Evolutionary Game Analysis of Power Battery Recycling in the Context of a Carbon Cap and Patent Licensing

Abstract

A wave of large-scale retirement of power batteries is gradually approaching, and the patent licensing conditions for remanufacturing retired power batteries present opportunities for third-party recycling manufacturers to emerge. Considering both the carbon-emission benefits of power battery recycling and the intellectual property disputes, this paper establishes an evolutionary game model with third-party recyclers and battery manufacturers as players. It examines the costs and utilities of stakeholders involved in the reverse logistics process of power battery recycling under carbon quotas, accounting for patent licensing, and analyzes key parameters and participant strategy choices. The research indicates: (1) when the volume of waste power batteries is significant, third-party recycling manufacturers tend to choose direct battery disassembly; (2) at higher carbon prices within the carbon market, third-party recycling manufacturers are more likely to adopt remanufacturing strategies; (3) lower patent licensing fees combined with higher patent maintenance costs help battery manufacturers secure greater profits and encourage third-party recycling manufacturers to engage in battery remanufacturing activities.

1. Introduction

In recent years, driven by the global carbon-neutrality transformation and energy security, the global new energy vehicle and power battery industry has achieved strong growth, leading to a significant expansion of the global consumer market and promoting the coordinated development of the industrial chain [1]. Relevant institutions predict that global electric vehicle ownership will reach 116 million in 2026. China continues to lead with 61% of the market share, and mature markets such as Europe and North America, as well as emerging markets such as Southeast Asia and India, are showing a differentiated development trend. At the same time, the continuous accumulation of global new energy vehicle ownership and the exponential growth of power battery installed capacity have made the management of decommissioned power batteries an urgent global challenge. Global decommissioning of power batteries reached 1.2 million tons in 2024 and is expected to exceed 4 million tons in 2030, with large-scale decommissioning becoming increasingly prominent [2].

According to the industry report, as the scale of decommissioning continues to rise, the global power battery recycling market is expected to reach $50 billion in 2030 [3]. This huge wave of decommissioning will not only promote standardization in the global battery recycling industry but will also promote high-value utilization and regional coordination. The direction of reconstruction will also bring unprecedented opportunities for the vigorous development of the global power battery echelon, including utilization and remanufacturing industries.

At present, the waste batteries in the retired power battery classification system have been tested and evaluated, and high-energy-density batteries are preferentially used in echelon utilization. After processing, the batteries are mostly used for home energy storage, backup power supply for communication base stations, and other scenarios with low battery life and power requirements [4]. Low-energy-density batteries are recycled by extracting their metal components using technical means. In recent years, the trend of power battery remanufacturing has emerged. Remanufacturing is also an effective way to utilize low-energy-density batteries. Due to production scale limitations, battery manufacturing enterprises often allow third-party enterprises to remanufacture batteries through patent licensing. Remanufactured batteries are often used in scenarios with high-performance requirements, such as low-speed electric vehicles and special vehicles [5]. The current power battery recycling process is shown in Figure 1.

On the one hand, the implementation of patent licensing agreements guarantees the interests of the original manufacturing enterprises; on the other hand, it also helps to improve the quality of remanufactured products. Many scholars have discussed and studied the product remanufacturing mode from the perspective of patent licensing. For example, Zhou et al. (2020) [6] constructed a Stackelberg game model between the original equipment manufacturer and two competing third-party manufacturers, and found that the original manufacturer’s high patent licensing fees are not always conducive to profit acquisition; Xia et al. (2020) [7] show that the original manufacturer can realize the transfer of remanufacturing revenue through outsourcing or patent licensing, and achieve an increase in overall revenue; Cao et al. (2025) [8] constructed a variety of patent licensing models to study the impact of different models on the optimal decision-making model of closed-loop supply chain enterprises; Zhao et al. (2020) [9] pointed out that the green innovation capability of the original equipment manufacturing enterprise can enable third-party manufacturers to obtain more profits, but when it is difficult for the refurbished product to be recognized by consumers, the original manufacturer will adopt an unauthorized patent strategy; Zhao et al. (2019) [10] found that the collection of patent fees for power battery manufacturers can effectively stimulate the enthusiasm of third-party battery-recycling manufacturers, thereby improving the service level of battery retailers; Cao et al. (2020) [11] found that consumers’ attention to remanufactured products was positively correlated with retail price; and Xia et al. (2026) [12] integrated and applied behavioral economics, game theory, and incomplete information theory to model irrational decisions in local energy markets to capture the diversity of trading behaviors and provide evidence-based insights for policy design to improve the efficiency and sustainability of local energy markets. Liu et al. (2018) [13] pointed out that remanufacturing patent licensing can help high-end manufacturers compete effectively through product differentiation and improve their profits. Therefore, it is necessary to consider further the role of third-party manufacturers’ decisions in the remanufacturing process.

