Evolutionary Game Analysis of Providers’ and Demanders’ Low-Carbon Cooperation in Cloud Manufacturing Mode

Abstract

The low-carbon cooperation between providers and demanders is one of the ways to achieve sustainable development in cloud manufacturing, which has become an important issue. However, the effective ways for the cloud platform to encourage such cooperation are unclear. Considering the low-carbon strategies of the supply and demand sides and the regulation of the cloud platform, an evolutionary game model involving service providers, service demanders, and the cloud platform is established, and the tripartite evolutionary stability is discussed. Further, the impacts of important factors, such as regulatory costs, on the tripartite strategies are analyzed through numerical simulation. The results illustrate that the cloud platform reasonably optimizes the rewards and penalties for low-carbon cooperation to promote the enthusiastic participation of service providers and demanders. The cloud platform can set penalties (rewards) for providers based on their low-carbon costs and rewards (penalties). Additionally, the low-carbon costs of service providers and the additional costs of demanders negatively affect the low-carbon cooperation between providers and demanders. Meanwhile, the low-carbon costs and additional costs for the providers and demanders to engage in low-carbon cooperation are affected by the rewards and penalties of the cloud platform. The results could provide insights into the game decisions of the supply and demand sides and the cloud platform, facilitating sustainable supply chain advancement.

1. Introduction

Cloud manufacturing has emerged to satisfy diverse user requirements; it allocates manufacturing resources on demand [1]. Specifically, this manufacturing mode integrates the dispersed manufacturing resources and capabilities for centralized management through advanced technologies [2]. Generally, cloud manufacturing involves three stakeholders: service providers, service demanders, and the cloud platform. Service providers publish idle manufacturing resources and capabilities on the industrial cloud platform through virtualization, while service demanders publish service demand information on the cloud platform [3]. The cloud platform matches services and demands and regulates service transactions between the supply and demand sides to ensure the healthy operation of the cloud manufacturing system.

As a sustainable manufacturing mode, cloud manufacturing improves resource utilization and reduces carbon emissions through manufacturing resource sharing [4]. Low-carbon cooperation in cloud manufacturing is important for environmental protection [5], and the active participation of providers and demanders in low-carbon cooperation produces environmental value. However, cloud manufacturing stakeholders adopting low-carbon strategies may not be able to reduce emissions independently due to the huge low-carbon costs. Therefore, cooperative emission reduction among stakeholders could be an effective way to reduce carbon emissions. Providers, demanders, and the platform could jointly develop low-carbon innovative technologies, share low-carbon costs, and reap carbon emission reduction benefits. Taking a cloud manufacturing service enterprise in Shanghai as an example, the enterprise provides machining manufacturing services and publishes information about idle machine tools on the cloud platform. Through the cloud platform, a demander with machining demands for hydraulic cylinder parts is matched to a provider with idle machine tools. After successful matching, the machine tool provider processes the hydraulic cylinder parts. During this process, the provider can optimize processing technology and processing parameters after evaluating processing energy consumption. It can also monitor its machine tools’ performance to suppress vibration (if present) in real time by developing intelligent machine tools, thus achieving low-carbon cooperation. In other words, the machine tool provider offers low-carbon manufacturing services, for which service demanders are willing to pay higher prices, i.e., low-carbon cooperation. In this study, low-carbon manufacturing services refer to more environmentally friendly machining services.

Meanwhile, service providers engaging in low-carbon cooperation must pay low-carbon costs, such as the costs for low-carbon technologies. Moreover, the low-carbon cooperation process might benefit other providers, rendering most service providers reluctant to provide low-carbon manufacturing services. Furthermore, demanders without strong environmental awareness may not be willing to pay extra for low-carbon manufacturing services, while their willingness to pay for low-carbon services increases with increasing environmental awareness [6]. Facing the increasingly complex and uncertain environment, providers and demanders have limited rationality and decision-making levels. Therefore, the low-carbon cooperation between providers and demanders is a long-term dynamic game with gains and losses. Providers and demanders cannot choose the optimal strategy at the beginning, but eventually achieve it to maximize their own benefits through continuous observation, learning, and imitation. To promote low-carbon cooperation among service providers and demanders, the cloud platform must provide rewards and penalties through regulation. This study mainly solves the following two problems: What are the equilibrium conditions for providers’ and demanders’ low-carbon cooperation? How do the rewards and penalties of the cloud platform affect the strategy evolution of providers and demanders?

Most of the existing studies on supply chain low-carbon cooperation are conducted from the perspective of carbon emission reduction coordination based on game theory [7,8], such as the Stackelberg game [9]. Yet, the game model assumes that stakeholders are completely rational, rendering it incapable of describing the long-term dynamic process of low-carbon cooperation [10]. Meanwhile, providers and demanders have bounded rationality and incomplete information, making it difficult to predict the strategies of other stakeholders. Providers and demanders learn through observation and imitation, gradually finding a suitable strategy, which is in line with the evolutionary game process [11]. Although scholars have analyzed the platform’s credit supervision through evolutionary game [12], few have investigated the evolution of the trend of low-carbon cooperation between service providers and demanders while considering the cloud platform’s regulation.

This paper analyzes the evolutionary trend of low-carbon cooperation in cloud manufacturing based on evolutionary game theory and investigates the impacts of various factors on the evolutionarily stable strategy (ESS) in different situations. The contributions of this research are three-fold. Firstly, an evolutionary game model is constructed to describe the low-carbon strategy evolution of providers and demanders instead of adopting the static low-carbon cooperation model in the supply chain, thus filling the research gap in the low-carbon cooperation under the cloud manufacturing mode. Secondly, this study considers the role of the platform and expands the two-party game between providers and demanders to a tripartite game to explore the strategy interactions among providers, demanders, and the cloud platform. Thirdly, the strategy evolution under different cloud platform rewards and penalties is discussed. Simulation results indicate that different rewards and penalties lead to different convergence directions and convergence speeds of the system, thus providing support for the cloud platform to encourage the participation of the supply and demand sides in low-carbon cooperation.

