Online Learning Management for Primary and Secondary Students during the COVID-19 Epidemic: An Evolutionary Game Theory Approach

Abstract

The purpose of this study is to explore the management of primary and secondary school students’ online learning during the COVID-19 pandemic and to analyze the impact of stakeholders’ behavioral choices on students’ online learning management. Based on evolutionary game theory, this paper constructs two-game models of "schools-students" and “schools-students-parents”, analyzes the influence of the behavioral interaction of game subjects on the game equilibrium in the two scenarios, and uses MATLAB 2018 software to carry out the numerical simulation. The results show significant differences in different game players’ strategy choices on students’ online learning management. Among them, the benefits brought by learning are the main factors affecting students’ strategic choices. Although the participation of parents has a positive effect on promoting students’ game strategy selection towards cooperation, there is a participation boundary to the involvement of parents. The school’s choice of punishment or reward has no significant effect on students’ online learning management. Compared with schools, punishments and rewards from parents have a substantial impact on promoting students’ strategic choices towards cooperation.

1. Introduction

The emergence of Corona Virus disease (COVID-19) has led the world to an unprecedented public health crisis. This rapid evolution on such a large scale has influenced students of all age groups [1]. According to data released by the United Nations Educational, Scientific and Cultural Organization in 2020, more than 1.5 billion students and youth on the planet are or have been affected by school and university closures due to the COVID-19 pandemic [2]. During the epidemic period, closing schools and migrating offline teaching to online teaching have become essential options for many countries to ensure the safety of students and maintain the smooth progress of teaching activities [3,4]. Taking China as an example, during the COVID-19 pandemic in China in 2020, under the requirement of the Chinese government’s education policy of “Suspending classes without stopping learning” in primary and secondary schools, 282 million school students in China generally turned to online learning [5]. Before the COVID-19 pandemic, primary and secondary schools rarely had online learning for such a long time, and schools have never experienced such large-scale online learning teaching and management. Therefore, during the 2020 COVID-19 pandemic, online student learning has no mature management experience to copy. After the COVID-19 pandemic in 2020, the issue of online learning management for primary and secondary school students has attracted the attention of scholars [6,7].

Why do scholars pay so much attention to the online learning management of primary and secondary school students? First of all, the online learning scale for primary and secondary school students is the largest. When the epidemic in China in 2020, about 282 million Chinese students turned to online learning, and the number of primary and secondary school students reached 190 million [8], accounting for more than 65% of the total number of students studying online. The vast learning group makes school management face significant challenges. Secondly, the student group is generally between 7 and 17 years old (Note: “student” in this article refers specifically to “primary and secondary school students”, i.e., “pupil.”) and is in physical and psychological development. Under the influence of COVID-19, long-term online learning is exceptionally prone to psychological problems [9]. Zhang et al. conducted an online survey of 22,380 middle school students in Jiangsu Province, China, during the COVID-19 pandemic and found that 25.6% of the tested middle school students had depressive symptoms, 26.9% had anxiety symptoms, and 20.6% had a combination of depressive and anxiety symptoms [10]. This is also the case in other countries. For example, Ravens-Sieberer et al. also found this problem in a study on the impact of the mental health of 7 to 17-year-old adolescents in Germany during the COVID-19 pandemic [11]. Finally, the online learning of primary and secondary school students is also inseparable from the help of parents [12,13]. However, in the practice of the 2020 pandemic, the role of parents in the management of online learning for primary and secondary school students has been ignored.

In the context of the current COVID-19 pandemic, it is significant to study the management of students’ online learning and improve the quality of students’ online learning [14]. This paper uses evolutionary game theory as an analytical framework to analyze the problem of online learning management for students in the COVID-19 epidemic from the changes between key stakeholders and their behavioral interactions. The stakeholder subjects in this paper include “schools” [15], “students” and “parents” [16]. This paper constructs two game systems of “school-student” and “school-student-parent.” By calculating the model and simulating the behavioral choices of the game subjects, the main factors affecting students’ online learning management are analyzed. The main problems solved in this study are in the following aspects:

• Reward or punishment, which method is more effective for students’ online learning management?
• Does parental involvement positively impact students’ online learning management compared to a scenario where parents are not involved? Are there participation boundaries for parent involvement in student online learning management?
• Parents or schools? Whose strategy is more effective in facilitating the management of students’ online learning?

The rest is structured as follows: Section 2 reviews and summarizes the relevant literature. Section 3 establishes the game model. Section 4 stability strategy analysis. Section 5 numerically simulates the model. Section 6 is the discussion part, and Section 7 is the conclusion.

2. Literature Review

2.1. Online Learning

Although distance education has developed for more than two centuries [17], it is still young in online learning development history and evolution [18]. Before the COVID-19 pandemic, never in human history had an entire student population suddenly transitioned from face-to-face teaching to online learning through the use of digital technology [19]. Therefore, online learning has not accumulated enough experience in large-scale applications. In previous studies, scholars such as Perna believed that online education could improve the quality of students’ knowledge and reduce learning costs and improve educational benefits [20]. Scholars such as Willett and Brown also believe that online teaching by teachers provides flexibility for students to participate in courses, reducing students’ need for face-to-face course guidance [18]. Of course, scholars are constantly reflecting on the problems of online learning. As early as 2014, Xu and other scholars found that online classes can affect students’ emotions [21], and he believed that online learning requires students’ self-discipline. Scholars such as Rueda and Benitez have studied the application of traditional educational technology to students’ satisfaction in education and found that online learning brings particular challenges to teachers’ online teaching and student management [22]. Under the influence of the COVID-19 epidemic, managing students’ long-term online education has become one of the main challenges of online learning faces [23,24]. Especially after the COVID-19 pandemic in 2021, management issues have become a focal issue for online learning scholars.

