Low-Carbon Collaboration in the Supply Chain under Digital Transformation: An Evolutionary Game-Theoretic Analysis

Abstract

In the face of the challenges posed by the need to drastically decrease carbon emissions, all agents in the supply chain need to strengthen low-carbon collaboration with the support of digital transformation. This study sets up a low-carbon collaboration evolutionary game model of the supply chain based on benefit sharing by introducing digital transformation. The equilibrium-point stability of the supply chain is then analyzed under two separate conditions—i.e., less and more government rewards and punishments compared to supply-chain agents’ strategic risk cost. Furthermore, based on the evolutionary game model, this study draws the system dynamics (SD) flow diagram to analyze the research problem quantitatively. The main results show that: (1) low-carbon benefit-driven effects promotes collaboration benefit sharing, thereby increasing the probability of low-carbon collaboration; (2) digital transformation is an essential regulator of low-carbon collaboration in the supply chain and can amplify the low-carbon benefit-driven effect; (3) collaboration benefit sharing can perfectly coordinate the vertical supply chain under low-carbon collaboration; and (4) government support and management are critical links in the low-carbon collaboration formation path of the supply chain. This research provides theoretical support for low-carbon collaboration in the supply chain under digital transformation.

1. Introduction

Global carbon dioxide emissions have increased significantly since the middle of the twentieth century [1]. In the last 20 years, a series of phenomena such as global warming, melting glaciers, rising sea levels and smog weather have indicated the severe impact of climate change brought about by the greenhouse effect on the survival of human beings [2]. Global energy consumption will undergo sharp rebound, further superimposed by severe weather, energy market shocks, etc., which will increase carbon emissions [3]. With countries worldwide paying more and more attention to global climate change, a series of low-carbon plans such as carbon peaking and carbon neutralization have been gradually implemented.

The acceleration of economic transformation, development of green technology industries, deepening of economic digitization, and active response of all relevant parties are essential for countries to achieve carbon neutrality [4]. The digital economy is expected to account for 22.5% of the global economy within the next few years, and the digital economy is increasingly becoming a driving force and engine for global economic recovery and low-carbon sustainable development [5]. Digital transformation has become a new norm and an important process tool for attaining competitive edge over countries or industries [6]. For companies, digital transformation is the integration of digital technology into all sectors of a business, fundamentally altering how they perform and bring value to customers [7]. It is also a process of major change in the business to enhance customer experience and innovate business models by leveraging new digital technologies [8]. Such definitions refer to applying digital technology in operation, business-model innovation, or digital strategy to create value for only a firm, not for a supply chain [9]. Therefore, our study follows the above definitions to define digital transformation in a supply chain as “Digital transformation in the supply chain is a deep-seated process of major changes where digital technologies alter value creation paths of whole chain and bring the structural changes and remove the interorganizational barriers of the positive outcomes”, which is partially consistent with the definition of Magistretti [10] and Vial [11]. Traditional supply-chain theory has been unable to meet the current low-carbon sustainable-development requirements of today’s highly developed digital economy, highlighiting the need for more theories and methods based on digital transformation to meet the development needs of the new era. Developing the low-carbon collaboration of the supply chain under digital transformation will soon become a critical problem for various countries.

Therefore, low-carbon development and digital transformation are two crucial development trends in today’s global supply chain. Theories of “low-carbon+”, which are of high practical significance and forward looking, are rapidly emerging. The low-carbon collaboration research into supply chains is becoming increasingly prominent as the supply chain begins to occupy a significant and memorable position in the low-carbon economy [12]. Simultaneously, driven by a new round of technological revolutions, digital transformation can promote the conversion of old and new kinetic energy from “factor-driven” to “innovation-driven” and is an important starting point to realize low-carbon sustainable-development goals in the supply chain. To form a digital economic system with close cooperation and achieve high-quality economic development, it is necessary to actively promote the digital transformation of enterprises, supply chains, and regions.

The supply chain is an entity based on multi-agent win–win collaboration. It is a dynamic, hyper-connected network where all stakeholders are inter-connected and inter-dependent [13]. Companies can even achieve zero-waste through circular supply-chain management [14]. Its governance mechanism is obviously different from the market mechanism and authoritative governance within a single enterprise. A supply chain is an autonomous organization whose mechanisms to maintain operation include trust mechanisms, information-sharing mechanisms, contract mechanisms, carbon-trading mechanisms, and so on [15]. Low-carbon collaboration is an essential subfield of supply-chain collaboration theory and one of the critical development ideas to deal with the current new reality of low-carbon sustainable development. The core of industrial-symbiosis theory is collaboration, which can improve the viability and profitability of enterprises and achieve resource conservation and the environmental protection of society [16]. All aspects of the supply chain involve energy consumption and carbon emissions in a complete business chain, from raw material suppliers, manufacturers, and distributors, to consumers. The low-carbon collaborative behaviors of different backgrounds, different links, and different agents are systematic.

Low-carbon collaboration can have substantial positive effects on the supply chain and its constitution enterprises. Above all, many scholars identify that low-carbon collaboration can improve not only low-carbon outcomes but also total performance in the supply chain, such as revenue increase, cost saving, efficiency, and profit [17]. Furthermore, low-carbon collaboration can improve indirect performances such as the environmental performance and sustainable development of participants [18,19]. However, these positive effects depend on multiple contextual factors such as relationship length, dependency, and supplier involvement in low-carbon collaboration [20], policies, technologies, knowledge exchange, and organization learning [21].

Some findings underscore the strategic significance of digital transformation as an important force in promoting the development of a low-carbon supply chain. It can facilitate low-carbon technology collaborative innovation in the supply chain. Taking the digital-transformation road can promote low-carbon-technology innovation and application promotion in essential process manufacturing fields [11]. For example, as a supplier of digital-transformation technology, Huawei launched the “Service Turbo Cloud” service platform and began providing collaboration services in 2018. Huawei embarked on a transformation path from “network construction and operation” to “focus on customer service” based on its digital practice. This platform helps customers with digital transformation and accelerates digital transformation in various industries, promoting low-carbon collaboration among industrial chain members and their customers. Another example is Haier, which has built an industrial Internet-of-Things platform named “COSMO plat” by integrating the new generation of information technology with industrial knowledge, technologies, and processes to better promote intelligent manufacturing. This platform can provide empowering functions from data, simulation, customization, and open-source software to provide instant modular overall solutions for user enterprises. It can be concluded that digital transformation can hasten low-carbon collaboration, including eco-design, green marketing, information sharing, and environmental management, etc. [22].

Existing game literature has paid considerable attention to low-carbon collaboration. There are usually three types of game participants in the game models: common participants from the vertical supply chain [23], from the horizontal supply chain, and the lateral supply chain; notably, game participant numbers have risen between the latter two [24,25]. However, relatively few studies have focused on the modes of collaboration brought about by digital transformation which, in most cases, are network-type collaborations. Extant game models reveal the optimal strategic decisions of each agent in the supply chain and identify the impacts of the cost-sharing coefficient, government incentives, default penalties, and income distribution coefficient on the evolution results of the government, enterprises, academic research institutions, and so on [26]. Digital transformation facilitates “altruism” and “mutual efforts or willingness of collaboration” in the contract collaboration mechanism and relational practices in low-carbon supply chains with a lower cost and in a convenient way. It improves motivation and willingness to initiate low-carbon collaboration. It has become the key external environment that plays an important role in initiating and implementing low-carbon collaboration [27]. Although all game models consider and analyze many forces, digital transformation as the main factor is seldomly discussed. In particular, few scholars have used evolutionary game and system dynamics (SD) methods to explore the correlation between digital transformation and low-carbon collaboration mechanisms in the supply chain or explore how optimal decision and collaboration mechanisms change under digital transformation.

Therefore, this study mainly explores the driving factors and formation path of low-carbon collaboration in the supply chain under digital transformation and attempts to answer the following questions: How do digital transformation and other factors affect supply-chain low-carbon collaboration? How to design an optimal analytical model for supply-chain low-carbon collaboration under digital transformation? What is the low-carbon collaboration path in the supply chain proposed in this study?

The scientific novelty of this study is to explore the regulative effect of digital transformation on the formation of low-carbon collaboration, based on a tripartite evolutionary game in the three-level supply chain consisting of suppliers, manufacturers and retailers. This study also innovatively combines the SD method into the research to present the formation pathway of low-carbon collaboration. Unlike previous studies, which regard the government as an important party in the game and focus on its gains and losses, our game model focuses on the effects of the government rewards and punishments on the formation pathway of low-carbon collaboration. Based on a new game model with the bounded-rationality and information-asymmetry conditions, we document that digital transformation can regulate low-carbon benefit-driven effects and promote the low-carbon collaboration formation of three-level supply chains. We verify that government support and management are the critical links in the low-carbon collaboration formation path of the three-level supply chain which can represent a novelty to some extent. This study also finds that low-carbon benefit distribution among the three parties in the vertical supply chain also plays an important role in the formation of low-carbon collaboration, which is consistent with some previous studies of two-level supply chains or horizontal-supply-chain structures.

