Tripartite Evolutionary Game Analysis of Power Battery Cascade Utilization Under Government Subsidies

The continued industrialization of new-energy vehicles has facilitated the rapid growth of the massive retired power battery drive recovery and cascade utilization industries. Improving the full lifecycle value of power batteries and recycling necessary materials has recently emerged as a hot issue. Cascade utilization, disassembly and recycling of power batteries are some key strategies to address this concern. In the context of government subsidies and extended producer responsibility, a tripartite evolutionary game model of manufacturers, third-party recyclers and cascade utilization enterprises is constructed in this study to enhance the entire lifecycle value of power batteries for the double closed-loop supply chain containing cascade utilization. Moreover, the stability of each subject’s strategy selection is analyzed, the effects of related factors on each subject’s strategy selection are examined, and the conditions for the stable evolution of the tripartite game to the equilibrium point are further discussed. The research demonstrates that: 1) increasing government subsidies and manufacturers’ reasonable formulation of internal incentive mechanisms in the supply chain are conducive to the coordinated development of the supply chain. In addition, relevant subjects jointly promote the healthy development of the cascade utilization industry; 2) the expected profits of various subjects are essential factors that affect their decision-making. Improving the profits of adopting recycled materials remanufacturing, high-level processing and large-scale cascade utilization are conducive to enhancing the comprehensive utilization level of industrial resources; 3) reducing the potential risks of the innovative development of recyclers and cascade utilization enterprises can increase the enthusiasm of both parties to promote the practical improvement of cascade utilization levels; and 4) raising the environmental treatment fees of professional battery disassembly enterprises is an effective approach to promote the resource utilization efficiency. Finally, Octave was applied for numerical simulation. Relevant countermeasures and suggestions were proposed for the coordinated and efficient development of power battery cascade utilization based on the influence relationship of various factors and equilibrium point stability conditions.


I. INTRODUCTION
The quick rise in the production and sales of new energy vehicles has skyrocketed the use of power batteries. Notably, the earlier batches of power batteries put into service have reached the retirement period. Gao Gong industry Research The associate editor coordinating the review of this manuscript and approving it for publication was Cheng Chin .
Institute (GGII) have predicted that the recovery amount of power batteries in China will reach 960,000 tonnes in 2025. This scale will continue to expand with the rapid diversification of the electric vehicle industry. The classification and reorganization of retired power batteries can benefit other fields with low requirements for battery performance. This cascade utilization can effectively enhance the utilization rate of resources and minimize large-scale idleness and waste of power batteries before their complete utilization. Following cascade utilization, disassembly and resource recovery can reduce the harm of harmful metal leakage in the battery to the environment and human body while ensuring the recycling of scarce resources. The state has issued a series of policies for developing power battery cascade utilization, among which the most typical strategies are government subsidies and producer responsibility extension. The state implements financial subsidy policies to manage resource recycling. In contrast, it executes the producer responsibility extension system to boost resource recycling levels. However, the cascade utilization industry is still in its infancy due to various specifications of retired power batteries, diverse testing standards and the lack of consensus on value evaluation. Effectively coordinating the relationship between relevant entities, balancing economic and environmental benefits, and improving the comprehensive utilization level of industrial resources have become prominent topics that are being discussed by society.
Research in the power battery cascade utilization field has also unfolded with the emergence of practical issues. From the perspective of industry operations, Li et al. sorted out the policies and standards related to the cascade utilization of retired power batteries [1]. Harper et al. summarized power battery recycling methods and policy measures, and made predictions for future recycling trends [2]. Dong et al. organized the recycling models of new energy vehicle power batteries at home and abroad, and compared the development advantages of different recycling models [3]. Hao et al. proposed a combination of forward and reverse logistics for recycling in the context of circular economy, and forwarded the improvement strategy to optimize the recycling methods and increase the recovery rate of scarce metals based on the actual situation in China [4]. Tang et al. took Beijing, China as an example, proposed the power battery recycling measures and relevant policy recommendations [5]. From the micro-operation perspective, Xie et al. constructed a multi-level recycling network for the closed-loop supply chain of power batteries, and explored the coordinated operation of the closed-loop supply chain of power batteries under different recycling modes [6]. Gu et al. explored the interrelationships between manufacturers and remanufacturers in the closed-loop supply chain of power batteries, and proposed relevant measures to improve remanufacturing profitability [7]. Guo et al. introduced valuable metal recycling stations to construct a supply chain model and analyzed the effectiveness of revenue sharing contracts [8]. Heydari et al. constructed a secondary reverse supply chain for a single manufacturer and retailer, increasing consumers' willingness to recycle through quantity discounts and payment fees, and designing cost contracts to achieve closed-loop supply chain coordination. [9]. Li et al. studied the joint recycling and coordination problem of upstream and downstream enterprises in the three-level closed-loop supply chain of power batteries, and explored the coordination effect of wholesale price discount contracts on the closed-loop supply chain [10]. Lin et al. aimed to maximize supply chain profits and analyzed the impact of different factors such as battery quality and processing costs on supply chain profitability [11]. Wang et al. used dynamic games to compare the optimal decisions of various members of the recycling and remanufacturing supply chain under different scenarios, focusing on factors such as remanufacturing costs and recovery rates [12]. From the government subsidy perspective, Gu et al. revealed that increasing government subsidies could effectively promote manufacturers' optimal production and expected utility based on uncertain market demands [13]. Liu et al. analyzed the effects of reverse subsidies on supply chain node variables and profit distribution [14]. Zhong et al. considered dismantling subsidies and explored the changes in the benefits and optimal decisions of supply chain members under the dominance of different subjects [15]. Zhao et al. compared and examined the optimal pricing decisions and profits of mixed competitive supply chain members that subsidize consumers, retailers, manufacturers and third-party recyclers [16]. Because of the proposal and implementation of the extended producer responsibility system, Xie et al. constructed a tripartite game model for vehicle enterprises that undertake product lifecycle resource and environmental responsibility, power battery production enterprises, and third-party disassembly enterprises. Moreover, they analyzed the impact of market factors and research and development subsidies on the strategies of various game subjects [17].
