less responsive in sectors such as the light and service industries [12]. Furthermore, there is a lack of universal participation, with responses primarily from large state-owned enterprises such as China National Petroleum Corporation and State Grid Corporation of China, and limited participation from private enterprises, small and medium-sized enterprises, and the public. Therefore, within the current context of low-carbon economic development, addressing this issue is an ongoing challenge to promote balanced development of carbon offsetting and ensure comprehensive social participation, while achieving low-carbon circulation within the industry chain and ensuring the healthy operation of economic entities. In this study, we constructed a three-party evolution game model of enterprises, governments, and the public to explore the dynamic strategic choices and mutual influence mechanisms of each stakeholder based on their own interests. We used numerical simulation visualization to present the impact of key parameters on the evolution trend, with the aim of providing a reference for multi-stakeholder emission reduction games under low-carbon policies such as carbon offsetting.

2. Literature Review

2.1. Research on Carbon Offsetting

Carbon offsetting can be defined as the allocation of emission reduction credits and carbon credit compensation [13]. The former refers to the government’s establishment of emission reduction policies and the allocation of carbon quotas to carbon-emitting entities, which can be traded in carbon markets. The latter refers to carbon-emitting entities maintaining their carbon credit by implementing investment projects aimed at reducing emissions. Currently, carbon offsetting research has encompassed numerous interdisciplinary fields, including forest carbon offsetting [14,15], transportation carbon offsetting [16], international trade carbon offsetting [17], ecological carbon offsetting [18], agricultural carbon offsetting [19], and fishery carbon offsetting [20]. Scholars have investigated the implementation of carbon offsetting policies from different research angles, utilizing various methods. The results demonstrate that both forest carbon compensation and fishery carbon compensation studies have confirmed the carbon absorption effect through comparative field experiments. The afforestation project was identified as the optimal emission reduction method, and the conclusion was drawn that iron elements can decrease the ocean’s carbon absorption capacity. The transportation carbon compensation study, based on past emission reduction practices of airlines, discusses the potential challenges faced under carbon compensation mechanisms and provides five countermeasures for airlines. The international trade carbon compensation study utilized an input–output model and estimated the carbon emissions avoided in other regions of the world based on the CO₂ emissions data produced by China’s import and export trade over the years. The study concluded that the scale effect and the technical effect have positive and negative impacts on carbon emissions, respectively. The ecological carbon compensation study established three compensation levels based on a company’s carbon emissions and utilized a production optimization decision model to determine the maximum compensation quota as the optimal compensation level. The agricultural carbon compensation study employed a literature review method and suggested that cost factors should be considered when promoting carbon compensation development through agricultural sources. The study emphasized the significance of improving the agricultural carbon compensation system.

Firstly, carbon offsetting at the government level is one of the important measures to address climate change. The government can limit high-carbon emissions by enacting policies while encouraging them to adopt low-carbon development. Wang et al. studied the effectiveness of carbon quotas and trading policies in regulating manufacturing carbon emissions [21]. Xu et al. researched the strategic planning of enterprises operating under a carbon trading system, highlighting how government policies facilitated their transition to environmentally friendly production methods by encouraging optimal decisions regarding production, pricing, and carbon emissions [22]. Cohen et al. conducted an analysis based on demand and pricing relationships, examining the effects of government subsidies on green technologies on manufacturers and consumers, using a two-stage Stackelberg model to identify these effects. Studies have demonstrated that the government must take into account the uncertainty of demand factors when formulating consumer subsidies. This consideration can aid in increasing consumers’ adoption targets. Regarding manufacturers, green subsidies can act as a synergistic mechanism, alleviating the pressure of production costs [23]. Meanwhile, Zhu et al. explored the benefits of green technology subsidies, carbon taxes, and carbon trading at various stages of green innovation development in the manufacturing sector. Their findings provide insights to the government on the adoption of appropriate measures based on the development goals of each stage [24].

Secondly, individual carbon offsetting is also an important measure to address climate change. Although the government has implemented a series of policies to encourage businesses and the public to actively participate in carbon offsetting, Whitmarsh and O’Neill argue that self-identity is the main factor affecting consumer response, more significant than the influence of moral constraints and objective norms [25]. Mackerron et al. proposed the concept of willingness to pay for individual carbon offsetting, and studies have shown that voluntary carbon offsetting behavior by individuals can help promote the development of green technologies and create a low-carbon market [26]. Mair studied the carbon offsetting behavior of consumers in the tourism and transportation sector [27], and subsequently, the use of contingent valuation methods to measure consumers’ willingness to pay for carbon offsetting has become a research hotspot [28,29]. In addition, Han and Hyun integrated the norm activation theory and the planned behavior theory into an integrated framework, and established a comprehensive prediction model by balancing intuitive measurement of social benefits and rational measurement of individual utility, which verified that individual norms and attitudes can effectively improve consumers’ awareness of carbon offsetting [30].

