How Can the Government Promote Sustainable Cooperation between Schools and Enterprises? A Quadrilateral Evolutionary Game Study

Abstract

Sustainable cooperation between schools and enterprises is crucial to maintaining a dynamic balance between the supply and demand of talents and driving the advancement of new quality productive forces. As a result, facilitating school–enterprise cooperation has become an important concern in many countries. However, there remains a gap in understanding the approaches taken by central and local governments to foster such cooperation through the lens of evolutionary game theory. Therefore, this paper develops a quadrilateral evolutionary game model involving schools, enterprises, the central government, and local governments by building the payoff matrix and calculating expected payoffs of different strategies to analyze the conditions under which governments can effectively promote school–enterprise cooperation. Our findings indicate that benefit is a decisive factor that affects the strategies of all parties. Increasing incentives and penalties from the central government and local governments can stimulate school–enterprise cooperation, but these measures are not sustainable in the long term. Additionally, the impact of the central government’s incentives and penalties on local governments in promoting school–enterprise cooperation is limited. Enhancing policy efficacy and the practical outcomes of school–enterprise cooperation is essential, which not only strengthens the bond between schools and enterprises but also ensures deep and enduring governmental involvement. Some suggestions are put forward at the end.

1. Introduction

The aim of school–enterprise cooperation is to align the operational mechanisms and job requirements of enterprises with the talent cultivation systems and objectives of schools. This cooperation is important for improving the quality of talent, facilitating scientific and technological innovation, promoting social responsibility, and driving economic development. Thus, school–enterprise cooperation has become a focal point of research. Various macroscopic models of school–enterprise cooperation, such as the Dual System [1,2,3,4], Cooperative Education [5,6,7,8], and Sandwich Courses [9,10,11,12], have been extensively studied, and many microcosmic training modes of school–enterprise cooperation [13,14,15,16] have also been widely discussed. Moreover, some scholars have analyzed the motivations [17,18,19], factors [20,21,22,23], and mechanisms [24,25,26] of school–enterprise cooperation in depth.

The cooperation between schools and enterprises has garnered significant government attention worldwide. For instance, Germany has implemented a comprehensive framework of laws, regulations, and management systems to ensure students acquire both theoretical knowledge in schools and practical skills in enterprises. Similarly, the United States’ the Strengthening Career and Technical Education (CTE) for the 21st Century Act was enacted to bolster partnerships between schools and enterprises. In China, the government also places significant emphasis on school–enterprise cooperation. Initiatives such as the Several Opinions on Deepening the Integration of Industry and Education highlight China’s commitment to align educational outcomes with workforce demands.

However, there are three shortcomings in the existing studies. Firstly, research on the role of governments remains insufficient. While some researchers have acknowledged the great influence of governments on school–enterprise cooperation [27,28], they have not considered the government as a main stakeholder in school–enterprise cooperation and conducted specialized research on it. Secondly, the distinctions between the central government and local governments have been almost ignored. It is well-known that the relationship between the central government and local governments is dynamic and complex. There is a phenomenon of information asymmetry between them [29]. Local governments have different goals and interests from the central government, so local governments submit to the unified leadership of the central government in some areas but have adversarial relationships with the central government in others [30,31,32]. Thus, it is necessary to explore the roles played by the central government and local governments in school–enterprise cooperation. Thirdly, the quadrilateral evolutionary game model has not been widely adopted and studied. There are few studies using evolutionary game theory to analyze the cooperation between schools and enterprises. The evolutionary game models in previous literature are mostly bilateral and tripartite evolutionary game models [33,34,35]. The quadrilateral evolutionary game model deserves more attention.

Considering the aforementioned limitations, this paper further divides the government into central and local governments and establishes a quadrilateral evolutionary game model of schools, enterprises, the central government, and local governments to explore the approach of the government to promote school–enterprise cooperation.

The remainder of this paper is structured as follows: Section 2 develops the theoretical framework and constructs a quadrilateral evolutionary game model of schools, enterprises, the central government, and local governments. Section 3 examines the stability of strategies adopted by the four parties. Section 4 analyzes the stability conditions pertaining to potential equilibrium points. Finally, Section 5 presents the main conclusions drawn from the study.

2. Methodology

In this section, we first explain the game relationship between schools, enterprises, the central government, and local governments in school–enterprise cooperation by building a theoretical framework. Thereafter, a quadrilateral evolutionary game model is constructed to explore the evolution of cooperative behavior among the four stakeholders.

2.1. Theoretical Framework

Schools and enterprises are the cores of this quadrilateral game. Schools benefit from cooperation with enterprises through enhancements of discipline construction, talent cultivation, and teacher development. Similarly, enterprises gain advantages such as access to cheap but high-quality labor, opportunities for technological innovation, and improved social standing through cooperation with schools. Governments play the roles of regulation and supervision in school–enterprise cooperation by promulgating relevant policies and providing financial support to push schools and enterprises to cooperate with each other. In return, the cooperation between schools and enterprises is more effective because of the government’s involvement, which will provide more outstanding talents, better economic development, and higher social reputations for governments. In addition, local governments are required to implement the policies of the central government and are managed by the central government. The relationship between schools, enterprises, the central government, and local governments is shown in Figure 1.

Specifically, in this four-party game between schools, enterprises, the central government, and local governments, all parties have bounded rationality. Each party has two strategies, cooperation strategy and noncooperation strategy, and each party chooses one of the two strategies with a certain probability. We assume that the probabilities of choosing a cooperation strategy by schools, enterprises, the central government, and local governments are w, x, y, and z, respectively, and the probabilities of choosing a noncooperation strategy by schools, enterprises, the central government, and local governments are 1 − w, 1 − x, 1 − y, and 1 − z, respectively.

For schools, when they choose a cooperation strategy, they incur some costs C₁. In this case, if enterprises also choose a cooperation strategy, schools will obtain many benefits B₁ from cooperating with enterprises. However, if enterprises choose a noncooperation strategy, schools will bear the losses L₁ caused by the defection of the enterprises. When schools choose a noncooperation strategy and enterprises choose a cooperation strategy, schools can still reap some benefits B₃; however, they also need to take some losses L₁ for defecting to enterprises. When both schools and enterprises choose a noncooperation strategy, schools will inevitably bear the losses L₁ resulting from unsuccessful cooperation with enterprises.

Similarly for enterprises, when they choose a cooperation strategy, they will incur some costs C₂. In this case, if schools also choose a cooperation strategy, enterprises will obtain many benefits B₂ from cooperating with schools. However, if schools choose a noncooperation strategy, enterprises will bear the losses L₂ caused by the defection of the schools. When enterprises choose a noncooperation strategy and schools choose a cooperation strategy, enterprises can still reap some benefits B₄, but they also need to take some losses L₂ of defecting to schools. When both schools and enterprises choose a noncooperation strategy, enterprises will inevitably bear the losses L₂ resulting from the unsuccessful cooperation with schools.

