Evolutionary Game Analysis of Three-Player for Low-Carbon Production Capacity Sharing

Abstract

In the era of the sharing economy, the rise of production capacity sharing has changed traditional manufacturing modes and broken the balance of original production systems. In addition to that environmental-friendly manufacturing enterprises are of great significance with regard to production capacity sharing and sustainable development of the ecology environment. To investigate the decision-making behaviors of the participants involved in low-carbon production capacity sharing, an evolutionary game model taking into account the platforms, manufacturing enterprises with idle production capacity, and those with demanding production capacity is constructed. Then, both evolutionary game theory and Lyapunov stability theorem are used to analyze the asymptotic stability of the equilibrium and evolutionary stability strategies of the system. Besides, the economic and managerial significance of the evolutionary stability strategy is given. Finally, the influence of low-carbon production capacity of enterprises on the stability of the dynamic system is discussed, such as the cost effect of low-carbon production capacity, the effect of transaction cost, and so on. Results indicate that they can provide theoretical reference for decision-making with respect to the platforms, manufacturing enterprises, and sustainable development of the dynamic system.

1. Introduction

With the development of information technology such as Cloud Computing (CC), Internet of Things (IoT) [1], Artificial Intelligence (AI), Industrial Internet (II), and so on, a new term ‘sharing economy’ has rapidly spread around the world [2]. Its typical characteristics include the sharing of access rights, as well as the integration of massive and decentralized resources. The sharing economy as a new business mode would provide new directions towards the transformation of manufacturing enterprises [3]. The proposal of Industry 4.0 and the development of collaborative production promoted upgrading the manufacturing industry, thus sufficiently utilizing production resources [4,5]. Particularly, production capacity sharing as one of the new directions has received increasing attention in recent years. It is the inevitable result of deepening the integration of production and the internet. It is obvious that it changes traditional methods to allocate resources and breaks the balance and sustainability of the original economic system [6]. Meanwhile, it has significant impacts on the ecological environment [7,8]. Besides, the emergence of production capacity sharing platforms would also change the nodes of the traditional supply chain, improve the utilization of production equipment, and reduce the idle and residual production capacity as much as possible. It brings new opportunities to manufacturing enterprises, but it also poses many challenges to be timely solved in its operations and management. For example, given environmental pressure or corporate social responsibility, the sharing mechanisms with respect to low-carbon production capacity is a challenge, as is how to make their choices in terms of the participants involved in the system of low-carbon production capacity sharing. In this sense, the focus of this paper on the low-carbon production capacity sharing problem is promising, urgent, and significant.

With regard to low-carbon production capacity, there is no unified definition. Given the goals of sustainable development [9,10], production capacity sharing characterized by resource access rights needs to consider the impact of resource allocation mode on the ecological environment. Climate change, global warming and greenhouse gas emission have been more and more concerned by many countries, as well as individuals in recent years [11]. Meanwhile, more and more customers prefer eco-friendly goods and services and they have begun to shift to low-carbon products [12]. More specifically, the manufacturing enterprises participating in production capacity sharing need to choose whether to produce low-carbon products out of the impact on the environment, consumer demand for low-carbon products and its own profits. In other words, production capacity can be divided into two groups based on the properties of products. The capacity to produce low-carbon products is defined as low-carbon production capacity (LCPC). Similarly, the capacity to produce ordinary products (Non-low carbon products) is named as ordinary production capacity (OPC). Such actions aim to respond to the environmental pressure of manufacturing enterprises. In this context, the goals of this paper are to provide an appropriate strategy regarding low-carbon production capacity sharing for the system consisting of sharing platforms, manufacturing enterprises with production capacity supply and those with production capacity demand, so as to achieve the sustainability of the whole system.

In recent years, although the production capacity sharing problem in manufacturing enterprises has received increasing attention, it is still in its early stage. Firstly, most of the existing literature assumes that the manufacturing enterprises participating in production capacity sharing are completely rational [13,14,15,16]. That is, they choose their own strategies for the purpose of maximizing profits. Nevertheless, practical cases indicate the behaviors of these manufacturing enterprises are always the bounded rational, which needs to be studied in depth. Secondly, most of the researchers only concentrate on the economic aspect with regard to the production capacity sharing problem [17,18,19,20]. The achievement concerning production capacity sharing taking environmental sustainability into consideration is relatively limited. In other words, they are keen on investigating OPC sharing problem. Thirdly, the existing literature indicates the researchers pay more attention to the two-player evolutionary game mechanism with respect to production capacity sharing problem. How to investigate three-player evolutionary game problems concerning LCPC sharing is an interesting and promising topic and need to be further studied.

In summary, according to an overview of the production capacity sharing problem, although researchers have made significant progress in this field, the following questions and challenges are still open: (1) what are the low-carbon production capacity? (2) How to formulate and solve the three-player evolutionary game model for LCPC sharing problem? (3) What are the optimal strategies for all participants, such as the platforms, manufacturing enterprises? (4) What is the cost effect of LCPC? What is the effect of transaction cost on the decisions of the platforms?

To address the above-mentioned issues, this paper focuses on the LCPC sharing system, which consists of platforms, manufacturing enterprises with production capacity supply, manufacturing enterprises with production capacity demand. Besides, all participants in this system are assumed to be bounded rational. Evolutionary game theory is adopted to analyze the stability of LCPC sharing system. As the emerging of LCPC sharing not only changes the traditional mode regarding production capacity transaction, but seriously affects the environmental sustainability. The aims of this paper expect to achieve the sustainability of the LCPC sharing system as well as analyze the impacts of LCPC on the environment. In addition, from the perspective of practice, this paper explores the impacts of LCPC on participants’ behaviors in decision-making, the economy, and the ecological environment.

The contributions of this paper include three points. Firstly, both the bounded rationality of participants and the environmental sustainability of manufacturing enterprises are all integrated into the production capacity sharing problem. In addition to that, the design of sharing mechanism regarding LCPC is the main focus of this paper. Secondly, an integrated decision framework regarding the LCPC sharing problem is proposed. Further, this issue is formulated as a three-player evolutionary game model. Additionally, the optimal control theory is also used to solve this model. Specifically, the platforms can provide high-level and low-level service. Manufacturing enterprises need to determine to share LCPC or OPC. Thirdly, this paper aims to not only validate the proposed model and its solution approach, but to underline the importance and urgency of incorporating low-carbon preference into the production capacity sharing problem.

The rest of this paper is organized as follows: The next section describes and analyzes the existing literature which concerns production capacity sharing, low-carbon preference and evolutionary game theory. Section 3 presents the problem definition in detail and designs an integrated decision framework regarding LCPC sharing. Section 4 formulates LCPC sharing problem as a three-player evolutionary game model. Section 5 analyses asymptotic stability of the equilibrium points and evolutionary stability strategies. The cost effect of LCPC and the effect of transaction cost on the strategy choice of platforms are given in Section 6. Finally, managerial insights and future directions are provided.

2. Literature Review

In recent years, production capacity sharing as one of the critical issues in manufacturing enterprises has garnered increasing attention. This paper contributes to the literature on production capacity sharing problem in manufacturing enterprises, low-carbon preference issue in supply chain management, as well as evolutionary game theory and its applications.

