Sustainable Cooperation between Schools, Enterprises, and Government: An Evolutionary Game Theory Analysis

Abstract

Promoting close and sustainable cooperation between schools, enterprises, and government has become an important concern in many countries. However, the factors that influence the cooperation between schools, enterprises, and government have not been sufficiently explored in previous studies. Therefore, based on evolutionary game theory, in this paper, we construct a tripartite evolutionary game model of schools, enterprises, and government in order to analyze the strategies of different players and their influencing factors. The results show that the main factor that influences the stability of the strategies of schools and enterprises is the reward for positive cooperation from sources other than the government, and the main factor that influences the stability of the strategy of the government is the benefits from a positive cooperation strategy under the scenario where schools cooperate with enterprises. Some suggestions to promote sustainable cooperation between schools, enterprises, and government are proposed at the end.

1. Introduction

In many models of economic development, the development of the economy depends on human capital (talent). Schools can be viewed as the producers of such capital and enterprises as the consumers. Thus, strengthening the relationship between schools and enterprises is crucial to maintaining a dynamic balance between the supply and demand of talent and promoting the growth of the economy. In addition, the government plays an important role in facilitating the connection between schools and enterprises. For example, the German government has established a series of laws, regulations, and management systems to ensure that students can acquire theoretical knowledge in school and develop practical skills in enterprise [1]. The U.S. government has also passed the Strengthening Career and Technical Education (CTE) for the 21st Century Act to support partnerships among various schools and enterprises [2], and the Chinese government has enacted Several Opinions on Deepening the Integration of Industry and Education to promote the supply side structural reform of talent and improve the synergy of educational resources and regional industries [3]. Promoting close and sustainable cooperation between schools, enterprises, and government has clearly become an important concern in these countries.

However, the reality is that the cooperation between schools, enterprises, and government has not been very effective. This is due to the differences in goals and culture between schools and enterprises, as well as disputes over the ownership of intellectual property. The differences in organizational attributes and social functions of schools and enterprises lead to two types of social division of labor, which results in differences in the goals of both of these types of institutions [4,5,6]. The goals of schools include cultivation, education, and theoretical research, and the primary goal of enterprises is to maximize profits. These differences in goals also lead to differences in culture between schools and enterprises, such as in organization and management, behavioral patterns, and approaches to schedules [7,8,9]. In addition, there are some differences regarding the ownership of intellectual property. For example, researchers at schools tend to publish research results in order to increase their influence and push the frontier of knowledge, but enterprises are incentivized to keep their core technology and know-how secret in order to monopolize the market [10]. All these factors can impede the cooperation between schools, enterprises, and government.

Therefore, this study set out to investigate the main factors that influence the strategies of schools, enterprises, and government and to provide some suggestions to improve the effectiveness of sustainable cooperation between them.

The rest of the paper is organized as follows: Section 2 reviews the relevant literature. Section 3 constructs the tripartite evolutionary game model. Section 4 analyzes the stability of different combinations of game strategies. Section 5 summarizes the main conclusions and provides some suggestions for sustainable cooperation between schools, enterprises, and government in the real world.

2. Literature Review

The cooperation between schools and enterprises has been explored extensively in the existing literature. Many scholars have already investigated various school–enterprise cooperation models, such as the Dual System, Cooperative Education, and Sandwich Courses.

The Dual System is considered to be the driving force of Germany’s post-war economic recovery and has become an exemplary case of school–enterprise cooperation. Theuerkauf and Weiner summarized five major characteristics of the Dual System, including a broad basic education, combined technical training and theory, training directed at acquiring key qualifications, a standardized system, and a planned change from schools to the training system [11]. The Federal Ministry of Education and Research of Germany (BMBF) has also published a book to introduce the origins, features, and training processes of the Dual System in detail [12]. Pleshakova analyzed the genesis and development of the Dual System in Germany from a historical perspective [13]. Given the advantages of the Dual System, some scholars have discussed the practice of Dual System in countries other than Germany, such as Russia [14,15], Ukraine [16], and China [17,18,19]. However, there are also some weaknesses with the Dual System. Pritchard pointed out that the Dual System is permeated with tensions emanating from individuals, schools, firms, and various influential interest groups [20] that can limit competition in the labor market, delay adult status in the labor market, and fail to guarantee employment [21].

