An Evolutionary Game Analysis of Decision-Making and Interaction Mechanisms of Chinese Energy Enterprises, the Public, and the Government in Low-Carbon Development Based on Prospect Theory

Abstract

The low-carbon development (LCD) of energy markets not only serves as a critical enabler in combating global climate change and advancing the green economy but also enhances global industrial competitiveness. Grounded in prospect theory, this study develops a tripartite evolutionary game model involving three core energy market stakeholders, i.e., energy enterprises, the public, and the government, to investigate the determinant factors and decision-making mechanisms underlying the LCD of energy enterprises, with subsequent simulation analyses conducted through MATLAB R2024a. The research findings indicate that loss aversion serves as the primary driver for energy enterprises’ adoption of LCD strategies. Public supervision demonstrates optimal effectiveness only under conditions of low risk and low loss, while risk sensitivity remains the dominant factor influencing the government’s strategic choices. Notably, government incentives combined with public supervision demonstrate significant synergistic effects in accelerating the corporate transition toward LCD. Accordingly, the government should actively promote LCD strategies to mitigate transformation risks for energy enterprises while concurrently optimizing regulatory frameworks to reduce public supervision costs and amplify incentive benefits, thereby fostering active public participation in LCD.

1. Introduction

Since the 21st century, the climate change crisis caused by carbon emissions and its solutions have increasingly attracted shared attention from the international community. Over the past decade, the share of coal in China’s energy consumption declined, from 65.8% in 2014 to 55.3% in 2023 [1]. However, according to data released by the National Bureau of Statistics in 2022, China’s energy enterprises consumed as much as 3070.86 million tons of standard coal, accounting for 56.77% of the country’s total energy consumption, exhibiting characteristics of high input, high pollution, low efficiency, and low output. As the largest developing country by total carbon emissions, China has never shied away from the challenge of LCD. In this context, promoting the LCD of the energy market, achieving high-quality development, and advancing economic and social decarbonization and greening are critical directions and key links for future socio-economic sustainable development, as well as an inevitable requirement for building a modern energy market system.

The LCD of China’s energy market is vital to shifting away from outdated development models and achieving sustainable growth. Success in this transition is important not only for building a low-carbon economy but also for gaining competitive advantage and ensuring long-term business growth [2]. Public supervision, as a typical social control mechanism, plays a significant role in the LCD process of energy enterprises [3]. In contrast to formal government regulation, public supervision is often seen as an effective means of regulating the exercise of government power, owing to the information advantage and pro-social behavior of the general public [4]. As a bottom-up approach to monitoring power, public supervision is broad and diverse. Public supervision not only encourages public engagement in social governance but also draws attention to the misconduct of entities such as enterprises and governments. Through public supervision, energy enterprises and governments can effectively improve service quality and operational efficiency, strengthen social cohesion, and safeguard citizens’ legitimate rights and interests [5]. Governments universally exercise regulatory oversight over energy markets through administrative instruments, employing mechanisms such as rewarding positive externalities and punishing negative externalities, thereby playing a pivotal role in advancing LCD within energy enterprises [6].

However, divergent goal expectations among energy enterprises, the public, and governments inevitably generate multi-stakeholder conflicts of interest during the LCD of energy markets. Moreover, their decision-making behaviors are demonstrably influenced by subjective cognitive biases and divergent value systems [7]. What factors influence the strategic choices of each stakeholder, and what evolutionary trends emerge from these factors? Specifically, how do risks, losses, and other variables affect the evolutionary outcomes of stakeholders? How can strategies be accelerated toward an optimal equilibrium state to achieve societal LCD? The key to addressing these challenges lies in clarifying the tripartite conflicts of interest, revealing the development laws of the dynamic evolutionary game paths of stakeholders, and establishing a coherent, stable, and implementable supervision and regulation system for sustainable LCD.

Prospect theory, as a valuable framework for aiding decision making in complex scenarios, can significantly support LCD by focusing on perceived benefits and losses [8]. It integrates decision-making factors from various stakeholders to provide insights tailored to different decision-making environments [9]. Evolutionary game theory serves as an effective framework for studying dynamic strategic changes among multiple boundedly rational actors in long-term repeated games. This approach provides profound insights into stakeholders’ behavioral patterns and strategic choice mechanisms while enabling the prediction of interactions and impacts between different strategies. The collaboration of these two approaches enables the integration of bounded rationality among energy enterprises, the public, and the government—thereby facilitating the formulation of optimal decisions under risks, losses, or other contextual factors while aiding in uncovering the mechanisms through which energy development strategies exert influence on stakeholders. Therefore, combining prospect theory with evolutionary games can optimize their application, providing a comprehensive analysis of risky decision-making processes and more effectively addressing the challenges faced by energy enterprises, the public, and the government during LCD decision making.

Building on this framework, this study constructs a tripartite evolutionary game model involving energy enterprises, the public, and the government to explore long-term complex interest interactions among these stakeholders. Through numerical simulations, the dynamic evolutionary game process and the stability state of the model are further observed and analyzed. First, by introducing public supervision as a key participant, this research study broadens the theoretical understanding of LCD and public supervision in energy enterprises through the integration of psychological and behavioral factors. Second, it enhances the design of government subsidy–punishment mechanisms, advancing game theory applications in LCD contexts. Third, it proposes targeted optimization strategies to address the challenges and critical issues in public supervision and government regulation during the implementation of LCD by energy enterprises, thereby providing a theoretical foundation and practical references for formulating and implementing low-carbon policies in energy markets.

2. Literature Review

LCD, as a comprehensive and systematic initiative, is empirically characterized by high implementation costs, prolonged timelines, elevated risks, and delayed outcomes in practice [10]. Nevertheless, LCD constitutes a critical component of socio-economic advancement, positing that reducing corporate environmental footprints through innovative systems, products, and processes can generate substantial economic value and competitive advantages for enterprises [11]. Consequently, serving as a cornerstone strategy for combating global climate change and attaining sustainable development objectives, LCD has attracted significant scholarly and policymaker engagement in recent years.