For the central government, when it chooses a cooperation strategy, it will incur some costs C₃. In this case, if schools, enterprises, and local governments also choose a cooperation strategy, the central government will give them some rewards, denoted by G₁, G₂, and G₃, respectively, and the central government will also gain some rewards R₁, in return, from their cooperation. If both schools and enterprises choose a cooperation strategy, but local governments choose a noncooperation strategy, the central government can still gain some rewards R₃ from the cooperation between schools and enterprises, although this is slightly less than R₁. However, if schools, enterprises, and local governments choose a noncooperation strategy, they will receive some punishments from the central government, denoted by P₁, P₂, and P₃, respectively. When the central government chooses a noncooperation strategy and schools, enterprises, and local governments choose a cooperation strategy, the central government can also reap some benefits R₅ from their cooperation. If both schools and enterprises choose a cooperation strategy but local governments choose a noncooperation strategy, the central government will gain some benefits R₇ from the cooperation between schools and enterprises, which is slightly less than R₅. Furthermore, no matter what strategy the central government chooses, as long as either schools or enterprises choose a noncooperation strategy, the central government will suffer certain losses L₃ because of the unsuccessful cooperation between schools and enterprises.

Similarly, when local governments choose a cooperation strategy, they will incur some costs C₄. In this case, if schools, enterprises, and the central government also choose a cooperation strategy, local governments will gain some benefits R₂ and will offer some rewards to schools and enterprises, respectively denoted by G₄ and G₅. However, if schools and enterprises choose a noncooperation strategy, they will receive some punishments from local governments, denoted by P₄ and P₅, respectively. If both schools and enterprises choose a cooperation strategy, but the central government chooses a noncooperation strategy, local governments can still gain some benefits R₆ from the cooperation between schools and enterprises, which is slightly less than R₂. When local governments choose a noncooperation strategy and schools, enterprises, and the central government choose a cooperation strategy, local governments can also reap some benefits R₄ from the cooperation of the other three parties. If both schools and enterprises choose a cooperation strategy, but the central government chooses a noncooperation strategy, local governments will gain some benefits R₈ from the cooperation between schools and enterprises, which is slightly less than R₄. Furthermore, no matter what strategy local governments choose, as long as either schools or enterprises choose a noncooperation strategy, local governments will suffer certain losses L₄ because of the unsuccessful cooperation between schools and enterprises.

The above parameters and their definitions for the four parties are listed in

Table 1. Parameter definitions.

Party	Parameter	Definition
schools	C1	The cost of schools’ cooperation strategy.
	B1	The benefit of schools’ cooperation strategy for cooperating with enterprises.
	B3	The benefit of schools’ noncooperation strategy due to the enterprises’ cooperation strategy.
	L1	The loss suffered by schools due to the unsuccessful cooperation between schools and enterprises.
enterprises	C2	The cost of enterprises’ cooperation strategy.
	B2	The benefit of enterprises’ cooperation strategy for cooperating with schools.
	B4	The benefit of enterprises’ noncooperation strategy due to the schools’ cooperation strategy.
	L2	The loss suffered by enterprises due to the unsuccessful cooperation between schools and enterprises.
the central government	C3	The cost of the central government’s cooperation strategy.
	G1	The reward from the central government for schools’ cooperation strategy.
	G2	The reward from the central government for enterprises’ cooperation strategy.
	G3	The reward from the central government for local governments’ cooperation strategy.
	P1	The punishment from the central government for schools’ noncooperation strategy.
	P2	The punishment from the central government for enterprises’ noncooperation strategy.
	P3	The punishment from the central government for local governments’ noncooperation strategy.
	R1	The reward for the central government’s cooperation strategy due to the cooperation of all parties.
	R3	The reward for the central government’s cooperation strategy due to the cooperation between schools and enterprises.
	R5	The reward for the central government’s noncooperation strategy due to the cooperation of the other three parties.
	R7	The reward for the central government’s noncooperation strategy due to the cooperation between schools and enterprises.
	L3	The loss suffered by the central government due to the unsuccessful cooperation between schools and enterprises.
local governments	C4	The cost of local governments’ cooperation strategy.
	G4	The reward from local governments for schools’ cooperation strategy.
	G5	The reward from local governments for enterprises’ cooperation strategy.
	P4	The punishment from local governments for schools’ noncooperation strategy.
	P5	The punishment from local governments for enterprises’ noncooperation strategy.
	R2	The reward for local governments’ cooperation strategy due to the cooperation of all parties.
	R4	The reward for local governments’ noncooperation strategy due to the cooperation of the other three parties.
	R6	The reward for local governments’ cooperation strategy due to the cooperation between schools and enterprises.
	R8	The reward for local governments’ noncooperation strategy due to the cooperation between schools and enterprises.
	L4	The loss suffered by local governments due to the unsuccessful cooperation between schools and enterprises.

. It should be pointed out that the values of all parameters in Table 1 are positive.

2.2. Model Construction

Based on the previous theoretical framework, the corresponding payoff matrix for schools, enterprises, the central government, and local governments is constructed and shown in

Table 2. Payoff matrix for the quadrilateral evolutionary game between schools, enterprises, the central government, and local governments.

				The Central Government
				Cooperation Strategy		Noncooperation Strategy
				Local Governments		Local Governments
				Cooperation Strategy	Noncooperation Strategy	Cooperation Strategy	Noncooperation Strategy
schools	cooperation strategy	enterprises	cooperation strategy	−C1+ B1+ G1+ G4, −C2+ B2+ G2+ G5, −C3− G1− G2− G3+ R1, −C4− G4− G5+ G3+ R2	−C1+ B1+ G1, −C2+ B2+ G2, −C3− G1− G2+ P3+ R3, −P3+ R4	−C1+ B1+ G4, −C2+ B2+ G5, R5, −C4− G4− G5+ R6	−C1+ B1, −C2+ B2, R7, R8
			noncooperation strategy	−C1− L1+ G1+ G4, B4− L2− P2− P5, −C3− L3− G1+ P2− G3, −C4− L4− G4+ P5+ G3	−C1− L1+ G1, B4− L2− P2, −C3− L3− G1+ P2+ P3, − L4− P3	−C1− L1+ G4, B4− L2− P5, − L3, −C4− L4− G4+ P5	−C1− L1, B4− L2, −L3, −L4
	noncooperation strategy	enterprises	cooperation strategy	B3− L1− P1− P4, −C2− L2+ G2+ G5, −C3− L3+ P1− G2− G3, −C4− L4+ P4− G5+ G3	B3− L1− P1, −C2− L2+ G2, −C3− L3+ P1− G2+ P3, − L4− P3	B3− L1− P4, −C2− L2+ G5, −L3, −C4− L4+ P4− G5	B3− L1, −C2− L2, −L3, −L4
			noncooperation strategy	− L1− P1− P4, − L2− P2− P5, −C3− L3+ P1+ P2− G3, −C4− L4+ P4+ P5+ G3	− L1− P1, − L2− P2, −C3− L3+ P1+ P2+ P3, −L4− P3	− L1− P4, − L2− P5, −L3, −C4− L4+ P4+ P5	−L1, −L2, −L3, −L4

.