From the perspective of stakeholder strategy selection, the above research works are primarily conducted based on rational assumptions. In reality, stakeholders are bounded rationally, meanwhile, previous research works have found that classical rational strategy selection methods cannot effectively describe actual system behavior. Based on this, Dirk Helbing, et al., studied by means of evolutionary games, and emphasized the importance of systemic influence and found through systematic analysis that complex science can better understand social systems and solve practical problems in society [18]. In the field of cascade utilization of power batteries, the research based on bounded rationality is still in the initial stage. Qiu et al. constructed a two-level closed-loop supply chain for vehicle factories and automobile sales service shop 4S. They studied the effects of subsidy recovery strategy using the evolutionary game theory [19]. Zhang et al. built a competitive closed-loop supply chain for dual-oligopoly manufacturers and remanufacturers based on evolutionary game and considering the third-party remanufacturing mode, and studied the strategic trends of each subject [20]. Wang et al. investigated the evolutionary stability strategy and benefits of a two-party game based on the impact of the investment model of power battery recyclers on the recycling rate of waste batteries [21]. Huo et al. used dynamic evolutionary game theory and optimal control strategies to explore the impact of recovery subsidies on selecting recovery methods [22]. Hao et al. built a tripartite game model of capacity-sharing platforms, manufacturing enterprises and demand companies based on the sharing economy. They discussed the effects of cost effects and network externalities on the stability of evolutionary systems [23]. You et al. took battery manufacturers and vehicle manufacturers as research objects, constructed a tripartite evolutionary game model, and discussed the impact of the application of blockchain technology on the power Battery recycling industry from the perspective of micro subject behavior [24]. Under the extended producer responsibility system, Ma et al. designed a tripartite evolutionary game model for power battery manufacturing enterprises (responsible subject), government departments, and consumers. In addition, they proposed relevant suggestions for promoting the efficient utilization of resources via the analysis of evolutionary stability strategies [25].
In summary, the cascade utilization of power batteries is still embryonic. Government control has not yet been detailed and implemented. Research on the closed-loop supply chain of power batteries mainly focusses on the battery recycling mode, remanufacturing supply chain coordination, cascade utilization policy interpretation, and the influence of government subsidy mode, object and degree. Considering the cascade utilization of power batteries, examining the effects of internal incentives in the supply chain on relevant stakeholders under government subsidies has become a crucial field of current research. In practice, the government provides specific grants to promote the efficient utilization of resources. Meanwhile, the government's implementation of the producer extension system allows manufacturers to promote improving resource utilization levels through a series of incentive measures as the leader in the supply chain. Based on this scenario, this paper conducted research. The specific research content is as follows. First, under the scheme of government subsidies to manufacturers, considering the cascade utilization of power batteries, the internal incentive mechanism for manufacturers to share the government subsidies of other members of the closed-loop supply chain of power batteries was designed to promote the coordinated development of the supply chain and effectively increase the cascade utilization level via the rational formulation of internal incentive mechanisms in the supply chain. Based on this reasoning, a tripartite evolutionary game model was constructed among manufacturers, third-party recyclers and cascade utilization enterprises. Moreover, the stability of each participant's strategy selection was analyzed, and the influence of main factors on each participant's strategy selection was studied. Second, the Lyapunov indirect method was used to analyze the stability of the pure strategy equilibrium point of the replicated dynamic system. The condition combination for the stable evolution of the tripartite game to the equilibrium point was obtained. Finally, Octave was used for simulation analysis to verify the effectiveness of the stability analysis of each equilibrium point in the model under different numerical conditions. Similarly, countermeasures and suggestions were proposed for the government and relevant enterprises to coordinate and jointly promote the steady improvement of the comprehensive utilization level of resources.

II. EVOLUTIONARY GAME MODEL
The closed-loop supply chain of power battery recycling and recycling comprises three node enterprises: manufacturer, third-party recycler and cascade utilization enterprise. Figure 1 depicts the basic structure and logic relationship of closed-loop supply chain of power battery constructed in this paper. The roles and functions of the three parties involved in the game, as well as the relationships between each part are as follows: The manufacturer is responsible for producing new power batteries and selling them in the electric vehicle market. At the same time, in the context of the extended producer responsibility system implemented by the government, the manufacturer is responsible for repurchasing retired batteries that after being processed by recyclers, which do not meet the standards for further processing into cascade utilization products, as well as retired batteries that have been recycled by the cascade utilization enterprise after cascade utilization. In addition, manufacturer has two strategic choices when producing new power batteries, namely remanufacturing of recycled materials or manufacturing of new materials. When manufacturer adopts the remanufacturing of recycled materials strategy, the manufacturer needs to use certain technical processing methods to purify the scrapped batteries recovered from third-party recycler and cascade utilization enterprise, extract electrode materials and other raw materials, then the manufacture remanufactures using recycled materials; When the manufacturer adopts the manufacturing of new materials strategy, the manufacturer should purchase brand new raw materials. However, due to the mandatory extension of producer responsibility system, when the manufacturer adopts the manufacturing of new materials strategy, the manufacturer needs to hand over the scrapped batteries recovered from third-party recycler and cascade utilization enterprise to professional dismantling enterprises for disposal to meet environmental requirements.
The third-party recycler is responsible for recycling retired power batteries in the electric vehicle market, and for testing, dismantling, and sorting them. At this time, the third-party recycler has two strategic choices, namely, high-level processing or low-level processing of recycled retired batteries. High-level processing refers to improving the technical level of detection, disassembly, and classification of recycled batteries through technological innovation, in order to more meticulously and accurately sort out retired batteries that meet the standards for further processing into cascade utilization products. Then, the third-party recycler will sell the processed retired batteries that meet the standards for further processing into cascade utilization products to cascade utilization enterprise, and the retired batteries that do not meet the standards for further processing into cascade utilization products will be returned to the manufacturer. It should be noted that the third-party recycler's treatment is only a simple preliminary treatment, and the retired batteries after being processed by the third-party recycler do not have the function of cascading utilization, and further processing and treatment by the cascade utilization enterprise are still needed to become cascading utilization products.
The cascade utilization enterprise purchases retired batteries that processed by the third-party recycler, which meet the standards for further processing into cascading utilization products, and conducts in-depth processing and restructuring to assemble them into cascade utilization products, after assembly, the cascade utilization products are put into the cascade utilization market for sales, and cascade utilization enterprise is responsible for recycling all cascade utilization products and handing them over to the manufacturer for disposal. At the same time, when processing cascade utilization products, cascade utilization enterprise has two strategic choices, namely large-scale cascade utilization or small-scale cascade utilization.
