Research on the Recycling Strategy of End-of-Life Power Battery for Electric Vehicles Based on Evolutionary Game

Abstract

The rapid growth of China’s electric vehicle (EV) market has led to a peak in end-of-life (EOL) power batteries, yet the recycling sector remains dominated by informal operations. This paper incorporates the formal and informal recycling participation behaviours of EV owners into the framework of evolutionary games, systematically examines the mechanism by which governmental incentive and disincentive mechanisms influence the evolutionary stability of each party, and constructs a tripartite evolutionary game model involving the government, recycling enterprises, and EV owners. Numerical simulation experiments conducted using PyCharm 2.3 provide an in-depth exploration of the strategic evolutionary trajectories of each participating agent. The findings indicate that (1) the stable strategy for the game-theoretic system of EOL power battery recycling is government non-regulation, recycling enterprises adopting formal recycling practices, and EV owners participating in formal recycling; (2) strengthening penalties against recycling enterprises will accelerate their transition towards formal recycling strategies, while increasing incentive levels can significantly enhance the steady-state probability of firms opting for formal recycling; (3) government subsidies for EV owners encourage both EV owners and recycling enterprises to adopt formal recycling, with recycling enterprises shifting first. This study enriches the application of evolutionary game theory in the field of EOL power battery recycling and further provides guidance for the healthy development of the recycling industry.

1. Introduction

Resource shortage and environmental pollution are the focus of global concern. Electric vehicles (EVs) have been widely promoted in various countries by virtue of their green, clean, energy-saving, and environmentally friendly advantages, and have gradually become the mainstream trend of the future automotive industry (LaMonaca and Ryan, 2022 [1]). Data from the International Energy Agency shows that China’s electric vehicle sales account for more than 70% of global sales, and it is expected that the total global stock will be more than 130 million by 2030 (International Energy Agency, 2024 [2]). The rapid development of EVs has led to a continuous increase in the demand for power batteries (Qiao et al., 2021 [3]), which have an average service life of 5–8 years (Skeete et al., 2020 [4]; Neumann et al., 2022 [5]) and are no longer suitable to EVs when their capacity drops to below 80 percent (Tao et al., 2022 [6]). It is predicted that the EOL power batteries will reach 708,000 metric tonnes in 2030 (Wu et al., 2020 [7]), and the recycling and disposal of EOL power batteries has become a major challenge for China. Disposal has become a major challenge for China, and EOL power batteries are about to usher in a “scrapping wave” (Sun et al., 2022 [8]). EOL power batteries contain rare metals such as nickel, cobalt, manganese, and lithium, which have very high economic value (Gratz et al., 2014 [9]; Sun et al., 2021 [10]). Recycling and remanufacturing them can extend the service life and residual value of EOL power batteries and reduce environmental hazards (Li et al., 2022 [11]). Without proper recycling and disposal, large-scale EOL power batteries can cause serious harm to the environment (Manzetti and Mariasiu, 2015 [12]; Martins et al., 2021 [13]).

Extended producer responsibility (EPR) regulations have been widely adopted in developed countries and regions such as the European Union, the United States, and Japan as the legislative foundation for environmental governance. Germany has implemented a nationwide recycling system through federal legislation, achieving a compliance recycling rate exceeding 85% for power batteries. Countries such as Norway and Sweden have attained recycling rates surpassing 90% (Yan et al., 2024 [14]). The European Union, through its new battery regulation, has elevated battery management from directives to regulations, driving the market-based and scaled operation of EOL power battery recycling. The United States rigorously enforces extended producer responsibility, utilising deposit schemes and additional environmental taxes to incentivise consumer participation in the recycling of EOL batteries.

Although China attaches great importance to the recycling of EOL power batteries for EVs, it has not yet established a comprehensive EOL power battery recycling system. First, China’s EOL power battery recycling market is chaotic, with a proliferation of informal recyclers. According to statistics, there are about 64,000 recyclers of EOL power batteries in China, but only about 5% of them have been authorised by the government (Li et al., 2023 [15]), resulting in nearly 80% of EOL power batteries ultimately flowing into informal recycling channels consisting of private recycling enterprises and small workshops (Liu et al., 2021 [16]). Informal recycling enterprises lack qualifications, have outdated technology, and operate in a non-standardised manner (Xie et al., 2022 [17]), and there are serious safety hazards in battery dismantling and reassembly (Xiao et al., 2024 [18]; Winslow et al., 2018 [19]). Thirdly, some EV owners have weak environmental awareness in EOL power battery recycling, low participation in formal recycling, and are sensitive to the recycling price (Huang et al., 2024 [20]), which leads to a large number of EOL batteries circulating to informal recycling channels, exacerbating the problems of environmental pollution and safety hazards.

With the surge in retired power batteries from China’s EV fleet, the recycling industry faces significant challenges: informal recyclers command nearly 80% of the market share due to lower costs, while formal recycling suffers from low corporate participation and insufficient owner engagement. This not only poses environmental threats but also squanders the economic value of rare metals within these spent batteries. Within the EV battery recycling system, the three key stakeholders—government, recycling enterprises, and EV owners—engage in both cooperation and competition. Their interactions exhibit characteristics of dynamic interplay, bounded rationality, and incremental strategic adjustments. The interplay among stakeholders in EV battery recycling is often intricate, and evolutionary game theory can capture the complexity of their interactions and decision-making processes.

To address this issue, this paper constructs a tripartite evolutionary game model centred on the government, recycling enterprises, and vehicle owners. It analyses the strategic interaction mechanisms among these stakeholders and identifies the key conditions for promoting the widespread adoption of formal recycling models.

The core research questions are as follows: (1) How do the strategic choices of government, recycling enterprises, and EV owners dynamically evolve? (2) What policy combinations can drive the system towards a stable state of formalised recycling? The evolutionary game model constructed herein focuses on analysing the dynamic strategic adjustment processes of stakeholders under conditions of bounded rationality, ultimately providing actionable insights for optimising the EOL power battery recycling system.

2. Literature Review

This section focuses on the current status of the application of government reward and punishment mechanisms in the field of EOL power battery recycling, and systematically reviews the research related to the evolutionary game around EOL power battery recycling of participants.

2.1. International EOL Power Battery Recycling

As the global adoption of EVs continues to accelerate, nations worldwide face the pressing challenge of a large-scale wave of power battery retirement (Antony et al., 2025 [21]). Regulations governing EV battery recycling across major markets are notably fragmented, localised, and typically of lower legal status, causing significant challenges for implementation and enforcement (Giosuè et al.,2021 [22]). The European Union has established a relatively comprehensive regulatory framework, such as Regulation (EU) 2023/1542, which harmonises battery collection, recycling efficiency, and extended producer responsibility (EPR) systems. In Germany, regulations require all parties along the battery value chain to take corresponding responsibilities to recycle batteries. In developing nations like China, most EV battery recycling regulations remain largely advisory in nature, lacking legal enforceability and yielding limited implementation outcomes (Shqairat et al., 2024 [23]), while countries like Laos lag further behind, with EOL battery management primarily involving mixed disposal. The absence of specialised regulations and recycling systems may trigger environmental risks as EVs proliferate (Noudeng et al., 2022 [24]). Overall, the implementation of recycling regulations is constrained by low public awareness, insufficient incentives, and inadequate regulatory coordination. Sustainable development urgently requires enhanced policy synergy, increased public participation, the introduction of economic incentives, and technological innovation.