The main contributions of this study are summed up as follows: first, this study creatively introduces digital transformation into the supply-chain low-carbon collaboration model, filling the gap between digital transformation and supply-chain research. We find that digital transformation and other factors can affect different links on the low-carbon collaboration formation path of the supply chain. Second, this study combines the evolutionary game model with SD; constructs a multi-agent game model including suppliers, manufacturers and retailers; and authenticates the optimal equilibrium solutions of low-carbon collaboration in three-level supply chains under the digital transformation. Third, this study offers insights into the effective path of realizing low-carbon supply-chain collaboration, that is, using low-carbon benefits especially with the additional benefit of driving low-carbon benefit sharing, and then promoting the formation of low-carbon collaboration in the supply chain.

The rest of this study is organized as follows: Section 2 provides a review of relevant literature. Section 3 builds an evolutionary game model of low-carbon collaboration in the supply chain and analyzes the different conditions of evolutionarily stable strategy (ESS). Section 4 draws the SD flow diagram and conducts numerical simulation analysis based on Section 3. Finally, Section 5 summarizes the conclusions and implications of this study.

2. Literature Review

Extant collaboration literature on the low-carbon supply chain has achieved fruitful results. This study reviews the literature from the following aspects: contract collaboration and relational practices, the effect of low-carbon collaboration and its influence factors, applications of the differential game, and the application of digital transformation in low-carbon cooperation.

2.1. Low-Carbon Collaboration in the Supply Chain

The existing research literature on low-carbon collaboration in the supply chain can be divided into contract collaboration, investment collaboration, cost and income sharing, and relational practices. The premise of all collaboration is a contract, and, thus, several scholars propose that all collaboration can be classified as a contract. Some researchers propose low-carbon collaboration for the supply chain. Agents in the supply chain should first develop product-development cooperation, exchange carbon knowledge, implement effective governance, and build a trusting relationship with their suppliers for low-carbon cooperation [28].

Some scholars explore the investment collaboration in regional low-carbon logistics network design, including non-budget-sharing, centralized budget sharing, and conditional budget sharing [29]. Others draw attention to price cooperation in low-carbon relationships, which is of interest in this context [30]. It needs to be agreed upon through the collaboration contract and is also the premise of other collaborations that extend to the collaboration of cost sharing and revenue sharing [31]. Low-carbon collaboration also depends on multiple contextual factors within these contracts, such as relationship length, dependency, and supplier involvement [32]. Participants should undertake relational practices between them, such as product-development cooperation, exchanging carbon knowledge, implementing effective governance, and building a trusting and long-lasting relationship to achieve a low-carbon outcome [28].

2.2. Effect and Influence Factors of Low-Carbon Collaboration

Most studies verify the outcomes of low-carbon collaboration from its effects on participants’ performance, including marketing performance, financial performance, and environmental performance. Carbon trading and carbon footprint are the focus of current low-carbon supply-chain research. Carbon trading could increase the unit retail price of ordinary products, so low-carbon manufacturers will also raise the unit retail price of low-carbon products to obtain greater revenue in the competitive market [33]. Reducing carbon emissions from the supply-chain perspective can improve an enterprise’s reputation and financial performance [34]. From another perspective, some scholars highlight that some supply-chain collaborations will bring sustainable-development performance. Although this kind of collaboration is not clearly defined as low-carbon collaboration, it does result in a similar performance [19]—such as collaboration in production, distribution, and transportation—leading to a substantial reduction in costs and carbon emissions in the supply chain [25]. Managing the carbon footprint of a product across the supply chain is another strategically crucial step for businesses to reduce carbon emissions and mitigate climate risks [35]. A collaboration, to some extent, depends on the policies. How governments and supply-chain members make low-carbon decisions to achieve carbon neutrality and recycle waste resources has become a prominent part of the present field. The priorities of government policies have substantial impacts on collaboration development, but these impacts are contingent on the variety in low-carbon outcomes. The effects of collaboration are affected by the level of collaboration or the coupling collaboration degree [36]. For example, high ratios in revenue-sharing contracts lead to low emission levels and prices, and eventually benefit consumers [37]. In addition, supply-chain collaboration is found to minimize the total system cost when emissions and penalty costs are considered [38]. However, there are few effect studies in the context of technology, such as digital technology.

2.3. Game Models of Low-Carbon Collaboration in the Supply Chain

Many scholars establish game models of low-carbon collaboration in the supply chain as consisting of different participants with or without some preferences. Some of them construct a simple model with a single manufacturer, retailer, or supplier to investigate the complex dynamic characteristics of pricing decisions and carbon abatement strategies in the supply chain [30]. The usual conclusion is that the collaboration strategies or relational practices between buyers and suppliers may be equally essential to achieving a low-carbon outcome [28]. There are two kinds of collaboration types—those led by the buyer or manufacturer or those led by the supplier or retailer [39] as different game backgrounds or actors [27,40]. Then, the researchers extend low-carbon games to more complex contexts, including more participants such as the government or third-party recyclers [41], two-level supply chains [42], closed-loop supply chains [40], vertical supply-chain structures, or horizontal supply-chain structures [23], or those outside the supply-chain structure (material exchange and resource recycling with other members outside the supply chain) [43]. Although the contexts and actors differ across the models, most studies draw the consistent conclusion that, in mutual low-carbon collaboration, the supply chain and its members are likely to benefit more from each other if they coordinate on their production, distribution, and transportation activities. This collaborative approach helps reduce the costs of operating the supply chain and minimize carbon emissions. In addition, a higher level of collaborative advantage can be achieved if there is collaboration between a greater number of participants at each level of the supply chain [25].

There are different preferences or references brought into game models. These models investigate the impacts of the manufacturers’ or retailers’ social preference on pricing decisions, carbon-emission abatement strategy, profits, supply-chain collaboration, and the complexity of dynamic models [30]. As a social preference, altruistic preference is an economically irrational factor that influences decision makers [17]. Many researchers establish dynamic game models to study social preference or low-carbon references [27,30,39,40]. The altruistic decisions of the manufacturer or retailers can promote emission reduction and recycling activities. They can increase total profit, but they will increase the manufacturers’ and retailers’ profits separately and contingently, while their altruistic intensities are maintained at a reasonable level. Compared with other altruistic modes, the manufacturers’ unilateral altruistic mode can achieve a higher level of emission reduction [40]. The introduction of low-carbon references can promote carbon reduction of the supply chain; however, interestingly, when the memory parameter of low-carbon preference tends to infinity or the sensitivity coefficient of a low-carbon reference approaches zero, the carbon-reduction effect of the supply chain will not change [39].

2.4. Low-Carbon Collaboration and Digital Transformation

Digital transformation is the use of digital technology to transform business models and improve users’ lives [44,45]. Digital transformation is fundamentally changing the way in which mature organizations innovate in services, and implementing digital transformation strategies will result in a shift from product-centric to service-centric business models [46]. Digital transformation will lead to new business-model opportunities for better interaction and collaboration with customers [47]. Therefore, digital transformation in the supply chain is bound to develop in the direction of service and collaboration.

Holistic co-evolution in the supply chain is essential for digital-transformation research [48]. Digital transformation can boost relationship performance through smart technologies [49]. The agile supply chain is a vital example of digital transformation, and it can cope with market instability, handle competitive pressures, and strengthen operational and organizational performance [50]. The agile supply chain requires the collaboration of all agents in the supply chain to gain competitive advantage [51]. The data-sharing profit coefficient, resource-integration coefficient, and trustworthiness coefficient from digital transformation increase the probability of enterprises participating in collaborative innovation in the supply chain [52].

Digital transformation provides new ways to measure and control environmental sustainability issues [53]. Survey data from 223 Chinese companies articulate that digital transformation has an inverse U-shaped relationship with environmental performance [54]. Digital transformation, together with connectivity and automation, is transforming traditional concepts of mobility, with a potential impact on transport decarbonization [55]. Some scholars have found opportunities for digital transformation to significantly reduce carbon emissions by substituting more environmentally intensive services [56]. High-quality carbon emissions can be achieved through digital transformation. Researchers found that continued economic growth and ICT penetration substantially decreased energy demand in Pakistan [57]. Therefore, digital transformation is imperative for achieving long-term sustainable growth and high-quality carbon emissions in a country.

In conclusion, the low-carbon collaboration process in the supply chain is a dynamic game process of multi-stakeholders, and altruistic collaboration is the result of this game. However, there is a dearth of game-theory research on low-carbon collaboration in the context where the costs and benefits of the supply chain are not clear, buyers and suppliers in the supply chain have no clear motivation nor willingness to initiate low-carbon collaboration, and where government policies and regulations as well as the external environment play an important role in the initiation and implementation of low-carbon synergy [27]. In addition, digital transformation promotes supply-chain collaboration, on the one hand, and reduces the carbon emissions of various organizations, on the other hand. This study explores the benefits of low-carbon collaboration strategies in the supply chain in terms of the actual situation and fills the research gap of supply-chain low-carbon synergy from the perspective of non-altruism under digital transformation.