2.1. Literature on Production Capacity Sharing Problem in Manufacturing Enterprises

The sharing economy creates new ways for individuals to acquire income from their excess capacity regarding goods and services [2,21], changed the nature of product and service accesses, and redefined the concept of ownership and employment [3]. Proponents of the sharing economy extol its positive impacts on environment, economy, and society [22]. Nevertheless, the sharing economy has also invoked a series of debates about whether this disruptive form of economy will negatively affect the equitable and sustainable development of the industry [23,24]. Some major concerns of the sharing economy include the unfair competition between traditional businesses (e.g., hotels and taxis) and the sharing economy companies (e.g., Airbnb and Uber).

Production capacity sharing is defined as the usage right of production resources. It is the inevitable results of the sharing economy incorporated into the field of production. Such a topic is also becoming more common in manufacturing sectors or enterprises. Competitive enterprises can establish joint ventures in production facilities and trade their capacity according to mutually agreed upon contract terms. For instance, Fiat and PSA equally share their production capacity under the alliance Sevelnord and have the option to trade each other’s capacity when they sell the vehicle through their respective brands and distribution networks [17]. DuPont and Honeywell jointly operate a world-scale factory to manufacture automotive refrigerants that they market and sell separately [18]. Other examples include the joint production of electric cars between BMW and PSA Peugeot Citroen [19]. Competitors may share the existing capacity without formal joint venture as well. For example, Mazda’s New Mexican assembly plant is planning to produce not only its automobile models but also a subcompact vehicle for Toyota from 2015 [20].

In recent years, production capacity sharing is an emerging topic. Although this issue is significantly important, it is still in its infancy stage. Fortunately, some researchers have started to follow this issue. For instance, Van Mieghem [13] considered capacity transfer problem after demand realization between a manufacturer and a subcontractor that operate in distinct markets. Meanwhile, they found that their ex-ante capacity investments could be coordinated only by state-dependent contracts. Wu et al. [14,15] concentrated on the optimal contracts design when a subcontractor could share capacity with a manufacturer that sells to the end market. Yu et al. [16] investigated the conditions under which the cooperation of capacity sharing in queueing systems could be beneficial for a set of independent firms. Park et al. [25] addressed the value of inventory sharing, as well as developed a structured trans-shipment price scheme that uses a linear combination of the spot and forward prices. Melo et al. [26] contended that capacity sharing could reduce emissions and mitigate the impacts of the transport sector on climate change. Karmarkar and Roels [27] used the gap quality model to manage the co-productive process. Guo and Wu [28] highlighted the optimal strategies and firm profitability with regard to capacity sharing between competing firms under both ex-ante and ex-post contracting.

Overall, although many scholars pay more attention to managerial problems in the context of sharing economy, production capacity sharing problems are still in the early stages. In addition, the aforementioned literature does not classify production capacity based on some specific criteria. That is, they regarded production capacity as the same. However, different types of production capacity may have different impacts on the whole system.

2.2. Literature on Low-Carbon Preference in Supply Chain Management

Climate change, global warming, and greenhouse gas emission have been more and more concerned by many countries as well as individuals in recent years [9]. In response to the concerns, more and more countries attempt to impose some regulations, such as cap-and-trade scheme and carbon tax policy [29]. In the meanwhile, consumers have begun to shift to low-carbon products. Therefore, they are willing to pay some premiums for those products [30]. Hartikainen et al. [31] indicated that a carbon footprint usually affected the consumers' buying decisions. Zhang et al. [32] claimed that the retailer's profit would increase when the manufacturer's profit was convex with low carbon preference. The aforementioned literatures demonstrated that consumers' purchase decisions would usually be influenced by their low-carbon preference. In this sense, it provides the possibility for the low-carbon manufacturer to charge price premiums for their low-carbon products.

Due to the increase in environmental awareness and stringent government regulations, companies simultaneously start to produce consumer goods with low-carbon emissions. Motivated by the regulations and customers' low-carbon preference, many manufacturers also come to invest in producing low-carbon products [33]. For instance, Du et al. [34] focused on the manufacturer's multi-product joint pricing and production problem on the prerequisite of consumers having low-carbon awareness. Results indicated that the consumers' low-carbon preference would affect business decision making. Gong and Zhou [35] proposed a dynamic production model with green technology consideration. They derived the optimal production strategy in a low-carbon supply chain. Benjaafar et al. [36] incorporated carbon emission concerns into operational decision-making for production and inventory management. Xu et al. [37] studied the joint production and pricing problem with carbon emission regulations consideration in manufacturing enterprises. Yang et al. [38] demonstrated that the vertical cooperation between two competitive supply chains would be beneficial for the reduction of carbon emission and the decrease of retail prices. Peng et al. [39] designed the quantity discount contract and the revenue sharing contract taking into account carbon emission carbon-trade scheme in a supply chain consisting of a supplier and a manufacturer. Hong et al. [40] employed dynamic programming to evaluate the trade-off between the total costs and the total carbon emission occurred during the extraction of the material, the production operations and assembly, and the required transportation. Zhang et al. [41] analyzed a monopoly manufacturer’s selection decision with the manufacturing and marketing model considerations under the mixed carbon policy. Li et al. [42] explored the impact of revenue-sharing and cost-sharing contracts offered by a retailer on emission reduction efforts and firms' profitability. Wang and Zheng [43] focused on a low-carbon diffusion problem from the perspective of network characteristics and consumers' environmental awareness. In view of these factors mentioned above, manufacturers may be incentivized to invest in sustainable or low-carbon technology to reduce carbon emissions, and use cleaner energy and eco-friendly materials [44]. Furthermore, Cao et al. [45] and Cao et al. [46] clarified that environmental sustainability (e.g., carbon emissions) should be integrated into the decisions (e.g., emergency organization allocation, relief distribution) regarding sustainable humanitarian supply chains.

In summary, there are plenty of achievements with respect to the integration of low-carbon preference and supply chain management. However, the integrated issue regarding production capacity sharing and environmental concerns is relatively limited. To fill this gap, this paper links production capacity sharing issue with environmental sustainability. Besides, production capacity sharing is divided into LCPC and OPC based on environmental concerns. Particularly, most of the existing literatures concentrate on OPC instead of the LCPC sharing problem. To deal with this issue, this paper aims to analyze the decision-making behavior of the participants of production capacity sharing, thus designing the mechanisms with respect to the LCPC sharing problem.

2.3. Literature on Evolutionary Game Theory and Its Applications

Classical game theory assumes that the players are completely rational [47,48]. However, it is difficult to capture the real characteristics of the players in practice. To deal with this issue, evolutionary game theory is proposed. It supposes that the players with bounded rationality have the ability to keep learning, adapt to the market environment and adjust their strategies [49]. It can provide effective decision-making guidelines for game players. In recent years, evolutionary game theory is widely used in the field of economy, management, and so on. In this context, evolutionary game theory is very suitable for modeling LCPC sharing problem. Particularly, evolutionary game theory can be used to capture the bounded rationality of manufacturing enterprises, platforms. Thus, it is widely used in the field of commercial and humanitarian supply chain management. A lot of relevant achievements are obtained.