The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. Section 3 establishes the evolutionary game model to analyze the low-carbon cooperation and regulation in cloud manufacturing. ESS is analyzed through the replicator dynamic equation in Section 4. Section 5 analyzes the impacts of important factors on the strategy evolution trend, such as cloud platform rewards and penalties. Section 6 presents the conclusions.

2. Related Works

With the promotion of national industrial Internet applications in China, over 100 cloud manufacturing service platforms of different sizes and characteristics have emerged successively, such as Aerospace Cloud Network and Intelligent Cloud “iSESOL Cloud”, which integrate a large number of service resources and transaction needs. At present, extensive research has been conducted on cloud manufacturing architecture [13,14], virtualization technology [15], manufacturing service combination, and service matching [16]. However, few studies have focused on low-carbon and green decision-making in cloud manufacturing. Tong et al. [17] designed a customer-oriented method to perform multi-task green scheduling for cloud manufacturing with the lowest total energy consumption as the goal. Wei et al. [4] constructed a manufacturing service evaluation system, considering environmental factors, and selected suppliers based on gray correlation. Li et al. [18] proposed a green supplier selection model for the cloud manufacturing platform, based on multi-criteria decision-making (MCDM). Gao et al. [19] constructed a Stackelberg game model comprising a supplier and a manufacturer and analyzed the government’s low-carbon subsidy strategy. Yang et al. [20] built a joint-decision-making model for the carbon emission reduction of cloud platform operators and service providers based on the Stackelberg game and introduced a side-payment self-executing contract to coordinate supply chain environmental governance. Zeng et al. [21] studied the coordination mechanism of the supply chain under the cloud manufacturing platform, considering consumer preferences through game theory. However, these studies did not consider the dynamic decision-making process of the cloud platform, providers, and demanders.

The evolutionary game considers the bounded rationality of the players and describes the continuous and revised decision-making process through constructing dynamic relations [22,23,24]. Players continuously improve their benefits by adjusting their strategies, finally achieving ESS. A player does not change their ESS if other players do not change their strategies. Evolutionary game theory has been widely applied to analyze the stakeholders’ interactions in cloud manufacturing, such as resources, capacity, knowledge sharing [25], and transaction trust [26], as shown in

Table 1. Research on the application of evolutionary game theory in cloud manufacturing.

Players	Strategies
enterprises	adopt cloud manufacturing/traditional mode [30]
providers	perform/violate non-intelligent contracts
demanders	perform/violate non-intelligent contracts [31]
providers	actively/negatively improve the quality
demanders	participate/not participate in quality collaboration [32]
cloud platform	provide better/basic services
service providers	share/not share resources
demanders	require better/basic services [25]
service platform	provide positive/negative service
providers	share/not share production capacity
demanders	expand/not expand production capacity [33]
service providers	share/not share knowledge
service integrators	incentivize/not incentivize knowledge sharing [34]
platform regulators	supervise/not supervise transaction process
providers	trust/distrust
demanders	trust/distrust [26]
cloud platform	supervise/not supervise
manufacturing enterprise	collaborative/general cooperation
demand enterprise	collaborative/general cooperation [35]
goverment	supervise/not supervise
enterprise A	collaborative/noncollaborative innovation
enterprise B	collaborative/noncollaborative innovation [28]

. Concerning the low-carbon and green decision-making in cloud manufacturing, Zeng et al. [27] analyzed the low-carbon strategy selection process of cloud manufacturing companies based on the evolutionary game theory, while the low-carbon game relationship between demanders and the cloud platform was not considered. Zhai et al. [28] analyzed enterprises’ green innovation cooperation in the cloud platform considering government supervision based on evolutionary game theory, but ignored the incentive effect of the cloud platform on providers’ low-carbon decisions. Wang et al. [29] constructed an evolutionary game model of innovation cooperation among manufacturing enterprises, platform enterprises, and governments.

In summary, few studies have considered the low-carbon cooperation evolution of cloud manufacturing stakeholders. Additionally, previous research has failed to consider the important role of the cloud platform in low-carbon cooperation and neglected the impact of the cloud platform’s regulation on the low-carbon cooperation between providers and demanders. Therefore, this paper establishes an evolutionary game model to discuss the low-carbon and regulation strategies of providers, demanders, and the cloud platform in cloud manufacturing and explores how the cloud platform can encourage the participation of providers and demanders in low-carbon cooperation, thus providing a reference for the low-carbon management of the cloud platform.

3. Tripartite Evolutionary Game

3.1. Problem Description and Assumptions

This paper views the cloud platform, service demanders, and providers as players in the evolutionary game, with bounded rationality and asymmetrical transaction information. The strategies of the players are optimal after trying different strategies in multiple games. Meanwhile, the cloud platform encourages providers and demanders to participate in low-carbon cooperation. The game framework for the cloud platform, providers, and demanders is shown in Figure 1.

The assumptions and the tripartite benefits are as follows:

(1) Service providers have two strategy options: {provide low-carbon manufacturing services (P), not provide low-carbon manufacturing services (NP)}. Service demanders also have two strategies: {adopt low-carbon manufacturing services (A), not adopt low-carbon manufacturing services (NA)}. The cloud platform has two strategy options: {regulate providers and demanders (R), not regulate providers and demanders (NR)}. When the cloud platform regulates service demanders and providers, it rewards the providers for providing low-carbon services and the demanders for adopting low-carbon services. Otherwise, it punishes the providers accordingly to press them to participate in low-carbon cooperation.

(2) The probability that a service provider provides low-carbon manufacturing services is $X(0\le X\le 1)$, with $X=1$ signifying the provider adopting strategy P and $X=0$ signifying the provider adopting strategy NP. Similarly, the probability of a demander adopting strategy A is $Y(0\le Y\le 1)$, and the probability of a demander adopting strategy NA is $1-Y$. The probability of the platform adopting strategy R is Z, $(0\le Z\le 1)$, with $Z=1$ signifying the platform adopting strategy R and $Z=0$ signifying the cloud platform not adopting strategy R.

(3) The basic benefit for service providers providing manufacturing services is $G_1$ [28]. Service providers providing low-carbon manufacturing services also bear low-carbon costs $C_1$, such as the costs for improving production processes [27], and receive low-carbon rewards $S_1$ from the cloud platform. Demanders’ benefit for adopting strategy NA is $G_2$ [28]. When choosing strategy A, they pay an extra price $C_2$ [7], but receive rewards $S_2$ from the cloud platform.

