Incentives in Matching Markets: Counting and Comparing Manipulating Agents

Vulnerability to manipulation is a threat to successful matching market design. However, some manipulation is often inevitable and the mechanism designer wants to compare manipulable mechanisms and pick the best. Real-life examples include reforms in the entry-level medical labor market in the US (1998), school admissions systems in New York (2004), Chicago (2009-2010), Denver (2012), some cities in Ghana (2007-2008), and England (2005-2010). We provide a useful criterion for these design decisions: we count the number of agents with an incentive to manipulate each mechanism under consideration during these reforms, and show that this number decreased as a result of the reforms. Our conclusion is robust to further additional strategic assumptions.

by removing the harm that well informed and sophisticated students could do to others (Pathak and Sönmez (2008)). 1 Manipulability is a practical concern that has motivated numerous reforms. The U.S. entry-level medical labor market and the Boston K-12 admissions systems are examples of matching systems that have reformed their matching rules, in part, to explicitly address manipulability. Numerous other changes have been observed worldwide (Pathak and Sönmez (2013)), arguably motivated by manipulability. Meanwhile, in many applications some manipulation is inevitable. Real markets are complex and involve constraints and policy goals that create opportunities for manipulation. In twosided matching, stability is considered important, yet any stable matching mechanism is manipulable (Roth (1982)). Similarly, many school and college admissions systems restrict the number of schools that students are allowed to apply to (Haeringer and Klijn (2009), Pathak and Sönmez (2013), Fack, Grenet, and He (2019)). Unfortunately, these constrained mechanisms are also manipulable.
We propose a notion to quantify manipulation and investigate the conjecture the that matching mechanisms adopted at many design decisions are less vulnerable to manipulation than those they have replaced. In our notion, we focus on the agents who can beneficially misreport their private information when others are truthful. We refer to them as manipulating agents. The number of these agents can be interpreted as a measure of the potential for manipulation since manipulation is more likely with more manipulating agents.
This number is arguably an arbitrary criterion and hard to justify. For example, in one important application it adds up agents with possibly different abilities and incentives to manipulate, such as students and schools. Nonetheless, it is intuitive and turns out to be useful. It has been occasionally used to measure incentives in matching markets: Roth and Peranson (1999), for example, conducted a simulation using data from the National Resident Matching Program (NRMP) and showed that a small number of medical students could have beneficially misreported their preferences when other agents were truthful.
Our analysis covers a wide range of settings that differ in the following aspects: the set of strategic agents, the strategies these agents can use, whether stability is required, and whether there are ranking constraints (see Table 1). Our results are twofold. First, we consider the college admissions problem where both students and schools are strategic agents (Gale and Shapley (1962)) and schools can misreport their preferences as well as their capacities. We show that when all manipulations (by students as well as by schools) are considered, the student-proposing Gale-Shapley (GS) mechanism has the smallest number of manipulating agents among all stable matching mechanisms (Theorem 1). Dubins and Freedman (1981) and Roth (1982) show that this mechanism is not manipulable by students. This result was one of the main arguments in favor of its choice for the NRMP. However, it also has the largest number of manipulating schools among all stable mechanisms (Pathak and Sönmez (2013)). Our result still supports its choice when all strategic agents are considered. What is more, it is still the best choice even when schools can only misreport their capacities, but not their preferences. All these conclusions carry over to the general model where, in addition, students face ranking constraints: although the student-proposing GS mechanism is now manipulable by students, it is still the least manipulable mechanism. Second, we consider the school choice problem (Abdulkadiroglu and Sönmez (2003)) where students are the only strategic agents and also face ranking constraints. Historically, many school choice systems have used the constrained immediate acceptance (Boston) mechanism, but over time shifted toward the constrained studentproposing GS mechanisms and relaxing the constraint. We demonstrate that the number of manipulating students (Theorem 2) weakly decreased as a result of these changes.

Related literature
The seminal approach by Pathak and Sönmez (2013) compares mechanisms by the set inclusion of the problems with no manipulating agent. 2,3 This approach considers one mechanism less manipulable than another if the former has a larger domain-by the inclusion of the problems where there is no manipulating agent-than the latter. This approach ignores the number of manipulating agents and regards two mechanisms as equal in manipulation at a problem where, for example, one mechanism has one manipulating agent and the other has numerous manipulating agents. Nonetheless, it became the state-of-art method for comparing manipulable matching mechanisms (Chen and Kesten (2017), Dur, Pathak, Song, and Sönmez (2022), Umut (2019), Dur, Hammond, and Morrill (2019)). In contrast, the counting approach is quantitative and enables subtle distinction of mechanisms in problems where they are manipulable. We will show that counting is useful in distinguishing stable matching mechanisms in two-sided matching, while the approach by Pathak and Sönmez (2013) cannot distinguish them (Proposition 1). However, for the school choice problem, both approaches have been able to similarly compare all manipulable mechanisms studied.