For instance, Barari et al. [50] established an evolutionary game model between producers and retailers to analyze their strategies to incentivize sustainable practices and to maximize economic profits. Ji et al. [51] developed an evolutionary game model to observe the long-term tendency for multiple stakeholders to cooperate in green purchasing. Zhao et al. [52] proposed an evolutionary game model to investigate the responses of enterprises to a carbon reduction labeling scheme. Tian et al. [53] utilized evolutionary game theory and system dynamics to explore the diffusion of sustainable supply chain management ideas between manufacturers. Wu et al. [54] contended that evolutionary game theory is an important method to investigate the dynamic impact of government incentive policies to reduce carbon emissions on the strategies and behaviors of enterprises. Aaron et al. [55] proposed a methodological toolkit comprised of evolutionary game theory and agent-based modeling to study OPEC and the Seven Sisters as they struggled for control over global petroleum markets. Babu and Mohan [56] evaluated the sustainability in supply chains by using evolutionary game theory. Zhang et al. [57] built an evolutionary game model for technology diffusion problems between enterprises in the context of a complex network. Li et al. [58] investigated the acceptance and performance of the recommendation incentive mechanism using evolutionary game theory. Mahmoudi and Rasti-Barzoki [59] designed the contrast between government and producer under different scenarios by using a two-population evolutionary game theory approach. Zhang et al. [60] used evolutionary game theory to analyze the impacts of government policy on the decisions of manufacturers and the dynamic tendency of cap-and-trade market. Chen and Hu [61] applied evolutionary game theory to examine the behavioral strategies of the manufacturers in response to various combinations of carbon taxes and subsidies. Tong et al. [62] established a retailer-led supply chain with the concern of low-carbon products under a cap-and-trade policy. In addition, they formulated this issue as an evolutionary game model to examine the evolution of behaviors of the powerful retailers and manufacturers.

According to the above-mentioned literature, evolutionary game theory provides an effective way to manage the relationships among multiple players by comparing with classical game theory. Most of the researchers concentrate on two game groups or two players. They only answer the question regarding whether to coordinate for players. Nevertheless, practical cases usually involve more than two participants. Regardless of its complexity, it has more meaning. In this context, this paper formulates LCPC sharing problem as a three-player evolutionary game model including platforms, manufacturing enterprises with production capacity supply and demand to design the corresponding sharing mechanisms.

3. Decision Framework Regarding Low-Carbon Production Capacity Sharing

3.1. Problem Definition

Production capacity sharing is the organic incorporation of production and internet technology, and it is characterized by access right. It integrates and configures scattered manufacturing resources. It shares this capacity to every link of the productive process. In this sense, the characteristics of production capacity sharing include multi-party, multi-element and multi-link, and so on. Specifically, manufacturing enterprises can realize technological collaboration during the process of production by joining the production capacity sharing platforms (denoted by “platforms”). Thus, these enterprises with idle production capacity have two choices: ①no actions; ②share or sell the idle production capacity through the platforms. Particularly, idle production capacity refers to the production capacity owned by manufacturing enterprises is not fully utilized. With respect to the manufacturing enterprises with production capacity demand (denoted by “D enterprises”), they can not only purchase the production capacity from traditional market, but obtain the access rights of production capacity through the platforms, so as to meet their demands. If manufacturing enterprises do not join the sharing platforms, the sustainability of the whole sharing system will not be achieved well. Furthermore, they can also hand over the production links that they are not good at to other enterprises. In terms of the manufacturing enterprises with production capacity supply (denoted by “S enterprises”), as S enterprises can reduce production costs, technical risks, opportunistic costs of idle capacity, and obtain profits, they are willing to participate in the production capacity sharing system through the platforms. With regard to D enterprises, they can either purchase production capacity through traditional channels or obtain the right to use it through the platforms. In this paper, it is assumed that the cost of obtaining the right to use production capacity through the platforms is lower than that of purchasing it through traditional channels. Therefore, D enterprises are willing to obtain the right to use production capacity through the platforms so as to meet their demands on production capacity. In this context, the process of production capacity sharing is depicted in Figure 1.

The platforms provide two kinds of services to the manufacturing enterprises participating in the system. It includes high-level and lower-level service. High-level service results in higher operating costs, but the increase of the matching rate with the concern of production capacity sharing would attract more manufacturing enterprises to join the platforms. With regard to the platforms, they charge transaction costs to enterprises that have successfully shared production capacity, thus guaranteeing that the platforms normally operate. It is worth noticing that the platforms have the characteristics of a typical bilateral market. Its revenue is affected by the number of the participated manufacturing enterprises.

To undertake the social responsibility regarding the environment as well as meet the demand of downstream consumers on low-carbon products, S enterprises can supply two types of production capacity. As mentioned in Section 1, one is low-carbon production capacity (LCPC). The other is ordinary production capacity (OPC). In this paper, D enterprises also need LCPC and OPC. Manufacturing enterprises participating in production capacity sharing can not only benefit from their activities, but also be affected by direct network externalities.

In order to analyze the stability of production capacity sharing system consisting of the platforms, S enterprises, and D enterprises, two kinds of services are provided to the manufacturing enterprises participating in production capacity sharing on the platforms. In addition to that, both S enterprises and D enterprises have two strategy choices, namely LCPC and OPC. In this sense, the above-mentioned parties form a typical evolutionary game problem.

3.2. An Integrated Decision Framework

According to the above analysis, a three-player evolutionary game decision framework regarding production capacity sharing is constructed. It is depicted in Figure 2. Additionally, a set of potential strategies of all participants are also presented in Figure 2.

In this paper, the players in the evolutionary game model include the platforms, S enterprises and D enterprises. All three players are bounded rational. Production capacity sharing is carried out through the platforms, which provides high-level and low-level services to both S and D enterprises. At the beginning, the platforms provide high-level service with probability $p_p$ and low-level service with probability ${\overline{p}}_P=1-p_P$. Furthermore, there are two strategies for S enterprises and D enterprises: LCPC and OPC. In terms of S enterprises and D enterprises, the probability of the strategy ‘LCPC’ for the initial state is $p_S$ and $p_D$, respectively. It is obvious that the probability of OPC for the initial state is ${\overline{p}}_S=1-p_S$ and ${\overline{p}}_D=1-p_D$, respectively.

4. Three-Player Evolutionary Game Model for Low-carbon Production Capacity Sharing

4.1. Assumptions

Assumption 1.

The set of strategies of the platforms is S={high-level service, low-level service}. The platforms encourage more manufacturing enterprises to join the system by making marketing efforts such as the increasing number of the participated enterprises, successfully matching the rate during the process of production capacity sharing. In this sense, the platforms can not only make the sharing business more rational and normative, but also increase the successfully matching rate of manufacturing enterprise. In terms of the platforms, the level of service provided by them has a positive correlation with its operating costs. In other words, the higher-level service would lead to the higher operating costs. The lower-level service would bring the lower operating costs. Hence, not all platforms are willing to adopt a high-level service strategy. Let${\mu }_S$and${\mu }_D$denote the direct network externality coefficient produced by the platforms to S and D enterprises, respectively. When the platforms choose low-level service, the number of S and D manufacturing enterprises participating in production capacity sharing is denoted by$n_S$and$n_D$, respectively. The operating cost is denoted by$c_P^L$. When the platforms provide high-level service,$k_S$and$k_D$$\left(1\le k_S,1\le k_D\right)$represent the influencing factors of the S and D enterprises, and $c_P^H$ $\left(0<c_P^L<c_P^H\right)$ is the operating cost. In addition to that, the platforms charge transaction cost to S and D enterprises, which can be written as $t_S$ and $t_D$ for each successful matching, respectively.

