Research on the unification of dominant and subjectivity of ideological and political education in colleges and universities based on big data technology

Game refers to all kinds of activities, interacting with each other to participate in the main body according to their own information and self-interest considerations so as to formulate the relevant decisions and programs. Game theory is in a particular environment or background, the participating subjects through a variety of comprehensive analysis of information and choose to benefit their own development of decision-making [15]. The implementation of game behavior usually includes the following basic elements: participating subjects, action information, action strategy, behavioral sequence and game results. The participating body is the key element of the game, where the participating body can be a specific actor, but also refers to the independent decision-making ability of the organization, such as enterprises, organizations, groups, institutions and so on. There are three main classifications of the game model, according to the behavioral characteristics of the participating subjects, it can be divided into static game and dynamic game. According to the game information situation of each participating subject, it can be divided into complete information game and incomplete information game. According to the way of implementing the game between subjects, it can be divided into cooperative game and non-cooperative game.

The study of game problems has developed into the study of two-player finite zero-sum game (FTZG) problems, also known as matrix game problems, which are affected by the limitations of people’s cognitive level, the incompleteness of information, the complexity of the system structure and randomness and other uncertainties, so that in advance and people can’t give an accurate estimation of their game payoff function, which leads to the optimal choice process of the two parties in the game in terms of the optimal strategy. There are certain difficulties in the process of choosing the optimal strategy. Therefore, how to conduct fuzzy matrix game (FMG) under the condition of fuzzy payoff function and clear strategy value and scientifically select the optimal strategy has become the hot direction of the current matrix game research [16].

Fuzzy matrix game payment function determination has been mainly intuitionistic fuzzy number, interval intuitionistic fuzzy number, triangular intuitionistic fuzzy number and intuitionistic trapezoidal fuzzy number several forms. This paper is based on the interval intuitionistic trapezoidal fuzzy number to express the matrix game problem payment function to construct the trapezoidal fuzzy payment matrix, to explore the optimal path of the dominant subjective unity of the optimal path of the Marxist anthropological field of view of the method of seeking, in order to achieve the purpose of scientific and reasonable selection of the optimal path.

Based on the inspiration of related literature, the estimation of some indicators or parameters can be roughly summarized into the following forms of expression: 1)

Deterministic form: i.e., a point estimate of an indicator or parameter is given, e.g., an estimate of the benefit to the person in the bureau is A = a, a ∈ R.

It is easy to know that among the above four forms, the trapezoidal fuzzy number form has the least loss of information due to the consideration of the subjective psychological factors of human beings, and it is the closest to the actual situation, so it is more scientific and reasonable to adopt the form of trapezoidal fuzzy number in calculating the payment function of the game problem.

Evolutionary game theory refers to a process in which the subject of the game reaches a long-term stable equilibrium through continuous attempts, learning and timely adjustment of his strategic behavior under the premise of limited rationality [17]. It can be traced back to Darwin’s theory of biological evolution, through the promotion of many outstanding scholars, after continuous development and improvement, it has become an important method for analyzing the social system, strategy formation, etc., and its application in the field of education has been relatively mature. Based on the perspective of higher education, this paper constructs an evolutionary game model from three types of subjects: students, teachers and schools, analyzes the relevant behavioral strategies and mutual influence of each subject, and explores the specific path of unification of dominance and subjectivity under the Marxist anthropological perspective, i.e., human-centered concept.

1)

Relevant Definitions In the decision-making process of the players in the game, both parties are accomplished under incomplete information, and both need to make trial and error, learn and then make decisions. In this process, the decision maker can not circumvent the limited rationality and psychological factors, therefore, the use of the exact value to construct the game payment matrix has some shortcomings. There are fuzzy numbers, interval numbers and triangular fuzzy numbers, etc. It is found that the trapezoidal fuzzy number form has the least loss of information and is the closest to the actual situation, so it is more scientific and reasonable to use the trapezoidal fuzzy number form in calculating the payment function of the game.