The paper that first formalizes the idea of counting manipulating agents in mechanism design is Andersson, Ehlers, and Svensson (2014a). They study the problem of allocating indivisible objects and money to agents, and compare fair and budget-balanced mechanisms by counting manipulating agents. The criterion has long been used in market design, but without a systematic theoretical treatment. For example, in studying incentives in the medical labor market, Roth and Peranson (1999) count the number of 2 They also introduced two other approaches: the approach comparing mechanisms by the inclusion of manipulating agents and the approach comparing mechanisms based on the magnitude of gain from a manipulation. 3 The social choice literature has suggested many other methods to compare manipulable mechanisms.
For example, voting rules can be compared by counting the manipulable instances in the entire domain (Kelly (1993), Aleskerov and Kurbanov (1999)), by finding the domains where some rules become strategyproof while others do not (Moulin (1980)), and by set inclusion of preference relations that admit dominant strategies (Arribillaga and Massó (2016)).
medical students who could have benefited from truncating their rankings (and separately the number of hospitals that could have benefited from reducing their capacities) and used it as a measure of the potential for manipulations. In experiments, this criterion is also used (see the surveys by Chen (2008) and by Hakimov and Kübler (2021)). Kojima and Pathak (2009) and Kojima, Pathak, and Roth (2013) study incentives in large markets by measuring the proportion of manipulating agents and their results support the student-proposing GS mechanism. Three papers subsequent to Pathak and Sönmez (2013) compared the constrained Boston and constrained GS mechanisms. Bonkoungou and Nesterov (2021a) used a criterion called strategic accessibility, and Decerf and Van der Linden (2021) used the notion of dominant preference inclusion introduced by Arribillaga and Massó (2016). These criteria and counting are logically independent. Independently from us, Imamura and Tomoeda (2022) also used the criteria of counting to compare these mechanisms, but their comparison between the constrained Boston and the constrained studentproposing GS is proven in the one-to-one setting.
Chen, Egesdal, Pycia, and Yenmez (2016) define a notion of manipulability that compares the set of outcomes that each agent can obtain via manipulations and show that manipulability comparisons of stable matching mechanisms are equivalent to preference comparisons. Since the preferences of agents on the two sides over stable matchings are opposed, stable matching mechanisms are not comparable for all agents. Andersson, Ehlers, and Svensson (2014b) also define a manipulability notion that compares each agent's maximal gain from manipulation, and find least manipulable budgetbalanced and envy-free mechanisms.
The rest of the paper is structured as follows. In Section 2, we introduce the general framework. In Section 3, we present our results. We present all proofs in the Appendix.

General framework
We consider the two-sided matching problem (Gale and Shapley (1962)). It consists of the following elements: • a finite set I of students • a finite set S of schools • a profile P = (P i ) ∈I∪S of preference relations for each student and each school • a vector q = (q s ) s∈S of capacities for each school.
The profile P is defined as follows. Being unmatched is denoted by ∅. For each student i, P i is a strict preference relation over the set S ∪ {∅} of schools and remaining unmatched. Then s P i ∅ means that school s is acceptable to student i. Let R i denote the "at least as good as" relation associated with P i . 4 For each school s, P s is a strict preference relation over 2 I ∪ {∅}, where 2 I is the set of all nonempty subsets of students and ∅ is the option of being unmatched. Let R s denote the "at least as good as" relation associated with P s .
In particular, P s induces a strict linear ordering over individual students that we denote by s , i.e., i s j if and only if {i} P s {j}. We assume that the preference relation P s over groups of students is responsive to s , meaning that (1) admitting any acceptable student when there is an empty seat is better than leaving the seat unfilled and (2) replacing any student with a more preferred student leads to a better student body. Formally, the preference relation P s of school s over groups of students is responsive (Roth (1985)) if (1) for each each N ∈ 2 I such that |N| < q s and each i / ∈ N, we have N ∪ {i} P s N ⇔ i s ∅, and (2) for each N ∈ 2 I and each i, j / ∈ N, we have N ∪ {i} P s N ∪ {j} ⇔ i s j. Given an agent v ∈ I ∪ S, let P −v denote the preference profile of agents other than v. Given a school s, let q −s denote the capacity vector of schools other than s. The tuple (I, S, P, q) is a college admissions problem, or simply a problem. We keep the sets I and S fixed, and simply denote a problem by (P, q).