Assumption 2.

The set of strategies of the two types of manufacturing enterprises are S={LCPC, OPC}. Here, the term “LCPC” and “OPC” of S enterprises and D enterprises refer to low-carbon and ordinary production capacity to be owned and required, respectively. With regard to S and D enterprises, the respective cost of producing low-carbon products can be denoted by$c_S\left(\tau \right)$and$c_D\left(\tau \right)$. It is related to the rate of emission reduction$\left(\tau \right)$of low-carbon products. Besides, the cost is the monotone increase function of the rate of emission. Compared with the case that the platforms provide low-level service, the income from high-level service in S and D enterprises is relatively higher. That is, the formula$R_i^L\left(\tau \right)<R_i^H\left(\tau \right)\left(i=S,D\right)$is true. The cost of producing ordinary products is denoted by$c_S$and$c_D$.

Assumption 3.

When S enterprises are unwilling to share their production capacity, the benefits from the low-carbon products and the ordinary products are denoted by$R_{S1}$and$R_{S2}$. When D enterprises are unwilling to share their production capacity, the benefits from the low-carbon products and the ordinary products are denoted as$R_{D1}$and$R_{D2}$. The sharing of production capacity improves the utilization rate of production capacity, reduces the production costs, and enhances the market competitiveness of the products. The extra income of S and D enterprises is denoted by$R_S^M$and$R_D^M$, respectively. Furthermore, when the platforms provide low-level and high-level service, the probability of successful matching is denoted$m_L$and$m_H$$\left(0\le m_L<m_H\le 1\right)$. In this context, let$m_L{R}_S^M$and$m_L{R}_D^M$respectively represent the additional benefits obtained by S and D enterprises, while the platforms provide the low-level service. Similarly, the additional benefits obtained by S and D enterprises under the high-level service are denoted by$m_H{R}_S^M$and$m_H{R}_D^M$.

Assumption 4.

When the platforms offer the high-level service, both S enterprises and D enterprises would gain more benefit from sharing low-carbon capacity than ordinary capacity. On the other hand, when the platforms provide low-level service, both S enterprises and D enterprises would gain less benefit from sharing low-carbon capacity than ordinary capacity. In other words, it follows$R_{i1}-R_{i2}+R_i^L\left(\tau \right)<c_i\left(\tau \right)-c_i<R_{i1}-R_{i2}+R_i^H\left(\tau \right)\left(i=S,D\right)$.

4.2. Return Matrix and Replicator Dynamic Equation

According to the aforementioned analysis, the revenue of the platforms is affected by transaction cost, service level and operating cost. On the other hand, the income of S and D enterprises is affected by the revenue of product sales, additional revenue, direct network externalities, transaction costs, production costs and low-carbon products. Accordingly, the return matrix of three-game groups in this model can be obtained, as shown in

Table 1. Return matrix.

Strategies		S Enterprises LCPC		S Enterprises OPC
		D Enterprises LCPC	D Enterprises OPC	D Enterprises LCPC	D Enterprises OPC
Platforms	High-level service	$\left[\begin{array}{c}{R}_P^H-c_P^H\\ R_{S1}^H+R_S^H\left(\tau \right)-c_S\left(\tau \right)\\ R_{D1}^H+R_D^H\left(\tau \right)-c_D\left(\tau \right)\end{array}\right]$	$\left[\begin{array}{c}{R}_P^H-c_P^H\\ R_{S1}^H+R_S^H\left(\tau \right)-c_S\left(\tau \right)\\ R_{D2}^H-c_D\end{array}\right]$	$\left[\begin{array}{c}{R}_P^H-c_P^H\\ R_{S2}^H-c_S\\ R_{D1}^H+R_D^H\left(\tau \right)-c_D\left(\tau \right)\end{array}\right]$	$\left[\begin{array}{c}{R}_P^H-c_P^H\\ R_{S2}^H-c_S\\ R_{D2}^H-c_D\end{array}\right]$
	low-level service	$\left[\begin{array}{c}{R}_P^L-c_P^H\\ R_{S1}^L+R_S^L\left(\tau \right)-c_S\left(\tau \right)\\ R_{D1}^L+R_D^L\left(\tau \right)-c_D\left(\tau \right)\end{array}\right]$	$\left[\begin{array}{c}{R}_P^L-c_P^H\\ R_{S1}^L+R_S^L\left(\tau \right)-c_S\left(\tau \right)\\ R_{D2}^L-c_D\end{array}\right]$	$\left[\begin{array}{c}{R}_P^L-c_P^H\\ R_{S2}^L-c_S\\ R_{D1}^L+R_D^L\left(\tau \right)-c_D\left(\tau \right)\end{array}\right]$	$\left[\begin{array}{c}{R}_P^L-c_P^L\\ R_{S2}^L-c_S\\ R_{D2}^L-c_D\end{array}\right]$

. For convenience, this paper assumes that $R_P^H=k_S{m}_S^H{n}_S{t}_S+k_D{m}_D^H{n}_D{t}_D$, $R_P^L=m_S^L{n}_S{t}_S+m_D^L{n}_D{t}_D$, $R_{ij}^H=R_{ij}+k_i{\mu }_i{n}_i+m_i^H\left(R_i^M-t_i\right)$, $R_{ij}^L=R_{ij}+{\mu }_i{n}_i+m_i^L\left(R_i^M-t_i\right)\left(i=S,D;j=1,2\right)$.

According to the return matrix presented in Table 1, the fitness of the platforms that adopt high-level service and low-level service are denoted by Formulas (1) and (2), respectively.

$$f\left(p_P\right)=R_P^H-c_P^H,$$

(1)

$$f\left({\overline{p}}_P\right)=R_P^L-c_P^L,$$

(2)

The fitness of S enterprises that adopt LCPC strategy and OPC strategy can be written as:

$$f\left(p_S\right)=R_{S1}^L+R_S^L\left(\tau \right)-c_S\left(\tau \right)+p_p\left[R_{S1}^H+R_S^H\left(\tau \right)-R_{S1}^L-R_S^L\left(\tau \right)\right],$$

(3)

$$f\left({\overline{p}}_S\right)=R_{S2}^L-c_S+p_p\left(R_{S2}^H-R_{S2}^L\right),$$

(4)

The fitness of D enterprises that adopt LCPC strategy and OPC strategy can be denoted by:

$$f\left(p_D\right)=R_{D1}^L+R_D^L\left(\tau \right)-c_D\left(\tau \right)+p_p\left[R_{D1}^H+R_D^H\left(\tau \right)-R_{D1}^L-R_D^L\left(\tau \right)\right],$$

(5)

$$f\left({\overline{p}}_D\right)=R_{D2}^L-c_D+p_p\left(R_{D2}^H-R_{D2}^L\right),$$

(6)