2.2. Research on Low-Carbon Policy in Multi-Agent Collaboration

During the implementation of low-carbon policies, there are usually multiple key stakeholders such as regulatory agencies, various levels of enterprises, and consumers, who engage in collaborative efforts, gaming with each other, mutually benefiting and limiting each other in the process. Hao and Li used a differential game model to analyze the green technology innovation strategy in the carbon trading policy supply chain, and found that the green innovation benefits from cooperative game among enterprises are greater than those in non-cooperative game scenarios [31]. Yao et al. found that the optimal cooperative strategy for promoting real estate developers to build low-carbon buildings is for the government to adopt static carbon taxes and dynamic subsidies [32]. Zhang and Liu used a master–slave game model to study the low-carbon cooperation between the government and non-state-owned enterprises. The findings indicate the presence of participation costs in low-carbon development and suggest that the government should implement subsidy policies to support the low-carbon development of non-state-owned enterprises. Moreover, punitive measures should be established to deter opportunistic behavior by the enterprises [33]. Tang et al. established a dual-objective optimization model that takes into account both carbon emissions and customer satisfaction, and solved the problem of large-scale product configuration for enterprises under the context of “dual carbon” [34]. Chen et al. believe that under government intervention, manufacturing enterprises will ultimately choose green and low-carbon innovation [35]. Li et al. conducted a study on the driving forces of low-carbon development in China, classifying them as either internal drivers stemming from the government’s political will for low-carbon development, or external drivers originating from public demand for green initiatives. The findings demonstrate a positive effect of external drivers on internal drivers, highlighting the inadequacy of a policy-driven approach alone to significantly bolster internal drivers. Additionally, economic and educational levels were shown to have an impact on external drivers [36].

After conducting a comprehensive review of the literature, it becomes evident that research on carbon offset has primarily centered on individual actors, with little attention paid to the collaborative mechanisms of carbon offsetting actors, and a dearth of research on the interaction mechanisms of multiple actors involved in the process of carbon offsetting. This paper makes a marginal contribution by elucidating the following aspects: (1) It clarifies the game relationship between the government and enterprises in carbon offsetting, which is conducive to exploring optimal emission reduction strategies. Moreover, by introducing public participation and supervision based on traditional government–business cooperation, the game model is enriched, providing novel ideas for the development and implementation of carbon offsetting. (2) Based on rational assumptions and fair principles, a three-party evolutionary game model is established, taking into account the interests and demands of all participants. This model provides comprehensive and objective development suggestions for every participant based on the results of the evolution. It is beneficial for the balanced evolution of carbon offsetting actions in multi-actor participation scenarios.

3. Construction and Analysis of Evolutionary Game Model

3.1. Model Description and Hypothesis

To accelerate the low-carbon transformation of industrial chains, carbon offsetting actions should adhere to a development system based on collaborative cooperation. Enterprises, as the main unit producing carbon emissions, have a direct impact on the environmental benefits based on their attitude towards carbon policies. Therefore, while the government is formulating reasonable policies, it also needs to increase management efforts to guide enterprises to actively participate in carbon compensation actions and achieve carbon reduction through carbon offsetting. Due to limited financial and human resources of local governments, it is difficult to achieve real-time monitoring, which may lead to neglect of opportunistic behavior by enterprises related to emissions reduction. As an important participant in restraining environmental pollution, the public can effectively improve the implementation efficiency of carbon compensation policies. Based on this, the model has three main stakeholders: enterprises, governments, and the public. The model assumptions, variables, and corresponding meanings are shown in

Table 1. Model variables and coincidence meanings.

Subject	Parameters	Explanations
enterprises	$E_a$	The basic benefits of the enterprise
	$C_a$	Enterprises participate in the cost of carbon offset
	$D_a$	The limited penalties and production limits imposed by enterprises when they participate negatively
	$G$	Government subsidies for companies to participate in carbon offsets
	$I_a$	Benefits (reputation, stock price, etc.) of enterprises actively responding to carbon offsetting policies
	$P_a$	Losses from passive participation (reputation, stock price, etc.)
government	$E_b$	Basic revenue of local government
	$C_b$	The cost of local government strategies
	$M$	The cost of weak local government management (publicity expenditure)
	$D_b$	Active participation of enterprises but weak government management leads to loss of development potential
	F	Enterprise passive participation, government environmental governance costs
	K	Government benefits when business is actively involved
	$I_b$	Benefits of government strategies (credibility, performance, etc.)
	$P_b$	Loss of weak government management (credibility, performance, etc.)
public	$E_c$	Public participation benefits (environmental quality improvement)
	$C_c$	The cost of public participation
	L	Additional benefits to the public from the active participation of enterprises or strategies adopted by the government
	$I_c$	Public participation in carbon offsets, incentives given by the government
	H	Losses caused by negative participation of enterprises or non-participation of the public

.

(i).

Game subject assumption: The local government is the issuer of carbon compensation policies, enterprises are the implementers of carbon compensation policies, and the public is the supervisor of carbon compensation policies.

(ii).

Each of the three parties involved has two strategic options available. Enterprises may choose to either adhere to carbon offset policies with a probability of x (0 ≤ x ≤ 1) or overlook them with a probability of 1 − x. Governments may choose to adopt carbon offset policies with a probability of y (0 ≤ y ≤ 1) or refrain from doing so with a probability of 1 − y. The public can choose to monitor the carbon offset behavior of both the government and enterprises with a probability of z (0 ≤ z ≤ 1), or abstain with a probability of 1 − z. Let the enterprise’s choice of carbon offsetting be defined as ${\theta }_1$ and the opposite as ${\theta }_2$; the adoption of carbon offset policies by the government be defined as ${\tau }_1$ and the opposite as ${\tau }_2$; and public participation in supervision be defined as ${\lambda }_1$ and the opposite as ${\lambda }_2$.

(iii).

Variable size assumption: According to the actual situation, the key to offsetting carbon emissions is for enterprises to actively responding carbon compensation policies, and for the public to spontaneously carry out environmental supervision behavior based on their own environmental awareness, setting $I_c<G$. Based on the principle of sustainable development, the losses caused by enterprises actively participating but the government being conservative are far greater than the government’s weak management costs, setting $M<D_b$. Based on the current environmental situation, both enterprises and the government bear high expectations from the public. If both parties actively participate, it will bring good social effects, setting $P_a<I_a$, $P_b<I_b$.

(iv).

Objectivity assumption: In reality, there is information asymmetry between various stakeholders, making it impossible to predict each other’s actions. The actions of the three parties also mutually influence each other, and each stakeholder adopts bounded rational behavior [37].