(4) The cloud platform adopting strategy R also bears a regulatory cost $C_r$ [35]. Additionally, it provides rewards $S_1$ to providers adopting strategy P and rewards $S_2$ to demanders adopting strategy A. When the providers do not participate in low-carbon cooperation, the cloud platform punishes the providers and obtains benefits $P_1$ [26].

(5) Environmental protection has no short-term economic benefit [36], and the cloud platform must pay regulatory costs. Therefore, continuous regulation is difficult without financial support. The government subsidizes the cloud platform adopting strategy R [35]. Taken together, there are four situations, as follows: With service providers adopting strategy P and the demanders adopting strategy A, the subsidy is $T_1$. With the providers adopting strategy P and the demanders adopting strategy NA, the subsidy is $T_2$. With the providers adopting strategy NP and the demanders adopting strategy A, the subsidy is $T_3$. With the providers adopting strategy NP and the demanders adopting strategy NA, the subsidy is $T_4$, where $T_1>T_2$, $T_3>T_4$. The main parameters and meanings are presented in

Table 2. Main parameters and meanings.

Stakeholders	Symbols	Meanings
Service	$G_1$	Providers’ benefit for adopting strategy NP
providers	$C_1$	Low-carbon cost of low-carbon services
Service	$G_2$	Demanders’ benefit for adopting strategy NA
demanders	$C_2$	Additional cost of adopting strategy A
	$G_3$	The cloud platform’s benefit for adopting strategy NR
	$C_r$	Regulatory cost of the cloud platform
Cloud	$S_1$	Reward of the cloud platform for adopting strategy P
platform	$S_2$	Reward of the cloud platform for adopting strategy A
	$P_1$	Penalty of the cloud platform for adopting strategy NP
	$T_i,i=1,2,3,4$	Government’s subsidy to the cloud platform

.

3.2. Model Building

$X=1$ denotes the providers providing low-carbon manufacturing services, and $X=0$ denotes otherwise. $Y=1$ denotes the demanders adopting low-carbon services, and $Y=0$ denotes otherwise. $Z=1$ denotes the cloud platform regulating providers and demanders, while $Z=0$ denotes otherwise. Therefore, the eight strategy combinations are presented in Figure 2. According to the above assumptions, a payoff matrix is derived, as shown in

Table 3. Payoff matrix of the evolutionary game model.

Providers and Demanders		The Cloud Platform
		Regulate $\left(\mathbf{Z}\right)$	Not Regulate $(\mathbf{1}-\mathbf{Z})$
	Demanders adopting	$G_1-C_1+S_1$	$G_1-C_1$
	strategy A	$G_2-C_2+S_2$	$G_2-C_2$
Providers adopting	$\left(Y\right)$	$G_3-C_{\mathrm{r}}-S_1-S_2+T_1$	$G_3$
strategy P $\left(X\right)$	Demanders adopting	$G_1-C_1+S_1$	$G_1-C_1$
	strategy NA	$G_2$	$G_2$
	$(1-Y)$	$G_3-C_{\mathrm{r}}-S_1+T_2$	$G_3$
	Demanders adopting	$G_1-P_1$	$G_1$
	strategy A	$G_2-C_2+S_2$	$G_2-C_2$
Providers adopting	$\left(Y\right)$	$G_3-C_{\mathrm{r}}+P_1-S_2+T_3$	$G_3$
	Demanders adopting	$G_1-P_1$	$G_1$
strategy NP $(1-X)$	strategy NA	$G_2$	$G_2$
	$(1-Y)$	$G_3-C_{\mathrm{r}}+P_1+T_4$	$G_3$

.

(1) Expected benefits for service providers

The expected benefits for providers adopting strategy P can be calculated with Equation (1):

$$\begin{array}{cc}\hfill & U_X=YZ\left(G_1-C_1+S_1\right)+Y(1-Z)\left(G_1-C_1\right)+\hfill \\ \hfill & (1-Y)Z\left(G_1-C_1+S_1\right)+(1-Y)(1-Z)\left(G_1-C_1\right)\hfill \\ \hfill & =Z{S}_1+\left(G_1-C_1\right)\hfill \end{array}$$

(1)

The expected benefits for providers adopting strategy NP can be expressed as Equation (2):

$$\begin{array}{cc}\hfill & U_{1-X}=YZ\left(G_1-P_1\right)+Y(1-Z)G_1+(1-Y)Z\left(G_1-P_1\right)+(1-Y)(1-Z)G_1\hfill \\ \hfill & =G_1-Z{P}_1\hfill \end{array}$$

(2)

The average benefits for the service providers are calculated in Equation (3):

$$\begin{array}{cc}\hfill & U_{X,1-X}=X{U}_X+(1-X)U_{1-X}=\mathrm{X}\left[Z{S}_1+\left(G_1-C_1\right)\right]+(1-X)\left(G_1-Z{P}_1\right)\hfill \\ \hfill & =G_1-Z{P}_1+XZ\left(S_1+P_1\right)-X{C}_1\hfill \end{array}$$

(3)

Therefore, the replicator dynamic equation for the providers can be written as Equation (4). In essence, the replicator dynamic equation determines how often a particular strategy is adopted within a population,

$$\begin{array}{cc}\hfill & F\left(X\right)=\frac{dX}{dt}=X\left(U_X-U_{X,1-X}\right)=X(1-X)\left(U_X-U_{1-X}\right)\hfill \\ \hfill & =X(1-X)\left[Z\left(S_1+P_1\right)-C_1\right]\hfill \end{array}$$

(4)

where $F\left(X\right)$ represents the change rate for the providers to adopt strategy P. When the expected benefits for the providers adopting strategy P are higher than the average benefits for the group, strategy P diffuses in the group, and the probability of adopting strategy P increases. With $F\left(X\right)>0$, service providers tend to provide low-carbon manufacturing services. With $F\left(X\right)<0$, service providers are unwilling to provide low-carbon services.

