Analysis of the harmonization of the layout and restructuring of basic education and the policy of proximity of students to schools

Game refers to the process that under the condition that competitors oppose each other and must obey certain rules, each competitor chooses his own strategy and implements it according to his own information, once or several times, simultaneously or sequentially, so as to get the result. Game theory is the study of how to make their own choices scientifically and rationally in this dynamic environment.

Learning game theory can deepen people’s understanding of social phenomena, deeply understand and grasp the essence of things, and guide people’s social practice and production activities, and has been widely used in various fields such as economics, political science, military, diplomacy, environmental protection, logistics and supply chain, and artificial intelligence.

If a binding agreement is reached between the participants of the game, then the game is a cooperative game, otherwise it is a non-cooperative game. Non-cooperative game focuses on how participants can make reasonable strategic choices to maximize their own or overall benefits, while cooperative game considers how to reasonably distribute the benefits of joint cooperation to each participant. In daily life, people often pursue the goal of maximizing their own interests, so the non-cooperative game started earlier, the research is more in-depth, occupies a dominant position, while the development of cooperative games is relatively lagging behind, and cooperation is usually regarded as the result of non-cooperative games. However, with the progress of social development, cooperation and win-win consciousness gradually penetrate into people’s hearts, when people realize that the practice of only caring for their own interests without considering the interests of others will ultimately lead to their own interests can not be guaranteed, they will consider the maximization of mutual cooperation for the pursuit of common interests and reach a unanimous agreement to regulate their respective behaviors, the advantages of the cooperative game will be highlighted.

The pursuit of common interests maximization of this goal prompts the cooperative game in the participants of mutual trust and through consultation and communication to ensure information symmetry, coordination of their respective strategies, so as to reach a common recognition of the binding agreement. Cooperative game can produce utility surplus, that is, the total benefit of cooperation is greater than the sum of the benefits of each participant acting individually, and the benefits of each participant should be at least higher than the benefits of the competition, that is, the Pareto improvement, otherwise the participants will not comply with the common agreement, so that the cooperative relationship rupture, transformed into a non-cooperative game, so the objective, fair and reasonable method of distributing the total benefit is related to the cooperation of the Therefore, an objective, fair and reasonable total benefit distribution method is related to the cooperation’s longevity and stability.

Shapley value allocation method is a classic method for allocating participants’ respective gains in cooperative games, which calculates the contribution of each participant to the total gains to allocate the gains, and it is more fair and reasonable than the average allocation method and the method of allocating according to the proportion of costs, and because it is an axiomatic allocation method, it has good operability. It has good operability and is an ideal method in static cooperative game revenue allocation.

The Shapley value assignment method is specified as follows:

Let the set of finite participants be N, and for any subset S of N corresponds to a function V(s) which represents the total benefit received by the coalition S and satisfies: V(∅) = 0, V(s1∪s2)≥V(s1)+V(s2)$V\left( {{s_1} \cup {s_2}} \right) \ge V\left( {{s_1}} \right) + V\left( {{s_2}} \right)$, where s₁ ∩ s₂ = ∅, s₁, s₂ ∈ N, the benefit to each member is: 1Xi=∑|s|=1nW(|s|)[V(s)−V(s\i)]${X_i} = \sum\limits_{|s| = 1}^n W (|s|)[V(s) - V(s\backslash i)]$

In equation W(|s|) = [(n − |s|)!(|s| − 1)!]/n!, n is the number of elements in set N, S is all the subsets of set N that contain cooperative subject i, |s| is the number of elements in subset S, V(s) is the coalition gain from subset S, and V(s\i) is the gain from subset S after removing cooperative member i. The formula can be understood from the perspective of probability expectation in mathematics: the number of sub-coalitions that can be formed is n!, and the order of s\i and n\i participants after the removal of i is (n − |s|)!(|s| − 1)!, so the probability of forming each coalition is W(|s|) = [(n − |s|)!(|s| − 1)!]/n!, or W(|s|), however, the marginal contribution of the participant i to the coalition S is V(s) − V(s\i), and the two are multiplied together to obtain the value of the distribution of benefits to the participant i.

The Shapley value assignment method also presents three basic theorems:

Symmetry Theorem: the order of the participants has no effect on the Shapley value;

The validity theorem: the total benefit of cooperative booting is equal to the sum of the Shapley values of each participant;

Addition theorem: the Shapley value of the merged coalition is the sum of the Shapley values of the two independent BoAs.

Although the goal of each participant in the cooperative game is to maximize the total gain, but if there is a lack of effective constraint mechanism, when there is a conflict of interest, it is possible that some members of the coalition will leave the cooperation and seek their own interests, which turns into a non-cooperative game. In order to better understand the final result of the competitive game, similar to the Shapley value in the cooperative game, the Nash equilibrium plays an important role in the non-cooperative game.

