Low-Carbon Policies and Power Generation Modes: An Evolutionary Game Analysis of Vertical Governments and Power Generation Groups

Abstract

Given the great proportion of CO₂ emissions from electricity generation in total energy-related CO₂ emissions, this article constructs a tripartite evolutionary game model consisting of vertical governments and power generation groups (PGGs), where the vertical governments include the central government (CG) and local governments (LGs), considering the externalities of different power generation modes on energy security and the environment. This article analyzes the stable strategies of the three players through replicator dynamics equations, draws the evolutionary phase diagrams, and analyzes the asymptotic stability of equilibrium points by using Jacobian matrices. To validate and broaden the results, this article also provides a numerical simulation. This article concludes that (1) a reduction in the supervision, enforcement, or low-carbonization costs of the CG, LGs, or PGGs motivates it or them to choose “supervision”, “enforcement”, or “low-carbonization” strategies; (2) an increase in penalty incomes or expenses encourages the CG or LGs to choose the “supervision” or “enforcement” strategies; (3) a rise in extra tax expenses motivates PGGs to choose the “low-carbonization” strategy; (4) a change in the externalities of energy security or the environment has no impact on the CG’s strategy. The above conclusions offer the CG and LGs with references for making effective low-carbon policies and provide PGGs with references for choosing an appropriate power generation mode.

1. Introduction

With the frequent occurrence of extreme weather events, the public is paying increasing attention to the issue of climate change, especially global warming, which brings about many consequences, such as rising sea levels and ecosystem destruction. Human activities are unequivocal causes of global warming, principally through the emissions of greenhouse gases [1]. In the year 2024, the global total energy-related CO₂ emissions were 37.8 Gt CO₂ [2], and the global CO₂ emissions from electricity generation were 13.8 Gt CO₂, which remains the highest of any sector [3]. As we can see, in the year 2024, the global CO₂ emissions from electricity generation accounted for 36.5% of the global total energy-related CO₂ emissions. Therefore, carbon abatement of the electricity generation sector is of great importance. As far as China is concerned, the total CO₂ emissions in the year 2024 were 12.6 Gt CO₂ [2], and the CO₂ emissions from electricity generation in the year 2024 were 5.7 Gt CO₂ [3]. The latter accounts for 45.6% of the former, being much higher than the global average level. For China, carbon abatement of the electricity generation sector is more important. Increasing the utilization of non-fossil energy is an effective way to reduce CO₂ emissions.

According to the “Transcript of the National Energy Administration’s Q2 2025 Press Conference” [4], up to the end of March 2025, the installed capacity of renewable energy power generation in China was 1966.0 MkW, accounting for approximately 57.3% of the total installed capacity. Among them, the installed capacities of hydropower, wind power, solar power, and biomass power generation were 438.0 MkW, 535.0 MkW, 946.0 MkW, and 46.0 MkW, accounting for approximately 12.8%, 15.6%, 27.6%, and 1.3% of the total installed capacity, respectively. During Q1 2025, the electricity production of renewable energy power was 816.0 GkW·h, accounting for approximately 35.9% of the total electricity production. Based on the “Working Guidance for Carbon Dioxide Peaking and Carbon Neutrality in Full and Faithful Implementation of the New Development Philosophy” [5], the goal of the share of non-fossil energy consumption in China in the year 2030 is about 25%, and the goal in the year 2060 is over 80%. The electricity generation industry in China can reduce CO₂ emissions by increasing the share of the installed capacity of non-fossil energy power generation and the consumption share of non-fossil energy.

Low-carbon supply chain management, especially games between firms [6,7,8,9,10,11], has been extensively studied. Considering consumers’ low-carbon preferences, Yu et al. [11] studied two games between one manufacturer and one retailer under a carbon tax policy. Based on prior research, several studies considered the government as a player to study the interaction between government and firms [12,13,14,15,16], and some studies focused on a vertical governmental perspective to study the interactive relationship between the central government (CG) and local governments (LGs) [17,18,19,20]. Considering environmental regulation, Sheng et al. [18] constructed a tripartite evolutionary game considering unemployment and the environment, and they analyzed the equilibrium points in different scenarios. Some research focused on this issue in the energy-related industry [21,22] or the electricity industry [23,24]. Wen et al. [23] studied a game model composed of a CG, LGs, and thermal power plants, and they analyzed the stable strategies and equilibrium points through replicator dynamics equations and Jacobian matrices. Unruh [25] came up with the concept of carbon lock-in, which means that industrial economies have been locked into fossil fuel-based energy systems through a process of technological and institutional co-evolution. Some research implied breaking carbon lock-in and achieving a low-carbon transition [26,27,28]. Su et al. [26] developed an evolutionary game between LGs and power enterprises and explored the impact of policy-related parameters on the transition to a hybrid energy system. Ou et al. [27] studied a tripartite evolutionary game model among the CG, LG, and coal power firms and mainly focused on the green technology innovation behavior. Yu et al. [28] developed an evolutionary game between the government and power plants and concluded that an increase in subsidies can encourage power plants to choose the coupled power generation of agricultural and forestry biomass and coal. Energy policies were considered in some research between the government and firms, such as carbon border adjustment mechanisms [29] and carbon-inclusive policies [30].