In addition, the government provides targeted subsidies to the manufacturer to promote industry development. At the same time, manufacturer develops an internal incentive mechanism in the supply chain to guide third-party recycler and cascade utilization enterprise to work together to promote the coordinated development of the power battery closed-loop supply chain and effectively increase the cascade utilization level.

A. VARIABLE DESCRIPTION AND MODEL ASSUMPTION
To clearly describe the problem, the variables in the model are defined and explained. Details are shown in TABLE 1.
To construct the game model, the strategic choice of each participant and the evolutionary stability of the system equilibrium point is analyzed, and the conditional combination of the three-party strategic selection is obtained. Based on the actual situation and referring to relevant literature, including references [7], [10], [14], and [26], etc., the following assumptions are made: Assumption 1: the manufacturer, third-party recycler and cascade utilization enterprise are three participants in the bureau. Each of the three participants is a bounded rational participant, and the strategy selection gradually evolves and stabilizes to the optimal strategy over time.
Assumption 3: The manufacturer's sales revenue from adopting the recycled materials remanufacturing strategy is M 11 , and the sales revenue from adopting the new materials manufacturing strategy is M 12 (M 12 > M 11 ). If the manufacturer adopts the remanufacturing of recycled materials, it is necessary to chemically disassemble the recycled battery at VOLUME 11, 2023  the cost of C 11 . Suppose the manufacturer adopts the strategy of new material manufacturing under the government's implementation of the extended producer responsibility system. In that case, recycling retired batteries and entrusting them to professional battery disassembly companies for disposal is mandatory. The payable environmental treatment fees are C 13 (C 11 > C 13 ).
Assumption 4: When the manufacturer adopts the strategy of recycled materials remanufacturing and if the third-party recycler adopts the high-level processing strategy, the disassembly costs of the manufacturer can be reduced to a certain extent, with an impact ratio of θ (θ ∈ (0, 1)). At this time, if the cascade utilization enterprise adopts the small-range cascade utilization strategy, the positioning of cascade products is at a relatively higher end and niche. The number of retired batteries received from the third-party recycler and put into the cascade utilization market is relatively small, which will lead to a reduction in the number of discarded batteries ultimately recovered and returned to the manufacturer, making it difficult to meet the number of materials required for the normal production by the manufacturer. The manufacturer must purchase additional new materials to meet production, with an additional procurement cost of C 12 . When the manufacturer adopts the strategy of new material manufacturing, the impact ratio of high-level processing of recycled batteries by a third-party recycler on the disassembly costs paid by manufacturers is ϕ (ϕ ∈ (0, 1)), and the procurement cost of new materials is C 14 (C 14 > C 12 ).
Assumption 5: If the manufacturer adopts the strategy of recycled materials remanufacturing, the government will grant subsidies S and the manufacturer will share the α proportion of subsidies with a third-party recycler adopting a high-level processing strategy and share β proportional of subsidies with the cascade utilization enterprise adopting the large-scale cascade utilization strategy to encourage relevant enterprises and to cooperate and improve the cascade utilization level of power batteries.
Assumption 6: When the third-party recycler adopts the high-level processing strategy and if the cascade utilization enterprise adopts the large-scale cascade utilization strategy (which will receive a large range of retired batteries processed by the recycler), the profit of the third-party recycler is R 21 . If the cascade utilization enterprise adopts the small-scale cascade utilization strategy (which will receive only a small portion of retired batteries processed by the recycler), the number of retired batteries sold and processed by the recycler will decrease, resulting in a loss of profits T . Similarly, when the third-party recycler adopts a low-level processing strategy and if the cascade utilization enterprise adopts the large-scale cascade utilization strategy, the profit will be R 22 (R 22 > R 21 ). If the cascade utilization enterprise adopts the small-scale cascade utilization strategy, the recycler will lose profits D (T > D) due to decreased sales volume. The additional cost of high-level processing by the third-party recycler is C 21 .
Assumption 7: If the cascade utilization enterprise adopts the large-scale cascade utilization strategy, the profit from selling the cascade products that are restructured after high-level processing by the third-party recycler is π 31 . In contrast, selling the cascade products that are restructured after low-level processing by the third-party recycler requires additional processing costs, resulting in a loss of profit e. Suppose the cascade utilization enterprise adopts the small-scale cascade utilization strategy. In that case, the profit from selling cascade products restructured after high-level processing by the third-party recycler is π 32 (π 32 > π 31 ) while selling the cascade products that are restructured following low-level processing by the third-party recycler also requires additional processing costs, resulting in a loss of profit L (e > L).

B. GAME MATRIX
Based on the above model assumptions, the game matrix for the manufacturer, the third-party recycler and the cascade utilization enterprise is shown in TABLE 2.

III. MODEL ANALYSIS A. MANUFACTURER STRATEGY STABILITY ANALYSIS
The expected benefits for manufacturers to adopt the strategy of recycled materials remanufacturing E 11 , the anticipated benefits for manufacturing are adopting the strategy of new material manufacturing E 12 , and the average expected benefits E 1 are, respectively, as follows: (1) The dynamic replication equation for manufacturer strategy selection is expressed as follows: The first derivative of x and the set G (y) are, respectively, represented as follows (5), as shown at the bottom of the next page, According to the stability theorem of differential equations, the manufacturer's probability adopts the recycled materials remanufacturing strategy in a stable state should meet the following requirements: subtractive function with respect to y. Therefore, when y, as shown at the bottom of the page.
, and all the x values are in an evolutionarily stable state. When y < y * , G (y) > 0, then d (F (x)) /dx | x=0 < 0, and x = 0 is the evolutionarily stable strategy of the manufacturer. In contrast, x = 1 is the evolutionarily stable strategy. Figure 2 depicts the evolution phase diagram of manufacturer's strategy selection. Figure 2 shows that the probability of the manufacturer stably adopting the strategy of new material manufacturing is the volume of A 1 , V A 1 , and the likelihood of stably adopting the approach of recycled materials remanufacturing is the volume of A 2 , V A 2 , which can be calculated as follows, (7), as shown at the bottom of the page, Theorem 1: The probability of the manufacturer adopting the strategy of recycled materials remanufacturing is positively correlated with government subsidies, environmental treatment fees for the end-of-life batteries, new material costs negatively associated with disassembly costs, additional procurement costs, and the income gap between new manufacturing and remanufacturing.
