# School Choice in Guangzhou: Why High-Scoring Students Are Protected?

## Abstract

**:**

## 1. Introduction

- The Chinese parallel (CP) mechanism (high school admission version):
- Students are ranked from top to bottom with respect to their test scores, and they are allowed to list at most e schools within each choice band. For example, if the preferences of a student are given by an ordering ${c}_{1},{c}_{2},{c}_{3},{c}_{4}$, and the length of choice band is $e=2$, then the first choice band contains ${c}_{1}$ (top choice) and ${c}_{2}$ (second choice), and the second choice band contains ${c}_{3}$ (third choice) and ${c}_{4}$ (fourth choice). The CP mechanism works in rounds. In each round i, based on students’ test scores and their e choices in the ith choice band, the clearinghouse uses a serial dictatorship (SD) mechanism which is defined in Section 2 to determine an allocation. Note that the allocation is finalized each e choices.

- High-scoring student protect policy:
- Based on students’ test scores, the clearinghouse divides the set of students into two groups: the “high-scoring group”and the “low-scoring group”. Members in the high-scoring group receive their allocations earlier than those in the low-scoring group.

- The Guangzhou (GZ) mechanism:
- Students submit their preferences for schools at the beginning of the mechanism, and these preference submissions are not allowed to be revised later. Given a high-scoring student protection policy, the GZ mechanism works in two rounds. The clearinghouse assigns the high-scoring group to schools in round 1, and then the low-scoring group to the remaining schools in round 2. In each round, based on students’ test scores and their preferences, the IA mechanism is used to determine an allocation. Note that the allocation is finalized at each step of the IA mechanism.

## 2. The Model and the Two Mechanisms

- a set of students, $S=\{{s}_{1},\cdots ,{s}_{n}\}$;
- a set of schools, $C=\{{c}_{1},\cdots ,{c}_{m}\}$;
- a vector of school quotas, $q=({q}_{{c}_{1}},\cdots ,{q}_{{c}_{m}})$;
- a list of strict student preferences, ${P}_{S}=({P}_{{s}_{1}},\cdots ,{P}_{{s}_{n}})$; and
- a strict common test score ordering, $\succ $.

**Definition**

**1.**

**Definition**

**2.**

- The SD mechanism:
- Step 1: The student with the highest test score is assigned their top choice.
- Step k, $k\ge 2$: The student with the kth highest test score is assigned their top choice among all schools except the ones whose quotas have been filled.
- End: The mechanism stops when all students have chosen a school or all schools have filled their quotas.

- The IA mechanism (under homogeneous priorities):
- Step 1: Each student proposes to their top choice. Each school (i) considers its applicants at this step; (ii) immediately accepts those applicants up to its quota, one at a time, following the test score ordering; and (iii) rejects the remaining applicants.
- Step k, $k\ge 2$: Each student that has been rejected in the previous step proposes their kth choice. Each school (i) considers its applicants at this step; (ii) immediately accepts those applicants up to its remaining quota, one at a time, following the test score ordering; and (iii) rejects the remaining applicants.
- End: The mechanism stops when no student is rejected or all schools have filled their quotas.

## 3. Guangzhou Mechanism

**Definition**

**3.**

- The GZ mechanism with a score partition $\mathbb{P}={\left\{{S}^{i}\left(\mathbb{P}\right)\right\}}_{i\in I}$:
- Round 1: The IA mechanism is applied to assign students in ${S}^{1}\left(\mathbb{P}\right)$ to schools.
- Round i, $i\ge 2$: The IA mechanism is applied to assign students in ${S}^{i}\left(\mathbb{P}\right)$ to those remaining schools for which the quotas have been not filled.
- End: The mechanism stops when all students have matched with a school or all schools have filled their quotas.

