# School Choice with Hybrid Schedules

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Basics

**School choice with hybrid schedules**is based on the standard school choice problem introduced by Balinski and Sönmez [8] and Abdulkadiroğlu and Sönmez [9]. We define this problem for a single school and focus on the assignment of students to the classroom-shift pairs in this single school. Shifts can be morning vs. afternoon or different days of a week. Usually, schools group students into two and allow the first group to come to school on two days of the week, say Monday and Tuesday, and allow the second group to come to school on the other two days of the week, say Thursday and Friday. See Table 1 in Introduction. Since the set of students enrolled to each school is predetermined, the assignment problem for each school is independent from the others. In particular, we have a single school and finite set of students denoted with s and I, respectively. Currently, all students in I are enrolled to the school s. We denote the classrooms in school s with C.

#### 2.2. Notions

**assignment**$Y\subset X$ is a subset of contracts such that

- $|{Y}_{i}|\le 1$ for all $i\in I$ and $|{Y}_{m,c}|\le \widehat{q}$ for all $c\in C$, and $m\in M$;
- $\gamma \left(\mathrm{i}\left(x\right)\right)=\gamma \left(\mathrm{i}\left({x}^{\prime}\right)\right)$ for all $x,{x}^{\prime}\in {Y}_{m,c}$, $c\in C$, and $m\in M$; and
- $|\{c\in C\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}x\in {Y}_{m,c}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\gamma \left(\mathrm{i}\left(x\right)\right)=g\}|\le {k}_{g,m}$ for all $g\in G$, and $m\in M$.

**mechanism**$\psi $ is a procedure which selects an assignment for any problem. We denote the assignment selected by mechanism $\psi $ for a given problem P with $\psi \left(P\right)$ and assignment of student i with ${\psi}_{i}\left(P\right)$.

**individually rational**if for each $i\in I$, ${Y}_{i}\phantom{\rule{0.277778em}{0ex}}{R}_{i}\phantom{\rule{0.277778em}{0ex}}{x}_{o}$ and whenever there exists $\tilde{x}\in {X}_{i}$ such that $\tilde{x}\phantom{\rule{0.277778em}{0ex}}{P}_{i}\phantom{\rule{0.277778em}{0ex}}{Y}_{i}$ then $\left|\right\{c\in {C}_{\mathrm{m}\left(\tilde{x}\right),\gamma \left(i\right)}\left(Y\right)\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{4.pt}{0ex}}|{Y}_{\mathrm{m}\left(\tilde{x}\right),c}|=\widehat{q}\left\}\right|\ge |{C}_{\gamma \left(i\right)}|$. Since students only care about the shift, $\tilde{x}\phantom{\rule{0.277778em}{0ex}}{P}_{i}\phantom{\rule{0.277778em}{0ex}}{Y}_{i}$ implies that i prefers shift $\mathrm{m}\left(\tilde{x}\right)$ to $\mathrm{m}\left({Y}_{i}\right)$, and therefore $\mathrm{m}\left(\tilde{x}\right)\ne \mathrm{m}\left({Y}_{i}\right)$. That is, whenever a student i strictly prefers contract $\tilde{x}$ to her assignment, then the number of classrooms fully filled with grade $\gamma \left(i\right)$ students at shift $\mathrm{m}\left(\tilde{x}\right)$ is at least the number of classrooms reserved for grade $\gamma \left(i\right)$. The second axiom requires no seat to be wasted.

**non-wasteful**if for each $i\in I$ whenever there exists $\tilde{x}\in {X}_{i}$ such that $\tilde{x}\phantom{\rule{0.277778em}{0ex}}{P}_{i}\phantom{\rule{0.277778em}{0ex}}{Y}_{i}$ then the following is true:

- There does not exist ${c}^{\prime}\in {C}_{\mathrm{m}\left(\tilde{x}\right),\gamma \left(i\right)}\left(Y\right)$ such that $|{Y}_{\mathrm{m}\left(\tilde{x}\right),{c}^{\prime}}|<\widehat{q}$; and
- If there exists ${c}^{\prime}\in C$ such that $|{Y}_{\mathrm{m}\left(\tilde{x}\right),{c}^{\prime}}|=0$, then $|{C}_{\mathrm{m}\left(\tilde{x}\right),\gamma \left(i\right)}\left(Y\right)|={k}_{\gamma \left(i\right),\mathrm{m}\left(\tilde{x}\right)}$.