Proof:According to the expression of the probability the manufacturer adopts the strategy of recycled materials remanufacturing V A 2 , the first partial derivative of each element can be obtained: Therefore, an increase in S, C 13 , C 14 or a decrease in C 11 , C 12 , (M 12 − M 11 ) could increase the probability of the manufacturer adopting the strategy of recycled materials remanufacturing.
Theorem 1 indicates that the government can increase manufacturers' enthusiasm for recycling the resources through measures, such as increasing subsidies for recycling and remanufacturing, increasing the environmental treatment fees for discarded batteries entrusted by manufacturers to external dismantling enterprises, and actively supporting and guiding relevant technological innovation to reduce the cost of dismantling discarded batteries. Rising raw material prices also positively affect enhancing the manufacturers' willingness to recycle and remanufacture. In addition, it is also possible to encourage manufacturers to adopt the strategy of recycled materials remanufacturing by strengthening consumers' green awareness, balancing consumers' consumption preferences for remanufactured and new products, increasing the price of remanufactured products, reducing the income gap between manufacturers' different strategies, expanding the market scale of cascade utilization, and reducing the cost of additional purchased materials.
Theorem 2: The probability of the manufacturer adopting the strategy of recycling and remanufacturing during the evolution process increases with the increase in the likelihood of third-party recyclers adopting the high-level processing strategy and cascade utilization enterprises adopting the large-scale cascade utilization strategy. Proof:According to the manufacturer's strategic stability analysis, , and x = 0 is the evolutionarily stable strategy of the manufacturer. In contrast, x = 1 is the evolutionarily stable strategy. Therefore, as y, z gradually increases, the manufacturer's stability strategy evolves from x = 0 (new materials manufacturing) to x = 1 (recycled materials remanufacturing). Theorem 2 shows that increasing the probability of third-party recyclers adopting the high-level processing strategy and cascade utilization enterprises adopting the large-scale cascade utilization strategy could increase the likelihood of manufacturers adopting the recycling and remanufacturing strategy. Improving the processing level of retired batteries by third-party recyclers could effectively improve the utilization rate of restructured batteries, promote the range of cascade utilization by cascade utilization enterprises, and expand the market share of cascade utilization. After the high-level processing, the enhanced safety and reliability of retired batteries could strengthen consumer trust, reduce the differential consumption level between new manufacturing and remanufacturing, improve resource utilization, and reduce environmental pollution.

B. THIRD-PARTY RECYCLER STRATEGY ANALYSIS
The expected benefits for the third-party recycler to adopt the strategy of high-level processing and low-level processing E 21 , E 22 , as well as the average expected benefits, E 2 are, respectively: The replicated dynamic equation and the first derivative of the third-party recycler can be obtained from the equations as follows: According to the stability theorem of differential equations, the probability that the third-party recycler adopts the strategy of high-level processing in a stable state should fulfil the following requirements: F (y) = 0 and d (F (y)) /dy < 0. Because of ∂J (z) /∂z < 0, J (z) is a subtractive function with respect to z. Therefore: when z = (R 22 + C 21 + T − R 21 − D − xαS) / (T − D) = z * , J (z) = 0, then d (F (y)) /dy ≡ 0, F (y) ≡ 0. All the y values are in an evolutionarily stable state. When z < z * and J (z) > 0, then d (F (y)) /dy y=0 < 0, and y = 0 is the evolutionarily stable strategy of the third-party recycler. In contrast, y = 1 is the evolutionarily stable strategy. Figure 3 demonstrates the evolution phase diagram of thirdparty recycler's strategy selection. Figure 3 shows that the probability of the third-party recycler stably adopting the strategy of low-level processing is the volume of B 1 , V B 1 . The likelihood of stably adopting the approach of high-level processing is the volume of B 2 , V B 2 , which can be calculated as follows: Theorem 3: The probability of third-party recyclers adopting the strategy of high-level processing is positively correlated with incentives provided by the manufacturer, the profits of adopting the approach of high-level processing, and the losses of adopting the strategy of small-scale cascade utilization and low-level processing. In contrast, it is negatively VOLUME 11, 2023 correlated with the profits of adopting the strategy of lowlevel processing, additional costs of high-level processing, and the losses of adopting the approach of small-scale cascade utilization and high-level processing.
Proof: According to the expression of the probability for the third-party recycler adopting the strategy of high-level processing V B 2 , the first partial derivative of each element can be obtained as follows: Therefore, an increase in αS, R 21 , D or a decrease in R 22 , C 21 , T could increase the probability of third-party recyclers adopting high-level processing.
Theorem 3 shows that an increase in government subsidies shared by the manufacturer to the third-party recyclers and the decrease in high-level processing costs of the third-party recyclers could effectively motivate the third-party recyclers to improve the processing level of recycled batteries. Government departments cannot only actively promote third-party recyclers' high-quality reconstituted batteries to be accepted by the market to increase their profits of adopting the strategy of high-level processing but also formulate clear cascade utilization standards to reduce the risk of profit losses by adopting high-level processing. In addition, the quality of the batteries processed at a low level by third-party recyclers is too low to meet the requirements of small-scale cascade utilization by the cascade utilization enterprise, resulting in greater losses, which will avoid the speculative psychology of the third-party recycler to a certain extent.
Theorem 4: The probability of a third-party recycler adopting the strategy of high-level processing during the evolution process increases with the increase of likelihood for the manufacturer to adopt the recycled materials remanufacturing strategy. Subsequently, the cascade utilization enterprise adopts the large-scale cascade utilization strategy.
Proof:According to the third-party recycler's strategic stability analysis, when x < [R 22 + C 21 + T − R 21 − D−z(T − D)]/(αS), z < z * , then J (z) > 0, d(F(y))/dy| y=0 < 0, and y = 0 is the evolutionarily stable strategy of the thirdparty recycler. In contrast, y = 1 is the evolutionarily stable strategy. Therefore, x, z gradually increases. The probability of the third-party recycler adopting the strategy of high-level processing increases from y = 0 to y = 1; therefore, y increases with an increase in x, z.
Theorem 4 indicates that with the increase in manufacturers' willingness to recycle and remanufacture, the impact of high-level treatment of batteries on its disassembly costs increases, which becomes the core factor restricting the manufacturer's remanufacturing profit. The manufacturer shares government subsidies with the recycler to encourage them to choose a high-level processing strategy, prompting them to improve battery treatment levels. Improving the cascade utilization rate could reduce the risk of profit loss suffered by third-party recyclers and guarantee the stable development of the cascade utilization industry. However, governmentrelevant departments and cascade utilization enterprises should improve the testing and screening level and listing standards of reconstituted batteries to avoid the ''free riding'' behaviour of third-party recyclers, which may lead to high-risk reconstituted batteries flowing into the market and causing social losses.