Each entry in Table 2 contains 4 values representing payoffs for schools, enterprises, the central government, and local governments. According to evolutionary game theory, when the expected payoff of a certain strategy for a population is higher than the average expected payoff of all other strategies for that population, the proportion of individuals who choose that strategy in that population will increase. That is, the strategy with the higher expected payoff is more likely to be imitated, replicated, and spread [36,37]. This mechanism can be expressed by the replicator dynamic equation, which is one of the most basic models in evolutionary game theory [38,39,40]. Therefore, we construct the replicator dynamic equation for schools, enterprises, the central government, and local governments by calculating expected payoffs and average expected payoffs of their cooperation and noncooperation strategies to investigate the evolution of cooperative behaviors among these four populations.

First, we use U_A1 and U_A2 to represent expected payoffs of schools’ cooperation strategy and noncooperation strategy, respectively, and use $\overline{U_A}$ to represent the average expected payoff. According to the above payoff matrix and the probabilities of selecting different strategies, U_A1, U_A2, and $\overline{U_A}$ can be expressed as follows:

$$\begin{gathered} U_{A1}=xyz\left(-C_1+B_1+G_1+G_4\right)+xy\left(1-z\right)\left(-C_1+B_1+G_1\right)+x\left(1-y\right)z\left(-C_1+B_1+G_4\right)+x\left(1-y\right)\left(1-z\right)\left(-C_1+B_1\right) \\ +\left(1-x\right)yz\left(-C_1-L_1+G_1+G_4\right)+\left(1-x\right)y\left(1-z\right)\left(-C_1-L_1+G_1\right) \\ +\left(1-x\right)\left(1-y\right)z\left(-C_1-L_1+G_4\right)+\left(1-x\right)\left(1-y\right)\left(1-z\right)\left(-C_1-L_1\right) \end{gathered}$$

(1)

$$\begin{gathered} U_{A2}=xyz\left(B_3-L_1-P_1-P_4\right)+xy\left(1-z\right)\left(B_3-L_1-P_1\right)+x\left(1-y\right)z\left(B_3-L_1-P_4\right)+x\left(1-y\right)\left(1-z\right)\left(B_3-L_1\right) \\ +\left(1-x\right)yz\left(-L_1-P_1-P_4\right)+\left(1-x\right)y\left(1-z\right)\left(-L_1-P_1\right)+\left(1-x\right)\left(1-y\right)z\left(-L_1-P_4\right) \\ +\left(1-x\right)\left(1-y\right)\left(1-z\right)\left(-L_1\right) \end{gathered}$$

(2)

$$\overline{U_A}=w{U}_{A1}+\left(1-w\right)U_{A2}$$

(3)

The replicator dynamic equation for schools ${\displaystyle \frac{dw}{dt}}$ can be calculated as:

$$F\left(w\right)={\displaystyle \frac{dw}{dt}}=w\left(U_{A1}-\overline{U_A}\right)=w\left(1-w\right)\left[x\left(B_1-B_3+L_1\right)+y\left(G_1+P_1\right)+z\left(G_4+P_4\right)-C_1\right]$$

(4)

In a similar way, we use U_B1 and U_B2 to represent the expected payoffs of enterprises’ cooperation strategy and noncooperation strategy, respectively, and use $\overline{U_B}$ to represent the average expected payoff. U_B1, U_B2, and $\overline{U_B}$ can be obtained as follows:

$$\begin{gathered} U_{B1}=wyz\left(-C_2+B_2+G_2+G_5\right)+wy\left(1-z\right)\left(-C_2+B_2+G_2\right)+w\left(1-y\right)z\left(-C_2+B_2+G_5\right) \\ +w\left(1-y\right)\left(1-z\right)\left(-C_2+B_2\right)+\left(1-w\right)yz\left(-C_2-L_2+G_2+G_5\right)+\left(1-w\right)y\left(1-z\right)\left(-C_2-L_2+G_2\right) \\ +\left(1-w\right)\left(1-y\right)z\left(-C_2-L_2+G_5\right)+\left(1-w\right)\left(1-y\right)\left(1-z\right)\left(-C_2-L_2\right) \end{gathered}$$

(5)

$$\begin{gathered} U_{B2}=wyz\left(B_4-L_2-P_2-P_5\right)+wy\left(1-z\right)\left(B_4-L_2-P_2\right)+w\left(1-y\right)z\left(B_4-L_2-P_5\right)+w\left(1-y\right)\left(1-z\right)\left(B_4-L_2\right) \\ +\left(1-w\right)yz\left(-L_2-P_2-P_5\right)+\left(1-w\right)y\left(1-z\right)\left(-L_2-P_2\right)+\left(1-w\right)\left(1-y\right)z\left(-L_2-P_5\right) \\ +\left(1-w\right)\left(1-y\right)\left(1-z\right)\left(-L_2\right) \end{gathered}$$

(6)

$$\overline{U_B}=x{U}_{B1}+\left(1-x\right)U_{B2}$$

(7)

The replicator dynamic equation for enterprises ${\displaystyle \frac{dx}{dt}}$ can thus be calculated as:

$$F\left(x\right)={\displaystyle \frac{dx}{dt}}=x\left(U_{B1}-\overline{U_B}\right)=x\left(1-x\right)\left[w\left(B_2-B_4+L_2\right)+y\left(G_2+P_2\right)+z\left(G_5+P_5\right)-C_2\right]$$

(8)

Next, we use U_C1 and U_C2 to represent the expected payoff of the central government’s cooperation strategy and noncooperation strategy, respectively, and use $\overline{U_C}$ to represent the average expected payoff. U_C1, U_C2, and $\overline{U_C}$ can be obtained as follows:

$$\begin{gathered} U_{C1}=wxz\left(-C_3-G_1-G_2-G_3+R_1\right)+wx\left(1-z\right)\left(-C_3-G_1-G_2+P_3+R_3\right) \\ +w\left(1-x\right)z\left(-C_3-L_3-G_1+P_2-G_3\right)+w\left(1-x\right)\left(1-z\right)\left(-C_3-L_3-G_1+P_2+P_3\right)+\left(1-w\right)xz\left(-C_3-L_3+P_1-G_2-G_3\right) \\ +\left(1-w\right)x\left(1-z\right)\left(-C_3-L_3+P_1-G_2+P_3\right)+\left(1-w\right)\left(1-x\right)z\left(-C_3-L_3+P_1+P_2-G_3\right) \\ +\left(1-w\right)\left(1-x\right)\left(1-z\right)\left(-C_3-L_3+P_1+P_2+P_3\right) \end{gathered}$$