2.2. Online Learning Management

After 2021, online learning management has attracted the attention of academics. Scholars who study from the perspective of schools believe that in the period of the COVID-19 pandemic, schools should change organizational management and innovate student management methods to meet the challenges brought by students’ online learning management [25,26]. Among them, scholars such as Hadriana studied the online learning management of students in Indonesian junior high schools during the Covid-19 pandemic. They found that the active participation of principals has a significant effect on promoting students’ online learning management [27]. Scholars such as Apak and Suki have studied the relationship between teachers’ creativity and remote classroom management and found that teachers’ improving the innovation of classroom teaching can help improve the students’ online learning management level [15]. Scholars who study from a family perspective believe that the family environment is one of the biggest challenges students face in online learning [28]. Especially under the influence of the COVID-19 epidemic, students who study online for a long time are prone to mental health problems such as anxiety and depression [23,29]. Parents must provide emotional management and learning assistance [30]. Therefore, scholars have called for schools and policymakers to invest energy and resources in responding to students’ prolonged online learning and encouraging family interventions [31].

Scholars’ existing related research has essential reference significance for improving students’ online learning management. However, the existing research still has room for further improvement. Firstly, online learning management is a systemic issue that should not be studied independently of “schools” or “parents”. At the same time, scholarly research has not analyzed the impact of specific parental and school behaviors. Finally, online learning management is dynamic and needs to be studied dynamically. Therefore, this paper uses evolutionary game theory to systematically and dynamically analyze this issue.

2.3. The Application of Evolutionary Game Theory

In 1973, Maynard Smith and Price combined Darwin’s theory of biological evolution and formally applied game theory to evolutionary biology [32], and evolutionary game theory was officially born. After Taylor and Jonker proposed the concept of “Replicator Dynamic” in 1978 [33], the evolutionary game theory began to receive widespread attention. Unlike classical game theory, evolutionary game theory explains the changes in the strategic choices of game subjects between systems from a dynamic process [34]. It is based on the assumption that the issue of the game is bounded rationality [35], which makes up for the limitations of the classical game theory assumption of an entirely rational person. Evolutionary game theory has been widely used in solving practical problems. For example, Cheng et al. studied the collaborative innovation of local engineering colleges and research institutes from game theory. They found that the two can evolve a collective innovation path during long-term cooperation [36]. Guo et al. used evolutionary games to study construction safety education and proved that increasing workers’ welfare subsidies are conducive to improving workers’ enthusiasm for participating in safety education [37]. Lipowski and Ferreira et al. used the evolutionary game model to study the dilemma of students’ further education and employment and put forward targeted suggestions for different students’ strategic choices [38]. Perc et al. used the evolutionary game model to study the interaction of individuals, groups, and social behaviors, and they analyzed the main factors affecting social cooperation and discussed the relationship between culture and individual cognition and individual sexual behavior [39]. In a word, evolutionary game theory is widely used to solve problems in all aspects of society and has achieved good results. Therefore, this paper applies evolutionary game theory to the online learning management of primary and secondary school students to provide a reference for students’ online learning management during the current COVID-19 pandemic.

3. Methodology

According to the research questions, the stakeholders selected in this paper are “schools” [15,27], “students”, and “parents” [16,30]. To truly simulate the management scenarios of students’ online learning in life, the authors modeled and discussed the two scenarios of “school-student” and “school-student-parent”, respectively.

3.1. Model Hypotheses

The evolutionary game model reveals a dynamic game process with incomplete information. The subjects participating in the game are all bounded rational [40]. We assume that there is such a rule among the groups participating in the game: First, each player has two options, cooperative or non-cooperative. x, y, and z represent the probability values of the game strategy choices of schools, parents, and students, respectively. $1-x$, $1-y$, $1-z$ represent the probability of the three strategies to be noncooperative, respectively, where $x,y,z\in \left[0,1\right]$. Second, there are reward and punishment rules in the game system. The players whose strategy is cooperative will be rewarded, and the players whose game strategy is non-cooperative will be punished. Some scholars have found that rewards can promote participants’ enthusiasm to cooperate [41], and appropriate punishment constrains participants’ choice of noncooperation [42]. Finally, each participant has a participation cost. When the school strategy chooses cooperation, the management cost is a, and the reward obtained is R. When the school strategy chooses to be non-cooperative, the penalty cost is b. To reflect the binding force of punishment, we stipulate $b>a+R$. When the student strategy chooses to cooperate, the student’s learning benefit is h, and the school’s reward is c. When the student’s game strategy chooses not to cooperate, the learning benefits $h=0$ and the penalty for obtaining a school is n. According to the game’s rules, it is assumed that the student’s payoff is m when the strategy choice is non-cooperation. In the same way, to reflect the binding force of punishment, we stipulate that $n>c$. When parents participate in the game, the parent’s income is d (we understand d as the parent’s sense of gain in cultivating children’s online learning, such as improving children’s academic performance). At this time, the cost paid by the parents is g (which can be understood as the time cost).

Under the condition of parental participation, students who choose the noncooperative strategy will be punished by their parents as k, and students will be rewarded by their parents as e for cooperation. In the same way, parent participation has a supervisory role in the school. When the school strategy chooses not to cooperate, the school will receive an additional penalty of $I^{\prime }$, and the parents will receive an additional reward of I. It should be noted that the meaning of cooperation between schools and parents means that schools and parents will seriously participate in students’ online learning management. Similarly, students’ strategic choice of cooperation means that students will actively cooperate with the school’s online learning management. For the convenience of readers, the major notations used in this paper are listed in

Table 1. Major notations.

Notations	Explanation
x, y, z	Probability of school, student, and parent strategy choice to be cooperative.
a	The management cost when the school game strategy is selected as cooperation.
R	The school game strategy chooses as the reward for cooperation.
b	The penalty cost when the school game strategy is chosen to be non-cooperative.
h	Learning benefits when students choose their game strategy as cooperation.
c	Students get the school’s reward when their game strategy chooses to cooperate.
n	Punishment cost of gaining school when student strategy chooses to cooperate.
m	Students sacrifice study time for gains.
d	Parents get benefits when they participate in the game.
g	Participation costs when parents participate in gaming.
k	Under the condition of parental participation, students are punished by parents when they choose not to cooperate strategically.
e	Under the condition of parental participation, students are rewarded by parents when they choose to cooperate strategically.
I’	Under the condition of parental participation, the additional penalty cost is obtained when the school strategy chooses to be non-cooperative.
I	Under the condition of parental participation, parents supervise the school and do not carefully supervise the reward income obtained.