2.2. Government Incentive and Penalty Mechanisms in Battery Recycling Research

With the gradual development of the recycling industry, government incentives, and penalties have gradually become a hot topic for scholars. Existing studies have confirmed that government incentives have a significant positive driving effect on the development of the EOL power battery recycling industry (Shen et al., 2022 [25]; Li et al., 2024 [26]). Government incentives and penalties can not only improve the recycling efficiency (Wang et al., 2015 [27]), but also promote the recycling industry to innovate and green transformation (Wang et al., 2022 [28]) and promote the formation of a green closed-loop supply chain for EOL power batteries (Li et al., 2023 [29]).

Research suggests that governments should focus on flexibility when formulating subsidy policies (Sun et al., 2025 [30]) and should combine subsidies with regulation or implement staged regulation (Lyu et al., 2021 [31]). The dynamic subsidy model can facilitate enterprises in achieving a stable operating state more quickly, thereby maximising the effectiveness of the policies (Sun et al., 2025 [30]). At the same time, incentives and penalties should be appropriate, neither too low nor too high (Du et al., 2023 [32]). For firms with low current profits but high future profit potential, the government should provide higher subsidies in the early stages and gradually reduce the incentives as the firms’ potential earnings grow (Wang et al., 2015 [27]). In addition, related studies reveal the differentiated effects of government incentives and penalties on different market players; manufacturers and recyclers are more sensitive to subsidies than consumers, but consumer decision-making is the key to market stability (Li et al., 2024 [33]), and government incentives still significantly increase consumers’ willingness to formally recycle their EOL power batteries (Li et al., 2023 [34]).

Constructing an efficient EOL power battery recycling system is regarded as a key link for the healthy development of the EV industry (Zhou et al., 2023 [35]). Existing studies have analysed the e-waste management system in depth from the perspective of the extended producer system (EPR) (Cao et al., 2016 [36]) and explicitly pointed out the necessity of recycling e-waste, including EOL power batteries (Li et al., 2016 [37]).

2.3. Evolutionary Game Theory in Battery Recycling Research

In the study of the operation mechanism, the game theory method has become an important tool for parsing the complex relationships in the recycling field. Existing studies use game theory methods to explore the recycling mechanism of EOL power battery from different perspectives and different participating subjects (He et al., 2021 [38]; Li et al., 2024 [33]; Li et al., 2022 [39]; Tang et al., 2018 [40]). Among them, related studies explore the optimal joint recycling strategy from multiple perspectives, including government-led (Zhang et al., 2022 [41]), and manufacturer-led (Adepetu et al., 2017 [42]). Some scholars have analysed the evolutionary characteristics of different scenarios by constructing an evolutionary game model of formal and informal recycling enterprises (Yu et al., 2022 [43]); relevant studies have analysed the pricing mechanism of recycling modes in different subsidy scenarios and found that subsidies can promote the inflow of EOL power batteries into the formal channel and form an inhibition to the development of informal recyclers (Zhu et al., 2020 [44]; Xiao et al., 2024 [18]). Studies further suggest that the recycling price advantage of informal firms is a key factor that interferes with the participation of supply-side groups in formal recycling, while consumers’ preferences for high recycling prices further exacerbates the attractiveness of informal recycling channels (He et al., 2021 [38]).

A review of existing research indicates that academia has established a preliminary theoretical framework for establishing EOL battery recycling mechanisms, encompassing closed-loop supply chains, deposit refund policies, and governmental incentive and penalty systems. Nevertheless, existing research exhibits two notable limitations: Firstly, while the policy effects of governmental incentive and penalty mechanisms have become a research focus, most of the literature concentrates solely on the impact analysis of individual policy instruments. Few studies incorporate governmental penalty mechanisms into game-theoretic frameworks or systematically examine the dual effects of reward and penalty mechanisms on the two core stakeholders—recycling enterprises and EV owners. Secondly, in terms of research dimensions concerning recycling participants, existing findings predominantly concentrate on core supply-side entities such as governments, producers, manufacturers, and recyclers, with insufficient exploration of EV owners as a critical group. Although some studies have noted the motivational impact of government incentive policies on EV owners’ recycling willingness, related analyses predominantly employ empirical research methods, making it challenging to depict the dynamic behavioural evolution patterns of recycling enterprises, EV owners, and governments within scenarios of information asymmetry.

3. Materials and Methods

3.1. Model Assumptions and Parameter Settings

In this section, a three-party evolutionary game model is constructed with the government, recycling enterprises, and EV owners. Participants in evolutionary game models achieve strategy optimisation through information acquisition and dynamic strategic adjustments. The government plays a guiding and supervisory role. Recycling enterprises include professional third-party recycling enterprises, recycling departments of the battery manufacturers, and recycling service outlets set up by EV manufacturers. EV owners act as EOL power battery suppliers.

Figure 1 shows the combinations of strategies available for the government, recycling enterprises, and EV owners.

Figure 2 illustrates the complete research framework from modelling analysis to numerical simulation.

In order to establish the evolutionary game model and to explore the strategic stability and equilibrium state of different participating subjects, the following relevant hypotheses are formulated:

Hypothesis 1:

The three main players of the game are the government, recycling enterprises, and EV owners, and all members are regarded as economic persons with bounded rationality.

Hypothesis 2:

The government implements the strategies of “regulation” and “nonregulation” with the probabilities of x and 1 − x, $x\in [0, 1]$. Recycling enterprises choose “formal recycling” or “informal recycling”, with probabilities of y and 1 − y, $y\in [0, 1]$. The probability that an EV owner adopts the recycling channels of “participating in formal recycling” and “not participating in formal recycling” z and 1 − z, $z\in [0, 1]$.

Hypothesis 3:

Informal recycling enterprises offer higher recycling prices than formal ones, ${ R}_1<R_2$. The cost incurred by formal recycling enterprises in recycling EOL power batteries is $C_1$, while that incurred by informal ones is $C_2$. The cost of formal recycling is higher than that of informal recycling, attributable to additional expenditures such as technological upgrades, ${\mathrm{C}}_1>{\mathrm{C}}_2$. When government incentives and penalties are not taken into account, the net profit of informal recycling exceeds the net profit of formal recycling, ${\mathrm{R}}_4-{\mathrm{C}}_2>{\mathrm{R}}_3-{\mathrm{C}}_1>0$.

Hypothesis 4:

The cost of regulation by the government is $C_g$, the incentive subsidy by the government to formal recycling enterprises is $T_1$, and the regulatory penalty by the government to informal recycling enterprises is $M_1$. The government’s incentive for EV owners to participate in formal recycling behaviour is T. Beyond the aforementioned subsidies and penalties, the government incurs other costs in the regulatory process, ${\mathrm{C}}_{\mathrm{g}}>{\mathrm{M}}_1+{\mathrm{T}}_1+\mathrm{T}$.

Hypothesis 5:

The formal recycling of EOL power batteries by recycling enterprises generates corporate reputation $E$ and social benefits for the government G. When the recycling company informally recycles EOL power batteries and EV owners choose not to participate in the formal recycling, it costs the government D.

The parameters are summarised in

Table 1. Parameter summary table.

Argument	Implication
${\mathrm{C}}_{\mathrm{g}}$	Regulatory cost of government involvement in promoting EOL power battery recycling
${\mathrm{T}}_1$	Government incentive subsidy for formal recycling enterprises
${\mathrm{M}}_1$	Administrative fines imposed by the government on informal recycling enterprises
$\mathrm{T}$	Government incentives for EV owners who participate in formal recycling
$\mathrm{G}$	Social benefits to the government from the formal recycling of EOL power batteries by recycling enterprises
$\mathrm{D}$	Costs to the government of regulating the recycling industry and promoting environmentally friendly behaviour
${\mathrm{R}}_1$	Economic benefits for EV owners participating in the formal recycling of EOL power batteries
${\mathrm{R}}_2$	Economic benefits to EV owners who participate in the informal recycling of EOL power batteries
${\mathrm{C}}_1$	Costs incurred by recycling enterprises choosing formal recycling of EOL power batteries
${\mathrm{C}}_2$	Costs incurred by recycling enterprises in choosing the informal recycling of EOL power batteries
${\mathrm{R}}_3$	Benefits of recycling enterprises for choosing formal recycling of EOL power batteries
${\mathrm{R}}_4$	Benefits to recycling enterprises of choosing informal recycling of EOL power batteries
$\mathrm{E}$	Reputational benefits for recycling enterprises choosing formal recycling

below.