(9)

$$\begin{gathered} U_{C2}=wxz{R}_5+wx\left(1-z\right)R_7+w\left(1-x\right)z\left(-L_3\right)+w\left(1-x\right)\left(1-z\right)\left(-L_3\right)+\left(1-w\right)xz\left(-L_3\right)+\left(1-w\right)x\left(1-z\right)\left(-L_3\right) \\ +\left(1-w\right)\left(1-x\right)z\left(-L_3\right)+\left(1-w\right)\left(1-x\right)\left(1-z\right)\left(-L_3\right) \end{gathered}$$

(10)

$$\overline{U_C}=y{U}_{C1}+\left(1-y\right)U_{C2}$$

(11)

The replicator dynamic equation for the central government ${\displaystyle \frac{dy}{dt}}$ can then be calculated as:

$$\begin{gathered} F\left(y\right)={\displaystyle \frac{dy}{dt}}=y\left(U_{C1}-\overline{U_C}\right) \\ =y\left(1-y\right)\left[wxz\left(R_1-R_3-R_5+R_7\right)+wx\left(R_3-R_7\right)-w\left(G_1+P_1\right)-x\left(G_2+P_2\right)-z\left(G_3+P_3\right)-C_3+P_1+P_2+P_3\right] \end{gathered}$$

(12)

Then, we use U_D1 and U_D2 to represent the expected payoff of local governments’ cooperation strategy and noncooperation strategy, respectively, and use $\overline{U_D}$ to represent the average expected payoff. U_D1, U_D2, and $\overline{U_D}$ can be obtained as follows:

$$\begin{gathered} U_{D1}=wxy\left(-C_4-G_4-G_5+G_3+R_2\right)+wx\left(1-y\right)\left(-C_4-G_4-G_5+R_6\right)+w\left(1-x\right)y\left(-C_4-L_4-G_4+P_5+G_3\right) \\ +w\left(1-x\right)\left(1-y\right)\left(-C_4-L_4-G_4+P_5\right)+\left(1-w\right)xy\left(-C_4-L_4+P_4-G_5+G_3\right) \\ +\left(1-w\right)x\left(1-y\right)\left(-C_4-L_4+P_4-G_5\right)+\left(1-w\right)\left(1-x\right)y\left(-C_4-L_4+P_4+P_5+G_3\right) \\ +\left(1-w\right)\left(1-x\right)\left(1-y\right)\left(-C_4-L_4+P_4+P_5\right) \end{gathered}$$

(13)

$$\begin{gathered} U_{D2}=wxy(-P_3+R_4)+wx\left(1-y\right)R_8+w\left(1-x\right)y\left(-L_4-P_3\right)+w\left(1-x\right)\left(1-y\right)\left(-L_4\right)+\left(1-w\right)xy\left(-L_4-P_3\right) \\ +\left(1-w\right)x\left(1-y\right)\left(-L_4\right)+\left(1-w\right)\left(1-x\right)y\left(-L_4-P_3\right)+\left(1-w\right)\left(1-x\right)\left(1-y\right)\left(-L_4\right) \end{gathered}$$

(14)

$$\overline{U_D}=z{U}_{D1}+\left(1-z\right)U_{D2}$$

(15)

The replicator dynamic equation for local governments ${\displaystyle \frac{dz}{dt}}$ can be calculated as:

$$\begin{gathered} F\left(z\right)={\displaystyle \frac{dz}{dt}}=z\left(U_{D1}-\overline{U_D}\right) \\ =z\left(1-z\right)\left[wxy\left(R_2-R_6-R_4+R_8\right)+wx\left(R_6-R_8\right)-w\left(G_4+P_4\right)-x\left(G_5+P_5\right)+y\left(G_3+P_3\right)-C_4+P_4+P_5\right] \end{gathered}$$

(16)

Finally, we let F(w) = 0, F(x) = 0, F(y) = 0, and F(z) = 0, and combine the replicator dynamic Equations (4), (8), (12) and (16) to establish a four-dimensional dynamic system for schools, enterprises, the central government, and local governments, as shown in Equation (17).

$$\left\{\begin{array}{l}w\left(1-w\right)\left[x\left(B_1-B_3+L_1\right)+y\left(G_1+P_1\right)+z\left(G_4+P_4\right)-C_1\right]=0 \\ x\left(1-x\right)\left[w\left(B_2-B_4+L_2\right)+y\left(G_2+P_2\right)+z\left(G_5+P_5\right)-C_2\right]=0 \\ y\left(1-y\right)\left[wxz\left(R_1-R_3-R_5+R_7\right)+wx\left(R_3-R_7\right)-w\left(G_1+P_1\right)-x\left(G_2+P_2\right)-z\left(G_3+P_3\right)-C_3+P_1+P_2+P_3\right]=0\\ z\left(1-z\right)\left[wxy\left(R_2-R_6-R_4+R_8\right)+wx\left(R_6-R_8\right)-w\left(G_4+P_4\right)-x\left(G_5+P_5\right)+y\left(G_3+P_3\right)-C_4+P_4+P_5\right]=0 \end{array}\right.$$

(17)

In the next section, we will solve this replicator dynamic system to analyze the stability of strategies for the four parties.

3. Results

Based on the quadrilateral evolutionary game model constructed in Section 2, we find 16 four-player pure strategy equilibrium points and build a Jacobian matrix of the model to evaluate the stability of these equilibrium points.