Definition 1 Let R be the set of real numbers, X ⊂ R be the set of nonempty numbers, and the domain U be the continuous set, then it is said: 1A˜=〈([a,b,c,d];μA˜),([a1,b,c,d1];νA˜)〉$$\tilde A = \langle ([a,b,c,d];{\mu_{\tilde A}}),([{a_1},b,c,{d_1}];{\nu_{\tilde A}})\rangle$$

A˜$$\tilde A$$ is the intuitionistic trapezoidal fuzzy number where: 0≤μλ¯≤1,0≤νλ¯≤1,0≤μλ¯+νλ¯≤1$$0 \leq {\mu_{\bar \lambda }} \leq 1,0 \leq {\nu_{\bar \lambda }} \leq 1,0 \leq {\mu_{\bar \lambda }} + {\nu_{\bar \lambda }} \leq 1$$; a₁, d₁ ∈ R; −∞ < a ≤ b ≤ c ≤ d < ∞; if the affiliation function μλ˜:R→[0,1]$${\mu_{\tilde \lambda }}:R \to [0,1]$$ of x ∈ X and the non-affiliation function vA¯:R→[0,1]$${v_{\bar A}}:R \to [0,1]$$ satisfy the following equations (2) and (3): 2μA˜(x)={ x−ab−aμA˜, a≤x<b μA˜, b≤x≤c d−xd−cμA˜, c<x≤d 0, otherwise$${\mu_{\tilde A}}(x) = \left\{ {\begin{array}{*{20}{l}} {\frac{{x - a}}{{b - a}}{\mu_{\tilde A}},}&{a \leq x < b} \\ {{\mu_{\tilde A}},}&{b \leq x \leq c} \\ {\frac{{d - x}}{{d - c}}{\mu_{\tilde A}},}&{c < x \leq d} \\ {0,}&{otherwise} \end{array}} \right.$$ 3vA˜(x)={ b−x+vA˜(x−a1)b−a1 a1≤x<b vA˜ b≤x≤c x−c+vA˜(d1−x)d1−c c<x≤d1 0 otherwise$${v_{\tilde A}}(x) = \left\{ {\begin{array}{*{20}{c}} {\frac{{b - x + {v_{\tilde A}}(x - {a_1})}}{{b - {a_1}}}}&{{a_1} \leq x < b} \\ {{v_{\tilde A}}}&{b \leq x \leq c} \\ {\frac{{x - c + {v_{\tilde A}}({d_1} - x)}}{{{d_1} - c}}}&{c < x \leq {d_1}} \\ 0&{otherwise} \end{array}} \right.$$

In (2) and Eq. (3) there is usually [a,b,c,d] = [a₁,b,c,d₁], and for this reason it can be noted A˜=([a,b,c,d];μA¯,vA¯)$$\tilde A = ([a,b,c,d];{\mu_{\bar A}},{v_{\bar A}})$$. If b = c, the intuitionistic trapezoidal fuzzy number degenerates into an intuitionistic triangular fuzzy number.

Definition 2: If μλ¯(x)$${\mu_{\bar \lambda }}(x)$$ and vA¯(x)$${v_{\bar A}}(x)$$ are not expressed numerically using exact real numbers, but are instead expressed interval-wise using closed subintervals on [0, 1], then the intuitionistic trapezoidal fuzzy number at this point is called the interval intuitionistic trapezoidal fuzzy number [18], and is denoted as: 4A˜=([a,b,c,d];[μ_,μ¯],[v_,v¯])$$\tilde A = ([a,b,c,d];[\underline \mu, \bar \mu],[\underline{v} ,\bar v])$$

Eq. (4) has: [μ_,μ¯]⊆[0,1],[v_,v¯]⊆[0,1]$$[\underline \mu, \bar \mu] \subseteq [0,1],[\underline{v} ,\bar v] \subseteq [0,1]$$ and 0≤μ¯+v¯≤1$$0 \leq \bar \mu + \bar v \leq 1$$. The interval intuitionistic trapezoidal fuzzy numbers degenerate to intuitionistic trapezoidal fuzzy numbers when μ¯=μ_=1$$\bar \mu = \underline \mu = 1$$ and v¯=v_=1$$\bar v = \underline v = 1$$.