C. CASCADE UTILIZATION ENTERPRISE STRATERY STABILITY ANALYSIS
The expected benefits for cascade utilization enterprises to adopt the strategy of large-range cascade utilization and small-range cascade utilization E 31 , E 32 , and the average expected benefits E 3 are, respectively, represented as follows: The replication dynamic for cascade utilization enterprise strategy selection, the first derivative of z, and the set H (y) are, respectively, represented as follows: (21) According to the stability theorem of differential equations, the probability that the cascade utilization enterprise adopts the strategy of large-range cascade utilization in a stable state should fulfil the following requirements: F (z) = 0 and d (F (z)) /dz < 0. Because of ∂H (y) /∂y < 0 H (y) is a subtractive function with respect to y. Therefore, when y = (π 32 + e − π 31 − L − xβS) / (e − L) = y * * , H (y) = 0, then d (F (z)) /dz ≡ 0, F (z) ≡ 0, and all the z values are in an evolutionarily stable state. When y < y * * , H (y) > 0, then d (F (z)) /dz | z=0 < 0, and z = 0 is the evolutionarily stable strategy of the cascade utilization enterprise. In contrast, z = 1 is the evolutionarily stable strategy. Figure 4 shows the evolution phase diagram of cascade utilization enterprise's strategy selection. Figure 4 shows that the probability of the cascade utilization enterprise stably adopting the strategy of small-range cascade utilization is the volume of C 1 , V C 1 . The probability of stability adopting the approach of large-range cascade utilization is the volume of C 2 , V C 2 , which can be calculated as follows: 66390 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  Theorem 5: The probability of the cascade utilization enterprise adopting the strategy of large-scale cascade utilization is positively correlated with incentives given by the manufacturer, the profits of adopting the large-scale cascade utilization strategy, and losses of adopting the approach of low-level processing and small-scale cascade utilization. In contrast, it is negatively correlated with profits of adopting a small-scale cascade utilization strategy and the failures of adopting the strategy of low-level processing and large-scale cascade utilization.
Proof: According to the expression of the probability for cascade utilization enterprise adopting the strategy of large-scale cascade utilization V C 2 , the first partial derivative of each element can be obtained as follows: ∂V C 2 /∂βS > 0, ∂V C 2 /∂π 31 > 0, ∂V C 2 /∂L > 0, ∂V C 2 /∂π 32 < 0, ∂V C 2 /∂e < 0. Therefore, an increase in βS, π 31 , L or a decrease in π 32 , e could increase the probability of the enterprise adopting the large-scale cascade utilization strategy.
Theorem 5 demonstrates that increasing the government subsidies shared by the manufacturer to cascade utilization enterprise and decreasing the potential losses of adopting the strategy of large-scale cascade utilization could effectively encourage the cascade utilization enterprise to enhance the efficiency of cascade utilization. The government departments should strictly supervise the recycling process of power batteries, emphasize the traceability of recycled batteries, use technologies such as the internet to comprehensively record and timely update the information of listed and traded restructured batteries, and advocate that cascade utilization enterprise should apply simple processed retired batteries in areas such as low-risk school backup power supply or home appliances to reduce potential risks in the industry and ensure the healthy development of the cascade utilization industry. Furthermore, pertinent publicity works could be conducted to increase the public recognition of cascade products, steadily increase the profits of cascade utilization enterprises adopting the large-scale cascade utilization strategy, and encourage cascade utilization enterprises to increase their enthusiasm for cascade utilization.
Theorem 6: The probability of cascade utilization enterprise adopting large-scale cascade utilization during the evolution process increases with the increase in the likelihood for the manufacturer adopting the recycled materials remanufacturing strategy and third-party recycler adopting the high-level processing strategy.
Proof:According to the cascade utilization enterprise's strategic stability analysis, when x < [π 32 + e − π 31 − L − y(e − L)]/(βS), y < y * * , then H (y) > 0, d(F(z))/dz| z=0 < 0, and z = 0 is the evolutionary stability strategy of the cascade utilization enterprise. In contrast, z = 1 is the evolutionarily stable strategy. Therefore, as x, y gradually increases, the probability of cascade utilization enterprise adopting the strategy of large-scale cascade utilization increases from z = 0 to z = 1; therefore, z increases with the rise in x, y.
Theorem 6 shows that the increased probability of the manufacturer adopting the recycled materials remanufacturing strategy strengthens the recycling effort of discarded batteries by the manufacturer, cascade utilization enterprise as the terminal enterprise in the power battery recycling chain. The depth and breadth of the cascade utilization will affect the scale and quality of the discarded batteries that ultimately flow to manufacturers. The increased willingness of third-party recyclers to process at a high level could reduce the risk of cascade utilization enterprises exploring new markets. The more accurate assessment of critical information, such as detection range and residual battery capacity, could improve restructured batteries' safety performance. It can also provide strong support for developing cascade utilization products in new fields, thus promoting the healthy development of the cascade utilization market.