Compared to the German Dual System, in which the government and enterprises are deeply involved, American Cooperative Education and British Sandwich Courses are driven by schools, with little responsibility from government or enterprises [22]. Cooperative Education is a model of school–enterprise cooperation in America that refers to an educational program that combines classroom learning with work experience, and Younis and Pierrakos et al. argued that Cooperative Education is essential for both students and society [23,24]. Cooperative Education has also been found to have a positive impact on students’ early career success and self-efficacy [25,26]. Students who enroll in Cooperative Education programs can learn professional knowledge and skills and gain practical on-the-job experience [27,28].

Sandwich Courses are a British model of school–enterprise cooperation that involve a pattern in which periods of school study alternate with periods of industrial training or experience [29]. Sandwich Courses have been recognized as an effective method for accumulating sustained, structured work experience and improving employment chances [30,31,32].

The motivations and factors of cooperation between schools and enterprises have also attracted the attention of scholars. Lee and Win (2004) summarized the motivations of schools in cooperating with enterprises, such as assessing the needs of the economy and developing talent accordingly, placing students in industry to connect classroom learning with practical experience, conducting both fundamental and applied research, accessing protected markets, enhancing the business stature, improving the implementation of new technology, developing new products and patents, and saving production costs [33]. Arza (2010) divided the motivations of schools into economic motivation and research motivation [34]. Similarly, Lam (2011) argued that “gold”, “ribbons”, and “puzzles” are the motivators of researchers in schools for cooperating with enterprises and found that few academic researchers are driven by economic motivation [35]. Reducing transaction costs [36,37]; obtaining human capital, technology, education, and equipment [38]; and establishing a network of cooperation [39] have all been argued to be motivators of enterprises to cooperate with schools. Moreover, the factors influencing the cooperation between schools and enterprises, such as the scale of schools and enterprises [40,41], trust and mutual benefits between schools and enterprises [42,43], and culture differences [44], have also been widely explored in the literature.

However, the previous literature on models, motivations, and factors has focused on schools and enterprises. Although it has been realized that the government plays a crucial role in school–enterprises cooperation and a few scholars have proposed that the government should formulate some policies to facilitate cooperation between schools and enterprises [11,12], the government has not been considered as the main party in the cooperation between schools and enterprises. The influence of the government on the cooperation between schools and enterprises has been ignored.

In addition, evolutionary game theory is an effective tool for analyzing the strategic interactions between different parties [45], and has been used in various disciplines, including economics [46,47], public policy [48], and environmental science [49,50,51]. Some scholars have also introduced evolutionary game theory into education. For example, Zhu and Wang (2022) built an evolutionary game model involving government, universities, and students to explore the development of the choice between innovation and entrepreneurship in education [52], and Li and Wang (2022) discussed the management of primary and secondary school students’ online learning during COVID-19 lockdowns by constructing two game models involving “schools and students” and “schools, students, and parents” [53]. Zhang and Zeng (2022) analyzed the manifestation of both the instrumental and human value of education for sustainable development, and proposed that a country’s curriculum on sustainable development should start from concrete education issues that urgently need to be solved within the theory of sustainable development [54]. However, evolutionary game theory has rarely been used to research the cooperation between schools and enterprises.

To sum up, although the above research has provided some theoretical and methodological support for the study of cooperation between schools, enterprises, and government, there are still some shortcomings. (1) There are very limited studies that use evolutionary game theory to analyze the cooperation between schools and enterprises. (2) The role government plays in this cooperation has yet to be sufficiently revealed. To address these shortcomings, in this paper, we construct a tripartite evolutionary game model in which schools, enterprises, and the government are all considered as main game players. Based on stability strategy analysis, we then put forward some effective ways to promote cooperation between schools, enterprises, and government.

3. Methodology

In this section, a tripartite evolutionary game model of schools, enterprises, and government is established to investigate the evolution of cooperative behavior among these three stakeholders.