Definition 3: The interval intuitionistic trapezoidal fuzzy set is: 5A≜{〈x,A˜〉|x∈X}={〈x,[a,b,c,d];[μ_,μ¯],[v_,v¯]〉|x∈X}$$A \triangleq \left\{ {\langle x,\tilde A\rangle |x \in X} \right\} = \left\{ {\langle x,[a,b,c,d];[\underline \mu, \bar \mu],[\underline v ,\bar v]\rangle |x \in X} \right\}$$

Definition 4: The hesitancy of the interval intuitionistic trapezoidal fuzzy number is: 6πA˜=1−μA˜−vA˜$${\pi_{\tilde A}} = 1 - {\mu_{\tilde A}} - {v_{\tilde A}}$$

The smaller πA˜$${\pi_{\tilde A}}$$ in Eq. (6), the more certain the fuzzy number is. 2)

Operational properties of interval intuitionistic trapezoidal fuzzy number and comparison regulation 1

Interval intuitionistic trapezoidal fuzzy number of arithmetic properties

Let any interval intuitionistic trapezoidal fuzzy number A˜i=([a1,b1,c1,d1];[μi,μ¯i]$${\tilde A_i} = ([{a_1},{b_1},{c_1},{d_1}];[{\mu_i},{\bar \mu_i}]$$, [v_i,v¯i])$$[\underline v_i,{\bar v_i}])$$ and another interval intuitionistic trapezoidal fuzzy number A˜2=([a2,b2,c2,d2];[μ_2,μ¯2],[v_2,v¯2])$${\tilde A_2} = ([{a_2},{b_2},{c_2},{d_2}];[\underline \mu_2,\bar \mu_2],[\underline v_2,{\bar v_2}])$$, the fuzzy algorithm is as follows: a)

Addition rule: 7A˜1+A˜2 = ([a1+a2,b1+b2,c1+c2,d1+d2,] [μ_1+μ_2−μ_1⋅μ_2,μ¯1+μ¯2−μ¯1⋅μ¯2] [v_1⋅v_2,v¯1⋅v¯2])$$\begin{array}{rcl} {{\tilde A}_1} + {{\tilde A}_2} &=& ([{a_1} + {a_2},{b_1} + {b_2},{c_1} + {c_2},{d_1} + {d_2},] \\ &&[{\underline{\mu}_1} + {\underline{\mu}_2} - {\underline{\mu}_1} \cdot {\underline{\mu}_2},{{\bar \mu }_1} + {{\bar \mu }_2} - {{\bar \mu }_1} \cdot {{\bar \mu }_2}] \\ &&[{{\underline v }_1} \cdot {{\underline v }_2},{{\bar v}_1} \cdot {{\bar v}_2}]) \\ \end{array}$$

In Eq. (7) and Eq. (8): a₁, λ ≥ 0. 2

Comparison rule for interval intuitionistic trapezoidal fuzzy numbers

Definition 5: In order to compare the size of two interval intuitionistic trapezoidal fuzzy numbers, firstly, the score function S(A˜)$$S(\tilde A)$$ of any interval intuitionistic trapezoidal fuzzy number A˜=([a,b,c,d];[μ_,μ¯],[v_,v¯])$$\tilde A = ([a,b,c,d];[\underline{\mu} ,\bar \mu ],[\underline{v} ,\bar v])$$ is defined as shown in Eq. (9) and the exact function H(A˜)$$H(\tilde A)$$ is shown in Eq. (10): 9S(A˜)=12E(μ_−v_+μ¯−v¯)$$S(\tilde A) = \frac{1}{2}E(\underline{\mu} - \underline v + \bar \mu - \bar v)$$ 10H(A˜)=12E(μ_+v_+μ¯+v¯)$$H(\tilde A) = \frac{1}{2}E(\underline{\mu} + \underline v + \bar \mu + \bar v)$$

In equations (9) and (10): E=14(a+b+c+d)$$E = \frac{1}{4}(a + b + c + d)$$ is the expected value function.

Let any interval intuitionistic trapezoidal fuzzy number A˜1=([a1,b1,c1,d1];[μ1,μ¯1],[v1,v¯1])$${\tilde A_1} = ([{a_1},{b_1},{c_1},{d_1}];[{\mu_1},{\bar \mu_1}],[{v_1},{\bar v_1}])$$ and another interval intuitionistic trapezoidal fuzzy number A˜2=([a2,b2,c2,d2];[μ2,μ¯2],[v2,v¯2])$${\tilde A_2} = ([{a_2},{b_2},{c_2},{d_2}];[{\mu_2},{\bar \mu_2}],[{v_2},{\bar v_2}])$$, then its comparison law is as follows: a)

If S(A˜1)>S(A˜2)$$S({\tilde A_1}) > S({\tilde A_2})$$, then A˜1>A˜2$${\tilde A_1} > {\tilde A_2}$$.