The Lyapunov indirect method is used to identify the property of the equilibrium point; for instance, if all eigenvalues of a Jacobian matrix have negative real parts, the equilibrium point is asymptotically stable. If at least one eigenvalue of a Jacobian matrix has a positive real part, the equilibrium  According to the eigenvalues of the Jacobian matrix corresponding to each equilibrium point in the table, considering the value range of each parameter in the eigenvalues and assumptions, eight scenarios are designed to explore the stability of each equilibrium point (TABLE 4). 14 , C 21 > αS, π 32 − π 31 > βS, the eigenvalues of the corresponding Jacobian matrix E 2 (1, 0, 0) satisfy the condition that all of them are less than 0, which is the evolutionarily stable point. With large government subsidies or low incentives from manufacturers, manufacturers still have a significant subsidy surplus after sharing some of the government subsidies with relevant node enterprises, which can greatly offset the expenditure of high disassembly costs and additional acquisition costs. Moreover, higher raw material costs reduce the extra benefits of new material ∂F (x) /∂x ∂F (x) /∂y ∂F (x) /∂z ∂F (y) /∂x ∂F (y) /∂y ∂F (y) /∂z ∂F (z) /∂x ∂F (z) /∂y ∂F (z) /∂z manufacturing. Therefore, manufacturers are more inclined to adopt the recycled materials' remanufacturing strategy. For third-party recyclers and cascade utilization enterprises, the subsidy incentives they receive are insufficient to offset higher costs; hence, they adopt the approach of low-level processing and small-scale cascade utilization. This scenario can only rely on continuous government support for manufacturers to ensure their remanufacturing of recycled materials. Still, the development of third-party recyclers and cascade utilization enterprises will be hit, prompting them to prioritize profits and survival, which is not conducive to the long-term healthy development of the battery recycling industry. 14 , C 21 > αS, π 31 + βS − e > π 32 − L, E 6 (1, 0, 1) is the evolutionarily stable point. Cascade utilization enterprises could increase the public acceptance of cascade products through advertising and other means, develop cascade products with different risk values and requirements in various fields with government support, expand the market scale, increase the profits by adopting a large-scale cascade utilization strategy, and reduce the risk of profit loss caused by third-party recyclers by simply handling recycled batteries. Meanwhile, benefiting from internal incentives from manufacturers, the probability of a cascade utilization enterprise adopting the large-scale cascade utilization strategy increases significantly, eventually evolving into a stable strategy. 1, 0) is the evolutionary stable point. When manufacturers adopt the recycled materials' remanufacturing strategy, their efforts to recycle end-of-life batteries have been strengthened. The profits of third-party recyclers use the high-level processing strategy. In addition, incentives provided by manufacturers can significantly compensate for their high-level processing costs and losses caused by the rejection of cascade utilization enterprises. The high net profit makes them more inclined to adopt the high-level processing strategy. When the profits from small-scale cascade utilization are higher than the sum of the profits from large-scale cascade utilization and incentive income, cascade utilization enterprises are more inclined to adopt a small-scale cascade utilization strategy, making it difficult to expand the cascade utilization market. Moreover, it is not conducive to improving the comprehensive utilization level of resources. This scenario should be avoided to some extent. (1, 1, 1) is the evolutionary stable point. With the government's support, manufacturers can set up a reasonable incentive mechanism to increase residual subsidies. The government subsidies share by the manufacturer to third-party recyclers can effectively offset the costs and risk losses of adopting a high-level processing strategy. That is, when third-party recyclers have high net profits from adopting a high-level processing strategy, they are more inclined to adopt it. It can also encourage cascade utilization companies to deepen and broaden the cascade utilization market. High net profits from adopting the large-scale cascade utilization strategy and the coordination and cooperation between supply chain enterprises can enhance the ability of cascade utilization companies to undertake the risk of exploring new markets so that they are more inclined to choose a large-scale cascade utilization strategy. This action prompts the joint construction of power battery closed-loop supply chain enterprises and jointly promotes the steady improvement of the cascade utilization level. (1, 0, 0) is the evolutionary stable point. The profits of manufacturers adopting the recycled materials remanufacturing strategy are greater than that of adopting a new material manufacturing strategy, which to some extent encourages manufacturers to undertake potential risks in the industry, implement the extended producer responsibility system and actively recycle discarded batteries. However, to ensure their profits, manufacturers share a low proportion of government subsidies to other supply chain nodes, which does not play an incentive role. The meagre earnings of third-party recyclers and cascade utilization enterprises are not conducive to balancing economic and social benefits and should be avoided. 21 > αS, π 31 +βS−e > π 32 − L, the system does not have a stable point. Manufacturers receiving government subsidies could effectively resist the risk of strategic selection. Still, the incentive for other node-point enterprises in the supply chain is relatively small, the cost of adopting a high-level processing strategy is greater than the incentive for third-party recyclers, and the enthusiasm for adopting a high-level processing strategy is low. The cascade utilization market is receiving much attention, with much government policy support and guidance. However, the quality of restructured batteries purchased by cascade utilization enterprises is uneven, and the potential risks are relatively large, which is not conducive to expanding the market. As a result, the scale of end-of-life batteries ultimately flowing to manufacturers is large, increasing manufacturers' recycling burden. At this time, the system's evolution cannot be stable, and the cascade utilization industry has always been in a relatively chaotic and disorderly state, which must be avoided.
π 32 − π 31 > βS, the system does not have a stable point. The manufacturer's incentives are relatively low, the cost of technology research and development and innovation for third-party recyclers is relatively low, the cost of developing new markets for cascade utilization enterprises is high and the risk of large-scale acceptance of restructured batteries is high. Similarly, the ability to assess the remaining value of batteries is insufficient, and the market profit margin is small, resulting in an imbalance between supply and demand in the market. This outcome is not conducive to providing high-quality restructured batteries by third-party recyclers. 22 − D and π 31 + βS − e > π 32 − L, the system does not have a stable point. The cost of dismantling and environmental treatment fees are reduced. The sales revenue from adopting a new material manufacturing strategy is high, which is not conducive to the stable adoption of recycled materials remanufacturing strategies by manufacturers. The enthusiasm of other nodal enterprises that fail to share government subsidies to coordinate the supply chain development also decreases.
In summary, the change of relevant parameters can theoretically affect the evolution process and the system results so that the system could be stabilized at different equilibrium points or unable to reach a stable state. The increase in government subsidies, the internal incentive set by manufacturers in the supply chain, potential market losses, environmental treatment fees for dismantling enterprises, new material costs, third-party recyclers' profits of adopting a high-level processing strategy and cascade utilization enterprises profits of adopting the large-scale cascade utilization strategy can positively promote the coordinated development of the power battery closed-loop supply chain and effectively improving the cascade utilization levels. The increase in the manufacturer disassembly costs, additional procurement costs, revenue differentials among different strategies, third-party recyclers' profits of adopting low-level processing strategy, high-level processing costs, and cascade utilization enterprises profits of adopting small-scale cascade utilization strategy is not conducive to the stable evolution of the system. Moreover, these factors strongly and negatively impact the coordinated operation within the supply chain and hinder the improvement of the comprehensive utilization level of resources.

IV. NUMERICAL ANALYSIS
To verify the effectiveness of evolutionary stability analysis, numerical values were assigned to the model under different scenarios based on the actual situation and related literature. The above theoretical analysis illustrate that factors such as government subsidies, internal incentives set by manufacturers in the supply chain, losses caused by potential market risks, and the expected profits of various subjects' strategy selection are of great significance to the system evolution process and results. Scenarios 4 and 2 are selected for analysis to cover more influencing factors. Then, Octave is used for numerical simulation.