In summary, the existing research mainly focuses on games between firms and overlooks the importance of the government. Some research introduced the CG, LGs, or both of them to study the role of government; however, most of them did not delve into the details of the electricity industry, failed to take into account the externalities of energy security and the environment for different power generation modes, or the enforcement of low-carbon polices. Based on this, the possible marginal contribution of this article lies in (1) considering the externalities of different power generation modes for energy security and the environment in the CG’s payoff; (2) model construction in which low-carbon policies are considered, with the CG imposing penalties on players who choose environmentally unfriendly strategies and LGs imposing extra taxes when power generation groups (PGGs) adopt non-low-carbon power generation mode; and (3) a comprehensive analysis conducted on the evolutionary stability strategies of three game players and the asymptotic stability of the equilibrium points of the system in different scenarios.

Based on prior research, this article analyzes a tripartite evolutionary game among a CG, LGs, and PGGs. During the process of model construction, this article takes into account low-carbon policies and the externalities of different power generation modes on fossil energy security, non-fossil energy security, and the environment. This article analyzes the stable strategies of the three players by using replicator dynamics equations, draws the phase diagrams, and analyzes the stability of equilibrium points by using Jacobian matrices in different scenarios. This paper studies the impacts of critical parameters on the system’s evolution process. Firstly, this article concludes that a decrease in supervision costs or a rise in penalty incomes motivates a CG to supervise the behaviors of LGs and PGGs; however, the change in externalities for energy security or the environment has no impact on the strategy of the CG. Secondly, a rise in the probability of a CG selecting to supervise another two players, a decrease in enforcement costs, or a rise in penalty expenses motivates LGs to enforce low-carbon policies. Thirdly, a reduction in low-carbonization costs or a rise in extra tax expenses encourages PGGs to select a low-carbon power generation mode. Fourthly, the equilibrium points differ in different scenarios.

The remainder of the paper is as follows. Section 2 shows the methodology. Section 3 presents the results and discussion, which solve the problem, analyze the stable strategies of the three players, analyze the stability of equilibrium points, and provide a numerical simulation. Section 4 includes the conclusions.

2. Methodology

2.1. Problem Statement and Model Variables

Figure 1 shows a tripartite evolutionary game model of a CG, LGs, and PGGs.

The CG can choose to supervise or not supervise the behavior of LGs and PGGs. LGs can choose to enforce or not enforce low-carbon policies. PGGs can choose a low-carbon power generation mode (i.e., take active measures to increase the proportion of electricity production via non-fossil energy) or choose a non-low-carbon power generation mode (i.e., no measures are taken to reduce the proportion of electricity production via thermal power). The specific explanation of the three players’ payoffs can be found in Appendix A.

Table 1. Explanation of symbols.

Parameter	Meaning	Note
$c_1$	Supervision costs of CG	$c_1>0$
$c_2$	Enforcement costs of LGs	$c_2>0$
$c_3$	Low-carbonization costs of PGGs	$c_3>0$
$p$	“No enforcement” or “no low-carbonization” penalties	$p>0$
$r$	Extra tax revenues or expenses of LGs or PGGs when PGGs carry out non-low-carbon power generation mode	$r>0$
$s_1$	Negative externalities of low-carbon power generation mode on fossil energy security	$s_1>0$
$s_2$	Positive externalities of low-carbon power generation mode on non-fossil energy security	$s_2>0$
$s_3$	Negative externalities of non-low-carbon power generation mode on non-fossil energy security	$s_3>0$
$s_4$	Positive externalities of non-low-carbon power generation mode on fossil energy security	$s_4>0$
$b$	Positive externalities of low-carbon power generation mode on environment	$b>0$
$l$	Negative externalities of non-low-carbon power generation mode on environment	$l>0$

shows an explanation of the used symbols.

2.2. Model Assumptions

(1)

The game players: This paper considers the tripartite evolutionary game model, where all the players possess bounded rationality. Over time, their strategic choices evolve and stabilize toward the optimal strategies. Among them, the CG is Game Player 1, the LGs are Game Player 2, and the PGGs are Game Player 3.

(2)

The strategies of game players: The CG designs and supervises low-carbon policies, with a strategy set of (supervision, no supervision). The probabilities of choosing the “supervision” and “no supervision” strategies are $x(x\in [0,1])$ and $1-x$, respectively. LGs enforce low-carbon policies and accept supervision from the CG, with a strategy set of (enforcement, no enforcement). The probabilities of choosing the “enforcement” and “no enforcement” strategies are $y(y\in [0,1])$ and $1-y$, respectively. PGGs decide the power generation mode and accept supervision from the CG, with a strategy set of (low-carbonization, no low-carbonization). The probabilities of choosing the “low-carbonization” and “no low-carbonization” strategies are $z(z\in [0,1])$ and $1-z$, respectively.

(3)

The payoffs of game players: The CG has a long-term vision and takes the externalities of different power generation modes for the environment and energy security into consideration. LGs have a short-term vision and do not consider these externalities.

The costs when a CG, LGs, or PGGs choose the “no supervision”, “no enforcement”, or “no low-carbonization” strategies are greater than $0$ in reality. However, because this article focuses on relative costs, we assume that these costs are all $0$. An alternative and more realistic way to understand this assumption may be to regard the costs of supervision, enforcement, and low-carbonization as additional costs compared with the costs of no supervision, no enforcement, and no low-carbonization.

Similarly, we assume that the tax expenses paid by PGGs for carrying out the low-carbon power generation mode are $0$, and the tax expenses paid by PGGs carrying out the non-low-carbon power generation mode are $r$.