Determine the subject of the educational game is the application of evolutionary game theory to carry out educational research the first premise, the subject of the educational game refers to the development of educational issues have an important impact on the limited rationality of the core subjects, these subjects are involved in a variety of interests, through the game behavior of each other to understand the information of all parties in order to pursue their respective interests of the maximization of the researcher needs to clarify the main players in the process of the occurrence of the educational issues, and to their The researcher needs to clarify the main players in the process of educational problems and define the subjects in order to sort out the inter-subjective interests and practical demands. It is worth noting that the number of subjects in the educational game varies from one educational problem to another. For example, the subjects of the educational game of multi-party cooperative education are schools, enterprises and students, while the subjects of the educational game of teacher teamwork are teachers. In this paper, the subjects of education game are schools, teachers and students.

On the basis of determining the subject of the educational game, the researcher needs to assume the behavioral strategies and interests of the subject of the educational game in different educational situations, which contains two phases of the assumption of the game strategy and the assumption of the educational situation. 1)

Game strategy assumption stage. In the game system, each game subject i contains a finite pure strategy set S_i. The finite pure strategy set refers to the strategy adopted by the game subject in the process of the game, which is denoted as S1={1,2,…,m1}$${S_1} = \left\{ {1,2, \ldots ,{m_1}} \right\}$$, integer m₁ ≥ 2. In this stage, the researcher needs to clarify the behavioral strategies of the educational game subjects, and make detailed explanation of it.

Based on the above model context assumptions can be constructed evolutionary game model. Standard evolutionary game can be expressed as a ternary G={I,S,π}$$G = \left\{ {I,S,\pi } \right\}$$, where I represents the game system of the game subject collection, S represents the game system behavior strategy space, π represents the game system behavioral benefit function, the function size of the main body of the educational game for the various benefits and costs of the difference. It is worth noting that, due to the different number of educational game subjects, the arrangement of the evolutionary game model is also different, which can be divided into two-party, three-party, four-party game model and so on. Taking the three-party game as an example, the three-party game model is shown in Table 1.

After the construction of the evolutionary game model is completed it needs to be solved. To this end, the researcher first needs to calculate the expected return function, average expected return function of each educational game subject. Then, the replication dynamic equation and Jacobi matrix are used comprehensively to judge the evolutionary stability of the subject’s behavior. Among them, the replication dynamic equation is a kind of dynamic differential equation reflecting the frequency or frequency of a particular strategy, which can deduce how the frequency of the group changes over time and its evolutionary path under different strategy perspectives. When the replication dynamic equation is 0, the equilibrium point of the game system can be obtained, i.e., the conditional combination of the subject’s behavior when things tend to equilibrate. It is worth noting that the value attribute of the equilibrium point does not mean that the conditions have positive or negative effects on the development of things, but only objectively presents the equilibrium state of the development of things, but the equilibrium point is not stable, and it is necessary to use Jacobi matrix to judge its stability, so as to obtain the evolutionary stability strategy.

In order to more intuitively describe and analyze the dynamic evolution process of the behavioral strategies of educational subjects, researchers can use simulation tools to numerically simulate and analyze the evolutionary stable strategies. Based on the preliminary literature research, this study found that the simulation and analysis tools mainly used by educational researchers include Matlab software, R language, Python and so on. Among them, Matlab is a commercial mathematical software that combines data analysis, formula calculation, programming, and plotting, which has the advantages of easy-to-understand code, advanced graphs, rich functions, and high interactivity, and thus is widely used by educational researchers.

The process of numerical simulation simulation and analysis is as follows: first, it is necessary to assume the value of the interest between educational subjects, which can be obtained through websites and questionnaires. Then, the code is inputted and compiled in Matlab software, so as to obtain the evolution trajectory diagram, which can intuitively and scientifically describe the trend of the behavioral strategies of educational subjects. Finally, in order to judge whether the size of the interest value among educational subjects has an impact on the choice of behavioral strategies of educational subjects and to what extent, the parameter sensitivity analysis can be carried out by adjusting the value of one of the interest values while the other values remain unchanged, so as to obtain the behavioral trend chart of educational subjects under different interest values, and then derive the conditional factors and the interaction mechanism affecting the behavioral strategies of educational subjects, and then put forward guiding suggestions to each educational subject accordingly. Accordingly, guiding suggestions are made for each educational subject to realize the benign development of the educational system.