More and more scholars around the world have begun integrating environmental factors into supply chain management and have studied in depth the problems that supply chains may face under carbon policy constraints. For example, Chai et al. (2018) [14] found that carbon cap-and-trade policies have positive benefits for both ordinary market remanufacturing and green market remanufacturing. Wang et al. (2025) [15] found that increases in patent licensing fees and carbon trading prices would hinder the echelon-level utilization of power batteries by examining the role of government subsidies in the context of dual-carbon goals. Wang et al. (2022) [16] studied the impact of a carbon cap-and-trade policy on manufacturers’ investment in emissions reductions, supply chain performance, and social welfare under an uncertain carbon price. The higher the surface carbon emission level, the smaller the impact of carbon price fluctuation on the supply chain. Mao et al. (2023) [17] analyzed the impact of a carbon cap-and-trade policy on product quality, sales volume, and price in detail. The study found that the cap-and-trade mechanism not only improved the quality of remanufacturing and expanded production but also helped reduce total carbon emissions. Cai et al. (2023) [18] analyzed low-carbon supply chain pricing and carbon emission-reduction decisions under cap-and-trade regulation. The results show that only under a certain cap and carbon-trading price threshold can cap-and-trade regulation improve the environment and supply chain performance. Yang et al. (2022) [19] explored the value of remanufacturing and the choice of recycling mode under the carbon cap-and-trade policy, and found that remanufacturing can effectively improve the level of carbon emission reduction and increase the profits of manufacturers and retailers. A low carbon price and high carbon intensity will lead to higher carbon emissions, according to Wang et al. (2023) [20], who constructed an evolutionary game model of battery manufacturers’ and automobile manufacturers’ decision-making on power battery recycling. The study found that the government’s incentive measures can significantly shorten the recycling time for waste batteries; therefore, it is necessary to analyze the evolution of carbon market prices, patent licensing, and decision-making in power battery recycling supply chain enterprises.

Considering the dual effects of carbon trading and patent licensing, and focusing on the production decision-making problems of upstream and downstream enterprises in the power battery recycling supply chain, a two-party evolutionary game model between the third-party recycling manufacturer and the battery production enterprise is constructed. The game’s evolutionary strategy—subject to the enterprise’s pursuit of the highest profit and the influence of the law promoting the development of the power battery remanufacturing industry—is analyzed. This study helps form a patent pool in the battery recycling industry, stimulates innovation in remanufacturing technology, aligns with the trend toward green manufacturing, and reduces environmental load.

2. Model Hypothesis and Construction

2.1. Model Hypothesis and Interpretation

The products produced by the original battery manufacturer are subject to patent protection. Given that recycling costs are high and the benefits of battery remanufacturing are usually lower than those of manufacturing new products, the original manufacturing enterprise does not typically directly engage in recycling and remanufacturing waste batteries. If a third-party recycling manufacturer plans to remanufacture waste batteries, it must obtain the original enterprise’s patent authorization. Third-party recycling manufacturers and battery manufacturers are the main players in the game. Third-party recycling manufacturers can choose the strategies of ‘remanufacturing’ and ‘disassembly’, and battery manufacturers can adopt the strategies of ‘authorized’ and ‘unauthorized’. To better describe the two-party subject game model, analyze the stability of each party’s strategy and equilibrium point, and the influence relationship of each factor, this paper makes the following assumptions and agreements:

Hypothesis 1.

Battery manufacturers and retailers in the closed-loop supply chain do not participate in the recycling of power batteries. The third-party recycling manufacturer is the only battery recycling party in the supply chain, and the market has a single battery manufacturing enterprise and a single third-party recycling manufacturer. In this model, third-party recycling manufacturers and battery production enterprises are the main players, and the game is bounded by rationality and incomplete information.

Hypothesis 2.

The third-party recycling manufacturer’s strategy set is {remanufacturing, disassembly}; the probability of taking ‘remanufacturing’ is $x(0\le x\le 1)$ and the probability of ‘disassembly’ is $1-x$. The battery recycling industry abides by the industrial logic of ‘resource scale determines operation strategy’. This paper assumes that the third-party recycling manufacturer is the only battery recycling party. The amount of waste battery recycling is directly related to the enterprise’s cost-sharing and income realization. It is also the material basis for the third-party recycling manufacturer to implement the battery recycling strategy. The amount of waste battery recycling is set to $M$. Battery remanufacturing revenue is the economic incentive for the third-party recycling manufacturer’s remanufacturing strategy. Remanufacturing, as a high-value-added battery recycling method, is more attractive to enterprises than disassembly battery revenue. The unit battery remanufacturing revenue is set to $E_1$, which means that the third-party recycling manufacturer completes remanufacturing and sales after deducting the processing and recycling costs. Battery patent licensing is the core institutional cost of third-party recycling manufacturers to carry out compliance remanufacturing, and it is also a key parameter affecting the interests of the two main players in the game. Battery manufacturers master the core patents of remanufacturing, and third parties can only obtain authorization by paying for no related technology. The patent licensing fee is not only the cost threshold for third-party compliance remanufacturing but also a source of income for battery manufacturers. The patent licensing fee in this paper adopts a charging model based on the volume of waste battery remanufacturing. The patent licensing fee per unit battery remanufacturing renewal payment is set to $h$. Unauthorized remanufacturing fines and risk factors serve as the constraint mechanism for third-party recycling manufacturers’ illegal remanufacturing. If the remanufacturing of waste batteries is carried out without patent authorization, the illegal behavior will be found by the battery production enterprise and will be fined by $U$. In reality, the probability of patent infringement being discovered is affected by multiple factors, such as patent holder market inspection and industry qualification review. In this paper, it is simplified as a fixed risk, and the risk factor is set to $\eta$. At the same time, in order to ensure that compliance remanufacturing is more economically reasonable than illegal remanufacturing, follow condition $\eta U\ge Mh$. Otherwise, third-party enterprises will ignore the risk of patent infringement and conduct compulsory remanufacturing.

Hypothesis 3.