Nash equilibrium is reached when all participants in the game have no incentive to come back and change their original strategies after proposing their own strategies, and this stable, relatively competitive combination of strategies is called a Nash equilibrium. In the same game there may exist more than one Nash equilibrium, specific strict Nash equilibrium mathematical expression is as follows:

If available for each participant i: 2ui(si*,s−i*)>ui(si,s−i*)si∈Si${u_i}\left( {s_i^*,s_{ - i}^*} \right) > {u_i}\left( {{s_i},s_{ - i}^*} \right){s_i} \in {S_i}$

Then (si*,s−i*)$\left( {s_i^*,s_{ - i}^*} \right)$ is said to be a Nash equilibrium solution of the game;

where i is a participant; s_i is a particular strategy of participant i; S_i is the set of strategies of participant i; S−i*$S_{ - i}^*$ is the set of strategies of participants other than i; and u_i is the payoff function of participant i.

This paper takes the layout structure of basic education and the adjustment of the policy of students’ proximity to schools as an example, and assumes that both of them can choose both cooperative and non-cooperative countermeasures under macro-control. Table 2 shows the payment matrix of the cooperative evolution game of the basic education layout structure and the adjustment of students’ proximity to school policy.

Where e_m and e_b are the benefit values when basic education and students’ nearness to school are adopted, Δπ_m and Δπ_b are the additional benefits of adopting cooperative operation, c is the wasted cost of cooperation when the other party adopts a non-cooperative attitude, and w is the penalty value of adopting a non-cooperative attitude. Assuming that the probability of adopting cooperation in the layout of basic education is x and the probability of adopting cooperation in students’ nearness to school is y, the payment matrix can be obtained from the structural adjustment of the layout of basic education and students’ nearness to school policy when adopting the cooperative and non-cooperative strategies are the benefits respectively: 3Um=x(em+Δπm)+(1−x)(em−c)${U_m} = x\left( {{e_m} + \Delta {\pi_m}} \right) + (1 - x)\left( {{e_m} - c} \right)$ 4Um′=x(em−w)+(1−x)em${U'_m} = x\left( {{e_m} - w} \right) + (1 - x){e_m}$ 5Ub=y(eb+Δπb)+(1−y)(eb−w)${U_b} = y\left( {{e_b} + \Delta {\pi_b}} \right) + (1 - y)\left( {{e_b} - w} \right)$ 6Ub′=y(eb−c)+(1−y)eb${U'_b} = y\left( {{e_b} - c} \right) + (1 - y){e_b}$

From equations (3)-(6), the average return for both parties is: 7U¯m=yUm+(1−y)Um′${\bar U_m} = y{U_m} + (1 - y){U'_m}$ 8U¯b=xUb+(1−x)Ub′${\bar U_b} = x{U_b} + (1 - x){U'_b}$

From Eqs. (7) and (8), the replicated dynamic equations of basic education layout restructuring and students’ proximity to school policy in the evolution of the game are: 9X=x(Um−U¯m)=x(1−x)((Δπm+c+w)y−w)$X = x\left( {{U_m} - {{\bar U}_m}} \right) = x(1 - x)\left( {\left( {\Delta {\pi_m} + c + w} \right)y - w} \right)$ 10Y=y(Ub−U¯b)=y(1−y)((Δπb+c+w)x−c)$Y = y\left( {{U_b} - {{\bar U}_b}} \right) = y(1 - y)\left( {\left( {\Delta {\pi_b} + c + w} \right)x - c} \right)$

The differential power Jacobi matrix is formed from Eqs. (9) and (10) as: 11J=( (1−2x)[y(Δπm+c+w)−c] x(1−x)(Δπm+c+w)) y(1−y)(Δπb+c+w) (1−2y)[x(Δπb+c+w)−c])$J = \left( {\begin{array}{*{20}{c}} {(1 - 2x)\left[ {y\left( {\Delta {\pi_m} + c + w} \right) - c} \right]}&{\left. {x(1 - x)\left( {\Delta {\pi_m} + c + w} \right)} \right)} \\ {y(1 - y)\left( {\Delta {\pi_b} + c + w} \right)}&{(1 - 2y)\left[ {x\left( {\Delta {\pi_b} + c + w} \right) - c} \right]} \end{array}} \right)$

The game model of basic education layout and students’ proximity to schools has two interrelated objective functions, which can be regarded as a multi-objective optimization problem. Obviously, the two objective functions before and after the strategy of passengers in the station to shorten the maximum time and the transportation company to pay the minimum conflict between each other, and the objective function of the composition of the complexity of the strategy variables, the problem is difficult to get the optimal solution, so this paper uses the Pareto solution set instead of the optimal solution, and the assumption that the equilibrium solution of the game in the Pareto solution set. For the Pareto solution has the following definition.