In recent studies, some scholars have constructed a Green and Low-Carbon Development Index (GLCDI), revealing that core cities create demonstration effects through technological innovation and policy coordination, yet persistent regional development disparities remain prominent [12]. Furthermore, research based on the Dynamic Spatial Durbin Model indicates that the effectiveness of LCD policies varies across cities at different development stages, emphasizing the need to design differentiated policies tailored to local resource endowments [13]. Scholars have shown that the carbon trading scheme can directly promote the upgrade of low-carbon technology and that this effect can be enhanced by establishing a market-oriented trading mechanism [14]. And carbon taxes serve as a critical tool for governments to regulate enterprises’ strategic choices. By integrating the advantages of different carbon tax policies and adopting a combination of input-oriented carbon tax and output-oriented tax has higher efficiency for the enhancement of generalized social LCD [15]. Meanwhile, research in Chinese listed enterprises reveals that LCD significantly enhances corporate value by improving market visibility, production efficiency, and financing capacity, thereby highlighting the positive correlation between LCD and corporate competitiveness [16]. Additionally, other studies reveal that digital environments serve as a core condition for advancing economic LCD; however, the divergent pathways observed across eastern, central, and western regions underscore the need to design phased goals aligned with regional digitalization levels [17].

By analyzing 14 industrial sectors in 17 OECD countries between 1975 and 2005, it was found that low-carbon technology can significantly reduce the energy intensity of energy markets, with a 1% increase in low-carbon technology patent applications correlating with a 0.03% decrease in energy intensity [18]. Some scholars have found that in general, government environmental regulation positively correlates with LCD to some extent; however, variations may exist across specific industries [19].

Some study using the Tobit model found that insufficient talent, weak supply–demand linkages, high sunk costs, insufficient capital, and weak managerial innovation impede enterprises’ low-carbon behavior transition [20]. In terms of the relationship between LCD and government behavior, LCD is divided into two modes, namely, internal independent R&D of low-carbon technology and external low-carbon technology introduction, and the government’s adoption of low-carbon development innovation subsidies or carbon tax affects the mode choice of LCD [21]. Other scholars have found that the increase in R&D investment, environmental protection, low-carbon investment, and environmental pressure on enterprises have a positive enhancement effect on LCD, while the two factors of local government environmental protection and low-carbon investment and environmental pressure have a significant effect on this transformation [22].

Public supervision is considered an effective way to prevent collusion between governments and enterprises while significantly improving their efforts in LCD [23]. Research indicates that public supervision positively impacts LCD in developed regions, but its effect is weaker in less developed areas. The effectiveness of this supervision is linked to the strength of government environmental regulations, with public supervision being the most effective when regulations surpass a certain threshold [24]. Researchers have used evolutionary games to study interactions among governments, businesses, and the public. Their findings show that public supervision is influenced by its costs and the psychological benefits of participation [25] and public scrutiny can reduce the supervision burden on local governments, lowering governance costs and increasing policy flexibility [26].

Some researchers have indicated that stronger government regulations could play a significant role in enabling enterprises to innovate in LCD [27]. Research suggests that government regulation can act as an intermediary in LCD, helping enterprises to adopt low-carbon technologies effectively and promoting LCD. Some scholars have found that the impact of government regulation varies depending on the level of economic development [28]. At low economic levels, such regulation has an inhibitory effect on low-carbon technological innovation. As development increases, the effect is relatively weak. However, at high economic levels, regulation can play a role in promoting advancements in low-carbon technology [29]. It is also worth noting that there are some notable regional differences in how government regulations affect LCD [30].

Prospect theory is a decision-making model rooted in psychology and behavior that has become widely used in fields such as economics, political science, and psychology [31]. In agriculture, research indicates that farmers who adhere to the principles of prospect theory may perceive greater risks associated with climate change [32]. Additionally, in the construction sector, the application of prospect theory has assisted governments in reevaluating the allocation of industry subsidies. This approach aims to provide targeted incentives for builders and has supported the government’s initiative to promote LCD practices [33]. Some scholars have suggested that based on prospect theory and evolutionary game theory, enterprises’ LCD decisions may be influenced by risk perception and loss aversion, in addition to the size of benefits and costs. It is thought that higher risk sensitivity and lower loss aversion may increase enterprises’ motivation for green technology innovation in the prospect evaluation stage [34]. Some scholars have analyzed the impact of the carbon tax rate on LCD by using prospect theory. Their findings suggest that while the government may not be very sensitive to carbon tax rates, energy enterprises show a certain degree of sensitivity to them. Furthermore, the public’s awareness of low-carbon practices significantly influences the selection of strategies [35].

As a tool for analyzing strategic interactions among multiple stakeholders, game theory is widely applied in energy policy research to simulate complex scenarios such as low-carbon transitions, carbon reduction cooperation, and technological innovation. Scholars have established a tripartite evolutionary game-theoretic model involving large steel enterprises, small-to-medium steel enterprises, and governments, providing theoretical support for steel industries and policymakers to address carbon border adjustment mechanisms [36]. Another study established a tripartite evolutionary game model of local government, power industry, and coal enterprises and showed that reducing costs drives coordinated development and access to low-carbon technologies and financial support is critical for coal firms to optimize production capacity [37]. Some researchers have developed an evolutionary game model which reveals that high penalties, low public supervision, and mid-level subsidy and carbon taxes form the optimal strategy combination to encourage collaborative green technology innovation between enterprises and financial institutions [38]. Integrating prospect theory, a tripartite game model involving governments, developers, and consumers reveals differences in risk perception and sensitivity to potential losses between consumers and developers, informing strategies for low-carbon building markets [39]. By using evolutionary game theory, scholars have simulated the dynamic decision-making behaviors of governments, communities, and individuals in low-carbon transportation systems, exploring how to achieve carbon reduction goals with minimal fiscal expenditure [40]. From the perspective of dynamic subsidies and taxation, a tripartite evolutionary game model involving enterprises, governments, and the public demonstrates that increased government subsidies encourage enterprises to adopt low-carbon innovation strategies, while public rewards and punishments exert stronger influence on governments than on enterprises [41].