(2) Expected benefits for service demanders

The expected benefits for the demanders adopting low-carbon services can be calculated with Equation (5):

$$\begin{array}{cc}\hfill & U_Y=XZ\left(G_2-C_2+S_2\right)+X(1-Z)\left(G_2-C_2\right)+\hfill \\ \hfill & (1-X)Z\left(G_2-C_2+S_2\right)+(1-X)(1-Z)\left(G_2-C_2\right)\hfill \\ \hfill & =G_2-C_2+Z{S}_2\hfill \end{array}$$

(5)

The expected benefits for the demanders choosing strategy NA can be calculated with Equation (6):

$$\begin{array}{cc}\hfill & U_{1-Y}=XZ{G}_2+X(1-Z)G_2+(1-X)Z{G}_2+(1-X)(1-Z)G_2\hfill \\ \hfill & =G_2\hfill \end{array}$$

(6)

The average benefits for the demanders are obtained in Equation (7):

$$\begin{array}{cc}\hfill & U_{Y,1-Y}=Y{U}_Y+(1-Y)U_{1-Y}=\mathrm{Y}\left(G_2-C_2+Z{S}_2\right)+(1-Y)G_2\hfill \\ \hfill & =G_2+\mathrm{Y}(Z{S}_2-C_2)\hfill \end{array}$$

(7)

The replicator dynamic equation for the service demanders can be expressed as Equation (8):

$$\begin{array}{cc}\hfill & H\left(Y\right)=\frac{dY}{dt}=Y\left(U_Y-U_{Y,1-Y}\right)=Y(1-Y)\left(U_Y-U_{1-Y}\right)\hfill \\ \hfill & =Y(1-Y)(Z{S}_2-C_2)\hfill \end{array}$$

(8)

where $F\left(Y\right)$ denotes the change rate for the demanders to choose strategy A. When $F\left(Y\right)>0$, the service demanders tend to adopt low-carbon manufacturing services. When $F\left(Y\right)<0$, the demanders are unwilling to adopt low-carbon services.

(3) Expected benefits for the cloud platform

The expected benefits for the cloud platform for choosing strategy R are expressed in Equation (9):

$$\begin{array}{cc}\hfill & U_Z=XY\left(G_3-C_{\mathrm{r}}-S_1-S_2+T_1\right)+X(1-\mathrm{Y})\left(G_3-C_{\mathrm{r}}-S_1+T_2\right)+\hfill \\ \hfill & (1-\mathrm{X})\mathrm{Y}\left(G_3-C_{\mathrm{r}}+P_1-S_2+T_3\right)+(1-\mathrm{X})(1-\mathrm{Y})\left(G_3-C_{\mathrm{r}}+P_1+T_4\right)\hfill \\ \hfill & =G_3-C_{\mathrm{r}}+P_1+T_4+X\left(T_2-T_4-P_1-S_1\right)+\hfill \\ \hfill & Y\left(T_3-T_4-S_2\right)+XY\left(T_1-T_2-T_3+T_4\right)\hfill \end{array}$$

(9)

The expected benefits for the platform choosing strategy NR are expressed as Equation (10):

$$\begin{array}{cc}\hfill & U_{1-Z}=XY{G}_3+X(1-\mathrm{Y})G_3+(1-\mathrm{X})\mathrm{Y}{G}_3+(1-\mathrm{X})(1-\mathrm{Y})G_3=G_3\hfill \end{array}$$

(10)

The average benefits for the platform are calculated in Equation (11):

$$\begin{array}{cc}\hfill & U_{Z,1-Z}=Z{U}_Z+(1-Z)U_{1-Z}=G_3+Z\left(P_1+T_4-C_{\mathrm{r}}\right)+\hfill \\ \hfill & ZX\left(T_2-T_4-P_1-S_1\right)+ZY\left(T_3-T_4-S_2\right)+ZXY\left(T_1-T_2-T_3+T_4\right)\hfill \end{array}$$

(11)

The replicator dynamic equation for the platform is derived in Equation (12):

$$\begin{array}{cc}\hfill & W\left(Z\right)=\frac{dZ}{dt}=Z\left(U_Z-U_{Z,1-Z}\right)=Z(1-Z)\left(U_Z-U_{1-Z}\right)\hfill \\ \hfill & =Z(1-Z)\left[-C_{\mathrm{r}}+P_1+T_4+\right.X\left(T_2-T_4-P_1-S_1\right)+Y\left(T_3-T_4-S_2\right)\hfill \\ \hfill & +XY\left(T_1-T_2-T_3+T_4\right)]\hfill \end{array}$$

(12)

where $F\left(Z\right)$ denotes the change rate for the platform regulating providers and demanders. $F\left(Z\right)>0$ indicates that the platform tends to regulate providers and demanders, while $F\left(Z\right)<0$ indicates that the platform is unwilling to regulate providers and demanders.

A three-dimensional dynamic system comprising the providers, demanders, and the platform is shown as Equation (13). The three parties adjust their strategies through continuous learning and trial and error, conforming to a dynamic process described by an evolutionary game:

$$\left\{\begin{array}{c}F\left(X\right)=\frac{dX}{dt}=X(1-X)\left[Z\left(S_1+P_1\right)-C_1\right]\hfill \\ H\left(Y\right)=\frac{dY}{dt}=Y(1-Y)(Z{S}_2-C_2)\hfill \\ W\left(Z\right)=\frac{dZ}{dt}=Z(1-Z)\left[-C_{\mathrm{r}}+P_1+T_4+\right.\hfill \\ X\left(T_2-T_4-P_1-S_1\right)+Y\left(T_3-T_4-S_2\right)+\hfill \\ XY\left(T_1-T_2-T_3+T_4\right)]\hfill \end{array}\right.$$

(13)

According to the stability theorem of the replicator dynamic equation, tripartite ESS is solved. The propositions and proofs are as follows.

Proposition 1.

(1) If $Z=Z_1^{*}=\frac{C_1}{S_1+P_1}$,$F\left(X\right)\equiv 0$, the probability of the providers adopting strategy P is stable and in [0, 1].

(2) If $Z\ne Z_1^{*}=\frac{C_1}{S_1+P_1}$ and $F\left(X\right)=0$, the probability of the providers adopting strategy P is $X=0$ or $X=1$.