.

3.2. Model Establishment

The rules of the game are described in detail in Section 3.1. To simulate the management scenarios of students’ online learning in realistic conditions to the greatest extent, this research built models according to the two situations of parent participation and non-participation. The combination of game strategies is shown in Figure 1.

Figure 1a shows the “school-student” game scenario, and there are four game strategy combinations for both sides. Figure 1b represents the “school-student-parent” game scenario, with a kind of 8 game strategy combinations. Next, we will model and solve them separately.

3.2.1. Construction of “School-Student” Game Model

It can be seen from Figure 1a, there are four possible game strategy combinations in the “school-student” game. This research constructs the profit matrix of the game subject based on these possible combinations of game strategies. It should be noted that in the game profit matrix, the profit of the participants in the game is calculated by subtracting the total cost from the total profit [43]. For example, in the $s_{21}$ game scenario, schools’ and students’ game strategy choices are cooperative. According to the game hypothesis proposed above, the school’s income R and school management costs are c and a. The benefits obtained by the students are h and c, and the cost paid is m. Therefore, we use $R-c-a$ and $h+c-m$ to represent the game benefits of the school and students in this situation, respectively. Similarly, $s_{22}$ indicates that the school’s game strategy is cooperation, and the students’ game strategy is non-cooperation. In this case, the school’s benefit can be expressed as $R-a$ (To simplify the formula, in this article, we use $R\Delta$ to represent $R-a$.), and the student’s benefit is $m-n$. $s_{23}$ indicates that the school’s policy choice is non-cooperation, and the student’s policy choice is cooperation. At this time, the school’s and student’s benefits are $-b$ and $h-m$, respectively. $s_{24}$ means that neither side cooperates. At this time, the school’s income is $-b$, and the student’s income is m. Combining the above discussion of the possible game scenarios of the “school-student” game, construct the “school-student” game profit matrix, as shown in

Table 2. “School-student” tripartite game profit matrix.

	Schools
Students	Cooperation x	Non-Cooperation $1-x$
Cooperation y	$R\Delta$ − c; $h+c-m$	$-b$; $h-m$
Non-cooperation $1-y$	$R\Delta$; $m-n$	$-b$; m

. Each entry in Table 2 contains two values, the former represents the school’s income, while the latter represents the student’s income.

According to Table 2, we can obtain the expected return formula $E_{{2SCH}_1}$ when the school game strategy is selected as cooperation, and the expected return formula $E_{{2SCH}_0}$ when the school game strategy is selected as non-cooperative as shown in Formula (1):

$$\left\{\begin{array}{c}{E}_{2SC{H}_1}=y\left(R\Delta -c\right)+\left(1-y\right)\left(R\Delta \right)=R\Delta -yc,\hfill \\ E_{2SC{H}_0}=y\left(-b\right)+\left(1-y\right)\left(-b\right)=-b.\hfill \end{array}\right.$$

(1)

In the same way, the expected return when the student’s strategy is chosen as cooperation can be expressed by the formula $E_{{2STU}_1}$. The expected return when the student’s strategy chooses to be non-cooperative can be expressed by the formula $E_{{2STU}_0}$, expressed as (2):

$$\left\{\begin{array}{c}{E}_{2ST{U}_1}=x\left(h+c-m\right)+\left(1-x\right)\left(h-m\right)=xc+h-m,\hfill \\ E_{2ST{U}_0}=x\left(m-n\right)+\left(1-x\right)\left(m\right)=m-xn.\hfill \end{array}\right.$$

(2)

The critical point of the evolutionary game is the dynamic change of strategy proportion. The replicator dynamics equation effectively explains strategy choices among different game agents [34]. The basic form of differential equation expression in evolutionary game model is $\frac{{dx}_n}{dt}=P(1-P)(E_{{strategy}_1}-E_{{strategy}_0})$. Among them, P is the frequency of making choices, $\frac{{dx}_n}{dt}$ represents the rate of change of the game subject’s strategy choice over the time, $E_{{strategy}_1}$ describes the game subject’s strategy choice is the expected return of cooperation, and the expected return of non-cooperation expressed as $E_{{strategy}_0}$. According to the differential equation expression, after simplification, the replicator dynamics equation for our model is then given by Formula (3):

$$\left\{\begin{array}{c}{\mathsf{\Pi }}_{2\text{-}1}=\frac{dx}{dt}=x\left(1-x\right)\left(E_{2SC{H}_1}-E_{2SC{H}_0}\right)=x\left(1-x\right)\left(R\Delta +b-yc\right),\hfill \\ {\mathsf{\Pi }}_{2\text{-}2}=\frac{dy}{dt}=y\left(1-y\right)\left(E_{2ST{U}_1}-E_{2ST{U}_0}\right)=y\left(1-y\right)\left(h+xc-2m+xn\right).\hfill \end{array}\right.$$

(3)

Among them, $\frac{dx}{dt}$ and $\frac{dy}{dt}$ denote the rate of change of the school’s and student’s policy choice over time.

3.2.2. Construction of “School-Student-Parent” Game Model

The purpose of constructing the “school-student-parent” game model is to compare it with the “school-student” game model to reflect parental participation’s impact on students’ online learning management. In the tripartite game of schools, students, and parents, $s_{31}$ indicates that the game strategies are all cooperative, and the game income are $R\Delta -c$, $h+c+e-m$, and $d-g-e$, respectively. $s_{32}$ means that parents’ passive participation and the game strategy of students and schools actively cooperate. At this time, the game income is $R\Delta -c$, $h+c-m$, and $-g$. $s_{33}$ indicates that the game strategy of the school and the parents are all cooperation, and the student’s game strategy chooses not to cooperate. At this time, the game income of the three can be expressed as $R\Delta$, $m-n-k$, and $d-g$. In the $s_{34}$ game situation, only the school manages carefully, and students and parents do not cooperate in the game strategy choices. The income of the three can be expressed as $R\Delta$, $m-n$, and $-g$. The $s_{35}$ game situation indicates that the school’s game strategy choice is not serious management, while the game strategy choices of students and parents are active cooperation. At this time, the three incomes are $-b-I^{\prime }$, $h+e-m$, and $d-g+I-e$, respectively. In the $s_{36}$ game situation, the school’s game strategy is negligent management, the parents choose to participate passively, and only the students’ game strategy is cooperation. At this time, the game gains of the three are $-b-I^{\prime }$, $m-k$, and $d-g+I$, respectively. In the $s_{37}$ game situation, the parents actively participate, and the game strategy choices of the schools and students are non-cooperative. $s_{38}$ is the worst game situation. The school chooses to neglect the management, the parents choose to participate passively, and the students choose not to cooperate with the management. At this time, the income of the three can be expressed as $-b$, m, and $-g$. Therefore, we clearly show the game payoffs of the players in different game situations in the game payoff matrix (

Table 3. “School-student-parent” tripartite game profit matrix.

