学校選択制. Spread of school choice around the globe

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2 学校選択制 Spread of school choice around the globe

3 学校選択制 Spread of school choice around the globe School authorities take into account preferences of students/parents

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6 National Governors Association Report

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9 respects improvements of school quality

10 respects improvements of school quality

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15 Criteria SOSM Boston TTC RI in General Environments RI for Desirable Students in General Environments RI in Large Environments RI for Desirable Students in Large Environments RI in Terms of Enrollment RI of Student Quality

16 Criteria SOSM Boston TTC RI in General Environments RI for Desirable Students in General Environments RI in Large Environments RI for Desirable Students in Large Environments RI in Terms of Enrollment RI of Student Quality

17 s1 >s1 c1 >c1 sm >sm cn >cn

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19 improvement for school c

20 improvement for school c c

21 respects improvements of school quality c c

22 respects improvements of school quality c c

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24 Student-Optimal Stable

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26 c :s, s,, c : s, s,, s: c, c,, s : c, c,,

27 c :s, s,, c : s, s,, s: c, c,, s : c, c,, c c

28 c :s, s,, c : s, s,, c c s: c, c,, s : c, c,, s s c c

29 c :s, s,, c : s, s,, c c s: c, c,, s : c, c,, s: c, c,. s s c c c

30 c :s, s,, c : s, s,, c c c c s: c, c,, s : c, c,, s: c, c,. s s s s c c c

31 c :s, s,, c : s, s,, c c c c s: c, c,, s : c, c,, s: c, c,. s s s s c c

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35 Boston

36 Boston

37 Top Trading Cycles

38 Top Trading Cycles

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46 c :s, s,, c : s, s,, c c c c s: c, c,, s : c, c,, s: c, c,. s s s s c c

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53 αc(φ) φ c

54 αc(φ) φ c φ approximately RI in large environments c αc(φ) 0.

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62 s c

63 s c c s

64 s c c s

65 s c c s

66 s c c s

67 s c

68 s c s c

69 s c

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72 s c

73 s c

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80 t ( 1) t-1

81 t ( 1) and students who are kept from Step t-1 together.

82 t ( 1) and students who are kept from Step t-1 together.

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88 SOSM Boston TTC RI in Large Environments

89 SOSM Boston TTC RI in Large Environments

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95 s1 >s1 c1 >c1 sm >s2 cn >c2

96 s1 >s1 c1 >c1 sm >sm cn >c2 ( ).

97 s1 >s1 c1 >c1 sm >sm cn >cm matching µ s µs

98 s1 >s1 c1 >c1 sm >sm cn >cn µ c µc

99 s1 >s1 c1 >c1 sm >sm cn >cn mechanism

100 µ individually rational s µs s.

101 µ individually rational s µs s. stable s, c c >s µs µc qc s µc s >c s.

102 Student-Optimal Stable

103 Student-Optimal Stable t ( 1) t-1

104 Student-Optimal Stable t ( 1) and students who are kept from Step t-1 together.

105 Student-Optimal Stable t ( 1) and students who are kept from Step t-1 together.

106 Student-Optimal Stable

107 Student-Optimal Stable

108 ( c : s, s,, c : s, s,, ( c ) c s : c,, s s s : c,. s : c, c,. c

109 c : s, s,, c : s, s,, ( c c ) ( c ) c s : c,, s s s s s : c, c,. s : c, c,. c

110 c : s, s,, c : s, s,, ( c c ) ( c ) c s : c,, s s s s s : c, c,. s : c, c,. c c

111 virtually homogeneous qc c

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114 Acyclicity (Ergin) x-acyclicity (Haeringer and Klijn) Virtual Homogeneity Essential Homogeneity (Kojima) Acyclicity (Kesten) Strong x-acyclicity (Haeringer and Klijn)

115 A random market is a tuple Γ =(C, S, k, D), where k is a positive integer and D is a pair (D C, D S ) of probability distributions. Each random market induces a market by randomly generating preferences of students. D S =(p c ) c C is a probability distribution on C. Preferences of each student s are drawn independently without replacement using probability distribution D S to form the preference list of students of length k. The preference distribution of schools is completely general: D C may be any distribution (or even degenerate).

116 Definition A sequence of random markets ( Γ n ) n N is regular if there exist positive integers k, q and ˆq such that 1 k n k for all n, 2 q c ˆq for all n and c C n, 3 S n qn for all n, and 4 for all n and c C n, every s S n is acceptable to c at any realization of preferences for c at D C n. We also impose the condition that the market is sufficiently thick, i.e. that there are no super-popular schools. For example, if p c p c T for some T R for all c, c C, the market is sufficiently thick.

117 Step t: Each student s S points to her most preferred school (if any); students who do not point at any school are assigned to. Each school c C points to its most preferred student. As there are a finite number of schools and students, there exists at least one cycle, i.e. a sequence of distinct schools and students (s 1, c 1, s 2, c 2,...,s K, c K )suchthatstudents 1 points at school c 1, school c 1 points to student s 2, student s 2 points to school c 2,..., student s K points to school c K, and, finally, school c K points to student s 1. Every student s k (k =1,...,K) isassignedtotheschoolsheispointing at.

118 s1 : c 3,c 1,, c1 : s 1,s 2,s 3,s 4,, s2 : c 2,c 1,, s3 : c 3,c 1,, s4 : c 2,c 4,, c2 : s 1,s 2,..., c3 : s 4,s 3,s 2,s 1, c4 : s 4,...,.

119 s1 : c 3,c 1,, c1 : s 1,s 2,s 3,s 4,, s2 : c 2,c 1,, s3 : c 3,c 1,, s4 : c 2,c 4,, c2 : s 1,s 2,..., c3 : s 4,s 3,s 2,s 1, c4 : s 4,...,. c1

120 c1 : c s1 3 1, : s c1 1,s 2,s 3,s 4,, s 1 1,c 3,. s2 : c 2,c 1,, s3 : c 3,c 1,, s4 : c 2,c 4,, c2 : s 1,s 2,..., c3 : s 4,s 3,s 2,s 1, c4 : s 4,...,. c1

121 c1 s1 : c s1 3 1, : s c1 1,s 2,s 3,s 4,, s 1 1,c 3,. s2 : c 2,c 1,, s3 : c 3,c 1,, s4 : c 2,c 4,, c2 : s 1,s 2,..., c3 : s 4,s 3,s 2,s 1, c4 : s 4,...,. c1

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124 respects improvements in terms of enrollment c c

125 SOSM Boston TTC RI in Terms of Enrollment

126 s1 : c 3,c 1,, c1 : s 1,s 2,s 3,s 4,, s2 : c 2,c 1,, s3 : c 3,c 1,, s4 : c 2,c 4,, c2 : s 1,s 2,..., c3 : s 4,s 3,s 2,s 1, c4 : s 4,...,. c1

127 c1 : c s1 3 1, : s c1 1,s 2,s 3,s 4,, s 1 1,c 3,. s2 : c 2,c 1,, s3 : c 3,c 1,, s4 : c 2,c 4,, c2 : s 1,s 2,..., c3 : s 4,s 3,s 2,s 1, c4 : s 4,...,. c1

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129 respecting improvements for desirable students

130 SOSM Boston TTC RI for Desirable Students in General Environments RI for Desirable Students in Large Environments

131 SOSM Boston TTC RI in General Markets RI by Desirable Students in General Markets RI in Large Markets RI for Desirable Students in Large Markets RI in Terms of Enrollment