Proof of Proposition 1. 

The first derivative of Equation (4) can be obtained as follows:

$$F^{\prime }\left(X\right)=(1-2X)\left[Z\left(S_1+P_1\right)-C_1\right]$$

(14)

when Equation (4) equals 0, $X=0$ or $X=1$ or $Z=Z_1^{*}=\frac{C_1}{S_1+P_1}>0$.

(1) If $1>Z>Z_1^{*}$, $F^{\prime }\left(1\right)<0$,$F^{\prime }\left(0\right)>0$. The probability of the providers adopting strategy P is $X=1$.

(2) If $0<Z<Z_1^{*}$, $F^{\prime }\left(1\right)>0$,$F^{\prime }\left(0\right)<0$. The probability of the providers adopting strategy P is $X=0$. □

The phase diagram of the replicator dynamic equation of the providers is shown in Figure 3.

Proposition 2.

(1) If $Z=Z_2^{*}=\frac{C_2}{S_2}$,$H\left(Y\right)\equiv 0$, $Y\in [0,1]$ and the probability of the demanders adopting strategy A is stable.

(2) If $Z\ne Z_2^{*}=\frac{C_2}{S_2}$ and $H\left(Y\right)=0$, the probability of the demanders adopting strategy A is $X=0$ or $X=1$.

Proof of Proposition 2. 

The first derivative of Equation (8) can be obtained as follows:

$$H^{\prime }\left(Y\right)=(1-2Y)\left[Z{S}_2-C_2\right]$$

(15)

When Equation (8) equals 0, $Y=0$ or $Y=1$ or $Z=\frac{C_2}{S_2}>0$.

(1) If $1>Z>Z_2^{*}$, $H^{\prime }\left(1\right)<0$,$H^{\prime }\left(0\right)>0$. The probability of the demanders adopting strategy A is $Y=1$.

(2) If $0<Z<Z_2^{*}$, $H^{\prime }\left(1\right)>0$,$H^{\prime }\left(0\right)<0$. The probability of the demanders adopting strategy A is $Y=0$. □

The phase diagram of the replicator dynamic equation of demanders is shown in Figure 4.

Proposition 3.

(1) If $Y=Y^{*}=\frac{C_r-P_1-T_4-X(T_2-T_4-P_1-S_1)}{T_3-T_4-S_2+X(T_1-T_2-T_3+T_4)}$, $W\left(Z\right)\equiv 0$. The probability that the cloud platform adopts strategy R is stable and in [0, 1].

(2) If $Y\ne Y^{*}=\frac{C_r-P_1-T_4-X(T_2-T_4-P_1-S_1)}{T_3-T_4-S_2+X(T_1-T_2-T_3+T_4)}$ and $W\left(Z\right)=0$, the probability of the cloud platform adopting strategy R is $Z=0$ or $Z=1$.

Proof of Proposition 3. 

The first derivative of Equation (12) can be obtained as follows:

$$\begin{array}{cc}\hfill W^{\prime }\left(Z\right)=& (1-2Z)\left[-C_{\mathrm{r}}+P_1+T_4+X\left(T_2-T_4-P_1-S_1\right)+\right.\hfill \\ \hfill & \left.Y\left(T_3-T_4-S_2\right)+XY\left(T_1-T_2-T_3+T_4\right)\right]\hfill \end{array}$$

(16)

When Equation (12) equals 0, $Z=0$ or $Z=1$ or $Y=Y^{*}=\frac{C_r-P_1-T_4-X(T_2-T_4-P_1-S_1)}{T_3-T_4-S_2+X(T_1-T_2-T_3+T_4)}$.

(1) If $1>Y>Y^{*}$, $W^{\prime }\left(1\right)<0$, $W^{\prime }\left(0\right)>0$. The probability of the cloud platform adopting strategy R is $Z=1$.

(2) If $0<Y<Y^{*}$, $W^{\prime }\left(1\right)>0$, $W^{\prime }\left(0\right)<0$. The probability of the cloud platform adopting strategy R is $Z=0$. □

Figure 5 shows the phase diagram of the replicator dynamic equation of the cloud platform.

4. Model Analysis

Accoring to Equation (13), the equilibrium points are $E_1$(0,0,0), $E_2$(0,1,0), $E_3$(0,0,1), $E_4$(0,1,1), $E_5$(1,0,0), $E_6$(1,1,0), $E_7$(1,0,1), and $E_8$(1,1,1). The asymptotic stability of pure strategy equilibria is only discussed in an asymmetric game [37]. If all eigenvalues of the Jacobian matrix of each equilibrium are below 0, the equilibrium is stable. If all eigenvalues are above 0, the equilibrium is unstable. If all eigenvalues are positive or negative, the equilibrium is a saddle point. The Jacobian matrix of the game system is expressed in Equation (17):

$$\begin{array}{cc}\hfill J& =\left[\begin{array}{ccc}\frac{\partial F\left(X\right)}{\partial X}\hfill & \frac{\partial F\left(X\right)}{\partial Y}\hfill & \frac{\partial F\left(X\right)}{\partial Z}\hfill \\ \frac{\partial H\left(Y\right)}{\partial X}\hfill & \frac{\partial H\left(Y\right)}{\partial Y}\hfill & \frac{\partial H\left(Y\right)}{\partial Z}\hfill \\ \frac{\partial W\left(Z\right)}{\partial X}\hfill & \frac{\partial W\left(Z\right)}{\partial Y}\hfill & \frac{\partial W\left(Z\right)}{\partial Z}\hfill \end{array}\right]\hfill \\ \hfill =& \left[\begin{array}{ccc}(1-2X)\left[Z\left(S_1+P_1\right)-C_1\right]& 0& X(1-X)\left(S_1+P_1\right)\\ 0& (1-2Y)Z_1& Y(1-Y)S_2\\ Z_2& Z_3& Z_4\end{array}\right]\hfill \end{array}$$

(17)

where $Z_1=Z{S}_2-C_2$, $Z_2=Z(Z-1)\left[T_2-T_4-P_1-S_1+Y\left(T_1-T_2-T_3+T_4\right)\right]$, $Z_3=Z(Z-1)\left[T_3-T_4-S_2+X\left(T_1-T_2-T_3+T_4\right)\right]$, and $Z_4=(1-2Z)\left[-C_{\mathrm{r}}+P_1+T_4+\right.$$X\left(T_2-T_4-P_1-S_1\right)$$\left.+Y\left(T_3-T_4-S_2\right)+XY\left(T_1-T_2-T_3+T_4\right)\right]$.