3.2. Modelling and Solution

A matrix of the benefits to the government, recycling enterprises, and EV owners under different combinations of strategies is constructed as shown in

Table 2. Tripartite game payment matrix.

Recycling Enterprise	EV Owner	Government
		Participate in Regulation	No Regulation
Regular Recycling	Participation	$-{\mathrm{C}}_{\mathrm{g}}-\mathrm{T}-{\mathrm{T}}_1+\mathrm{G}$; ${\mathrm{R}}_3-{\mathrm{C}}_1+{\mathrm{T}}_1+\mathrm{E}; {\mathrm{R}}_1+\mathrm{T}$	$\mathrm{G}$; ${\mathrm{R}}_3-{\mathrm{C}}_1+\mathrm{E}$; ${\mathrm{R}}_1$
	Non-Participation	$-{\mathrm{C}}_{\mathrm{g}}-{\mathrm{T}}_1+\mathrm{G}$; $-{\mathrm{C}}_1+{\mathrm{T}}_1+\mathrm{E}; {\mathrm{R}}_2$	$\mathrm{G}$; $-{\mathrm{C}}_1+\mathrm{E}$; ${\mathrm{R}}_2$
Informal Recycling	Participation	$-{\mathrm{C}}_{\mathrm{g}}-\mathrm{T}+{\mathrm{M}}_1$; $-{\mathrm{C}}_2-{\mathrm{M}}_1$; $R_1$ + T	0; $-{\mathrm{C}}_2$; ${\mathrm{R}}_1$
	Non-Participation	$-{\mathrm{C}}_{\mathrm{g}}+{\mathrm{M}}_1; {\mathrm{R}}_4-{\mathrm{C}}_2-{\mathrm{M}}_1$; $R_2$	$-\mathrm{D}; {\mathrm{R}}_4-{\mathrm{C}}_2$; ${\mathrm{R}}_2$

.

According to the benefit matrix, the expected benefits when the government conducts regulation are as follows:

$$\begin{array}{cc}{U}_{11}& =yz\left(-C_g-T-T_1+G\right)+y\left(1-z\right)\left(-C_g-T_1+G\right)\hfill \\ & +(1-y)z\left(-C_g-T+M_1\right)+(1-y)(1-z)(-C_g+M_1)\hfill \end{array}$$

(1)

The expected benefits when the government conducts non-regulation are as follows:

The government’s average expected benefits are as follows:

$$U_{12}=yzG+y\left(1-z\right)G-(1-y)(1-z)D$$

(2)

$$U_1=x{U}_{11}+(1-x)U_{12}$$

(3)

Construct a replicator dynamic equation for government strategies based on the principles and methods of evolutionary stability. Based on Equations (1) and (2), the replication dynamic equation for the government strategy is as follows:

$$\begin{array}{cc}F\left(x\right)={\displaystyle \frac{dx}{dt}}=& x{(U}_{11}-U_1)=x(1-x)(U_{11}-U_{12})\hfill \\ & =x(1-x)[yzD+y\left(-T_1-M_1-D\right)-z\left(T+D\right)+M_1-C_g+D]\hfill \end{array}$$

(4)

$\mathrm{F}(\mathrm{x})={\displaystyle \frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}}$ represents the rate of change over time in the probability of the government adopting the “regulatory participation” strategy.

Similarly, the replicator dynamics equation for recycling enterprises is as follows:

$$\begin{array}{cc}F\left(y\right)& ={\displaystyle \frac{dy}{dt}}=y{(U}_{21}-U_2)=y(1-y)(U_{21}-U_{22})\hfill \\ & =y(1-y)[xz(C_1-C_2)+x\left(T_1+M_1\right)+z\left(R_3+R_4\right)-C_1+R_4-C_2]\end{array}$$

(5)

The replicator dynamics equation for EV owners is as follows:

$$\begin{array}{cc}F\left(z\right)& ={\displaystyle \frac{dz}{dt}}=z{(U}_{31}-U_3)\hfill \\ & =z(1-z)(U_{31}-U_{32})=z(1-z)(-xyT+xT+yT+R_1-R_2)\hfill \end{array}$$

(6)

4. Results

4.1. Evolutionary Stability Analysis of Government Strategies

From the stability theorem of the replicator dynamics equation, the stability point of the government’s ESS requires satisfying both $\mathrm{F}\left(\mathrm{x}\right)=0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{x})}{\mathrm{d}\mathrm{x}}}<0$. When F′(x) < 0, this ensures that the function F(x) is monotonically decreasing within its domain, thereby guaranteeing the convergence of the control strategy.

$$F^{\prime }\left(x\right)={\displaystyle \frac{dF(x)}{dx}}=\left(1-2x\right)[yzD+y\left(-T_1-M_1-D\right)-z\left(T+D\right)+M_1-C_g+D]$$

(7)

Let G(y,z) = $[yzD+y\left(-T_1-M_1-D\right)-z\left(T+D\right)+M_1-C_g+D]$, so that Equation (4) can be written as $F\left(x\right)=x(1-x)G(y,z)$, Equation (7) can be written as $F^{\prime }\left(x\right)=\left(1-2x\right)G(y,z)$. Let G(y,z) = 0 have ${\mathrm{y}}^{\mathrm{*}}={\displaystyle \frac{\mathrm{z}\left(\mathrm{T}+\mathrm{D}\right)-{\mathrm{M}}_1+{\mathrm{C}}_{\mathrm{g}}-\mathrm{D}}{\mathrm{z}\mathrm{D}-\mathrm{T}-{\mathrm{M}}_1-\mathrm{D}}}$ and ${\mathrm{z}}^{\mathrm{*}}={\displaystyle \frac{\mathrm{y}\left({\mathrm{T}}_1+{\mathrm{M}}_1+\mathrm{D}\right)-{\mathrm{M}}_1+{\mathrm{C}}_{\mathrm{g}}-\mathrm{D}}{\mathrm{y}\mathrm{D}-\mathrm{T}-\mathrm{D}}}$.

When ${\mathrm{y}=\mathrm{y}}^{\mathrm{*}}={\displaystyle \frac{\mathrm{z}\left(\mathrm{T}+\mathrm{D}\right)-{\mathrm{M}}_1+{\mathrm{C}}_{\mathrm{g}}-\mathrm{D}}{\mathrm{z}\mathrm{D}-\mathrm{T}-{\mathrm{M}}_1-\mathrm{D}}}$ or ${\mathrm{z}=\mathrm{z}}^{\mathrm{*}}={\displaystyle \frac{\mathrm{y}\left({\mathrm{T}}_1+{\mathrm{M}}_1+\mathrm{D}\right)-{\mathrm{M}}_1+{\mathrm{C}}_{\mathrm{g}}-\mathrm{D}}{\mathrm{y}\mathrm{D}-\mathrm{T}-\mathrm{D}}}$, then $\mathrm{F}\left(\mathrm{x}\right)\equiv 0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{x})}{\mathrm{d}\mathrm{x}}}$ ≡ 0, the government cannot determine an ESS.