A review of extant literature reveals substantial scholarly advancements in examining the roles of energy markets and policies in advancing LCD. However, current research on energy market LCD predominantly concentrates on transformation drivers and pathway selection. Although the facilitative effects of public oversight and governmental regulatory frameworks on corporate LCD have been well documented, limited scholarly attention has been devoted to analyzing the decision-making rationales and strategic mechanisms of interaction among the public, governments, and enterprises during this transition. To address this gap, this study integrates prospect theory to develop a tripartite evolutionary game model encompassing energy enterprises, the public, and governmental entities. The model enables the in-depth analysis of stakeholders’ strategic choices under behavioral constraints such as risk sensitivity and loss aversion during LCD implementation.

3. Model Assumptions and Payoff Matrix

3.1. Model Assumptions

In the evolutionary game process of LCD, energy enterprises, the public, and the government interact in the following ways: (1) Energy enterprises consider multiple factors when selecting LCD strategies, including anticipated future benefits, public supervision intensity, and government regulations, rewards, and punishments [42]. (2) The public, as a key component of social supervision, serves as an important complement to government regulation. When the public fails to actively supervise energy enterprises’ LCD practices, it increases the likelihood of corporate opportunistic behavior [43]. (3) The government can promote LCD adoption through policy guidance, financial subsidies [28], and regulation. Additionally, the government can enhance public supervision motivation through targeted incentives.

In summary, these stakeholders engage in dynamic evolutionary games based on cost–benefit perceptions, forming the interaction framework illustrated in Figure 1.

3.2. Basic Model Assumptions

3.2.1. Assumption 1

The process of promoting LCD involves the willingness of energy enterprises to pursue LCD, the willingness of the public to supervise, and the willingness of the government to enforce regulations. Energy enterprises, the public, and the government operate under conditions of limited rationality and information asymmetry. Starting from this premise, as the main players in LCD, they make continuous adjustments to their strategic choices within the dynamic evolutionary game process. Their decisions are based on the goal of maximizing their own interests and take into account initial evolutionary conditions, evolutionary paths, and various influencing factors. This adjustment process continues until a particular strategy yields the highest return and the system reaches a stable state.

3.2.2. Assumption 2

Energy enterprises, the public, and the government adjust their strategies based on the decisions made by the other two parties. Enterprises have two strategic options: positive LCD and negative LCD. These strategies are represented as {positive, negative}, where the probability of choosing positive is denoted by $x$, and the probability of opting for negative is $1-x$. Similarly, the public has two strategic choices: positive supervision and negative supervision, represented as {positive, negative}. The probability of the public choosing positive supervision is $y$, while the probability of negative supervision is $1-y$. The government also has two strategic options: positive supervision and negative supervision, denoted by {positive, negative}. The probability of the government selecting positive supervision is $z$, and the probability of choosing negative supervision is $1-z$. In summary, the probabilities for each strategy choice are defined in the following, where $0\le x,y,z\le 1$.

3.2.3. Assumption 3

The three stakeholders make strategic choices influenced by their own perceptions, and their decision-making behavior exhibits characteristics described by prospect theory [44]. Stakeholders tend to avoid risk when there is a certain prospect of return, exhibit risk aversion when faced with potential gains, and seek risk when confronted with losses. At the same time, they prefer to minimize losses rather than maximize profits and tend to ignore small probabilities, despite the potential for adverse consequences such as risk accumulation, opportunity cost mismatch, regulatory lag, etc. Moreover, stakeholders evaluate outcomes based on current situations rather than final status, as assumed in expected utility theory [45,46].

For this reason, prospect theory is introduced to portray the decision-making behavior under limited rationality [47]. In this paper, the psychological perception of the gains of the three subjects is represented by the prospect value $V$, as shown in Equation (1).

$$\left\{\begin{array}{c} V=\sum \pi \left(p_i\right) v\left(xi\right)\hfill \\ \pi (p_i)={\displaystyle \frac{p_i^{\gamma }}{p_i^{\gamma }+{\left(1-p_i^{\gamma }\right)}^{\frac{1}{\gamma }}}}\hfill \\ v(xi)=\left\{\begin{array}{c}{x}_i^{\alpha } (xi\ge 0)\\ -\lambda {(xi)}^{\alpha } (xi<0)\hfill \end{array}\right.\hfill \end{array}\right.$$

(1)

where $p_i$ denotes the actual probability of event $i$ occurring, $\pi (p_i)$ is the probability weighting function, $xi$ denotes the difference between the decision maker’s actual gain and the value of the reference point after the event occurs, $v(xi)$ is the value function, γ is the gain perception probability coefficient, $\alpha$ is the coefficient of risk sensitivity (0 < $\alpha$ < 1), and λ (λ > 1) is the coefficient of loss aversion.

3.2.4. Assumption 4

Benefits and costs associated with energy enterprises: When energy enterprises opt for positive LCD, they have the opportunity to benefit from LCD advantages $R_1$, while also bearing the associated LCD costs $C_1$. If energy enterprises engage in negative LCD, they may be able to obtain benefits $R_2$. However, they may also be required to pay government penalties $P$ in instances where they are subject to active regulation by the government or reported due to active public supervision. $R_1>R_2$.

Benefits and costs associated with the public: When the public adopts a positive supervision strategy, it costs $C_2$ in time to supervise the energy enterprises. At this time, the public receives government incentives $D$ for active participating in the construction of society. When the enterprises adopt negative LCD, positive public supervision will incur a loss $S_3$, and negative public supervision will result in a loss $S_1$. $S_1>S_3$.

Benefits and costs associated with the government: When the government implements positive regulations, it provides subsidies $B$ to energy enterprises that positively participate in LCD. However, this also requires the government to incur regulatory costs $C_3$, such as manpower and capital, while gaining reputational benefits $R_3$, which can enhance the government’s credibility and overall image. In this scenario, if enterprises engage in LCD, it can also generate social benefits $R_4$. Conversely, if the government adopts a negative regulatory strategy, it will face costs $C_4$ and a reputational loss $S_2$ due to a decline in credibility. $C_3>C_4$.

By prospect theory, there is no discrepancy between actual utility and perceived value when benefits or losses are certain. Conversely, psychological perceived utility only arises when the decision maker is uncertain about benefits and costs [44]. The costs of implementing active regulation policies by the government are related only to themselves and are deterministic expenditures. Government subsidies, penalties, and incentives for energy enterprises and the public are determined by the government and are deterministic expenditures. The costs of LCD are only related to themselves and are deterministic, and the benefits of negative LCD are the same as before and are deterministic benefits. Therefore, $C_1$, $C_3$, $C_4$,$R_2$, $B$, $P$, and $D$ do not have perceptual bias, while $R_1$,$R_3$, $R_4$, $C_2$, $S_1$, $S_2$, and $S_3$ have uncertainty and perceptual features. The related parameters are described in

Table 1. Variables and explanation.