For the sake of model simplification, assume that under the strategy combination of (supervision, no enforcement, and no low-carbonization), both LGs and PGGs need to pay the penalty expenses $p$ (hereinafter referred to as the bilateral penalty expenses) to the CG, and under the strategy combination of (supervision, no enforcement, low-carbonization) or (supervision, enforcement, no low-carbonization), LGs or PGGs need to pay the penalty expenses $2p$ (hereinafter referred to as the unilateral penalty expenses) to the CG. In the first kind of scenario, LGs and PGGs do not choose environmentally friendly strategies, and their behavior will be supervised by the CG. The CG imposes penalties of $2p$, and the LGs and PGGs share the penalties equally. In the second kind of scenario, either the LGs and PGGs choose environmentally friendly strategies while the others do not, and the player who does not choose environmentally friendly strategies will need to bear the full penalties of $2p$.

2.3. Model Construction

Table 2. Payoff matrix of the three players.

LGs			PGGs
			Low-Carbonization $z$	No Low-Carbonization $1-z$
CG	Supervision $x$	Enforcement $y$	$(-c_2,-c_3,-c_1-s_1+s_2+b)$	$(-c_2,-c_3-2p,-c_1-s_1+s_2+b+2p)$
		No enforcement $1-y$	$(-2p,-c_3,-c_1-s_1+s_2+b+2p)$	$(r-p,-r-p,-c_1-s_3+s_4-l+2p)$
	No supervision $1-x$	Enforcement $y$	$(-c_2,-c_3,-s_1+s_2+b)$	$(-c_2,-c_3,-s_1+s_2+b)$
		No enforcement $1-y$	$(0,-c_3,-s_1+s_2+b)$	$(r,-r,-s_3+s_4-l)$

shows the payoff matrix of the CG, LGs, and PGGs.

3. Results and Discussion

3.1. Strategic Stability Analysis of the Three Players

3.1.1. Strategic Stability Analysis of the CG

According to the above payoff matrix, we conclude that the expected return of the CG for selecting the “supervision” strategy is $U_x$, and the average expected return is ${\overline{U}}_1$.

The replicator dynamics equation of the CG is (see further details in Appendix B)

$$F_1(x)={\displaystyle \frac{dx}{dt}}=x(U_x-{\overline{U}}_1)=x(1-x)(-c_1+2p-2yzp)$$

(1)

The first-order derivative of $F_1(x)$ with respect to $x$ is

$${\displaystyle \frac{d{F}_1(x)}{dx}}=(-c_1+2p-2yzp)(1-2x)$$

(2)

Let $G_1(z)=-c_1+2p-2yzp$. Based on the stability theorem of differential equations, the probability of the CG selecting the “supervision” strategy in a stable state must meet $F_1(x)=0$ and ${\displaystyle \frac{d{F}_1(x)}{dx}}<0$. Because of ${\displaystyle \frac{d{G}_1(z)}{dz}}=-2yp<0$, $G_1(z)$ is a decreasing function of $z$. When $z={\displaystyle \frac{-c_1+2p}{2yp}}=z_1^{*}$ is met, $G_1(z)=0$, and ${\displaystyle \frac{d{F}_1(x)}{dx}}\equiv 0$, we are unable to determine the stable strategy of the CG. When $z<z_1^{*}$ is met, $G_1(z)>0$, and ${\left.{\displaystyle \frac{d{F}_1(x)}{dx}}\right|}_{x=1}<0$, $x=1$ is the stable strategy of the CG. As shown in Figure 2b, the CG will tend to select the “supervision” strategy. When $z>z_1^{*}$ is met, $G_1(z)<0$, and ${\left.{\displaystyle \frac{d{F}_1(x)}{dx}}\right|}_{x=0}<0$, $x=0$ is the stable strategy of the CG. As shown in Figure 2c, the CG will tend to select “no supervision” strategy. The evolutionary phase diagrams of the CG’s strategy are shown in Figure 2.

According to prior analysis, this article proposes the following:

Proposition 1.

The lower the supervision costs of a CG, the more inclined it is to choose the “supervision” strategy.

Proof. 

$z_1^{*}={\displaystyle \frac{-c_1+2p}{2yp}}$, and ${\displaystyle \frac{\partial z_1^{*}}{\partial c_1}}=-{\displaystyle \frac{1}{2yp}}<0$. We have $z_1^{*}$ increases with the reduction in $c_1$, and $z<z_1^{*}$, then the strategy will be stable when $x=1$; conversely, the opposite is true. Therefore, the higher the supervision costs of the CG, the more inclined it is to choose the “no supervision” strategy. On the contrary, the lower the supervision costs of the CG, the more inclined it is to choose the “supervision” strategy. □

Proposition 1 indicates that reducing the supervision costs of a CG can motivate it to choose the “supervision” strategy.

Proposition 2.

The higher the penalties imposed by a CG, the more inclined it is to choose the “supervision” strategy.

The proof process of Proposition 2 is similar to that of Proposition 1, and due to length limitations, it has been omitted. Similarly, the proof processes of Propositions 3–7 have been omitted. They are available on request from the corresponding author.

Proposition 2 indicates that increasing the penalties imposed by a CG can encourage it to supervise LGs’ and PGGs’ behaviors.

