Moderation Under Limited Reporting: Hidden Instability in Osaka’s High School Admissions

Abstract

Osaka’s public high school admissions system uses a common score-based priority order and schools’ preferences, while restricting each student to reporting a single school. We show that the Osaka mechanism is neither priority-stable nor preference-stable, is not strategy-proof for students, and can conceal substantial instability in the admission records. Allowing students to report more schools weakly reduces underlying instability and the number of manipulable students, while making more of the remaining blocking pairs visible. Simulations show that tight reporting compresses quantitative differences across mechanisms. In the unstructured environment, the Osaka mechanism occupies an intermediate position between serial dictatorship and deferred acceptance, but this position disappears in the structured environment. The results show the benefits of broader reporting and the limits of Osaka’s compromise between score-based priority and school discretion.

1. Introduction

Public high school admissions in Osaka constitute a large centralized matching problem involving tens of thousands of students each year.1 Standardized exam scores provide a common basis for ranking applicants, while schools retain limited discretion to select students in line with their own admission policies. At this scale, however, it is impractical for schools to review every applicant in detail. The design challenge is therefore to preserve a common score-based order while giving schools a limited role in student selection.

Osaka addresses this challenge through a two-stage admission procedure.2 Each student may apply to only one school. At each school, applicants are ranked by exam score, and only the highest-ranked applicants, up to 110% of the school’s capacity, enter the review pool. The school then fills 90% of its seats in score order and allocates the remaining 10% among the other applicants in the review pool according to its own preferences. School discretion is thus limited to applicants who pass the initial score-based screening.

The procedure gives a role to both the common priority order and schools’ preferences rather than relying exclusively on either ranking. It is intended to moderate the tension between score-based priority and school discretion. This raises the question: how does this design perform when students can report only a limited number of schools?

To address this question, we formulate a many-to-one matching model with two rankings: exam scores induce a common priority order, while each school has its own preferences over students. These rankings give rise to two notions of stability. Priority stability captures score-based fairness: no student should be denied a preferred school that has a vacant seat or admits a lower-priority student. Preference stability is the classical notion based on schools’ preferences. A matching is double-stable if it satisfies both. We also consider students’ incentives to misreport their preferences. The Osaka mechanism guarantees neither form of stability, is not strategy-proof for students, and may fail to select a double-stable matching even when one exists (Proposition 1). The use of both rankings therefore does not by itself reconcile the two stability requirements or eliminate incentives to misreport.

These stability notions are defined using students’ full preferences, whereas the admissions authority observes only the school each student reports. This information gap motivates our distinction between underlying and visible instability: a blocking pair is visible if the student reports the school involved. Under the Osaka mechanism, visible priority blocking at a school is bounded by the gap between its review cap and capacity, and visible double blocking cannot arise. Visible preference blocking, by contrast, remains unbounded (Proposition 2). The mechanism therefore does not eliminate instability; single-choice reporting filters which conflicts appear in the application records.

How does this picture change when students are allowed to report more schools? To answer this question, we embed the current Osaka mechanism in a K-choice family, $O{S}^K$, in which each student reports up to K schools while the school-side admission rule remains unchanged. As K increases, the number of unmatched students weakly decreases, and the underlying sets of priority, preference, and double blocking pairs weakly shrink, while the corresponding visible sets weakly expand (Proposition 3). Broader reporting therefore moves underlying and visible instability in opposite directions: it leaves fewer students unmatched and reduces underlying instability, but brings more of the remaining conflicts into view. Broader reporting also reduces manipulability: every problem at which $O{S}^L$ is manipulable is also manipulable under $O{S}^K$ for $K<L$, while the converse need not hold (Proposition 4). Nevertheless, instability and manipulability can persist even with complete reporting (Proposition 5).

These results establish the direction of the changes, but not their magnitude. We complement the theory with simulations in both unstructured and structured environments. The simulations confirm these qualitative patterns and show that the gap between underlying and visible instability can be substantial under limited reporting. They also allow us to compare $O{S}^K$ with the K-choice versions of serial dictatorship, $S{D}^K$, and student-proposing deferred acceptance, $D{A}^K$, under the same reporting limit.3

In the unstructured environment, where student preferences are randomly generated, tight reporting compresses quantitative differences across the three mechanisms. As students are allowed to report more schools, their outcomes diverge, and $O{S}^K$ occupies an intermediate position between $S{D}^K$ and $D{A}^K$: on average, it generates fewer preference blocking pairs than $S{D}^K$ and fewer priority blocking pairs than $D{A}^K$. This stability compromise does not survive in the structured environment, where demand is shaped by geography and school quality. There, $O{S}^K$ no longer lies between $S{D}^K$ and $D{A}^K$ along either stability dimension, a finding that persists throughout the parameter variations examined. Taken together, the results point to the value of broader reporting, but also reveal the limits of Osaka’s moderation: the stability compromise observed in the unstructured environment is not robust across market environments.

Institutional Background and Related Literature

Osaka’s current admissions system dates back to a reform implemented for the 2016 admissions cycle. Before the reform, admissions were based on students’ preferences and exam scores, while schools had no formal role in selecting among applicants. The reform altered this structure by allowing schools to select a limited number of students according to their own admission policies.4 The current Osaka mechanism thus retains its score-based structure while adding a limited school-side selection channel.

Another defining feature is that each student may apply to only one public high school. Under this restriction, the admissions authority observes each student’s demand only at the school they report. Osaka is now reconsidering this restriction and exploring the introduction of a second-choice option while retaining school discretion.5

This restriction connects the Osaka mechanism to the literature on limited reporting. In two-sided matching markets, truncation has been studied as a strategic restriction of submitted preferences (Coles & Shorrer, 2014; Ehlers, 2008; Roth & Rothblum, 1999). In school choice, related work examines institutional limits on how many schools students may list or apply to (Calsamiglia et al., 2010; Haeringer & Klijn, 2009). Such limits make submitted lists less informative and affect students’ reporting incentives. These concerns also arise in Japanese high school admissions, where limited application opportunities have been discussed as a source of unfairness and strategic application behavior (Aoi et al., 2021; Noda, 2025; Tanaka et al., 2020). Our analysis highlights a different implication of limited reporting: it affects what the admissions authority can observe. Substantial instability may persist in the matching while much of it remains hidden in the submitted applications.