3.1. Model Assumptions

According to the evolutionary game theory, each player is an individual with limited rationality and has two alternative strategies (cooperation and defection) with different probabilities and payoff [55,56]. Each player updates their strategy based on their payoff [57]. Many scholars argue that when schools cooperate with enterprises (i.e., when both schools and enterprises choose a cooperative strategy), they can get a lot of benefits, such as outstanding talent and advanced technology [33,36,37,38,39]. Moreover, the government will also promulgate policies to reward or punish schools and enterprises for their strategies of cooperation or defection [1,2,3]. Therefore, considering evolutionary game theory; previous research on the cooperation between schools, enterprises, and the government; and practical experience, we make the following assumptions.

Assumption 1.

There are three populations, schools, enterprises, and government, all of which have bounded rationality.

Assumption 2.

Each population has two strategies: a positive cooperation strategy and a negative cooperation strategy. x, y, and z represent the probability of selecting the positive cooperation strategy for schools, enterprises, and government, respectively, and 1 − x, 1 − y, and 1 − z represent the probability of selecting the negative cooperation strategy of schools, enterprises, and government, respectively.

Assumption 3.

We assume that schools can receive some benefits in terms of discipline construction, knowledge innovation, and talent cultivation, denoted by M, before cooperating with enterprises. When schools choose the positive cooperation strategy, some costs will be spent on the development of teachers and the adjustment of talent cultivation to meet the needs of enterprises, denoted by C₁. In return, schools obtain some rewards from the government such as political and financial support, denoted by G₁. Beyond that, if enterprises also choose the positive cooperation strategy, schools will produce more high-quality talent. This talent will not only bring direct economic benefits to schools, such as alumni donations, but will also bring indirect benefits, such as improving the social reputation of the schools. We use R₁ to denote the additional benefits to schools from cooperating with enterprises. However, when schools choose the negative cooperation strategy, they receive some punishment from the government, denoted by P₁. In order to reflect the binding force of punishment, we assume P₁ > C₁ + G₁ + R₁.

Assumption 4.

We assume that enterprises can receive some benefits from the cooperation process, denoted by N, before cooperating with schools. When enterprises choose the positive cooperation strategy, some costs will be spent on providing internships and salaries for students, as well as training and managing students, denoted by C₂. In return, enterprises obtain some rewards from the government such as political and financial support, denoted by G₂. Beyond that, if schools choose the positive cooperation strategy, enterprises will have access to cheap labor and technical support from schools, which can improve their social reputation. We use R₂ to denote the additional benefits to enterprises from cooperating with schools. However, when enterprises choose the negative cooperation strategy, they receive some punishment from the government, denoted by P₂, and we assume P₂ > C₂ + G₂ + R₂.

Assumption 5.

When the government chooses the positive cooperation strategy, some costs will be incurred on aligning the interests of various parties and issuing relevant policies to regulate and facilitate the cooperative behavior of schools and enterprises, denoted by C₃. In return, if both schools and enterprises choose the positive cooperation strategy, the government will gain more outstanding talent, better development of the economy, and a higher social reputation, denoted by B₁. Moreover, when the government chooses the negative cooperation strategy, and both schools and enterprises choose the positive cooperation strategy, the government will still receive the benefits of talent cultivation, economic development, and social reputation as a result of the cooperation between schools and enterprises, denoted by B₂. In order to reflect the effect of the government’s support, we assume B₁ > B₂. However, when the government chooses the negative cooperation strategy, and at least one of schools or enterprises chooses the negative cooperation strategy, the government suffers losses in talent cultivation, economic development, and social reputation, denoted by P₃. Consistent with the previous assumption, we also stipulate P₃ > C₃ + G₃ + R₃.

In order to understand the model assumptions clearly, we list the meanings of the aforementioned symbols in

Table 1. Notation of symbols.

Symbol	Meaning
M	The benefit of schools before cooperating with enterprises.
N	The benefit of enterprises before cooperating with schools.
C1	The cost of schools’ positive cooperation strategy.
C2	The cost of enterprises’ positive cooperation strategy.
C3	The cost of the government’s positive cooperation strategy.
G1	The reward for schools’ positive cooperation strategy from the government.
G2	The reward for enterprises’ positive cooperation strategy from the government.
R1	The additional rewards of schools’ positive cooperation strategy from sources other than the government.
R2	The additional rewards of enterprises’ positive cooperation strategy from sources other than the government.
B1	The benefits of the government’s positive cooperation strategy.
B2	The benefits of the government’s negative cooperation strategy.
P1	The punishment for schools’ negative cooperation strategy from the government.
P2	The punishment for enterprises’ negative cooperation strategy from the government.
P3	The punishment for the government’s negative cooperation strategy.