3.1.2. Strategic Stability Analysis of LGs

Similar to Section 3.1.1, we have

$$F_2(y)={\displaystyle \frac{dy}{dt}}=y(U_y-{\overline{U}}_2)=y(1-y)(-c_2+xp+xzp-r+zr)$$

(3)

and

$${\displaystyle \frac{d{F}_2(y)}{dy}}=(-c_2+xp+xzp-r+zr)(1-2y)$$

(4)

Let $G_2(z)=-c_2+xp+xzp-r+zr$. Because of ${\displaystyle \frac{d{G}_2(z)}{dz}}=xp+r>0$, $G_2(z)$ is an increasing function of $z$. When $z={\displaystyle \frac{c_2-xp+r}{xp+r}}=z_2^{*}$ is met, $G_2(z)=0$, and ${\displaystyle \frac{d{F}_2(y)}{dy}}\equiv 0$, we are unable to determine the stable strategy of LGs. When $z<z_2^{*}$ is met, $G_2(z)<0$, and ${\left.{\displaystyle \frac{d{F}_2(y)}{dy}}\right|}_{y=0}<0$, $y=0$ is the stable strategy of LGs. As shown in Figure 3b, LGs will tend to select “no enforcement” strategies. When $z>z_2^{*}$ is met, $G_2(z)>0$, and ${\left.{\displaystyle \frac{d{F}_2(y)}{dy}}\right|}_{y=1}<0$, $y=1$ is the stable strategy of LGs. As shown in Figure 3c, LGs will tend to select “enforcement” strategies. Figure 3 shows the evolutionary phase diagrams of LGs’ strategies.

Thus, we have the following:

Proposition 3.

The more CG tends to supervise LGs’ and PGGs’ behaviors, the more inclined LGs tend to be to select the “enforcement” strategies.

Proposition 3 shows that an increase in the probability of a CG supervising LGs’ and PGGs’ behaviors can incentivize LGs to choose the “enforcement” strategies.

Proposition 4.

The lower the enforcement costs for the LGs, the more likely they are to select the “enforcement” strategies.

Proposition 4 indicates that a reduction in the enforcement costs for LGs can incentivize them to choose the “enforcement” strategies.

Proposition 5.

The higher the penalties imposed by a CG, the more likely LGs are to enforce low-carbon policies.

Proposition 5 indicates that an increase in the penalties imposed by a CG will incentivize LGs to enforce low-carbon policies.

3.1.3. Strategic Stability Analysis of PGGs

Similar to Section 3.1.1 and Section 3.1.2, we have

$$F_3(z)={\displaystyle \frac{dz}{dt}}=z(U_z-{\overline{U}}_3)=z(1-z)(-c_3+y{c}_3+xp+xyp+r-yr)$$

(5)

and

$${\displaystyle \frac{d{F}_3(z)}{dz}}=(-c_3+y{c}_3+xp+xyp+r-yr)(1-2z)$$

(6)

Let $G_3(x)=-c_3+y{c}_3+xp+xyp+r-yr$. Because of ${\displaystyle \frac{d{G}_3(x)}{dx}}=p+yp>0$, $G_3(x)$ is an increasing function of $x$. When $x={\displaystyle \frac{c_3-y{c}_3-r+yr}{yp+p}}=x^{*}$ is met, $G_3(x)=0$, and ${\displaystyle \frac{d{F}_3(z)}{dz}}\equiv 0$, we are unable to determine the stable strategy of PGGs. When $x<x^{*}$ is met, $G_3(x)<0$, and ${\left.{\displaystyle \frac{d{F}_3(z)}{dz}}\right|}_{z=0}<0$, $z=0$ is the stable strategy of PGGs. As shown in Figure 4b, PGGs will tend to select “no low-carbonization” strategies. When $x>x^{*}$ is met, $G_3(x)>0$, and ${\left.{\displaystyle \frac{d{F}_3(z)}{dz}}\right|}_{z=1}<0$, $z=1$ is the stable strategy of PGGs. As shown in Figure 4c, PGGs will tend to select “low-carbonization” strategies. Figure 4 shows the evolutionary phase diagrams of PGGs’ strategies.

Thus, we have the following:

Proposition 6.

The lower the low-carbonization costs of PGGs, the more inclined they are to select the “low-carbonization” strategies.

Proposition 6 indicates that a reduction in the low-carbonization costs of PGGs will motivate them to choose the “low-carbonization” strategies.

Proposition 7.

The higher the extra tax expenses of PGGs for carrying out a non-low-carbon power generation mode, the more likely they are to select the “low-carbonization” strategies.

Proposition 7 indicates that an increase in the extra tax expenses of PGGs can incentivize them to choose the “low-carbonization” strategies.

In summary, reducing the supervision costs and increasing the penalties can motivate a CG to supervise LGs’ and PGGs’ behaviors. Increasing the probability of a CG supervising LGs’ and PGGs’ behaviors, reducing the enforcement costs, and increasing the penalties can motivate LGs to select the “enforcement” strategy. Reducing the low-carbonization costs and increasing the extra tax expenses can motivate PGGs to select the “low-carbonization” strategy.