This paper selects undergraduate education as the specific research object, constructs the evolutionary game model of school, tutor and student, explores the factors affecting students’ active learning, and analyzes the evolutionary strategies of school, tutor and student under limited rationality. The above three parties are finite rationality, and the three parties will be affected by information asymmetry, can not make the optimal decision at one time, they will gradually adjust their own strategies, until the state of the two sides to reach an equilibrium.

The strategy set of undergraduate students is {active learning, passive learning}, the strategy set of supervisors is {conscientious cultivation, non-conscientious cultivation}, and the strategy set of the university is {effective incentives, ineffective incentives}.

The main model parameters of this paper are shown below:

s,t,u - represent students, mentors and schools, respectively.

C_i - i = s,t,u, represent the cost invested by the student, the mentor and the school, respectively, C_i ≥ 0.

A_i - i = s,t,u, represent the basic benefits for students, mentors and schools, respectively, A_i ≥ 0.

q - the percentage of the student’s subsidy allocation, 0 ≤ q ≤ 1.

r - The percentage of additional earnings allocated to students, 0 ≤ r ≤ 1.

B - The total amount of education subsidies allocated by the government, B ≥ 0.

B_u - Effective government incentive subsidies to schools, B_u ≥ 0.

L - Additional academic gains for tutors and students for academic achievements, L ≥ 0.

L_u - Additional academic gains for the school created by student-initiated learning and serious mentor development, B_u ≥ 0.

E_t - Ride-along gains for the instructor when the student is actively learning and the instructor is not seriously cultivating, E_t ≥ 0.

E_u - Ride-along gains for the school from ineffective incentives for student-initiated learning and serious mentor training, E_u ≥ 0.

In order to formulate the relationship between the interests of undergraduate students, instructors and schools in undergraduate training, this paper proposes the following hypotheses:

Assumption 1: For undergraduate students, no matter what strategy the school or the supervisor chooses, the benefit of undergraduate students’ active learning A_s is greater than the cost of their learning C_s. For supervisors, the basic benefit A_t is greater than the cost of training students C_t. For schools, the basic benefit A_u is greater than the cost of their inputs C_u.

Assumption 2: The school, as a third party, can appropriately make use of the educational subsidies allocated by the higher authorities or the government to cover the costs of educational inputs of undergraduate students and instructors, and the educational subsidies B include not only the Ministry of Education subsidies B_u for students or instructors, but also funds for school construction, etc., i.e., B > B_u. If we make the allocation ratio of subsidies to students be q, then the costs paid by students are C_s − Bq, and that by instructors are C_t − B(1 − q).

Assumption 3: When undergraduate student takes the initiative and the mentor chooses a cooperative strategy, there is a common goal that drives both parties to work hard to achieve certain academic results and thus gain additional academic benefits. If the student’s share of the additional academic benefit is r, the undergraduate student receives an additional academic benefit of rL₄, and the mentor receives an additional academic benefit of (1 − r)L. At the same time, both the undergraduate student and the mentor receive an additional academic benefit to the university, which is denoted as L_u.

Assumption 4: Since the incentives given by the university to undergraduate students or supervisors are limited, but the additional academic gains obtained by supervisors or undergraduate students through behaviors such as publishing articles are greater than the university’s incentives, i.e., (1 − r)L > B(1 − q), and rL > Bq.

According to the above model assumptions and strategy analysis, the benefit matrices of the three parties in the effective incentive and ineffective incentive strategies of the school are shown in Table 2 and Table 3 respectively, taking the “active learning” of undergraduate students and the “serious training” of supervisors in Table 2 as an example, the income of undergraduate students refers to the basic income A_s and additional academic income rL minus the input cost C_s − Bq. A mentor’s benefit is the sum of the basic benefit A_t and the additional academic benefit (1 − r)L, minus his or her input costs C_s − B(1 − q). A school’s benefit is the sum of the basic benefit A_u, its additional academic benefit L_u, and the government’s effective incentive subsidy to the school B_u, minus its input costs C_u.