School-choice models emphasize students’ priorities at schools (Abdulkadiroğlu & Sönmez, 2003; Balinski & Sönmez, 1999), whereas college-admissions models emphasize schools’ preferences over students (Gale & Shapley, 1962; Roth, 1985). Our model incorporates both: exam scores generate a common priority order, while schools have their own preferences over students.6 Because the two rankings need not select the same students, priority-based fairness and preference-based stability may conflict.

Westkamp (2013) studies a closely related problem in German university admissions, where different admission quotas use grade-based priorities and universities’ own selection criteria. He develops choice protocols and establishes conditions for a student-optimal, procedurally stable mechanism that is strategy-proof for students. When the review cap is nonbinding, Osaka’s within-school selection rule can be represented as a two-quota choice protocol. The notions of stability differ, however: his procedural stability evaluates blocking according to the ranking governing each quota, whereas our two stability notions evaluate the entire matching under each ranking, irrespective of the admission quota. Our research question also differs. Motivated by the proposed second-choice option, we study broader reporting while retaining Osaka’s school-side admission rule. The $O{S}^K$ family allows us to examine how this change affects underlying and visible instability and students’ incentives to misreport.

• Roadmap. Section 2 sets up the many-to-one matching model, defines the stability and incentive notions, and introduces the benchmark mechanisms. Section 3 formalizes the Osaka mechanism and analyzes its stability, manipulability, visibility, and reporting-limit properties. Section 4 uses simulations to quantify these properties across reporting limits and market environments. Section 5 concludes. All proofs are collected in Appendix A.

2. The Model and Benchmark Mechanisms

This section sets up a many-to-one matching model with two rankings. We define the relevant stability and incentive notions and introduce serial dictatorship (SD) and deferred acceptance (DA) as benchmark mechanisms.

2.1. Environment and Stability

Let $S=\{s_1,\dots ,s_n\}$ be the set of students and $C=\{c_1,\dots ,c_m\}$ the set of schools. Each school $c\in C$ has capacity $q_c\in \mathbb{N}$. Each student $s\in S$ has a strict preference $P_s$ over $C\cup \{Ø\}$, where Ø denotes being unmatched. We write $c{P}_s{c}^{\prime }$ when s strictly prefers c to $c^{\prime }$, and let $R_s$ denote the induced weak relation. Let $P_S={\left(P_s\right)}_{s\in S}$ denote the students’ preference profile.

Students are evaluated according to two distinct rankings. Standardized exam scores induce a common strict priority order ≻ on S, where $s\succ s^{\prime }$ means that s has a strictly higher score than $s^{\prime }$. In addition, each school $c\in C$ has a strict preference $P_c$ over feasible admission sets, which we assume to be responsive (Roth, 1985).7 Let $R_c$ denote the induced weak relation. Consistent with the admissions setting we study, we assume universal acceptability on the school side: $s{P}_cØ$ for all $s\in S$ and $c\in C$. Responsiveness and universal acceptability imply that $P_c$ is monotonic on feasible sets: if $A\subseteq B\subseteq S$ and $\left|B\right|\le q_c$, then $B{R}_{c}A$. Let $P_C={\left(P_c\right)}_{c\in C}$ denote the schools’ preference profile.

A matching problem, or simply a problem, is represented by8

$$G=(S,C,q,P_S,P_C,\succ ).$$

A matching $\mu$ assigns each student $s\in S$ to $\mu \left(s\right)\in C\cup \{Ø\}$ and each school $c\in C$ to $\mu \left(c\right)\subseteq S$, such that $\left|\mu \left(c\right)\right|\le q_c$ and $\mu \left(s\right)=c$ if and only if $s\in \mu \left(c\right)$. Throughout, we restrict attention to individually rational matchings: $\mu \left(s\right)R_sØ$ for every $s\in S$.9

We distinguish three types of blocking pairs. Given a matching $\mu$, a pair $(s,c)\in S\times C$ is a priority blocking pair (SBP) if $c{P}_s\mu \left(s\right)$ and either $\left|\mu \left(c\right)\right|<q_c$ or there exists $s^{\prime }\in \mu \left(c\right)$ with $s\succ s^{\prime }$. Similarly, $(s,c)$ is a preference blocking pair (PBP) if $c{P}_s\mu \left(s\right)$ and either $\left|\mu \left(c\right)\right|<q_c$ or there exists $s^{\prime }\in \mu \left(c\right)$ with $s{P}_c{s}^{\prime }$. Finally, $(s,c)$ is a double blocking pair (DBP) if $c{P}_s\mu \left(s\right)$ and either $\left|\mu \left(c\right)\right|<q_c$ or there exists $s^{\prime }\in \mu \left(c\right)$ such that $s\succ s^{\prime }$ and $s{P}_c{s}^{\prime }$. Thus, when a school is full, a DBP requires the same admitted student to witness both the priority and preference components of blocking.

These blocking notions lead to three corresponding stability concepts.

Definition 1.

A matching μ is priority-stable if it is individually rational and admits no SBPs, preference-stable if it is individually rational and admits no PBPs, and double-stable if it is both priority-stable and preference-stable.

Priority stability captures score-based fairness (Abdulkadiroğlu & Sönmez, 2003; Balinski & Sönmez, 1999), whereas preference stability coincides with classical stability under schools’ preferences (Gale & Shapley, 1962; Roth, 1985). Double stability imposes both requirements.10 Since every DBP is both an SBP and a PBP, any priority-stable or preference-stable matching, and hence any double-stable matching, admits no DBPs.

2.2. Mechanisms and Incentives

A mechanism $\varphi$ assigns a matching $\varphi \left(G\right)$ to each problem G. We write ${\varphi }_s\left(G\right)=\varphi \left(G\right)\left(s\right)$ for student s’s assignment and ${\varphi }_c\left(G\right)=\varphi \left(G\right)\left(c\right)$ for school c’s intake. We consider strategic deviations only by students. A mechanism $\varphi$ is strategy-proof for students if, for every problem G, every student $s\in S$, and every misreport $P_s^{\prime }$,

$${\varphi }_s(P_s,P_{-s},P_C,\succ )R_s{\varphi }_s(P_s^{\prime },P_{-s},P_C,\succ ).$$

We use two canonical strategy-proof mechanisms as benchmarks: serial dictatorship (SD) and student-proposing deferred acceptance (DA). They are defined as follows.