3.2. Model Analysis

Based on the model variables mentioned above, the behavior strategy sets of the subjects were derived, and the pay-off matrix of the three-party evolutionary game was established. The pay-off arrays of the enterprise, local government, and the public are shown in

Table 2. Income matrix of evolutionary game among enterprises, governments and public.

	Strategy Combination	Payoff Matrix
		Enterprises	Government	Public
1	$\left({\theta }_1,{\tau }_1,{\lambda }_1\right)$	$E_a+G+I_a-C_a$	$E_b+K+I_b-C_b-G-I_c$	$E_c+L+I_c-C_c$
2	$\left({\theta }_1,{\tau }_1,{\lambda }_2\right)$	$E_a+G-C_a$	$E_b+K-C_b-G$	$L-H$
3	$\left({\theta }_1,{\tau }_2,{\lambda }_1\right)$	$E_a+I_a-C_a$	$E_b+K-D_b-P_b-M$	$L+E_c-C_c$
4	$\left({\theta }_1,{\tau }_2,{\lambda }_2\right)$	$E_a-C_a$	$E_b+K-M-D_b$	$L-H$
5	$\left({\theta }_2,{\tau }_1,{\lambda }_1\right)$	$E_a-D_a-P_a$	$E_b-C_b-F+I_b-I_c$	$E_c+L+I_c-H-C_c$
6	$\left({\theta }_2,{\tau }_1,{\lambda }_2\right)$	$E_a-D_a$	$E_b-C_b-F$	$L-H$
7	$\left({\theta }_2,{\tau }_2,{\lambda }_1\right)$	$E_a-P_a$	$E_b-F-M-P_b$	$E_c-C_c-H$
8	$\left({\theta }_2,{\tau }_2,{\lambda }_2\right)$	$E_a$	$E_b-M-F$	$-H$

.

3.3. Model Construction and Solution

3.3.1. Replication Dynamic Equation

If we assume that the expected benefit gained by the enterprise when it chooses to respond to carbon offset policies is $E_{X1}$.

$$\begin{array}{c}{E}_{X1}=y\left[z\left(E_a+G+I_a-C_a\right)+\left(1-z\right)\left(E_a+G-C_a\right)\right]\\ +\left(1-y\right)\left[z\left(E_a+I_a-C_a\right)+\left(1-z\right)\left(E_a-C_a\right)\right]\end{array}$$

(1)

If we assume that the expected benefit gained by the enterprises when it chooses not to respond to carbon offset policies is $E_{X2}$.

$$E_{X2}=y\left[z\left(E_a-D_a-P_a\right)+\left(1-z\right)\left(E_a-D_a\right)\right]+\left(1-y\right)\left[z\left(E_a-P_a\right)+\left(1-z\right)E_a\right]$$

(2)

The dynamic equation of enterprises replication balance is:

$$\begin{array}{c}F\left(x\right)=\frac{dx}{dt}=x\left(E_{X1}-E_x\right)=x\left(1-x\right)\left(E_{X1}-E_{X2}\right)\\ =x\left(1-x\right)\left[y\left(G+D_a\right)+z\left(I_a+P_a\right)-C_a\right]\end{array}$$

(3)

Similarly, using $E_{y1}$ and $E_{y2}$ to represent the expected benefits of the government adopting and not adopting carbon offset policies, respectively, we can express them as follows:

$$\begin{array}{c}{E}_{y1}=x\left[z\left(E_b+K+I_b-C_b-G-I_c\right)+\left(1-z\right)\left(E_b+K-C_b-G\right)\right]\\ +\left(1-x\right)\left[z\left(E_b-C_b-F+I_b-I_c\right)+\left(1-z\right)\left(E_b-C_b-F\right)\right]\end{array}$$

(4)

$$\begin{array}{c}{E}_{y2}=x\left[z\left(E_b+K-D_b-P_b-M\right)+\left(1-z\right)\left(E_b+K-M-D_b\right)\right]\\ +\left(1-x\right)\left[z\left(E_b-F-P_b-M\right)+\left(1-z\right)\left(E_b-M-F\right)\right]\end{array}$$

(5)

The dynamic equation of government replication balance is:

$$\begin{array}{c}F\left(y\right)=\frac{dy}{dt}=y\left(E_{y1}-E_y\right)=y\left(1-y\right)\left(E_{y1}-E_{y2}\right)\\ =y\left(1-y\right)\left[x\left(D_b-G\right)+z\left(I_b+P_b-I_c\right)-C_b+M\right]\end{array}$$

(6)

Using $E_{z1}$ and $E_{z2}$ to represent the expected benefits of public supervision of carbon offset behavior and not supervising it, respectively, we can express them as follows:

$$\begin{array}{c}{E}_{z1}=x\left[y\left(E_c+L+I_c-C_c\right)+\left(1-y\right)\left(E_c+L-C_c\right)\right]\\ +\left(1-x\right)\left[y\left(E_c+L+I_c-H-C_c\right)+\left(1-y\right)\left(E_c-C_c-H\right)\right]\end{array}$$

(7)

$$E_{z2}=x\left[y\left(L-H\right)+\left(1-y\right)\left(L-H\right)\right]+\left(1-x\right)\left[y\left(L-H\right)+\left(1-y\right)\left(-H\right)\right]$$

(8)

The dynamic equation of public replication balance is:

$$F\left(z\right)=\frac{dz}{dt}=z\left(E_{z1}-E_z\right)=z\left(1-z\right)\left(E_{z1}-E_{z2}\right)=z\left(1-z\right)\left(xH+y{I}_c+E_c-C_c\right)$$

(9)