	Schools
	Cooperation x		Non-cooperation $1-x$
Students	Parents		Parents
	Cooperation z	Non-cooperation $1-z$	Cooperation z	Non-cooperation $1-z$
Cooperation y	$R\Delta -c$; $h+c+e-m$; $d-g-e$	$R\Delta -c$; $h+c-m$; $-g$	$-b-I^{\prime }$; $h+e-m$; $d-g+I-e$	$-b-I^{\prime }$; $m-k$; $d-g+I$
Non-cooperation $1-y$	$R\Delta$; $m-n-k$; $d-g$	$R\Delta$; $m-n$; $-g$	$-b$; $h-m$; $-g$	$-b$; m; $-g$

). Each entry in Table 3 contains three values, the former represents the school’s income, the middle represents the student’s income, and the latter represents the parents’ income.

Combining Table 3 above, $E_{{3SCH}_1}$ and $E_{{3SCH}_0}$ represent the expected benefits when the school strategy is selected as cooperation and non-cooperation. After simplification, it is as Formula (4):

$$\left\{\begin{array}{cc}\hfill E_{3SC{H}_1}& =\mathrm{yz}(R\Delta -c)+y(1-z)(R\Delta -c)+(1-y)z(R\Delta )+(1-y)(1-z)(R\Delta )\hfill \\ \hfill & =R\Delta -yc,\hfill \\ \hfill E_{3SC{H}_0}& =(-b-I^{{}^{\prime }})+y(1-z)(-b-I^{{}^{\prime }})+(1-y)z(-b)+(1-y)(1-z)(-b)\hfill \\ \hfill & =-b-z{I}^{{}^{\prime }}.\hfill \end{array}\right.$$

(4)

$E_{{3STU}_1}$ and $E_{{3STU}_0}$ represent the expected benefits of cooperation and non-cooperation, respectively, which are simplified as Formula (5):

$$\left\{\begin{array}{cc}\hfill E_{3ST{U}_1}& =xz(h+c+e-m)+x(1-z)(h+c-m)+(1-x)z(h+e-m)+(1-x)(1-z)(m-k)\hfill \\ \hfill & =ze+cx+h-m,\hfill \\ \hfill E_{3ST{U}_0}& =xz(m-n-k)+x(1-z)(m-n)+(1-x)z(h-m)+(1-x)(1-z)\left(m\right)\hfill \\ \hfill & =m-zk-xn.\hfill \end{array}\right.$$

(5)

$E_{{3PAR}_1}$ and $E_{{3PAR}_1}$ represent the expected benefits of parental strategy choice as cooperation and non-cooperation. Simplified as Formula (6):

$$\left\{\begin{array}{cc}\hfill E_{3ST{U}_1}& =xz(h+c+e-m)+x(1-z)(h+c-m)+(1-x)z(h+e-m)+(1-x)(1-z)(m-k)\hfill \\ \hfill & =ze+cx+h-m,\hfill \\ \hfill E_{3ST{U}_0}& =xz(m-n-k)+x(1-z)(m-n)+(1-x)z(h-m)+(1-x)(1-z)\left(m\right)\hfill \\ \hfill & =m-zk-xn.\hfill \end{array}\right.$$

(6)

Consistent with the solution process in Section 3.2.1, according to Formulas (4)–(6), The replicator dynamics equation for explaining the behavioral strategy choices of schools, students, and parents can be obtained, which is simplified as a Formula (7):

$$\left\{\begin{array}{c}{\mathsf{\Pi }}_{3\text{-}1}=\frac{dx}{dt}=x\left(1-x\right)\left(E_{SC{H}_1}-E_{SC{H}_0}\right)=x\left(1-x\right)\left(R\Delta +b-yc+z{I}^{\prime }\right),\hfill \\ {\mathsf{\Pi }}_{3\text{-}2}=\frac{dy}{dt}=y\left(1-y\right)\left(E_{ST{U}_1}-E_{ST{U}_0}\right)=y\left(1-y\right)\left(x\left(c+n\right)+z\left(e+k\right)+h-2m\right),\hfill \\ {\mathsf{\Pi }}_{3\text{-}3}=\frac{dz}{dt}=z\left(1-z\right)\left(E_{PA{R}_1}-E_{PA{R}_0}\right)=z\left(1-z\right)\left(d+I-xI-ye\right).\hfill \end{array}\right.$$

(7)

Among them, $\frac{dx}{dt},\frac{dy}{dt}$, and $\frac{dz}{dt}$ represent the rate of change over schools, students, and parents policy choices, respectively.