A matching is a function μ : I → S ∪ {∅} mapping the set of students to the set of schools as well as the unmatched option such that no school is assigned to more students than it has seats for, that is, for each school s, |μ −1 (s)| ≤ q s . The student i finds matching μ at least as good as matching μ if and only if μ(i) R i μ (i). The school s finds matching μ at least as good as matching μ if and only if μ −1 (s) R s μ −1 (s). A mechanism ϕ is a function that maps each problem to a matching. If ϕ(P, q) = μ for a problem (P, q), then we denote by ϕ i (P, q) = μ(i) the assignment of student i and by ϕ s (P, q) = μ −1 (s) the set of students assigned to school s. We introduce two useful definitions.
Definition 1 (Manipulation via preferences and capacities). (I) We say that student i is a manipulating student of mechanism ϕ at (P, q) if there isP i such that ϕ i (P i , P −i , q) P i ϕ i (P, q).
(II) We say that school s is a manipulating school of mechanism ϕ at (P, q) if there is (P s ,q s ) such thatq s ≤ q s and ϕ s P s , P −s , (q s , q −s ) P s ϕ s (P, q).
(III) We say that school s is a manipulating school via capacities under mechanism ϕ at problem (P, q)-and a manipulating school-if there isq s < q s such that ϕ s P, (q s , q −s ) P s ϕ s (P, q).
(IV) We say that mechanism ϕ is manipulable at (P, q) if there is a manipulating agent of ϕ at (P, q).
We model ranking constraints where students cannot list all schools. There is a maximum number k ∈ {1, , |S|} of schools that each student can list. For each student i, the truncation after the kth acceptable school (if any) of P i with x acceptable schools is the preference relation P k i with min(x, k) acceptable schools such that all schools are ordered as in P i .
, |S|}. The constrained version ϕ k of the mechanism ϕ assigns to each problem (P, q) the matching ϕ k (P, q) = ϕ(P k I , P S , q), where P k I is the profile of truncated preferences after the kth acceptable school.
A mechanism ϕ is constrained stable if for each problem (P, q), ϕ(P, q) is stable under (P k , q).

Results
We first present our results for the college admissions problem where agents on both sides are strategic. We next present our results for the school choice problem where students are strategic but schools are not.

Model and results
We consider the college admissions problem (Gale and Shapley (1962)) where students as well as schools are strategic. Students can misreport their preferences, but schools may misreport their preferences or their capacities. The following notion of stability turns out to be important for the design of such a market. A matching μ is stable at the problem (P, q) if (1) it is individually rational-every student is assigned to an acceptable school and every school is assigned to acceptable students-and (2) it is not blocked-no student prefers a school that has an empty seat or has admitted a less preferred student. That is, we have the following situations: • Matching μ is individually rational at (P, q): for each s ∈ S and each i ∈ μ −1 (s), we have s P i ∅ and i s ∅.
• Matching μ is not blocked at (P, q): there exists no school s and student i / ∈ μ −1 (s) such that s P i μ(i) and either [|μ −1 (s)| < q s and i s ∅] or [i s j for some j ∈ μ −1 (s)].
A mechanism ϕ is stable if for each problem (P, q), its outcome ϕ(P, q) is stable at (P, q). Gale and Shapley (1962) show that for any problem, there exists a stable matching. The set of stable matchings has a lattice structure such that there is an element, called student-optimal stable matching, where, for each student, it is at least as good as any other stable matching. Gale and Shapley (1962) develop an algorithm called student-proposing deferred acceptance for producing the student-optimal stable matching. We denote this mechanism as GS. Similarly, for each problem, there is a school-optimal stable matching that can be obtained by applying the school-proposing deferred acceptance algorithm.
Stable matching mechanisms are subject to various kinds of manipulations by both students and schools. Every stable matching mechanism is manipulable by students and schools. Interestingly, the student-proposing GS mechanism is not manipulable by students (Dubins and Freedman (1981), Roth (1982)), while any stable matching mechanism is manipulable by schools (see, e.g., Sönmez (1997)). The school-proposing GS mechanism is manipulable by both students and schools. These results constitute the standard argument supporting the student-proposing GS: since schools can manipulate any stable mechanism, let us remove all manipulations by students. At the same time, the student-proposing GS mechanism has weakly more manipulating schools than any other stable matching mechanism (Pathak and Sönmez (2013)) and it is unclear how to resolve this trade-off. In the following proposition, we show that the approach by Pathak and Sönmez (2013) about comparing mechanisms by the inclusion of problems with no manipulating agent cannot distinguish stable matching mechanisms.