3.3.2. Model Equilibrium Point and Strategy Stability Analysis

Due to the limited rationality of the government, enterprises, and the public, it is difficult for each game player to make the best strategic choice in a single decision. Therefore, Equations (3), (6) and (9) can be seen as an evolutionary process, forming a 3D replication dynamic system, as shown in Equation (10).

$$\{\begin{array}{l}{F}_x\left(x,y,z\right)=x\left(1-x\right)\left[y\left(G+D_a\right)+z\left(I_a+P_a\right)-C_a\right]\\ F_y\left(x,y,z\right)=y\left(1-y\right)\left[x\left(D_b-G\right)+z\left(I_b+P_b-I_c\right)-C_b+M\right]\\ F_z\left(x,y,z\right)=z\left(1-z\right)\left(xH+y{I}_c+E_c-C_c\right)\end{array}$$

(10)

By letting $F_x\left(x,y,z\right)=0$, $F_y\left(x,y,z\right)=0$, $F_z\left(x,y,z\right)=0$, we obtain that the three-dimensional dynamic system has eight pure strategy equilibrium points (ESS) [38]: $E_1\left(0,0,0\right)$, $E_2\left(1,0,0\right)$, $E_3\left(0,1,0\right)$, $E_4\left(0,0,1\right)$, $E_5\left(1,1,0\right)$, $E_6\left(0,1,1\right)$, $E_7\left(1,0,1\right)$, and $E_8\left(1,1,1\right)$. Using the method proposed by Friedman [39], we can calculate the local stability of the eight equilibrium points using the eigenvalues of the Jacobian matrix (denoted as J).

$$\begin{array}{l}J=\left[\begin{array}{ccc}\frac{\partial F_x\left(x,y,z\right)}{\partial x}& \frac{\partial F_x\left(x,y,z\right)}{\partial y}& \frac{\partial F_x\left(x,y,z\right)}{\partial z}\\ \frac{\partial F_y\left(x,y,z\right)}{\partial x}& \frac{\partial F_y\left(x,y,z\right)}{\partial y}& \frac{\partial F_y\left(x,y,z\right)}{\partial z}\\ \frac{\partial F_z\left(x,y,z\right)}{\partial x}& \frac{\partial F_z\left(x,y,z\right)}{\partial y}& \frac{\partial F_z\left(x,y,z\right)}{\partial z}\end{array}\right]\\ =\left[\begin{array}{ccc}\left(1-2x\right)\left[\begin{array}{c}y\left(G+D_a\right)\\ +z\left(I_a+P_a\right)-C_a\end{array}\right]& x\left(1-x\right)\left(G+D_a\right)& x\left(1-x\right)\left(I_a+P_a\right)\\ y\left(1-y\right)D_b& \left(1-2y\right)\left[\begin{array}{c}x\left(D_b-G\right)\\ +z\left(I_b+P_b-I_c\right)-C_b+M\end{array}\right]& y\left(1-y\right)\left(I_b+P_b\right)\\ z\left(1-z\right)H& z\left(1-z\right)I_c& \left(1-2z\right)\left(xH+y{I}_c+E_c-C_c\right)\end{array}\right]\end{array}$$

By substituting the eight pure strategy equilibria into the Jacobian matrix, we obtain 24 eigenvalues. Stability analysis of these eigenvalues reveals the evolutionary stability of each equilibrium point in the pure strategy evolutionary game. Specifically, if the real parts of all eigenvalues in each group are negative, then the equilibrium point is considered evolutionarily stable. The eigenvalues and stability analysis results of each equilibrium point in the pure strategy evolutionary game are shown in

Table 3. Analysis of the eigenvalues and stability of each equilibrium point in Jacobian matrix.

Equilibrium Point	Eigenvalue	Stability Condition
$E_1\left(0,0,0\right)$	${\lambda }_1=-C_a$
	${\lambda }_2=M-C_b$	$M<C_b$
	${\lambda }_3=E_c-C_c$	$E_c<C_c$
$E_2\left(1,0,0\right)$	${\lambda }_1=C_a$	Unstable point
	${\lambda }_2=D_b+M-G-C_b$
	${\lambda }_3=H+E_c-C_c$
$E_3\left(0,1,0\right)$	${\lambda }_1=G+D_a-C_a$	$D_a<C_a-G$
	${\lambda }_2=C_b-M$	$C_b<M$
	${\lambda }_3=I_c+E_c-C_c$	$I_c+E_c<C_c$
$E_4\left(0,0,1\right)$	${\lambda }_1=I_a+P_a-C_a$	$I_a+P_a<C_a$,
	${\lambda }_2=I_b+P_b+M-C_b-I_c$	$I_b+P_b+M<C_b+I_c$
	${\lambda }_3=C_c-E_c$	$C_c<E_c$
$E_5\left(1,1,0\right)$	${\lambda }_1=C_a-D_a-G$	$C_a-G<D_a$
	${\lambda }_2=C_b+G-D_b-M$	$C_b+G<D_b+M$
	${\lambda }_3=H+I_c+E_c-C_c$	$H<C_c-I_c-E_c$
$E_6\left(0,1,1\right)$	${\lambda }_1=G+D_a+I_a+P_a-C_a$	$G+D_a+I_a+P_a<C_a$
	${\lambda }_2=C_b+I_c-M-I_b-P_b$	$C_b+I_c<M+I_b+P_b$
	${\lambda }_3=C_c-I_c-E_c$	$C_c<I_c+E_c$
$E_7\left(1,0,1\right)$	${\lambda }_1=C_a-I_a-P_a$	$C_a<I_a+P_a$
	${\lambda }_2=D_b+I_b+P_b+M-G-I_c-C_b$	$D_b+I_b+P_b+M<G+I_c+C_b$
	${\lambda }_3=C_c-E_c-H$	$C_c<E_c+H$
$E_8\left(1,1,1\right)$	${\lambda }_1=C_a-G-D_a-I_a-P_a$	$C_a-G-I_a<D_a+P_a$
	${\lambda }_2=G+I_c+C_b-D_b-I_b-P_b-M$	$C_b+G+I_c-I_b<M+D_b+P_b$
	${\lambda }_3=C_c-I_c-H-E_c$	$C_c-I_c-E_c<H$

.