**within-grade-fair**if whenever there exists $\tilde{x}\in {X}_{i}$ such that $\tilde{x}\phantom{\rule{0.277778em}{0ex}}{P}_{i}\phantom{\rule{0.277778em}{0ex}}{Y}_{i}$, then $j\phantom{\rule{0.277778em}{0ex}}{\succ}_{\gamma \left(i\right)}\phantom{\rule{0.277778em}{0ex}}i$ for every $j\in \left(\mathrm{i}\left({Y}_{\mathrm{m}\left(\tilde{x}\right)}\right)\cap {I}_{\gamma \left(i\right)}\right)$.

**strategy-proof**if a student i cannot be better off by misreporting her true preferences in any problem P, i.e., ${\psi}_{i}\left(P\right)\phantom{\rule{0.277778em}{0ex}}{R}_{i}\phantom{\rule{0.277778em}{0ex}}{\psi}_{i}({P}_{i}^{\prime},{P}_{-i})$ for all ${P}_{i}^{\prime}$.

## 3. Results

**Step A: Assigning students to reserved classrooms:**

- Step A.1:
- If $\overline{I}\cap {I}_{\kappa \left({c}_{1}\right)}\ne \varnothing $, then we select up to $\widehat{q}$ students from $\overline{I}\cap {I}_{\kappa \left({c}_{1}\right)}$ for classroom ${c}_{1}$ according to ${\succ}_{\kappa \left({c}_{1}\right)}$. Each selected student i for ${c}_{1}$ is added to $C{h}^{m}\left(\overline{I}\right)$ and removed from $\overline{I}$. We remove ${c}_{1}$ if some student is assigned to it.

- Step A.k:
- If $\overline{I}\cap {I}_{\kappa \left({c}_{k}\right)}\ne \varnothing $, then we select up to $\widehat{q}$ students from $\overline{I}\cap {I}_{\kappa \left({c}_{k}\right)}$ for classroom ${c}_{k}$ according to ${\succ}_{\kappa \left({c}_{k}\right)}$. Each selected student i for ${c}_{k}$ is added to $C{h}^{m}\left(\overline{I}\right)$ and removed from $\overline{I}$. We remove ${c}_{k}$ if some student is assigned to it.

**Step B: Assigning students to unfilled classrooms:**

- Step B.0
- If the number of classrooms filled with grade $g\in G$ students is ${k}_{g,m}$, then all remaining students in $\overline{I}\cap {I}_{g}$ are removed.
- Step B.1:
- If ${c}_{1}$ is removed, then we move to the next step. Otherwise, we consider grade g, if there is any, such that ${I}_{g}\cap \overline{I}\ne \varnothing $ and $g\phantom{\rule{0.277778em}{0ex}}{\pi}^{m}\phantom{\rule{0.277778em}{0ex}}{g}^{\prime}$ for all ${g}^{\prime}\ne g$ with ${I}_{{g}^{\prime}}\cap \overline{I}\ne \varnothing $. Then, we select up to $\widehat{q}$ students from $\overline{I}\cap {I}_{g}$ for ${c}_{1}$ according to ${\succ}_{g}$. Each selected student i is added to $C{h}^{m}\left(\overline{I}\right)$ and removed from $\overline{I}$. If the number of classrooms filled with grade g students is ${k}_{g,m}$, then all students in $\overline{I}\cap {I}_{g}$ are removed.

- Step B.k:
- If ${c}_{k}$ is removed, then we move to the next step. Otherwise, we consider grade g, if there is any, such that ${I}_{g}\cap \overline{I}\ne \varnothing $ and $g\phantom{\rule{0.277778em}{0ex}}{\pi}_{m}\phantom{\rule{0.277778em}{0ex}}{g}^{\prime}$ for all ${g}^{\prime}\ne g$ with ${I}_{{g}^{\prime}}\cap \overline{I}\ne \varnothing $. Then, we select up to $\widehat{q}$ students from $\overline{I}\cap {I}_{g}$ for ${c}_{k}$ according to ${\succ}_{g}$. Each selected student i is added to $C{h}^{m}\left(\overline{I}\right)$ and removed from $\overline{I}$. If the number of classrooms filled with grade g students is ${k}_{g,m}$, then all students in $\overline{I}\cap {I}_{g}$ are removed.

- a.
- There does not exist a student $j\in C{h}^{m}\left(\overline{I}\right)$ such that $\gamma \left(j\right)=\gamma \left(i\right)$ and $i{\succ}_{\gamma \left(i\right)}j$;
- b.
- Existence of a classroom c with no filled seat implies that the number of classrooms filled with grade $\gamma \left(i\right)$ students is ${k}_{\gamma \left(i\right),m}$;
- c.
- No classroom, in which some grade $\gamma \left(i\right)$ student assigned, has fewer grade $\gamma \left(i\right)$ students than $\widehat{q}$; and
- d.
- There are at least $|{C}_{\gamma \left(i\right)}|$ classrooms filled with grade $\gamma \left(i\right)$ students.