If $0<\mathrm{y}<{\mathrm{y}}^{\mathrm{*}}$ or $0<\mathrm{z}<{\mathrm{z}}^{\mathrm{*}}$, $G(y,z)>0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{x})}{\mathrm{d}\mathrm{x}}}{|}_{\mathrm{x}=0}>0$, ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{x})}{\mathrm{d}\mathrm{x}}}{|}_{\mathrm{x}=1}<0$, then $\mathrm{x}=1$ is the only evolutionarily stable state, and the government chooses the “regulate” strategy.

If ${\mathrm{y}}^{\mathrm{*}}<\mathrm{y}<1$ or ${\mathrm{z}}^{\mathrm{*}}<\mathrm{z}<1$, $G(y,z)<0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{x})}{\mathrm{d}\mathrm{x}}}{|}_{\mathrm{x}=0}<0$, ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{x})}{\mathrm{d}\mathrm{x}}}{|}_{\mathrm{x}=1}>0$, then $\mathrm{x}=0$ is the only evolutionarily stable state, and the government chooses the “no regulation” strategy.

4.2. Evolutionary Stability Analysis of Recycling Enterprises’ Strategies

Similarly, the stability point of the recycling enterprise’s ESS requires satisfying both $\mathrm{F}\left(\mathrm{y}\right)=0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{y})}{\mathrm{d}\mathrm{y}}}<0$.

$$\begin{array}{cc}{ F}^{\prime }\left(y\right)& {\displaystyle =\frac{dF\left(y\right)}{dy}}\hfill \\ & =(1-2y)[xz(C_1-C_2)+x\left(T_1+M_1\right)+z\left(R_3+R_4\right)-C_1+R_4-C_2]\hfill \end{array}$$

(8)

Let G(x,z) $=[xz(C_1-C_2)+x\left(T_1+M_1\right)+z\left(R_3+R_4\right)-C_1+R_4-C_2]$, so that Equation (5) can be written as $F\left(y\right)=y(1-y)G(x,z)$, Equation (8) can be written as $F^{\prime }\left(y\right)=\left(1-2y\right)G(x,z)$. Let G(x,z) = 0 have ${\mathrm{x}}^{\mathrm{*}}={\displaystyle \frac{-\mathrm{z}\left({\mathrm{R}}_3+{\mathrm{R}}_4\right)+{\mathrm{C}}_1-{\mathrm{R}}_4+{\mathrm{C}}_2}{\mathrm{z}\left({\mathrm{C}}_1-{\mathrm{C}}_2\right)+{\mathrm{T}}_1+{\mathrm{M}}_1}}$ and ${\mathrm{z}}^{\mathrm{*}}={\displaystyle \frac{-\mathrm{x}\left({\mathrm{T}}_1+{\mathrm{M}}_1\right)+{\mathrm{C}}_1-{\mathrm{R}}_4+{\mathrm{C}}_2}{\mathrm{x}\left({\mathrm{C}}_1-{\mathrm{C}}_2\right)+{\mathrm{R}}_3+{\mathrm{R}}_4}}$.

When ${\mathrm{x}=\mathrm{x}}^{\mathrm{*}}={\displaystyle \frac{-\mathrm{z}\left({\mathrm{R}}_3+{\mathrm{R}}_4\right)+{\mathrm{C}}_1-{\mathrm{R}}_4+{\mathrm{C}}_2}{\mathrm{z}\left({\mathrm{C}}_1-{\mathrm{C}}_2\right)+{\mathrm{T}}_1+{\mathrm{M}}_1}}$ or ${\mathrm{z}=\mathrm{z}}^{\mathrm{*}}={\displaystyle \frac{-\mathrm{x}\left({\mathrm{T}}_1+{\mathrm{M}}_1\right)+{\mathrm{C}}_1-{\mathrm{R}}_4+{\mathrm{C}}_2}{\mathrm{x}\left({\mathrm{C}}_1-{\mathrm{C}}_2\right)+{\mathrm{R}}_3+{\mathrm{R}}_4}}$, then $\mathrm{F}\left(\mathrm{y}\right)\equiv 0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{y})}{\mathrm{d}\mathrm{y}}}$ ≡ 0, the recycling enterprises cannot determine an ESS.

If ${\mathrm{x} }^{\mathrm{*}}<\mathrm{x}<1$ or ${\mathrm{z}}^{\mathrm{*}}<\mathrm{z}<1$, $G(x,z)>0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{y})}{\mathrm{d}\mathrm{y}}}{|}_{\mathrm{y}=0}>0$, ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{y})}{\mathrm{d}\mathrm{y}}}{|}_{\mathrm{y}=1}<0$, then $\mathrm{y}=1$ is the only evolutionarily stable state, and the recycling enterprises chooses the “formal recycling” strategy.

If $0<\mathrm{x}<{\mathrm{x}}^{\mathrm{*}} \mathrm{o}\mathrm{r} 0<\mathrm{z}<{\mathrm{z}}^{\mathrm{*}}$, $G(x,z)<0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{y})}{\mathrm{d}\mathrm{y}}}{|}_{\mathrm{y}=0}<0$, ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{y})}{\mathrm{d}\mathrm{y}}}{|}_{\mathrm{y}=1}>0$, then $\mathrm{y}=0$ is the only evolutionarily stable state, and the recycling enterprises chooses the “informal recycling” strategy.

4.3. Evolutionary Stability Analysis of the EV Owner’s Strategy

Similarly, the stability point of the EV owner’s ESS requires satisfying both $\mathrm{F}\left(\mathrm{z}\right)=0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{z})}{\mathrm{d}\mathrm{z}}}<0$.

$$F^{\prime }\left(z\right)={\displaystyle \frac{dF(z)}{dz}}=\left(1-2z\right)(-xyT+xT+yT+R_1-R_2)$$

(9)

Let G(x,y) $=(-xyT+xT+yT+R_1-R_2)$, so that Equation (6) can be written as $F\left(z\right)=z(1-z)G(x,y)$, and Equation (9) can be written as $F^{\prime }\left(z\right)=\left(1-2z\right)G(x,y)$. Let G(x,y) = 0 have ${\mathrm{x}}^{\mathrm{*}}={\displaystyle \frac{-\mathrm{y}\mathrm{T}-{\mathrm{R}}_1+{\mathrm{R}}_2}{-\mathrm{y}\mathrm{T}+\mathrm{T}}}$ and ${\mathrm{y}}^{\mathrm{*}}={\displaystyle \frac{-\mathrm{x}\mathrm{T}-{\mathrm{R}}_1+{\mathrm{R}}_2}{-\mathrm{x}\mathrm{T}+\mathrm{T}}}$.

When ${\mathrm{x}=\mathrm{x}}^{\mathrm{*}}={\displaystyle \frac{-\mathrm{y}\mathrm{T}-{\mathrm{R}}_1+{\mathrm{R}}_2}{-\mathrm{y}\mathrm{T}+\mathrm{T}}}$ or ${\mathrm{y}=\mathrm{y}}^{\mathrm{*}}={\displaystyle \frac{-\mathrm{x}\mathrm{T}-{\mathrm{R}}_1+{\mathrm{R}}_2}{-\mathrm{x}\mathrm{T}+\mathrm{T}}}$, then $\mathrm{F}\left(\mathrm{z}\right)\equiv 0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{z})}{\mathrm{d}\mathrm{z}}}$ ≡ 0, the EV owner cannot determine an ESS.