3. Evolutionary Game-Model Building

The optimal background for low-carbon collaboration in the supply chain is that all agents are fully rational and have complete information transparency. However, in the complex market environment, the agents in the supply chain cannot fully maintain full rationality and complete information transparency. The evolutionary game model is based on the bounded rationality and incomplete information transparency of each agent in the game [58]. Compared with classic game theory, evolutionary games pay more attention to the changing trend of strategies [59]. Therefore, we choose the evolutionary game model which can provide the most suitable realistic research framework for studying low-carbon collaboration in the supply chain.

3.1. Model Assumptions

The selection of low-carbon collaboration strategies for each supply-chain agent under digital transformation is an evolutionary game process. Under digital transformation, this study assumes three agents in the low-carbon collaboration evolutionary game in the supply chain: suppliers, manufacturers, and retailers. The tripartite agents build a low-carbon collaboration based on their low-carbon development and form a complete supply, production and sales channels for low-carbon products. This study has made some assumptions to facilitate the theoretical study and establishment of the model, as shown in

Table 1. Basic variables in the evolutionary game model.

Variables	Meaning	Range
$x$	The probability that suppliers choose low-carbon collaboration strategies	$0 \le x \le 1$
$1-x$	The probability that suppliers choose non-low-carbon collaboration strategies	$0 \le 1-x \le 1$
$y$	The probability that manufacturers choose low-carbon collaboration strategies	$0 \le y \le 1$
$1-y$	The probability that manufacturers choose non-low-carbon collaboration strategies	$0 \le 1-y \le 1$
$z$	The probability that retailers choose low-carbon collaboration strategies	$0 \le z \le 1$
$1-z$	The probability that retailers choose non-low-carbon collaboration strategies	$0 \le 1-z \le 1$

and

Table 2. Basic parameters in the evolutionary game model.

Parameters	Meaning	Range
${\alpha }_1$	Suppliers’ digital-transformation degree	${\alpha }_1>0$
${\alpha }_2$	Manufacturers’ digital-transformation degree	${\alpha }_2>0$
${\alpha }_3$	Retailers’ digital-transformation degree	${\alpha }_3>0$
${\pi }_1$	The fundamental benefits of suppliers	${\pi }_1>0$
${\pi }_2$	The fundamental benefits of manufacturers	${\pi }_2>0$
${\pi }_3$	The fundamental benefits of retailers	${\pi }_3>0$
$R$	Additional benefits from tripartite agents’ low-carbon collaboration in the supply chain	$R>0$
$E$	Additional benefits from two agents’ low-carbon collaboration in the supply chain	$E>0$$, R>E$
${\lambda }_1$	The tripartite agents’ low-carbon collaboration additional-benefits-distribution coefficient of suppliers	$0 \le {\lambda }_1 \le 1$
${\lambda }_2$	The tripartite agents’ low-carbon collaboration additional-benefits-distribution coefficient of manufacturers	$0 \le {\lambda }_2 \le 1$
${\lambda }_3$	The tripartite agents’ low-carbon collaboration additional-benefits-distribution coefficient of retailers	$$\begin{gathered} 0 \le {\lambda }_3 \le 1, \\ {\lambda }_1+{\lambda }_2+{\lambda }_3=0 \end{gathered}$$
${\theta }_1$	The two agents’ low-carbon collaboration additional-benefits-distribution coefficient of suppliers	$0 \le {\theta }_1 \le 1$
${\theta }_2$	The two agents’ low-carbon collaboration additional-benefits-distribution coefficient of manufacturers	$0 \le {\theta }_2 \le 1$
${\theta }_3$	The two agents’ low-carbon collaboration additional-benefits-distribution coefficient of retailers	$$\begin{gathered} 0 \le {\theta }_3 \le 1, \\ {\theta }_1+{\theta }_2=0 \\ \mathrm{or} {\theta }_1+{\theta }_3=0 \\ \mathrm{or} {\theta }_2+{\theta }_3=0 \end{gathered}$$
$P_1$	Government rewards for suppliers choosing low-carbon collaboration	$P_1>0$
$P_2$	Government rewards for manufacturers choosing low-carbon collaboration	$P_2>0$
$P_3$	Government rewards for retailers choosing low-carbon collaboration	$P_3>0$
$U_1$	Government punishments for suppliers not choosing low-carbon collaboration	$U_1>0$
$U_2$	Government punishments for manufacturers not choosing low-carbon collaboration	$U_2>0$
$U_3$	Government punishments for retailers not choosing low-carbon collaboration	$U_3>0$
$C_1$	The strategic risk cost of suppliers choosing low-carbon collaboration	$C_1>0$
$C_2$	The strategic risk cost of manufacturers choosing low-carbon collaboration	$C_2>0$
$C_3$	The strategic risk cost of retailers choosing low-carbon collaboration	$C_3>0$
$T_1$	The input cost of suppliers choosing low-carbon collaboration	$T_1>0$
$T_2$	The input cost of manufacturers choosing low-carbon collaboration	$T_2>0$
$T_3$	The input cost of retailers choosing low-carbon collaboration	$T_3>0$
$D_1$	The exclusion loss of suppliers not choosing low-carbon collaboration	$D_1>0$
$D_2$	The exclusion loss of manufacturers not choosing low-carbon collaboration	$D_2>0$
$D_3$	The exclusion loss of retailers not choosing low-carbon collaboration	$D_3>0$

:

3.2. Strategy Combinations and Payoff Matrix

From the above assumptions, this study obtains the strategy combinations and payoff matrix, as shown in

Table 3. Strategy combinations and payoff matrix of the evolutionary game.

Strategies	Suppliers	Manufacturers	Retailers
$x, y, z$	$$\begin{gathered} {\alpha }_1\left({\pi }_1+{\lambda }_{1}R\right) \\ +P_1-C_1-T_1 \end{gathered}$$	$$\begin{gathered} {\alpha }_2\left({\pi }_2+{\lambda }_{2}R\right) \\ +P_2-C_2-T_2 \end{gathered}$$	$$\begin{gathered} {\alpha }_3\left({\pi }_3+{\lambda }_{3}R\right) \\ +P_3-C_3-T_3 \end{gathered}$$.
$x, y, 1-z$	$$\begin{gathered} {\alpha }_1\left({\pi }_1+{\theta }_{1}E\right) \\ +P_1-C_1-T_1 \end{gathered}$$	$$\begin{gathered} {\alpha }_2\left({\pi }_2+{\theta }_{2}E\right) \\ +P_2-C_2-T_2 \end{gathered}$$	${\pi }_3-D_3-U_3$
$x, 1-y, z$	$$\begin{gathered} {\alpha }_1\left({\pi }_1+{\theta }_1{E}_1\right) \\ +P_1-C_1-T_1 \end{gathered}$$	${\pi }_2-D_2-U_2$	$$\begin{gathered} {\alpha }_3\left({\pi }_3+{\theta }_3{E}_3\right) \\ +P_3-C_3-T_3 \end{gathered}$$
$x, 1-y, 1-z$	${\pi }_1+P_1-C_1$	${\pi }_2-U_2$	${\pi }_2-U_2$
$1-x, y, z$	${\pi }_1-D_1-U_1$	$$\begin{gathered} {\alpha }_2\left({\pi }_2+{\theta }_{2}E\right) \\ +P_2-C_2-T_2 \end{gathered}$$	$$\begin{gathered} {\alpha }_3\left({\pi }_3+{\theta }_{3}E\right) \\ +P_3-C_3-T_3 \end{gathered}$$
$1-x, y, 1-z$	${\pi }_1-U_1$	${\pi }_1+P_2-C_2$	${\pi }_3-U_3$
$1-x, 1-y, z$	${\pi }_1-U_1$	${\pi }_2-U_2$	${\pi }_3+P_3-C_3$
$1-x, 1-y, 1-z$	${\pi }_1-U_1$	${\pi }_2-U_2$	${\pi }_2-U_2$

.