In fact, the duplicator dynamics equation is a dynamic differential equation to be used to describe the frequency or the frequency of such strategies in a population [47]. According to the dynamic equation given by Selten [63], the frequency of change of the strategy is equal to its fitness, so the differential equations regarding the dynamic process of gene replication in the evolutionary game model including platforms, S enterprises and D enterprises can be denoted by:

$$\{\begin{array}{l}F\left(p_P\right)=p_P\left(1-p_P\right)\left(R_P^H-R_P^L-c_P^H+c_P^L\right)\\ F\left(p_S\right)=p_S\left(1-p_S\right)\{R_{S1}^L+R_S^L\left(\tau \right)-c_S\left(\tau \right)-R_{S2}^L+c_S\\ +p_p\left[R_{S1}^H+R_S^H\left(\tau \right)-R_{S1}^L-R_S^L\left(\tau \right)-R_{S2}^H+R_{S2}^L\right]\}\\ F\left(p_D\right)=p_D\left(1-p_D\right)\{R_{D1}^L+R_D^L\left(\tau \right)-c_D\left(\tau \right)-R_{D2}^L+c_D\\ +p_p\left[R_{D1}^H+R_D^H\left(\tau \right)-R_{D1}^L-R_D^L\left(\tau \right)-R_{D2}^H+R_{D2}^L\right]\}\end{array},$$

(7)

5. Asymptotic Stability of the Equilibrium Points and Evolutionary Stability Strategies

5.1. Asymptotic Stability Analysis Concerning the Equilibrium Points

To analyze the asymptotic stability of the equilibrium points, the following Lemma is given.

Lemma 1.

With respect to the asymmetric games, the equilibrium E must be a strict Nash equilibrium if the state of evolutionary game equilibrium E is asymptotically stable. In addition, the strict Nash equilibrium is the pure strategic equilibrium, thus the equilibrium E is also the pure strategic equilibrium.

According to Lemma 1, to analyze the asymptotic stability of the equilibrium points, this Subsection only needs to discuss the asymptotic stability of the equilibrium points concerning pure strategy in the replicator dynamics equation. In this context, in terms of replicator dynamics Equation (7), the equilibrium points of the pure strategy include $E_1\left(0,0,0\right)$, $E_2\left(0,0,1\right)$, $E_3\left(1,0,0\right)$, $E_4\left(1,0,1\right)$, $E_5\left(0,1,0\right)$, $E_6\left(0,1,1\right)$, $E_7\left(1,1,0\right)$, and $E_8\left(1,1,1\right)$.

According to Lyapunov stability theorem [64], when all eigenvalues ($\lambda$) of the Jacobian matrix satisfy $\lambda <0$, the equilibrium point is the asymptotically stable. That is the confluence. When all eigenvalues of the Jacobian matrix follow $\lambda >0$, the equilibrium point is unstable. That is the source. Furthermore, with regard to Jacobian matrix, if the eigenvalues ($\lambda$) are mixed. In other words, some are positive, and others are negative. In this context, the equilibrium point, also called the saddle point, is unstable.

In order to analyze the asymptotic stability of the equilibrium points, the Jacobian matrix and its eigenvalue are firstly calculated.

The Jacobian matrix is presented by Formula (8).

$$J=\left[\begin{array}{ccc}\frac{\partial F\left(p_P\right)}{\partial p_P}& \frac{\partial F\left(p_P\right)}{\partial p_S}& \frac{\partial F\left(p_P\right)}{\partial p_D}\\ \frac{\partial F\left(p_S\right)}{\partial p_P}& \frac{\partial F\left(p_S\right)}{\partial p_S}& \frac{\partial F\left(p_S\right)}{\partial p_D}\\ \frac{\partial F\left(p_D\right)}{\partial p_P}& \frac{\partial F\left(p_D\right)}{\partial p_S}& \frac{\partial F\left(p_D\right)}{\partial p_D}\end{array}\right],$$

(8)

Thus, the following Formulas (9)–(17) are obtained.

$$\frac{\partial F\left(p_P\right)}{\partial p_P}=\left(1-2p_P\right)\left(R_P^H-R_P^L-c_P^H+c_P^L\right),$$

(9)

$$\frac{\partial F\left(p_P\right)}{\partial p_S}=0,$$

(10)

$$\frac{\partial F\left(p_P\right)}{\partial p_D}=0,$$

(11)

$$\frac{\partial F\left(p_S\right)}{\partial p_P}=p_S\left(1-p_S\right)\left[R_{S1}^H+R_S^H\left(\tau \right)-R_{S1}^L-R_S^L\left(\tau \right)-R_{S2}^H+R_{S2}^L\right],$$

(12)

$$\begin{array}{l}\frac{\partial F\left(p_S\right)}{\partial p_S}=\left(1-2p_S\right)\{R_{S1}^L+R_S^L\left(\tau \right)-c_S\left(\tau \right)-R_{S2}^L+c_S\\ +p_p\left[R_{S1}^H+R_S^H\left(\tau \right)-R_{S1}^L-R_S^L\left(\tau \right)-R_{S2}^H+R_{S2}^L\right]\}\end{array},$$

(13)

$$\frac{\partial F\left(p_S\right)}{\partial p_D}=0,$$

(14)

$$\frac{\partial F\left(p_D\right)}{\partial p_P}=p_D\left(1-p_D\right)\left[R_{D1}^H+R_D^H\left(\tau \right)-R_{D1}^L-R_D^L\left(\tau \right)-R_{D2}^H+R_{D2}^L\right],$$

(15)

$$\frac{\partial F\left(p_D\right)}{\partial p_A}=0,$$

(16)

$$\begin{array}{l}\frac{\partial F\left(p_D\right)}{\partial p_D}=\left(1-2p_D\right)\{R_{D1}^L+R_D^L\left(\tau \right)-c_D\left(\tau \right)-R_{D2}^L+c_D\\ +p_p\left[R_{D1}^H+R_D^H\left(\tau \right)-R_{D1}^L-R_D^L\left(\tau \right)-R_{D2}^H+R_{D2}^L\right]\}\end{array},$$

(17)

According to Lyapunov stability theorem, the stability of eight equilibrium points regarding the pure strategy is analyzed. The corresponding results are depicted in

Table 2. The stability of the equilibrium points regarding the pure strategy.