When the third-party recycling manufacturer adopts the ‘disassembly’ strategy, this paper is an economic indicator that compares with the remanufacturing strategy. When the battery manufacturer’s patent is not authorized, the battery raw materials obtained by the third-party recycling manufacturer for disassembling the waste battery will be sold to the battery production enterprise as the production material of the new power battery. The profit $E_2$ $(E_1>E_2)$, obtained by selling the disassembly material, is generally less than the high-value-added remanufacturing profit. In the case of patent licensing, in order to obtain the initial processing of battery raw materials, reduce the market competition of remanufactured products, or obtain patent licensing income, battery manufacturers will adjust the recycling price of disassembly materials. At this time, the third party’s profit from selling battery disassembly materials is set to $E_3$ $(E_1>E_3)$, and the disassembly income is generally less than the high-value-added remanufacturing income. Enterprise carbon emissions are constrained by the government’s carbon limit, and these emissions can be converted into economic costs or benefits through the carbon trading market. This paper takes the free trading of carbon quotas in the carbon trading market as the research background. The carbon quota of enterprises is allocated by the government free of charge. The upper limit of the carbon limit constraint of the third-party recycling manufacturer is $T_0$. At the same time, the carbon emissions of unit battery remanufacturing are set to $T_1$, and the carbon emissions of unit battery disassembly are set to $T_2$ $(T_2>T_1)$, which is in line with the industrial reality of remanufacturing as a low-carbon process. When the enterprise’s carbon emissions exceed the permitted level, it is necessary to purchase the difference in the carbon market. If the enterprise has a carbon surplus in a given year, it can sell the surplus in the carbon market to earn income; the carbon price fluctuates with market supply and demand. The marginal emission-reduction cost, energy price, and enterprise economic cycle all affect the carbon price. The carbon price in the carbon market is set to $P$.

Hypothesis 4.

The battery production enterprise strategy set is {authorized, unauthorized}; the probability of taking ‘authorized’ is $y(0\le y\le 1)$ and the probability of ‘unauthorized’ is $1-y$. When the battery production enterprise chooses not to authorize the patent, it must pay the patent maintenance costs for technical protection. As a patent holder, if the battery production enterprise chooses not to authorize, it will incur high costs for technical confidentiality, infringement monitoring, and legal rights protection to maintain the patent monopoly. The maintenance cost paid for protecting the core remanufacturing technology and preventing third-party infringement is set to $I$. It is easy for consumers to attribute the inferior products that are not authorized to be remanufactured to the original battery production enterprise, which leads to the damage of the brand reputation of the enterprise, a decline in the market share, the brand loss and potential market loss borne by the enterprise, and the brand reputation loss of the original battery production enterprise is set to $L$. The disassembly behavior of the third-party recycling manufacturer can provide a large number of primary processing raw materials for battery production enterprises. Compared with procuring primary mineral resources, recycled raw materials can greatly reduce smelting and processing costs. The battery production enterprise saves the processing cost set to $K$.

Hypothesis 5.

As the supplier of the system, the government’s carbon quota, patent protection, and other regulatory functions have been reflected through the external constraint framework, which is not the main body of the game in this paper. It is assumed that retailers and scrap automobile disassembly enterprises serve only as recycling channel connectors, and have no core strategic options such as remanufacturing and patent licensing. The influence of consumers’ behavior has been internalized into practical considerations, such as corporate remanufacturing revenue and brand reputation loss, and is not used as an influencing factor in system evolution. The parameter descriptions are shown in

Table 1. Parameter descriptions.

Parameter	Implication	Range
$x$	Third-party recycling manufacturer’s remanufacturing probability	$0\le x\le 1$
$y$	Patent authorization probability of the battery production enterprises	$0\le y\le 1$
$M$	Third-party recycling manufacturer’s waste battery recycling volume	$0<M$
$E_1$	Third-party recycling manufacturer’s unit battery remanufacturing revenue	$0<E_1$
$E_2$	Third-party recycling manufacturer’s disassembly revenue per unit battery without patent authorization	$0<E_2<E_1$
$E_3$	Third-party recycling manufacturer’s disassembly income under the unit battery patent authorization	$0<E_3<E_1$
$h$	Unit battery remanufacturing patent licensing fees	$0<h$
$U$	Amount of penalty for third-party recycling manufacturer’s unauthorized remanufacturing	$0<U$
$\eta$	Risk factors of remanufacturing without patent authorization	$0\le \eta \le 1$
$T_0$	Third-party recycling manufacturer’s carbon limit	$0<T_0$
$T_1$	Carbon emissions per unit battery remanufactured	$0<T_1$
$T_2$	Carbon emissions per unit battery disassembled	$0<T_1<T_2$
$P$	Carbon price of carbon trading market	$0<P$
$I$	Battery production enterprise patent maintenance cost	$0<I$
$L$	Brand reputation loss caused by inferior unit products	$0<L$
$K$	Battery production enterprises save raw material processing costs	$0<K$

.

2.2. Model Construction

In this paper, the evolutionary game structure of battery manufacturing enterprises and third-party recycling manufacturers is constructed. As the intellectual property center and value implementer, the two roles are irreplaceable, and their strategic choices are the core driving force for system evolution. The asymmetric characteristics of the two in the dominant position restore the essence of the interest game of the power battery recycling system. Based on the above assumptions, the cooperative income matrix for the third-party recycling manufacturer and the battery production enterprise is presented in

Table 2. Cooperation income matrix.

Game Players	Third-Party Recycling Manufacturers
	Remanufacturing	Disassembly
authorized	$M{E}_1+P\left(T_0-M{T}_1\right)-Mh$	$M{E}_3+P\left(T_0-M{T}_2\right)$
	$Mh$	$KM$
unauthorized	$M{E}_1+P\left(T_0-M{T}_1\right)-\eta U$	$M{E}_2+P\left(T_0-M{T}_2\right)$
	$\eta U-ML-I$	$KM-I$

.