3.3. Replicator Dynamics Equations

Let the expected benefits of suppliers under low-carbon collaboration be $E{S}_1$ and the expected benefits of suppliers under non-low-carbon collaboration be $E{S}_2$. Then:

$$\begin{array}{l}E{S}_1=yz\left. [{\alpha }_1\left({\pi }_1+{\lambda }_{1}R\right)+P_1-C_1-T_1\right]+y(1-z) [{\alpha }_1\left({\pi }_1+{\theta }_{1}E\right)\left.+P_1-C_1-T\right]\\ \hspace{1em}\hspace{1em} +(1-y)z\left. [{\alpha }_1\left({\pi }_1+{\theta }_{1}E\right)+P_1-C_1-T_1\right]\\ \hspace{1em}\hspace{1em} +(1-y)(1-z)\left({\pi }_1+P_1-C_1\right)\\ \hspace{1em}\hspace{1em}={\alpha }_1{\lambda }_{1}Ryz+\left({\alpha }_1{\pi }_1-T_1-{\pi }_1\right)(y+z-yz)+{\alpha }_1{\theta }_{1}E(y+z-2yz)\\ \hspace{1em}\hspace{1em} +{\pi }_1+P_1-C_1\end{array}$$

(1)

$$\begin{array}{l}E{S}_2=yz\left({\pi }_1-D_1-U_1\right)+y\left(1-z\right)\left({\pi }_1-U_1\right)+\left(1-y\right)z\left({\pi }_1-U_1\right)\\ \hspace{1em}\hspace{1em} +\left(1-y\right)\left(1-z\right)\left({\pi }_1-U_1\right)\\ \hspace{1em}\hspace{1em}={\pi }_1-yz{D}_1-U_1\end{array}$$

(2)

Therefore, the suppliers’ dynamics equation of regulators can be written as:

$$\begin{array}{l}F(x)=\frac{dx}{dt}=xE{S}_1+(1-x)E{S}_2=x(1-x)\left(E{S}_1-E{S}_2\right)\\ \hspace{1em}\hspace{1em}=x(1-x) [{\alpha }_1{\lambda }_{1}Ryz+\left({\alpha }_1{\pi }_1-T_1-{\pi }_1\right)(y+z-yz)\\ \hspace{1em}\hspace{1em}\hspace{1em}\left.+{\alpha }_1{\theta }_{1}E(y+z-2yz)+P_1-C_1+yz{D}_1+U_1\right]\end{array}$$

(3)

Let the expected benefits of manufacturers under low-carbon collaboration be $E{M}_1$ and the expected benefits of manufacturers under non-low-carbon collaboration be $E{M}_2$. Then:

$$\begin{array}{l}E{M}_1=xz [{\alpha }_2\left.\left({\pi }_2+{\lambda }_{2}R\right)+P_2-C_2-T_2\right]+x(1-z) [{\alpha }_2\left({\pi }_2+{\theta }_{2}E\right)\\ \hspace{1em}\hspace{1em} \left.+P_2-C_2-T_2\right]+(1-x)z\left. [{\alpha }_2\left({\pi }_2+{\theta }_{2}E\right)+P_2-C_2-T_2\right]\\ \hspace{1em}\hspace{1em} +(1-x)(1-z)\left({\pi }_1+P_2-C_2\right)\\ \hspace{1em}\hspace{1em} ={\alpha }_2{\lambda }_{2}Rxz+\left({\alpha }_2{\pi }_2-T_2-{\pi }_2\right)(x+z-xz)+{\alpha }_2{\theta }_{2}E(x+z-2xz)\\ \hspace{1em}\hspace{1em} +{\pi }_2+P_2-C_2\end{array}$$

(4)

$$\begin{array}{l}E{M}_2=xz\left({\pi }_2-D_2-U_2\right)+x(1-z)\left({\pi }_2-U_2\right)+(1-x)z\left({\pi }_2-U_2\right)\\ \hspace{1em}\hspace{1em} +(1-x)(1-z)\left({\pi }_2-U_2\right)\\ \hspace{1em}\hspace{1em}={\pi }_2-xz{D}_2-U_2\end{array}$$

(5)

Therefore, the manufacturers’ dynamics equation of regulators can be written as:

$$\begin{array}{l}F(y)=\frac{dy}{dt}=yE{M}_1+(1-y)E{M}_2=y(1-y)\left(E{M}_1-E{M}_2\right)\\ \hspace{1em}\hspace{1em}=y(1-y) [{\alpha }_2{\lambda }_{2}Rxz+\left({\alpha }_2{\pi }_2-T_2-{\pi }_2\right)(x+z-xz)\\ \hspace{1em}\hspace{1em}\hspace{1em}\left.+{\alpha }_2{\theta }_{2}E(x+z-2xz)+P_2-C_2+xz{D}_2+U_2\right]\end{array}$$

(6)

Let the expected benefits of retailers under low-carbon collaboration be $E{R}_1$ and the expected benefits of retailers under non-low-carbon collaboration be $E{R}_2$. Then:

$$\begin{array}{l}E{R}_1=xy\left. [{\alpha }_3\left({\pi }_3+{\lambda }_{3}R\right)+P_3-C_3-T_3\right]+x(1-y) [{\alpha }_3\left({\pi }_3+{\theta }_{3}E\right)\\ \hspace{1em}\hspace{1em} \left.+P_3-C_3-T_3\right]+(1-x)y\left. [{\alpha }_3\left({\pi }_3+{\theta }_{3}E\right)+P_3-C_3-T_3\right]\\ \hspace{1em}\hspace{1em} +(1-x)(1-y)\left({\pi }_3+P_3-C_3\right)\\ \hspace{1em}\hspace{1em}={\alpha }_3{\lambda }_{3}Rxy+\left({\alpha }_3{\pi }_3-T_3-{\pi }_3\right)(x+y-xy)+{\alpha }_3{\theta }_{3}E(x+y-2xy)\\ \hspace{1em}\hspace{1em} +{\pi }_3+P_3-C_3\end{array}$$

(7)

$$\begin{array}{l}E{R}_2=xy\left({\pi }_3-D_3-U_3\right)+x(1-y)\left({\pi }_2-U_2\right)+(1-x)y\left({\pi }_3-U_3\right)\\ \hspace{1em}\hspace{1em} +(1-x)(1-y)\left({\pi }_3-U_3\right)\\ \hspace{1em}\hspace{1em}={\pi }_3-xy{D}_3-U_3\end{array}$$

(8)

Therefore, the manufacturers’ dynamics equation of regulators can be written as:

$$\begin{array}{l}F(z)=\frac{dz}{dt}=zE{R}_1+(1-z)E{R}_2=z(1-z)\left(E{R}_1-E{R}_2\right)\\ \hspace{1em}\hspace{1em}=z(1-z) [{\alpha }_3{\lambda }_{3}Rxy+\left({\alpha }_3{\pi }_3-T_3-{\pi }_3\right)(x+y-xy)\\ \hspace{1em}\hspace{1em}\hspace{1em}\left.+{\alpha }_3{\theta }_{3}E(x+y-2xy)+P_3-C_3+xy{D}_3+U_3\right]\end{array}$$

(9)

3.4. Agents Stability Analysis

Firstly, take the first-order derivative of $F\left(x\right)$:

$$\begin{array}{l}{F}^{\prime }(x)=(1-2x) [{\alpha }_1{\lambda }_{1}Ryz+\left({\alpha }_1{\pi }_1-T_1-{\pi }_1\right)(y+z-yz)\\ \hspace{1em}\hspace{1em} \left.+{\alpha }_1{\theta }_{1}E(y+z-2yz)+P_1-C_1+yz{D}_1+U_1\right]\end{array}$$

(10)

According to Friedman’s evolutionary game theory, if $F\left(x\right)=0$, $F^{\prime }\left(x\right)<0$, then x is the evolutionarily stable strategy (ESS) adopted by suppliers. The suppliers’ behavior strategies stability analysis is as follows:

Proposition 1.

• When $y={\eta }_1=\left. [-z\left({\alpha }_1{\pi }_1+{\alpha }_1{\theta }_{1}E-T_1-{\pi }_1\right)-U_1-P_1+C_1\right]/\left. [\left({\alpha }_1{\lambda }_{1}R-{\alpha }_1{\pi }_1-2{\alpha }_1{\theta }_{1}E+T_1+{\pi }_1+D_1\right)+{\alpha }_1{\pi }_1+{\alpha }_1{\theta }_{1}E-T_1-{\pi }_1\right]$, all game strategies are at a steady state.
• When $y \ne {\eta }_1$, supposing$F\left(x\right)=0$, then$x=1$are the stable points of$x$.

Proof 1.

Then, two conditions are discussed separately according to Equation (10):

• If ${\alpha }_1{\lambda }_{1}Ryz+{\alpha }_1{\pi }_1-T_1-{\pi }_{1}y+z-yz+{\alpha }_1{\theta }_{1}Ey+z-2yz+P_1-C_1+yz{D}_1+U_1=0$, under the restraints of $0 \le x \le 1$, $0 \le y \le 1,$ and $0 \le z \le 1$, then when $y=\left. [-z\left({\alpha }_1{\pi }_1+{\alpha }_1{\theta }_{1}E-T_1-{\pi }_1\right)-U_1-P_1+C_1\right]/\left. [\left({\alpha }_1{\lambda }_{1}R-{\alpha }_1{\pi }_1-2{\alpha }_1{\theta }_{1}E+T_1+{\pi }_1+D_1\right)+{\alpha }_1{\pi }_1+{\alpha }_1{\theta }_{1}E-T_1-{\pi }_1\right]$ (denoted as $y={\eta }_1$), irrespective of what the suppliers choose for low-carbon collaboration, it is still a stable strategy and will not evolve.
• If ${\alpha }_1{\lambda }_{1}Ryz+{\alpha }_1{\pi }_1-T_1-{\pi }_{1}y+z-yz+{\alpha }_1{\theta }_{1}Ey+z-2yz+P_1-C_1+yz{D}_1+U_1 \ne 0$, then let $F\left(x\right)=0$, get $x=0$ and $x=1,$ two equilibrium points. This study assumes that the collaboration benefits obtained by each agent are more significant than the low-carbon input cost and strategic risk cost, that is: ${\alpha }_1{\lambda }_{1}Ryz+{\alpha }_1{\pi }_1-T_1-{\pi }_{1}y+z-yz+{\alpha }_1{\theta }_{1}Ey+z-2yz+P_1-C_1+yz{D}_1+U_1>0$. Therefore, when $x=0$, $F^{\prime }\left(x\right)>0$; and when $x=1$, $F^{\prime }\left(x\right)<0$. At this time, $x=1$ is the evolutionary stable point, which means that suppliers are in a stable state only when they choose the low-carbon collaboration strategies. □

The evolution roadmap of suppliers’ low-carbon collaboration strategies is shown in Figure 1:

Then, take the first-order derivative of $F\left(y\right)$:

$$\begin{array}{l}{F}^{\prime }(y)=(1-2y) [{\alpha }_2{\lambda }_{2}Rxz+\left({\alpha }_2{\pi }_2-T_2-{\pi }_2\right)(x+z-xz)\\ \hspace{1em}\hspace{1em} \left.+{\alpha }_2{\theta }_{2}E(x+z-2xz)+P_2-C_2+xz{D}_2+U_2\right]\end{array}$$

(11)

Similarly, if $F\left(y\right)=0$, $F^{\prime }\left(y\right)<0$, $y$ is the ESS adopted by the manufacturers. The manufacturers’ behavior-strategies stability analysis is as follows:

Proposition 2.