Proposition 1. Consider the two-sided matching problem and suppose that schools can misreport their preferences. Let ϕ and φ be two stable matching mechanisms. For any problem, either ϕ and φ are both manipulable or neither is manipulable.
All stable matching mechanisms are manipulable via preferences on the same domain of problems. One of the approaches that can differentiate stable matching mechanisms in this domain is counting the number of manipulating agents. The approach is still useful in differentiating them when there are ranking constraints. Note that with ranking constraints, the standard argument above cannot guide the choice because these constraints make the student-proposing GS manipulable by students. 5, 6 Constrained mechanisms are very common. The reasons behind these constraints are not yet fully understood, but they appear to be crucial for practitioners.
Theorem 1. Let k ≥ 2 and let ϕ be a stable matching mechanism. Suppose that students can only rank up to k schools.
(i) Suppose that schools can misreport their preferences. Then the constrained student-proposing GS mechanism GS k has fewer or an equal number of manipulating agents compared to the constrained stable matching mechanism ϕ k .
(ii) Suppose that schools can only misreport their capacities. Then the constrained student-proposing GS mechanism GS k has fewer or an equal number of manipulating agents compared to the constrained stable matching mechanism ϕ k .

Discussion
In Table 1, we illustrate practical design decisions that the above theorem explains. The NRMP and the New York City high school match are examples. In the New York City high school match, in particular, students face ranking constraints and schools can only misreport their capacities since their rankings of students were determined by students' place of residence or whether they have siblings attending the school. (Abdulkadiroglu, Pathak, Roth, and Sönmez (2005)). The intuition behind the result is as follows. First, we show that for any problem (P, q), manipulating students of the constrained student-proposing GS are unmatched 5 Every sensible constrained mechanism is manipulable by students; see Proposition 2 in Bonkoungou and Nesterov (2021b). 6 Note that if the constraint k is larger than the number of schools, then we are dealing with unconstrained mechanisms, and we compare all stable matching mechanisms.
at the matching GS k (P, q). The rural hospital theorem (Roth (1986)) implies that these students are also unmatched at the constrained stable matching ϕ k (P, q). The main part of the argument is to show that they are also manipulating students of the constrained stable matching mechanism ϕ k at (P, q). The reason is that the strategy for manipulating the constrained student-proposing GS mechanism GS k can be replicated to constitute a manipulating strategy of ϕ k -again due to the rural hospital theorem. Therefore, every (unmatched) manipulating student of the constrained student-proposing GS mechanism is also an unmatched manipulating student of any constrained stable matching mechanism. Thus, we essentially need to prove the result for unconstrained mechanisms.
The intuition for why the unconstrained student-proposing GS has fewer manipulating agents compared to any stable matching mechanism is that students and schools have opposing interests over stable matching mechanisms. To see this, consider a problem (P, q) and let ϕ be a stable matching mechanism. Note that every school finds ϕ(P, q) at least as good as GS(P, q). By implementing the matching ϕ(P, q) instead of GS(P, q), some schools receiving their more preferred stable matching do not have any interest in misreporting their preferences or capacities. These schools are matched with different students between these two stable matchings. The rural hospital theorem (Roth (1986)) implies that each such school has filled all its seats under any stable matching. Therefore, some students were matched with this school under GS(P, q), but are matched to different schools under ϕ(P, q). Because GS(P, q) is the student-optimal stable matching, these students are worse off under ϕ(P, q) compared to GS(P, q). Finally, these students are manipulating agents of ϕ at (P, q), as each of them can truncate their preferences and get the same school as under GS(P, q).
Note that only schools that have filled all their seats can manipulate the studentproposing GS mechanism via capacities (Ehlers (2010)). However, the proof is similar.
One of the implications of the theorem is that in a marriage market (where each school has one seat), the optimal stable matching mechanisms have the same number of manipulating agents. Corollary 1. Consider the marriage market where every school has one seat. For any problem, the student-proposing and the school-proposing GS mechanisms have the same number of manipulating agents.
Finally, when the problem has a unique stable matching, the notion is still useful. Roth and Peranson (1999) observed that, in the NRMP, the core tends to be relatively small. This core "convergence" can be explained by the large size of the market, competition, and interview requirements that restrict the number of hospitals students can rank (Roth and Peranson (1999), Pathak (2009), Ashlagi, Kanoria, andLeshno (2017)). When there is a unique stable matching, students cannot manipulate, but schools can still manipulate via preferences as well as via capacities (Ehlers (2010)).