Equilibrium	Eigenvalue	Stability
$E_1\left(0,0,0\right)$	${\lambda }_1=R_P^H-R_P^L-c_P^H+c_P^L$	If $R_P^H-R_P^L<c_P^H-c_P^L$, $E_1\left(0,0,0\right)$ is the asymptotically stable point (confluence). Otherwise, $E_1\left(0,0,0\right)$ is saddle point.
	${\lambda }_2=R_{S1}^L+R_S^L\left(\tau \right)-c_S\left(\tau \right)-R_{S2}^L+c_S<0$
	${\lambda }_3=R_{D1}^L+R_D^L\left(\tau \right)-c_D\left(\tau \right)-R_{D2}^L+c_D<0$
$E_2\left(0,0,1\right)$	${\lambda }_1=R_P^H-R_P^L-c_P^H+c_P^L$	Saddle point
	${\lambda }_2=R_{S1}^L+R_S^L\left(\tau \right)-c_S\left(\tau \right)-R_{S2}^L+c_S<0$
	${\lambda }_3=-\left[R_{D1}^L+R_D^L\left(\tau \right)-c_D\left(\tau \right)-R_{D2}^L+c_D\right]>0$
$E_3\left(1,0,0\right)$	${\lambda }_1=-\left(R_P^H-R_P^L-c_P^H+c_P^L\right)$	If $R_P^H-R_P^L<c_P^H-c_P^L$, $E_3\left(1,0,0\right)$ is the unstable point (source). Otherwise, $E_3\left(1,0,0\right)$ is the saddle point.
	${\lambda }_2=R_{S1}^H+R_S^H\left(\tau \right)-R_{S2}^H-c_S\left(\tau \right)+c_S>0$
	${\lambda }_3=R_{D1}^H+R_D^H\left(\tau \right)-R_{D2}^H-c_D\left(\tau \right)+c_D>0$
$E_4\left(1,0,1\right)$	${\lambda }_1=-\left(R_P^H-R_P^L-c_P^H+c_P^L\right)$	Saddle point
	${\lambda }_2=R_{S1}^H+R_S^H\left(\tau \right)-R_{S2}^H-c_S\left(\tau \right)+c_S>0$
	${\lambda }_3=-\left[R_{D1}^H+R_D^H\left(\tau \right)-R_{D2}^H-c_D\left(\tau \right)+c_D\right]<0$
$E_5\left(0,1,0\right)$	${\lambda }_1=R_P^H-R_P^L-c_P^H+c_P^L$	Saddle point
	${\lambda }_2=-\left[R_{S1}^L+R_S^L\left(\tau \right)-c_S\left(\tau \right)-R_{S2}^L+c_S\right]>0$
	${\lambda }_3=R_{D1}^L+R_D^L\left(\tau \right)-c_D\left(\tau \right)-R_{D2}^L+c_D<0$
$E_6\left(0,1,1\right)$	${\lambda }_1=R_P^H-R_P^L-c_P^H+c_P^L$	If $R_P^H-R_P^L<c_P^H-c_P^L$, $E_6\left(0,1,1\right)$ is the saddle point. Otherwise, $E_6\left(0,1,1\right)$ is the unstable point (source).
	${\lambda }_2=-\left[R_{S1}^L+R_S^L\left(\tau \right)-c_S\left(\tau \right)-R_{S2}^L+c_S\right]>0$
	${\lambda }_3=-\left[R_{D1}^L+R_D^L\left(\tau \right)-c_D\left(\tau \right)-R_{D2}^L+c_D\right]>0$
$E_7\left(1,1,0\right)$	${\lambda }_1=-\left(R_P^H-R_P^L-c_P^H+c_P^L\right)$	Saddle point
	${\lambda }_2=-\left[R_{S1}^H+R_S^H\left(\tau \right)-R_{S2}^H-c_S\left(\tau \right)+c_S\right]<0$
	${\lambda }_3=R_{D1}^H+R_D^H\left(\tau \right)-R_{D2}^H-c_D\left(\tau \right)+c_D>0$
$E_8\left(1,1,1\right)$	${\lambda }_1=-\left(R_P^H-R_P^L-c_P^H+c_P^L\right)$	If $R_P^H-R_P^L<c_P^H-c_P^L$, $E_8\left(1,1,1\right)$ is the saddle point. Otherwise, $E_8\left(1,1,1\right)$ is the asymptotically stable point (confluence).
	${\lambda }_2=-\left[R_{S1}^H+R_S^H\left(\tau \right)-R_{S2}^H-c_S\left(\tau \right)+c_S\right]<0$
	${\lambda }_3=-\left[R_{D1}^H+R_D^H\left(\tau \right)-R_{D2}^H-c_D\left(\tau \right)+c_D\right]<0$

.

The following Theorem 1 can be obtained based on the results presented in Table 2 and the Lyapunov stability theorem.

Theorem 1.

When$R_P^H-R_P^L<c_P^H-c_P^L$, the equilibrium point$E_1\left(0,0,0\right)$is asymptotically stable. Otherwise, the equilibrium point$E_8\left(1,1,1\right)$is asymptotically stable.

To clearly describe the results, the corresponding phase diagrams are shown in Figure 3 and Figure 4, respectively.

The Jacobian matrix of the equilibrium points indicates that the velocity of $p_P,p_S,p_D$ deviated from the corresponding equilibrium state. The eigenvalue of the Jacobian matrix represents the scale transformation of the feature vector. If the real part of the eigenvalue is greater than zero, it indicates that the velocity deviated from the equilibrium point is positive. On the contrary, if it is less than zero, it indicates that the velocity deviated from the equilibrium point is negative. That is, it is stable. In this context, according to the results in Table 2, Figure 3 and Figure 4, the following conclusions can be summarized. When $R_P^H-R_P^L<c_P^H-c_P^L$, the three eigenvalues of the equilibrium point $E_1\left(0,0,0\right)$ are all less than zero. It indicates that the equilibrium point $E_1\left(0,0,0\right)$ is asymptotically stable (confluence). With regard to the equilibrium point $E_2\left(0,0,1\right)$, its eigenvalues hold that ${\lambda }_1,{\lambda }_2<0$, and ${\lambda }_3>0$, which is mixed. Such case demonstrates that the equilibrium point $E_2\left(0,0,1\right)$ is unstable and the saddle point. In terms of the equilibrium point $E_3\left(1,0,0\right)$, the corresponding three eigenvalues are all more than zero, which shows that this point is unstable (source). Other equilibrium points can be analyzed by a similar approach. On the other hand, when $R_P^H-R_P^L>c_P^H-c_P^L$, the equilibrium point $E_8\left(1,1,1\right)$ is asymptotically stable (confluence), and $E_6\left(0,1,1\right)$ is an unstable point (source).

5.2. Evolutionary Stability Strategies regarding Low-Carbon Production Capacity

To analyze the evolutionary stability strategies of this three-player evolutionary game model, the following theorem is firstly presented.

Theorem 2.

In the dynamic replication system of multi-player evolutionary game, the evolutionary stability strategy (ESS) is asymptotically stable, and the asymptotically stable state must be the evolutionary stability strategy (ESS).

Proof. 

X is the evolutionary stability strategy (ESS) of multi-player evolutionary game if and only if X is strictly a Nash equilibrium. If and only if X is strictly Nash equilibrium, the strategy combination X is asymptotically stable in the dynamic replication system of multi-player evolutionary game. Thus, Theorem 2 is proved.□

In this sense, the evolutionary stability strategies of this paper can be obtained based on Theorems 1 and 2.

Theorem 3.

When$R_P^H-R_P^L<c_P^H-c_P^L$, the evolutionary stability strategy of the whole system is (low-level service, OPC, OPC). When$R_P^H-R_P^L>c_P^H-c_P^L$, the evolutionary stability strategy of the whole system is (high-level service, LCPC, LCPC).