2.3. Model Solving

The expected revenue and average expected revenue of the third-party recycling manufacturer adopting the ‘remanufacturing’ and ‘disassembly’ strategies are set as $E_{11}$, $E_{12}$, and ${\overline{E}}_1$:

$$E_{11}=y\left(M{E}_1+P{T}_0-PM{T}_1-Mh\right)+\left(1-y\right)\left(M{E}_1+P{T}_0-PM{T}_1-\eta U\right)$$

(1)

$$E_{12}=y(M{E}_3+P{T}_0-PM{T}_2)+\left(1-y\right)(M{E}_2+P{T}_0-PM{T}_2)$$

(2)

$${\overline{E}}_1=x{E}_{11}+\left(1-x\right)E_{12}$$

(3)

By substituting Formulas (1) and (2) into Formula (3), the replication dynamic equation of the third-party recycling manufacturer can be obtained as follows:

$$F\left(x\right)=x\left(x-1\right)\left(\eta U-E_{1}M+E_{2}M+MP{T}_1-MP{T}_2-E_{2}My+E_{3}My-\eta Uy+Mhy\right)$$

The expected revenue and average expected revenue of battery manufacturers adopting ‘authorized’ and ‘unauthorized’ strategies are set as $E_{21}$, $E_{22}$, and ${\overline{E}}_2$:

$$E_{21}=xMh+(1-x)KM$$

(4)

$$E_{22}=x\left(\eta U-ML-I\right)+\left(1-x\right)\left(KM-I\right)$$

(5)

$${\overline{E}}_2=y{E}_{21}+\left(1-y\right)E_{22}$$

(6)

Substituting Formulas (4) and (5) into (6), the replication dynamic equation of the battery production enterprise can be obtained as follows:

$$F\left(y\right)=y\left(y-1\right)\left(\eta Ux-I-LMx-Mhx\right)$$

3. Equilibrium Point and Evolution Path Analysis

3.1. Equilibrium Point Analysis

In order to study the evolutionary stable state of the game system, let the strategy set represented by $\left(1,1\right)$ be {remanufacturing, authorization}. According to the dynamic formula of evolutionary game replication, the evolutionary game system of third-party recycling manufacturers and battery manufacturers is:

$$\left\{\begin{array}{l}F\left(x\right)=x\left(x-1\right)\left(\eta U-E_{1}M+E_{2}M+MP{T}_1-MP{T}_2-E_{2}My+E_{3}My-\eta Uy+Mhy\right)\\ F\left(y\right)=y\left(y-1\right)\left(\eta Ux-I-LMx-Mhx\right)\end{array}\right.$$

Let $F\left(x\right)=0,F\left(y\right)=0$ on the plane $M=\left\{\left(x,y\right)\left|0\le x\le 1,0\le y\le 1\right.\right\}$, then the evolutionary game system has five strategic equilibrium points, which are $(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$, $\left(x_0,y_0\right)$, $0\le x_0,y_0\le 1$, and $x_0={\displaystyle \frac{I}{\eta U-LM-Mh}},y_0={\displaystyle \frac{\eta U+\left(E_2-E_1\right)M-(T_2-T_1)MP}{\left(E_2-E_3\right)M+\eta U-Mh}}$.

According to the local stability analysis method proposed by Friedman to test the properties of the equilibrium point, the stability of the equilibrium point is obtained by the local stability of the Jacobian matrix of the system [21]. The specific method of judgment is: if the determinant $Det\left(J\right)>0$ of the Jacobian matrix of the equilibrium point and the trace $Tr\left(J\right)<0$, the property that the corresponding equilibrium point can be judged to be asymptotically stable is called $ESS$; if $Det\left(J\right)>0$ and $Tr\left(J\right)>0$, the corresponding equilibrium point can be judged to be unstable; if $Det\left(J\right)<0$ and $Tr\left(J\right)=0$ or uncertain, it can be judged that the corresponding equilibrium point is a saddle point [22]. Therefore, by copying the dynamic equations, the Jacobian matrix of the power battery recovery evolution system is:

$$J\left(x,y\right)=\left[\begin{array}{cc}(2x-1)\left(\begin{array}{c}\eta U-E_{1}M+E_{2}M+MP{T}_1-MP{T}_2\\ -E_{2}My+E_{3}My-\eta Uy+Mhy\end{array}\right)& x(x-1)(Mh+E_{3}M-\eta U-E_{2}M)\\ y(y-1)(\eta U-Mh-LM)& (2y-1)(\eta Ux-I-LMx-Mhx)\end{array}\right]$$

Furthermore, the expressions of the Jacobian matrices $Det\left(J\right)$ and $Tr\left(J\right)$ of each equilibrium point are shown in

Table 3. Determinant and trace of the Jacobian matrix.

Equilibrium Point	Det(J)	Tr(J)
$E_1(0,0)$	$-\left[\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)\right]\cdot I$	$-\left[\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)\right]+I$
$E_2(0,1)$	$\left[Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)\right]\cdot I$	$-\left[Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)\right]-I$
$E_3(1,0)$	$\begin{array}{c}\left[\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)\right]\cdot \\ \left[-\left(\eta U-I-LM-Mh\right)\right]\end{array}$	$\begin{array}{c}\left[\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)\right]-\\ \left(\eta U-I-LM-Mh\right)\end{array}$
$E_4(1,1)$	$\begin{array}{c}\left[Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)\right]\cdot \\ \left(\eta U-I-LM-Mh\right)\end{array}$	$\begin{array}{c}\left[Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)\right]+\\ \left(\eta U-I-LM-Mh\right)\end{array}$
$E_5(x_0,y_0)$	$\ast \ast$	$0$

Note: $\left(x_0,y_0\right)$ is substituted into the matrix solution, $Tr\left(J\right)$ must be 0, the $Det\left(J\right)$ result is meaningless, and $\ast \ast$ indicates that the $Det\left(J\right)$ result is not obtained.