Model and results
We consider the school choice problem (Abdulkadiroglu and Sönmez (2003)) where students are strategic with respect to reporting their preferences but not schools. That is, each school reports its priorities and capacities truthfully. In contrast to the college admissions problem above, we assume that each student is acceptable to each school. That is, for each student i and each school s, i s ∅.
In their seminal paper, Abdulkadiroglu and Sönmez (2003) describe a matching procedure that was used in Boston and is still used in many places around the world. The mechanism is called the Boston mechanism. It is also called an immediate acceptance mechanism to highlight the difference from the "deferred" acceptance of the GS mechanism. Broadly, students apply to acceptable schools one at a time in decreasing order of preferences. Schools immediately accept the highest priority applicants and reduce their capacities accordingly. This mechanism is individually rational but does not always produce a stable matching. Worse, it is manipulable.
In recent years, strategic concerns have motivated many school districts to reform their admissions systems (Pathak and Sönmez (2013)). Some school districts replaced the Boston mechanism with the student-proposing GS mechanism but maintained the ranking constraints, other reforms allowed students to apply to more schools, and some reforms did both. None of these reforms eliminated manipulation, but, as we show next, they reduced the number of manipulating agents.
Theorem 2. Let k > ≥ 1. Then, for any problem, the following statements are true.
(i) The constrained GS mechanism GS k has fewer or an equal number of manipulating students compared to the constrained Boston mechanism Boston k .
(ii) The constrained GS mechanism GS k has fewer or an equal number of manipulating students compared to the constrained GS mechanism GS .
(iii) The constrained GS mechanism GS k has fewer or an equal number of manipulating students compared to the constrained Boston mechanism Boston .

Discussion
In Table 1, we document reforms that Theorem 2 explains. Statement (iii) is a straightforward corollary of the first two statements. Statements (i) and (ii) require a strong argument each. The main and novel part of the proof of statement (i) is to construct a one-to-one function between manipulating students of GS k and a subset of manipulating students of Boston k . Replacing the manipulable Boston mechanism with the non-manipulable student-proposing GS is an obvious improvement. However, for constrained mechanisms, the comparison is not straightforward because there may be some students who cannot manipulate the constrained Boston mechanism, but can manipulate the constrained GS. To see this, consider the following example.
Example 1. There are five students i 1 , i 2 , , i 5 and five schools s 1 , s 2 , , s 5 . Let (P, q) be a problem such that each school has one seat and the remaining components are specified as Consider replacing Boston 2 with GS 2 . The outcome of Boston 2 is Students i 2 and i 4 are manipulating students: i 2 could benefit by top-ranking s 2 and being matched to it, while i 4 could benefit by top-ranking s 1 and being matched to it. Each of the remaining students received her most preferred school and, thus, cannot manipulate Boston 2 at (P, q). But under GS 2 , student i 5 becomes a manipulating student. To see this, consider the outcome of GS 2 : Student i 5 is unmatched. However, she is the highest priority student at s 2 . If she top-ranks s 2 (or even ranks it second), then she is matched to it under the new problem: GS 2 i 5 (P s 2 i 5 , P −i 5 , q) = s 2 . Therefore, i 5 is a manipulating student of GS 2 , but not Boston 2 . Note that student i 2 is also a manipulating student of GS 2 at (P, q). ♦ Recall, that Boston 2 has two manipulating students, i 2 and i 4 , and GS 2 has two manipulating students, i 2 and i 5 . In the proof, we show that if a manipulating student of Boston k is unmatched under GS k , which is the case for student i 2 , then this student remains a manipulating student of GS k . Let us focus on i 4 and i 5 . We show that by replacing Boston 2 with GS 2 , student i 4 's and i 5 's incentives to manipulate are changed correspondingly. Note that under Boston 2 (P, q), student i 5 is matched to school s 3 , which was assigned to student i 3 under GS 2 (P, q). Student i 3 is matched to school s 2 , which was assigned to student i 1 under GS 2 (P, q). Finally, student i 1 is matched to school s 1 , which was assigned to student i 4 under GS 2 (P, q) and student i 4 is unmatched. We draw a sequence of these links as where every student is pointing at the student who was assigned under GS 2 (P, q) to the school that she is assigned to under Boston 2 (P, q). The last student, i 4 , is not assigned under Boston 2 (P, q) to any school that was assigned under GS 2 (P, q) to any student and, thus, does not point at any student. Student i 4 is a manipulating student of Boston 2 at (P, q). Thus, the number of manipulating students of GS 2 is not greater than the number of manipulating students of Boston 2 . The steps of the proof involve showing the following points: • Each manipulating student of GS 2 at (P, q) who is unmatched under Boston 2 (P, q) is also a manipulating student of Boston 2 at (P, q).