• The SD mechanism
  • Step 1: The student with the highest priority is assigned to their most-preferred acceptable school; if no such school exists, they remain unmatched.
  • Step $t\ge 2$: The student with the t-th highest priority is assigned to their most-preferred acceptable school with remaining capacity; if no such school exists, they remain unmatched. The mechanism terminates after all students have been processed.

• The student-proposing DA mechanism
  • Step 1: Each student proposes to their most-preferred acceptable school. For each school c, let $A_c^1$ denote the set of students who propose to c in Step 1. School c tentatively accepts its $\min \{q_c,|A_c^1\left|\right\}$ most-preferred students in $A_c^1$ according to $P_c$ and rejects the rest.
  • Step $t\ge 2$: Each student rejected in Step $t-1$ proposes to their most-preferred acceptable school that has not rejected them. For each school c, let $A_c^t$ denote the set consisting of its tentatively accepted students from Step $t-1$ together with the new proposers in Step t. School c tentatively accepts its $\min \{q_c,|A_c^t\left|\right\}$ most-preferred students in $A_c^t$ according to $P_c$ and rejects the rest. The mechanism terminates when no rejection occurs, at which point all tentative acceptances become final.

Both mechanisms are strategy-proof for students. Moreover, SD is priority-stable by construction, whereas DA is preference-stable by the standard deferred-acceptance argument. Neither mechanism combines the two rankings: SD depends only on ≻, while DA depends only on $P_C$. This motivates the following distinction.

Definition 2.

A mechanism ϕ is extreme if either (i) schools’ preferences are irrelevant: $\varphi (P_S,P_C^{\prime },\succ )=\varphi (P_S,P_C,\succ )$ for every problem $(P_S,P_C,\succ )$ and every $P_C^{\prime }$, or (ii) the priority order is irrelevant: $\varphi (P_S,P_C,{\succ }^{\prime })=\varphi (P_S,P_C,\succ )$ for every problem $(P_S,P_C,\succ )$ and every ${\succ }^{\prime }$. A mechanism is moderate if it is not extreme.

Under Definition 2, SD and DA are extreme in opposite directions. The current Osaka mechanism, by contrast, is moderate: it uses both rankings under a single-choice reporting constraint. The next section examines the implications of this moderation for stability, incentives, and the structure of blocking pairs.

3. The Osaka Mechanism Under Limited Reporting

This section analyzes the Osaka mechanism and the role of limited reporting. We first examine the current single-choice mechanism and characterize the instability visible in the admission records. We then introduce a K-choice extension and study how broader reporting affects instability, visibility, and manipulability.

3.1. The Current Osaka Mechanism

For each school $c\in C$, let ${\widehat{q}}_c\in \mathbb{N}$ denote its review cap, $q_c^S\in \mathbb{N}$ its score quota, and $q_c^P\in \mathbb{N}$ its discretionary quota, where

$$q_c=q_c^S+q_c^P\mathrm{and}{\widehat{q}}_c\ge q_c.$$

Under the current mechanism, each student may apply to at most one school. Given a report profile $m={\left(m\left(s\right)\right)}_{s\in S}$, where $m\left(s\right)\in C\cup \{Ø\}$, the mechanism proceeds as follows.

• The Osaka mechanism
  • Step 1: Review and score-based admission. For each school c, let $A_c\left(m\right)=\{s\in S:m\left(s\right)=c\}$ denote its set of applicants. School c retains the $\min \{{\widehat{q}}_c,\left|A_c\left(m\right)\right|\}$ highest-priority applicants according to the common priority order ≻. Let $R_c\left(m\right)\subseteq A_c\left(m\right)$ denote the resulting review pool. Among the students in $R_c\left(m\right)$, school c admits the $\min \{q_c^S,\left|R_c\left(m\right)\right|\}$ highest-priority students. Let $S_c\left(m\right)$ denote the set of students admitted at this stage.
  • Step 2: Discretionary admission. From the remaining reviewed applicants $R_c\left(m\right)\setminus S_c\left(m\right)$, school c admits its $\min \{q_c^P,|R_c\left(m\right)\setminus S_c\left(m\right)|\}$ most-preferred students according to $P_c$. Let $D_c\left(m\right)$ denote the set of students admitted at this stage. The final intake of school c is

    $$O{S}_c^1\left(m\right)=S_c\left(m\right)\cup D_c\left(m\right).$$

Let $O{S}^1\left(m\right)$ denote the matching induced by report profile m. For a problem G, write $O{S}^1\left(G\right)$ for the matching induced by truthful reports.11 By Definition 2, $O{S}^1$ is moderate because its outcome can depend on both rankings. The following example illustrates how the mechanism operates and reveals several of its limitations.

Example 1.

Consider a market with $S=\{s_1,s_2,s_3,s_4,s_5\}$ and $C=\{a,b\}$, where

$$q=(2,2),\widehat{q}=(3,3),q^S=(1,1),q^P=(1,1).$$

Students’ preferences, schools’ preferences, and the common priority order are as follows:

$$\begin{array}{cccccccc}{P}_{s_1}& P_{s_2}& P_{s_3}& P_{s_4}& P_{s_5}& P_a& P_b& \succ \\ a& a& a& b& b& s_3& s_2& s_1\\ b& b& b& a& a& s_1& s_4& s_3\\ & & & & & s_2& s_5& s_2\\ & & & & & s_4& s_1& s_4\\ & & & & & s_5& s_3& s_5\end{array}$$

Under truthful reporting, students $s_1,s_2,s_3$ apply to a, whereas $s_4,s_5$ apply to b. At school a, all three applicants are reviewed; $s_1$ receives the score seat, while $s_3$ receives the discretionary seat because a prefers $s_3$ to $s_2$. At school b, both applicants are reviewed; $s_4$ receives the score seat and $s_5$ the discretionary seat. Hence,

$$O{S}^1\left(G\right)=\left(\begin{array}{ccccc}{s}_1& s_2& s_3& s_4& s_5\\ a& Ø& a& b& b\end{array}\right).$$

The resulting matching is neither priority-stable nor preference-stable. In particular, $(s_2,b)$ is a double blocking pair: $s_2$ prefers b to being unmatched, has higher priority than $s_5$, and is preferred by b to $s_5$. The mechanism is also manipulable: by applying to b rather than a, $s_2$ obtains a score seat at b, which they strictly prefer to being unmatched. Moreover, although a double-stable matching exists, the mechanism does not select one. The matching that assigns $s_1$ and $s_3$ to a, assigns $s_2$ and $s_4$ to b, and leaves $s_5$ unmatched is double-stable.