.

3.2. Model Construction

Based on the above assumptions, we drew a tree diagram to show all the combinations of strategies among schools, enterprises, and government, shown in Figure 1.

As shown in Figure 1, there are eight strategy combinations, labeled as S₁ (schools’ positive cooperation strategy, enterprises’ positive cooperation strategy, government’s positive cooperation strategy), S₂ (schools’ positive cooperation strategy, enterprises’ positive cooperation strategy, government’s negative cooperation strategy), S₃ (schools’ positive cooperation strategy, enterprises’ negative cooperation strategy, government’s positive cooperation strategy), S₄ (schools’ positive cooperation strategy, enterprises’ negative cooperation strategy, government’s negative cooperation strategy), S₅ (schools’ negative cooperation strategy, enterprises’ positive cooperation strategy, government’s positive cooperation strategy), S₆ (schools’ negative cooperation strategy, enterprises’ positive cooperation strategy, government’s negative cooperation strategy), S₇ (schools’ negative cooperation strategy, enterprises’ negative cooperation strategy, government’s positive cooperation strategy), and S₈ (schools’ negative cooperation strategy, enterprises’ negative cooperation strategy, government’s negative cooperation strategy).

According to the previous assumptions and strategy combinations, we construct the tripartite evolutionary game payoff matrix of schools, enterprises, and government, as shown in

Table 2. Payoff matrix of the tripartite evolutionary game between schools, enterprises, and government.

				Government
				Positive Cooperation Strategy	Negative Cooperation Strategy
schools	positive cooperation strategy	enterprises	positive cooperation strategy	M − C1 + G1 + R1, N − C2 + G2 + R2, −C3 − G1 − G2 + B1	M − C1 + R1, N − C2 + R2, B2
			negative cooperation strategy	M − C1 + G1, N − P2, −C3 − G1 + P2	M − C1, N, −P3
	negative cooperation strategy	enterprises	positive cooperation strategy	M − P1, N − C2 + G2, −C3 − G2 + P1	M, N − C2, −P3
			negative cooperation strategy	M − P1, N − P2, −C3 + P1 + P2	M, N, −P3

.

Each entry in Table 2 contains three values; the first represents the schools’ payoff, the second represents the enterprises’ payoff, and the last represents the government’s payoff. We adopt a replicator dynamics equation, one of the most fundamental methods in evolutionary game theory, to indicate the evolution mechanism of cooperative behavior [58]. Here, we set U_A1 and U_A2 as the expected payoff of schools’ positive and negative cooperation strategies, respectively. According to the tripartite evolutionary game payoff matrix and the probability of different strategies among schools, enterprises, and government, U_A1 and U_A2 can be calculated as follows:

$$\begin{array}{l}{U}_{A1}=yz \left(M-C_1+G_1+R_1\right)+y\left(1-z\right)\left(M-C_1+R_1\right)+\left(1-y\right)z\left(M-C_1+G_1\right)\\ +\left(1-y\right)\left(1-z\right)\left(M-C_1\right)\end{array}$$

(1)

$$U_{A2}=yz \left(M-P_1\right)+y\left(1-z\right)M+\left(1–y\right)z\left(M-P_1\right)+\left(1-y\right)\left(1-z\right)M$$

(2)

$\overline{U_A}$ denotes the average expected payoff of schools, which can be simplified to:

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

(3)

Similarly, we use U_B1 and U_B2 to represent the expected payoff of enterprises’ positive and negative cooperation strategies, respectively, and use $\overline{U_B}$ to represent the average expected payoff of enterprises, which are simplified below.

$$\begin{array}{l}{U}_{B1}=xz \left(N-C_2+G_2+R_2\right)+x\left(1-z\right)\left(N-C_2+R_2\right)+\left(1-x\right)z\left(N-C_2+G_2\right)\\ +\left(1-x\right)\left(1-z\right)\left(N-C_2\right)\end{array}$$