Subject of Decision	Variables	Description
Energy enterprises	$R_1$	Benefits of positive LCD
	$R_2$	Benefits of negative LCD
	$C_1$	Costs of LCD
	$P$	Penalties charged for negative LCD when reported by the public or positive regulation by the government
The public	$C_2$	Cost of time and effort in proactive supervision
	$D$	Government incentives
	$S_1$	Loss of negative public supervision when energy enterprises adopt negative LCD
	$S_3$	Loss of positive public supervision when energy enterprises adopt negative LCD
The government	$R_3$	Reputational benefits when regulation is positive
	$S_2$	Reputational loss when regulation is negative
	$R_4$	Social benefits when energy enterprises adopt positive LCD and there is positive regulation
	$C_3$	Costs under positive regulation
	$C_4$	Costs under negative regulation
	$B$	Subsidies for energy enterprises adopting positive LCD

.

According to the above assumptions, the variables and parameters are summarized in Table 1.

Based on the above assumptions, a tripartite game payoff matrix is constructed, as shown in

Table 2. Tripartite game payment matrix.

The Government	The Public	Energy Enterprises
		Positive			Negative
		Energy Enterprises	Public	Government	Energy Enterprises	Public	Government
Positive	Positive	$V\left(R_1\right)-C_1+B$	$D-C_2$	$V\left(R_3\right)+V\left(R_4\right)-C_3-D-B$	$R_2-P$	$D-V\left(C_2\right)-V\left(S_3\right)$	$V\left(R_3\right)+P-C_3-D$
	Negative	$V\left(R_1\right)-C_1+B$	0	$V\left(R_3\right)+V\left(R_4\right)-C_3-B$	$R_2-P$	$-V\left(S_1\right)$	$V\left(R_3\right)+P-C_3$
Negative	Positive	$V\left(R_1\right)-C_1$	$-C_2$	$-V\left(S_2\right)-C_4$	$R_2-P$	$-V\left(C_2\right)-V\left(S_3\right)$	$P-V\left(S_2\right)-C_4$
	Negative	$V\left(R_1\right)-C_1$	0	$-V\left(S_2\right)-C_4$	$R_2$	$-V\left(S_1\right)$	${-C}_4$

.

4. Model Solution

4.1. Stability Analysis of Energy Enterprises

The expected prospect values for the two strategies of the energy enterprises in LCD are represented as $E_1^{\left(1\right)}$ and $E_1^{\left(2\right)}$, and the average prospect value is denoted by $\overline{E_1}$:

$$E_1^{\left(1\right)}=z[y\left(V\left(R_1\right)-C_1+B\right)+\left(1-y\right)\left(V\left(R_1\right)-C_1+B\right)]+(1-z)[y\left(V\left(R_1\right)-C_1+B\right)+(1-y)(V\left(R_1\right)-C_1+B)]$$

(2)

$$E_1^{\left(2\right)}=z[y\left(R_2-P\right)+\left(1-y\right)\left(R_2-P\right)]+\left(1-z\right)[y\left(R_2-P\right)+(1-y)R_2]$$

(3)

$$\overline{E_1}=x{E}_1^{(1)}+(1-x)E_1^{(2)}$$

(4)

The replicated dynamic equation can be calculated as follows:

$$F\left(x\right)=dx/dt=x\left(E_1^{\left(1\right)}-{\overline{E}}_1\right)=x\left(1-x\right)\left(E_1^{\left(1\right)}-E_1^{\left(2\right)}\right)=x(1-x)[V\left(R_1\right)-C_1-R_2+By+P(z+y-zy)]$$

(5)

We find the first derivative of Equation (5) as

$${\displaystyle \frac{dF(x)}{dx}}=(1-2x)[V\left(R_1\right)-C_1-R_2+By+P(z+y-zy)]$$

We set $G\left(z^{*}\right)$ to

$$G\left(z^{\mathrm{*}}\right)=V\left(R_1\right)-C_1-R_2+y(B+P)+zP(1-y)$$

According to the stability theorem of differential equations, the probability that energy enterprises are in a steady state of positive LCD must be satisfied: $F\left(x\right)=0$ and $dF(x)/dx<0$. Let $G\left(z^{*}\right)=0$; then,

$$z=z^{\mathrm{*}}={\displaystyle \frac{V\left(R_1\right)-C_1-R_2+y(B+P)}{-P(1-y)}}$$

When $z>z^{*}$, $G\left(z\right)>0$, $F\left(1\right)=0$, and $dF(x)/dx{│}_{x=1}<0$,$x=1$ is an evolutionary stabilization point for energy enterprises which have a strategy of positive LCD. When $z=z^{*}$, $G\left(z\right)=0$, $F\left(x\right)=0$, and $dF(x)/dx\equiv 0$, all $x$ are in equilibrium, i.e., the behavior adopted by energy enterprises does not change the stabilization strategy regardless of positive or negative LCD. When $z<z^{*}$, $G\left(z\right)<0$, $F\left(0\right)=0$, and $dF(x)/dx{│}_{x=0}<0$,$x=0$ is an evolutionary stabilization point for energy enterprises which have a strategy of negative LCD. In summary, this shows that the probability of LCD increases from $x=0$ to $x=1$ as $z$ gradually increases.