• Starting from each manipulating student of GS 2 at (P, q) who is matched under Boston 2 (P, q), the pointing sequence ends at a manipulating student of Boston 2 at (P, q).
• Two distinct sequences lead to distinct manipulating students of Boston 2 at (P, q).
More generally, the function in question is constructed as follows (see Figure 1). The set of manipulating students of GS k is partitioned into those who are matched under Boston k , M, and those who are unmatched under Boston k , M ∅ . Our function returns each student in M ∅ to herself via an identity relation I d and each student in M (initiator of a sequence) to the student closing this sequence via a relation h. The set of manipulating students of Boston k includes M ∅ ∪ h(M ) and possibly others. For statement (ii), the main and novel part of the proof involves the construction of intermediary mechanisms, in which the constraint changes for only one student. For each subset N of students, we construct a mechanism GS N that assigns to each problem (P, q) the matching GS(P N , P k I\N , P S , q). That is, the constraint applies to students in N, while the constraint k applies to the remaining students. Thus, GS ∅ = GS k and GS I = GS . For each problem (P, q), we count and compare the number of manipulating students of GS ∅ , GS {i} , , GS I at (P, q). The following examples illustrate the comparison.
Example 2. Consider the problem (P, q) in Example 1 and let GS 1 be replaced by GS 2 . The outcome of GS 1 at the problem (P, q) is GS 1 (P, q) = i 1 i 2 i 3 i 4 i 5 s 1 ∅ s 2 ∅ s 3 .
Student i 5 received her most preferred school, s 3 , and, thus, cannot manipulate GS 1 at (P, q). But, as we saw in Example 1, student i 5 can manipulate GS 2 at (P, q). ♦ The point of this example is to show that by extending the constraint in the studentproposing GS mechanism, some students may become manipulating students and, thus, there is no inclusion order relation. However, as the following example illustrates, the number of manipulating students does not increase.
Example 3. Consider the same problem as in Example 1. At problem (P, q), we compare the number of manipulating students of GS ∅ = GS 2 , where all students have an extended constraint k = 2, and GS {i1} = GS(P 1 i 1 , P 2 −i 1 ), where student i 1 has a smaller constraint = 1.
Student i 1 is unmatched at the matching Student i 2 is matched at GS {i1} (P, q) and, thus, is not a manipulating student of GS {i1} at (P, q). However, she was a manipulating student of GS 2 at (P, q). Student i 1 is a manipulating student of GS {i1} at (P, q). Indeed, if she misreports her preferences by ranking school s 2 first, she will be matched to it: Student i 5 also remains unmatched under GS {i1} (P, q) and is a manipulating student of GS {i1} at (P, q). To sum up, there are two manipulating students, i 2 and i 5 , of GS 2 at (P, q). One student, i 2 , is no longer a manipulating student of GS {i1} at (P, q). However, one new manipulating student, i 1 , of GS {i1} at (P, q) appears. Thus, when we replace GS 2 by GS {i1} , the number of manipulating students in the example does not decrease. ♦ To prove the theorem, we first prove that for each proper subset N of students and each i / ∈ N, there are weakly more manipulating students of GS N∪{i} at (P, q) compared to GS N . The most difficult steps involve showing the following situations.
• There is at most one student j, who is a manipulating student of GS N at (P, q) and who is not a manipulating student of GS N∪{i} at (P, q), and, if such student j exists, then the following situation holds: • Student i is a manipulating student of GS N∪{i} , but not a manipulating student of GS N at (P, q), thereby "compensating" for the removal of the manipulating student j.
We conclude that there are weakly more manipulating students of GS {i1} at (P, q) compared to GS ∅ . Similarly, there are weakly more manipulating students of GS {i1,i2} at (P, q) compared to GS {i1} . By a repeated application of this argument, there are weakly more manipulating students of GS I = GS at (P, q) compared to GS ∅ = GS k .

Appendix
We need the following result that is very much used in this paper. The stable set has an interesting property called the rural hospital theorem. It says that (i) each agent is matched with the same number of partners across all stable matchings and (ii) every agent that is not matched or has unfilled seats is matched to the same set of partners across all stable matchings.