. Taking $\left(x_0,y_0\right)$ into the post-matrix $Tr\left(J\right)=0$, the mixed strategy equilibrium in the asymmetric evolutionary game must not be $ESS$.

3.2. Path Evolution Analysis

According to the initial state and evolutionary phase diagram of the evolutionary game system in

Table 4. Local stability of the initial state ①~⑧ equilibrium point of the evolutionary game system.

Equilibrium point	Initial state ①			Initial state ②
	$\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)>0$			$\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)>0$
	$Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)>0$			$Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)>0$
	$\eta U-I-LM-Mh>0$			$\eta U-I-LM-Mh<0$
	$Det\left(J\right)$	$Tr\left(J\right)$	Stability	$Det\left(J\right)$	$Tr\left(J\right)$	Stability
$E_1(0,0)$	$-$	$\pm$	Saddle point	$-$	$\pm$	Saddle point
$E_2(0,1)$	$+$	$-$	$ESS$	$+$	$-$	$ESS$
$E_3(1,0)$	$-$	$\pm$	Saddle point	$+$	$+$	Unstable
$E_4(1,1)$	$+$	$+$	Unstable	$-$	$\pm$	Saddle point
$E_5(x_0,y_0)$	Non-existence			Non-existence
Evolutionary phase diagram
Equilibrium point	Initial state ③			Initial state ④
	$\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)>0$			$\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)>0$
	$Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)<0$			$Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)<0$
	$\eta U-I-LM-Mh>0$			$\eta U-I-LM-Mh<0$
	$Det\left(J\right)$	$Tr\left(J\right)$	Stability	$Det\left(J\right)$	$Tr\left(J\right)$	Stability
$E_1(0,0)$	$-$	$\pm$	Saddle point	$-$	$\pm$	Saddle point
$E_2(0,1)$	$-$	$\pm$	Saddle point	$-$	$\pm$	Saddle point
$E_3(1,0)$	$-$	$\pm$	Saddle point	$+$	$+$	Unstable
$E_4(1,1)$	$-$	$\pm$	Saddle point	$+$	$-$	$ESS$
$E_5(x_0,y_0)$	Non-existence			Non-existence
Evolutionary phase diagram
Equilibrium point	Initial state ⑤			Initial state ⑥
	$\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)<0$			$\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)<0$
	$Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)>0$			$Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)>0$
	$\eta U-I-LM-Mh>0$			$\eta U-I-LM-Mh<0$
	$Det\left(J\right)$	$Tr\left(J\right)$	Stability	$Det\left(J\right)$	$Tr\left(J\right)$	Stability
$E_1(0,0)$	$+$	$+$	Unstable	$+$	$+$	Unstable
$E_2(0,1)$	$+$	$-$	$ESS$	$+$	$-$	$ESS$
$E_3(1,0)$	$+$	$-$	$ESS$	$-$	$\pm$	Saddle point
$E_4(1,1)$	$+$	$+$	Unstable	$-$	$\pm$	Saddle point
$E_5(x_0,y_0)$	Center point			Non-existence
Evolutionary phase diagram
Equilibrium point	Initial state ⑦			Initial state ⑧
	$\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)<0$			$\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)<0$
	$Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)<0$			$Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)<0$
	$\eta U-I-LM-Mh>0$			$\eta U-I-LM-Mh<0$
	$Det\left(J\right)$	$Tr\left(J\right)$	Stability	$Det\left(J\right)$	$Tr\left(J\right)$	Stability
$E_1(0,0)$	$+$	$+$	Unstable	$+$	$+$	Unstable
$E_2(0,1)$	$-$	$\pm$	Saddle point	$-$	$\pm$	Saddle point
$E_3(1,0)$	$+$	$-$	$ESS$	$-$	$\pm$	Saddle point
$E_4(1,1)$	$-$	$\pm$	Saddle point	$+$	$-$	$ESS$
$E_5(x_0,y_0)$	Non-existence			Non-existence
Evolutionary phase diagram

, the evolutionary trends of the third-party recycling manufacturer, the battery production enterprise, and the strategy combination of the two sides of the game are discussed, respectively.

(1)

Analysis of the Evolution Trend of the Third-Party Recycling Manufacturer’s Strategy Space: For third-party recycling manufacturers, as long as the amount of unlicensed remanufacturing fines or patent licensing fees is greater than the sum of the remanufacturing lead revenue and the carbon emission difference, $\eta U>M\left(E_1-E_2\right)+MP\left(T_2-T_1\right)$ and $Mh>M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)$ (initial state ① ②). Under this condition, the expected cost and risk of the third party’s choice of remanufacturing will exceed the revenue. Even if the resources and carbon benefits of remanufacturing are greater, disassembly remains a rational choice from the perspective of maximizing corporate profits. Therefore, no matter what strategy the battery production enterprise adopts, the third party will converge on the disassembly strategy.

Conversely, when $\eta U<M\left(E_1-E_2\right)+MP\left(T_2-T_1\right)$ and $Mh<M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)$ (initial state ⑦ ⑧), no matter how the initial state and the strategy of the battery production enterprise evolve, the result is that the third-party recycling manufacturer chooses the cooperative strategy of battery remanufacturing. In this state, the net income from remanufacturing is significantly higher than that from disassembly, and the low-carbon income advantage can be leveraged. The third party will firmly adopt the remanufacturing strategy and serve as the core driving force to promote the evolution of the system toward a non-disassembly equilibrium.