**Remark**

**1.**

**Example**

**1.**

${P}_{{s}_{1}}$: | ${c}_{1},{c}_{2},{c}_{3},$ | ≻: | ${s}_{1},{s}_{2},{s}_{3},{s}_{4},$ |

${P}_{{s}_{2}}$: | ${c}_{2},{c}_{3},{c}_{1},$ | ||

${P}_{{s}_{3}}$: | ${c}_{2},{c}_{3},{c}_{1},$ | ||

${P}_{{s}_{4}}$: | ${c}_{3},{c}_{1},{c}_{2}$. |

**Definition**

**4.**

**Theorem**

**1.**

**Example**

**2.**

${P}_{{s}_{1}}$: | ${c}_{1},$ | ≻: | ${s}_{1},{s}_{2},{s}_{3},{s}_{4},{s}_{5},$ |

${P}_{{s}_{2}}$: | ${c}_{1},{c}_{2},{c}_{3},$ | ||

${P}_{{s}_{3}}$: | ${c}_{2},$ | ||

${P}_{{s}_{4}}$: | ${c}_{1},{c}_{3},$ | ||

${P}_{{s}_{5}}$: | ${c}_{3}.$ |

**Lemma**

**1.**

**Corollary**

**1.**

**Theorem**

**2.**

**Remark**

**2.**

**Remark**

**3.**

**Example**

**3.**

${P}_{{s}_{1}}$: | ${c}_{1},$ | ≻: | ${s}_{1},{s}_{2},{s}_{3},{s}_{4},{s}_{5},$ |

${P}_{{s}_{2}}$: | ${c}_{1},{c}_{2},$ | ||

${P}_{{s}_{3}}$: | ${c}_{1},{c}_{2},{c}_{3},{c}_{4},$ | ||

${P}_{{s}_{4}}$: | ${c}_{1},{c}_{3},$ | ||

${P}_{{s}_{5}}$: | ${c}_{4}.$ |

**Theorem**

**3.**

**Remark**

**4.**

**Example**

**4.**

${P}_{{s}_{1}}$: | ${c}_{1},$ | ≻: | ${s}_{1},{s}_{2},{s}_{3},{s}_{4},{s}_{5},{s}_{6},$ |

${P}_{{s}_{2}}$: | ${c}_{1},{c}_{2},$ | ||

${P}_{{s}_{3}}$: | ${c}_{1},{c}_{2},{c}_{3},$ | ||

${P}_{{s}_{4}}$: | ${c}_{1},{c}_{4},{c}_{2},{c}_{3},{c}_{5},{c}_{6},$ | ||

${P}_{{s}_{5}}$: | ${c}_{1},{c}_{2},{c}_{3},{c}_{4},{c}_{5},{c}_{6},$ | ||

${P}_{{s}_{6}}$: | ${c}_{1},{c}_{2},{c}_{3},{c}_{5},{c}_{4},{c}_{6}.$ |

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Theorem 1

**Proof.**

#### Appendix A.2. Proof of Lemma 1

**Proof.**

#### Appendix A.3. Proof of Theorem 2

**Proof.**

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Score Partition | Assignment |
---|---|

${\mathbb{P}}^{0}=\{\{{s}_{1},{s}_{2},{s}_{3}\},\left\{{s}_{4}\right\}\}$ | ${\mu}^{0}=\left(\genfrac{}{}{0pt}{}{{s}_{1}\phantom{\rule{4pt}{0ex}}{s}_{2}\phantom{\rule{4pt}{0ex}}{s}_{3}\phantom{\rule{4pt}{0ex}}{s}_{4}}{{c}_{1}\phantom{\rule{4pt}{0ex}}{c}_{2}\phantom{\rule{4pt}{0ex}}{c}_{3}\phantom{\rule{4pt}{0ex}}\varnothing}\right)$ |

${\mathbb{P}}^{1}=\{\{{s}_{1},{s}_{2}\},\{{s}_{3},{s}_{4}\}\}$ | ${\mu}^{1}=\left(\genfrac{}{}{0pt}{}{{s}_{1}\phantom{\rule{4pt}{0ex}}{s}_{2}\phantom{\rule{4pt}{0ex}}{s}_{3}\phantom{\rule{4pt}{0ex}}{s}_{4}}{{c}_{1}\phantom{\rule{4pt}{0ex}}{c}_{2}\phantom{\rule{4pt}{0ex}}\varnothing \phantom{\rule{4pt}{0ex}}{c}_{3}}\right)$ |