4.2. Stability Analysis of the Public

The expected prospect values for the two strategies of the public in LCD are represented as $E_2^{\left(1\right)}$ and $E_2^{\left(2\right)}$, respectively, and the average prospect value is denoted by $\overline{E_2}$:

$$E_2^{\left(1\right)}=z\left[x\left(D-V\left(C_2\right)\right)+\left(1-x\right)\left(D-V\left(C_2\right)-V\left(S_3\right)\right)\right]+(1-z)[x\left(-V\left(C_2\right)\right)+(1-x)(D-V\left(C_2\right))]$$

(6)

$$E_2^{\left(2\right)}=-(1-x)V\left(S_1\right)$$

(7)

$$\overline{E_2}=y{E}_2^{(1)}+(1-y)E_2^{(2)}$$

(8)

The replicated dynamic equation can be calculated as follows:

$$\begin{array}{c}F\left(y\right)=dy/dt=y\left(E_2^{\left(1\right)}-{\overline{E}}_2\right)=y\left(1-y\right)\left(E_2^{\left(1\right)}-E_2^{\left(2\right)}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=y\left(y-1\right)[V\left(C_2\right)-D-V\left(S_1\right)+x(D+V\left(S_1\right))+zV\left(S_3\right)-xz(D-V\left(S_3\right))]\end{array}$$

(9)

We find the first derivative of Equation (9) as

$${\displaystyle \frac{dF\left(y\right)}{dy}}=\left(2y-1\right)[V\left(C_2\right)-D-V\left(S_1\right)+x(D+V\left(S_1\right))+zV\left(S_3\right)-xz(D-V\left(S_3\right))]$$

We set $M\left(x^{*}\right)$ to

$$M\left(x^{\mathrm{*}}\right)=V\left(C_2\right)-D-V\left(S_1\right)+x[D+V\left(S_1\right)-z(D-V\left(S_3\right))]+zV\left(S_3\right)$$

Let $M\left(x^{*}\right)=0$; then,

$$x=x^{\mathrm{*}}={\displaystyle \frac{V\left(C_2\right)-D-V\left(S_1\right)+zV\left(S_3\right)}{-D-V\left(S_1\right)+z(D-V\left(S_3\right))}}$$

When $x>x^{*}$, $M\left(x\right)>0$, $F\left(0\right)=0$, and $dF(y)/dy{│}_{y=0}<0$,$y=0$ is an evolutionary stabilization point for the public which has a strategy of negative supervision. When $x=x^{*}$, $M\left(x\right)=0$, $F\left(y\right)=0$, and $dF(y)/dy\equiv 0$, all $y$ are in equilibrium, i.e., the behavior adopted by the public does not change the stabilization strategy regardless of positive or negative supervision. When $x<x^{*}$, $M\left(x\right)<0$, $F\left(1\right)=0$, and $dF(y)/dy{│}_{y=1}<0$,$x=1$ is an evolutionary stabilization point for the public which has a strategy of positive supervision. In summary, this shows that as $x$ increases, the probability of the public choosing positive supervision strategy decreases from $y=1$ to $y=0$.

4.3. Stability Analysis of the Government

The expected prospect values for the two strategies of the government in LCD are represented as $E_3^{\left(1\right)}$ and $E_3^{\left(2\right)}$, respectively, and the average prospect value is denoted by $\overline{E_3}$:

$$\begin{array}{c}{E}_3^{\left(1\right)}=x\left[y\left(V\left(R_3\right)+V\left(R_4\right)-C_3-B\right)+\left(1-y\right)\left(V\left(R_3\right)+V\left(R_4\right)-C_3-B\right)\right]\\ +\left(1-x\right)[y\left(V\left(R_3\right)+P-C_3\right)+\left(1-y\right)\left(V\left(R_3\right)+P-C_3\right)]\end{array}$$

(10)

$$\begin{gathered} E_3^{\left(2\right)}=x\left[y\left(-V\left(S_2\right)-C_4\right)+\left(1-y\right)\left(-V\left(S_2\right)-C_4\right)\right] \\ +\left(1-x\right)\left[y\left({P-S}_2-D-C_4\right){-C}_4\left(1-y\right)\right] \end{gathered}$$

(11)

$$\overline{E_3}=z{E}_3^{(1)}+(1-z)E_3^{(2)}$$

(12)

The replicated dynamic equation can be calculated as follows:

$$\begin{gathered} F\left(z\right)=dz/dt=z\left(E_3^{\left(1\right)}-{\overline{E}}_3\right)=z\left(1-z\right)\left(E_3^{\left(1\right)}-E_3^{\left(2\right)}\right) \\ =z\left(1-z\right)[C_4{-C}_3+P+V\left(R_3\right)+x\left(V\left(S_2\right)+V\left(R_4\right)-P-B\right) \\ +y(S_2-P)+xy\left(P-D{-S}_2\right)] \end{gathered}$$

(13)

We find the first derivative of Equation (13) as

$$\begin{gathered} {\displaystyle \frac{dF\left(z\right)}{dz}}=\left(1-2z\right)[C_4{-C}_3+P+V\left(R_3\right)+x\left(V\left(S_2\right)+V\left(R_4\right)-P-B\right) \\ +y(V\left(S_2\right)-P)+xy\left(P-D-V\left(S_2\right)\right)] \end{gathered}$$

We set $J\left(y^{*}\right)$ to

$$\begin{gathered} J\left(y^{\mathrm{*}}\right)=C_4{-C}_3+P+V\left(R_3\right)+x\left(V\left(S_2\right)+V\left(R_4\right)-P-B\right)+ \\ y(V\left(S_2\right)-P)+xy\left(P-D-V\left(S_2\right)\right) \end{gathered}$$

Let $J\left(y^{*}\right)=0$; then,

$$y=y^{\mathrm{*}}={\displaystyle \frac{C_4{-C}_3+P+V\left(R_3\right)+x\left(V\left(S_2\right)+V\left(R_4\right)-P-B\right)}{-V\left(S_2\right)+P-x\left(P-D-V\left(S_2\right)\right)}}$$

When $y>y^{*}$, $J\left(y\right)>0$, $F\left(1\right)=0$, and $dF(z)/dz{│}_{z=1}<0$, $z=1$ is an evolutionary stabilization point for the government which has a strategy of positive regulation. When $y=y^{*}$, $M\left(y\right)=0$, $F\left(z\right)=0$, and $dF(z)/dz\equiv 0$, all $z$ are in equilibrium, i.e., the behavior adopted by the government does not change the stabilization strategy regardless of positive or negative regulation. When $y<y^{*}$, $J\left(y\right)<0$, $F\left(0\right)=0$, and $dF(z)/dz{│}_{\mathrm{z}=0}<0$,$z=0$ is an evolutionary stabilization point for government which has a strategy of negative regulation. In summary, this shows that the probability of the government choosing the positive regulation increases from $z=0$ to $z=1$ as $y$ gradually increases.