The above-mentioned theorem demonstrates that no matter which strategy the platforms adopts, the combination of the strategies including (OPC, OPC) and (LCPC, LCPC) will exist in the whole system for a long time in terms of S enterprises and D enterprises. In addition, this theorem indicates when $R_P^H-R_P^L<c_P^H-c_P^L$, the LCPC sharing system is sustainable. Meanwhile, manufacturing enterprises do not consider the impacts of their own behaviors on the ecological environment during the production process. Another conclusion is that when $R_P^H-R_P^L>c_P^H-c_P^L$, the LCPC sharing system is sustainable, and the manufacturing enterprise is environmental-friendly.

6. Parameter Analysis

According to the established three-player evolutionary game model, it is obvious that the evolutionary stability strategy of the whole system is affected by the parameters of the return matrix. In this context, this section aims to examine the impacts of parameters on the evolutionary stability strategy. Particularly, the main impacts result from the following constraint conditions: $c_i\left(\tau \right)<R_{i1}-R_{i2}+R_i^L\left(\tau \right)+c_i$ and $R_{i1}-R_{i2}+R_i^H\left(\tau \right)+c_i<c_i\left(\tau \right)\left(i=S,D\right)$. That is, the evolutionary stability strategy (ESS) would change with the value of these constraints.

6.1. Cost Effect of Low-Carbon Production Capacity

In this paper, Theorem 3 is obtained if $R_{i1}-R_{i2}+R_i^L\left(\tau \right)+c_i<c_i\left(\tau \right)<R_{i1}-R_{i2}+R_i^H\left(\tau \right)+c_i\left(i=S,D\right)$. In this Subsection, the evolutionary stability strategy of the whole system is analyzed in the context of different scenarios where the cost of low-carbon production capacity is too low or too high.

(1) Taking into consideration the cost of low-carbon production capacity is too low, that is $c_i\left(\tau \right)<R_{i1}-R_{i2}+R_i^L\left(\tau \right)+c_i\left(i=S,D\right)$, the stability analysis of the equilibrium points is shown in

Table 3. Stability analysis of the equilibrium points when $c_i\left(\tau \right)<R_{i1}-R_{i2}+R_i^L\left(\tau \right)+c_i\left(i=S,D\right)$.

Equilibrium Points		${\mathit{E}}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{3}}\mathbf{\left(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{4}}\mathbf{\left(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{5}}\mathbf{\left(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{6}}\mathbf{\left(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{7}}\mathbf{\left(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{8}}\mathbf{\left(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{\right)}$
Eigenvalues	${\lambda }_1$	--	--	--	--	--	--	--	--
	${\lambda }_2$	$>0$	$>0$	$>0$	$>0$	$<0$	$<0$	$<0$	$<0$
	${\lambda }_3$	$>0$	$<0$	$>0$	$<0$	$>0$	$<0$	$>0$	$<0$
Stability		$E_1\left(0,0,0\right)$ is the saddle point when $R_P^H-R_P^L$ $<c_P^H-c_P^L$. Otherwise, $E_1\left(0,0,0\right)$ is the unstable point (source)	Saddle point	$E_3\left(1,0,0\right)$ is unstable point (source) when $R_P^H-R_P^L$ $<c_P^H-c_P^L$. Otherwise, $E_3\left(1,0,0\right)$ is the saddle point.	Saddle point	Saddle point	$E_6\left(0,1,1\right)$ is asymptotically stable point (confluence) when $R_P^H-R_P^L$ $<c_P^H-c_P^L$. Otherwise, $E_6\left(0,1,1\right)$ is the saddle point.	Saddle point	$E_8\left(1,1,1\right)$ is the saddle point when $R_P^H-R_P^L$ $<c_P^H-c_P^L$. Otherwise, $E_8\left(1,1,1\right)$ is asymptotically stable point (confluence)

.

To clearly analyze the results in Table 3, the corresponding phase diagrams are presented in Figure 5 and Figure 6, respectively.

According to the results depicted in Table 3, the following theorem is also obtained.

Theorem 4.

When$c_i\left(\tau \right)<R_{i1}-R_{i2}+R_i^L\left(\tau \right)+c_i\left(i=S,D\right)$, if$R_P^H-R_P^L<c_P^H-c_P^L$, the evolutionary stability strategy of the whole system is (low-level service, LCPC, LCPC). Otherwise, the evolutionary stability strategy should be (high-level service, LCPC, LCPC).

According to Theorem 4, it can be seen that the stable strategies of S enterprises and D enterprises are located in LCPC after a long-term evolution, when the cost of low-carbon production capacity is too low. If the benefits of low-level service provided by the platforms are higher, the platforms will not change their strategies. That is, they will always adopt a low-level service strategy. In this case, the equilibrium point $E_6\left(0,1,1\right)$ is the only asymptotically stable point in terms of the whole system. The evolutionary stability strategy is (low-level service, LCPC, LCPC). On the contrary, when the benefits of high-level service are higher, the platforms will always adopt such a strategy. In this case, the equilibrium point $E_8\left(1,1,1\right)$ is the only asymptotically stable point in the whole system. The corresponding evolutionary stability strategy is (high-level service, LCPC, LCPC). The aforementioned theorem also indicates that when the cost of low-carbon production capacity is too low, no matter which strategy the platforms adopts, S enterprises and D enterprises are all environmental-friendly manufacturing enterprises. In other words, the manufacturing enterprises pay high attention to the ecological and environmental factors. In this context, LCPC sharing system would promote environmental sustainability.

(2) When the cost of low-carbon production capacity is too high, that is $R_{i1}-R_{i2}+R_i^H\left(\tau \right)+c_i<c_i\left(\tau \right)\left(i=S,D\right)$, and the stability analysis of the equilibrium points is presented in

Table 4. Stability analysis of the equilibrium points when $R_{i1}-R_{i2}+R_i^H\left(\tau \right)+c_i<c_i\left(\tau \right)\left(i=S,D\right)$.

Equilibrium Points		${\mathit{E}}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{3}}\mathbf{\left(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{4}}\mathbf{\left(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{5}}\mathbf{\left(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{6}}\mathbf{\left(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{7}}\mathbf{\left(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{\right)}$	${\mathit{E}}_{\mathbf{8}}\mathbf{\left(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{\right)}$
Eigenvalues	${\lambda }_1$	--	--	--	--	--	--	--	--
	${\lambda }_2$	$<0$	$<0$	$<0$	$<0$	$>0$	$>0$	$>0$	$>0$
	${\lambda }_3$	$<0$	$>0$	$<0$	$>0$	$<0$	$>0$	$<0$	$>0$
Stability		$E_1\left(0,0,0\right)$ is the asymptotically stable point (confluence) when $R_P^H-R_P^L$ $<c_P^H-c_P^L$. Otherwise, $E_1\left(0,0,0\right)$ is the saddle point.	Saddle point	$E_3\left(1,0,0\right)$ is the saddle point when $<c_P^H-c_P^L$ $R_P^H-R_P^L$. Otherwise, $E_3\left(1,0,0\right)$ is the asymptotically stable point (confluence).	Saddle point	Saddle point	$E_6\left(0,1,1\right)$ is the saddle point when $<c_P^H-c_P^L$ $R_P^H-R_P^L$. Otherwise, $E_6\left(0,1,1\right)$ is the unstable point (source).	Saddle point	$E_8\left(1,1,1\right)$ is the unstable point (source) when $<c_P^H-c_P^L$ $R_P^H-R_P^L$. Otherwise, $E_8\left(1,1,1\right)$ is the saddle point.