(4)

$$U_{B2}=xz \left(N-P_2\right)+x\left(1-z\right)N+\left(1-x\right)z\left(N-P_2\right)+\left(1-x\right)\left(1-z\right)N$$

(5)

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

(6)

Finally, we use U_C1 and U_C2 in the same way to represent the expected payoff of the government’s positive and negative cooperation strategies, respectively, and use $\overline{U_C}$ to represent the average expected payoff of the government:

$$\begin{array}{l}{U}_{C1}=xy \left(-C_3-G_1-G_2+B_1\right)+x\left(1-y\right)\left(-C_3-G_1+P_2\right)+\left(1-x\right)y\left(-C_3-G_2+P_1\right)\\ +\left(1-x\right)\left(1-y\right)\left(-C_3+P_1+P_2\right)\end{array}$$

(7)

$$U_{C2}=xy{B}_2+x\left(1-y\right)\left(-P_3\right)+\left(1-x\right)y\left(-P_3\right)+\left(1-x\right)\left(1-y\right)\left(-P_3\right)$$

(8)

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

(9)

Thus, the replicator dynamics equations of schools, enterprises, and government, which are denoted by $\frac{dx}{dt}$, $\frac{dy}{dt}$, and $\frac{dz}{dt}$, respectively, can be obtained as follows:

$$F\left(x\right)=\frac{dx}{dt}=x\left(U_{A1}-\overline{U_A}\right)=x\left(1-x\right)\left[y{R}_1+z\left(G_1+P_1\right)-C_1\right]$$

(10)

$$F\left(y\right)=\frac{dy}{dt}=y\left(U_{B1}-\overline{U_B}\right)=y\left(1-y\right)\left[x{R}_2+z\left(G_2+P_2\right)-C_2\right]$$

(11)

$$\begin{array}{l}F\left(z\right)=\frac{dz}{dt}=z\left(U_{C1}-\overline{U_C}\right)\\ =z\left(1-z\right)[xy\left(B_1-B_2-P_3\right)+x\left(-G_1-P_1\right)+y\left(-G_2-P_2\right)-C_3+P_1\\ +P_2+P_3]\end{array}$$

(12)

In the next section, we analyze the stability of the strategies of each population by solving the replicator dynamics equations, constructing a Jacobian matrix, and calculating the eigenvalues of the Jacobian matrix at different equilibrium points.

4. Strategy Stability Analysis

First of all, we let F(x) = 0, F(y) = 0, and F(z) = 0 and define the system of simultaneous equations as:

$$\left\{\begin{array}{l}F\left(x\right)=x\left(1-x\right)\left[y{R}_1+z\left(G_1+P_1\right)-C_1\right]=0\\ F\left(y\right)=y\left(1-y\right)\left[x{R}_2+z\left(G_2+P_2\right)-C_2\right]=0\\ F\left(z\right)=z\left(1-z\right)\left[xy\left(B_1-B_2-P_3\right)+x\left(-G_1-P_1\right)+y\left(-G_2-P_2\right)-C_3+P_1+P_2+P_3\right]=0\end{array}\right.$$

(13)

By solving the equations in (13), 14 equilibrium points can be obtained: E₁ (0, 0, 0), E₂ (0, 0, 1), E₃ (0, 1, 0), E₄ (0, 1, 1), E₅ (1, 0, 0), E₆ (1, 0, 1), E₇ (1, 1, 0), E₈ (1, 1, 1), E₉ (0, $\frac{C_3-P_1-P_2-P_3}{-G_2-P_2}$, $\frac{C_2}{G_2+P_2}$), E₁₀ (1, $\frac{G_1+C_3-P_2-P_3}{B_1-B_2-P_2-P_3-G_2}$, $\frac{C_2-R_2}{G_2+P_2}$), E₁₁ ($\frac{C_3-P_1-P_2-P_3}{-G_1-P_1}$, 0, $\frac{C_1}{G_1+P_1}$), E₁₂ ($\frac{G_2+C_3-P_1-P_3}{B_1-B_2-P_1-P_3-G_1}$, 1, $\frac{C_1-R_1}{G_1+P_1}$), E₁₃ ($\frac{C_2}{R_2}$, $\frac{C_1}{R_1}$, 0), and E₁₄ ($\frac{C_2-G_2-P_2}{R_2}$, $\frac{C_1-G_1-P_1}{R_1}$, 1). However, the solution of the replicator dynamics system in a multi-agent evolutionary game must be a strict Nash equilibrium solution [58]. Hence, we selected only the pure strategy combinations E₁, E₂, E₃, E₄, E₅, E₆, E₇, and E₈ (1, 1, 1) as the possible stable equilibrium points.