3.2. Stability Analysis of the Equilibrium Points in the Game

Selten [31] and Ritzberger and Weibull [32] concluded that in the dynamic replication system of a multi-party evolutionary game, it is asymptotically stable when and only when a strategy combination is a pure-strategy Nash equilibrium. Therefore, the asymptotically stable equilibrium point must be an evolutionary stable strategy (ESS) (Wainwright [33]). This article only considers the asymptotic stability of the pure-strategy Nash equilibrium:

$$\left\{\begin{array}{l}{F}_1(x)=x(1-x)(-c_1+2p-2yzp)=0\\ F_2(y)=y(1-y)(-c_2+xp+xzp-r+zr)=0\\ F_3(z)=z(1-z)(-c_3+y{c}_3+xp+xyp+r-yr)=0\end{array}\right.$$

(7)

As shown in Equation (7), when combining $F_1(x)=0$, $F_2(y)=0$ and $F_3(z)=0$, we have eight pure-strategy equilibrium points for the game, which are $E_1(0,0,0)$, $E_2(1,0,0)$, $E_3(0,1,0)$, $E_4(0,0,1)$, $E_5(1,1,0)$, $E_6(1,0,1)$, $E_7(0,1,1)$ and $E_8(1,1,1)$.

Based on Friedman [34], the Jacobian matrix of the game is

$$\begin{array}{l}J=\left[\begin{array}{ccc}{J}_{11}& J_{12}& J_{13}\\ J_{21}& J_{22}& J_{23}\\ J_{31}& J_{32}& J_{33}\end{array}\right]=\left[\begin{array}{ccc}{\displaystyle \frac{\partial F_1(x)}{\partial x}}& {\displaystyle \frac{\partial F_1(x)}{\partial y}}& {\displaystyle \frac{\partial F_1(x)}{\partial z}}\\ {\displaystyle \frac{\partial F_2(y)}{\partial x}}& {\displaystyle \frac{\partial F_2(y)}{\partial y}}& {\displaystyle \frac{\partial F_2(y)}{\partial z}}\\ {\displaystyle \frac{\partial F_3(z)}{\partial x}}& {\displaystyle \frac{\partial F_3(z)}{\partial y}}& {\displaystyle \frac{\partial F_3(z)}{\partial z}}\end{array}\right]\\ =\left[\begin{array}{ccc}(-c_1+2p-2yzp)(1-2x)& -2zpx(1-x)& -2ypx(1-x)\\ (1+z)py(1-y)& (-c_2+xp+xzp-r+zr)(1-2y)& (xp+r)y(1-y)\\ (1+y)pz(1-z)& (c_3+xp-r)z(1-z)& (-c_3+y{c}_3+xp+xyp+r-yr)(1-2z)\end{array}\right]\end{array}$$

(8)

The Jacobian matrix at $E_1(0,0,0)$ is

$$E_1(0,0,0): J_1=\left[\begin{array}{ccc}-c_1+2p& 0& 0\\ 0& -c_2-r& 0\\ 0& 0& -c_3+r\end{array}\right]$$

(9)

Based on Lyapunov [35], the equilibrium point is asymptotically stable only when all eigenvalues of the Jacobian matrix are negative. For example, when $-c_1+2p<0$ and $-c_3+r<0$ are met ($c_2>0$ and $r>0$, and thus $-c_2-r<0$), $E_1(0,0,0)$ is asymptotically stable. The Jacobian matrices at the other seven equilibrium points are shown in Appendix C.

As shown in

Table 3. Stability of equilibrium points.

Equilibrium Point	Eigenvalues of the Jacobian Matrices		Conclusion	Condition	Scenario
	${\lambda }_1,{\lambda }_2,{\lambda }_3$	Symbols
$E_1(0,0,0)$	$-c_1+2p,-c_2-r,-c_3+r$	$(\times ,-,\times )$	ESS	①	Scenario 1
$E_2(1,0,0)$	$-(-c_1+2p),-c_2+p-r,-c_3+p+r$	$(\times ,\times ,\times )$	ESS	②	Scenario 2
$E_3(0,1,0)$	$-c_1+2p,-(-c_2-r),0$	$(\times ,+,0)$	Unstable point
$E_4(0,0,1)$	$-c_1+2p,-c_2,-(-c_3+r)$	$(\times ,-,\times )$	ESS	③	Scenario 3
$E_5(1,1,0)$	$-(-c_1+2p),-(-c_2+p-r),2p$	$(\times ,\times ,+)$	Unstable point
$E_6(1,0,1)$	$-(-c_1+2p),-c_2+2p,-(-c_3+p+r)$	$(\times ,\times ,\times )$	ESS	④	Scenario 4
$E_7(0,1,1)$	$-c_1,c_2,0$	$(-,+,0)$	Unstable point
$E_8(1,1,1)$	$-(-c_1),-(-c_2+2p),-2p$	$(+,\times ,-)$	Unstable point

Note: “$\times$”, “$-$”, or “$+$” means the symbol is uncertain, negative, or positive, respectively. The equilibrium points that satisfy the corresponding conditions are ESSs, where Condition ① is $-c_1+2p<0$ and $-c_3+r<0$, denoted as Scenario 1; Condition ② is $-(-c_1+2p)<0$, $-c_2+p-r<0$, and $-c_3+p+r<0$, denoted as Scenario 2; Condition ③ is $-c_1+2p<0$ and $-(-c_3+r)<0$, denoted as Scenario 3; and Condition ④ is $-(-c_1+2p)<0$, $-c_2+2p<0$, and $-(-c_3+p+r)<0$, denoted as Scenario 4.

, since the Jacobian matrices at $E_3(0,1,0)$, $E_5(1,1,0)$, $E_7(0,1,1)$, and $E_8(1,1,1)$ all have at least a positive eigenvalue, they are not asymptotically stable.