4.4. Analysis of System Evolutionary Stability Points

In order to provide a more comprehensive analysis of the equilibrium strategies of the tripartite evolutionary game system, a system of replicated dynamic equations, as shown in Equation (14), is developed based on Equations (5), (9) and (13).

$$\left\{\begin{array}{c}F\left(x\right)=x(1-x)[V\left(R_1\right)-C_1-R_2+By+P(z+y-zy)]=0\hfill \\ F\left(y\right)=y\left(y-1\right)[V\left(C_2\right)-D-S_1+x\left(D+S_1\right)+z{S}_3-xz\left(D{-S}_3\right)]=0\hfill \\ F\left(z\right)=z\left(1-z\right)[C_4{-C}_3+P+V\left(R_3\right)+x\left(V\left(S_2\right)+V\left(R_4\right)-P-B\right)\hfill \\ +y(V\left(S_2\right)-P)+xy\left(P-D-V\left(S_2\right)\right)]=0\end{array}\right.$$

(14)

Let $F\left(x\right)=0$, $F\left(y\right)=0$, and $F\left(z\right)=0$; then, $S_1\left(\mathrm{0,0,0}\right)$, $S_2\left(\mathrm{0,1,0}\right)$, $S_3\left(\mathrm{1,0,0}\right)$, $S_4\left(\mathrm{0,0,1}\right)$, $S_5\left(\mathrm{1,1,0}\right)$, $S_6\left(\mathrm{1,0,1}\right)$, $S_7\left(\mathrm{0,1,1}\right)$, and $S_8(\mathrm{1,1,1})$ are strategic partial equilibrium points for the tripartite evolutionary game system.

Friedman [48] argued that an evolutionary stability strategy (ESS) can be obtained by the stability analysis of the system’s Jacobian matrix. According to the equations in (14), the Jacobian matrix can be expressed as follows:

$$J=\left[\begin{array}{c}{\displaystyle \frac{dx/dt}{dx}}{\displaystyle \frac{dx/dt}{dy}}{\displaystyle \frac{dx/dt}{dz}}\\ {\displaystyle \frac{dy/dt}{dx}}{\displaystyle \frac{dy/dt}{dy}}{\displaystyle \frac{dy/dt}{dz}}\\ {\displaystyle \frac{dz/dt}{dx}}{\displaystyle \frac{dz/dt}{dy}}{\displaystyle \frac{dz/dt}{dz}}\end{array}\right]$$

According to the first theorem of Lyapunov, it is known that this equilibrium point is evolutionarily stable if all three eigenvalues are negative [49]. The eight equilibrium points of the system are brought into the Jacobi matrix to calculate the eigenvalues of each equilibrium point and its evolutionary stability conditions, as shown in

Table 3. The stability analysis of the equilibrium point.

Equilibrium Point	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$
$S_1(\mathrm{0,0,0})$	$V\left(R_1\right)-C_1-R_2$	$D-V\left(C_2\right)+V\left(S_1\right)$	$P+C_4-C_3+V\left(R_3\right)$
$S_2(\mathrm{0,1,0})$	$B-C_1+P+V\left(R_1\right)-R_2$	$V\left(C_2\right)-D-V\left(S_1\right)$	$C_4-C_3+V\left(R_3\right)+S_2$
$S_3(\mathrm{1,0,0})$	$C_1-V\left(R_1\right)+R_2$	$-V\left(C_2\right)$	$C_4-C_3-B+V\left(R_4\right)+V\left(R_3\right)+S_2$
$S_4(\mathrm{0,0,1})$	$P-C_1+V\left(R_1\right)-R_2$	$D-V\left(C_2\right)+V\left(S_1\right)-V\left(S_3\right)$	$C_3-C_4-P-V\left(R_3\right)$
$S_5(\mathrm{1,1,0})$	$C_1-B-P-V\left(R_1\right)+R_2$	$V\left(C_2\right)$	$C_4-C_3-B-D+V\left(R_4\right)+V\left(R_3\right)+S_2$
$S_6(\mathrm{1,0,1})$	$C_1-P-V\left(R_1\right)+R_2$	$D-V\left(C_2\right)$	${B+C}_3-C_4-S_2-V\left(R_4\right)-V\left(R_3\right)$
$S_7(\mathrm{0,1,1})$	$B-C_1+P+V\left(R_1\right)-R_2$	$V\left(C_2\right)-D-V\left(S_1\right)+V\left(S_3\right)$	$C_3-C_4-V\left(R_3\right)-S_2$
$S_8(\mathrm{1,1,1})$	$C_1-B-P-V\left(R_1\right)+R_2$	$V\left(C_2\right)-D$	$C_3-C_4+B+D-V\left(R_4\right)-V\left(R_3\right)-S_2$

.

To encourage energy enterprises to positively adopt LCD, the ideal state of positive LCD by energy enterprises, positive monitoring by the public, and positive regulation by the government is constructed, i.e., the evolutionary game model is finally evolved to $S_8\left(\mathrm{1,1,1}\right)$, and the set of strategies of the three stakeholders is {positive LCD, positive supervision, positive regulation}.

Table 3 shows that in order to achieve the ideal state $S_8\left(\mathrm{1,1,1}\right)$, the following need to be satisfied: $R_2-P<R_1{-C}_1+B,$ $V\left(C_2\right)<D$, and $-S_2-C_4<R_3+R_4-C_3-B-D$. That is say, the benefits of positive government regulation exceed their costs. The government’s incentives for positive public supervision are greater than the costs of positive public supervision. Moreover, government subsidies and penalties ensure that energy enterprises gain more from positive LCD than from negative LCD.

Regarding the evolutionary trajectory of the ESS and its stabilization conditions, the pivotal factors influencing the evolution of energy enterprises towards an ideal state include the benefits and costs of positive LCD, the benefits of negative LCD, and the strength of government subsidies and penalties. For the public, the benefits of positive supervision are the main factors driving the evolution towards the ideal state. For the government, the reputational and social benefits of implementing positive regulation, as well as the reputational losses associated with negative regulation, are the key factors driving the system towards the ideal state. In practice, the government can reduce the cost of LCD for energy enterprises by updating infrastructure, providing tax incentives, and implementing other supportive policies. On the other hand, the government should enhance the level of digitalization in society to reduce the cost of public supervision. Concurrently, the government can increase incentives and penalties to amplify the benefits of positive public supervision. Furthermore, it can diminish the benefits of negative LCD, thereby increasing the likelihood of positive LCD.