Next, we adopted Friedman’s replication dynamics system stability analysis method [59] to construct a Jacobian matrix of the “schools–enterprises–government” tripartite game.

$$J=\left(\begin{array}{ccc}\frac{\mathit{\partial }F\left(x\right)}{\mathit{\partial }x}& \frac{\mathit{\partial }F\left(x\right)}{\mathit{\partial }y}& \frac{\mathit{\partial }F\left(x\right)}{\mathit{\partial }z}\\ \frac{\mathit{\partial }F\left(y\right)}{\mathit{\partial }x}& \frac{\mathit{\partial }F\left(y\right)}{\mathit{\partial }y}& \frac{\mathit{\partial }F\left(y\right)}{\mathit{\partial }z}\\ \frac{\mathit{\partial }F\left(z\right)}{\mathit{\partial }x}& \frac{\mathit{\partial }F\left(z\right)}{\mathit{\partial }y}& \frac{\mathit{\partial }F\left(z\right)}{\mathit{\partial }z}\end{array}\right)=\left(\begin{array}{ccc}{J}_{11}& J_{12}& J_{13}\\ J_{21}& J_{22}& J_{23}\\ J_{31}& J_{32}& J_{33}\end{array}\right)$$

(14)

where

J₁₁ = (1 − 2x) [y R₁ + z (G₁ + P₁) − C₁]

J₁₂ = x (1 − x) R₁

J₁₃ = x (1 − x) (G₁ + P₁)

J₂₁ = y (1 − y) R₂

J₂₂ = (1 − 2y) [x R₂ + z (G₂ + P₂) − C₂]

J₂₃ = y (1 − y) (G₂ + P₂)

J₃₁ = z (1 − z) [y (B₁ − B₂ − P₃) + (− G₁ − P₁)]

J₃₂ = z (1 − z) [x (B₁ − B₂ − P₃) + (− G₂ − P₂)]

J₃₃ = (1 − 2z) [x y (B₁ − B₂ − P₃) + x (− G₁ − P₁) + y (− G₂ − P₂) − C₃ + P₁ + P₂ + P₃]

According to Lyapunov stability theory, the stability at an equilibrium point can be judged by analyzing the eigenvalues of the Jacobian matrix [60]. Specifically, for any equilibrium point, if there are no positive eigenvalues, that point is stable. If there are any positive eigenvalues, the equilibrium point is unstable, and if the eigenvalues cannot be estimated, the equilibrium point is a saddle point. The results of substituting each of the eight equilibrium points into the Jacobian matrix and calculating the eigenvalues of the Jacobian matrix at different equilibrium points are shown in

Table 3. Eigenvalues and stability of equilibrium points.

Equilibrium Point	λ1	Symbol	λ2	Symbol	λ3	Symbol	State
E1 (0, 0, 0)	−C1	−	−C2	−	P1 + P2 + P3 − C3	+	unstable
E2 (0, 0, 1)	G1 + P1 − C1	+	G2 + P2 − C2	+	C3 − P1 − P2 − P3	−	unstable
E3 (0, 1, 0)	R1 − C1	N	C2	+	P1 + P3 − G2 − C3	+	unstable
E4 (0, 1, 1)	R1 + G1 + P1 − C1	+	C2 − G2 − P2	−	G2 + C3 − P1 − P3	N	unstable
E5 (1, 0, 0)	C1	+	R2 − C2	N	P2 + P3 − G1 − C3	N	unstable
E6 (1, 0, 1)	C1 − G1 − P1	−	R2 + G2 + P2 − C2	+	G1 + C3 − P2 − P3	N	unstable
E7 (1, 1, 0)	C1 − R1	N	C2 − R2	N	B1 − B2 − G1 − G2 − C3	N	saddle
E8 (1, 1, 1)	C1 − R1 − G1 − P1	−	C2 − R2 − G2 − P2	−	B2+ G1 + G2 + C3 − B1	N	saddle

.