3.1. Equilibrium Points of the Evolutionary Game Model

By solving Equation (17), 41 equilibrium points can be obtained, including 16 four-player pure strategy solutions, 24 two-player pure strategy solutions, and 1 mixed strategy solution, shown in Equations (18)–(20).

$$\left\{\begin{array}{l}{E}_1(w=0, x=0, y=0, z=0) \\ E_2(w=0, x=0, y=0, z=1) \\ E_3(w=0, x=0, y=1, z=0) \\ E_4(w=0, x=0, y=1, z=1) \\ E_5(w=0, x=1, y=0, z=0) \\ E_6(w=0, x=1, y=0, z=1) \\ E_7(w=0, x=1, y=1, z=0) \\ E_8(w=0, x=1, y=1, z=1) \\ E_9(w=1, x=0, y=0, z=0) \\ E_{10}(w=1, x=0, y=0, z=1)\\ E_{11}(w=1, x=0, y=1, z=0)\\ E_{12}(w=1, x=0, y=1, z=1)\\ E_{13}(w=1, x=1, y=0, z=0)\\ E_{14}(w=1, x=1, y=0, z=1)\\ E_{15}(w=1, x=1, y=1, z=0)\\ E_{16}(w=1, x=1, y=1, z=1)\end{array}\right.$$

(18)

$$\left\{\begin{array}{l}{E}_{17}\left(w=0, x=0, y={\displaystyle \frac{C_4-P_4-P_5}{G_3+P_3}}, z={\displaystyle \frac{-C_3+P_1+P_2+P_3}{G_3+P_3}}\right) \\ E_{18}\left(w=0, x=1, y={\displaystyle \frac{G_5+C_4-P_4}{G_3+P_3}}, z={\displaystyle \frac{-G_2-C_3+{P_1+P}_3}{G_3+P_3}}\right)\\ E_{19}\left(w=1, x=0, y={\displaystyle \frac{G_4+C_4-P_5}{G_3+P_3}}, z={\displaystyle \frac{-G_1-C_3+{P_2+P}_3}{G_3+P_3}}\right)\\ E_{20}(w=1, x=1, y={\displaystyle \frac{-R_6+{R_8+G_4+G}_5+C_4}{R_2-R_6-R_4+{R_8+G_3+P}_3}}, z={\displaystyle \frac{-R_3+{R_7+G_1+G}_2+C_3-P_3}{R_1-R_3-R_5+{R_7-G_3-P}_3}})\\ E_{21}(w=0, x={\displaystyle \frac{-C_4+P_4+P_5}{G_5+P_5}}, y=0, z={\displaystyle \frac{C_2}{G_5+P_5}})\\ E_{22}(w=0, x={\displaystyle \frac{G_3+P_3-C_4+P_4+P_5}{G_5+P_5}}, y=1, z={\displaystyle \frac{C_2-G_2-P_2}{G_5+P_5}}) \\ E_{23}(w=1, x={\displaystyle \frac{G_4+C_4-P_5}{R_6-{R_8-G_5-P}_5}}, y=0, z={\displaystyle \frac{C_2-B_2+B_4-L_2}{G_5+P_5}})\\ E_{24}(w=1, x={\displaystyle \frac{G_4-G_3-P_3+C_4-P_5}{R_2-{R_4-G_5-P}_5}}, y=1, z={\displaystyle \frac{C_2-B_2+B_4-L_2-G_2-P_2}{G_5+P_5}}) \\ E_{25}(w=0, x={\displaystyle \frac{-C_3+P_1+P_2+P_3}{G_2+P_2}}, y={\displaystyle \frac{C_2}{G_2+P_2}}, z=0)\\ E_{26}(w=0, x={\displaystyle \frac{-G_3-C_3+{P_1+P}_2}{G_2+P_2}}, y={\displaystyle \frac{C_2-G_5-P_5}{G_2+P_2}}, z=1) \\ E_{27}(w=1, x={\displaystyle \frac{G_1+G_3-P_2-P_3}{R_3-{R_7-G_2-P}_2}}, y={\displaystyle \frac{C_2-B_2+B_4-L_2}{G_2+P_2}}, z=0)\\ E_{28}(w=1, x={\displaystyle \frac{{G_1+G}_3+C_3-P_2}{R_1-R_5-G_2-P_2}}, y={\displaystyle \frac{C_2-B_2+B_4-L_2-G_5-P_5}{G_2+P_2}}, z=1)\\ E_{29}(w={\displaystyle \frac{-C_4+P_4+P_5}{G_4+P_4}}, x=0, y=0, z={\displaystyle \frac{C_1}{G_4+P_4}})\\ E_{30}(w={\displaystyle \frac{G_3+P_3-C_4+P_4+P_5}{G_4+P_4}}, x=0, y=1, z={\displaystyle \frac{C_1-G_1-P_1}{G_4+P_4}})\\ E_{31}(w={\displaystyle \frac{G_5+C_4-P_4}{R_6-{R_8-G_4-P}_4}}, x=1, y=0, z={\displaystyle \frac{C_1-B_1+B_3-L_1}{G_4+P_4}})\\ E_{32}(w={\displaystyle \frac{G_5-G_3-P_3+C_4-P_4}{R_2-{R_4-G_4-P}_4}}, x=1, y=1, z={\displaystyle \frac{C_1-B_1+B_3-L_1-G_1-P_1}{G_4+P_4}})\\ E_{33}(w={\displaystyle \frac{-C_3+P_1+P_2+P_3}{G_1+P_1}}, x=0, y={\displaystyle \frac{C_1}{G_1+P_1}}, z=0) \\ E_{34}(w={\displaystyle \frac{-G_3-C_1+{P_1+P}_2}{G_1+P_1}}, x=0, y={\displaystyle \frac{C_1-G_4-P_4}{G_1+P_1}}, z=1)\\ E_{35}(w={\displaystyle \frac{G_2+G_3-P_1-P_3}{R_3-{R_7-G_1-P}_1}}, x=1, y={\displaystyle \frac{C_1-B_1+B_3-L_1}{G_1+P_1}}, z=0)\\ E_{36}(w={\displaystyle \frac{G_2+G_3+C_3-P_1}{R_1-{R_5-G_1-P}_1}}, x=1, y={\displaystyle \frac{C_1-B_1+B_3-L_1-G_4-P_4}{G_1+P_1}}, z=1)\\ E_{37}(w={\displaystyle \frac{C_2}{B_2-B_4+L_2}}, x={\displaystyle \frac{C_1}{B_1-B_3+L_1}}, y=0, z=0) \\ E_{38}\left(w={\displaystyle \frac{C_2-G_5-P_5}{B_2-B_4+L_2}}, x={\displaystyle \frac{C_1-G_4-P_4}{B_1-B_3+L_1}}, y=0, z=1\right) \\ E_{39}(w={\displaystyle \frac{C_2-G_2-P_2}{B_2-B_4+L_2}}, x={\displaystyle \frac{C_1-G_1-P_1}{B_1-B_3+L_1}}, y=1, z=0)\\ E_{40}(w={\displaystyle \frac{C_2-G_2-P_2-G_5-P_5}{B_2-B_4+L_2}}, x={\displaystyle \frac{C_1-G_1-P_1-G_4-P_4}{B_1-B_3+L_1}}, y=1, z=1)\end{array}\right.$$

(19)

$$\left\{\begin{array}{l}F\left(w\right)=0\\ \begin{array}{c}F\left(x\right)=0\\ F\left(y\right)=0\\ F\left(z\right)=0\end{array}\end{array}\right. and w^{*}, x^{*},y^{*},z^{*}\in (0, 1)$$

(20)

The strategy solutions for all players are certain values of 0 or 1 in Equation (18), but there are uncertain results of players’ strategy solutions in Equations (19) and (20). According to game theory, the strategy solution in a multi-agent evolutionary game must be a Nash equilibrium in pure strategies [38]. Thus, we only choose the 16 pure strategy solutions E₁, E₂, E₃, E₄, E₅, E₆, E₇, E₈, E₉, E₁₀, E₁₁, E₁₂, E₁₃, E₁₄, E₁₅, and E₁₆ as the possible stable equilibrium points.