3.3. Model Solution

3.3.1. “School-Student” Game Equilibrium Solution

In the “student-school” game system, let ${\mathsf{\Pi }}_{2\text{-}1}=0$, ${\mathsf{\Pi }}_{2\text{-}2}=0$, it can be seen that there are 5 equilibrium points. Both $P_1(0,0)$, $P_1(0,1)$, $P_1(1,1)$, $P_1(1,0)$. When $0<x^{*}=\frac{2m-h}{c+n}<1$, $0<y^{*}=\frac{R-a+b}{c}<1$, $P^{*}(x^{*},y^{*})$ is also the equilibrium point of the game system. Whether these equilibrium points are stable or not, this paper is based on the stability discrimination method of the Jacobian matrix proposed by Friedman [34] to solve. As Formula (8):

$$\begin{array}{cc}\hfill & J=\left(\begin{array}{cc}\begin{array}{c}\frac{d{\mathsf{\Pi }}_{2\text{-}1}}{dx}\\ \frac{d{\mathsf{\Pi }}_{2\text{-}2}}{dx}\end{array}& \begin{array}{c}\frac{d{\mathsf{\Pi }}_{2\text{-}1}}{dy}\\ \frac{d{\mathsf{\Pi }}_{2\text{-}2}}{dy}\end{array}\end{array}\right)=\left(\begin{array}{cc}{\lambda }_{11}& {\lambda }_{12}\\ {\lambda }_{21}& {\lambda }_{22}\end{array}\right)=\hfill \\ \hfill & \left(\begin{array}{cc}\left(1-2x\right)\left(R\Delta +b-yc\right)& x(1-x)(-c)\\ y(1-y)(c+n)& \left(1-2y\right)\left(c+xh-2m+xn\right)\end{array}\right)\hfill \end{array}$$

(8)

Bringing $P_1\sim P^{*}$ into Formula (8), if there is an equilibrium point such that ${det}_J={\lambda }_{11}{\lambda }_{22}-{\lambda }_{21}{\lambda }_{12}>0$, ${tr}_J={\lambda }_{11}+{\lambda }_{22}<0$ of the Jacobian matrix, we consider this equilibrium point to be stable. The results are shown in

Table 4. “School-student” equilibrium point calculation results.

Equilibrium	$\mathit{det}\left(\mathit{J}\right)$	Symbol	$\mathit{tr}\left(\mathit{J}\right)$	Symbol	State
(0,0)	$(R\Delta +b)(c-2m)$	−	$R\Delta$ + b + c − 2m	N	Instability point
(0,1)	$(R\Delta +b-c)(2m-c)$	N	$R\Delta$ + b + 2m − 2c	+	Instability point
(1,0)	$-(R\Delta +b)(c+n+h-2m)$	N	$-(R\Delta +b)$ + (c + n + h − 2m)	N	Uncertain
(1,1)	$(R\Delta +b-c)(c+n+h-2m)$	N	$(R\Delta +b-c)$ + (c + n + h − 2m)	N	Uncertain
$(x*,y*)$	0	0	0	0	Instability point

:

In Table 4, “N” indicates that the sign of the $de{t}_J,t{r}_J$ value cannot be discriminated during the solution process, and further discussion is required.

3.3.2. “School-Student-Parent” Tripartite Game Equilibrium Solution

Repeat the solution steps in Section 3.3.1 to obtain the Jacobian matrix $J_1$ of the “school-student-parent” tripartite game, such as Formula (9):

$$\begin{array}{cc}\hfill & J_1=\left(\begin{array}{ccc}\frac{d{\Pi }_{3\text{-}1}}{dx}& \frac{d{\Pi }_{3\text{-}1}}{dy}& \frac{d{\Pi }_{3\text{-}1}}{dz}\\ \frac{d{\Pi }_{3\text{-}2}}{dx}& \frac{d{\Pi }_{3\text{-}2}}{dy}& \frac{d{\Pi }_{3\text{-}2}}{dy}\\ \frac{d{\Pi }_{3\text{-}3}}{dx}& \frac{d{\Pi }_{3\text{-}3}}{dy}& \frac{dd{\Pi }_{3\text{-}3}}{dz}\end{array}\right)=\left(\begin{array}{ccc}{\tau }_{11}& {\tau }_{12}& {\tau }_{13}\\ {\tau }_{21}& {\tau }_{22}& {\tau }_{23}\\ {\tau }_{31}& {\tau }_{32}& {\tau }_{33}\end{array}\right)=\hfill \\ \hfill & \left(\begin{array}{ccc}\left(1-2x\right)\left(R\Delta +b-yc+z{I}^{\prime }\right)& x\left(1-x\right)\left(-c\right)& x(1-x)I^{\prime }\\ y\left(1-y\right)\left(c+n\right)& \left(1-2y\right)\left(x\left(c+n\right)+z\left(e+k\right)+h-2m\right)& y\left(1-y\right)\left(e+k\right)\\ z\left(z-1\right)\left(I\right)& z\left(1-z\right)\left(-e\right)& \left(1-2z\right)\left(d+I-xI-ye\right)\end{array}\right)\hfill \end{array}$$

(9)

We will equalize points $(0,0,0)$, $(0,0,1)$,$(0,1,1)$,$(1,0,0)$,$(1,0,1)$, $(1,1,1)$, $(0,1,1)$,$(0,1,0)$ into the Jacobian matrix $J_1$, except for ${\tau }_{11}$,${\tau }_{22}$,${\tau }_{33}$, the rest of the mean values are 0. Therefore, ${\tau }_{11}$,${\tau }_{22}$,${\tau }_{33}$ are the eigenvalues of the Jacobian matrix $J_1$. As long as these three eigenvalues are all less than 0, the equilibrium point can be considered stable [44]. The results obtained by calculation are shown in

Table 5. Calculation results of the equilibrium point of the game “school-student-parent”.

Equilibrium	${\mathit{\tau }}_{11}$	Symbol	${\mathit{\tau }}_{22}$	Symbol	${\mathit{\tau }}_{33}$	Symbol	State
$(1,1,1)$	$-(R\Delta +b-c+I^{\prime })$	−	$-(c+n+e+k+h-2m)$	N	$e-d$	N	Uncertain
$(1,1,0)$	$-(R\Delta +b-c+I^{\prime })$	−	$-(c+n+h-2m)$	N	$d-e$	N	Uncertain
$(1,0,0)$	$-(R\Delta +b)$	−	$(c+n+h-2m)$	N	d	+	Instability point
$(0,0,0)$	$R\Delta +b$	+	$h-2m$	N	$d+I$	+	Instability point
$(0,0,1)$	$R\Delta +b+I^{\prime }$	+	$e+k+h-2m$	N	$-d-I$	−	Instability point
$(0,1,1)$	$R\Delta +b-c+I^{\prime }$	+	$-(e+k+h-2m)$	N	$e-d$	N	Instability point
$(1,0,1)$	$-(R\Delta +b+I^{\prime })$	−	$c+n+e+k+h-2m$	N	$-d$	−	Uncertain
$(0,1,0)$	$R\Delta +b-c$	+	$h-2m$	N	$d+I-e$	+	Instability point

.