The example illustrates three distinct limitations of the Osaka mechanism. These are not peculiar to this particular market but reflect general properties of the mechanism.

Proposition 1.

The current Osaka mechanism $O{S}^1$ guarantees neither priority stability nor preference stability and is not strategy-proof for students. Even when a double-stable matching exists, $O{S}^1$ need not select one.

Despite these limitations, the Osaka mechanism has remained in use for about a decade. One possible explanation lies in the distinction between the underlying instability of the resulting matching and the instability revealed by the admission records. We turn to this distinction next.

3.2. Visibility Under Single-Choice Reporting

Under single-choice reporting, the admissions authority observes each student’s demand only at the school they report. Whether a student–school pair blocks a matching, however, depends on students’ preferences, schools’ preferences, and the common priority order. The admission records may therefore reveal only part of the underlying instability.

We formalize this distinction through the notion of visibility. Given a report profile m, a blocking pair $(s,c)$ under $O{S}^1\left(m\right)$ is visible if $m\left(s\right)=c$. A visible SBP, PBP, or DBP is a visible blocking pair of the corresponding type. Example 1 illustrates the distinction. The resulting matching contains exactly one blocking pair, $(s_2,b)$, which is a DBP. Since $s_2$ applies to a rather than b, this blocking pair is not visible. Thus, the matching is unstable even though no blocking pair is visible in the admission records.

More generally, the three types of blocking pairs exhibit distinct visibility properties under the current Osaka mechanism.

Proposition 2.

Under truthful reporting, the current Osaka mechanism $O{S}^1$ satisfies the following properties:

(i) 

For each school $c\in C$, the number of visible SBPs involving c is at most ${\widehat{q}}_c-q_c$, and hence, the total number of visible SBPs is at most ${\sum }_{c\in C}({\widehat{q}}_c-q_c)$.

(ii) 

No visible DBP can arise.

(iii) 

Even with a fixed school set C and fixed policy parameters $(\widehat{q},q^S,q^P)$, the number of visible PBPs can be arbitrarily large as the number of students increases.

Proposition 2 shows that single-choice reporting selectively restricts the visibility of instability. Visible SBPs are bounded by the difference between the review caps and school capacities, and this bound can be attained (see Example A1 in Appendix A). No DBP can be visible, whereas visible PBPs can be arbitrarily numerous. The instability revealed by the admission records may therefore differ from the underlying instability of the matching. In particular, limited visible instability need not imply that the resulting matching is close to stable.

These results suggest that the reporting constraint shapes not only the resulting matching but also the relationship between underlying and visible instability. Allowing students to report more schools may change both the matching itself and which blocking relationships are revealed in the admission records. This raises a natural question: how do the matching and the visibility of instability change as the reporting constraint is relaxed?

3.3. Relaxing the Reporting Constraint

We extend the Osaka mechanism by allowing each student to report up to K schools while leaving the school-side admission rule unchanged. Fix $K\in \{1,\dots ,|C\left|\right\}$. Each student $s\in S$ may submit an ordered list $m_s=(m_s^1,\dots ,m_s^{{\mathsf{\ell }}_s})$ of distinct acceptable schools, where ${\mathsf{\ell }}_s\le K$. Let $m={\left(m_s\right)}_{s\in S}$ denote the report profile. Given m, the mechanism proceeds as follows.

• The K-choice Osaka mechanism
  • Let $U^{r-1}\left(m\right)$ denote the set of students who remain unmatched at the beginning of round r, with $U^0\left(m\right)=S$. In each round $r=1,\dots ,K$, every student $s\in U^{r-1}\left(m\right)$ with ${\mathsf{\ell }}_s\ge r$ applies to $m_s^r$. For each school $c\in C$, its round-r applicants are

    $$A_c^r\left(m\right)=\left\{s\in U^{r-1}\left(m\right):{\mathsf{\ell }}_s\ge r\mathrm{and}{m}_s^r=c\right\}.$$
  • At the beginning of round r, school c’s remaining review capacity, score quota, and discretionary quota are

    $$\begin{array}{c}{\widehat{q}}_c^r\left(m\right)={\widehat{q}}_c-{\displaystyle \sum _{\tau <r}}\left|R_c^{\tau }\left(m\right)\right|,\\ q_c^{S,r}\left(m\right)=q_c^S-{\displaystyle \sum _{\tau <r}}\left|S_c^{\tau }\left(m\right)\right|,q_c^{P,r}\left(m\right)=q_c^P-{\displaystyle \sum _{\tau <r}}\left|D_c^{\tau }\left(m\right)\right|,\end{array}$$

    where an empty sum is zero and the sets $R_c^{\tau }\left(m\right)$, $S_c^{\tau }\left(m\right)$, and $D_c^{\tau }\left(m\right)$ are defined recursively below.
  • Step 1: Review and score-based admission. School c ranks the students in $A_c^r\left(m\right)$ according to the common priority order ≻ and reviews the top $\min \{{\widehat{q}}_c^r\left(m\right),\left|A_c^r\left(m\right)\right|\}$ applicants, who form the review pool $R_c^r\left(m\right)$. It then admits the top $\min \{q_c^{S,r}\left(m\right),\left|R_c^r\left(m\right)\right|\}$ students in $R_c^r\left(m\right)$ according to ≻, forming $S_c^r\left(m\right)$.
  • Step 2: Discretionary admission. School c admits the top $\min \{q_c^{P,r}\left(m\right),|R_c^r\left(m\right)\setminus S_c^r\left(m\right)|\}$ students in $R_c^r\left(m\right)\setminus S_c^r\left(m\right)$ according to $P_c$, forming $D_c^r\left(m\right)$.
  • Termination. Admissions are final. At the end of round r, the unmatched students are

    $$U^r\left(m\right)=U^{r-1}\left(m\right)\setminus \bigcup _{c\in C}\left(S_c^r\left(m\right)\cup D_c^r\left(m\right)\right).$$
  • Each student $s\in U^r\left(m\right)$ proceeds to round $r+1$ if $r<K$ and ${\mathsf{\ell }}_s\ge r+1$. Unused review capacity and unfilled score and discretionary quotas carry over to the next round. The mechanism terminates after round K, or earlier if no unmatched student has another reported school. The final intake of school c is

    $$O{S}_c^K\left(m\right)={\displaystyle \bigcup _{r=1}^K}\left(S_c^r\left(m\right)\cup D_c^r\left(m\right)\right).$$

For each student $s\in S$, let $P_s^K$ denote the truthful K-truncation of $P_s$: the ordered list of their K most-preferred acceptable schools, or all acceptable schools if there are fewer than K. Let $O{S}^K\left(G\right)$ denote the outcome under the report profile ${\left(P_s^K\right)}_{s\in S}$. When $K=1$, $O{S}^1$ coincides with the current Osaka mechanism, whereas $K=\left|C\right|$ allows students to report all acceptable schools. Across the family ${\left\{O{S}^K\right\}}_{K=1}^{\left|C\right|}$, the common priority order, schools’ preferences, review capacities, score quotas, discretionary quotas, and the within-round admission rule remain unchanged; only the reporting limit varies.12

The following example illustrates how the extension operates as the reporting limit is relaxed.