According to Tables 2 and 3, if the probability of active learning is x, the probability of serious training by the supervisor is y, and the probability of effective motivation is z, 0 ≤ x, y, z ≤ 1, then the expected return of the undergraduate student choosing the “active learning” strategy U_s1, the expected return U_s2 and the average expected benefit U¯s$${\bar U_s}$$ of the “passive learning” strategy are respectively as follows: 11Us1 = yz(As+Bq+rL−Cs) +y(1−z)(As+rL−Cs) +z(1−y)(As+Bq−Cs) +(1−y)(1−z)(As−Cs) Us2 = 0 U¯s = xUs1+(1−x)Us2$$\begin{array}{rcl} {U_{s1}} &=& yz({A_s} + Bq + rL - {C_s}) \\ &&+ y(1 - z)({A_s} + rL - {C_s}) \\ &&+ z(1 - y)({A_s} + Bq - {C_s}) \\ &&+ (1 - y)(1 - z)({A_s} - {C_s}) \\ {U_{s2}} &=& 0 \\ {{\bar U}_s} &=& x{U_{s1}} + (1 - x){U_{s2}} \\ \end{array}$$

Similarly, the expected return of the mentor choosing the “serious training” strategy is U_t1, the expected return of the mentor choosing the “not serious training” strategy is U_t2, and the average expected return is U¯t$${\bar U_t}$$, respectively: 12Ut1 = xz[At+B(1−q)+(1−r)L−Ct]+x(1−z)[At+(1−r)L−Ct] +z(1−x)[At+B(1−q)−Ct]+(1−x)(1−z)(At−Ct) Ut2 = xz(At+Et)+x(1−z)(At+Et)+z(1−x)At+(1−x)(1−z)At Ut¯ = yUt1+(1−y)Ut2$$\begin{array}{rcl} {U_{t1}} &=& xz[{A_t} + B(1 - q) + (1 - r)L - {C_t}] + x(1 - z)[{A_t} + (1 - r)L - {C_t}] \\ &&+ z(1 - x)[{A_t} + B(1 - q) - {C_t}] + (1 - x)(1 - z)({A_t} - {C_t}) \\ {U_{t2}} &=& xz({A_t} + {E_t}) + x(1 - z)({A_t} + {E_t}) + z(1 - x){A_t} + (1 - x)(1 - z){A_t} \\ \overline {{U_t}} &=& y{U_{t1}} + (1 - y){U_{t2}} \\ \end{array}$$

Similarly, the school’s expected payoff U_u1 for choosing the “effective incentive” strategy, the expected payoff U_u2 for choosing the “ineffective incentive” strategy, and the average expected payoff U¯u$${\bar U_u}$$ at the time of the game are: 13Uu1 = xy(Au+Lu+Bu−Cu)+x(1−y)(Au+Bu−Cu) +y(1−x)(Au+Bu−Cu)+(1−x)(1−y)(Au+Bu−Cu) Uu2 = xy(Au+Lu+Eu)+x(1−y)Au+y(1−x)Au+(1−x)(1−y)Au U¯u = zUu1+(1−z)Uu2$$\begin{array}{rcl} {U_{u1}} &=& xy({A_u} + {L_u} + {B_u} - {C_u}) + x(1 - y)({A_u} + {B_u} - {C_u}) \\ &&+ y(1 - x)({A_u} + {B_u} - {C_u}) + (1 - x)(1 - y)({A_u} + {B_u} - {C_u}) \\ {U_{u2}} &=& xy({A_u} + {L_u} + {E_u}) + x(1 - y){A_u} + y(1 - x){A_u} + (1 - x)(1 - y){A_u} \\ {{\bar U}_u} &=& z{U_{u1}} + (1 - z){U_{u2}} \\ \end{array}$$

From the above analysis, the equation for the replication dynamics of undergraduate students is obtained as: 14F(x) = dxdt=x(Us1−U¯s) = x(1−x)(As+zBq+yrL−Cs)$$\begin{array}{rcl} F(x)&=&\frac{{dx}}{{dt}}=x({U_{s1}} - {{\bar U}_s}) \\&=&x(1 - x)({A_s} + zBq + yrL - {C_s}) \\ \end{array}$$

Similarly, the equation for the replication dynamics of the tutor is: 15F(y) = dydt=y(Ut1−U¯t) = y(1−y)[x(1−r)L+z(1−q)B−Ct−xEt]$$\begin{array}{rcl} F(y)&=&\frac{{dy}}{{dt}}=y({U_{t1}} - {{\bar U}_t}) \\&=&y(1 - y)[x(1 - r)L + z(1 - q)B - {C_t} - x{E_t}] \\ \end{array}$$