3.4. Dynamic Evolution Trend of Tripartite Game

Based on the stability requirements of the tripartite strategies of enterprises, government, and the public, it is evident that participation in profits constitutes the fundamental influence on stakeholder strategic action selection. Against the backdrop of low-carbon development, the evolution of carbon compensation-driven, multi-stakeholder low-carbon cooperation can be classified into four stages: theoretical, promotional, developmental, and mature stages. Additionally, an analysis of the equilibrium state at each stage of this process is provided, based on the principles of circular economy [40].

3.4.1. Theoretical Stage

The fundamental objective of enterprise operations is profitability, and during the early stages of evolution, most enterprises are still in a wait-and-see state with a low level of response to carbon offset policies. At this stage, they tend towards a “no compensation” strategy. Due to the recent issuance of the central government’s policy guidelines, local governments are in a transitional period of implementing policies based on their economic conditions and regulatory policies. During this period, the tendency is towards a “no action” strategy. As both the government and enterprises are dominant players at the societal level and have not yet participated in carbon offset actions, the public, as followers, generally selects a “no monitoring” strategy. Therefore, under the impetus of carbon offsetting, collaborative low-carbon initiatives among multiple stakeholders are currently in a theoretical stage, corresponding to equilibrium point $E_1\left(0,0,0\right)$, as shown in Table 3. According to Table 3, reaching equilibrium point $E_1\left(0,0,0\right)$ requires that $M<C_b$, that is, the cost of the government’s “no action” strategy is less than the cost of its “action” strategy. It also requires that $E_c<C_c$, that is, the benefits of the public’s “supervision” are less than the costs of their “supervision” strategy. Therefore, both the government and the public will choose not to act. The dynamic evolutionary path is shown in Figure 1. The system evolution curves of enterprises, government, and the public eventually converge at the point (0,0,0). In Figure 1, Figure 2, Figure 3 and Figure 4, the x-, y-, and z-axes represent the evolutionary paths of enterprises, government, and the public, respectively. The point of zero on the axis indicates inaction, while a point of one denotes action.

3.4.2. Promotion Stage

Due to enterprises’ profit-oriented mentality, they often neglect environmental protection in their production and operations, resulting in increasingly severe emissions issues. In response to these environmental issues, the government has initiated relevant carbon policies to urge enterprises to reform. Meanwhile, due to limited access to information and the absence of advanced examples at the time, the public’s level of concern regarding these issues was relatively low. Therefore, under the driving force of carbon offsetting, a variety of stakeholders involved in low-carbon initiatives have entered a phase of promotion, corresponding to equilibrium point $E_3\left(0,1,0\right)$, as shown in Table 3. According to Table 3, achieving equilibrium point $E_3\left(0,1,0\right)$ requires that $D_a<C_a-G$, that is, the cost of “compensation” for enterprises is greater than the punishment for “no compensation”, and enterprises tend to choose a “no compensation” strategy. It also requires that $C_b<M$, that is, the cost of the government’s “action” strategy is less than the cost of its “no action” strategy, and the government tends to choose to implement carbon compensation policies. Finally, it requires that $I_c+E_c<C_c$, that is, the benefits of the public’s “supervision” are less than the costs of their “supervision” strategy, and the public tends to choose “non-supervision”. The path of state evolution is shown in Figure 2. The system evolution curves of enterprises, government, and the public eventually converge at the point (0,1,0).

3.4.3. Development Stage

In response to the dual carbon policy and to achieve sustainable development of resources while establishing a positive corporate image, enterprises have gradually shifted from neglecting carbon emissions to actively responding to carbon compensation policies and proactively neutralizing carbon emissions under the government’s active advocacy. At this stage, the focus of work is on mutual cooperation between the government and enterprises, and the supervision channels are not yet sound, which may easily overlook the issues reflected by the public. Furthermore, the public is often in a passive acceptance state, and in the initial stage of development where results have not been achieved, the public is more likely to adopt a wait-and-see attitude. Therefore, under the impetus of carbon offsetting, collaborative low-carbon initiatives among multiple stakeholders have transitioned to a developmental stage, corresponding to equilibrium point $E_5\left(1,1,0\right)$. According to Table 3, to reach equilibrium point $E_5\left(1,1,0\right)$, it is necessary to satisfy $C_a-G<D_a$, the cost of “compensation” for enterprises is less than the penalty of “non-compensation”, and enterprises tend to choose the “compensation” strategy. It is also necessary to satisfy $C_b+G<D_b+M$, the cost of “implementation” for the government is less than the cost and loss of “non-implementation”, and the government tends to choose the “implementation” strategy. Furthermore, it is necessary to satisfy $H<C_c-I_c-E_c$, the loss of “non-supervision” for the public is less than the cost of “supervision”, and the public chooses the “non-supervision” strategy. The evolutionary path is illustrated in Figure 3. The system evolution curves of enterprises, government, and the public eventually converge at the point (1,1,0).