If ${\mathrm{x}}^{\mathrm{*}}<\mathrm{x}<1$ or ${\mathrm{y}}^{\mathrm{*}}<\mathrm{y}<1$, $G(x,y)>0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{z})}{\mathrm{d}\mathrm{z}}}{|}_{\mathrm{z}=0}>0$, ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{z})}{\mathrm{d}\mathrm{z}}}{|}_{\mathrm{z}=1}<0$, then $\mathrm{z}=1$ is the only evolutionarily stable state, and the EV owner chooses to “participate in formal recycling”.

If $0<\mathrm{x}<{\mathrm{x}}^{\mathrm{*}} \mathrm{o}\mathrm{r} 0<\mathrm{y}<{\mathrm{y}}^{\mathrm{*}}$, $G(x,y)<0$ and ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{z})}{\mathrm{d}\mathrm{z}}}{|}_{\mathrm{z}=0}<0$, ${\displaystyle \frac{\mathrm{d}\mathrm{F}(\mathrm{z})}{\mathrm{d}\mathrm{z}}}{|}_{\mathrm{z}=1}>0$, then $\mathrm{z}=0$ is the only evolutionarily stable state, and the EV owner chooses not to participate in formal recycling.

4.4. Evolutionary Stability Analysis of the System

The replication dynamic equations are constructed by combining Equations (4)–(6), as shown in Equation (10). Let $\mathrm{F}\left(\mathrm{x}\right)=0$,$\mathrm{F}\left(\mathrm{y}\right)=0$,$\mathrm{F}\left(\mathrm{z}\right)=0$, the eight pure strategy equilibrium points can be obtained as ${\mathrm{E}}_1(0,0,0)$, ${\mathrm{E}}_2(1,0,0)$, ${\mathrm{E}}_3(0,1,0)$, ${\mathrm{E}}_4(0,0,1)$, ${\mathrm{E}}_5(0,1,1)$, ${\mathrm{E}}_6(1,0,1)$, ${\mathrm{E}}_7(1,1,0)$, ${\mathrm{E}}_8(1,1,1)$.

$$\left\{\begin{array}{l}F\left(x\right)={\displaystyle \frac{dx}{dt}}=x\left(1-x\right)[yzD+y\left(-T_1-M_1-D\right)-z\left(T+D\right)+M_1-C_g+D]\\ F\left(y\right)={\displaystyle \frac{dy}{dt}}=y\left(1-y\right)[xz(C_1-C_2)+x\left(T_1+M_1\right)+z\left(R_3+R_4\right)-C_1+R_4-C_2]\\ F\left(z\right)={\displaystyle \frac{dz}{dt}}=z(1-z)(-xyT+xT+yT+R_1-R_2)\end{array}\right.$$

(10)

According to the system stability criterion proposed by Friedman (Friedman, 1991 [45]), the Jacobian matrix ${\mathrm{J}}^{\prime }$ of this game system is as follows:

$$J^{\prime }=\left[\begin{array}{ccc}{\displaystyle \frac{\partial F(x)}{\partial x}}& {\displaystyle \frac{\partial F(x)}{\partial y}}& {\displaystyle \frac{\partial F(x)}{\partial z}}\\ {\displaystyle \frac{\partial F(y)}{\partial x}}& {\displaystyle \frac{\partial F(y)}{\partial y}}& {\displaystyle \frac{\partial F(y)}{\partial z}}\\ {\displaystyle \frac{\partial F(z)}{\partial x}}& {\displaystyle \frac{\partial F(z)}{\partial y}}& {\displaystyle \frac{\partial F(z)}{\partial z}}\end{array}\right]=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$$

Then,

$${ a}_{11}=\left(1-2x\right)\left[{-C}_g+yzD+D+M_1-z\left(T+D\right)-y\left(D+M_1+T_1\right)\right]$$

$${ a}_{12}=x(x-1)(-zD+D+M_1+T_1)$$

$${ a}_{13}=x(x-1)(-yD+D+T)$$

$${ a}_{21}=y\left(1-y\right)[M_1+T_1+z(C_1-C_2)]$$

$${ a}_{22}=\left(1-2y\right)[-C_1-C_2+R_4+xz\left(C_1-C_2\right)+x\left(M_1+T_1\right)+z(R_3+R_4)]$$

$${ a}_{23}=y\left(1-y\right)[R_3+R_4+x(C_1-C_2)]$$

$${ a}_{31}=z(z-1)(y-1)T$$

$${ a}_{32}=zT(x-1)(z-1)$$

$${ a}_{33}=\left(1-2z\right)[R_1-R_2-(xy-x-y)T]$$

According to the Lyapunov stability theorem, when all the eigenvalues in the Jacobi matrix have negative real parts, the equilibrium point is asymptotically stable; if there is at least one positive real part in the eigenvalues of the Jacobi matrix, the equilibrium point is an unstable point (Yang et al., 2020 [46]).

According to Lyapunov’s law, if the eigenvalues only have negative real parts, i.e., ${\mathsf{\lambda }}_{\mathbf{i}}$ < 0, then the equilibrium point is the asymptotically stable point of the evolutionary game system; if the eigenvalues only have positive real parts, i.e., ${\mathsf{\lambda }}_{\mathbf{i}}$ > 0, then the equilibrium point is the unstable point; and if the eigenvalues have both negative and positive real parts, then the equilibrium point is the saddle point.

The main diagonal elements ${ a}_{11}$, ${ a}_{22}$, ${ a}_{33}$, respectively, correspond to the self-influences of the government, recycling enterprises, and EV owners’ strategies. For pure strategy equilibrium points (where x, y, z are 0 or 1), the Jacobian matrix simplifies to diagonal form, with eigenvalues ${\mathsf{\lambda }}_{\mathbf{1}}$, ${\mathsf{\lambda }}_{\mathbf{2}}$, ${\mathsf{\lambda }}_{\mathbf{3}}$ representing the values of ${ a}_{11}$, ${ a}_{22}$, and ${ a}_{33}$.

The eigenvalues of each equilibrium point in the evolutionary game system are shown in

Table 3. Eigenvalues of the Jacobi matrix of equilibrium points.