(2)

Analysis of the Evolution Trend of the Battery Production Enterprise Strategy Space: For battery manufacturers, only the amount of fines for unlicensed battery remanufacturing is greater than the sum of patent licensing fees, regulatory costs, and brand reputation losses, that is, $\eta U>Mh+I+LM$ (initial state ⑤ ⑦). The evolution results converge on the cooperation strategy in which battery manufacturers choose not to license patents. In this case, the battery production enterprise chooses to use unauthorized patent monopoly income to cover its maintenance costs and potential brand loss, and unauthorized becomes the rational choice for the enterprise. If the patent licensing revenue can cover the above costs and losses, the licensing strategy can not only enable enterprises to obtain stable patent revenue but also avoid the brand reputation risk caused by unauthorized remanufacturing and promote the coordinated development of the industrial chain. Enterprises will tend to choose the licensing strategy.

(3)

The disassembly only realizes the primary recycling of battery raw materials, and the resource utilization efficiency is much lower than the high-value-added conversion of remanufacturing. The overall profit is at the lowest level of the industrial chain, and the carbon emission intensity of the disassembly process is significantly higher than that of remanufacturing, which cannot give full play to the low-carbon advantage of power battery recycling, so E₂(0,1) is the worst equilibrium. Third-party recycling manufacturers choose remanufacturing to realize the high-value-added resource utilization of waste batteries. However, due to the unauthorized option of battery manufacturers, third-party remanufacturing lacks compliance with technical support, which can easily lead to patent infringement, insufficient product quality control, and other problems. Moreover, there is no interest coordination among enterprises, and the profit of the industrial chain is not maximized. Therefore, E₃(1,0) is a sub-optimal equilibrium. The third party mitigates the risk of infringement and improves product quality through compliance remanufacturing, and the battery production enterprise obtains stable income through patent authorization, avoiding the loss of brand reputation. The two achieve win–win benefits and maximize the overall profit of the industrial chain. Therefore, E₄(1,1) is the optimal equilibrium.

(4)

Analysis of the Evolution Trend of the Strategy Combination of the Two Sides of the Game: According to the above analysis, the evolution path of the whole evolution system is from E₂(0,1) to E₃(1,0) and then to E₄(1,1) and the evolution of the three is progressive. It is derived from the continuous cost–benefit trade-off, strategy trial and error, and interactive adjustment of the game subject under the bounded rationality. This gradually transitions from the low-efficiency, low-synergy strategy combination to the high-yield, high-synergy optimal combination convergence, and, due to the industry shift from primary disassembly to compliance remanufacturing, the evolution from disorderly competition to the coordinated development of the reality of the law is highly consistent. Only when the leading revenue of remanufacturing is large enough, and the supervision cost of battery manufacturing enterprises and the loss of brand reputation are large enough, can the ‘lock’ of the optimal equilibrium state be achieved in the game evolution process of the whole system.

4. Sensitivity Analysis of Initial State and Key Factors of Game System

4.1. Initial Simulation of Evolutionary Game System

To more intuitively reflect the evolution mechanism of the main strategy selection between the two parties, the corresponding parameters in the model are assigned across different scenarios, revealing the sensitivity of the initial-state evolution process and its key factors. Referring to the idea of parameter assignment by Wang et al. [15] and Wang et al. [20], and satisfying the interaction between parameters, it is assumed that the probability of third-party recycling manufacturers and battery manufacturers choosing each strategy in the initial state is 0.5, which satisfies the initial state ⑤ in Table 4: $\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)<0$, $Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)>0$, $\eta U-I-LM-Mh>0$. The relevant assignments are shown in

Table 5. Simulation parameter assignment.

Parameter	Assignment	Parameter	Assignment	Parameter	Assignment
$M$	20	$T_0$	350	$U$	1000
$E_1$	27	$T_1$	14	$h$	8
$E_2$	17	$T_2$	18	$I$	10
$E_3$	25	$K$	7.2	$L$	1.4
$P$	1	$\eta$	0.2

.

The simulation results of the strategy evolution of the third-party recycling manufacturer and the battery manufacturing enterprise under this initial condition are shown in Figure 2a. The sensitivity of the key factors is shown in Figure 2b. The strategy selection of the two parties will converge to the two equilibrium points of $(0,1)$ and $(1,0)$, which is used as the basis for comparative analysis of evolution. The patent costs and carbon-emission income of third-party remanufacturing offset each other. The income from patent authorization and the costs of patent maintenance for battery production enterprises are essentially the same. Both sides have no clear tendency to adopt strategies and, in the end, form a double-equilibrium feature of strategy evolution. In Figure 2a, different color lines indicate that the two agents in the game system evolve with different initial strategy probabilities and eventually converge to a stable state. In Figure 2b, the blue line represents the evolution trajectory of the strategy choice of the third-party recycling manufacturer after the parameter changes, and the red line represents the evolution trajectory of the strategy choice of the battery production enterprise after the parameter changes.

4.2. The Influence of Waste Battery Recovery on System Evolution

When the amount of waste battery recovery increases—that is, $M$ increases from 20 to 24—and other parameters remain unchanged, the initial state ① in Table 4 is satisfied: $\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)>0$, $Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)>0$, $\eta U-I-LM-Mh>0$.

In this case, the behavioral strategies of third-party recycling manufacturers and battery manufacturers have changed significantly. By comparing Figure 2a and Figure 3a, it can be intuitively found that the two equilibrium points existing in the original strategy selection disappear, and instead, the strategies of both parties converge to the worst equilibrium point $(0,1)$. In order to more clearly show the changes in the evolution path of the strategy, the evolution path after the recovery amount is increased is drawn in Figure 3b for comparative analysis. The results show that with the increase in the amount of recycling, the rate of convergence of the strategy of the third-party recycling manufacturer and the battery manufacturing enterprise to point $(0,1)$ is obviously accelerated. In particular, the third-party recycling manufacturer reaches a stable equilibrium state around $t=0.1$, and this convergence time node is significantly earlier than the corresponding time of Figure 2b in the initial state. In contrast, when the amount of waste battery recovery is reduced, the strategy selection of both parties will eventually converge to the sub-optimal equilibrium point $(1,0)$ stably. Based on the above analysis, there is a significant negative incentive relationship between the amount of waste battery recycling and the equilibrium state of cooperation between the two sides: reducing the amount of recycling leads the two sides to a sub-optimal cooperative equilibrium. At the same time, by comparing the sensitivity of strategy convergence, it is found that the response sensitivity of third-party recycling manufacturers to the change in waste battery recycling volume is significantly lower than that of battery manufacturing enterprises, which means that in the scenario of a change in recycling volume, the strategy adjustment of battery manufacturing enterprises is more rapid and significant.