Proposition 8.

When $c_1>2p$ and $c_3>r$ are met, $E_1(0,0,0)$ is the only ESS.

Proof. 

When Condition ① holds—that is, when $c_1>2p$ and $c_3>r$ are met—all the eigenvalues of the Jacobian matrix at $E_1(0,0,0)$ are negative, and then $E_1(0,0,0)$ is the ESS. Meanwhile, due to $c_1>2p$, $E_2(1,0,0)$ or $E_6(1,0,1)$ is not the ESS, and because of $c_3>r$, $E_4(0,0,1)$ is not the ESS. Therefore, when $c_1>2p$ and $c_3>r$ are met, $E_1(0,0,0)$ is the only ESS. □

Proposition 8 indicates that when the supervision costs of a CG are greater than the penalty incomes it may receive from choosing the “supervision” strategy, and the low-carbonization costs of PGGs are greater than the extra tax expenses, $E_1(0,0,0)$ is the only ESS. The CG, LGs, and PGGs choose the “no supervision”, “no enforcement”, and “no low-carbonization” strategies, respectively, which bring about the externalities of $-s_3+s_4-l$ for the entire society.

Proposition 9.

When $c_1<2p$, $c_2>p-r$, and $c_3>p+r$ are met, $E_2(1,0,0)$ is the only ESS.

The proof process of Proposition 9 is similar to that of Proposition 8, and due to length limitations, it is omitted. Similarly, the proof processes of Propositions 10 and 11 are omitted. They are available on request from the corresponding author.

Proposition 9 indicates that when the supervision costs of a CG are less than the penalty incomes it may obtain, the enforcement costs of LGs are greater than the difference between the bilateral penalty expenses and extra tax revenues, the low-carbonization costs of PGGs are greater than the sum of the bilateral penalty expenses and extra tax expenses, and $E_2(1,0,0)$ is the only ESS. The CG, LGs, and PGGs choose the “supervision”, “no enforcement”, and “no low-carbonization” strategies, respectively, and the entire society bears the externalities of $-s_3+s_4-l$.

Proposition 10.

When $c_1>2p$ and $c_3<r$ are met, $E_4(0,0,1)$ is the only ESS.

Proposition 10 indicates that when the supervision costs of a CG are greater than the penalty incomes it may receive, and the low-carbonization costs of PGGs are less than the extra tax expenses, $E_4(0,0,1)$ is the only ESS. The CG, LGs, and PGGs choose the “no supervision”, “no enforcement”, and “low-carbonization” strategies, respectively, and the entire society bears the externalities of $-s_1+s_2+b$.

Proposition 11.

When $c_1<2p$, $c_2>2p$, and $c_3<p+r$ are met, $E_6(1,0,1)$ is the only ESS.

Proposition 11 indicates that when the supervision costs of a CG are less than the penalty incomes it may obtain, the enforcement costs of LGs are greater than the unilateral penalty expenses, and the low-carbonization costs of PGGs are less than the sum of the bilateral penalty expenses and extra tax expenses, $E_6(1,0,1)$ is the only ESS. The CG, LGs and PGGs choose the “supervision”, “no enforcement”, and “low-carbonization” strategies, respectively, and the entire society bears the externalities of $-s_1+s_2+b$.

To sum up, when the supervision costs of a CG are greater than the penalty incomes it may receive from supervising LGs’ and PGGs’ behaviors, the CG chooses the “no supervision” strategy. At this point, LGs choose the “no enforcement” strategies. In this case, first, when the low-carbonization costs of PGGs are greater than the extra tax expenses, PGGs choose the “no low-carbonization” strategies, with $E_1(0,0,0)$. Second, when the low-carbonization costs of PGGs are less than the extra tax expenses, PGGs choose the “low-carbonization” strategies, with $E_4(0,0,1)$.

When the supervision costs of a CG are less than the penalty incomes it may receive, the CG chooses the “supervision” strategy. At this point, first, when the difference between the bilateral penalty expenses and extra tax revenues is less than LGs’ enforcement costs, the LGs choose the “no enforcement” strategies. In this case, when the low-carbonization costs of PGGs are greater than the sum of the bilateral penalty expenses and extra tax expenses, the PGGs choose the “no low-carbonization” strategies, with $E_2(1,0,0)$. Second, when the enforcement costs of LGs exceed the unilateral penalty expenses, the LGs choose the “no enforcement” strategies. In this case, when the low-carbonization costs of PGGs are less than the sum of the bilateral penalty expenses and extra tax expenses, the PGGs choose the “low-carbonization” strategies, with $E_6(1,0,1)$.

3.3. Numerical Simulation

To verify the effectiveness of the evolutionary stability analysis for each player’s strategy and further demonstrate the impacts of various parameters on the strategy evolution process of each player, this article used MATLAB R2018a for numerical simulations. To ensure the generality of the conclusions, the simulated values only represent the relative sizes between the parameters and do not represent the actual values. Due to space limitations, as in Scenario 4 of $E_6(1,0,1)$, the CG and PGGs selected the “supervision” and “low-carbonization” strategies, respectively, which was beneficial for the low-carbon development. This section takes Scenario 4 as an example for analysis.

3.3.1. Parameter Settings

Referring to Table 3, we set up a set of arrays where $c_1=0.5$, $c_2=1.2$, $c_3=0.9$, $p=0.5$, and $r=0.6$ to simulate Scenario 4.