3.2. The Stability of the Equilibrium Points

We adopt Friedman’s replication dynamics system stability analysis method [41] to evaluate the stability of the 16 equilibrium points by constructing the Jacobian matrix and calculating its eigenvalues. The Jacobian matrix of the quadrilateral evolutionary game is:

$$J=\left[\begin{array}{cc}\begin{array}{cc}{\displaystyle \frac{\partial F\left(w\right)}{\partial w}}& {\displaystyle \frac{\partial F\left(w\right)}{\partial x}}\\ {\displaystyle \frac{\partial F\left(x\right)}{\partial w}}& {\displaystyle \frac{\partial F\left(x\right)}{\partial x}}\end{array}& \begin{array}{cc}\begin{array}{c}{\displaystyle \frac{\partial F\left(w\right)}{\partial y}}\\ {\displaystyle \frac{\partial F\left(x\right)}{\partial y}}\end{array}& \begin{array}{c}{\displaystyle \frac{\partial F\left(w\right)}{\partial z}}\\ {\displaystyle \frac{\partial F\left(x\right)}{\partial z}}\end{array}\end{array}\\ \begin{array}{cc}{\displaystyle \frac{\partial F\left(y\right)}{\partial w}}& {\displaystyle \frac{\partial F\left(y\right)}{\partial x}}\\ {\displaystyle \frac{\partial F\left(z\right)}{\partial w}}& {\displaystyle \frac{\partial F\left(z\right)}{\partial x}}\end{array}& \begin{array}{cc}{\displaystyle \frac{\partial F\left(y\right)}{\partial y}}& {\displaystyle \frac{\partial F\left(y\right)}{\partial z}}\\ {\displaystyle \frac{\partial F\left(z\right)}{\partial y}}& {\displaystyle \frac{\partial F\left(z\right)}{\partial z}}\end{array}\end{array}\right]=\left[\begin{array}{cc}\begin{array}{cc}{J}_{11}& J_{12}\\ J_{21}& J_{22}\end{array}& \begin{array}{cc}{J}_{13}& J_{14}\\ J_{23}& J_{24}\end{array}\\ \begin{array}{cc}{J}_{31}& J_{32}\\ J_{41}& J_{42}\end{array}& \begin{array}{cc}{J}_{33}& J_{34}\\ J_{43}& J_{44}\end{array}\end{array}\right]$$

(21)

where

$$\begin{array}{l}{J}_{11}=(1-2w)\left[x\left(B_1-B_3+L_1\right)+y\left(G_1+P_1\right)+z\left(G_4+P_4\right)-C_1\right]\\ J_{12}=w(1-w)(B_1-B_3+L_1)\\ J_{13}=w(1-w)(G_1+P_1)\\ J_{14}=w(1-w)(G_4+P_4)\\ J_{21}=x(1-x)(B_2-B_4+L_2)\\ J_{22}=\left(1-2x\right)\left[w\left(B_2-B_4+L_2\right)+y\left(G_2+P_2\right)+z\left(G_5+P_5\right)-C_2\right]\\ J_{23}=x(1-x)(G_2+P_2)\\ J_{24}=x(1-x)(G_5+P_5)\\ J_{31}=y(1-y)\left[xz\left(R_1-R_3-R_5+R_7\right)+x\left(R_3-R_7\right)-G_1-P_1\right]\\ J_{32}=y(1-y)\left[wz\left(R_1-R_3-R_5+R_7\right)+w\left(R_3-R_7\right)-G_2-P_2\right]\\ J_{33}=\left(1-2y\right)\left[wxz\left(R_1-R_3-R_5+R_7\right)+wx\left(R_3-R_7\right)-w\left(G_1+P_1\right)-x\left(G_2+P_2\right)-z\left(G_3+P_3\right)-C_3+P_1+P_2+P_3\right]\\ J_{34}=y(1-y)\left[wx\left(R_1-R_3-R_5+R_7\right)-G_3-P_3\right]\\ J_{41}=z(1-z)\left[xy\left(R_2-R_6-R_4+R_8\right)+x\left(R_6-R_8\right)-G_4-P_4\right]\\ J_{42}=z(1-z)\left[wy\left(R_2-R_6-R_4+R_8\right)+w\left(R_6-R_8\right)-G_5-P_5\right]\\ J_{43}=z(1-z)\left[wx\left(R_2-R_6-R_4+R_8\right)+G_3+P_3\right]\\ J_{44}=\left(1-2z\right)\left[wxy\left(R_2-R_6-R_4+R_8\right)+wx\left(R_6-R_8\right)-w\left(G_4+P_4\right)-x\left(G_5+P_5\right)+y\left(G_3+P_3\right)-C_4+P_4+P_5\right]\end{array}$$

According to Lyapunov’s stability theory, the stability of the equilibrium point can be evaluated by analyzing the eigenvalues of the Jacobian matrix [42]. If all the eigenvalues are negative, the corresponding equilibrium point is a stable point; if there are any positive eigenvalues, the corresponding equilibrium point is an unstable point; if the signs of the eigenvalues cannot be determined, the corresponding equilibrium point is a saddle point; and if all the eigenvalues are 0, the stability of the corresponding equilibrium point is of no significance. We thus substitute each of the 16 equilibrium points into the Jacobian matrix and calculate the corresponding eigenvalues, which are shown in

Table 3. The eigenvalues and stability of the equilibrium points.