Example 2.

Consider a market with $S=\{s_1,s_2,s_3,s_4,s_5\}$ and $C=\{a,b\}$, where

$$q=(2,2),\widehat{q}=(3,3),q^S=(1,1),q^P=(1,1).$$

Students’ preferences, schools’ preferences, and the common priority order are given by

$$\begin{array}{cccccccc}{P}_{s_1}& P_{s_2}& P_{s_3}& P_{s_4}& P_{s_5}& P_a& P_b& \succ \\ a& a& a& a& b& s_3& s_5& s_1\\ b& b& b& b& a& s_1& s_4& s_2\\ & & & & & s_2& s_2& s_3\\ & & & & & s_4& s_3& s_4\\ & & & & & s_5& s_1& s_5\end{array}$$

Under $K=1$, students $s_1,\dots ,s_4$ apply to a, whereas $s_5$ applies to b. School a admits $s_1$ through the score channel and $s_3$ through the discretionary channel, while school b admits only $s_5$. Hence,

$$O{S}^1\left(G\right)=\left(\begin{array}{ccccc}{s}_1& s_2& s_3& s_4& s_5\\ a& Ø& a& Ø& b\end{array}\right).$$

Under $K=2$, the first round is unchanged. In the second round, the unmatched students $s_2$ and $s_4$ apply to b. School b has two units of review capacity and one discretionary seat remaining. It reviews both students and admits $s_4$, whom it prefers to $s_2$. Hence,

$$O{S}^2\left(G\right)=\left(\begin{array}{ccccc}{s}_1& s_2& s_3& s_4& s_5\\ a& Ø& a& b& b\end{array}\right).$$

Thus, the additional round allows $s_4$, who remains unmatched under $O{S}^1$, to apply to and be admitted by their second-choice school b, while all first-round assignments are preserved.

The notion of visibility extends naturally to K-choice reporting. A blocking pair $(s,c)$ under $O{S}^K\left(m\right)$ is visible if c appears on student s’s submitted list $m_s$. For $K=1$, this definition coincides with that in Section 3.2. For each problem G, let ${\mathrm{SBP}}^K\left(G\right)$, ${\mathrm{PBP}}^K\left(G\right)$, and ${\mathrm{DBP}}^K\left(G\right)$ denote the sets of SBPs, PBPs, and DBPs under $O{S}^K\left(G\right)$, respectively, and let ${\mathrm{VSBP}}^K\left(G\right)$, ${\mathrm{VPBP}}^K\left(G\right)$, and ${\mathrm{VDBP}}^K\left(G\right)$ denote their visible subsets.

Example 2 also illustrates how broader reporting affects underlying and visible instability. Under $O{S}^1$, the SBPs are $\{(s_2,a),(s_2,b),(s_4,b)\}$, while the PBPs and DBPs are both $\{(s_2,b),(s_4,b)\}$. Because $s_2$ and $s_4$ report only a, the only visible blocking pair is $(s_2,a)$. Under $O{S}^2$, the SBPs are $\{(s_2,a),(s_2,b)\}$, while no PBP or DBP remains. Both SBPs are visible because $s_2$ reports both a and b. Thus, in this example, broader reporting reduces underlying instability while making more of the remaining instability visible. The following proposition establishes these set-inclusion relations generally.

Proposition 3.

Fix a problem G. Under truthful reporting, for every $K<\left|C\right|$, the following statements hold.

(i) 

For every school $c\in C$, $O{S}_c^K\left(G\right)\subseteq O{S}_c^{K+1}\left(G\right)$. Moreover, every student assigned under $O{S}^K\left(G\right)$ receives the same assignment under $O{S}^{K+1}\left(G\right)$. Hence, the number of unmatched students is weakly decreasing in K.

(ii) 

For each $B\in \{\mathrm{SBP},\mathrm{PBP},\mathrm{DBP}\}$, $B^{K+1}\left(G\right)\subseteq B^K\left(G\right)$. Hence, the numbers of priority, preference, and double blocking pairs are weakly decreasing in K.

(iii) 

For each $B\in \{\mathrm{VSBP},\mathrm{VPBP},\mathrm{VDBP}\}$, $B^K\left(G\right)\subseteq B^{K+1}\left(G\right)$. Hence, the numbers of visible priority, preference, and double blocking pairs are weakly increasing in K.

Increasing the reporting limit from K to $K+1$ allows one additional round without changing the first K rounds. Students already admitted therefore keep their assignments, while previously unmatched students may obtain a school through an additional choice. This assignment-preservation property underlies Proposition 3: the underlying blocking sets weakly shrink, while their visible subsets weakly expand.

The same assignment-preservation property holds under the K-choice immediate acceptance mechanism ($I{A}^K$).13 We consider two variants: $I{A}_{\succ }^K$, which uses the common priority order ≻, and $I{A}_P^K$, which uses each school’s preference ranking $P_c$. Proposition A1 in Appendix A shows that both variants satisfy the same three reporting-limit monotonicities as $O{S}^K$. By contrast, $S{D}^K$ and $D{A}^K$ need not preserve earlier assignments, and their visible blocking sets need not expand as the reporting limit increases; see Example A2 in Appendix A.

Remark 1 (Assignment preservation).

The three reporting-limit monotonicities hold for $O{S}^K$ and both IA variants, reflecting their common assignment-preservation property. By contrast, $S{D}^K$ and $D{A}^K$ need not preserve earlier assignments.