Further, the equation for the replication dynamics of the school is: 16F(z) = dzdt=z(Uu1−U¯u) = z(1−z)(Bu−Cu−xyEu)$$\begin{array}{rcl} F(z)&=&\frac{{dz}}{{dt}}=z({U_{u1}} - {{\bar U}_u}) \\&=&z(1 - z)({B_u} - {C_u} - xy{E_u}) \\ \end{array}$$

The evolutionary stabilization strategy (ESS) of a system of differential equations can be determined by the local stability of the Jacobi matrix of this system, which is therefore obtained from the above equation: 17J=( (1−2x)(As+zBq x(1−x)rL x(1−x)Bq +yrL−Cs) (1−2y)[x(1−r)L y(1−y)(1−q)B y(1−y)[(1−r)L−Et] +z(1−q)B−Ct−xEt] (1−2z)(Bu−Cu z(1−z)yEu z(1−z)xEu −xyEu)$$\left. {J = \left( {\begin{array}{*{20}{c}} {(1 - 2x)({A_s} + zBq}&{x(1 - x)rL}&{x(1 - x)Bq} \\ { + yrL - {C_s})}&{(1 - 2y)[x(1 - r)L}&{y(1 - y)(1 - q)B} \\ {y(1 - y)[(1 - r)L - {E_t}]}&{ + z(1 - q)B - {C_t} - x{E_t}]}&{(1 - 2z)({B_u} - {C_u}} \\ {z(1 - z)y{E_u}}&{z(1 - z)x{E_u}}&{ - xy{E_u}} \end{array}} \right.} \right)$$

When the solution of the replicated dynamic equation is equal to 0, the evolutionary game strategy is in a steady state. Copying F(x) = F(y) = F(z) = 0 in the dynamic equation above yields the local equilibrium points, which represent the strategies of the undergraduate student, the supervisor, and the school, respectively. 1 represents the strategy of “active learning, serious training, and effective motivation”, and 0 represents “passive learning, not serious training, and ineffective motivation”, and 9 equilibrium points are obtained: 18E1(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=(x*,y*,z*)$$\begin{array}{l} {E_1}(0,0,0),{E_2}(0,0,1),{E_3}(0,1,0),{E_4}(0,1,1),{E_5}(1,0,0) \\ {E_6}(1,0,1),{E_7}(1,1,0),{E_8}(1,1,1),{E_9} = ({x^*},{y^*},{z^*}) \\ \end{array}$$

where x*=Ct−z(1−q)B(1−r)L−Et,y*=Bu−CuxEu,z*=Cs−As−yrLBq$${x^*} = \frac{{{C_t} - z(1 - q)B}}{{(1 - r)L - {E_t}}},{y^*} = \frac{{{B_u} - {C_u}}}{{x{E_u}}},{z^*} = \frac{{{C_s} - {A_s} - yrL}}{{Bq}}$$.

The stability of the point is judged according to the positivity and negativity of the 9 equilibrium points, and the local equilibrium point is evolutionarily stable when F(x) = 0 and F′(x) < 0, or F(y) = 0 and F′(y) < 0, or F(z) = 0 and F′(z) < 0. In asymmetric games, if the evolutionary game equilibrium E is an evolutionary stable equilibrium, then E must be a strict Nash equilibrium, and a strict Nash equilibrium is a pure strategy equilibrium i.e., in asymmetric games, a mixed strategy equilibrium must not be an evolutionary stable equilibrium, and thus only the asymptotic stability of pure strategy equilibria need be discussed. Since E₉ is a mixed-strategy Nash equilibrium, E₉ is not an ESS, and the asymptotic stability of point E₉ will not be discussed later, and the eigenvalues of the other eight equilibrium points are shown in Table 4.