• When $z={\eta }_2=\left. [-x\left({\alpha }_2{\pi }_2+{\alpha }_2{\theta }_{2}E-T_2-{\pi }_2\right)-U_2-P_2+C_2\right]/\left. [x\left({\alpha }_2{\lambda }_{2}R-{\alpha }_2{\pi }_2-2{\alpha }_2{\theta }_{2}E+T_2+{\pi }_2+D_2\right)+{\alpha }_2{\pi }_2+{\alpha }_2{\theta }_{2}E-T_2-{\pi }_2\right]$, all game strategies are at a steady state.
• When $z \ne {\eta }_2$, supposing$F\left(y\right)=0$, then$y=1$are the stable points of$y$.

Proof 2.

Then, two conditions are discussed separately according to Equation (11):

• If ${\alpha }_2{\lambda }_{2}Rxz+{\alpha }_2{\pi }_2-T_2-{\pi }_{2}x+z-xz+{\alpha }_2{\theta }_{2}Ex+z-2xz+P_2-C_2+xz{D}_2+U_2=0$, under the restraints of $0 \le x \le 1$, $0 \le y \le 1,$ and $0 \le z \le 1$, then when $z=\left. [-x\left({\alpha }_2{\pi }_2+{\alpha }_2{\theta }_{2}E-T_2-{\pi }_2\right)-U_2-P_2+C_2\right]/\left. [x\left({\alpha }_2{\lambda }_{2}R-{\alpha }_2{\pi }_2-2{\alpha }_2{\theta }_{2}E+T_2+{\pi }_2+D_2\right)+{\alpha }_2{\pi }_2+{\alpha }_2{\theta }_{2}E-T_2-{\pi }_2\right]$ (denoted as $z={\eta }_2$), no matter what the manufacturers choose for low-carbon collaboration, it is still a stable strategy and will not evolve.
• If ${\alpha }_2{\lambda }_{2}Rxz+{\alpha }_2{\pi }_2-T_2-{\pi }_{2}x+z-xz+{\alpha }_2{\theta }_{2}Ex+z-2xz+P_2-C_2+xz{D}_2+U_2 \ne 0$, then let $F\left(y\right)=0$, get $y=0$ and $y=1,$ two equilibrium points. Due to ${\alpha }_2{\lambda }_{2}Rxz+{\alpha }_2{\pi }_2-T_2-{\pi }_{2}x+z-xz+{\alpha }_2{\theta }_{2}Ex+z-2xz+P_2-C_2+xz{D}_2+U_2>0$, when $y=0$, $F^{\prime }\left(y\right)>0$; and when $y=1$, $F^{\prime }\left(y\right)<0$. At this time, $y=1$ is the evolutionary stable point, which means that manufacturers are in a stable state only when they choose low-carbon collaboration strategies. □

The evolution roadmap of manufacturers’ low-carbon collaboration strategies is shown in Figure 2:

Finally, take the first-order derivative of $F\left(z\right)$:

$$\begin{array}{l}{F}^{\prime }(z)=(1-2z) [{\alpha }_3{\lambda }_{3}Rxy+\left({\alpha }_3{\pi }_3-T_3-{\pi }_3\right)(x+y-xy)\\ \hspace{1em}\hspace{1em} \left.+{\alpha }_3{\theta }_{3}E(x+y-2xy)+P_3-C_3+xy{D}_3+U_3\right]\end{array}$$

(12)

Similarly, if $F\left(z\right)=0$, $F^{\prime }\left(z\right)<0$, then $z$ is the ESS adopted by retailers. Retailers’ behavior-strategies stability analysis is as follows:

Proposition 3.

• When$x={\eta }_3=\left. [-y\left({\alpha }_3{\pi }_3+{\alpha }_3{\theta }_{3}E-T_3-{\pi }_3\right)-U_3-P_3+C_3\right]/\left. [y\left({\alpha }_3{\lambda }_{3}R-{\alpha }_3{\pi }_3-2{\alpha }_3{\theta }_{3}E+T_3+{\pi }_3+D_3\right)+{\alpha }_3{\pi }_3+{\alpha }_3{\theta }_{3}E-T_3-{\pi }_3\right]$, all game strategies are at a steady state.
• When $x \ne {\eta }_3$, supposing$F\left(z\right)=0$, then$z=1$are the stable points of$z$.

Proof 3.

Then, two conditions are discussed separately according to Equation (12):

• If ${\alpha }_3{\lambda }_{3}Rxy+{\alpha }_3{\pi }_3-T_3-{\pi }_{3}x+y-xy+{\alpha }_3{\theta }_{3}Ex+y-2xy+P_3-C_3+xy{D}_3+U_3=0$, under the restraints of $0 \le x \le 1$, $0 \le y \le 1,$ and $0 \le z \le 1$, then when $x=\left. [-y\left({\alpha }_3{\pi }_3+{\alpha }_3{\theta }_{3}E-T_3-{\pi }_3\right)-U_3-P_3+C_3\right]/\left. [y\left({\alpha }_3{\lambda }_{3}R-{\alpha }_3{\pi }_3-2{\alpha }_3{\theta }_{3}E+T_3+{\pi }_3+D_3\right)+{\alpha }_3{\pi }_3+{\alpha }_3{\theta }_{3}E-T_3-{\pi }_3\right]$ (denoted as $x={\eta }_3$), irrespective of what the retailers choose for low-carbon collaboration, it is still a stable strategy and will not evolve.
• If ${\alpha }_3{\lambda }_{3}Rxy+{\alpha }_3{\pi }_3-T_3-{\pi }_{3}x+y-xy+{\alpha }_3{\theta }_{3}Ex+y-2xy+P_3-C_3+xy{D}_3+U_3 \ne 0$, get $z=0$ and $z=1,$ two equilibrium points. Due to ${\alpha }_3{\lambda }_{3}Rxy+{\alpha }_3{\pi }_3-T_3-{\pi }_{3}x+y-xy+{\alpha }_3{\theta }_{3}Ex+y-2xy+P_3-C_3+xy{D}_3+U_3>0$, when $z=0$, $F^{\prime }\left(z\right)>0$; when $z=1$, $F^{\prime }\left(z\right)<0$. At this time, $z=1$ is the evolutionary stable point, which means that retailers are in a stable state only when they choose the low-carbon collaboration strategies. □

The evolution roadmap of retailers’ low-carbon collaboration strategies is shown in Figure 3:

3.5. Tripartite Stability Analysis

The replicator dynamics equation calculated in the preceding content is a mathematical expression of the behavioral tendencies of all parties in the evolutionary game system. When the entire evolutionary game model tends to a convergent solution, it shows that the system is stable on this combination of strategies. Let the replicated dynamic system be treated as the combination of Equations (3), (6) and (9), resulting in Proposition 4.

Proposition 4.

Eight replicated dynamic equilibrium points exist in a replicated dynamic system, which are (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1), if and only if the desired condition $x\in \left. [0,1\right]$,$y\in \left. [0,1\right]$and$z\in \left. [0,1\right]$is established.

Proof 4.