Definition 1: Pareto dominate relationship of individuals. Let α and β be any two individuals, if individual α has at least one objective better than β, and all objectives of individual β are inferior to α, then individual α is said to dominate individual β, or individual α is non-inferior to individual β. Denoted as β ≺ α.

Definition 2: Pareto optimal solution set. For a multi-objective optimization problem, S is the space of policy variables, if α ∈ S, while there is no β ∈ S, so that β ≺ α, say α for the multi-objective optimization of the Pareto optimal solution, all the Pareto optimal solution constitutes the Pareto optimal solution set.

Currently, the main methods for solving multi-objective optimization problems are ideal point method, linear weighting method, ant colony algorithm, particle swarm optimization algorithm and genetic algorithm. Compared with other algorithms, genetic algorithm has the advantages of strong operability, high solution accuracy, strong global optimization ability and fast convergence speed. In this paper, the second generation of non-dominated sorting genetic algorithm (NSGA-II) is used to solve this multi-objective optimization problem, NSGA-II is an improved genetic algorithm based on NSGA, which adopts the same selection, crossover, and mutation operations as the Simple Genetic Algorithm (SGA), but in the process of selection, NSGA also performs non-dominated sorting based on the dominance and non-inferiority relationships between individuals. In addition, NSGA also performs non-dominated sorting according to the dominance and non-inferiority relationships between individuals during the selection operation, which ensures the diversity of the optimal solution set well. However, NSGA algorithm has the disadvantages of high complexity of the non-dominated hierarchical sorting process and the lack of elite strategy, NSGA-II is proposed to solve these shortcomings of NSGA. NSGA-II adopts the fast non-dominated sorting algorithm, which greatly reduces the complexity of NSGA, introduces the elite strategy, enlarges the sample space, prevents losing the optimal individuals, and improves the speed of the computation and robustness. NSGA-II has more prominent advantages than SGA and NSGA in terms of global optimization search, computational efficiency, and diversity of solutions, etc. Therefore, this paper adopts NSGA-II to solve the game model of the parties in the layout of basic education and the students’ proximity to school.

The following assumptions are given to better analyze the structural adjustment of the layout of basic education and the coordination of the policy of students’ proximity to schools:

The probability of adjusting the layout of basic education according to the policy of proximity of students to schools is x, and the probability of not adjusting the layout of basic education according to the policy of proximity of students is 1 − x. The probability of not adjusting the layout of basic education according to the policy of proximity of students is y, and the probability of not adjusting the layout of basic education is 1 − y, x, and y is satisfied by 0 ≤ x and y ≤ 1.

Let the initial cost of basic education layout restructuring be C_i and the operating cost be C₀, and the cooperative behavior of joining the policy of students’ nearness to school will affect the earnings of education layout. When the investor does not adjust the education layout structure, the investor’s revenue is P_c. If the investor adjusts the education layout structure, its revenue is (1 + j)P_c, where i is the growth rate of the investor’s profit and j > 0. On the contrary, if the investor abandons the adjustment of the education layout structure in accordance with the policy of proximity of students to schools, it will incur a loss C_p. Assuming that, −C_P > P_c − C_i − C₀ for the investor to give up the adjustment of the investor in the case of the two parties failing to reach a cooperative strategy is more economic efficiency. The benefit matrix of the evolutionary game between the adjustment of basic education layout and the policy of proximity of students to schools is shown in Table 3.

The local stability analysis method of Jacobi matrix is applied to solve the stable strategy of the evolutionary game. Where the Jacobi matrix J is shown in Equation (11).

The type of the five equilibrium points can be determined by calculating Det(J) and Tr(J) of the Jacobi matrix, where Det(J) = a₁₁a₂₂ − a₁₂a₂₁, Tr(J) = a₁₁ + a₂₂. If Det(J) > 0 and Tr(J) > 0, the equilibrium point is unstable; if Det(J) > 0 and Tr(J) < 0, the equilibrium point is stable. Under other conditions, the equilibrium point is a saddle point.

Then, the above five equilibrium points are substituted into Eq. (11) to obtain Det(J) and Tr(J), and the equilibrium points of the basic model evolution game are shown in Table 4. According to Table 4, the local stability analysis of the system is then carried out.

After finding the equilibrium solution of the basic game model, the evolutionary stabilization strategies and processes of this two-dimensional dynamic game system are analyzed, and four scenarios are obtained:

Scenario 1: When Ci+C0−CP>(1+j)PC${C_i} + {C_0} - {C_P} > \left( {1 + j} \right){P_C}$ and C_e < kP_i are satisfied, the participants in the system gradually converge to the equilibrium point (0, 0). In this case, the strategy choice of both parties is (no investment, no adjustment). The specific results are analyzed as shown in Table 5.