**substitutable**if for any $\overline{I}\subset I$ and $i,j\notin \overline{I}$$i\notin C{h}^{m}(\overline{I}\cup \left\{i\right\})$ implies $i\notin C{h}^{m}(\overline{I}\cup \{i,j\})$. We say a choice function for shift m, $C{h}^{m}$, satisfies

**law of aggregate demand (LAD)**if $|C{h}^{m}\left(\overline{I}\right)|\le |C{h}^{m}\left(\widehat{I}\right)|$ for any $\overline{I}\subset \widehat{I}\subseteq I$.

**Proposition**

**1.**

**Proof.**

**Generalized DA Mechanism:**

- Step 1:
- Each student $i\in I$ proposes to the best shift under ${P}_{i}$, possibly ${m}_{o}$. Let ${A}_{1}^{m}$ be the set of applicants for shift $m\in M$ in Step 1. Shift $m\in M$ tentatively holds students in $C{h}^{m}\left({A}_{1}^{m}\right)$ and rejects all students in ${A}_{1}^{m}\setminus C{h}^{m}\left({A}_{1}^{m}\right)$.

- Step h:
- Each student $i\in I$ proposes to the best shift under ${P}_{i}$, possibly ${m}_{o}$, which has not rejected her yet. Let ${A}_{h}^{m}$ be the set of applicants for shift $m\in M$ in Step h. Shift $m\in M$ tentatively holds students in $C{h}^{m}\left({A}_{h}^{m}\right)$ and rejects all students in ${A}_{h}^{m}\setminus C{h}^{m}\left({A}_{h}^{m}\right)$.

**Proposition**

**2.**

**Proof.**

- (i)
- $C{h}^{m}\left({A}^{m}\right)={A}^{m}$, and
- (ii)
- if $m\phantom{\rule{0.277778em}{0ex}}{P}_{i}\phantom{\rule{0.277778em}{0ex}}\mathrm{m}\left({\mu}_{i}\right)$, then $i\notin C{h}^{m}({A}^{m}\cup \left\{i\right\})$.

**Individual rationality:**On the contrary, suppose $\mu $ is not individually rational. Since no student proposes to an unacceptable shift, all students are assigned to either ${x}_{o}$ or some acceptable contract. Then, there exist $i\in I$ and $\tilde{x}\in {X}_{i}$ such that $\tilde{x}\phantom{\rule{0.277778em}{0ex}}{P}_{i}\phantom{\rule{0.277778em}{0ex}}{\mu}_{i}$ and $\left|\right\{c\in {C}_{\mathrm{m}\left(\tilde{x}\right),\gamma \left(i\right)}\left(\mu \right)\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{4.pt}{0ex}}|{\mu}_{c,\mathrm{m}\left(\tilde{x}\right)}|=\widehat{q}\left\}\right|<|{C}_{\gamma \left(i\right)}|$. During the application of the generalized DA mechanism, i has applied to shift $\mathrm{m}(\tilde{x})$ and she has been rejected. We consider ${A}^{\mathrm{m}\left(\tilde{x}\right)}\cup \left\{i\right\}$ under choice function $C{h}^{m\left(\tilde{x}\right)}$. Condition $\left|\right\{c\in {C}_{\mathrm{m}\left(\tilde{x}\right),\gamma \left(i\right)}\left(\mu \right)\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{4.pt}{0ex}}|{\mu}_{c,\mathrm{m}\left(\tilde{x}\right)}|=\widehat{q}\left\}\right|<|{C}_{\gamma \left(i\right)}|$ implies that either there exists ${c}^{\prime}\in {C}_{\mathrm{m}\left(\tilde{x}\right),\gamma \left(i\right)}\left(\mu \right)$ such that $\phantom{\rule{4.pt}{0ex}}|{\mu}_{c,\mathrm{m}\left(\tilde{x}\right)}|<\widehat{q}$ or $|{C}_{\mathrm{m}\left(\tilde{x}\right),\gamma \left(i\right)}\left(\mu \right)|<|{C}_{\gamma \left(i\right)}|$. In either case, by the definition of choice function $C{h}^{\mathrm{m}\left(\tilde{x}\right)}$, we have $i\in C{h}^{\mathrm{m}\left(\tilde{x}\right)}({A}^{\mathrm{m}\left(\tilde{x}\right)}\cup \left\{i\right\})$. However, this contradicts condition (ii) above.