The decline in total blocking does not reveal whether blocking falls only because vacant seats are filled, or whether fewer pairs also block at filled schools. We therefore distinguish blocking through vacant seats from blocking through displacement at filled schools. Let

$$W^K\left(G\right)=\{(s,c)\in S\times C:c{P}_{s}O{S}_s^K\left(G\right)\mathrm{and}|O{S}_c^K\left(G\right)|<q_c\}$$

denote the vacant-seat blocking pairs. For each $B\in \{\mathrm{SBP},\mathrm{PBP},\mathrm{DBP}\}$, let

$${\overline{B}}^K\left(G\right)=B^K\left(G\right)\setminus W^K\left(G\right)$$

denote the displacement blocking pairs at filled schools.

The vacant-seat blocking sets weakly shrink as reporting expands: $W^{K+1}\left(G\right)\subseteq W^K\left(G\right)$.14 By contrast, displacement blocking sets need not be nested: for each $B\in \{\mathrm{SBP},\mathrm{PBP},\mathrm{DBP}\}$, neither

$${\overline{B}}^{K+1}\left(G\right)\subseteq {\overline{B}}^K\left(G\right)\mathrm{nor}{\overline{B}}^K\left(G\right)\subseteq {\overline{B}}^{K+1}\left(G\right)$$

holds in general. Filling a vacant seat can leave a pair blocking through displacement: in Example 2, $(s_2,b)$ blocks through a vacancy under $O{S}^1$ but remains an SBP through displacement of $s_4$ under $O{S}^2$. Displacement blocking pairs can also disappear. Example A3 in Appendix A establishes both directions of non-inclusion for all three blocking types.

Remark 2 (Composition of underlying blocking).

The total blocking sets weakly shrink as K increases, but individual displacement blocking pairs can emerge and disappear.

Next, we turn from instability to manipulability. For a problem G and a reporting limit $K\le \left|C\right|$, $O{S}^K$ is manipulable at G if there exist a student $s\in S$ and a feasible K-choice report $m_s^{\prime }$ such that

$$O{S}_s^K(m_s^{\prime },P_{-s}^K,P_C,\succ )P_{s}O{S}_s^K(P_s^K,P_{-s}^K,P_C,\succ ).$$

Thus, some student can profitably deviate from truthful K-choice reporting while all other students report truthfully.

Following Pathak and Sönmez (2013) and Chen and Kesten (2017), we compare mechanisms by the set of problems at which they are manipulable.

Proposition 4.

For any $1\le K<L$, $O{S}^K$ is more manipulable than $O{S}^L$. That is, over the common domain of problems with at least L schools, every problem at which $O{S}^L$ is manipulable is also manipulable under $O{S}^K$, while the converse fails for at least one problem.

Lemma A2 shows that any school obtainable through a deviation can also be obtained by reporting only that school as acceptable. Such a report is considered in the first round, when the other students’ applications are unchanged across reporting limits. Its outcome therefore does not depend on the reporting limit. Broader truthful reporting, however, may improve a student’s assignment. Thus, any student who can profitably manipulate under a longer reporting limit can also do so under a shorter one, while a previously profitable deviation may cease to be profitable as truthful assignments improve.

Example 2 illustrates this point. Under $O{S}^1$, $s_4$ remains unmatched when reporting truthfully but obtains b by reporting it instead of a. Under $O{S}^2$, they obtain b truthfully in the second round, so this deviation is no longer profitable. The proof of Proposition 4 also yields the following student-level implication.

Remark 3 (Manipulable students).

For a problem G, let $M^K\left(G\right)$ denote the set of students who can profitably manipulate $O{S}^K$. Then, for every $1\le K<L\le \left|C\right|$,

$$M^L\left(G\right)\subseteq M^K\left(G\right).$$

Hence, the number of manipulable students is weakly decreasing in the reporting limit.

The preceding results show that relaxing the reporting limit weakly reduces both underlying instability and manipulability. This raises a natural question: can sufficiently broad reporting eliminate the failures identified for the current mechanism in Proposition 1? The answer is no. These failures can persist even when students can report all acceptable schools and the review constraint is nonbinding.

Proposition 5.

For every reporting limit $K\ge 1$, there exists a problem at which $O{S}^K$ is neither priority-stable nor preference-stable, is manipulable, and fails to select a double-stable match even though one exists. Moreover, for every $K\ge 2$, such a problem can be constructed with

$$K=\left|C\right|\mathit{and}{\widehat{q}}_c=\left|S\right|\mathit{for}\mathit{every}c\in C.$$

The theoretical results establish the effects of relaxing the reporting constraint and their limitations, but not their quantitative magnitude. They do not indicate how rapidly blocking pairs or manipulability decline as the reporting limit expands, how much of the remaining instability becomes visible, or how the Osaka mechanism compares with SD and DA under the same reporting limit. We therefore turn to simulations.

4. Simulation Analysis

This section uses simulations to quantify the effects established in Section 3 and to compare the Osaka mechanism with SD and DA under a common reporting limit. We examine how outcomes vary with the reporting limit and market environment, and then study how differences across the three mechanisms change with effective demand for public-school seats.

4.1. Simulation Design

Each simulated market consists of $N=1000$ students and $M=10$ schools. Every school c has capacity $q_c=100$, so aggregate capacity equals the number of students. This provides a balanced baseline in which differences across mechanisms are not driven by aggregate excess demand or excess capacity. To mirror the current Osaka system, we set the review cap to ${\widehat{q}}_c=110$, the score quota to $q_c^S=90$, and the discretionary quota to $q_c^P=10$ at every school. In all simulations, every school is acceptable to every student.

For each $K\in \{1,\dots ,M\}$, every student truthfully reports their K most-preferred schools. We apply $O{S}^K$, $S{D}^K$, and $D{A}^K$ to the same submitted lists.15 Within each simulated market, the three mechanisms face the same student preferences, school preferences, and common priority order. We also hold the underlying market fixed across values of K. Thus, variation across K isolates the effect of relaxing the reporting constraint, while comparisons at a given K isolate differences across allocation rules.

We evaluate each assignment along four dimensions. First, we count SBPs, PBPs, and DBPs under students’ full preferences. Second, we count the visible blocking pairs of each type, using the submitted lists to determine visibility. Third, we record the number of unmatched students. Finally, we measure strategic vulnerability by the number of students who can obtain a strictly preferred assignment through a unilateral misreport.

Directly checking all feasible reports is computationally costly. The following result reduces this problem to a sequence of single-school deviations.