.

To explicitly analyze the results presented in Table 4, the corresponding phase diagrams are depicted in Figure 7 and Figure 8, respectively.

According to the results presented in Table 4, the following Theorem 5 is also obtained.

Theorem 5.

When$R_{i1}-R_{i2}+R_i^H\left(\tau \right)+c_i<c_i\left(\tau \right)\left(i=S,D\right)$, if$R_P^H-R_P^L<c_P^H-c_P^L$, the evolutionary stability strategy of the whole system is (low-level service, OPC, OPC). Otherwise, the evolutionary stability strategy should be (high-level service, OPC, OPC).

The Theorem 5 demonstrates that when the cost of low-carbon production capacity is too high, the stable strategies of S enterprises and D enterprises are located in OPC sharing after a long-term evolution. When the benefits of low-level service provided by the platforms are higher, the platforms will not change their strategies. In this case, the equilibrium point $E_1\left(0,0,0\right)$ is the only asymptotically stable point in the whole system. In addition to that the evolutionary stability strategy is (low-level service, OPC, OPC). On the contrary, when the benefits of high-level service are higher, the platforms will always adopt such a strategy. In this case, the equilibrium point $E_3\left(1,0,0\right)$ is the only asymptotically stable point. The corresponding evolutionary stability strategy is (high-level service, OPC, OPC). The aforementioned theorem also shows that when the cost of low-carbon production capacity is too high, manufacturing enterprises will always produce the ordinary products. They aim to maximize their own utilities, while they ignore their impacts on the environment. That is, the LCPC sharing system does not concern environmental sustainability.

6.2. Effect of Transaction Cost on Strategy Choice of the Platforms

In this paper, it is assumed that the benefits of the platforms come from the transaction cost charged to the manufacturing enterprises participating in production capacity sharing. Therefore, this subsection takes into account the impacts of transaction cost on the decisions of the platforms.

Let $\Delta R$ denote the relative revenue of the platforms. It can be measured by the difference between the income of high-level service and that of low-level service. Besides, it can be denoted by $\Delta R=\left(k_S{m}_S^H-m_S^L\right)n_S{t}_S+\left(k_D{m}_D^H-m_D^L\right)n_D{t}_D-c_P^H+c_P^L$. The tendency of the relative revenue of the platforms with the variation of transaction costs $t_S$ and $t_D$ considerations is presented in Figure 9.

The results depicted in Figure 9 indicate that when the transaction cost $t_S\left(t_D\right)$ is fixed, the relative revenue ($\Delta R$) of the platforms has an increasing trend with an increasing transaction cost $t_D\left(t_S\right)$. It is obvious that the color plane is divided into three sub-areas by the lattice plane. In this context, the following conclusions can be summarized.

Firstly, the coordinates $\left(t_S,t_D\right)$ of the points located on the intersection line L between the lattice plane ($\Delta R=0$) and the color plane ($\Delta R=\left(k_S{m}_S^H-m_S^L\right)n_S{t}_S+\left(k_D{m}_D^H-m_D^L\right)n_D{t}_D-c_P^H+c_P^L$) satisfy $\left(k_S{m}_S^H-m_S^L\right)n_S{t}_S+\left(k_D{m}_D^H-m_D^L\right)n_D{t}_D-c_P^H+c_P^L=0$. When $\left(t_S,t_D\right)\in L$, there is no difference whether the platforms adopt high-level service or low-level service. The reason for this is that evolutionary stability strategies are pure strategies.

Secondly, the coordinates $\left(t_S,t_D\right)$ of the points located on the color plane below line L meet $\Delta R<0$. In this case, the benefits of the strategy ‘low-level service’ are higher in terms of the platforms. The platforms will always adopt low-level service after a long-term evolution.

Thirdly, the coordinates $\left(t_S,t_D\right)$ of the points located on the color plane above line L satisfy $\Delta R>0$. In this case, the benefits of the strategy ‘high-level service’ are higher for the platforms. The platforms will not change its strategy. It will adopt high-level service after a long-term evolution.

7. Conclusions

The focus of this paper is to design sharing mechanisms with regard to low-carbon production capacity. In this paper, production capacity is divided into LCPC and OPC according to environmental concerns. The participants with bounded rationality of production capacity sharing system include the platforms, S and D enterprises. In this context, this issue is formulated as a three-player evolutionary game model. Then, the stability of the whole system as well as evolutionary stability strategy of the participants is discussed. Finally, the cost effect of LCPC and the effect of transaction cost on the strategy of the platforms are investigated.

The results show that: (1) The platforms will always adopt the strategy with a higher income. (2) When the cost of low-carbon production capacity is too low, the manufacturing enterprises participating in the system always choose the combination of strategies (LCPC, LCPC), whatever the platforms adopt. (3) When the cost of low-carbon production capacity is too high, the stable strategies of S enterprises and D enterprises are located in OPC sharing after a long-term evolution whatever the platforms adopt. (4) Given a fixed transaction cost of S (D) enterprise, the relative revenue of the platforms has an increasing trend with an increasing transaction cost of D (S) enterprises.

Computational results provide several insights on the theory and practice of the production capacity sharing problem. Firstly, a theoretical link between production capacity sharing and environmental concerns is considered to achieve the sustainable development of the whole system. Second, it provides a new viewpoint for decision makers who design the mechanisms of production capacity sharing. Third, a multiplayer evolutionary game theory is linked with the low-carbon production capacity sharing problem, which is a promising and interesting topic.

In practice, in terms of the platforms, they can supervise the behavior of manufacturing enterprises participating in production capacity sharing, and provide a better service, thus attracting more manufacturing enterprises to join the system. Secondly, with regard to manufacturing enterprises, D enterprises can purchase the idle capacity to produce the products from the platforms, thus obtaining extra income. S enterprises can improve the utilization of the equipment as well as reduce production costs, technical risks, and opportunistic costs of idle capacity. Thirdly, it is beneficial in improving the sustainability of the system by incorporating multiple players, low-carbon preference, and other factors into the production capacity sharing problem. Nevertheless, it also poses more challenges to cope with such complex issues. So, the need to develop and improve the multi-agent modeling software in disaster management sectors is urgent. Fourth, this paper expects to not only validate the proposed methodology, but it also addresses the importance and urgency of incorporating low-carbon preference or environmental concerns into the production capacity sharing problem. Fifth, for policy-makers, it is an efficient way to reduce energy consumption by strengthening low-carbon policy propaganda and support. In terms of manufacturing enterprises, they should adopt the energy-saving technologies and equipment, thus improving the utilization of resources, so as to promote energy conservation and emission reduction. In addition to that, policy-makers and manufacturing enterprises should jointly achieve the sustainability of economy and environment, as well as promote the sustainable development of the LCPC sharing system.

Valuable topics remain for further study. Firstly, this paper focus on a single platform, the topic regarding the influence of competitive strategies among multiple platforms on production capacity sharing system is promising. Secondly, the integrated issue that incorporates government policies into low-carbon production capacity mechanism is the follow-up direction. Thirdly, since practical cases are more complicated, evolutionary game regarding LCPC sharing problem with multi-participant and multi-strategy need to be further studied.