3.3.2. Results

(1)

Impacts of changes in each player’s initial strategy on the evolution of the system.

As shown in Figure 5a–d, in Scenario 4, when the supervision costs of the CG were less than the penalty incomes it might have obtained from choosing the “supervision” strategy, the enforcement costs of LGs were greater than the unilateral penalty expenses, and the low-carbonization costs of PGGs were less than the sum of the bilateral penalty expenses and extra tax expenses, regardless of the initial strategies adopted by the CG, LGs, and PGGs. The system would eventually converge to the (supervision, no enforcement, low-carbonization) strategy combination, which is consistent with Proposition 11.

As shown in Figure 5e, with the initial probability of the CG supervising LGs’ and PGGs’ behaviors increasing, the rate at which the probability of LGs enforcing low-carbon policies trended toward zero slowed down (consistent with Proposition 3), and the rate at which the probability of PGGs selecting the low-carbon power generation mode trended toward one accelerated. We conclude that an increase in $x$ encouraged LGs and PGGs to choose environmentally friendly strategies. As shown in Figure 5f, with the initial probability of LGs enforcing low-carbon policies increasing, the rate at which the probability of the CG supervising LGs’ and PGGs’ behaviors trended toward one slowed down, and the rate at which the probability of PGGs selecting the low-carbon power generation mode trended toward one accelerated. We conclude that the increase in $y$ motivated the CG to choose the “no supervision” strategy and motivated the PGGs to choose “low-carbonization” strategies. In Figure 5g, as the initial probability of the PGGs selecting low-carbon power generation mode increased, the rate at which the probability of the CG supervising LGs’ and PGGs’ behaviors trended toward one slowed down, and the rate at which the probability of LGs enforcing low-carbon policies trended toward zero slowed down. We conclude that the increase in $z$ motivated the CG to choose the “no supervision” strategy and motivates the LGs to choose “enforcement” strategies. This may be because as $z$ increased, the CG no longer needed to choose a “supervision” strategy, and the LGs’ returns for choosing “no enforcement” strategies were reduced.

(2)

Impacts of changes in the CG’s supervision costs on the evolution of the system.

Figure 6 shows that the rise in the CG’s supervision costs slowed down the rate at which the probability of the CG supervising LGs’ and PGGs’ behaviors trended toward one (consistent with Proposition 1), accelerated the rate at which the probability of LGs enforcing low-carbon policies trended toward zero, and slowed down the rate at which the probability of PGGs selecting the low-carbon power generation mode trended toward one. We conclude that the increase in supervision costs led the CG to tend to choose a “no supervision” strategy, led the LGs to tend to choose “no enforcement” strategies, and led the PGGs to tend to choose “no low-carbonization” strategies.

(3)

Impacts of changes in LGs’ enforcement costs on the evolution of the system.

Figure 7 shows that the rise in the LGs’ enforcement costs accelerated the rate at which the probability of LGs enforcing low-carbon policies trended toward zero (consistent with Proposition 4), accelerated the rate at which the probability of the CG supervising LGs’ and PGGs’ behaviors trended toward one, and slowed down the rate at which PGGs choosing the “low-carbonization” strategy trended toward one. We conclude that the increase in enforcement costs led the LGs to tend to choose “no enforcement” strategies, led the CG to tend to choose a “supervision” strategy, and led the PGGs to tend to choose “no low-carbonization” strategies.

(4)

Impacts of changes in PGGs’ low-carbonization costs on the evolution of the system.

Figure 8 shows that the increase in the PGGs’ low-carbonization costs slowed down the rate at which the probability of PGGs selecting a low-carbon power generation mode trended toward one (consistent with Proposition 6), accelerated the rate at which the probability of the CG supervising LGs’ and PGGs’ behaviors trended toward one, and accelerated the rate at which the probability of LGs enforcing low-carbon policies trended toward zero. We conclude that the increase in low-carbonization costs led the PGGs to tend to choose “no low-carbonization” strategies, led the CG to tend to choose a “supervision” strategy, and led the LGs to tend to choose “no enforcement” strategies. This may be because as $c_3$ increased, the CG needed to choose a “supervision” strategy more, and the LGs’ returns for choosing “no enforcement” strategies increased.

(5)

Impacts of changes in the penalties on the evolution of the system.

Figure 9 shows that an increase in the penalties accelerated the rate at which the probability of the CG supervising LGs’ and PGGs’ behaviors trended toward one (consistent with Proposition 2), slowed down the probability of LGs enforcing the low-carbon policies trending toward zero (consistent with Proposition 5), and accelerated the rate at which the probability of PGGs selecting the low-carbon power generation mode trended toward one. We conclude that the increase in penalties led the CG to tend to choose a “supervision” strategy, led the LGs to tend to choose “enforcement” strategies, and led the PGGs to tend to choose “low-carbonization” strategies.

(6)

Impacts of changes in the extra taxes on the evolution of the system.

Figure 10 shows that the rise in the extra tax expenses of the PGGs accelerated the rate at which the probability of PGGs selecting a low-carbon power generation mode trended toward one (consistent with Proposition 7), while the increase in the LGs’ extra tax revenues first accelerated and then slowed down the rate at which the probability of LGs enforcing low-carbon policies trended toward zero. In addition, the increase in extra taxes slowed down the rate at which the probability of the CG supervising LGs’ and PGGs’ behaviors trended toward one, which may have been due to changes in taxes affecting the strategies of the LGs and PGGs, thereby affecting the strategies of the CG. We conclude that the increase in extra taxes led the PGGs to tend to choose “low-carbonization” strategies, led the CG to tend to choose a “no supervision” strategy, and led the LGs to at first tend to choose “no enforcement” strategies and then tend to choose “enforcement” strategies (the reason for this requires further studies).