Equilibrium Point	Eigenvalues
	${\mathsf{\lambda }}_{\mathbf{1}}$	${\mathsf{\lambda }}_{\mathbf{2}}$	${\mathsf{\lambda }}_{\mathbf{3}}$
${\mathbf{E}}_{\mathbf{1}}(\mathbf{0},\mathbf{0},\mathbf{0})$	${-\mathbf{C}}_{\mathbf{g}}+{\mathbf{M}}_{\mathbf{1}}+\mathbf{D}$	$-{\mathbf{C}}_{\mathbf{1}}-{\mathbf{C}}_{\mathbf{2}}+{\mathbf{R}}_{\mathbf{4}}$	${\mathbf{R}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{2}}$
${\mathbf{E}}_{\mathbf{2}}(\mathbf{1},\mathbf{0},\mathbf{0})$	${\mathbf{C}}_{\mathbf{g}}-{\mathbf{M}}_{\mathbf{1}}-\mathbf{D}$	$-{\mathbf{C}}_{\mathbf{1}}-{\mathbf{C}}_{\mathbf{2}}+{\mathbf{M}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{4}}+{\mathbf{T}}_{\mathbf{1}}$	${\mathbf{R}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{2}}+\mathbf{T}$
${\mathbf{E}}_{\mathbf{3}}(\mathbf{0},\mathbf{1},\mathbf{0})$	${-\mathbf{C}}_{\mathbf{g}}-{\mathbf{T}}_{\mathbf{1}}$	${\mathbf{C}}_{\mathbf{1}}+{\mathbf{C}}_{\mathbf{2}}-{\mathbf{R}}_{\mathbf{4}}$	${\mathbf{R}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{2}}+\mathbf{T}$
${\mathbf{E}}_{\mathbf{4}}(\mathbf{0},\mathbf{0},\mathbf{1})$	$-{\mathbf{C}}_{\mathbf{g}}+{\mathbf{M}}_{\mathbf{1}}-\mathbf{T}$	$-{\mathbf{C}}_{\mathbf{1}}-{\mathbf{C}}_{\mathbf{2}}+{\mathbf{R}}_{\mathbf{3}}{+\mathbf{2}\mathbf{R}}_{\mathbf{4}}$	${-\mathbf{R}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{2}}$
${\mathbf{E}}_{\mathbf{5}}(\mathbf{0},\mathbf{1},\mathbf{1})$	$-{\mathbf{C}}_{\mathbf{g}}-\mathbf{T}-{\mathbf{T}}_{\mathbf{1}}$	${\mathbf{C}}_{\mathbf{1}}+{\mathbf{C}}_{\mathbf{2}}-{\mathbf{R}}_{\mathbf{3}}-{\mathbf{2}\mathbf{R}}_{\mathbf{4}}$	${-\mathbf{R}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{2}}-\mathbf{T}$
${\mathbf{E}}_{\mathbf{6}}(\mathbf{1},\mathbf{0},\mathbf{1})$	${\mathbf{C}}_{\mathbf{g}}-{\mathbf{M}}_{\mathbf{1}}+\mathbf{T}$	$-\mathbf{2}{\mathbf{C}}_{\mathbf{2}}+{\mathbf{M}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{3}}+\mathbf{2}{\mathbf{R}}_{\mathbf{4}}+{\mathbf{T}}_{\mathbf{1}}$	${-\mathbf{R}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{2}}-\mathbf{T}$
${\mathbf{E}}_{\mathbf{7}}(\mathbf{1},\mathbf{1},\mathbf{0})$	${\mathbf{C}}_{\mathbf{g}}+{\mathbf{T}}_{\mathbf{1}}$	${\mathbf{C}}_{\mathbf{1}}+{\mathbf{C}}_{\mathbf{2}}-{\mathbf{M}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{4}}-{\mathbf{T}}_{\mathbf{1}}$	${\mathbf{R}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{2}}+\mathbf{T}$
${\mathbf{E}}_{\mathbf{8}}(\mathbf{1},\mathbf{1},\mathbf{1})$	${\mathbf{C}}_{\mathbf{g}}+\mathbf{T}+{\mathbf{T}}_{\mathbf{1}}$	$\mathbf{2}{\mathbf{C}}_{\mathbf{2}}-{\mathbf{M}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{3}}-\mathbf{2}{\mathbf{R}}_{\mathbf{4}}-{\mathbf{T}}_{\mathbf{1}}$	${-\mathbf{R}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{2}}-\mathbf{T}$

.

Under the assumed preconditions, all variables are greater than 0, and $R_2>R_1>0, C_1>C_2>0, R_4-C_2>R_3-R_1>0, C_g>M_1+T_1+T.$ By observation, a preliminary assessment of the characteristic value ${\mathsf{\lambda }}_{\mathbf{1}}$, ${\mathsf{\lambda }}_{\mathbf{2}}$, ${\mathsf{\lambda }}_{\mathbf{3}}$ a can be made. We could determine that the eigenvalues ${\mathsf{\lambda }}_{\mathbf{1}}$ of ${\mathrm{E}}_7(1,1,0)$, ${\mathrm{E}}_8(1,1,1)$ are positive, the eigenvalues ${\mathsf{\lambda }}_{\mathbf{2}}$ of ${\mathrm{E}}_4(0,0,1)$, ${\mathrm{E}}_6(1,0,1)$ are positive under the assumption $R_4-C_2>R_3-R_1>0$, and the eigenvalues ${\mathsf{\lambda }}_{\mathbf{3}}$ of ${\mathrm{E}}_4(0,0,1)$ are positive under the assumption $R_2>R_1>0$. Therefore, it is just required to analyse the conditions when the rest of the four points are ESS.

The stability of each equilibrium point is judged by analysing the positivity and negativity of all the eigenvalues in Table 3, as shown in

Table 4. Equilibrium stability point analysis.

Equilibrium Point	Stability Condition	Stability
${\mathrm{E}}_1(0,0,0)$	${-\mathbf{C}}_{\mathbf{g}}+{\mathbf{M}}_{\mathbf{1}}+\mathbf{D}<0$, $-{\mathbf{C}}_{\mathbf{1}}-{\mathbf{C}}_{\mathbf{2}}+{\mathbf{R}}_{\mathbf{4}}<0$	Scenario 1
${\mathrm{E}}_2(1,0,0)$	${\mathbf{C}}_{\mathbf{g}}-{\mathbf{M}}_{\mathbf{1}}-\mathbf{D}<\mathbf{0}$, $-{\mathbf{C}}_{\mathbf{1}}-{\mathbf{C}}_{\mathbf{2}}+{\mathbf{M}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{4}}+{\mathbf{T}}_{\mathbf{1}}<\mathbf{0}$, ${\mathbf{R}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{2}}+\mathbf{T}<\mathbf{0}$	Scenario 2
${\mathrm{E}}_3(0,1,0)$	${\mathbf{C}}_{\mathbf{1}}+{\mathbf{C}}_{\mathbf{2}}-{\mathbf{R}}_{\mathbf{4}}<\mathbf{0}$, ${\mathbf{R}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{2}}+\mathbf{T}<\mathbf{0}$	Scenario 3
${\mathrm{E}}_5(0,1,1)$	${-\mathbf{R}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{2}}-\mathbf{T}<\mathbf{0}$	Scenario 4

.

There may exist four kinds of equilibrium stability points in this evolutionary system, which are ${\mathrm{E}}_1(0,0,0)$, ${\mathrm{E}}_2(1,0,0)$, ${\mathrm{E}}_3(0,1,0)$, ${\mathrm{E}}_5(0,1,1)$, and the equilibrium stability points are analysed next.

Scenario 1: When ${-\mathbf{C}}_{\mathbf{g}}+{\mathbf{M}}_{\mathbf{1}}+\mathbf{D}<\mathbf{0}$, $-{\mathbf{C}}_{\mathbf{1}}-{\mathbf{C}}_{\mathbf{2}}+{\mathbf{R}}_{\mathbf{4}}<\mathbf{0}$, then ${\mathbf{C}}_{\mathbf{g}}>{\mathbf{M}}_{\mathbf{1}}+\mathbf{D}, {\mathbf{C}}_{\mathbf{2}}>{\mathbf{R}}_{\mathbf{4}}-{\mathbf{C}}_{\mathbf{1}}, {\mathrm{E}}_1(0,0,0)$ is a stable point. The evolution strategy is (non-regulation, adopt an informal recycling strategy, choose informal recycling channels). The government chooses not to regulate because its net regulatory benefit is negative; recycling enterprises opt for informal recycling due to the constraint of high costs associated with formal recycling; and vehicle owners gain greater profits from participating in informal recycling.