${\mathbb{P}}^{2}=\{{s}_{1},{s}_{2},{s}_{3},{s}_{4}\}$ | ${\mu}^{2}=\left(\genfrac{}{}{0pt}{}{{s}_{1}\phantom{\rule{4pt}{0ex}}{s}_{2}\phantom{\rule{4pt}{0ex}}{s}_{3}\phantom{\rule{4pt}{0ex}}{s}_{4}}{{c}_{1}\phantom{\rule{4pt}{0ex}}{c}_{2}\phantom{\rule{4pt}{0ex}}\varnothing \phantom{\rule{4pt}{0ex}}{c}_{3}}\right)$ |

${\mathbb{P}}^{3}=\{\{{s}_{1},{s}_{2}\},\left\{{s}_{3}\right\},\left\{{s}_{4}\right\}\}$ | ${\mu}^{3}=\left(\genfrac{}{}{0pt}{}{{s}_{1}\phantom{\rule{4pt}{0ex}}{s}_{2}\phantom{\rule{4pt}{0ex}}{s}_{3}\phantom{\rule{4pt}{0ex}}{s}_{4}}{{c}_{1}\phantom{\rule{4pt}{0ex}}{c}_{2}\phantom{\rule{4pt}{0ex}}{c}_{3}\phantom{\rule{4pt}{0ex}}\varnothing}\right)$ |

Score Partition | Assignment |
---|---|

${\mathbb{P}}^{1}=\left\{\{{s}_{1},{s}_{2},{s}_{3},{s}_{4},{s}_{5}\}\right\}$ | ${\mu}^{1}=\left(\genfrac{}{}{0pt}{}{{s}_{1}\phantom{\rule{4pt}{0ex}}{s}_{2}\phantom{\rule{4pt}{0ex}}{s}_{3}\phantom{\rule{4pt}{0ex}}{s}_{4}\phantom{\rule{4pt}{0ex}}{s}_{5}}{{c}_{1}\phantom{\rule{4pt}{0ex}}\varnothing \phantom{\rule{4pt}{0ex}}{c}_{2}\phantom{\rule{4pt}{0ex}}\varnothing \phantom{\rule{4pt}{0ex}}{c}_{3}}\right)$ |

${\mathbb{P}}^{2}=\{\{{s}_{1},{s}_{2},{s}_{3}\},\{{s}_{4},{s}_{5}\}\}$ | ${\mu}^{2}=\left(\genfrac{}{}{0pt}{}{{s}_{1}\phantom{\rule{4pt}{0ex}}{s}_{2}\phantom{\rule{4pt}{0ex}}{s}_{3}\phantom{\rule{4pt}{0ex}}{s}_{4}\phantom{\rule{4pt}{0ex}}{s}_{5}}{{c}_{1}\phantom{\rule{4pt}{0ex}}{c}_{3}\phantom{\rule{4pt}{0ex}}{c}_{2}\phantom{\rule{4pt}{0ex}}\varnothing \phantom{\rule{4pt}{0ex}}\varnothing}\right)$ |

${\mathbb{P}}^{3}=\{\{{s}_{1},{s}_{2}\},\left\{{s}_{3}\right\},\{{s}_{4},{s}_{5}\}\}$ | ${\mu}^{3}=\left(\genfrac{}{}{0pt}{}{{s}_{1}\phantom{\rule{4pt}{0ex}}{s}_{2}\phantom{\rule{4pt}{0ex}}{s}_{3}\phantom{\rule{4pt}{0ex}}{s}_{4}\phantom{\rule{4pt}{0ex}}{s}_{5}}{{c}_{1}\phantom{\rule{4pt}{0ex}}{c}_{2}\phantom{\rule{4pt}{0ex}}\varnothing \phantom{\rule{4pt}{0ex}}\varnothing \phantom{\rule{4pt}{0ex}}{c}_{3}}\right)$ |

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**MDPI and ACS Style**

Fang, Y. School Choice in Guangzhou: Why High-Scoring Students Are Protected? *Games* **2021**, *12*, 31.
https://doi.org/10.3390/g12020031

**AMA Style**

Fang Y. School Choice in Guangzhou: Why High-Scoring Students Are Protected? *Games*. 2021; 12(2):31.
https://doi.org/10.3390/g12020031

**Chicago/Turabian Style**

Fang, Yuanju. 2021. "School Choice in Guangzhou: Why High-Scoring Students Are Protected?" *Games* 12, no. 2: 31.
https://doi.org/10.3390/g12020031