Equilibrium Point	Eigenvalues								State
	λ1	Symbol	λ2	Symbol	λ3	Symbol	λ4	Symbol
E1 (0,0,0,0)	−C1	0	−C2	0	−C3 + P1 + P2 + P3	0	−C4 + P4 + P5	0	×
E2 (0,0,0,1)	G4 + P4 − C1	+	G5 + P5 − C2	+	−G3 − C3 + P1 + P2	0	C4 − P4 − P5	N	unstable
E3 (0,0,1,0)	G1 + P1 − C1	+	G2 + P2 − C2	+	C3 − P1 − P2 − P3	N	G3 + P3 − C4 + P4 + P5	+	unstable
E4 (0,0,1,1)	G1 + P1 + G4 + P4 − C1	+	G2 + P2 + G5 + P5 − C2	+	G3 + C3 − P1 − P2	N	−G3 − P3 + C4 − P4 − P5	N	unstable
E5 (0,1,0,0)	B1 −  B3 + L1 −  C1	N	C2	+	−G2 −  C3 + P1 + P3	0	−G5 −  C4 + P4	0	unstable
E6 (0,1,0,1)	B1 −  B3 + L1 + G4 + P4 − C1	N	−G5 − P5 + C2	N	−G2 −  G3 −  C3 + P1	0	G5 + C4 − P4	N	×/unstable
E7 (0,1,1,0)	B1 −  B3 + L1 + G1 + P1 −  C1	N	−G2 − P2 + C2	N	G2 + C3 − P1 − P3	N	−G5 + G3 + P3 −  C4 + P4	+	unstable
E8 (0,1,1,1)	B1 −  B3 + L1 + G1 + P1 + G4 + P4 −  C1	N	−G2 − P2 −  G5 − P5 + C2	N	G2 + G3 + C3 − P1	N	G5 −  G3 − P3 + C4 − P4	N	saddle
E9 (1,0,0,0)	C1	+	B2 −  B4 + L2 −  C2	N	−G1 −  C3 + P2 + P3	0	−G4 −  C4 + P5	0	unstable
E10 (1,0,0,1)	−G4 − P4 + C1	N	B2 − B4 + L2 + G5 + P5 − C2	N	−G1 −  G3 −  C3 + P2	0	G4 + C4 − P5	N	×/unstable
E11 (1,0,1,0)	−G1 − P1 + C1	N	B2 −  B4 + L2 + G2 + P2 − C2	N	G1 + C3 − P2 − P3	N	−G4 + G3 + P3 −  C4 + P5	+	unstable
E12 (1,0,1,1)	−G1 − P1 −  G4 − P4 + C1	N	B2 −  B4 + L2 + G2 + P2 + G5 + P5 − C2	N	G1 + G3 + C3 − P2	N	G4 −  G3 − P3 + C4 − P5	N	saddle
E13 (1,1,0,0)	−B1 + B3 −  L1 + C1	N	−B2 + B4 −  L2 + C2	N	R3 −  R7 −  G1 −  G2 −  C3 + P3	−	R6 −  R8 −  G4 −  G5 −  C4	−	saddle
E14 (1,1,0,1)	−B1 + B3 −  L1 −  G4 − P4 + C1	N	−B2 + B4 − L2 − G5 − P5 + C2	N	R1 − R5 − G1 − G2 − G3 − C3	−	−R6 + R8 + G4 + G5 + C4	N	saddle
E15 (1,1,1,0)	−B1 + B3 −  L1 −  G1 − P1 + C1	N	−B2 + B4 −  L2 −  G2 − P2 + C2	N	−R3 + R7 + G1 + G2 + C3 − P3	N	R2 −  R4 −  G4 −  G5 + G3 + P3 −  C4	N	saddle
E16 (1,1,1,1)	−B1 + B3 −  L1 −  G1 − P1 −  G4 − P4 + C1	N	−B2 + B4 −  L2 −  G2 − P2 −  G5 − P5 + C2	N	−R1 + R5 + G1 + G2 + G3 + C3	N	−R2 + R4 + G4 + G5 −  G3 − P3 + C4	N	saddle

.

Here, “N” means that the sign of the eigenvalue cannot be determined, “+” means that the eigenvalue is positive, and “−” means that the eigenvalue is negative. As shown in Table 3, there is no stable point, which indicates that there is no stable state of cooperation between schools, enterprises, the central government, and local governments. However, there are six saddle points at which the quadrilateral evolutionary game has the possibility of reaching a stable state; that is, stable cooperation between the four parties requires conditions. Therefore, in the following section, we will discuss the conditions required for the cooperation between schools, enterprises, the central government, and local governments, especially the approaches that should be taken by governments to promote sustainable cooperation between schools and enterprises.

4. Discussion

As shown in Table 3, there are six saddle points, E₈ (0,1,1,1), E₁₂ (1,0,1,1), E₁₃ (1,1,0,0), E₁₄ (1,1,0,1), E₁₅ (1,1,1,0), E₁₆ (1,1,1,1). Next, we will assess the stability conditions for each saddle point separately.

Scenario 1 (E₈): When −B₃ + L₁ + P₁ + P₄ < 0, −G₂ − G₅ + C₂ < 0, G₂ + G₃ + C₃ − P₁ < 0, and G₅ − G₃ + C₄ − P₄ < 0, the quadrilateral evolutionarily game reaches a stable state. In this case, enterprises, the central government, and local governments all choose a cooperation strategy, while only schools choose a noncooperation strategy. Thus, this point is not ideal. The stability conditions for this point are not expected to be achieved. It requires P₁ + P₄ > B₃ − L₁, or P₁ > G₂ + G₃ + C₃, or P₄ > G₅ − G₃ + C₄, which indicates that strengthening the punishment from the central government and local governments for choosing a noncooperation strategy is an important measure to prevent the emergence of schools’ noncooperative behavior.

Scenario 2 (E₁₂): This is similar to Scenario 1. When −G₁ − G₄ + C₁ < 0, −B₄ + L₂ + P₂ + P₅ < 0, G₁ + G₃ + C₃ − P₂ < 0, and G₄ − G₃ + C₄ − P₅ < 0, the quadrilateral evolutionarily game reaches a stable state. In this case, schools, the central government, and local governments all choose a cooperation strategy, while only enterprises choose a noncooperation strategy. This point is also not ideal. The stability conditions for this point are also not expected to be achieved. It requires P₂ + P₅ > B₄ − L₂, or P₂ > G₁ + G₃ + C₃, or P₅ > G₄ − G₃ + C₄, which also indicates that strengthening the punishment from the central government and local governments for choosing a noncooperation strategy is an important measure to prevent the emergence of enterprises’ noncooperative behavior.

Scenario 3 (E₁₃): When −B₁ + C₁ < 0 and −B₂ + C₂ < 0, the quadrilateral evolutionarily game reaches a stable state. In this case, schools and enterprises choose a cooperation strategy, while the central government and local governments choose a noncooperation strategy. The stability conditions can be transformed into B₁ > C₁ and B₂ > C₂. That is, when the benefits obtained by schools and enterprises through cooperating with each other are greater than their costs, even without the participation of the government, the cooperation between schools and enterprises can also reach a stable state.