The Jacobian matrix of the equilibrium point $E_1$(0,0,0) is expressed in Equation (18), and the eigenvalues of the matrix are $-C_1$, $-C_2$, and $-C_r+P_1+T_4$. Similarly, the eigenvalues and stability of other equilibrium points are illustrated in

Table 4. Stability analysis of equilibrium points.

Equilibrium	${\mathit{\lambda }}_1,{\mathit{\lambda }}_2,{\mathit{\lambda }}_3$	Stability
	${\lambda }_1=-C_1<0$	${\lambda }_3<0$, stable point
$E_1$(0,0,0)	${\lambda }_2=-C_2<0$
	${\lambda }_3=-C_{\mathrm{r}}+P_1+T_4$	${\lambda }_3>0$, saddle point
	${\lambda }_1=-C_1>0$
$E_2$(0,1,0)	${\lambda }_2=-C_2<0$	saddle point
	${\lambda }_3=P_1-C_{\mathrm{r}}-S_2+T_3$
	${\lambda }_1=S_1+P_1-C_1$	${\lambda }_1<0,{\lambda }_2<0,{\lambda }_3<0$, stable point
$E_3$(0,0,1)	${\lambda }_2=S_2-C_2$	${\lambda }_1>0,{\lambda }_2>0,{\lambda }_3>0$, unstable point
	${\lambda }_3=C_r-P_1-T_4$	Others, saddle point
	${\lambda }_1=P_1-C_1+S_1$	${\lambda }_1<0,{\lambda }_2<0,{\lambda }_3<0$, stable point
$E_4$(0,1,1)	${\lambda }_2=C_2-S_2$	${\lambda }_1>0,{\lambda }_2>0,{\lambda }_3>0$, unstable point
	${\lambda }_3=C_r+S_2-P_1-T_3$	Others, saddle point
	${\lambda }_1=C_1>0$
$E_5$(1,0,0)	${\lambda }_2=-C_2<0$	saddle point
	${\lambda }_3=-C_{\mathrm{r}}-S_1+T_2$
	${\lambda }_1=C_1>0$	${\lambda }_3>0$, unstable point
$E_6$(1,1,0)	${\lambda }_2=C_2>0$
	${\lambda }_3=T_1-C_{\mathrm{r}}-S_2-S_1$	Others, saddle point
	${\lambda }_1=C_1-S_1-P_1$	${\lambda }_1<0,{\lambda }_2<0,{\lambda }_3<0$, stable point
$E_7$(1,0,1)	${\lambda }_2=S_2-C_2$	${\lambda }_1>0,{\lambda }_2>0,{\lambda }_3>0$, unstable point
	${\lambda }_3=C_r+S_1-T_2$	Others, saddle point
	${\lambda }_1=C_1-S_1-P_1$	${\lambda }_1<0,{\lambda }_2<0,{\lambda }_3<0$, stable point
$E_8$(1,1,1)	${\lambda }_2=C_2-S_2$	${\lambda }_1>0,{\lambda }_2>0,{\lambda }_3>0$, unstable point
	${\lambda }_3=S_2+S_1+C_r-T_1$	Others, saddle point

.

$$J=\left[\begin{array}{ccc}-C_1& 0& 0\\ 0& -C_2& 0\\ 0& 0& -C_{\mathrm{r}}+P_1+T_4\end{array}\right]$$

(18)

As shown in Table 4, $E_2$(0,1,0) and $E_5$(1,0,0) are saddle points, and $E_6$(1,1,0) is not stable. Consequently, $E_1$(0,0,0), $E_3$(0,0,1), $E_4$(0,1,1), $E_7$ (1,0,1), and $E_8$(1,1,1) are analyzed as follows. The phase diagram is shown in Figure 6.

(1) $E_1$(0,0,0): with $P_1+T_4<C_{\mathrm{r}}$, $E_1$(0,0,0) is an ESS. The strategies are {NP, NA, NR}. The regulatory cost of the cloud platform is above the extra benefits composed of penalties and government subsidies, and the benefit for the cloud platform to adopt strategy NR is higher than that of the cloud platform to adopt strategy R. Therefore, the cloud platform tends to adopt strategy NR.

(2) $E_3$(0,0,1): with $P_1<C_1-S_1,S_2<C_2,P_1+T_4>C_{\mathrm{r}}$, $E_3$(0,0,1) is an ESS. Providers do not provide low-carbon manufacturing services, and the demanders do not adopt low-carbon services. The cloud platform chooses to regulate providers and demanders. When adopting strategy P, the providers bear low-carbon costs and receive rewards from the cloud platform. When adopting strategy NP, the providers are punished by the cloud platform. As the benefit of adopting strategy NP is higher than that of adopting strategy P ($P_1<C_1-S_1$), the providers are unwilling to provide low-carbon services. The analysis of the demanders is the same as that of the providers. As the regulatory cost of the cloud platform is lower than the extra benefits ($P_1+T_4>C_{\mathrm{r}}$), the cloud platform is willing to regulate the supply and demand sides.

(3) $E_4$(0,1,1): with $P_1<C_1-S_1,S_2>C_2,P_1+T_3>C_{\mathrm{r}}+S_2$, $E_4$(0,1,1) is the stable point. The strategies are {NP, A, R}. As the benefit of adopting strategy NP is higher than that of choosing strategy P ($P_1<C_1-S_1$), the providers adopt strategy NP. Since the benefit of adopting strategy A is higher than that of adopting strategy NA ($S_2>C_2$), the demanders prefer to adopt low-carbon services. Since the extra benefits for the platform are higher than the regulatory cost of the cloud platform ($P_1+T_3-S_2>C_{\mathrm{r}}$), the cloud platform tends to adopt strategy R.