Lemma 1 (Rural hospital theorem (Roth (1986))). Suppose that schools have responsive preferences. Let (P, q) be a problem, and let ν and μ be two stable matchings.
(i) Each agent is matched with the same number of partners under ν and μ.
Proof of Proposition 1. We show that for any problem where there is more than one stable matching, any stable matching mechanism is manipulable. For any problem where there is one stable matching, stable matching mechanisms have the same set of manipulating agents. Thus, stable matching mechanisms are (non-)manipulable on the same set of problems. Let (P, q) be a problem.
Case 1: There is more than one stable matching. Let ϕ be a stable matching mechanism and let μ = ϕ(P, q). Since there is more than one stable matching, there is a stable matching ν = μ. We consider two subcases.
Case 1.1: There is a student i such that ν(i) P i μ(i). Since μ is individually rational, ν(i) ∈ S. Let s = ν(i) and let P s i be a preference relation where school s is acceptable and every other school is unacceptable. Clearly, the matching ν is stable under (P s i , P −i , q). This is because if it is blocked under (P s i , P −i , q), it is also blocked under (P, q). By Lemma 1(i), student i is matched with the same number of partners across all stable matchings. Thus, she is matched under ϕ(P s i , P −i , q). Since ϕ is individually rational and no school other than s is acceptable under P s i , ϕ i (P s i , P −i , q) = s. Since s = ν(i) P i μ(i), student i is a manipulating agent of ϕ at (P, q). Therefore, ϕ is manipulable at (P, q).
Case 2: There is one stable matching μ. Let ϕ and φ be two stable matchings. By Kojima and Pathak (2009), we focus on preference misreports. Suppose that v is a manipulating agent of ϕ at (P, q). There is a preference relation P v such that ϕ v (P v , P −v , q) P v ϕ v (P, q). Similarly, we construct a preference relation P v and show that φ(P v , P −v , q) = ϕ(P v , P −v , q). Since φ(P, q) = ϕ(P, q), agent v is also a manipulating agent of φ at (P, q).
We prove the following lemmas and use them in the proof of Theorem 1 below.
Lemma 2. Let (P, q) be a problem. Suppose that schools can misreport their preferences and capacities or only their capacities. Let ϕ be a stable mechanism. Let M 2 be the subset of schools that can manipulate GS but not ϕ at (P, q). Then for each school s ∈ M 2 , ϕ s (P, q) = GS(P, q).
Proof. The proof is different depending on whether schools can misreport their preferences or capacities or only their capacities.
Case 1: Schools can misreport their preferences and their capacities. Let s ∈ M 2 . We prove it by contradiction. Suppose that GS s (P, q) = ϕ s (P, q). Because s is a manipulating school of GS at (P, q), there is (P s , q s ) such that q s ≤ q s and GS s P s , P −s , q s , q −s P s GS s (P, q).
( 1 ) Kojima and Pathak (2009, Lemma 1) show that the following manipulation strategy, called dropping strategy, is exhaustive in any stable matching mechanism; in the sense that it can be used to improve upon, according to the true preferences, the outcome of any manipulation, a dropping strategy is any strategy that declares a subset of acceptable students as not acceptable, but keeps the remaining acceptable students ranked as in the original strategy. In particular, they constructed a dropping strategy P d s such that the acceptable students is the set of students in GS s (P s , P −s , q s , q −s ) who are acceptable under P s . By Kojima and Pathak (2009, Lemma 1), we have GS s P d s , P −s , q R s GS s P s , P −s , q s , q −s .
Lemma 1(i) implies that school s is matched with the same number of students under both GS s (P d s , P −s , q) and ϕ(P d s , P −s , q). Since |GS s (P s , P −s , q s , q −s )| ≤ q s ≤ q s , there are less than q s or an equal number of acceptable students under P d s . Therefore, since ϕ and GS are individually rational, they match s to the same set of students, By (2) and (3), we have ϕ(P d s , P −s , q) R s GS s (P s , P −s , q s , q −s ). Now, because the preference relation P s is transitive, this equation and (1) imply that ϕ s (P d s , P −s , q) P s GS s (P, q). Finally, because GS s (P, q) = ϕ s (P, q) by assumption, we have ϕ s P d s , P −s , q P s ϕ s (P, q).
This equation means that school s is a manipulating agent of ϕ at (P, q) and, thus, contradicts our assumption that school s is not a manipulating agent of ϕ at (P, q). Therefore, GS s (P, q) = ϕ s (P, q).