5. Simulation

5.1. Simulation Analysis Data

Herein, MATLAB R2024a was used to simulate the relevant model parameters and analyze the influence of each parameter on the evolutionary game behavior of the three stakeholders in the process of LCD. Due to the large number of influencing factors, when discussing how a single factor affects the change in the stable strategy, it is necessary to set the remaining factors to meet the specified conditions. This makes it easier to observe the impact of the change in a particular factor on the ideal stable state. In order for the system to eventually evolve to $S_8\left(\mathrm{1,1,1}\right)$, the stability conditions $C_1-B-P-R_1+R_2<0,C_2-D<0$, and $C_3-C_4+B+D-R_3-R_4-S_2<0$ need to be satisfied. Given the considerable number of variables at play, when examining the impact of a specific factor on the stabilization strategy, the remaining variables are set to achieve the ideal state, thereby facilitating the observation of the effect of the change in one factor on the ideal state. Considering that the LCD of the economy and society is essential to modern development, the government is becoming increasingly aware of the benefits and challenges associated with this transformation. As a result, government departments are more likely to take a positive approach to regulating LCD. Let $x=0.5$, $y=0.5$, and $z=0.6$; according to the results of previous research [50,51], the behavioral preferences of decision makers can be indicated when $\alpha =0.88$, $\lambda =2.25$, and $\gamma =0.61$. Other parameters, such as the strength of incentives, subsidies, and penalties, are referenced in past empirical studies, government documents, and previous research results [22,52]. The initial value of each parameter is set as follows: $C_1=43$ [17], $C_3=35$ [39], $C_4=20$ [41], $B=15$ [37], $P=20$ [36], $D=12$ [53], $R_1=57$ [54], $R_2=30$ [52], $R_3=25$ [40], $R_4=14$ [55], $S_2=30$ [40], and $C_2=10$ [38].

5.2. Simulation Analysis of Initial Values Affecting Evolution

By setting the initial values to 0.5, 0.5, and 0.6 for energy enterprises, the public, and the government, the evolutionary path of the tripartite game behavior is obtained as shown in Figure 2. It can be seen that although the three stakeholders evolve at different rates, they eventually evolve to the ideal state $S_8\left(\mathrm{1,1,1}\right)$. The government is the fastest to reach 1, while the public evolves more quickly than the energy enterprises in the early stages. However, this trend gradually slows down and ultimately reaches 1 last. This result is consistent with the phase diagram change in the equilibrium point analysis mentioned above.

To analyze the influence of different initial values of each of the three stakeholders on the overall evolution path, the initial values of energy enterprises, the public, and the government are selected cyclically in the interval [0, 1] with a unit of 0.2. The resulting evolution is presented in Figure 3. It can be found that although different initial values lead to different evolution rates, the system will reach $S_8\left(\mathrm{1,1,1}\right)$. In the Chinese context, in the initial stage of the LCD of energy enterprises, it is necessary for the public and the government to implement positive supervision and regulation in order to motivate energy enterprises to engage in LCD. As the LCD of energy enterprises progresses, it becomes appropriate for the public and the government to relax supervision and regulation.

5.3. Simulation Analysis of Factors Affecting Evolution

On the basis of the initial value setting of each parameter, the initial values of $x$, $y$, and $z$ are set to (0.5, 0.5, 0.6). Next, the coefficient of loss aversion is kept constant, while the coefficient of risk sensitivity takes on values of 0.8, 0.88, and 1. In contrast, the coefficient of risk sensitivity remains constant, with the coefficient of loss aversion assuming values of 2, 2.25, and 2.5. This approach aims to examine how the risk sensitivity and loss aversion sensitivity of energy enterprises, the public, and the government influence their evolutionary trajectories. The results are illustrated in Figure 4 and Figure 5.

According to the simulation results in Figure 4, it can be seen that when $\alpha$ is 0.8 and 0.88, the system is stable at $S_8\left(\mathrm{1,1,1}\right)$, and when $\alpha =1$, the public evolves towards negative supervision. As the value of $\alpha$ increases, the government becomes less sensitive to the costs of positive regulation. This leads to an acceleration of the rate at which the government chooses to implement positive regulation. However, as $\alpha$ increases, the public becomes more sensitive to the costs of positive supervision and gradually evolves towards negative supervision, and there is an absence of a clear change in the energy enterprises’ choice strategy.

According to the simulation results in Figure 5, it can be seen that when $\lambda$ is 2, 2.25, and 2.5, the system is stable at $\left(\mathrm{1,1,1}\right)$. As the value of $\lambda$ increases, the loss aversion sensitivity of energy enterprises and the public to the costs of positive LCD and positive supervision increases. This results in a slowing down of the rate at which they choose positive LCD and positive supervision and in the absence of a clear change in the government’s choice strategy.

The above results indicate that the choices of energy enterprises are strongly influenced by the coefficient of loss aversion, while both the coefficient of risk sensitivity and the coefficient of loss aversion have a strong influence on the public’s choices, and the government’s choices are strongly influenced by the coefficient of risk sensitivity.

5.4. Strategy Evolution for Different Values of $P$, $D$, and $B$

As illustrated in Figure 6, when the government’s penalties for the negative LCD of energy enterprises reach a certain threshold, these enterprises start to shift towards positive LCD. As the penalties increase, the pace at which energy enterprises adopt positive low-carbon practices also accelerates. In other words, as government penalties rise for negative LCD, the willingness of energy enterprises to pursue positive LCD increase.

Figure 7a illustrates that when government subsidies are low, energy enterprises tend to pursue negative LCD. However, as government subsidies increase, energy enterprises evolve towards positive LCD. Figure 7b demonstrates that once the subsidy reaches a certain threshold, the government initially chooses positive regulation, but this rapidly declines until it evolves into negative regulation. It has been demonstrated that the government is encouraging the positive LCD of energy enterprises by reducing the cost of such transformation by providing subsidies. Nevertheless, while a robust subsidy policy bolsters the motivation of energy enterprises to pursue positive LCD, it may also precipitate an uptick in the cost of government regulation, thereby dampening the appetite for such support. It is, therefore, incumbent upon the government to exercise due diligence in determining the quantum and duration of subsidies and implement a long-term subsidy mechanism.