(4) $E_7$(1,0,1): when $P_1>C_1-S_1,S_2<C_2,T_2-S_1>C_{\mathrm{r}}$, $E_7$(1,0,1) is the stable point. The strategies are {P, NA, R}. Since the benefit of adopting strategy NP is less than that of adopting strategy P ($P_1>C_1-S_1$), the providers tend to provide low-carbon services. As the benefit of adopting strategy NA is higher than that of adopting strategy A ($0>S_2-C_2$), the demanders are unwilling to adopt low-carbon services. Since the regulatory cost of the cloud platform is lower than the extra benefits ($T_2-S_1>C_{\mathrm{r}}$), the cloud platform is inclined to regulate the supply and demand sides.

(5) $E_8$(1,1,1): with $P_1>C_1-S_1,S_2>C_2,T_1-S_2-S_1>C_{\mathrm{r}}$, $E_8$(1,1,1) is an ESS of the dynamic system. The strategies are {P, A, R}. As the benefits of adopting strategy NP are less than those of adopting strategy P ($P_1>C_1-S_1$), the providers prefer to provide low-carbon services. Since the benefit of adopting strategy A is higher than that of adopting strategy NA ($S_2-C_2>0$), the demanders are inclined to adopt low-carbon services. As the regulatory cost of the cloud platform is lower than the extra benefits ($T_1-S_2-S_1>C_{\mathrm{r}}$), it is willing to regulate providers and demanders.

5. Simulation Analysis and Discussion

Based on the replicator dynamic equations of the system, a simulated evolutionary game model was conducted to explore each game player’s stable strategy and analyze the impacts of important factors on tripartite evolution trends through Matlab.

According to the literature [38], the initial parameters of the payoff matrix are set as $G_1=1000$, $G_2=1500$, and $G_3=250$. Additionally, $E_8(1,1,1)$ is the expected state in this study. Therefore, other parameters of the payoff matrix were assigned according to stability conditions ($C_1-S_1-P_1<0,C_2-S_2<0,S_2+S_1+C_r-T_1<0$) of $E_8(1,1,1)$ in Table 4. Other parameters of the payoff matrix were assigned as $C_1=150,C_2=50,C_{\mathrm{r}}=90,$$T_1=240,$$T_2=180,T_3=120,T_4=60,S_1=70,S_2=60,$ and $P_1=90$. Stakeholders of the game had bounded rationality, and the initial probability was set as $O(0.5,0.5,0.5)$.

5.1. The Impacts of Penalties $P_1$ of the Cloud Platform

Considering that the strategy evolution of the demanders and the cloud platform is not significantly affected by $P_1$, the impact of $P_1$ on the strategy evolution of the providers was analyzed.

The impact of $P_1$ on the providers’ strategy evolution is shown in Figure 7a. When $P_1$ is 60 and 70, i.e., below $C_1-S_1$, the strategy equilibrium state of the providers is “0”, i.e., the providers are unwilling to adopt strategy P. As $P_1$ increases, the stabilization toward equilibrium state “1” accelerates.

The impact of $P_1$ on the tripartite strategy evolution is shown in Figure 7b. With $P_1=60$ and $P_1=70$, the system finally converges to $E_4$(0,1,1), i.e., the providers adopt strategy NP, the demanders adopt strategy A, and the platform chooses strategy R. When $P_1$ is above $C_1-S_1$, the stability condition of ESS $E_8$(1,1,1) is met. Therefore, the system eventually converges to $E_8$(1,1,1). Through simulation analysis, high penalties were found to entice the providers to provide low-carbon services.

5.2. The Impacts of Rewards $S_1$ and $S_2$ of the Cloud Platform

As the cloud platform reward $S_1$ to the service providers has little impact on the strategy evolution of the demanders and the cloud platform, only the impact of $S_1$ on the strategy evolution of the providers was analyzed. Similarly, only the impact of $S_2$ on the strategy evolution of the demanders was analyzed.

The impact of $S_1$ on the strategy evolution of the providers is plotted in Figure 8a. With a smaller $S_1$, providers tend to evolve toward the equilibrium state “0”, which takes longer as $S_1$ increases. With $S_1$ above $C_1-P_1$, the providers tend to evolve toward the equilibrium state “1”, i.e., the providers are willing to adopt strategy P, and the time to equilibrium state "1" shortens with an increasing $S_1$.

The impact of $S_1$ on the tripartite strategy evolution is shown in Figure 8b. With $S_1=40$ and $S_1=50$, the system finally converges to ESS $E_4$(0,1,1). As $S_1$ increases, the system finally converges to ESS $E_8$(1,1,1), and the final strategy is (P, A, R).

The impact of $S_2$ on the strategy evolution of the demanders is plotted in Figure 8c. As $S_2$ increases, the demanders tend to evolve from equilibrium state “0” to equilibrium state “1”. The reward demarcation point of the demanders adopting different strategies is $C_2$. With a higher reward $S_2$, the demanders are more willing to adopt strategy A.

The effect of $S_2$ on the tripartite strategy evolution is illustrated in Figure 8d. With $S_2=30$ and $S_2=40$, the system finally converges to ESS $E_7$(1,0,1), i.e., the demanders are unwilling to choose a low-carbon strategy, while the providers are willing to adopt low-carbon strategy and the platform tends to regulate them. When $S_2$ is above $C_2$, the system eventually converges to ESS $E_8$(1,1,1). According to the above analysis, $S_2$ promotes the demanders to adopt low-carbon strategies.

5.3. The Impacts of $C_1,C_2$, and $C_r$ on the Strategy Evolution

As the low-carbon cost $C_1$ has little impact on the strategy evolution of the demanders and the cloud platform, only the impact of $C_1$ on the strategy evolution of the providers was analyzed. Similarly, only the impact of $C_2$ on the strategy evolution of the demanders was analyzed.

The impact of $C_1$ on the strategy evolution of the providers is shown in Figure 9a. When $C_1$ is above $S_1+P_1$, the evolution time to equilibrium state “0” shortens as $C_1$ increases. When the stability condition of $E_8$(1,1,1) is met, the providers evolve toward equilibrium state “1”, i.e., the providers are willing to adopt strategy P.