The amount of waste battery recycling affects the system evolution by reconstructing the cost-sharing and revenue logic between the two parties in the game, and there are differences in the revenue characteristics of the remanufacturing and disassembly processes. When the recovery amount is large, the fixed-cost allocation pressure in the remanufacturing fine process increases sharply, and the unit marginal cost offsets the income advantage, while the disassembly rough-machining process reduces the unit cost by relying on scale effects and becomes the third-party optimal choice. At the same time, the cost savings from the battery production enterprise’s use of recycled raw materials far exceed the patent authorization income, so the authorization strategy is adopted, and the two sides jointly profit to promote system convergence. When the recycling amount is reduced, the cost-sharing pressure on remanufacturing decreases, the profit advantage is restored, and the disassembly scale income disappears. Battery production enterprises are turning their focus to patent licensing income, and the interests of both parties converge to promote the balanced evolution of the system.

4.3. The Influence of Carbon Price on System Evolution

When the carbon price increases—that is, $P$ increases from 1 to 1.6—and other parameters remain unchanged, the initial state ⑦ in Table 4 is satisfied: $\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)<0$, $Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)<0$, $\eta U-I-LM-Mh>0$.

In this case, due to the impact of changes in cost structure and revenue expectation caused by carbon price adjustment, the cooperative stabilization strategy between third-party recycling manufacturers and battery manufacturers has changed significantly, from Figure 2a to Figure 4a; the original double-equilibrium point strategy selection pattern of the two sides is broken, and there are no longer two independent stable equilibrium states, but gradually converge to the sub-optimal cooperation equilibrium point $(1,0)$. Compared with the strategy evolution path under the initial carbon price level, the strategy evolution trajectory of the two parties after the carbon price is significantly increased is shown in Figure 4b. The third-party recycling manufacturer is constrained by the carbon cost and will converge to the sub-optimal equilibrium point $(1,0)$ at a faster rate, and it has reached a stable equilibrium state when the evolution process is advanced to $t=0.1$. This time node is significantly earlier than the corresponding setting in the initial carbon price scenario in Figure 2b. Similarly, when the carbon price shows a downward adjustment trend, the carbon cost-constraint effect is weakened, resulting in insufficient cooperation incentives, and the strategic choices of both parties will eventually converge to the worst equilibrium point $(0,1)$. It can be seen that a reasonable increase in the carbon price can promote convergence of the two sides toward the sub-optimal cooperative equilibrium state through the cost transmission and income adjustment mechanism, indicating a significant positive incentive effect. Further observation shows that battery manufacturers are less sensitive to changes in carbon prices than third-party recycling manufacturers due to their strong position in the industrial chain and cost-sharing capabilities.

Carbon price affects system evolution by reconstructing the economic feasibility of remanufacturing and disassembly of third-party recycling manufacturers. There are differences in carbon emission intensity between remanufacturing and disassembly. When the carbon price increases, disassembly must bear high carbon purchase costs due to high carbon emissions, while remanufacturing can sell the remaining carbon quota based on low carbon emissions to obtain substantial benefits. The comprehensive income advantage is significant, thereby promoting third parties to choose remanufacturing. Because the carbon market transaction income and cost are borne by the third party, and the battery production enterprise cannot share the carbon income, the patent authorization income does not change and still faces the risk of brand reputation loss; this results in a lack of positive incentive and choosing not to authorize the strategy, which ultimately promotes system convergence. Overall, when the carbon price decreases, the carbon incentive effect weakens, and the system evolves to the worst equilibrium point.

4.4. The Impact of Patent Fees and Maintenance Costs on System Evolution

When the patent fee is reduced, and the monitoring cost is greatly increased—that is, $h$ and $I$ change from $h=8,I=10$ to $h=5,I=80$—and other parameters remain unchanged, the initial state ⑧ in Table 4 is satisfied: $\eta U-M\left(E_1-E_2\right)-MP\left(T_2-T_1\right)<0$, $Mh-M\left(E_1-E_3\right)-MP\left(T_2-T_1\right)<0$, $\eta U-I-LM-Mh<0$.

In this case, driven by the increase in cooperation benefits from reduced patent licensing costs and the rise in violation risk constraints due to increased maintenance intensity, the cooperation stability strategy of third-party recycling manufacturers and battery manufacturing enterprises has been significantly adjusted. The former is more active in laying out the compliance remanufacturing business, while the latter tends to relax the patent licensing restrictions, and the corresponding evolution diagram is switched from Figure 2a to Figure 5a; the original double-equilibrium point pattern of the two sides is broken, and they converge to the optimal equilibrium point $(1,1)$ towards the goal of unified interests. Compared with the evolution path of the initial state, the evolution trajectory of the strategy after the adjustment of patent fees and monitoring costs is shown in Figure 5b. The convergence rate of the two types of subjects to point $(1,1)$ is significantly faster than that of the initial state. The simulation image intuitively shows that, under the incentive of cost–benefit structure reconstruction, the two sides have evolved more rapidly toward battery remanufacturing scale and patent authorization standardization, and the willingness to cooperate and the efficiency of achievement have been improved simultaneously, indicating a positive incentive effect with both stability and sustainability. It can be seen that while increasing the cost of patent maintenance to curb violations, lowering the patent fee to reduce the threshold of cooperation can effectively break the deadlock of the game and promote the rapid convergence of the two sides to the optimal cooperative equilibrium. Moreover, battery manufacturers are more sensitive to the two cost changes above than third-party recycling manufacturers, as they are directly related to patent income distribution and patent maintenance costs, and their response to strategy is faster.