Similarly, in Table 5, “N” means that the eigenvalue sign cannot be judged. The possible stable equilibrium points in the three-way game are $(1,1,1)$, $(1,1,0)$, $(1,0,1)$. However, the satisfaction conditions of equilibrium point stability still need further analysis.

4. Strategy Stability Analysis

Stability analysis of the “student-school” game equilibrium point (see Table 4) shows that when the stable equilibrium point is at $(1,1)$, we know from the game rules that $R-a+b>0$ is permanently established. At this time, the condition $\frac{1}{2}\left(c+n+h\right)>m$ still needs to be satisfied; when the stable equilibrium point is at $(1,0)$, the condition $\frac{1}{2}\left(c+n+h\right)<m$ needs to be satisfied at this time. Only when the equilibrium point satisfies the conditions is it found that when the school’s reward c increases and the punishment n increases, or when the student’s online learning harvest h value increases, the student’s strategy choice tends to increase toward cooperation.

The stability analysis of the equilibrium point of the “school-student-parent” game: The results in Table 5 show that when the stable equilibrium point is at $(1,1,1)$, the condition $\frac{1}{2}(c+n+e+k+h)>m$ and $d>e$. When the stable equilibrium point is at $(1,1,0)$, the conditions $\frac{1}{2}(c+n+h)>m$ and $d<e$ need to be satisfied at the same time. When the stable equilibrium point is at $(1,0,1)$, at this time, the condition $\frac{1}{2}(c+n+e+k+h)<m$ needs to be satisfied. Therefore, in the possible, stable equilibrium point of the tripartite game between the school, students, and parents, the game strategy of the school is cooperation. Judging from the above-established conditions, the main factors affecting students’ choice of game strategies are the school’s reward c and punishment n and their online learning gains h. The main influencing factors influencing parents’ choice of strategy come from the value of harvest d and parents’ reward e for children’s online learning.

However, only relying on the analysis of the stable equilibrium point cannot draw the influence of the behavioral choice of the game subject on the formation process of the system equilibrium. To analyze the impact of the behavior of the game subject on the operation of the formation of the system stability equilibrium, we need to make further use of computer simulation.

5. Numerical Experiments

Computer simulation technology provides a feasible method for revealing the impact of stakeholders’ behavioral choices on system equilibrium [43]. However, in practice, most variables are challenging to quantify. Of course, the data of relevant variables are even less likely to be obtained directly from real life. Simulation with experimental values is a method commonly used by scholars for such research [45,46,47,48]. The idea of this research method is that the researcher sets initial experimental values based on their own research experience. These set experimental values are used to simulate changes in game-scenario-related variables. Then, changes in stakeholder strategy choices are represented by adjusting the size of the preliminary experimental value. Finally, the relationship between stakeholders’ strategy changes and the system’s game equilibrium is analyzed through computer simulation technology [49]. Of course, it has to be acknowledged that this research method is subject to the subjective influence of the researchers and has certain limitations. However, it is worth affirming that the application of computer simulation to the evolutionary game provides an analytical perspective for explaining the relationship between the changes in the game strategies of stakeholders and the equilibrium of the system game.

This research adopts the method of experimental value simulation. However, different from the experimental value settings of scholars, this research’s initial experimental value settings draw on the opinions of “experts”. This can bring research closer to real life. Among these experts, four graduate students are studying for doctoral degrees, three are teachers from primary and secondary schools, and the other three are parents and students involved in managing students’ online learning during the epidemic. The specific operation is shown in Figure 2.

Figure 2 contains three parts. The first part is to explain to the expert the question studied in this paper and explain to the expert the meaning of the relevant variables. The second part is the process of assigning variables value. The condition for judging the consistency of experts’ assignment to variables is that the difference in the assignment of any two experts to the same variable does not exceed 20% (Note: the expert assignment process is anonymous, and the study does not involve any privacy and ethical issues). The third part combines the expert’s scoring results and the author’s research experience to set experimental values and conduct simulation experiments. Although there are also limitations in setting the initial experimental values in this paper, the above operations can make the relevant value settings closer to real life and can reduce the subjectivity of the experimental value method to a certain extent. Based on some of the above routines, the initial values of the variables studied in this paper are shown in

Table 6. Initial parameter setting.

Parameters	Values	Parameters	Values
a	$0.6$	m	$0.7$
R	$0.8$	n	$1.5$
b	$2.0$	d	$2.0$
k	$0.2$	$I^{\prime }$	$1.0$
h	$1.0$	I	$0.5$
c	$0.3$	e	$1.0$

. It should be noted that the numerical simulation is realized by computer programming of MATLAB2018(b) software.

5.1. Simulation of the “School-Student” Game

It can be seen from the solution of the model that the possible, stable equilibrium points in the “school-student” game system are $(1,1)$ and $(1,0)$. Section 4 gives the conditions that need to be met when it is established. On this basis, set the evolution time of the replicator dynamics Equation (3) to 20, combined with the experimental values in Table 6, and take the initial probability value of 0.2 for model simulation (Note: In this paper, the selection of the initial probability value does not affect the final simulation results. any effect), the results are shown in Figure 3.

Figure 3A shows the simulation when the condition $c+n+h>2m$ is satisfied. Among them, the probability value curves reflecting the strategy choices of schools and students all converge to 1. It shows that both parties’ strategic choices are cooperation under this condition. Figure 3B indicates the simulation performed under $c+n+h<2m$. The results show that the probability value curve reflecting the school’s strategy choice converges to 1, and the probability value curve representing the student’s strategy choice converges to 0. This shows that the school’s final strategy choice is cooperation under this condition, and the students’ final choice is non-cooperation.