The stable strategies combination of the entire evolutionary game can be obtained by solving the Jacobian matrix. From Equations (3), (6) and (9), it can be deduced that the Jacobian matrix of the low-carbon collaboration replicator dynamics system in the supply chain is:

$$J=\left. [\begin{array}{ccc}\frac{\partial F\left(x\right)}{\partial x}& \frac{\partial F\left(x\right)}{\partial y}& \frac{\partial F\left(x\right)}{\partial z}\\ \frac{\partial F\left(y\right)}{\partial x}& \frac{\partial F\left(y\right)}{\partial y}& \frac{\partial F\left(y\right)}{\partial z}\\ \frac{\partial F\left(z\right)}{\partial x}& \frac{\partial F\left(z\right)}{\partial y}& \frac{\partial F\left(z\right)}{\partial z}\end{array}\right]$$

(13)

Let $F\left(x\right)=F\left(y\right)=F\left(z\right)=0$, the following pure strategic equilibrium points are: $E_1$(0,0,0), $E_2$(1,0,0), $E_3$(0,1,0), $E_4$(0,0,1), $E_5$(1,1,0), $E_6$(1,0,1), $E_7$(0,1,1), $E_8$(1,1,1). Weibull proved in his monograph on evolutionary games that the ESS of multi-population (two or more) evolutionary games must be a strict Nash equilibrium and pure strategic equilibrium [60]. Since the mixed strategic equilibrium does not conform to a strict Nash equilibrium, this study only selects the pure strategic equilibrium points for discussion. Then, the pure strategic equilibrium points $E_1$~$E_8$ are incorporated into Formula (13), and the eigenvalues of the Jacobian matrix are solved, as shown in

Table 4. Equilibrium points eigenvalues.

Equilibrium Point	Eigenvalues
$E_1$(0,0,0)	$P_1-C_1+U_1; P_2-C_2+U_2; P_3-C_3+U_3$
$E_2$(1,0,0)	$$\begin{gathered} C_1-P_1-U_1; P_2-C_2-T_2+U_2-{\pi }_2+{\alpha }_2{\pi }_2+{\alpha }_2{\theta }_{2}E; \\ {P}_3-C_3-T_3+U_3-{\pi }_3+{\alpha }_3{\pi }_3+{\alpha }_3{\theta }_{3}E \end{gathered}$$
$E_3$(0,1,0)	$$\begin{gathered} C_2-P_2-U_2; P_1-C_1-T_1+U_1-{\pi }_1+{\alpha }_1{\pi }_1+{\alpha }_1{\theta }_{1}E; \\ {P}_3-C_3-T_3+U_3-{\pi }_3+{\alpha }_3{\pi }_3+{\alpha }_3{\theta }_{3}E \end{gathered}$$
$E_4$(0,0,1)	$$\begin{gathered} C_3-P_3-U_3; P_1-C_1-T_1+U_1-{\pi }_1+{\alpha }_1{\pi }_1+{\alpha }_1{\theta }_{1}E; \\ {P}_2-C_2-T_2+U_2-{\pi }_2+{\alpha }_2{\pi }_2+{\alpha }_2{\theta }_{2}E \end{gathered}$$
$E_5$(1,1,0)	$$\begin{gathered} C_1-P_1+T_1-U_1+{\pi }_1-{\alpha }_1{\pi }_1-{\alpha }_1{\theta }_{1}E; \\ {C}_2-P_2+T_2-U_2+{\pi }_2-{\alpha }_2{\pi }_2-{\alpha }_2{\theta }_{2}E; \\ {D}_3-C_3+P_3-T_3+U_3-{\pi }_3+{\alpha }_3{\pi }_3+{\alpha }_3{\lambda }_{3}R \end{gathered}$$
$E_6$(1,0,1)	$$\begin{gathered} C_1-P_1+T_1-U_1+{\pi }_1-{\alpha }_1{\pi }_1-{\alpha }_1{\theta }_{1}E; \\ {C}_3-P_3+T_3-U_3+{\pi }_3-{\alpha }_3{\pi }_3-{\alpha }_3{\theta }_{3}E; \\ {D}_2-C_2+P_2-T_2+U_2-{\pi }_2+{\alpha }_2{\pi }_2+{\alpha }_2{\lambda }_{2}R \end{gathered}$$
$E_7$(0,1,1)	$$\begin{gathered} C_2-P_2+T_2-U_2+{\pi }_2-{\alpha }_2{\pi }_2-{\alpha }_2{\theta }_{2}E; \\ {C}_3-P_3+T_3-U_3+{\pi }_3-{\alpha }_3{\pi }_3-{\alpha }_3{\theta }_{3}E; \\ {D}_1-C_1+P_1-T_1+U_1-{\pi }_1+{\alpha }_1{\pi }_1+{\alpha }_1{\lambda }_{1}R \end{gathered}$$
$E_8$(1,1,1)	$$\begin{gathered} C_1-D_1-P_1+T_1-U_1+{\pi }_1-{\alpha }_1{\pi }_1-{\alpha }_1{\lambda }_{1}R; \\ {C}_2-D_2-P_2+T_2-U_2+{\pi }_2-{\alpha }_2{\pi }_2-{\alpha }_2{\lambda }_{2}R; \\ {C}_3-D_3-P_3+T_3-U_3+{\pi }_3-{\alpha }_3{\pi }_3-{\alpha }_3{\lambda }_{3}R \end{gathered}$$

. □

According to the Lyapunov rule and references [61,62], when all the eigenvalues of the Jacobian matrix are negative natural numbers, the corresponding equilibrium point is the stable point; when all the eigenvalues of the Jacobian matrix are positive real numbers, the corresponding equilibrium point is the unstable point; and when the eigenvalues of the Jacobian matrix contain both negative and positive real numbers, the corresponding equilibrium point is the saddle point. Based on each equilibrium point’s positive and negative, Proposition 5 is obtained as below.

Proposition 5.

• When$P_i+U_i>C_i$,$E_8$(1,1,1) are the ESS of$J$.
• When $P_i+U_i<C_i$,$E_1$(0,0,0) and$E_8$(1,1,1) are the ESS of$J$.

Proof 5.

This study assumes that the low-carbon collaboration benefits and costs of agents in the supply chain under the digital transformation have the following relationship: $P_i+U_i+{\alpha }_i{\pi }_i+{\alpha }_i{\theta }_{i}E>C_i+T_i+{\pi }_i$, $D_i+P_i+U_i+{\alpha }_i{\pi }_i+{\alpha }_i{\lambda }_{i}R>C_i+T_i+{\pi }_i$. If this assumption does not exist, the tripartite agents in the supply chain will inevitably be unable to form low-carbon collaboration due to a too-little low-carbon collaboration benefit and high low-carbon input cost. Even if the tripartite agents in the supply chain choose low-carbon collaboration in this condition, the total benefits obtained by each agent after collaboration cannot be higher than their original fundamental benefits. The low-carbon collaboration will not have any practical significance at this time. Under this assumption, two conditions still need to be classified and discussed. □

• If $P_i+U_i>C_i$, that is, the government’s reward and punishment are more than the strategic risk cost of suppliers, manufacturers, and retailers, the stability of each equilibrium point is as shown in

  Table 5. Equilibrium points stability analysis.

  Equilibrium Points	Condition (1)		Condition (2)
  	Eigenvalue Sign	Stability	Eigenvalue Sign	Stability
  $E_1$	(+,+,+)	unstable point	(−,−,−)	ESS
  $E_2$	(−,+,+)	saddle point	(+,+,+)	unstable point
  $E_3$	(−,+,+)	saddle point	(+,+,+)	unstable point
  $E_4$	(−,+,+)	saddle point	(+,+,+)	unstable point
  $E_5$	(−,−,+)	saddle point	(−,−,+)	saddle point
  $E_6$	(−,−,+)	saddle point	(−,−,+)	saddle point
  $E_7$	(−,−,+)	saddle point	(−,−,+)	saddle point
  $E_8$	(−,−,−)	$\mathrm{ESS}$	(−,−,−)	ESS

  , Condition (1). In this condition, there is only one equilibrium point, $E_8$(1,1,1), that has all negative eigenvalues. Currently, the corresponding evolutionarily stable strategy (ESS) in the supply chain are {collaborate, collaborate, collaborate}.
• If $P_i+U_i<C_i$, that is, the government’s reward and punishment are less than the strategic risk cost of suppliers, manufacturers, and retailers, the stability of each equilibrium point is as shown in Table 5, Condition (2). There are two equilibrium points, $E_1\left(0,0,0\right)$ and $E_8\left(1,1,1\right),$ that have all negative eigenvalues. At this time, the corresponding evolutionarily stable strategy in supply chain are {non-collaborate, non-collaborate, non-collaborate}, {collaborate, collaborate, collaborate}.

4. System Dynamics Simulation Analysis

4.1. System Dynamics Assumptions

An evolutionary game is only a qualitative analysis, and the relationship between variables cannot be gauged intuitively. Thus, this study uses SD to analyze the evolutionary game problem further and tease out causality. The main variables that constitute the SD flow diagram can be determined by establishing the previous tripartite evolutionary game model. This study uses Vensim PLE to construct the SD flow diagram. Through the SD simulation, the strategy selection of each agent can be analyzed by changing the auxiliary variables. Then, it can provide advice for forming low-carbon collaboration under digital transformation. The SD flow diagram is shown in Figure 4.

There are three level variables, three rate variables, and 35 auxiliary variables in the SD flow diagram in this study. In the SD flow diagram, the level variable “suppliers’ low-carbon collaboration probability (x)” changing denotes the “supplier’s low-carbon collaboration strategies”. The rate of “suppliers’ low-carbon collaboration probability (x)” changing is represented by the rate variable “dx/dt”. The related variables of manufacturers and retailers are similar to suppliers. Most initial probabilities in system dynamics simulations are set to 0.5, to better observe the changing trend [63,64].