**Non-wastefulness:**On the contrary, suppose $\mu $ is wasteful. Then, there exist $i\in I$ and $\tilde{x}\in {X}_{i}$ such that $\tilde{x}\phantom{\rule{0.277778em}{0ex}}{P}_{i}\phantom{\rule{0.277778em}{0ex}}{\mu}_{i}$ and either

- a.
- There exists ${c}^{\prime}\in {C}_{\mathrm{m}\left(\tilde{x}\right),\gamma \left(i\right)}\left(\mu \right)$ such that $|{\mu}_{{c}^{\prime},\mathrm{m}\left(\tilde{x}\right)}|<\widehat{q}$, or
- b.
- There exists ${c}^{\prime}\in C$ such that $|{\mu}_{{c}^{\prime},\mathrm{m}\left(\tilde{x}\right)}|=0$ and $|{C}_{\mathrm{m}\left(\tilde{x}\right),\gamma \left(i\right)}\left(\mu \right)|<{k}_{\gamma \left(i\right),\mathrm{m}\left(\tilde{x}\right)}$.

**Within-grade-fairness:**On the contrary suppose $\mu $ is not within-grade-fair. Then, there exist $i,j\in I$ and $\tilde{x}\in {X}_{i}$ such that $\tilde{x}\phantom{\rule{0.277778em}{0ex}}{P}_{i}\phantom{\rule{0.277778em}{0ex}}{\mu}_{i}$, $\gamma \left(i\right)=\gamma \left(j\right)$, $i{\succ}_{\gamma \left(i\right)}j$ and $\mathrm{m}\left({\mu}_{j}\right)=\mathrm{m}\left(\tilde{x}\right)$. Then, $\mathrm{m}\left(\tilde{x}\right)\phantom{\rule{0.277778em}{0ex}}{P}_{i}\phantom{\rule{0.277778em}{0ex}}\mathrm{m}\left({\mu}_{i}\right)$, $j\in {A}^{\mathrm{m}\left(\tilde{x}\right)}$, $i\notin {A}^{\mathrm{m}(\tilde{x})}$ and $i\in C{h}^{\mathrm{m}\left(\tilde{x}\right)}({A}^{\mathrm{m}\left(\tilde{x}\right)}\cup \left\{i\right\})$. This follows from the fact that choice function $C{h}^{m}$ selects students from grade $\gamma \left(i\right)$ according to ${\succ}_{\gamma \left(i\right)}$. However, this contradicts condition (ii). □

**Proposition**

**3.**

**Proof.**

${P}_{{i}_{1}}$ | ${P}_{{i}_{2}}$ | ${P}_{{i}_{3}}$ | ${P}_{{i}_{4}}$ | ${P}_{{i}_{5}}$ | ${P}_{{i}_{6}}$ | ${P}_{{j}_{1}}$ | ${P}_{{j}_{2}}$ | ${P}_{{j}_{3}}$ | ${P}_{{j}_{4}}$ | ${\succ}_{1}$ | ${\succ}_{2}$ |

${m}_{1}$ | ${m}_{1}$ | ${m}_{1}$ | ${m}_{2}$ | ${m}_{1}$ | ${m}_{1}$ | ${m}_{3}$ | ${m}_{3}$ | ${m}_{3}$ | ${m}_{3}$ | ${i}_{1}$ | ${j}_{1}$ |

${m}_{2}$ | ${m}_{2}$ | ${m}_{2}$ | ${m}_{1}$ | ${m}_{2}$ | ${m}_{3}$ | ${m}_{2}$ | ${m}_{1}$ | ${m}_{2}$ | ${m}_{1}$ | ${i}_{2}$ | ${j}_{2}$ |

${m}_{3}$ | ${m}_{3}$ | ${m}_{3}$ | ${m}_{3}$ | ${m}_{3}$ | ${m}_{2}$ | ${m}_{1}$ | ${m}_{2}$ | ${m}_{1}$ | ${m}_{2}$ | ${i}_{3}$ | ${j}_{3}$ |

${m}_{0}$ | ${m}_{0}$ | ${m}_{0}$ | ${m}_{0}$ | ${m}_{0}$ | ${m}_{0}$ | ${m}_{0}$ | ${m}_{0}$ | ${m}_{0}$ | ${m}_{0}$ | ${i}_{4}$ | ${j}_{4}$ |