Scenario 2: When PC<Ci+C0−CP<(1+j)PC${P_C} < {C_i} + {C_0} - {C_P} < \left( {1 + j} \right){P_C}$ and C_e > kP_i are satisfied, the participants in the system gradually converge to the equilibrium point (0, 0), and the strategy choices of the two sides are (no investment, no adjustment). The specific results are analyzed as shown in Table 6.

Scenario 3: When Ci+C0−CP>(1+j)PC${C_i} + {C_0} - {C_P} > \left( {1 + j} \right){P_C}$ and C_e > kP_i are satisfied, the participants in the system gradually converge to the equilibrium point (0, 0), and the strategy choices of the two sides are (no investment, no adjustment). The specific results are analyzed as shown in Table 7.

Scenario 4: When PC<Ci+C0−CP<(1+j)PC${P_C} < {C_i} + {C_0} - {C_P} < \left( {1 + j} \right){P_C}$ and C_e < kP_i are satisfied, there are two stabilization points (0, 0) and (1, 1) in the system, and the two parties will choose between (pitch, adjust) and (no pitch, adjust). The results are analyzed in Table 8.

In Case 1 to Case 3, (0, 0) is a stable point, i.e., the strategic choices of both parties in the system are (no construction, no adjustment), which indicates that cooperation will not be reached between the structural adjustment of the layout of basic education and the coordination of the strategy of students’ access to nearby schools. In case 4, the probability that (cast construction, adjustment) is a stable point is: 12Z=1−12(CekPr)−12(Ci+C0−PC−CPjPc)$Z = 1 - \frac{1}{2}\left( {\frac{{{C_e}}}{{k{\operatorname{P}_{\rm r}}}}} \right) - \frac{1}{2}\left( {\frac{{{C_i} + {C_0} - {P_C} - {C_P}}}{{j{\operatorname{P}_c}}}} \right)$

From equation (12), after analyzing the basic game model of basic education layout and students’ proximity to school strategy, the conditions for both sides to choose (investment and construction, adjustment) as a strategy are PC<Ci+C0−CP<(1+j)PC${P_C} < {C_i} + {C_0} - {C_P} < \left( {1 + j} \right){P_C}$ and C_e < kP_i.

As a result of population decline and mobility, “pocket schools” have appeared in some old urban areas and rural villages, and the problem of overcrowded classes has arisen in urban and rural areas, and the education administrations of cities and counties have re-planned the layout of their schools on the basis of full research, in accordance with instructions from higher levels. Through mergers, new construction, expansion and zoning adjustments, existing school sizes have been basically optimized, and the overall change in the size of elementary school in Province A over the years since 2015 is shown in Figure 3.

Figure 3.

The overall change of primary school size in Guangdong Province

As can be seen from Figure 3, the total number of elementary school and the number of students enrolled in elementary school have been increasing over the past 10 years, and the average school size has been expanding through the integration of educational resources. The average school size in 2024 is 101.59, which is a 16.74% year-on-year increase over the average school size of 87.02 in 2015.

The number of small-sized schools and teaching points has been decreasing since 2015, and the number of oversized classes has been basically controlled, and the ratio of small-sized schools and oversized classes in Province A since 2015 is shown in Figure 4.

Figure 4.

Proportion of small schools and large classes in Guangdong Province

As can be seen from Figure 4, the proportion of small-sized schools and large class sizes in Province A declined from 2015 to 2024, with the proportion of small schools with fewer than 100 students at 8.39% by 2024, the proportion of junior high schools with fewer than 300 students at 10.21%, and the proportion of large junior high school class sizes dropping to 38.82%.

Despite the remarkable achievements of school layout adjustment over the past 10 years or so, a certain percentage of schools are still small-scale, which is a realistic option for ensuring that children in remote areas can attend school close to their homes. The coexistence of small-sized schools and large class sizes in junior middle schools, with most small-sized schools located in townships and most large junior middle schools concentrated in counties, is not only due to the relative concentration of the urban population as a result of urbanization, but also, and more importantly, to the imbalance between urban and rural education, which has led to a large number of students from rural areas choosing to go to secondary schools in the city, resulting in a tightening of the number of places available in urban secondary schools.

After the layout adjustment of basic education in conjunction with the policy of students going to schools close to their homes and the integration of educational resources, the conditions of primary and secondary schools have been improved year by year. The basic conditions of primary and secondary schools in Province A are shown in Table 10.

In 2024, the average floor area of primary and secondary school students in Province A was 105.79 m², the average school building area was 10.36 m², the average teaching instrument and equipment for students was 514.91 yuan, the average books for students was 39.41, and the number of computers for 100 students was 10.45.