First analyze the case where the equilibrium point is E₁(0, 0, 0), at which point the Jacobi matrix is: 19J=( As−Cs 0 0 0 −Ct 0 0 0 Bu−Cu)$$J = \left( {\begin{array}{*{20}{c}} {{A_s} - {C_s}}&0&0 \\ 0&{ - {C_t}}&0 \\ 0&0&{{B_u} - {C_u}} \end{array}} \right)$$

It can be seen that at this time the eigenvalues of the Jacobi matrix λ₁ = A_s − C_s, λ₂ = −C_t, λ₃ = B_u − C_u and so on will be 7 equilibrium points respectively into the Jacobi matrix, can be respectively obtained equilibrium points corresponding to the eigenvalues of the Jacobi matrix as shown in Table 4. By analyzing the magnitude of the eigenvalues, it is concluded that the stability of the eight equilibrium points is affected by the cost-sharing ratio q and the benefit-sharing ratio r as shown in Table 5.

As can be seen from Table 5, the evolutionary stability point of the three parties is obtained under different values of the cost-sharing ratio q and benefit-sharing ratio r. Based on the final evolutionary strategy selection of the three parties, there are four kinds of strategies: (active learning, no serious cultivation, ineffective incentives), (active learning, no serious cultivation, effective incentives), (active learning, serious cultivation, ineffective incentives), and (active learning, serious cultivation, effective incentives). To reach the final stable point (active learning, serious cultivation, effective incentive), first of all, undergraduate students, as the key target, should take the initiative to learn and get more benefits for themselves, while supervisors, as the bridge between students and schools, should take the initiative to take the responsibility of education and promote the good development of students.

In the case of meeting the equilibrium point (0,1,1), all the eigenvalues of the comparable matrix are negative, and the values of the benefits of the mentor’s “conscientious cultivation” of the student A_t are set to 0.3, 3, 6. The results of the analysis of the benefits of the mentor’s “conscientious cultivation” of the student are shown in Fig. 2. The results of the analysis are shown in Figure 2, where (a) to (c) denote the decision-making situations of the student, the mentor, and the school, respectively, at the time of A_t = 0.3, 3, 6.

Figure 2.

Income analysis of tutors conscientiously cultivating students

Figure 2(b) shows that as the benefits of mentoring students increase, mentors’ decisions will converge to “serious training” more quickly, so increasing the benefits of mentors can positively promote their “serious training” of students. The three lines in Figures 2(a) and (c) almost overlap, reflecting the fact that the increase in mentors’ earnings has almost no effect on students’ and schools’ decisions.

When all eigenvalues of the Jacobian matrix satisfying the equilibrium point (0,1,1) are negative, the values of the cost of “serious training” of the tutor C_t are set to 2, 4, and 6, respectively. Figure 3 shows the results of the cost analysis of the tutor’s “serious training” of students, and (a) ~ (c) represent the decision-making situation of students, tutors, and schools at C_t = 2, 4, 6 o’clock.

Figure 3.

Cost analysis of tutors conscientiously cultivating students

Figure 3(b) reflects that tutors are more inclined to “train seriously” when the cost of mentoring students is lower. When the cost of tutors to train students continues to rise and exceed the threshold, tutors turn to “not seriously developing” students. Figure 3(c) shows that the school’s decision to choose “effective incentive” or “ineffective incentive” teacher-student cooperation was made by the tutor. Figure 3(a) shows that students are more inclined to implement a “passive learning” strategy when tutors are “serious about training” and schools choose “effective motivation”. When tutors and schools shifted to “not serious training” and “ineffective incentives”, respectively, students were more inclined to choose to implement “active learning” strategies.

Analysis of the additional academic benefits of the mentor’s “serious cultivation” of students in the case of meeting the equilibrium point (0,1,1) Jacobian matrix all eigenvalues are negative, the mentor’s “serious cultivation” of students to obtain the additional academic benefits of the value of L(1−r)$$L\left( {1 - r} \right)$$ were set to 0.5. 1.2, 4, the results of the analysis of the additional academic benefits of the mentor’s “conscientious cultivation” of students are shown in Fig. 4, with (a) to (c) indicating the decision-making situations of students, mentors and schools at the time of L(1−r)=0.5,1.2,4$$L\left( {1 - r} \right) = 0.5,1.2,4$$, respectively.

Figure 4.

Additional income analysis of tutors conscientiously cultivating students

Figure 4(b) shows that when the mentor gains additional academic benefits, the mentor’s decision converges more quickly to “seriously cultivate” students. Figure 4(c) reflects that the probability that the school will “effectively incentivize” student-teacher collaboration increases as the mentor receives additional academic benefits. Figure 4(a) reflects that the additional academic benefits of mentor-led research programs have relatively little impact on student decision-making.