In Table 3, “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 can be seen from Table 3, there are two saddle points (E₇ and E₈) in the system. When the equilibrium point is E₇(1, 1, 0), when both schools and enterprises choose the positive cooperation strategy and the government chooses the negative cooperation strategy, the conditions for stability of the system are R₁ > C₁, R₂ > C₂ and C₃ > B₁ − B₂ − G₁ − G₂. When this occurs, the positive cooperation strategy rewards for both schools and enterprises received from sources other than the government are greater than their cost, and the cost of the government’s positive cooperation strategy is greater than the difference between the benefit of the government’s positive cooperation strategy, the benefit of the government’s negative cooperation strategy, and the incentives given to schools and enterprises by the government.

When the equilibrium point is E₈(1, 1, 1), when all populations cooperate, the condition for stability of the system is C₃ < B₁ − B₂ − G₁ − G₂. When this occurs, the cost of the government’s positive cooperation strategy is lower than the difference between the benefit of the government’s positive cooperation strategy, the benefit of the government’s negative cooperation strategy, and the incentives given to schools and enterprises by the government. Comparing the two equilibrium points shows that the main factor that influences the stability of the strategies of schools and enterprises is the reward for the positive cooperation strategy from sources other than the government, and the main factor that influences the stability of the strategy of the government is the benefit from the positive cooperation strategy under the condition of schools cooperating with enterprises.

5. Conclusions

In this paper, we studied cooperation between schools, enterprises, and government from the perspective of evolutionary game theory by establishing a tripartite evolutionary game model. Based on our strategy stability analysis, we found several effective ways to promote the cooperation between schools, enterprises, and government, and reached the following conclusions.

(1)

There are two equilibrium points, E₇ (1, 1, 0) and E₈ (1, 1, 1), at which the system can reach a stable state.

(2)

The main factor that influences the stability of the strategies of schools and enterprises is the reward for positive cooperation from sources other than the government. When schools and enterprises receive enough rewards from sources other than the government, the system can reach a stable state regardless of whether the government chooses the positive or negative cooperation strategy. The nongovernmental effects of the cooperation between schools and enterprises are greater than the governmental effects. That is, the current influence of the government on school–enterprise cooperation is not obvious. Therefore, the government should focus on implementing more effective policies, such as increasing incentives and penalties, improving the mechanism for managing conflicts, ensuring the fairness of benefit distribution, and clarifying the responsibilities of different departments, in order to provide the best possible environment to facilitate the cooperation between schools and enterprises.

(3)

The main factor that influences the stability of the strategy of the government is the benefit from the positive cooperation strategy under the condition of schools cooperating with enterprises. When the government receives enough benefits from the cooperation of schools and enterprises as a result of a positive cooperation strategy, the system can reach a stable state. Therefore, the effectiveness of schools cooperating with enterprises should be emphasized. Schools should focus on providing more practical curricula and programs for students, training high-quality teachers, and perfecting talent cultivation to meet the needs of enterprises. Similarly, enterprises should focus on providing job experiences for students and transforming the results of schools’ teaching and theoretical research into practical productivity.

In summary, this study reveals the factors that influence the sustainable cooperation between schools, enterprises, and government, and emphasizes the critical role of the government in the connection between schools and enterprises, which is lacking in the existing literature. The tripartite evolutionary game model provides a new framework for analyzing the cooperation between schools, enterprises, and government, and the results deepen our understanding of sustainable cooperation between schools, enterprises, and government. Admittedly, the process by which different factors influence the strategy evolution of schools, enterprises, and government has not been explored in this study. How various parameters impact strategy selection and evolution deserves more attention. Therefore, a future study with numerical simulations is needed to investigate the strategy evolution mechanism of schools, enterprises, and government.