(7)

Impacts of changes in the energy security externalities and environmental externalities on the evolution of the system.

Figure 11 shows that the energy security externalities and environmental externalities had no effect on the evolution of the system, which may have been due to the assumptions that only the CG had a long-term vision to consider these externalities in its payoff, and the PGGs had to carry out the low-carbon generation mode no matter what strategy they chose, while the LGs chose to enforce the low-carbon policies. This situation may occur when LGs have strong policy enforcement, but their short-term vision is not enough to consider these externalities. In future research, we may break from these assumptions and consider other situations.

4. Conclusions

According to the bounded rationality hypothesis, this article constructed the tripartite evolutionary game model composed of a CG, LGs, and PGGs. First, in the process of model construction, the impacts of different power generation modes of PGGs on the national comprehensive energy security and environment were considered. Second, by utilizing replicator dynamic equations, the stable strategy of each player in the game was analyzed, and evolution phase diagrams of each player’s strategy were drawn. Meanwhile, using Jacobian matrices, the stability of the equilibrium points in the tripartite game system in different scenarios was analyzed. Third, numerical simulations were conducted to analyze the impact of the initial strategies of the three players and parameter changes on the dynamic evolution process of the tripartite game system in Scenario 4, where the CG supervised LGs’ and PGGs’ behaviors, LGs did not enforce low-carbon policies, and PGGs carried out a low-carbon power generation mode. The results show the following:

(1)

For the CG, the reduction in supervision costs or the increase in penalty incomes that could be obtained from choosing the “supervision” strategy motivated it to supervise LGs’ and PGGs’ behaviors, while parameters related to energy security externalities or environment externalities had no impact on its decision making.

(2)

For the LGs, a rise in the probability of the CG initially supervising LGs’ and PGGs’ behaviors, a decrease in the enforcement costs, or an increase in the penalty expenses motivated them to enforce low-carbon policies.

(3)

For the PGGs, a reduction in low-carbonization costs or an increase in extra tax expenses motivated them to choose the “low-carbonization” strategy.

(4)

When the supervision costs of the CG were less than the penalty incomes it could receive from choosing the “supervision” strategy, the enforcement costs of the LGs exceeded the unilateral penalty expenses, and the low-carbonization costs of the PGGs were less than the sum of the bilateral penalty expenses and extra tax expenses, the CG chose a “supervision” strategy, the LGs chose a “no enforcement” strategy, and the PGGs chose a “low-carbonization” strategy.

Based on the above conclusions, in order to better promote low-carbon development of the electricity industry, the following suggestions are proposed:

(1)

The CG should reduce supervision costs. For example, in order to simplify the supervision process, a CG could establish a real-time public database for LGs’ low-carbon policy enforcement and PGGs’ CO₂ emissions and build an effective evaluation system to help evaluate their behaviors. When it comes to LGs, a CG could analyze the low-carbon policy enforcement news on the LGs’ official websites by using artificial intelligence tools and upload the analysis results to the database. When it comes to PGGs, a CG could build some CO₂ emission monitoring points around PGGs and upload real-time CO₂ emission data to the database.

(2)

The CG should increase the level of penalties, which will increase the costs of LGs choosing the “no enforcement” strategies and PGGs choosing the “no low-carbonization” strategies, thereby reducing the occurrence of environmentally unfriendly strategies.

(3)

LGs should decrease enforcement costs. For instance, LGs could enhance promotion of the importance of low-carbon production and life, leverage the collaborative effectiveness of multiple stakeholders, and encourage consumers to force PGGs to engage in a low-carbon power generation mode.

(4)

LGs should raise the carbon tax level within a reasonable range. An increase in the carbon tax will reduce the CO₂ emissions of PGGs, but it may also affect the profits of PGGs [11], and a trade-off should be made.

(5)

PGGs should reduce low-carbonization costs. At present, PGGs could use mature low-carbon power generation technologies, such as increasing the installed capacity of wind and solar power on a large scale and implementing clean transformation of thermal power. In the future, PGGs could allocate research and development investments for new technologies reasonably, avoiding investing heavily in low returns and high-cost technologies.

This article provides certain guidance on whether a CG chooses to supervise the behaviors of LGs and PGGs, whether LGs choose to enforce low-carbon policies, and whether PGGs choose a low-carbon power generation mode. It is beneficial for the design and enforcement of vertical governments’ low-carbon policies, and it also offers useful references for the decision making of PGGs, thus contributing to low-carbon development. However, there are some limitations:

(1)

We only considered the strategies of a CG, LGs, and PGGs and failed to take the strategies of consumers into consideration.

(2)

For simplification of the analysis, we assumed that the “no supervision” costs of the CG, the “no enforcement” costs of the LGs, and the “no low-carbonization” costs of the PGGs were all zero, which is not realistic.

Referring to the above limitations, we may expand our research in the following directions:

(1)

We will take the consumers into consideration and study the interactions among the CG, LGs, PGGs, and consumers through a four-party evolutionary game model in future studies.

(2)

In future research, we may try to discard this unrealistic assumption and come up with some new and interesting conclusions.