Proposition 6.

Fix K and a mechanism ${\varphi }^K\in \{O{S}^K,S{D}^K,D{A}^K\}$, and let μ denote the assignment under truthful K-choice reporting. Student s is manipulable if and only if there exists a school $c^{\prime }$ such that $c^{\prime }{P}_s\mu \left(s\right)$ and they are assigned to $c^{\prime }$ when reporting only $c^{\prime }$ as acceptable.

Proposition 6 implies that, for each school that a student prefers to their truthful assignment, it is enough to check whether reporting that school alone assigns them to it. The simulation therefore requires at most M counterfactual checks per student rather than a search over all feasible K-choice reports.

4.2. The Unstructured Environment

We begin with an unstructured environment in which students’ preferences, schools’ preferences, and the common priority order are generated independently from uniform draws and converted into strict rankings. With no common school-quality component or geographic sorting, student demand is dispersed across schools. This environment provides a benchmark in which the three rankings are otherwise unrelated.

Figure 1 reports average outcomes over 200 simulated markets as the reporting limit K varies from 1 to 10. Panels (a)–(f) report total and visible SBPs, PBPs, and DBPs, while panels (g) and (h) report manipulable and unmatched students. Three findings emerge.

Figure 1. Outcomes in the unstructured environment. Panels (a–f) report total and visible SBPs, PBPs, and DBPs; panels (g,h) report manipulable and unmatched students. Theory guarantees zero visible SBPs under SD, visible PBPs under DA, and visible DBPs under both for all K, as well as zero visible DBPs under OS at $K=1$. At $K=10$, theory also guarantees zero total SBPs under SD, total PBPs under DA, and total DBPs and manipulable students under both. All mechanisms have zero unmatched students at $K=10$. In all 200 simulated markets, DA also has exactly zero total PBPs, total DBPs, and manipulable and unmatched student counts at $K=8,9$.

F1. 

Tight reporting compresses quantitative differences across mechanisms. When students report only one school, differences across the three mechanisms are relatively small across most outcomes. As K increases, their outcomes diverge and the differences become more pronounced. Tight reporting therefore compresses the quantitative differences generated by the three mechanisms.

F2. 

Broader reporting reduces underlying instability while increasing its visibility. As K increases, total SBPs, PBPs, and DBPs decline; fewer students remain unmatched; and visible blocking pairs generally increase before leveling off. For $O{S}^K$, these movements accord with Proposition 3. Although the corresponding monotonicity results do not generally hold for $S{D}^K$ and $D{A}^K$, the same qualitative pattern emerges for all three mechanisms in this environment. Thus, tighter reporting is associated with greater underlying instability but reveals less of it in the admission records.

F3. 

The Osaka mechanism occupies an intermediate position between SD and DA along the two stability dimensions. As reporting becomes broader, $O{S}^K$ generates fewer SBPs than $D{A}^K$ and fewer PBPs than $S{D}^K$, placing it between the two benchmark mechanisms along these dimensions. This intermediate position does not extend to double blocking or incentives: DBPs and manipulability remain substantial under $O{S}^K$, whereas both decline toward zero under $S{D}^K$ and $D{A}^K$.

To examine how OS’s blocking changes with K, we use the decomposition in Section 3.3 into vacant-seat blocking, $W^K\left(G\right)$, and displacement blocking, ${\overline{B}}^K\left(G\right)$. Figure A1 in Appendix A shows that the mean counts of both components decline as K increases. The decline in total blocking therefore includes a reduction in displacement blocking, not just vacant-seat blocking. This finding is consistent with Remark 2: a decline in mean displacement blocking counts does not imply that the corresponding sets are nested. With complete reporting, vacant-seat blocking is exactly zero, and all remaining blocking involves displacement at filled schools.

4.3. The Structured Environment

Student demand is now shaped by geography and school quality: students favor nearby schools, while elite schools enjoy a common quality advantage.16 The locations of students and schools are drawn independently and uniformly on the unit square, and $d(s,c)$ denotes the Euclidean distance between student s and school c. Two schools are designated elite, with quality $v_c=5$, while the remaining eight have quality $v_c=3$. Student s’s utility from school c is

$$u_{s,c}=-\delta d(s,c)+v_c+{\epsilon }_{s,c},{\epsilon }_{s,c}\sim U(0,\gamma ),$$

where the taste shocks are independent across student–school pairs. Students rank schools in decreasing order of $u_{s,c}$. The parameters $\delta$ and $\gamma$ control the weight on distance and the magnitude of taste shocks, respectively.

In the baseline, we set $\delta =15$ and $\gamma =1.5$. With equal taste shocks, the elite-school quality advantage offsets $2/15\simeq 0.133$ additional distance units, approximately 13.3% of the unit square’s side length. Since the taste-shock range of $1.5$ is smaller than the quality gap of 2, tastes cannot reverse the elite–regular ordering at equal distances but can generate different rankings of schools with the same quality.

Schools’ preferences are correlated with the common priority order. Let ${\pi }_s\in \{1,\dots ,N\}$ denote student s’s score-based priority, with N assigned to the highest-priority student and 1 to the lowest. School c evaluates student s according to

$$e_{c,s}={\pi }_s+{\eta }_{c,s},{\eta }_{c,s}\sim U(0,\lambda N),$$

where the school-specific shocks are independent across student–school pairs. Schools rank students in decreasing order of $e_{c,s}$. The parameter $\lambda$ controls departures from the common priority order; when $\lambda =0$, every school’s ranking coincides with that order.

In the baseline, we set $\lambda =0.2$. School-specific evaluations can reverse two students’ relative order only if they are fewer than $0.2N$ priority positions apart—fewer than 200 positions when $N=1000$. Thus, schools can reorder students who are close in priority, while larger priority differences are preserved.

Figure 2 reports average outcomes over 200 simulated markets under the baseline specification $(\delta ,\gamma ,\lambda )=(15,1.5,0.2)$, as the reporting limit K varies from 1 to 10. Market size, capacities, and the policy parameters remain as specified in Section 4.1. Panels (a)–(f) report total and visible SBPs, PBPs, and DBPs, while panels (g) and (h) report manipulable and unmatched students. Two findings emerge.