The impact of $C_1$ on the tripartite strategy evolution is presented in Figure 9b. As $C_1$ decreases, the system trends toward ESS $E_8$(1,1,1), and the final strategy is (P, A, R).

The effect of $C_2$ on the evolution process of the demanders is shown in Figure 9c. With lower $C_2$, the demanders tend to evolve towards equilibrium state “1” from equilibrium state “0”, indicating that the demanders are more enthusiastic about adopting strategy A. As $C_2$ decreases, the evolution time to equilibrium state “1” shortens, and the demanders tend to adopt low-carbon services.

The impact of $C_2$ on the tripartite strategy evolution is shown in Figure 9d. As $C_2$ decreases, i.e., below $S_2$, the system eventually converges to $E_8$(1,1,1) from ESS $E_7$(1,0,1), and the final strategy is (P, A, R).

The effect of $C_r$ on the evolutionary path of the cloud platform is shown in Figure 9e. When $C_r$ is below $T_1-S_1-S_2$, the cloud platform tends to evolve towards equilibrium state “1”. As $C_r$ increases, the evolution speed decreases.

The effect of $C_r$ on the tripartite evolutionary path is shown in Figure 9f. When $C_r$ is below $T_1-S_1-S_2$, the stability condition of ESS $E_8$(1,1,1) is met. Therefore, the system stabilizes to ESS $E_8$(1,1,1).

5.4. Discussion

To achieve low-carbon cooperation between providers and demanders, $E_8$(1,1,1) is the expected state, i.e., {the providers providing low-carbon manufacturing services, the demanders adopting low-carbon services, and the cloud platform choosing to regulate the supply and demand sides}.

(1) As the penalties for the providers increase, the system eventually converges to $E_8$(1,1,1) from ESS $E_4$(0,1,1). The penalty demarcation point of $E_8$(1,1,1) and $E_4$(0,1,1) is $C_1-S_1$. The cloud platform improves its penalties for the providers, promoting the entire system to converge to $E_8$(1,1,1). (2) As the cloud platform rewards for service providers increase, the system eventually converges to $E_8$(1,1,1) from ESS $E_4$(0,1,1). The reward demarcation point of $E_8$(1,1,1) and $E_4$(0,1,1) is $C_1-P_1$. Increasing cloud platform rewards for the service demanders prompts the system to converge to $E_8$(1,1,1) from ESS $E_7$(1,0,1), and the reward demarcation point is $C_2$. The cloud platform sets high rewards for the providers and demanders, thus stabilizing the system to $E_8$(1,1,1). (3) As the low-carbon costs of the providers decrease, i.e., below $S_1+P_1$, the system is stabilized to $E_8$(1,1,1) from $E_4$(0,1,1). As the additional costs of the demanders decrease, i.e., below $S_2$, the system eventually converges to $E_8$(1,1,1) from ESS $E_7$(1,0,1). Additionally, lower regulatory costs accelerate the evolutionary rate of the cloud platform toward the equilibrium state "1." The providers decrease low-carbon costs, the demanders reduce additional costs, and the cloud platform reduces regulatory costs, which can promote the system to converge to $E_8$(1,1,1).

6. Conclusions

To analyze the low-carbon cooperation between providers and demanders in cloud manufacturing, this paper established a tripartite evolutionary game among the service providers, service demanders, and the cloud platform and discussed evolutionary stability strategies based on replicator dynamics equations. The research can effectively promote the low-carbon and sustainable development of cloud manufacturing, which has practical significance for environmental improvement.

The main conclusions are as follows: (1) As the platform’s penalties increase, the providers tend to provide low-carbon services. (2) With the cloud platform providing high rewards to the service providers and demanders, they tend to choose low-carbon strategies. High rewards promote the participation of providers and demanders in low-carbon cooperation. (3) With lower low-carbon costs for the providers and lower additional costs for the demanders, the providers and demanders are more enthusiastic about choosing low-carbon strategies. Additionally, the cloud platform becomes willing to regulate the providers and demanders as the regulatory cost decreases.

Some suggestions can be proposed through the above analysis: (1) The cloud platform plays a leading role in the low-carbon cooperation between providers and demanders. The rewards and penalties of the cloud platform have a positive impact on the low-carbon cooperation between providers and demanders, and the cloud platform encourages them to participate in low-carbon cooperation through appropriate penalties and rewards. The cloud platform can set penalties (rewards) for the providers based on their low-carbon costs and their rewards (penalties). The cloud platform can determine the rewards for the demanders according to their additional costs for choosing a low-carbon strategy. (2) The low-carbon costs of the providers and the additional costs of the demanders negatively impact their low-carbon cooperation, which should, therefore, be reduced through measures such as innovating low-carbon technology. Additionally, the regulatory cost has a negative impact on the regulatory strategy of the cloud platform. Thus, the cloud platform should reduce its regulatory costs through measures such as online regulation. The low-carbon costs that the providers are willing to pay for low-carbon cooperation are affected by the rewards and penalties of the cloud platform. The additional costs for low-carbon cooperation that the demanders are willing to pay are closely related to the rewards of the cloud platform. The cloud platform determines its regulatory costs according to government subsidies and its rewards for low-carbon cooperation.

Future work may be conducted based on the following aspects: (1) An evolutionary game model was established to analyze the strategy evolution of providers, demanders, and the cloud platform. However, the government also plays an important role in low-carbon cooperation, and an evolutionary game model involving the government can be established in the future. (2) Some simulation parameters were assumed according to related references, and future research can verify the game model based on actual data on cloud manufacturing. However, it is possible that the providers and demanders are unwilling to disclose or falsely disclose some parameters. Therefore, it is necessary to further eliminate the information asymmetry between the cloud platform and the supply and demand sides through mechanism design. (3) In practice, many factors affect the low-carbon cooperation between providers and demanders. Based on the investigation of low-carbon cooperation between actual supply and demand enterprises, the model can be further improved. (4) This study only discusses whether the cloud platform chooses regulation and non-regulation strategies, whereas the regulation mode should also be analyzed in future studies, including positive regulation and negative regulation.