We hope that student online learning management can achieve the simulation results in Figure 3A in the future. Therefore, based on the condition of $c+n+h>2m$, we further simulate the main factor variables that affect the students’ game strategy choice. It can be seen from the conditions that the main variables that affect students’ choice of game strategies include $c,h,m,n$. Therefore, we use the control variable method to simulate the impact of a numerical change on the formation of the game equilibrium by fixing the values of the three variables. The results are shown in Figure 4.

Figure 4A shows that when the values of h, m, and n remain unchanged, the experimental values $c=0.3,c=0.5,c=0.7$, and $c=0.9$ are simulated. When the value of c increases, the probability value curve of student cooperation to converge to 1 is accelerated. The simulation results show that school rewards positively affect student management. Figure 4D shows the effect of school penalty value n on students’ strategy choices. Keep the values of $c,h,m$ unchanged, and set $n=1.4$, $n=1.5$, $n=1.6$, $n=1.7$ for simulation. It can be seen from the results in Figure 4B that when the value of n increases, the probability of students choosing to cooperate in the game strategy increases, but the increase is not obvious. Figure 4C shows changes in the student’s return value h. The results show that the greater the student’s learning benefit, the shorter the time for the probability value curve to converge to 1. In Figure 4D, keep the values of c, n, and h unchanged, and set the values of m be 0.5, 0.7, 1, and 1.3, respectively. As indicated by the arrow F1 in the picture above(see Figure 4D), when the value of m is larger, the rate of convergence of the probability value curve representing the student’s choice of game strategy is slower. The greater the degree of convexity of the probability value curve to 0 in the initial stage, the longer the time for the probability value curve of student cooperation to converge to 1. The results show that when students invest their time in learning to gain more benefits, their strategy of choosing to be cooperative is lower.

5.2. Simulation of the “School-Student-Parent” Game

According to the calculation results in Section 3, in the “school-student-parent” game system, the possible stable equilibrium points are $(1,1,1)$, $(1,1,0)$, and $(1,0,1)$. Section 4 gives the conditions for the existence of a stable equilibrium point. On this basis, the value of the evolution time of the replicator dynamics Equation (7) is set to 20, the initial probability value is 0.2, and the initial experimental values in Table 6 are brought into the numerical simulation. The results are shown in Figure 5.

Figure 5A shows the simulation results of time-varying game strategy choices of schools, students and parents when the conditions $(c+n+e+k+h)>2m$ and $d>e$ are satisfied. In Figure 5A, the probability value curves of schools, parents, and students all converge to 1, indicating that the game strategies of the three are cooperative under this condition. Figure 5B is the simulation result when the conditions $(c+h+n)>2m$ and $d<e$ are satisfied. Among them, the probability value curve of school and student strategy selection converges to 1, and the parents’ strategy selection probability value curve converges to 0. The results show that the final strategic choice of students and schools is cooperation, while parents are not cooperative. Figure 5C shows the simulation results when $(c+n+e+k+h)<2m$ is satisfied. Among them, the probability value curve of school and parents converges to 1, which is cooperation.

To sum up, comparing the results in Figure 3 and Figure 5, it is found that in the three-way game involving parents, the probability value curve representing the strategy choice of schools and students converges to 1 at a significantly faster rate. Therefore, the study’s findings suggest that parental involvement can impact students’ online learning management. We hope that the online learning management of students can reach the equilibrium shown in Figure 5 in the future. Therefore, under the conditions of $(c+n+e+k+h)>2m$ and $d>e$, this research further analyzes the influence factors of parental participation on the game subject. Since the parameters $e,k$, and d have been simulated in Figure 5A, the subsequent work needs to act on the parameters $e,k,d$.

The first study is on the influence of parental reward value e on the system equilibrium. Keep the values of $c,n,k,h,m$, and d unchanged, and set $e=1,e=1.5,e=0.5$, and $e=0.1$ for simulation. The result is shown in Figure 6.

The simulation results in Figure 6 show that the change of e value does not impact the school’s strategy choice but significantly impacts the student’s strategy choice. From Figure 6A–D, it can be seen from the curve that reflects the student’s strategy choice, the faster the curve converges to 1 as the value of e increases. This shows that the rise of parental reward is helpful for students’ game strategy choice to tend to cooperate. At the same time, we found an interesting phenomenon in the comprehensive comparison Figure 6A–D. When the parent’s reward value e for the child is higher, the longer the parent’s probability value curve converges to 1. This indicates that parents who give more rewards to students may have more negative game strategy choices in students’ online learning management.

Secondly, we simulate the changes of variables k and d in the same way to explain the effect of changes in the parent’s benefit value d and the parent’s penalty value k on the system game equilibrium. The result is shown in Figure 7.

Figure 7A is the simulation result of taking $k=0.2$, $k=0.6$, $k=0.8$, and $k=1.2$ under the condition of keeping the values of $c,n,h,m$, and d unchanged. As the value of k increases, the probability value curve of student cooperation to converge to 1 is shortened, but from the enlarged part of Figure 7A, the change in the curve convergence speed is not obvious. The results show that parents’ increased punishment has a certain role in promoting students’ online learning management, but the impact is insignificant. Figure 7B is the simulation result of changing the simulated value d while keeping the values of c, n, h, m, and e unchanged. Take the experimental value $d=1.5$, $d=2$, $d=2.3$, $d=2.8$, respectively. From the convergence of the probability value curve in the figure, as the value d increases, the speed of the probability value curve of parents’ cooperation converges to 1 sooner. Therefore, combined with the real situation, it can be further concluded that when the student’s academic performance is better, the enthusiasm for parents’ participation may be higher.

6. Discussion

This paper uses evolutionary game theory to analyze the problem of online learning management for primary and secondary school students during the COVID-19 pandemic. A comprehensive discussion of the results follows through model building, solving, and numerical simulations.