A. SCENARIO 4 SIMULATION ANALYSIS
Referring to relevant research, including references [17], [19], and [20], etc., system parameters are set to meet the require-  Figure 5 depicts the impact of profit changes of cascade utilization enterprises adopting the large-scale cascade utilization strategy on the evolution process and results. Next, the values of π 31 = 125, 135, 145 are assigned, respectively, and the replication dynamic equations are set to evolve 200 times over time.  Figure 5 shows that during the evolution process, the probability of cascade utilization enterprises adopting a large-scale cascade utilization strategy increases with the increased probability of third-party recyclers adopting a high-level processing strategy. As the profit of large-scale cascade utilization increases, the likelihood of cascade utilization enterprises adopting large-scale cascade utilization strategy increases, and the probability of third-party recyclers adopting high-level processing strategy decreases. The government should actively encourage the development of the cascade utilization industry, increase public welfare publicity and improve public recognition. Meanwhile, relevant enterprises should deepen customer loyalty, improve the net profit of adopting a large-scale cascade utilization strategy, and accelerate the evolution process of large-scale cascade utilization by cascade utilization enterprises.
Considering potential market risks, the effects of potential losses on the cascade utilization enterprises under different strategies are analyzed. Figure 6 depicts the impact of potential losses changes of cascade utilization enterprises adopting the large-scale cascade utilization strategy on the evolution process and results. The value of e = 30, 40, 50 are, respectively, assigned. At the same time, the impact of potential losses changes of cascade utilization enterprises adopting the small-scale cascade utilization strategy on the evolution process and results are shown in Figure 7. The value of L = 5, 15, 25 are, respectively, assigned.   Figure 6 shows that during evolutionary stability, the speed of system evolution decreases with the increase in the potential loss of adopting a large-scale cascade utilization strategy. Moreover, the probability of cascade utilization enterprises adopting a large-scale cascade utilization strategy decreases. The low-level processing strategy adopted by third-party recyclers will increase the risk cost of the large-scale cascade utilization of cascade utilization enterprises and decrease the enthusiasm of the cascade utilization of cascade utilization enterprises. Government-relevant departments should clearly implement the delivery standards of restructured batteries in all dimensions. Relevant enterprises should strengthen information sharing, reduce potential risks and improve the reliability of cascade utilization. Figure 7 shows that as the possible loss of adopting a small-scale cascade utilization strategy increase, the probability of cascade utilization enterprises adopting a large-scale cascade utilization strategy and the speed of system evolution increase. When cascade utilization enterprises adopt a small-scale cascade utilization strategy, the requirements for the reconstituted battery are higher. If the third-party recycler has a ''free rider'' behaviour and provides a reconstituted battery of uneven quality, it will lead to safety accidents and huge economic and social losses. At this time, government regulation and control of the industry is critical. The government departments should strengthen regulatory efforts, guide relevant enterprises to coordinate and cooperate by setting reasonable rewards and punishment mechanisms, and jointly assume the responsibility of improving the efficiency of cascade utilization and the comprehensive utilization level of industrial resources.
Similarly, to analyze the impact of market potential losses on third-party recyclers under different strategies, (where T = 20, 30, 40 are assigned), the impact of potential losses changes of third-party recyclers adopting the high-level processing strategy on the evolution process and results are shown in Figure 8. At the same time, the impact of potential losses changes of third-party recyclers adopting the low-level processing strategy on the evolution process and results for D = 5, 15, 25 are shown in Figure 9.   Figure 8 shows that the increase in the potential loss of adopting a high-level processing strategy by third-party recyclers is accompanied by the increased probability of adopting a large-scale cascade utilization strategy by cascade utilization enterprises. In contrast, the likelihood of adopting a high-level processing strategy by third-party recyclers decreases. Figure 9 shows that when the potential loss of adopting a low-level processing strategy by third-party recyclers increases, the probability of adopting a large-scale cascade utilization strategy by cascade utilization enterprises decreases. In contrast, the probability of adopting a high-level processing strategy by third-party recycler recyclers increases. Under the strengthened implementation of the extended producer responsibility system by the government, manufacturers should actively assist third-party recyclers and cascade utilization enterprises in establishing and improving battery recycling platforms and network systems, clarify the roles and responsibilities of enterprises at various nodes of the supply chain in battery recycling, screening, restructuring and other aspects, and formulate the standards of the evaluation grading and screening for power batteries of various specifications, establish a quality control system, and support corresponding reward and punishment mechanisms or licensing systems. Meanwhile, more innovative business models should be tried, such as the cooperation between recyclers and manufacturers to experiment on electricity exchange mode, targeted development of cascade utilization battery products, among others, to reduce the operating costs of recyclers, improve the economy of cascade utilization and steadily improve the level of cascade utilization.

B. SCENARIO 2 SIMULATION ANALYSIS
Referring to relevant research, including references [17], [20], and [25], etc., and the basic assumptions, to meet the requirements of Scenario 2, relevant parameters are adjusted based on simulation parameters in scenario 4:α = 0.25, β = 0.35, R 21 = 155, C 21 = 70, T = 40, π 32 = 160, e = 55. Other parameters shall be set according to the parameters of the simulation analysis in Scenario 4 and remain unchanged. The effects of S, α, β, M 11 , R 21 on the process and result of the evolutionary game are further analyzed. Figure 10 depicts the impact of government subsidy changes on the evolution process and results, S = 170, 220, 270 respectively, were assigned, and the replication dynamic equation are set to evolve 200 times over time. Figure 10 shows that increased government subsidies also raised the probability of manufacturers adopting recycled materials remanufacturing and low-level processing strategies by third-party recyclers. In evolutionary stability, the rise in government subsidies could accelerate the evolution speed of the system. Therefore, the government should encourage and support the development of new industries, thoroughly investigate the development status of the cascade utilization industry, coordinate with industry associations and other relevant departments to formulate precise policies, regulate the entire chain production and application process of recycling batteries, encourage relevant node enterprises to establish a sharing platform, use internet to achieve traceability and real-time tracking of recycled batteries, and create a recycling network system information database to improve the feasibility, reliability, and economy of cascade utilization, promote the long-term healthy development of the industry, and ensure the effective improvement of the recycling level of industrial resource.