Let the initial simulation time be INITIAL TIME = 0, the simulation termination time be FINAL TIME = 100, and the simulation step size be TIME STEP = 1. According to [65,66,67] and other research, we obtained the initial-value assignment rules based on these references: (1) the benefits of each agent are more significant than the costs in the supply chain; and (2) the parameters of the manufacturers are greater than retailers, and retailers are greater than suppliers. The initial value of variables in the SD flow diagram are set as shown in

Table 6. The initial value of parameters in the system dynamics.

Parameters	Initial Value	Parameters	Initial Value	Parameters	Initial Value
${\alpha }_1$	1	${\lambda }_1$	0.3	$C_1$	80
${\alpha }_2$	1	${\lambda }_2$	0.4	$C_2$	120
${\alpha }_3$	1.2	${\lambda }_3$	0.3	$C_3$	100
${\pi }_1$	120	${\theta }_1$	0.5	$T_1$	30
${\pi }_2$	180	${\theta }_2$	0.5	$T_2$	45
${\pi }_3$	150	${\theta }_3$	0.5	$T_3$	40
$R$	150	$P_1$	60	$D_1$	40
$E$	80	$P_2$	70	$D_2$	50
$U_1$	70	$P_3$	50	$D_3$	45
$U_2$	70	$U_3$	60

.

4.2. Fundamental Simulation

The fundamental simulation results in this study are shown in Figure 5.

In Figure 5a, the initial low-carbon collaboration probability of agents under digital transformation are all set to 0.5; in Figure 5b, the initial probability of suppliers is set to 0.8, and the initial probability of manufacturers and retailers is set to 0.1; in Figure 5c, the initial probability of manufacturers is set to 0.8, and the initial probability of suppliers and retailers is set to 0.1; in Figure 5d, the initial probability of retailers is set to 0.8, and the initial probability of suppliers and manufacturers is set to 0.1.

As can be seen from Figure 5a–d, the probability of low-carbon collaboration among suppliers, manufacturers, and retailers is gradually increasing. The tripartite agents finally evolve into a steady strategies state {collaborate, collaborate, collaborate}. It can also be seen that, among the tripartite agents in the supply chain, when the initial collaboration probability is low, the manufacturers’ low-carbon collaboration probability increases most slowly. In contrast, the suppliers’ low-carbon collaboration probability increases faster than the other agents. A reasonable explanation for this state is that, under the ideal low-carbon collaboration situation, that is, with good benefit-driven and benefit-sharing, even if the initial low-carbon collaboration probability is low, the low-carbon collaboration can eventually be formed. At the same time, if the tripartite agents in the supply chain have a high initial probability of low-carbon collaboration, the speed of the low-carbon collaboration will be faster. In addition, the manufacturers are the core agents in the supply chain, since they usually dominate and drive low-carbon behaviors [68]. Thus, the manufacturers take over the essential connecting role of upstream suppliers and downstream retailers; they have more consideration for low-carbon collaboration and will pay more for it. Therefore, the probability of low-carbon collaboration for manufacturers will lag behind other agents in the supply chain.

4.3. Digital Transformation Simulation

In this study, digital transformation is the first parameter of the simulation analysis. The study assumes the average digital transformation to be 1 and the low-carbon collaboration probability in the supply chain to be 0.5.

In Figure 6a–c, this study sets the digital transformation α of suppliers, manufacturers, and retailers to 0.5, respectively, and that of the other agents are set to 1. That is, only one supply-chain agent lags behind the other agents in digital transformation. It can be seen from Figure 6a–c that when the digital transformation of one supply-chain agent lags behind that of other agents, this agent’s low-carbon collaboration probability in the supply chain will be significantly reduced. The other agents in the supply chain will still increase their low-carbon collaboration probability.

In Figure 6d, this study sets the digital transformation of suppliers and manufacturers as 0.5 and the digital transformation of retailers as 1. In Figure 6e, this study sets the digital transformation of suppliers and retailers as 0.5 and that of the manufacturers as 1. In Figure 6f, this study sets the digital transformation of manufacturers and retailers as 0.5 and that of the suppliers as 1. It can be seen from Figure 6d–f that when the two supply-chain agents lag behind the other agent in digital transformation, the low-carbon collaboration probability of these two supply-chain agents will be reduced more quickly. At the same time, when the two agents in the supply chain are too low in digital transformation, it will significantly hinder the other agent’s low-carbon collaboration probability, which has a higher digital transformation. As per Figure 6g, this study sets the digital transformation of each agent in the supply chain to 1.5. It can be seen from Figure 6g that, when all agents in the supply chain have a high digital transformation, their low-carbon collaboration probability will be significantly improved and will reach a stable strategic state {collaborate, collaborate, collaborate} in a short period.

A comprehensive analysis of the digital-transformation parameters simulation results reveals that the more agents there are with a low digital transformation, the more the overall low-carbon collaboration probability of the supply chain will be reduced significantly. Then, a stable low-carbon collaboration cannot be formed within a limited time. Additionally, the manufacturers’ low-carbon collaboration probability is most sensitive to digital transformation, and the low digital transformation will prompt the manufacturers to choose non-collaboration quickly.

A reasonable explanation for this state is that, firstly, digital transformation helps organizations shorten product development, generate innovation, and remain competitive in the Internet age [69]. Thus, a single agent’s low digital transformation in the supply chain will negatively impact product development and innovation, such as making it lag behind in fierce market-competition conditions and unable to participate in the low-carbon collaboration immediately. Secondly, a developed digital supply chain helps organizations increase responsiveness, flexibility, and agility [70]. Low digital transformation will make this supply-chain agent unable to accurately judge the direction of market development and quickly respond to low-carbon collaboration invitations from other agents in the supply chain.

4.4. Additional Benefits and Benefit-Sharing Simulation

The additional benefits of low-carbon collaboration are the core of the benefit-driven approach.

In Figure 7a, the additional benefit of low-carbon collaboration is $R$ = 400, $E$ = 300; in Figure 7b, the additional benefit of low-carbon collaboration is $R$ = 60, $E$ = 50. In Figure 7c, this study sets the digital transformation of each agent in the supply chain to 0.75; in Figure 7d, this study sets the digital transformation of each agent in the supply chain to 1.25. Figure 7c,d are both based on the high additional benefit of low-carbon collaboration, that is, $R$ = 400, $E$ = 300.

According to the simulation of different additional benefits of low-carbon collaboration, it can be seen from Figure 7a,b that the tremendous additional benefits significantly promote low-carbon collaboration probability in the supply chain. Thus, the tripartite agents in the supply chain quickly adopt the strategies of {collaborate, collaborate, collaborate} in this situation. In contrast, the lack of additional benefits will significantly inhibit the low-carbon collaboration probability of each agent in the supply chain. It can be seen from Figure 7c,d that even as the additional benefit of low-carbon collaboration changes, digital transformation still has a considerable influence.

A reasonable explanation for this state is a positive relationship between supply-chain collaboration and collaboration benefits [71]. Enterprise collaboration costs and benefits directly affect gaming behavior [72]. Therefore, the additional benefits of low-carbon collaboration in the supply chain directly affect the low-carbon collaboration probability in the supply chain. This is an intuitive display of a benefit-driven effect. Therefore, combined with the above simulation results, we believe that digital transformation is a regulator between a low-carbon benefit-driven approach and low-carbon collaboration in the supply chain.

After benefit-driven considerations, benefit sharing is also crucial for low-carbon collaboration.

In Figure 8a,b, this study sets ${\lambda }_1$ = ${\theta }_1$ = 0.8, ${\lambda }_2$ = ${\lambda }_3$ = 0.1, and ${\theta }_2={\theta }_3=$ 0.2; that is, suppliers take most of the additional benefits of low-carbon collaboration in the entire supply chain, and manufacturers and retailers can only obtain a small part of the additional benefits. As shown in Figure 8a, this study sets the initial low-carbon collaboration probability of agents in the supply chain as 0.5, and in Figure 8b, the initial probability is set as 0.1.

In Figure 8c,d, this study sets ${\lambda }_2$ = ${\theta }_2$ = 0.8, ${\lambda }_1$ = ${\lambda }_3$ = 0.1, and ${\theta }_1={\theta }_3=$ 0.2. That is, manufacturers take most of the additional benefits of low-carbon collaboration in the entire supply chain, whereas suppliers and retailers can only obtain a small part of the additional benefits. In Figure 8c, this study sets agents’ initial low-carbon collaboration probability in the supply chain to 0.5, and in Figure 8d, the initial probability is set to 0.1.

In Figure 8e,f, this study sets ${\lambda }_3$ = ${\theta }_3$ = 0.8, ${\lambda }_1$ = ${\lambda }_2$ = 0.1, and ${\theta }_1={\theta }_2=$ 0.2. That is, retailers take most of the additional benefits of low-carbon collaboration in the entire supply chain, and suppliers and manufacturers can only obtain a small part of the additional benefits. In Figure 8e, this study sets agents’ initial low-carbon collaboration probability in the supply chain to 0.5, and in Figure 8f, the initial probability is set to 0.1.