Scenario 2: When ${\mathbf{C}}_{\mathbf{g}}-{\mathbf{M}}_{\mathbf{1}}-\mathbf{D}<\mathbf{0}$, $-{\mathbf{C}}_{\mathbf{1}}-{\mathbf{C}}_{\mathbf{2}}+{\mathbf{M}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{4}}+{\mathbf{T}}_{\mathbf{1}}<\mathbf{0}$, ${\mathbf{R}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{2}}+\mathbf{T}<\mathbf{0}, {\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} {\mathbf{C}}_{\mathbf{g}}<{\mathbf{M}}_{\mathbf{1}}+\mathbf{D}, {\mathbf{C}}_{\mathbf{2}}>-{\mathbf{M}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{4}}-{\mathbf{T}}_{\mathbf{1}}+{\mathbf{C}}_{\mathbf{1}}, {\mathbf{R}}_{\mathbf{2}}>{\mathbf{R}}_{\mathbf{1}}+\mathbf{T}, \mathrm{E}}_{\mathbf{2}}(1,0,0)$ is a stable point. The evolution strategy is (regulation, adopt an informal recycling strategy, choose informal recycling channels). When the government’s net regulatory benefit is positive, it will choose to regulate; if the profit of recycling enterprises from engaging in informal recycling—after accounting for government rewards and penalties—remains higher than the costs of formal recycling, the enterprises will stably opt for informal recycling; meanwhile, vehicle owners’ benefits from informal channels are higher than the sum of their benefits from formal channels and subsidies.

Scenario 3: When ${\mathbf{C}}_{\mathbf{1}}+{\mathbf{C}}_{\mathbf{2}}-{\mathbf{R}}_{\mathbf{4}}<\mathbf{0}$, ${\mathbf{R}}_{\mathbf{1}}-{\mathbf{R}}_{\mathbf{2}}+\mathbf{T}<\mathbf{0}$, then ${\mathbf{C}}_{\mathbf{1}}<{\mathbf{R}}_{\mathbf{4}}-{\mathbf{C}}_{\mathbf{2}}, {\mathbf{R}}_{\mathbf{2}}>{\mathbf{R}}_{\mathbf{1}}+\mathbf{T},{\mathrm{E}}_{\mathbf{3}}(0,1,0)$ is a stable point. The evolution strategy is (non-regulation, adopt a formal recycling strategy, choose informal recycling channels).When recycling enterprises face lower costs in choosing formal recycling, they will take the initiative to opt for it; meanwhile, if EV owners’ benefits from informal channels are higher than the sum of their benefits from formal channels and subsidies, they will also choose informal recycling.

Scenario 4: When ${-\mathbf{R}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{2}}-\mathbf{T}<\mathbf{0}$, then ${\mathbf{R}}_{\mathbf{1}}>{\mathbf{R}}_{\mathbf{2}}-\mathbf{T}, {\mathrm{E}}_5(0,1,1)$ is a stable point. The evolution strategy is (non-regulation, adopt a formal recycling strategy, choose formal recycling channels). When EV owners gain higher benefits from participating in formal recycling, they will take the initiative to choose to engage in it.

4.5. Analysis of Evolutionary Game Instability Points

E₄(0, 0, 1) reflects the disconnect between EV owners’ intentions to participate in formal recycling and their actual actions when recycling enterprises adopt informal recycling strategies. Despite some EV owners possessing environmental awareness, the absence of formal recycling infrastructure combined with the price advantage of informal recycling channels creates an incentive structure unfavourable to formal recycling.

E₆(1, 0, 1) reflects the “transitional dilemma” within the recycling industry. Despite government campaigns fostering EV owners’ willingness to participate in formal recycling, the profitability of informal recycling coupled with excessive regulatory costs renders oversight ineffective. The recycling market thus remains in an unstable state characterised by “EV owners desiring formality, recycling enterprises operating informally, and government struggling to regulate”.

E₇(1, 1, 0) illustrates a scenario where government regulation drives recycling enterprises towards formal recycling, yet EV owners persist with informal channels due to short-term profit preferences. This instability demonstrates the limitations of policy interventions targeting recycling enterprises alone; stimulating owner participation is equally crucial for systemic stability.

E₈(1, 1, 1) reveals that the system’s sustainability hinges on a specific cost–benefit structure. Any imbalance in returns for any party may destabilise the system: excessively high government management costs may lead to lax or inadequate regulation; recycling enterprises’ core motivation for formal recycling is profit, and they will only persist with formal recycling if its net returns exceed those of informal recycling; EV owners will likewise make strategic choices based on their own returns.

5. Discussion

In order to further verify the validity and applicability of the model, this section selects the actual research situation of EV EOL power battery recycling in Shenzhen, China, as well as the public government data for numerical simulation.

5.1. Data Sources and Model Parameter Settings

This section calculates and assigns model parameters based on the actual research situation of EV EOL power battery recycling in Shenzhen, China, central and local government data, and relevant departmental statistics.

According to CATARC statistics, it is estimated that the total amount of EOL EV batteries in China will reach 137.4 GWh by 2025, approximately 820,000 tonnes. The EOL power batteries in 2025 primarily originate from EVs put into use in 2017. It is estimated that the quantity of it in Shenzhen in 2025 will be approximately 7431 tonnes. According to market research on the recycling industry in Shenzhen, the recycling profit for lithium iron phosphate batteries is CNY 11,400 per tonne, with formal recycling enterprises potentially earning approximately CNY 84.7134 million, ${\mathrm{R}}_3$ = 85. The recycling price for lithium iron phosphate batteries recovered through non-formal channels is approximately CNY 12,100 per tonne, earning approximately CNY 89.9151 million, ${\mathrm{R}}_4$ = 90. The Shenzhen Municipal Government’s annual total expenditure on environmental protection supervision averages approximately CNY 40.25 million,${\mathrm{C}}_{\mathrm{g}}$ = 40; investment in environmental pollution publicity is approximately CNY 1 million, D = 1. Setting model parameters are shown in

Table 5. Parameter settings.

Conditions	${\mathbf{C}}_{\mathbf{g}}$	D	T	${\mathit{T}}_1$	${\mathit{M}}_1$	${\mathbf{C}}_1$	${\mathbf{C}}_2$	${\mathbf{R}}_1$	${\mathbf{R}}_2$	${\mathbf{R}}_3$	${\mathbf{R}}_4$
Scenario 1	40	1	5	10	10	80	70	30	50	85	90
Scenario 2	40	1	5	5	50	80	70	30	50	85	90
Scenario 3	40	1	5	10	10	45	35	30	50	85	90
Scenario 4	40	1	30	10	10	60	50	49	50	85	90

.

Parameters Cg, D, ${\mathrm{R}}_3$, ${\mathrm{R}}_4$ are set based on Shenzhen survey data. The remaining parameters are subject to reasonable assumptions, provided they satisfy the conditions for establishing stable points, in order to explore the evolutionary pathways towards achieving a stable state.

5.2. Evolutionary Path Analysis of Stabilisation Strategies

In the system exists three stable strategy combinations of (0, 0, 0), (0, 1, 0), (1, 0, 0), and (0, 1, 1) and the evolution paths of the strategy choices of the government, recycling enterprises, and EV owners are analysed to explore the influence of different initial strategies on the system evolution results.

Combined with Table 5, the (0, 0, 0) evolution path is shown 50 times in Figure 3a.

Under the conditions of Scenario 1, regardless of the initial strategy combination, the system will eventually stabilise rapidly at the state of (0, 0, 0). The evolution strategy is (non-regulation, adopt an informal recycling strategy, choose informal recycling channels). As can be clearly observed from the evolutionary path, recycling enterprises show a tendency to evolve toward formal recycling, yet ultimately stabilise at informal recycling.

Combined with Table 5, the (1, 0, 0) evolution path is shown 50 times in Figure 3b.

Under Scenario 2, regardless of the initial policy combination, the system rapidly stabilises at the state (1, 0, 0).The evolution strategy is (regulation, adopt an informal recycling strategy, choose informal recycling channels). The evolutionary trajectory clearly demonstrates that recycling enterprises exhibit a tendency towards formal recycling, yet ultimately stabilise in informal recycling. The government exhibits a tendency to evolve towards non-regulation, yet ultimately stabilises in a regulatory participation state. Regardless of the initial strategy combination, vehicle owners’ strategy choices rapidly evolve towards participating in informal recycling.