3.4.4. Mature Stage

With the comprehensive deepening of reforms, both the government and enterprises are increasingly engaging in low-carbon collaborations, driven by policies that advocate for austerity, the development of a circular economy, the promotion of a green and low-carbon lifestyle, and the enhancement of public awareness of conservation. Simultaneously, the regulatory system is being refined, and information channels are becoming increasingly abundant, leading to an increase in the public’s enthusiasm, environmental consciousness, and regulatory awareness to participate in such collaborations. Therefore, the collaborative efforts of multiple stakeholders in the low-carbon sector have matured under the impetus of carbon offsetting, corresponding to equilibrium point $E_8\left(1,1,1\right)$. According to Table 3, to reach equilibrium point $E_8\left(1,1,1\right)$, it is necessary to satisfy $C_a-G-I_a<D_a+P_a$, the cost of “compensation” for enterprises is less than the loss of “non-compensation”, and enterprises tend to choose the “compensation” strategy. It is also necessary to satisfy $C_b+G+I_c-I_b<M+D_b+P_b$, the cost of “implementation” for the government is less than the cost and loss of “non-implementation”, and the government tends to choose the “implementation” strategy. Furthermore, it is necessary to satisfy $C_c-I_c-E_c<H$, the cost of “supervision” for the public is less than the loss of “non-supervision”, and the public tends to choose the “supervision” strategy. The evolutionary path is illustrated in Figure 4. The system evolution curves of enterprises, government, and the public eventually converge at the point (1,1,1).

4. Numerical Simulation

Considering the limitations of evolutionary game theory in elucidating the evolutionary path trends and impacts of key parameters, it is evident from the aforementioned stage-wise stability analysis results that the stable strategies of enterprises, local governments, and the public are influenced by key parameters. To further explore the evolutionary paths and the degree of influence of key parameters on the strategies of the three parties, this section conducts a visualized simulation analysis of the initial participation rate and relevant parameters of the three groups.

The fundamental expectation of the behaviors of the three parties, i.e., companies, local governments, and the general public, is cooperation. In order to achieve Evolutionarily Stable Strategy (ESS) and reach a stable state of the system model, the stability condition should be satisfied, which is $C_a-G-I_a<D_a+P_a,C_b+G+I_c-I_b<M+D_b+P_b,C_c-I_c-E_c<H$. Therefore, based on the previous research and simulation analysis and in consideration of the actual situation [35,41,42], we can set the initial parameter values as follows: $\left(x,y,z\right)=(0.5,0.5,0.5),C_a=12,D_a=15,G=3,I_a=12,P_a=4,M=8,C_b=28,D_b=25,I_b=15,P_b=6,$ $E_c=5,C_c=23,I_c=2,H=30$.

4.1. Strategy Simulation

The initial willingness of enterprises, local governments, and the public to participate influences the time it takes for other stakeholders to reach an equilibrium point in their strategic evolution. A higher initial participation proportion closer to the equilibrium point leads to a faster attainment of an equilibrium evolutionary strategy in the evolution model.

As shown in Figure 5a,b, by changing the initial participation level proportion x value of enterprises adopting a “compensation” strategy, the strategy evolution of local governments and the public can be positively influenced, and the time for the game model to reach an equilibrium point can be shortened. Similarly, changing the initial participation level y or z values of local governments or the public will also have a positive impact on the evolutionary outcomes of the other two parties, resulting in the game model eventually reaching a state of evolutionary stability.

The role of initial intentions in multi-stakeholder low-carbon collaborations is paramount. A heightened willingness to participate in the outset can exert a positive influence on the cooperation enthusiasm of other participants, leading to a shortened timeframe for the system to attain optimal cooperation status. This can facilitate the low-carbon transformation of the industrial chain and ultimately, the realization of carbon neutrality objectives.

4.2. Parameter Simulation

Drawing on the stable point conditions of evolutionary game theory, it is inferred that when the benefits of active participation in carbon compensation actions for enterprises, local governments, and the public outweigh those of conservative actions, the behavioral strategies of all stakeholders will progressively evolve towards “compensation, adoption, and supervision.” Consequently, adjusting critical parameters and analyzing the degree of influence of external variables on the systemic evolutionary trends through the combined effect of influencing factors and objective environments can enable an investigation of the system’s evolutionary patterns.

4.2.1. The Impact of the Degree of Constraint Imposed by Local Governments on the System’s Evolutionary Trend

The impact of local government constraints on the evolutionary trends of the system is demonstrated in Figure 6. When the degree of constraint shows fluctuations of up to 50%, the trends of local governments and the public tend to stabilize. Enterprises are particularly affected since they possess a profit-oriented characteristic. In the context of weak governmental constraints, enterprises exhibit low response rates and tend to adopt a “no compensation” strategy for their evolutionary trend. Nevertheless, as the degree of governmental constraint increases, enterprises become more sensitive to losses and converge faster towards a stable point of “compensation” strategy.

In summary, enterprises tend to prioritize maintaining their economic benefits when the degree of local government constraint is weak, resulting in a negative response towards carbon compensation policies. However, as the level of constraint increases, enterprises gradually shift towards a “compensation” strategy to improve their production and operational environment and pursue profitability. The evolutionary rate is positively correlated with the degree of constraint.

4.2.2. The Impact of the Subsidy Intensity of Local Governments on the System’s Evolutionary Trend

The impact of local government subsidies on the evolutionary trend of the system is demonstrated in Figure 7. Specifically, the subsidies fluctuate by “50%” for companies that actively respond to carbon offset policies, as depicted in Figure 7a. An increase in local government subsidies reduces the cost of companies participating in low-carbon actions. Thus, these subsidies can be leveraged to optimize industrial structures, enhance product sustainability, and improve core competitiveness, ultimately leading to a significant increase in the evolutionary trend of companies that respond to carbon offset policies.

As shown in Figure 7b, the subsidies offered by local governments fluctuate by 50% in terms of cost. The increase in subsidies provided to companies and individuals by local governments results in continuous investment costs. However, in times of tight financial constraints, local governments may consider reducing the implementation of carbon offset policies, which could slow down their own evolutionary trend.