It can be seen from Figure 8a–f that, in this simulated condition, the additional benefit sharing in the low-carbon collaboration of the supply chain is unreasonable. One agent takes most of the low-carbon collaboration additional benefits in the supply chain. However, if the agents’ initial low-carbon collaboration probability is high (all greater than 0.5), they will also reach a stable strategies state, {collaborate, collaborate, collaborate}; that is, they have formed a low-carbon collaboration. However, if agents’ initial low-carbon collaboration probability is low, the stable strategies state {collaborate, collaborate, collaborate} cannot be achieved within a limited game time; that is, a low-carbon collaboration cannot be formed. A reasonable explanation for this state is that benefit-sharing agreements could increase the motivation of organization members under certain conditions [73]. An illogical additional benefit sharing of low-carbon collaboration will not only not improve the probability of the low-carbon collaboration of individual supply-chain agents but also inhibit the formation of whole low-carbon collaboration in the supply chain.

4.5. Government Rewards and Punishments Simulation

According to the analysis of the evolutionary game equilibrium point above, government rewards and punishments correlate closely with the equilibrium of the low-carbon collaboration game in the supply chain. Therefore, the simulation in this section will also be according to the two conditions mentioned above.

Condition 1: Government rewards and punishments are less than the supply-chain agents’ strategic risk cost, as shown in Figure 9a,b. In Figure 9a, the government rewards and punishments $P_i$ and $U_i$ are both set to 0; that is, the government does not provide any support and management for low-carbon collaboration in the supply chain. In Figure 9b, the government rewards and punishments $P_i$ and $U_i$ are both set to 30; that is, the government supports and manages low-carbon collaboration in the supply chain slightly, and does not cover the strategic risk cost of supply-chain agents. According to the simulation results, whether there is no reward and punishment or little rewards and punishments, as long as the sum of rewards and punishments does not cover the strategic risk cost of the agents in the supply chain, the low-carbon collaboration probability of the tripartite agents will ultimately tend to (0,0,0); that is, they will adopt {non-collaborate, non-collaborate, non-collaborate} strategies. Suppose the government does not have any rewards and punishments, and compared with little rewards and punishments, the agents in the supply chain will tend to choose the strategies of {non-collaborate, non-collaborate, non-collaborate} more quickly.

Condition 2: Government rewards and punishments are greater than the supply-chain agents’ strategic risk cost, as shown in Figure 9c,d.

In Figure 9c, the government rewards and punishments $P_i$ and $U_i$ are both set to 60; that is, the government’s support and management for low-carbon collaboration in the supply chain exceed their strategic risk costs. They will eventually tend to (1,1,1); that is, to adopt the {collaborate, collaborate, collaborate} strategies. In Figure 9d, the government rewards and punishments $P_i$ and $U_i$ are both set to 100; that is, the government strongly supports and manages low-carbon collaboration in the supply chain. At this time, the trend of increasing probability is more apparent and faster; that is, they tend to adopt the strategies of {collaborate, collaborate, collaborate} within a short period.

A reasonable explanation for this state is that, in the green supply chain, to gain long-term benefits, governments should enact and enforce increasingly strict environmental regulations and increase relevant subsidies and penalties [74]. At the same time, the government’s active rewards and punishments can also provide confidence for agents in the supply chain to choose low-carbon development strategies and form collaboration. The long-term benefits from active government rewards and punishments incentivize the probability of low-carbon collaboration among the supply chain and create a stable virtuous cycle. It is not easy to form a stable low-carbon collaboration in the long term without active government rewards and punishments.

4.6. Other Parameters Simulation

Finally, this study conducts a simulation analysis of the low-carbon input cost $T_i$ and the exclusion cost loss $D_i$. In Figure 10a, this study sets the low-carbon input costs of suppliers, manufacturers, and retailers as 10, 20, and 15; in Figure 10b, this study sets the low-carbon input costs of 80, 100, and 90. In Figure 11a, this study sets the exclusion loss of suppliers, manufacturers, and retailers to 10, 15, and 12; in Figure 11b, this study sets the exclusion loss of 100, 150, and 120.

It can be seen from Figure 10 and Figure 11 that input cost and exclusion loss are a pair of parameters with opposite impacts. High input costs will significantly hinder the formation of a low-carbon collaboration in the supply chain. On the other hand, excessive exclusion loss will rapidly increase the probability of low-carbon collaboration in the supply chain. Among supply-chain agents, high low-carbon input costs and little exclusion losses have the most significant inhibitory effect on the probability of manufacturers’ participation in low-carbon collaboration. A reasonable explanation for this state is that, in supply-chain collaboration, the probability of collaboration between agents is affected by the cost, and a reasonable cost will better facilitate collaboration between agents [75]. A lower input cost can drop the threshold for each supply-chain agent to participate in the low-carbon collaboration. At the same time, a higher exclusion loss makes it mandatory for each supply-chain agent to participate in the low-carbon collaboration, even forcibly, to maintain their benefits.

5. Conclusions and Implications

5.1. Conclusions

The low-carbon collaboration in the supply chain under digital transformation is an essential extension of low-carbon research. This study constructs a tripartite evolutionary game model of the supply chain, consisting of suppliers, manufacturers, and retailers. Then, this study obtained ESS for two different situations by solving the replicator dynamics equations and Jacobian matrix. Through the evolutionary game in the supply chain, this study obtained the driving factors and formation logic of low-carbon collaboration in the supply chain. Subsequently, this study used the SD method to simulate the strategies for the above-mentioned game situations and derive conclusions from quantitative perspectives. The primary conclusions are as follows.

First, based on the evolutionary game model and SD, this research obtained a clear path for low-carbon collaboration in the supply chain, that is, low-carbon benefit-driven effect promote the collaboration benefit-sharing, thereby increasing the probability of low-carbon collaboration.

Second, digital transformation is an essential regulator of low-carbon collaboration in the supply chain. The digital transformation of each agent in the supply chain can amplify the low-carbon benefit-driven effect and affects the formation of the tripartite low-carbon collaboration. All agents can quickly form the tripartite low-carbon collaboration in the supply chain through deeper digital transformation.

Third, collaboration benefit sharing can perfectly coordinate the vertical supply chain under the low-carbon collaboration, thereby preventing damage to low-carbon collaboration probability and supply-chain profit. Driven by low-carbon benefits, benefit sharing is the core of low-carbon collaboration in the supply chain.

Fourth, government support and management are critical links in the low-carbon collaboration formation path of the supply chain. If government rewards and punishments cannot cover the strategic risk cost of supply-chain agents, it will not be possible to form low-carbon collaboration in a supply chain.

Finally, some other factors also have important impacts on low-carbon collaboration in the supply chain. For instance, low-carbon exclusion losses are positively related to low-carbon collaboration probability in a supply chain, and input costs are negatively related to low-carbon collaboration probability.

5.2. Implications

This study proposes several management implications based on the above research and conclusions. First, for the average agents in the supply chain, low-carbon collaboration and digital transformation have become equally important under the dual requirements of low-carbon development and digital economy. Previous research has been limited to the field of low-carbon development and has largely overlooked digital transformation, which is a significant global development trend. The average agents in the supply chain should accurately grasp the opportunities of the times, accelerate the low-carbon collaboration and digital transformation, and improve the benefits of low-carbon collaboration, to realize the coordinated development of the economy and environment.

Secondly, government or regulatory agencies should combine market laws and digital-transformation trends to promote the formation of a low-carbon collaboration in the supply chain within the scope of reasonable rewards and punishments. A meager rewards and punishments policy hinders the formation of a low-carbon collaboration in the supply chain, whereas a very high rewards and punishments policy will create significant financial pressure on the government. Therefore, government or regulatory agencies should draw up reasonable low-carbon guide policies and establish a market-supervision system, such as actively combining digital means supervising carbon emissions. This could provide sufficient low-carbon development confidence to all agents in the supply chain.

Finally, the core agents of the supply chain are strong promoters of low-carbon collaboration. They are also pioneering in digital transformation, and, thus, they should consciously and actively undertake greater obligations to achieve low-carbon collaboration and digital transformation in the supply chain. The core agents and enterprises of the supply chain may be makers of the allocation mechanism or takers of most of the collaboration benefits. Thus, they should take the lead in building benefits and cost-sharing regulations in the whole supply chain and promote low-carbon collaboration to develop benignly.

5.3. Limitations

We recognize and acknowledge the existence of some significant limitations due to basic model assumptions, which may provide avenues for future research. Firstly, due to the limitation of the game model, this study does not set the government game agent, and the government’s benefits and costs of low-carbon collaboration cannot be measured. Therefore, it is impossible to further determine the optimal and reasonable range of government rewards and punishments. Secondly, the low-carbon collaboration model in this study only considers benefit sharing in the supply chain without considering cost sharing in the supply chain. The combination of the benefit and cost sharing of each agent can make the supply chain low-carbon collaboration stronger. Finally, low-carbon supply chain collaboration requires the participation of more supply-chain agents such as recyclers, channels, and consumers. Thus, further research on low-carbon collaboration is needed to propose a different model that could provide more insights.