Combined with Table 5, the (0, 1, 0) evolution path is shown 50 times in Figure 3c.

Under the conditions of Scenario 3, regardless of the initial policy combination, the system ultimately stabilises rapidly at the state (0, 1, 0). The evolution strategy is (non-regulation, adopt a formal recycling strategy, choose informal recycling channels). The evolutionary trajectory clearly demonstrates that recycling enterprises rapidly evolve towards formal recycling practices.

Combined with Table 5, the (0, 1, 1) evolution path is shown 50 times in Figure 3d. In Scenario 4, regardless of the initial strategy combination, the system rapidly stabilises at the state (0, 1, 1), The evolution strategy is (non-regulation, adopt a formal recycling strategy, choose formal recycling channels). The evolutionary pathway clearly demonstrates that recycling enterprises rapidly evolve towards formal recycling practices.

5.3. Parameter Sensitivity Analysis

This section differentially assigns values to key parameters with a steady state of (1, 1, 1) and analyses their impact on the system evolution results. The initial strategies are set as $\mathrm{x}=0.5$, $\mathrm{y}=0.5$, $\mathrm{z}=0.5$, $\mathrm{x},\mathrm{y},\mathrm{z}\in [0,1]$, the initial time is set to 0, the termination time is set to 50, and the step size is 0.005 (Zhang et al., 2022 [47]).

5.3.1. Influence of Government-Related Parameters on the Subject Strategy Evolution

The key parameters for the government are regulatory costs and environmental publicity campaign costs. The initial values of the above parameters are adjusted individually for simulation when the rest of the parameters remain unchanged.

Figure 4 demonstrates that variations in government regulatory costs do not influence the strategies of the three stakeholders. Both recycling enterprises and EV owners ultimately gravitate towards participating in the formal recycling of EOL batteries. However, as government regulatory costs continue to rise, the rate at which the government adopts a non-regulatory strategy progressively slows.

Figure 5 demonstrates that variations in government environmental publicity costs do not influence the strategies of the three participating entities. Both recycling enterprises and vehicle owners ultimately gravitate towards formal recycling channels for EOL batteries. However, as government regulatory costs progressively increase, the rate at which the government adopts non-regulatory strategies gradually diminishes.

5.3.2. Influence of Recycling Enterprise-Related Parameters on the Evolution of Subject Strategies

In the EOL power battery recycling system, the factors affecting the strategy choices of recycling enterprises include the amount of government penalties and incentives.

Observing Figure 6 reveals that fluctuations in the government’s penalty levels for recycling enterprises do not influence the ultimate recycling strategy choices of participating entities. However, as the government increases penalty levels for recycling enterprises, the pace at which these enterprises evolve towards formal recycling strategies accelerates accordingly. This demonstrates that the government’s regulatory measures imposing penalties on informal recycling enterprises are reasonable, aiming to steer recycling enterprises towards formalisation.

As shown in Figure 7, According to policy documents issued by the Shenzhen Industry Battery Association, the maximum funding available is CNY 10 million. Variations in government incentive levels influence recycling enterprises’ strategies, specifically manifested in a significant increase in the probability of recycling enterprises adopting formal recycling strategies as government incentives rise. When $T_1=0$, i.e., when the government provides no subsidy incentives to recycling enterprises, the probability of recycling enterprises choosing formal recycling is 0. This reflects that, in the absence of external incentives, recycling enterprises tend to favour informal recycling strategies that are lower in cost and higher in profit. As government incentives for recycling enterprises gradually increase, these enterprises proactively adopt formal recycling strategies, and vehicle owners also opt for formal recycling approaches. This phenomenon demonstrates that government incentive policies play a crucial role in optimising market competition and ensuring the formal recycling of EOL batteries. For recycling enterprises driven by economic returns, government incentive policies constitute a significant factor influencing their choice of recycling strategy.

5.3.3. Influence of EV Owner-Related Parameters on the Evolution of Subject Strategies

The evolutionary path depicted in Figure 8 illustrates how variations in government incentive levels for EV owners influence strategic choices during the recycling process of EOL batteries from EVs. It is evident that government incentive schemes exert a significant impact on the strategic decisions of both EV owners and recycling enterprises throughout this recycling process. When $T=5$, both EV owners and recycling enterprises tend to adopt strategies of non-participation in formal recycling and selection of informal recycling, respectively. This phenomenon can be attributed to the psychological drive of both parties towards maximising economic benefits. For EV owners, if the government incentives prove insufficient to attract them into formal recycling activities for EOL batteries, they will opt for the more lucrative informal recycling strategy. When $T=20$, recycling enterprises naturally favour informal strategies when their profits exceed those from formal recycling. Around the threshold value, recycling enterprises’ willingness to participate in formal recycling begins to shift positively. When $T=25$, both recycling enterprises and EV owners change their strategies, opting to engage in formal recycling.

When the government incentive exceeds this threshold, the willingness of both recycling enterprises and EV owners to participate in formal recycling gradually increases with the scale of the incentive. This phenomenon highlights the significance of government incentive schemes. When the incentive is sufficiently high, both EV owners and recycling enterprises share a similar mindset: the greater the economic benefit derived from formal recycling, the more willing they are to proactively engage in the formal recycling of EOL batteries.

6. Conclusions

6.1. Conclusions

In this section, the results of the previous study are discussed and analysed.

(1)

The evolutionary game analysis in this study indicates that the ESS for the EOL power battery recycling system is characterised by government non-regulation, recycling enterprises adopting formal recycling practices, and EV owners participating in formal recycling. This strategy combination is regarded as an ideal state in practice, as it minimises government intervention costs while achieving synergistic recycling efficiency and environmental protection through endogenous market dynamics.

(2)

The strategic choices of recycling enterprises and EV owners depend on their benefits and costs in the recycling process. Government subsidies to EV owners positively encourage both EV owners and recycling enterprises to adopt formal recycling practices, with recycling companies shifting to formal recycling strategies before EV owners.

(3)

Strengthening penalties against recycling enterprises will accelerate their transition towards formal recycling strategies, whilst increasing incentive levels can significantly enhance the steady-state probability of firms opting for formal recycling.

6.2. Policy Recommendations

In response to the above conclusions, recommendations are made from the perspectives of the government, recycling enterprises, and EV owners.

(1)

The government should establish a flexible reward and punishment mechanism, increase the tax burden for enterprises that choose informal recycling and force them to pay environmental protection fees; give financial subsidies to recycling enterprises that choose formal recycling on their own initiative, improve supporting facilities and policies, and enhance their market competitiveness. Government subsidies may be structured as a tiered subsidy policy based on the formal recycling scale of recycling enterprises, thereby encouraging their active participation in formal recycling. A cap should be set on government regulatory costs to prevent unnecessary financial wastage.

(2)

Recycling enterprises should strengthen their own management, actively introduce advanced recycling and processing technology and equipment, and increase the investment in technology research and development to enhance the level of formalised recycling of EOL power batteries for EVs. Establish a whistleblower reward scheme for reporting informal recycling operations. Where reports of informal recycling enterprises are verified, a portion of the fines collected may be allocated to reward the informants.

(3)

EV owners should strengthen their own environmental and legal awareness, recognise the importance of formal recycling, avoid choosing informal recycling channels due to short-term interests, and also increase the supervision and reporting of the recycling market. Establish a traceable digital identity system for power batteries and implement a deposit incentive scheme when EV owners purchase new EVs.

There are some limitations in this study, and the factors to be considered and paid attention to vary in different development stages of the EOL power battery recycling market, which should seek to include more relevant factors and study the impact of these factors on the EOL power battery recycling system.