${i}_{5}$ | |||||||||||

${i}_{6}$ | |||||||||||

- a.
- ${i}_{1}$, ${i}_{2}$, ${i}_{3}$ and ${i}_{6}$ are matched with contracts including ${m}_{1}$, ${i}_{4}$ and ${i}_{5}$ are matched with contracts including ${m}_{2}$, and ${j}_{1}$, ${j}_{2}$, ${j}_{3}$ and ${j}_{4}$ are matched with contracts including ${m}_{3}$; or
- b.
- ${i}_{1}$, ${i}_{2}$ and ${j}_{4}$ are matched with contracts including ${m}_{1}$, ${i}_{3}$, ${i}_{4}$ and ${j}_{3}$ are matched with contracts including ${m}_{2}$, and ${j}_{1}$, ${j}_{2}$, and ${i}_{6}$ are matched with contracts including ${m}_{3}$.

**Proposition**

**4.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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School District/Country | State | AM/PM | A/B | 3 Week Rotation | |
---|---|---|---|---|---|

Chico Unified School District | CA | X | |||

University High School ^{a} | IN | X | |||

Mid-Del Public School ^{a} | OK | X | |||

McDowell County School District ^{a} | NC | X | |||

Madison City Schools ^{b} | TN | X | |||

Charlotte-Mecklenburg Schools ^{b} | NC | X | |||

Pender County Public Schools ^{b} | NC | X | |||

Bullitt County Public Schools ^{b} | KY | X | |||

Cumberland County Public Schools ^{b} | NC | X | |||

Hamilton County Schools ^{b} | TN | X | |||

Edmonds School district ^{b} | WA | X | |||

New York City Public Schools ^{cjk} | NY | X | X | ||

Hilliard City Schools ^{c} | OH | X | |||

Toledo Public Schools ^{d} | OH | X | |||

Dobbs Ferry Schools ^{d} | NY | X | |||

Rockingham County Public Schools ^{d} | VA | X | |||

Herricks Public Schools ^{e} | NY | X | |||

Washoe County School District ^{e} | NV | X | |||

Valley Stream Central High School District ^{e} | NY | X | |||

Wilks County School ^{e} | NC | X | |||

Bellmore-Merrick Central High School District ^{d} | NY | X | |||

Greenville City Schools ^{e} | TN | X | |||

Poudre School District ^{f} | CO | X | |||

Mill Valley ^{g} | KS | X | |||

Olentangy Schools ^{h} | OH | X | |||

Columbia County School district ^{i} | GA | X | |||

Wake County Public Schools | NC | X | |||

Clark County | NV | X | |||

Turkey | X | ||||

Germany | X | ||||

Denmark | X | ||||

Norway | X | ||||

Scotland | X | ||||

Belgium ^{c} | X | ||||

Greece | X | ||||

Switzerland ^{e} | X | ||||

Japan ^{e} | X | X |

^{a}One cohort attends in-person for a full week and thually for the following week.

^{b}Cohort A attends in-person on Monday and Tuesday, while Cohort B attends in-person learning on Thursday and Friday. All students are virtual on Wednesday to allow for the school to be thoroughly cleaned.

^{c}Each cohort attends three days a week every other week.

^{d}Cohort A attends in-person on Monday and Thursday, while cohort B is in-person Tuesday and Friday. All students are virtual on Wednesday.

^{e}Cohorts alternate attending school two to three times a week.

^{f}Cohort A attends in person Monday andWednesday. While cohort B attends in person Tuesday and Thursday. All students are remote Friday.

^{g}Students attend two consecutive days in-person then the next two virtual.

^{h}Group A attends in-person on Monday, Thursday and every otherWednesday. Group B attends Tuesday, Friday and every other Wednesday.

^{i}Alternate cohort days except Friday all students have the option to attend with any earning below a C mandatory.

^{j}Students are divided into three cohorts and each cohort attends in-person twice a week on different days over a three week rotation.

^{k}New York City Public Schools’ principles choose between two plans.

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## Share and Cite

**MDPI and ACS Style**

Afacan, M.O.; Dur, U.; Harris, W.
School Choice with Hybrid Schedules. *Games* **2021**, *12*, 37.
https://doi.org/10.3390/g12020037

**AMA Style**

Afacan MO, Dur U, Harris W.
School Choice with Hybrid Schedules. *Games*. 2021; 12(2):37.
https://doi.org/10.3390/g12020037

**Chicago/Turabian Style**

Afacan, Mustafa Oğuz, Umut Dur, and William Harris.
2021. "School Choice with Hybrid Schedules" *Games* 12, no. 2: 37.
https://doi.org/10.3390/g12020037