Figure 8a shows that when government incentives for the public are low, the public is more likely to opt for negative supervision due to the associated supervision costs. However, as incentives increase, the public tends to shift towards positive supervision. In contrast, as shown in Figure 8b, when the government chooses to provide high incentives, although it initially prefers to encourage active public supervision through these measures, the sustained provision of high incentives imposes escalating financial burdens over time, ultimately driving governments to revert to negative regulation. Research has shown that government incentives are vital to promoting positive public supervision. However, it is important to be moderate, as providing too many incentives may reduce the government’s willingness to offer support. In the early stages of development, the government can effectively use higher incentives to encourage public participation in positive supervision. As the public’s understanding of supervision grows, it is essential to gradually decrease these incentives to ensure long-term sustainability.

6. Discussion and Suggestions

6.1. Conclusions

This study employs an evolutionary game approach grounded in prospect theory to investigate LCD. It constructs a tripartite evolutionary game model comprising energy enterprises, the public, and the government, systematically analyzing their evolutionary trajectories and examining how risk sensitivity and loss aversion propensity shape game equilibria. The analytical outcomes are detailed in the following sections.

This study identifies critical factors shaping strategic choices among energy enterprises, the public, and governments. Specifically, energy enterprises are more affected by LCD costs and government penalties. Increasing penalties for negative LCD practices and enhancing subsidies for positive LCD adoption can encourage energy enterprises to adopt LCD. The public’s decision to adopt active supervision is influenced by perceived risks and costs simultaneously. The government can encourage active public supervision through measures such as providing regulatory incentives. However, it is crucial to ensure that subsidies and rewards remain within reasonable bounds to avoid becoming a long-term financial burden on the government. Moreover, the government values its reputation and is sensitive to risks; increasing reputational losses from adverse regulations can further incentivize the government to adopt positive regulations.

6.2. Discussion

Numerous scholars have explored how effective government energy policies can promote LCD in energy enterprises [56], focusing on areas such as setting appropriate policy intensity across different development stages [57] and the impact of government regulation on competitive dynamics among enterprises [54], the interplay between government energy policies and market mechanisms [58], and the role of government taxation in driving LCD for energy enterprises [53].

These studies primarily emphasize the influence of government policies on energy enterprises’ LCD. In contrast, this research study investigates the factors affecting strategic choices among energy enterprises, the public, and governments, as well as their interactive relationships. Key findings reveal the following.

Perceived loss serves as a critical driver for energy companies in transitioning to LCD, while governments are more significantly influenced by perceived risk. This divergence stems from distinct motivations. Energy enterprises, constrained by large-scale investments and singular revenue models, often require several years or even decades to achieve profitability [59], making them highly sensitive to cost-related risks. Considering the government’s social welfare characteristics, the main purpose of its policies is to reduce risks and keep social development stable. Thus, the government can bear the greater cost [60]. Regarding the public, due to its limited ability to bear losses and risks, it is necessary for the government to actively provide a transparent monitoring platform and stable guarantees for the public.

6.3. Policy Suggestions

Based on this, suggestions are proposed for LCD under public supervision and government regulation.

Firstly, under green development principles, the LCD of China’s energy market is essential to modernizing industries. Leading global enterprises, like those in Europe and the U.S., prioritize R&D in low-carbon technologies and industrial decarbonization, securing competitive edges through innovation [61]. Chinese energy enterprises, historically focused on imitation, lag in LCD awareness [62]. To bridge this gap, they should adopt international technical standards and accelerate low-carbon technology upgrades.

Secondly, the government should strengthen carbon emission laws and regulations, and drive enterprises toward LCD. The EU’s Emissions Trading Scheme (ETS) uses market mechanisms to cut emissions, while the U.S. IRA supports clean energy with tax incentives [63]. China should offer initial fiscal aid (e.g., tax breaks and R&D grants) during the early stages of LCD to ease energy enterprises’ development costs and then shift to market-based tools like carbon trading platforms as LCD matures.

Thirdly, the government should invest in digital infrastructure and build platforms for sharing low-carbon technologies and experiences. Europe and the U.S. have advanced carbon management through international platforms like the International Energy Agency (IEA) and Clean Energy Ministerial (CEM), breaking down technical barriers [64]. China should enhance shared LCD platforms, optimize resource and talent allocation, and accelerate the adoption of low-carbon energy, materials, and equipment.

Fourthly, the government should enhance public supervision. The EU’s participatory budgeting model for climate projects and the U.S. EPA’s public comment system demonstrate effective mechanisms for societal oversight [65], fostering green innovation and policy accountability. China lags in public engagement due to its long-standing energy-intensive development model [66]. To address this, China should adopt ‘Carbon Inclusion’ mechanisms to encourage public participation in emission cuts, aligning with global practices.

Fifthly, government subsidies and incentives for energy enterprises and the public must be maintained within a reasonable range. Excessive subsidies and incentives can lead to unsustainable fiscal burdens, and they may not effectively encourage energy enterprises to focus on LCD or public supervision. To find the right balance, policymakers should create dynamic adjustment mechanisms that align financial support with economic feasibility and policy objectives, ensuring long-term sustainability.

This study makes important contributions by integrating prospect theory into evolutionary game theory to enhance the LCD of energy markets. This unique combination provides fresh insights into the strategic decisions of the public and government in this process, enriching the understanding of public supervision and government regulation. Additionally, this study uses prospect theory to examine the decision-making behaviors of stakeholders, focusing on how risk sensitivity and loss aversion influence outcomes. It also identifies mechanisms that affect the decisions of energy enterprises, the public, and the government, offering recommendations to motivate LCD efforts and improve public engagement in supervision.

However, in practical contexts, the decision-making processes of these three stakeholders are not confined to bounded rationality but are subject to multiple uncertainties, while cognitive biases and external shocks also substantially influence strategic choices. Therefore, future research should refine the existing game-theoretic models to incorporate more sophisticated frameworks that account for these multifaceted factors, thereby exploring their dynamic impacts on stakeholder decision making under real-world complexity.