As indicated in Figure 7c, local governments offer rewards for public oversight with a fluctuation of 50%. Such incentives for public monitoring promote the active engagement of individuals in carbon offset actions and lead to a slight improvement in the system’s evolutionary trend.

4.2.3. The Impact of Future Development Potential Factors on the System’s Evolutionary Trend

Figure 8 depicts the impact of potential future development factors on the system’s evolutionary trajectory. When businesses opt to participate in carbon offset programs, they experience additional operating costs, which increase their financial burden. The cost of the “subsidy” strategy pursued by businesses can vary by up to 50%, as shown in Figure 8a. As the cost of “compensation” continues to escalate, the evolutionary trajectory of the business undergoes a pronounced decline, culminating in the eventual adoption of the “no compensation” strategy and a refusal to collaborate with the government.

The local government’s adoption of a “non-adoption” strategy with regard to business participation in carbon offset programs results in a potential loss of up to 50% in development potential, as depicted in Figure 8b. If local governments disregard business “compensation” activities, it will dampen their enthusiasm for participating in carbon offset programs and responding to related low-carbon policies, thus undermining future collaboration opportunities between government and enterprise. This in turn results in a decline in government reputation and affects investment expectations for businesses outside the region. When the loss continues to expand, local governments will become aware of their misbehavior and opt for the “adoption” strategy, leading to a significant increase in their evolutionary trajectory.

Future development potential factors result in a fluctuation of up to 90% in public benefits, as shown in Figure 8c. Regardless of whether the environmental benefits for the public significantly increase or decrease, the change in the evolutionary trend is not substantial, and the public’s evolutionary pathway will ultimately reach a stable point of “supervision”. Despite a 90% increase in the Ec variable, no significant effect was observed on the evolutionary trend curve of the public towards environmental protection. This could be explained by the fundamental role of the environment as a prerequisite for survival, which renders every citizen responsible for its conservation. Additionally, the public’s bottom line towards environmental issues may not be influenced by material conditions. Facing the future social development, the public will choose to support environmental protection policies and take practical actions.

4.2.4. The Impact of Public Supervision on the System’s Evolutionary Trend

The influence of public monitoring on system evolution trends is shown in Figure 9. When the public monitors, the benefits of the “compensation” strategy and the losses of the “non-compensation” strategy fluctuate by 50%, affecting the evolution outcomes of the enterprise. As shown in Figure 9a, when the benefits of the “compensation” strategy and losses of the “non-compensation” strategy increase by 50%, the evolutionary trajectory slightly rises. Conversely, when the benefits of the “compensation” strategy and losses of the “non-compensation” strategy decrease by 50%, the evolutionary trajectory decreases significantly, and the trend quickly shifts towards the “non-compensation” strategy. Therefore, the decrease of income and loss significantly affects the low-carbon behavior of enterprises.

During public monitoring, the benefits of implementing a carbon offset policy and the losses of not implementing such a policy fluctuate by 50%, which affects the evolution outcomes of the local government, as shown in Figure 9b. When the benefits of implementing a carbon offset policy and the losses of not implementing it both increase by 50%, the evolution path quickly moves towards the stable point of “implementation.” This indicates that public monitoring can promote the implementation of carbon offset policies by the government. When the benefits of implementing a carbon offset policy and the losses of not implementing it both decrease by 50%, the evolution path still reaches the stable point of “implementation,” albeit in a relatively slower process. This shows that public monitoring only affects the execution level of the local government, but does not change their environmental attitudes.

Figure 9c illustrates the impact of environmental issues on public awareness of supervision. As environmental concerns continue to grow, public participation in low-carbon behavior becomes more pronounced, bolstering the social supervision force and promoting the evolutionary trend of stable strategies in the system.

5. Discussion

The present study offers a comprehensive analysis of the evolutionary trends and influencing factors associated with multi-stakeholder, low-carbon collaboration under the impetus of carbon offset policies. The research findings demonstrate that:

(a)

During the initial implementation of low-carbon policies, businesses, local governments, and public have adopted a more conservative approach due to social environment and cost considerations. This has resulted in an equilibrium in the game where “no compensation, no action, no monitoring” is the norm, limiting the effectiveness of carbon offsetting in energy-conservation and emission-reduction practices. Therefore, it is essential for the government to take a leadership role by implementing subsidy policies that incentivize corporate enthusiasm and extend the “window period” for business operations, while maintaining policy flexibility. In response to the low-carbon policies, businesses have joined the carbon reduction and carbon neutrality movements. Both the government and enterprises have opted for active carbon offsetting strategies, and frequent interactions have led the carbon market to enter a benign development stage. As the government continues to improve its reward and punishment policies, the public has taken the initiative to fulfill their supervisory obligations, ultimately achieving the optimal and stable strategy of “compensation, action, and monitoring” for the system.

(b)

In multi-agent evolutionary game processes, the initial intention plays a crucial role. A higher initial participation rate can affect the cooperative intentions of other participants, subsequently reducing the time required for the system to achieve an optimal cooperation status.

(c)

As practitioners of carbon offsetting, the attitude of enterprises towards this practice is primarily influenced by local government strategies. Their response to government reward and punishment policies reveals a greater fear of restrictions and penalties, a reaction largely driven by external factors such as risk aversion. In light of this situation, enterprises can take the initiative to select carbon offsetting actions and lead the industry. Such action will not only proactively avoid the penalties resulting from passive participation, but also garner support from the public and establish an image of an advanced enterprise.