Figure 2. Outcomes in the structured baseline environment. Panels (a–f) report total and visible SBPs, PBPs, and DBPs; panels (g,h) report manipulable and unmatched students. Vertical scales differ across panels and from the corresponding panels in Figure 1. Theory guarantees zero visible SBPs under SD, visible PBPs under DA, and visible DBPs under both for all K, as well as zero visible DBPs under OS at $K=1$. At $K=10$, theory also guarantees zero total SBPs under SD, total PBPs under DA, and total DBPs and manipulable students under both. All mechanisms have zero unmatched students at $K=10$.

F4. 

The structured environment magnifies the consequences of limited reporting. Relative to the unstructured environment, total blocking pairs, manipulability, and the number of unmatched students are substantially higher, especially when reporting is tight. Broader reporting reduces all three, but substantial blocking and manipulability remain under $O{S}^K$ even for large values of K. In particular, manipulability under $O{S}^K$ falls sharply from $K=1$ to $K=2$ but changes little thereafter, even as the number of unmatched students continues to decline toward zero.

F5. 

Osaka’s intermediate position disappears in the structured environment. The stability ordering observed in the unstructured environment no longer holds. For every $K>1$, $O{S}^K$ generates, on average, more SBPs than $D{A}^K$ and more PBPs than $S{D}^K$. It therefore no longer lies between the two benchmark mechanisms along either stability dimension. The intermediate position observed in Section 4.2 is thus not robust to the structured environment considered here.

To examine whether F4 and F5 hold when parameter values change, we vary one parameter at a time: $\delta \in \{10,15,20\}$, $\gamma \in \{0.75,1.5,3\}$, and $\lambda \in \{0.1,0.2,0.4\}$, holding all others at baseline. This yields seven environments, including the baseline, each with 200 simulated markets.17 Figure A2 and Figure A3 and Table A1 in Appendix A confirm F5’s average ordering in all seven environments: for every $K>1$, $O{S}^K$ generates more SBPs than $D{A}^K$ and more PBPs than $S{D}^K$. F4’s higher blocking and manipulability also persist across all seven environments under tight reporting (e.g., $K=1,2$).

With broader reporting, F4’s increase in blocking persists under OS but has exceptions under DA and SD. At $K=10$, OS’s mean counts across the seven structured environments range from 458 to 513 SBPs and from 431 to 554 PBPs, compared with about 79 and 94, respectively, in the unstructured environment. The mean number of manipulable students ranges from 138 to 157, compared with about 18. At $\lambda =0.1$, however, DA generates fewer SBPs on average than its unstructured counterpart at $K=8,9,10$, and SD generates slightly fewer PBPs at $K=10$.18 These checks vary one parameter at a time; simultaneous parameter changes and weakly aligned school rankings remain unexplored.

4.4. Demand Comparative Statics

We finally examine whether differences across mechanisms depend on the level of demand relative to available capacity. This comparative static is also relevant to Osaka, where tuition-support policies may make private schools more attractive and thereby reduce demand for public-school seats.19 We take these demand changes as given and examine their implications for the three mechanisms.

We retain the structured environment of Section 4.3 and vary the number of students across $N\in \{400,500,\dots ,1000\}$. The number of schools and their capacities remain fixed at $M=10$ and $q_c=100$, so aggregate capacity remains at 1000 seats. We simulate 200 markets at each demand level. For each simulated market, lower-demand markets are generated by randomly removing students from the corresponding 1000-student market. Priority ranks are recomputed among the retained students, and the scale of school-specific shocks is adjusted with N as in Section 4.3.

At each demand level, students truthfully report their top K schools. We consider two reporting limits: $K=1$, corresponding to single-choice reporting, and $K=M=10$, corresponding to complete reporting. Figure 3 reports total SBPs, PBPs, DBPs, and manipulable students under single-choice reporting.

Figure 3. Outcomes under single-choice reporting ($K=1$). The panels report total SBPs, PBPs, DBPs, and manipulable students as N varies from 400 to 1000. All reported means are strictly positive.

F6. 

Single-choice reporting keeps mechanism differences compressed across demand levels. Lower demand reduces blocking and manipulability under all three mechanisms, but differences across mechanisms remain small throughout the demand range. Thus, the compression associated with single-choice reporting persists across substantial changes in demand relative to capacity.

Figure 4 repeats the demand comparison under complete reporting.

Figure 4. Outcomes under complete reporting ($K=M=10$). The panels report total SBPs, PBPs, DBPs, and manipulable students as N varies from 400 to 1000. At every demand level, SD’s SBP counts, DA’s PBP counts, and both mechanisms’ DBP and manipulable-student counts are exactly zero, as guaranteed by stability and strategy-proofness under complete reporting.

F7. 

Under complete reporting, mechanism differences widen as demand approaches capacity. The three mechanisms diverge as demand approaches aggregate capacity. $S{D}^{10}$ eliminates SBPs, while $D{A}^{10}$ eliminates PBPs, and both eliminate DBPs and manipulability. By contrast, $O{S}^{10}$ continues to generate all four types of distortion, with the differences becoming increasingly pronounced as demand rises.

5. Conclusions

This paper studies public high school choice in Osaka as a moderate mechanism operating under limited reporting. The analysis focuses on two features of the current system: the use of both exam scores and school discretion in admissions, and the restriction on how many schools students may report.

Single-choice reporting affects not only the resulting matching but also which conflicts are visible in the admission records. Under the current mechanism, substantial underlying instability may remain hidden even when visible instability is limited. As the reporting constraint is relaxed, the underlying sets of priority, preference, and double blocking pairs weakly shrink, while their visible counterparts weakly expand. Broader reporting also weakly reduces the set of manipulable students, although instability and manipulability can persist even with complete reporting. The simulations further show that tight reporting compresses quantitative differences across mechanisms.

Allowing both exam scores and school discretion to influence admissions does not by itself produce a robust compromise between the two stability criteria. In the unstructured environment, $O{S}^K$ occupies an intermediate position between $S{D}^K$ and $D{A}^K$: it generates fewer preference blocking pairs than $S{D}^K$ and fewer priority blocking pairs than $D{A}^K$. This intermediate position disappears in the structured environment. The stability compromise observed under dispersed demand therefore depends on the market environment rather than on the use of both rankings alone.

These findings are directly relevant to Osaka’s ongoing reform. The results support expanding the number of schools that students may report: broader reporting reduces underlying instability, even though it makes more of the remaining conflicts visible. Support for broader reporting should not be interpreted as support for the current mechanism itself: its compromise between score-based priority and school discretion is not robust across market environments.