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

Figure 1.

Figure 2.

Figure 3.

Figure 4.

The payment matrix of the cooperative evolutionary game between them

Basic education layout structure	Cooperation	Not cooperation
Adjustment of students’ nearby admission policy
Cooperation	em + Δπm, eb + Δπb	em − c, eb − w
Not cooperation	em − w, eb − c	em, eb

The game matrix of proximity admission policy and school layout

		Implement a nearby enrollment policy
		Good to execute	Difficult to execute
School layout structure adjustment	Good to execute	1	0
		1	8
	Difficult to execute	8	3
		0	3

Evolutionary game payoff matrix

Investor and builder	Take into account student proximity admission policy
	Adjust	Not adjust
Put into construction(x)	(1 + j)Pc − Ci − C0	Pc − Ci − C0, Pc − Ci
Not put into construction(1-x)	−Cy, −Ci	Cy, 0

Stability analysis results of basic model in case 2

Equalization point	Det(J)	Tr(J)	Stability
E1(0,0)	+	-	Stable point
E2(0,1)	+	+	Unstable point
E3(1,0)	-	*	Saddle point
E4(1,1)	-	*	Saddle point
E5(x2,y2)	+	0	Saddle point

Stability analysis results of basic model in case 3

Equalization point	Det(J)	Tr(J)	Stability
E1(0,0)	+	-	Stable point
E2(0,1)	-	*	Saddle point
E3(1,0)	-	*	Saddle point
E4(1,1)	+	+	Unstable point
E5(x2,y2)	-	0	Saddle point

Standard table of the size and class size of compulsory education schools

Scale	Appropriate scale			Class size
School type	Number of classes per year	Class size	Student size
Full primary school	≤6	12-36	400-1000	≤40
Independent junior high school	4-12	20-40	650-2000	≤55
Nine-year system school	4-12	36-72	900-2400	Primary School ≤40Junior High ≤50
Full middle school	4-8	12-24	500-1200	≤50

The equilibrium point of the basic model evolutionary game

Equalization point	Det(J)	Tr(J)
E1(0,0)	(Pc+Cy−Ci−C0)(−C0)$\left( {{P_c} + {C_y} - {C_i} - {C_0}} \right)\left( { - {C_0}} \right)$	Pc + Cy − Ci − C0 − CP
E2(0,1)	(Pc+jPc+Cp−Ci−C0)(C0)$\left( {{P_c} + j{P_c} + {C_p} - {C_i} - {C_0}} \right)\left( {{C_0}} \right)$	jPc + Pi + Cy − Ci − C0 + CP
E3(1,0)	(Ci+C0−Pc−Cp)(kPC−Ci)$\left( {{C_i} + {C_0} - {P_c} - {C_p}} \right)\left( {k{P_C} - {C_i}} \right)$	Ci + C0 − Pc − Cy + kPC − Cy
E4(1,1)	(Pc+jPc+Cp−Ci−C0)(kPC−C0)$\left( {{P_c} + j{P_c} + {C_p} - {C_i} - {C_0}} \right)\left( {k{P_C} - {C_0}} \right)$	−(jPc+PC+Cy−Ci−C0+kPC−CP)$ - \left( {j{P_c} + {P_C} + {C_y} - {C_i} - {C_0} + k{P_C} - {C_P}} \right)$
E5(x2,y2)	(−CikPi)(kPC−Ci)Ci+Cp-Cy-PijPi (jPc+Pi−Ci−C0+Cy)$\begin{array}{rcl} \left( { - \frac{{{C_i}}}{{k{P_i}}}} \right)\left( {k{P_C} - {C_i}} \right)\frac{{{C_i} + {C_p} - {C_y} - {P_i}}}{{j{P_i}}} \\ \left( {j{P_c} + {P_i} - {C_i} - {C_0} + {C_y}} \right) \\ \end{array}$	0

Stability analysis results of basic model in case 1

Equalization point	Det(J)	Tr(J)	Stability
E1(0,0)	+	-	Stable point
E2(0,1)	-	*	Saddle point
E3(1,0)	+	+	Unstable point
E4(1,1)	-	*	Saddle point
E5(x2,y2)	+	0	Saddle point

Basic conditions of schools in Guangdong Province

Year	Per capita floor area (m2)	School area per student (m2)	Teaching equipment per student (Yuan)	Average book (volume)	Number of computers per 100 students (units)
2015	90.41	8.11	453.49	30.83	7.09
2016	91.53	8.47	457.12	32.78	7.18
2017	92.78	8.87	457.5	33.12	7.35
2018	97.63	9.18	464.85	34.65	7.84
2019	97.67	9.25	466.48	35.88	8.4
2020	98.01	9.38	470.94	36.16	8.89
2021	98.95	9.48	477.96	36.95	9.08
2022	99.89	9.82	484.97	37.68	9.58
2023	100.15	10.13	495.52	38.59	9.93
2024	105.79	10.36	514.91	39.41	10.45

Stability analysis results of basic model in case 4

Equalization point	Det(J)	Tr(J)	Stability
E1(0,0)	+	-	Stable point
E2(0,1)	+	+	Unstable point
E3(1,0)	+	+	Unstable point
E4(1,1)	+	-	Stable point
E5(x2,y2)	-	0	Saddle point