Scenario 4 (E₁₄): When −B₁ − G₄ + C₁ < 0, −B₂ − G₅ + C₂ < 0, and −R₆ + G₄ + G₅ + C₄ < 0, the quadrilateral evolutionarily game reaches a stable state. In this case, schools, enterprises, and local governments all choose a cooperation strategy, while only the central government chooses a noncooperation strategy. The stability conditions can be transformed into B₁ + G₄ > C₁, B₂ + G₅ > C₂, and R₆ > G₄ + G₅ + C₄, which shows that increasing rewards offered by local governments to encourage the cooperation strategy between schools and enterprises can promote their cooperation, even without direct intervention from the central government. However, just improving the incentives may not suffice. It is paramount to focus on improving the practical effects of the cooperation between schools and enterprises.

Scenario 5 (E₁₅): When −B₁ − G₁ + C₁ < 0, −B₂ − G₂ + C₂ < 0, −R₃ + G₁ + G₂ + C₃ − P₃ < 0, and −R₄ + P₃ < 0, the quadrilateral evolutionarily game reaches a stable state. In this case, schools, enterprises, and the central government all choose a cooperation strategy, while only local governments choose a noncooperation strategy. The stability conditions can be transformed into B₁ + G₁ > C₁, B₂ + G₂ > C₂, R₃ > G₁ + G₂ + C₃ − P₃, and P₃ < R₄, which shows that increasing rewards provided by the central government to incentivize the cooperation strategy between schools and enterprises can facilitate their cooperation, even in the absence of active involvement from local governments. In addition, the effectiveness of punishments imposed by the central government on local governments has minimal impact on fostering school–enterprise cooperation. What really matters is improving the effectiveness of policies on school–enterprise cooperation.

Scenario 6 (E₁₆): When −B₁ − G₁ − G₄ + C₁ < 0, −B₂ − G₂ − G₅ + C₂ < 0, −R₁ + G₁ + G₂ + G₃ + C₃ < 0, and −R₂ + G₄ + G₅ − G₃ + C₄ < 0, the quadrilateral evolutionarily game reaches a stable state. In this case, all the four parties choose a cooperation strategy. This point is the most ideal. The stability conditions can be transformed into B₁ + G₁ + G₄ > C₁, B₂ + G₂ + G₅ > C₂, R₁ > G₁ + G₂ + G₃ + C₃, and R₂ > G₄ + G₅ − G₃ + C₄. Compared with Scenarios 4 and 5, in this scenario, increasing the rewards from the central government and local governments for schools and enterprises choosing a cooperation strategy can also have positive effects, and beyond that, since the rewards are provided jointly by the central government and local governments, their spending on participating in school–enterprise cooperation will be decreased. Moreover, the reward from the central government to local governments does not have much influence on promoting the cooperation between schools and enterprises. What really matters is improving the effectiveness of policies on school–enterprise cooperation.

5. Conclusions

This paper explores the stability of cooperation between schools, enterprises, the central government, and local governments, and analyzes the approaches that can be taken by central and local governments to promote school–enterprise cooperation. The main conclusions are as follows:

(1) There is no stable equilibrium point in the quadrilateral game between schools, enterprises, the central government, and local governments; instead, there are six saddle points, E8 (0,1,1,1), E12 (1,0,1,1), E13 (1,1,0,0), E14 (1,1,0,1), E15 (1,1,1,0), E16 (1,1,1,1), at which the quadrilateral game has the potential to be stable. Facilitating stable and sustainable cooperation between schools and enterprises requires conditions.

(2) The primary condition for school–enterprise cooperation is benefit rather than governmental intervention. When the benefits of cooperation outweigh the costs, schools and enterprises are inclined to cooperate with each other independently of governmental intervention. Therefore, creating a robust platform for school–enterprise cooperation is crucial, and include providing informational support, communication channels, and ongoing evaluation.

(3) Rewards and punishments from both the central government and local governments have a profound impact on strategy selection by schools and enterprises. Hence, appropriate enhancements to incentives and penalties by the central and local governments can effectively promote cooperation between schools and enterprises.

(4) Simply increasing rewards and punishments addresses symptoms rather than the underlying issues. Enhancing the actual outcomes of cooperation is paramount. Improving policy effectiveness, rather than merely formulating policies, is critical for promoting sustainable school–enterprise cooperation. Making effective policies should become an essential issue for the central government and local governments to facilitate school–enterprise cooperation in the future.

(5) Benefits also decisively influence the strategy choice of both the central government and local governments. When the benefits of participating in school–enterprise cooperation outweigh the costs, both the central government and local governments will choose a cooperation strategy. Thus, fostering tangible outcomes derived from school–enterprise cooperation not only strengthens the relationship between schools and enterprises but also encourages deep and enduring government involvement. Moreover, the rewards and punishments from the central government to local governments have a limited effect on promoting school–enterprise cooperation.

These conclusions underscore the importance of aligning incentives with tangible benefits to foster effective cooperation between schools, enterprises, and governments. On the one hand, reward and punishment are important means for governments to promote school–enterprise cooperation. Both the central government and local governments could appropriately increase the incentives for cooperative behaviors of schools and enterprises, such as encourage schools to adjust the structure of disciplines according to market supply and demand and employment quality assessment, encourage high-level professionals from enterprises to teach in schools, etc. Additionally, governments could appropriately increase the penalties, especially financial penalties, for non-cooperative behavior of schools and enterprises. Optimizing the central government’s rewards and punishments for local governments could also raise the enthusiasm of local governments to participate in school–enterprise cooperation. On the other hand, enhancing policy effectiveness and promoting meaningful outcomes will be more critical for advancing school–enterprise cooperation. It is necessary to establish a dynamic evaluation mechanism to monitor the quality of the implementation of policies and the performance of school–enterprise cooperation. In addition, establishing and optimizing cooperation platforms for school–enterprise cooperation will also be of great significance in raising the efficiency of the cooperation between schools and enterprises. The cooperation platforms can provide supply and demand information, such as talent cultivation and industrial development, for closer and higher quality cooperation between schools and enterprises.

In summary, this study focuses on the role played by the central and local governments in the process of school–enterprise cooperation from the perspective of quadrilateral evolutionary games, which has not been sufficiently explored in previous studies. Compared with the study that we published in 2023, this study has developed the model from tripartite game to quadrilateral game. This model not only divides governments into the central government and local governments, but also resets the costs, benefits, and losses of each party, especially the benefits under different strategy combinations, which is closer to real life. However, the extent do which policy, funding, and supervision of the central and local governments affect school–enterprise cooperation; whether different cooperation models and levels have an impact on the costs, benefits and losses of school–enterprise cooperation in realistic situations; whether the findings be applied to different regional contexts or specific educational situations; and how to better integrate cooperative initiatives into existing education systems have not been fully studied. Moreover, the potential irrationality of the four-player populations and simulation of the evolution mechanism were also not explored in this study. These issues deserve further attention in future work.