To analyze the effects of changes in the proportion of government subsidies shared by manufacturers on the evolutionary game process, the impact of the proportion changes of government subsidies shared by manufacturers to third-party recycler (which adopting a high-level processing strategy) on the evolution process and results are shown in Figure 11, and the values of α = 0.2, 0.25, 0.30 are assigned, respectively. Later, the impact of the proportion changes of government subsidies shared by manufacturers to cascade utilization enterprises (which adopting a large-scale cascade utilization strategy) on the evolution process and results are shown in Figure 12, and β = 0.30, 0.35, 0.40 were assigned, respectively. FIGURE 11. Impact of α changes on the evolution process and results. Figures 11 and 12 demonstrate that an increased proportion of government subsidies shared by manufacturers could accelerate the speed of system evolution to the stable point in the system evolution process. The change in the proportion of government subsidies shared by manufacturers exerts little influence on the probability of third-party recyclers' strategy selection. However, it significantly affects the likelihood of cascade utilization enterprises' strategy selection. With the increased proportion of manufacturers sharing government subsidies with cascade utilization enterprises, the probability of adopting a large-scale cascade utilization strategy by cascade utilization enterprises will significantly increase in the subsequent stages of the evolution process. The likelihood of adopting recycled materials' remanufacturing strategy by manufacturers and high-level processing strategy by third-party recyclers will slightly decrease. Figure 13 demonstrates the impact of the profit changes of manufacturers adopting recycled materials remanufacturing strategy on the evolution process and results, and M 11 = 150, 170, 190 were assigned, respectively. Figure 14 shows the impact of the profit changes of third-party recyclers adopting a high-level processing strategy on the evolutionary game process and results, when R 21 = 135, 155, 175 is assigned.  Figure 13 depicts that increasing manufacturers' revenues from adopting recycled materials' remanufacturing strategy could accelerate the evolution process and improve the probability of a high-level processing strategy by third-party recyclers and large-scale cascade utilization strategy by cascade utilization enterprises. The cascade utilization industry is still in its early stages of development. Manufacturers still have significant room for progress in innovating and developing critical technologies to improve resource utilization efficiency. The public has an insufficient understanding and acceptance of reproductions, and manufacturing new material products is still the mainstream process of market consumption. Manufacturers should rely on government policy support to increase recycling and remanufacturing revenues through measures such as strengthening the promo-tion of green consumption concepts and transforming public consumption concepts to narrow the income gap between the two strategies and improve the enthusiasm for recycling resources.  Figure 14 demonstrates that the increase in third-party recycler profits from adopting the high-level processing strategy raised the increased probability of recycled materials remanufacturing strategy by manufacturers and large-scale cascade utilization strategy by cascade utilization enterprises. The evolution speed of the two sides to the equilibrium point is accelerated. Meanwhile, it can effectively slow the evolution speed of recyclers' selection of low-level processing strategy, which is conducive to the benign development of the power battery's closed-loop supply chain.

V. CONCLUSION
This paper considered the government subsidy and producer responsibility extension system, introduced cascade utilization into the power battery closed-loop supply chain, and constructed a tripartite evolutionary game model for manufacturers, third-party recyclers, and cascade utilization enterprises. Next, this paper demonstrated the stability of each entity's strategy selection, the equilibrium strategy combination of the game system and the impact of various factors. Finally, the Octave was used for numerical simulation to verify the validity of the analysis results. From a theoretical perspective, this paper provides essential theoretical guidance for the stable and efficient operation of the power battery cascade utilization industry, which, to some extent, enriches relevant academic research in this field. From a practical perspective, this paper provides ideas for improving and innovating the cascade utilization mode. It puts forward countermeasures and suggestions for the government and relevant nodal enterprises to improve the comprehensive utilization level of resources. The primary conclusions of this paper are as follows: (1) Boosting government subsidies could help encourage the socio-economic behaviour of manufacturers adopting the remanufacturing of recycled materials strategy, thirdparty recyclers adopting a high-level processing strategy and cascade utilization enterprises adopting a large-scale cascade utilization strategy. The government subsidy incentives shared by manufacturers to third-party recyclers and cascade utilization enterprises would further impact the strategic selection of supply chain node enterprises. Manufacturers should reasonably establish internal incentive mechanisms in the supply chain under the guidance of government policy support, contribute fully to incentive effectiveness, promote the coordinated development of the supply chain and jointly assume the responsibility of improving the recycling level of industrial resources.
(2) The expected profits of various subjects are essential factors that affect their decision-making and significantly impact the results of system evolution. The government should actively guide and enable the relevant enterprises to establish and improve the power battery recycling system, utilize internet technology to achieve battery data sharing and traceability, vigorously support crucial technology improvements and innovations, expand the market scale to enhance the profitability of relevant enterprises, promote the healthy development of the market, and ensure the effective improvement of cascade utilization level.
(3) Reducing the potential losses of adopting a large-scale cascade utilization strategy by cascade utilization enterprises and a high-level processing strategy by third-party recyclers could improve the probability of both parties making decisions conducive to enhancing the cascade utilization levels. Relevant node enterprises should clarify their respective roles and responsibilities in production, recycling, restructuring, and other aspects. Moreover, they improve the detection system, value evaluation criteria for recycled batteries to reduce risk and losses, and ensure the reliability and safety of cascade utilization products.
(4) Increasing the environmental treatment fees of professional battery dismantling enterprises, effectively executing government functions, forcefully implementing the extended producer responsibility system, reforming relevant laws and regulations, strictly controlling the entire process of power battery production, recycling, and reuse, and urging relevant enterprises to fulfil their primary responsibilities are also effective ways to improve resource recycling efficiency.
In addition, this paper only considers the issue of enterprises' strategy selection for retired power battery cascade utilization in the single-channel supply chain under bounded rationality. It does not consider the effects of online and offline dual-channel recycling and sales, competitive cooperative games, and other situations. The following research direction is to establish a dynamic, collaborative game model considering factors such as consumers and different recycling and sales channels and investigate the effects of multi-channel sales and recycling on the power batteries' cascade utilization. On the other hand, incorporating the government as a participant into the game process and exploring in depth how the government adjusts subsidy policies based on specific market conditions, as well as its impact on the closed-loop supply chain of power batteries are also the key direction for future research.

CONFLICTS OF INTEREST
The authors declare no conflicts of interest regarding the publication of this paper.