In the “school-student” game system, the simulation results in Figure 4C show that the probability curve representing the student’s strategy choice converges to 1 in less time as the school’s reward c increases.Therefore, the authors conclude that school rewards have a positive effect on facilitating student management of online learning. This finding is in line with scholar Szolnoki’s study that specific reward mechanisms can promote the motivation of game participants [41]. Comparing the convergence results of the probability value curves in Figure 4A,B increases the proofreading student’s reward value c and penalty value n. As arrows $C1$ and $D1$ point to some of the results, it can be seen that the condition for the increase of c, the probability value curve representing student policy choices, converges faster. This shows that the school’s reward is better than the management effect brought by the punishment. The results in Figure 4C show that the greater the student’s online learning benefits, the faster the probability value curve converges. There is a significant positive correlation between students’ online learning benefits and students’ game strategy choices. The simulation results in Figure 4D are noteworthy. As indicated by the arrow $F1$, if the student spends the study time on other aspects to obtain a larger value m, the longer it takes for the probability value curve representing the student’s strategy choice to converge to 1. We can further extend that under the influence of the COVID-19 epidemic, students’ learning benefits will inevitably be affected, and the benefits of non-learning activities may continue to increase. In this case, the difficulty of online learning management for students will increase. From another perspective, students who are less fond of learning may be less fond of learning during online learning. Therefore, in student online learning management, students with average learning ability should receive more attention.

In the “school-student-parent” game system, it can be seen from the results in Figure 6 that the more parents reward students, the faster the probability value curve of students converges. Therefore, there is a positive correlation between the parent’s reward and the convergence of the student’s probability value curve. A particular reward mechanism can promote the enthusiasm of participants [41], and this conclusion also applies to parents managing students’ online learning. From the comparison of the simulation results in Figure 6A–D, it is found that as the parental reward value e increases, the probability value curve representing the parental strategy choice converges more slowly to 1.The simulation results show that parents who reward their children more may be more passive in their gaming strategy choices. Figure 7A reflects the simulation of the impact of parental punishment on students’ strategy choices. The results show that parents’ increased punishment is conducive to students’ strategic choice and cooperation. However, comparing the convergence of the probability value curves in Figure 6 and Figure 7, it is found that the effect of the parent’s reward is more evident under the same application. Figure 7B is an optimistic simulation of the impact on parental involvement. The results show that the more benefits parents get, the more active parents participate. Combined with real-life extrapolation, parents’ enthusiasm for participation may increase when students’ academic performance improves.

Comprehensive analysis comparing the simulation results in Figure 3A and Figure 5A, it is found that when parents participate in students’ online learning management, the rate of convergence of the probability value curves of schools and students accelerates. It shows that the participation of parents can promote students’ online learning management. This finding is similar to the research conclusion of scholar Kowalski [50]. At the same time, comparing the parts pointed to by arrows $A1$ and $G1$ in Figure 3A and Figure 5A, it is found that the active participation of parents promotes the initial convergence speed of the probability value curve of students’ gaming. However, as indicated by the line-pointed arrows $G1$ and $H1$ in Figure 5A,B, it is found that the initial probability value curve reflecting students’ strategy choice converges faster when parental involvement decreases. The striking contrast in this result shows an engagement boundary for parents to participate in students’ online learning. Parental involvement is not entirely positively correlated with students’ online learning management. This finding is the same as the scholar Deslandes’ last point of view that parental involvement in student learning management is borderline [51]. Parental involvement in managing boundaries also exists in online student learning. However, how to measure this boundary still needs further research.

6.1. Limitations and Future Directions

However, we recognize that the present study has some deficiencies and limitations. First of all, the research perspective has limitations. Student online learning management is a complex project. This paper only analyzes the problem of student online learning management from the perspective of evolutionary game theory. In the future, research should be carried out from more views, which is conducive to better promoting the sustainable development of online education. Secondly, there are limitations in the research data. In the process of evolutionary game numerical simulation, the practice of using experimental values to simulate is widely used by researchers in related fields [45,46]. Although this research method has been improved to some extent, there are still limitations. In the future, we should consider using accurate panel data for simulation. This will make the results of the model more actionable and the conclusions of the model simulation more convincing. Finally, the research results have theoretical and practical significance. In the simulation results, this paper found that “parents’ participation in students’ online learning management is not the more, the better, but there are limitations”. This is a meaningful discovery. However, what are the limitations of parental involvement? How to achieve measurement of this limitation will be a significant, exciting study in the future.

6.2. Theoretical and Practical Implications

Despite the above limitations, our findings have significant practical and theoretical implications. Above all, the research has significant practical implications. On the one hand, the COVID-19 epidemic is still not over, and the sustainability of education development is still affected by the COVID-19 epidemic. Online learning is an essential option in the current response to the COVID-19 outbreak. From this view, it is of great significance to study the management of students’ online learning to promote the sustainable development of online education. On the other hand, online learning for students and online teaching for teachers will be an essential part of future education. Student management is still an inescapable reality problem. On the whole, whether it is to deal with the challenges brought by the COVID-19 epidemic or to promote the sustainable development of online education in the future, the research in this paper has specific theoretical significance and practical value. Secondly, this research is of academic importance. In terms of theoretical application, the development of evolutionary game theory has been very mature, but its reference in education is still young. From this point of view, the research in this paper has a particular contribution to expanding the application of evolutionary game theory. Second, there are theoretical contributions to the understanding ents online learning management. How to manage students’ online learning has always been a common concern of schools, parents, and society. This paper conducts mathematical modeling based on evolutionary game theory and uses computer technology to simulate. The research conclusions have both theoretical and practical significance for guiding students’ online learning management and promoting the sustainable development of online learning.

7. Conclusions

This study explores online learning management for primary and secondary school students during the COVID-19 epidemic from an evolutionary game perspective. By modeling the two-game scenarios of “school-student” and “school-student-parent”, computer simulation technology is used to analyze the influence of the stakeholder’s behavior choice on the game system. The study conclusions: First, by comparing the simulation results of the two-game scenarios of “school-student” and “school-student-parent”, it is found that parental participation can significantly impact students’ online learning management. Second, the school’s reward to students has a more significant management effect than punishment under the same effect. Third, comparing the results of behavioral strategies of schools and parents, parents have a more substantial impact on students’ rewards than schools. Fourth, through the comparison of computer simulation results, it is found that there is a boundary for parents to participate in students’ online learning management, and it is not that the higher the degree of participation, the better.