(d)

As the primary driver of the carbon market, the government is expected to persist in its implementation of carbon offsetting policies in pursuit of profit. By augmenting incentives and penalties, the government endeavors to stimulate responses from enterprises and the general public alike. Nevertheless, there exists a boundary for the extent of subsidies that the government can furnish. Over-subsidization has the potential to negatively impact the entire financial system and impede the government’s attainment of optimal stability.

(e)

The involvement of the public in monitoring can serve as a significant driver of public opinion, promoting their willingness to participate. Furthermore, such involvement can create a feedback mechanism that influences governmental and corporate practices. This includes monitoring the implementation of governmental policies, enhancing governmental efficiency, identifying inactive participating businesses, and enhancing the reputation of exemplary businesses. These activities play a crucial role in promoting low-carbon policies and facilitating multi-stakeholder low-carbon cooperation, thereby underscoring their importance in the quest for sustainability.

Based on the aforementioned conclusion and considering the developmental traits of divergent phases of carbon offset actions, as well as the multi-stakeholder collaborative trend, we propose the following recommendations:

(a)

In the initial phases of carbon offset development, local governments should establish a permissive policy environment by actively endorsing the initiative, providing elevated levels of subsidies, extending the “window period” for company preparations, and implementing other measures to alleviate corporate doubts and stimulate emissions reductions. As the carbon market evolves, the interactions between government and businesses become more complex. At this stage, government entities should adopt a targeted approach and employ dynamic reward and penalty strategies to encourage optimal behavior and deter opportunistic actions among different enterprises. Specifically, those who demonstrate positive collaboration could receive incentives such as increased carbon quotas, lower carbon taxes, or financial rewards. Meanwhile, those who act opportunistically could face consequences such as production limitations, higher carbon taxes, or mandated carbon neutrality targets. By taking appropriate measures during critical periods, the government can promote a positive feedback loop, eliminate periodic obstacles to cooperation, and propel the system towards sustainability. In the mature stage of the carbon market, with a stable environment, pilot carbon trading programs can be expanded, and overall reduction targets can be optimized by developing green credit mechanisms.

To enhance cooperation and participation, a range of measures can be implemented. Firstly, incorporating annual achievements in environmental governance into the government’s evaluation system as a key performance indicator would increase accountability and promote greater willingness to participate. Secondly, policy guidance could be provided to incentivize green enterprises through tax reductions, financing methods, and project approval, thereby encouraging their initial participation. Finally, information channels may be broadened by establishing supervisory platforms, rewarding effective reporting, disclosing information, and purifying the public opinion environment, all of which would promote initial willingness to participate among the general public. By adopting these measures, enthusiasm can be stimulated among all parties, who can then work together to promote the development of environmental protection.

(b)

As direct victims of environmental pollution, the public has typically found themselves in a passive role with regards to carbon offset activities, largely due to a lack of awareness and sense of responsibility. However, with the recent push towards a “dual carbon” policy and increased collaboration between government and industry, it is imperative that the public takes an active role in collaborative efforts, playing a positive role in public opinion supervision, reporting companies exhibiting negative behavior, working to reduce government regulatory costs, providing feedback on government oversights, and assisting in the establishment of a sustainable, multi-stakeholder low-carbon cooperation system. These actions are critical in ensuring a more environmentally conscious and sustainable future.

(c)

As the primary driving force behind carbon offset actions, enterprises bear a social responsibility to mitigate carbon emissions while seeking economic benefits. Corporate development is subject to policy and public scrutiny. Therefore, it is imperative for companies to take decisive action in adopting transformative strategies, proactively implementing carbon reduction measures, and developing low-carbon economies.

To this end, companies should firstly develop periodic low-carbon development strategies that incorporate a strong sense of responsibility for emission reduction enhancement. This involves prioritizing compliance with low-carbon policy requirements and integrating the advancement of low-carbon product innovation as an integral element of their business strategies.

The second measure involves enhancing the internal management level of enterprises, which can be achieved by strengthening the decision-making capabilities of management, refining low-carbon transformation strategies, enhancing supervision and assessment of emissions reduction processes, and leveraging talent pool to maximize its effect. It is recommended to appoint experts with expertise in green credit financing, resource and environmental assessment, and energy-saving production technology as consultants to provide technical support and increase enterprise value.

Thirdly, companies should refine their low-carbon transformation risk management system. They should not only actively respond to low-carbon policy, increase green investment, and expand green projects but also develop emergency plans to prevent risks such as “funding chain break” and “public opinion storms” during the low-carbon transformation process. Strengthening the carbon emissions reduction risk assessment system will improve the risk management effect and enable sustainable development through multi-stakeholder low-carbon collaboration.

6. Conclusions

This article presents a comprehensive analysis of the evolving trends and sensitive factors associated with multi-party low-carbon cooperation, which is driven by carbon offset policies. By employing evolutionary game theory, we establish an evolutionary game model for the participation of enterprises, local governments, and the general public in carbon offset behaviors. We examine the overall evolutionary stable tendencies, evolutionary strategies, and relevant stability conditions of each stakeholder in the system. Furthermore, through numerical simulations, we provide visual representations of the impact of various factors, including each stakeholder’s initial participation level, government restrictions and subsidies, future development potential, and public participation. We investigate the underlying game mechanism and evolutionary pathway. This study offers a reference for low-carbon collaboration among enterprises, governments, and the public under carbon offset policies, based on rational assumptions and the interests of all participating stakeholders. For example, Europe would benefit from such a model, given its legislative framework, as well as the missions and initiatives of the European Union, which promote policies and approaches of this kind.

Please note that this study has several limitations. Firstly, the game theory used in model construction assumes that all parties operate under complete information. However, the participation conditions of each stakeholder require further refinement based on the specific application field and policy background. Furthermore, although the data model meets the relationship requirements of ESS, the subjective nature of the simulation of key parameter values necessitates further empirical research to obtain more precise estimates.