The patent fee and maintenance cost affect the system evolution by adjusting the cost–benefit structure of both sides of the game in a two-way manner, breaking the deadlock of the patent licensing game and promoting the formation of positive incentives for both strategies. Reducing patent costs lowers the institutional cost threshold for third-party compliance remanufacturing and increases remanufacturing’s net profit. The profit advantage is significant, and the risk of violation can be avoided, making it the rational choice for third parties. The patent maintenance costs of battery production enterprises increase, which greatly raises the patent monopoly cost, and it is difficult for the monopoly income to cover the cost increment, while the licensing strategy can generate stable income and avoid loss of brand reputation, thereby forming licensing incentives. The two jointly promote the system’s convergence to the optimal equilibrium; by contrast, a game deadlock is difficult to break, and the system evolves toward a disadvantage-related equilibrium.

5. Conclusions

Given the development status of the battery recycling industry, considering the enterprise carbon limit and the power battery remanufacturing patent authorization, the evolutionary game model between the third-party recycling manufacturer and the battery production enterprise is constructed. The influence of the main factors on the strategic choice of the parties is analyzed, and the following conclusions and suggestions for improvement are drawn:

(1)

The amount of waste battery recycling is the core material factor affecting the strategic choice of both parties. When the amount of recycling is too high, the third-party recycling manufacturer tends to adopt the battery disassembly strategy due to diseconomies of scale, and the system converges on the worst equilibrium. A reasonable reduction in recycling quantity promotes the evolution of the system toward a remanufacturing-related sub-optimal equilibrium, and the battery production enterprise’s strategic response sensitivity to changes in recycling quantity is significantly higher than that of the third party.

(2)

The carbon price has a significantly low-carbon incentive-oriented effect on the evolution of the game system. The high carbon price in the carbon market will encourage third-party recycling manufacturers to adopt the remanufacturing strategy through carbon cost transmission and carbon asset income realization, and the system will converge on a sub-optimal equilibrium. The decrease in the carbon price will weaken the low-carbon incentive effect and lead the system to revert to the worst equilibrium of disassembly.

(3)

When the patent licensing fee is at a reasonably low level, and the patent licensing income of the battery manufacturer can cover the sum of patent maintenance cost and brand reputation loss, the two parties will form the optimal strategy combination of remanufacturing and licensing, and promote the system to converge to the optimal equilibrium state.

According to the above research conclusions, the power battery recycling supply chain enterprises can obtain the best benefits, promote the efficient recycling of waste batteries, and encourage the renewal of remanufacturing technology from the following aspects:

(1)

Led by the government, a dynamic monitoring platform for the whole life cycle of waste power batteries could be built by combining battery manufacturers, third-party recycling manufacturers, and new energy vehicle enterprises. Based on the terminal data, the stock, flow, and flow direction of retired batteries in the region could be tracked in real time, and the recycling scale threshold of remanufacturing adaptation in each region could be accurately measured. Based on the threshold, the third-party recycling manufacturers could be guided to differentiate the layout of recycling outlets to avoid excessive concentration of recycling in local areas. At the same time, the battery production enterprises and the third party could be promoted to establish a collaborative adjustment mechanism for recycling volume, so as to realize the accurate matching of waste battery recycling scale and remanufacturing process, and avoid the system evolving to the worst equilibrium caused by the imbalance in recycling volume from the source.

(2)

It is recommended to further improve the connection mechanism between the carbon trading market and the power battery recycling industry, establish carbon emission accounting standards for power battery recycling processes, clarify carbon emission measurement rules for remanufacturing and disassembly processes, and improve the stability and transparency of carbon price formation. Third-party recycling manufacturers can accurately measure the carbon asset income generated by remanufacturing and establish stable, low-carbon income expectations. At the same time, this would raise the threshold for obtaining carbon quotas for high-carbon emission processes such as disassembly, moderately increase the punishment of ultra-emission carbon prices, amplify the carbon costs of disassembly and the carbon benefits of remanufacturing through carbon price leverage, strengthen the economic incentives for third parties to choose remanufacturing strategies, and promote the system to evolve to a remanufacturing-related equilibrium.

(3)

The battery manufacturing enterprises could build a step-by-step patent licensing fee mechanism linked to the scale of remanufacturing and the stage of industrial development. During the cultivation period of the remanufacturing industry, the unit patent licensing fee could be moderately reduced, so that the patent licensing fee is lower than the sum of the leading revenue of remanufacturing and the cost of carbon emission difference, and the system cost threshold of third-party compliance remanufacturing would be reduced. At the same time, the power battery industry association could build a platform for patent rights protection and brand reputation co-governance, clarify the joint and several liability of battery production enterprises for unauthorized remanufacturing of inferior products, link the quality problems caused by unauthorized remanufacturing with the brand rating and industry qualification of the original production enterprises, improve the patent maintenance and infringement supervision costs of battery production enterprises, ensure that their patent licensing benefits are higher than the sum of infringement fines, maintenance costs, and reputation losses, and force enterprises to actively choose patent licensing strategies—ultimately promoting both parties to form an optimal combination of remanufacturing and licensing strategies to achieve win–win benefits and low-carbon development of the industrial chain.