Case 2: Schools can only misreport their capacities. Let s ∈ M 2 . We also prove it by contradiction. Suppose that ϕ s (P, q) = GS s (P, q). Because school s is a manipulating agent of GS at (P, q), then there is q s < q s such that GS s P, q s , q −s P s GS s (P, q).
( 4 ) Because GS(P, q s , q −s ) is the school-pessimal stable matching at (P, q s , q −s ), we have ϕ s P, q s , q −s R s GS s P, q s , q −s .
Since R s is transitive, (4) and (5), and the fact that ϕ s (P, q) = GS s (P, q) imply that ϕ s P, q s , q −s P s ϕ s (P, q).
This equation contradicts the assumption that school s is not a manipulating agent (via capacities) of ϕ at (P, q). Therefore, ϕ s (P, q) = GS s (P, q).
To proceed to the next lemma we first define intermediary mechanisms. Note that under GS the ranking constraint is the same for all students, as well as under GS k . We define intermediate mechanisms where the constraint is for some students and k for the remaining students. Let N ⊆ I be a subset of students. We define the mechanism GS N that assigns to each problem (P, q) the matching This is the mechanism where the constraint is for students in N and the constraint is k for students in I \ N. Then GS k = GS ∅ and GS = GS I . We now establish that manipulating students are unmatched and any manipulating strategy can be replicated via top-ranking schools.
Lemma 3. Let (P, q) be a problem, i ∈ I, and s ∈ S.
(i) Suppose that student i is a manipulating student of GS N at (P, q). Then she is unmatched under GS N (P, q).
(ii) Suppose that GS N i (P, q) = s and let P s i be a preference relation where i has ranked only school s acceptable. Then Proof. We prove (i) by contradiction. Suppose that there is a student i and a school s such that GS N i (P, q) = s, and there is a preference relation P i such that Because GS N is individually rational, there is a school s such that GS N i (P i , P −i , q) = s . LetP = (P N , P k −N , P S ). Then, by definition, GS N (P, q) = GS(P, q). Suppose that i ∈ N. Then schools s and s are among the top acceptable schools under P i . Thus, s P i s and This means that student i can manipulate GS atP, contradicting the fact that GS is not manipulable.
Suppose that i / ∈ N. The proof is the same. Schools s and s are among the top k schools at P i ; thus, s P i s. We have and GS is manipulable atP, which is a contradiction.
To prove (ii), letP = (P N , P k −N , P S ). Then GS i (P, q) = s. As shown by Roth (1982), GS i (P, q) = s implies that GS i (P s i ,P −i , q) = s. Since k > ≥ 1, the truncation of P s i at k or is nothing but P s i . Thus, Proof of Theorem 1. The proof has three steps. Let M 1 denote the set of manipulating students of GS k and let M 2 denote the set of manipulating schools of GS k at (P, q).
Step 1: Every student in M 1 is a manipulating student of ϕ k at (P, q). Let i ∈ M 1 . By Lemma 3, student i is unmatched under GS k (P, q) and there is an acceptable school s under P i such that GS k i (P s i , P −i , q) = s, where school s is the only acceptable school under P s i . Recall that GS(P k I , P S , q) is stable at (P k I , P S , q). By Lemma 1, student i is also unmatched under ϕ k i (P, q) = ϕ(P k I , P S , q). That is, ϕ k i (P, q) = ∅. Since student i is matched under GS k i (P s i , P −i , q) = s, then by Lemma 1, she is also matched under ϕ k i (P s i , P −i , q). Since ϕ k is individually rational and s is the only acceptable school under P s i , we have Therefore, i is a manipulating student of ϕ at (P, q).
To formulate the second step of the proof we need more notation. Divide the set of manipulating schools M 2 intoM 2 -the subset of schools that are also manipulating schools of ϕ k at (P, q)-andM 2 -the subset of schools that are not manipulating schools of ϕ k at (P, q). Then M 2 =M 2 ∪M 2 andM 2 ∩M 2 = ∅.
Step 2: For every school s ∈M 2 , there is a subset I(s) of manipulating students of ϕ k at (P, q) such that no student in I(s) is in M 1 . Consider the problem (P k I , P S , q). By Lemma 2, for each school s ∈M 2 , we have ϕ s (P k I , P S , q) = GS s (P k I , P S , q). By Lemma 1, |GS s (P k I , P S , q)| = q s . Let I(s) = ϕ s (P k I , P S , q) \ GS s (P k I , P S , q). Then I(s) = ∅. Let i ∈ I(s). We claim that i is a manipulating student of ϕ k at (P, q). Because student i is matched under ϕ(P k I , P S , q), then Lemma 1 implies that she is also matched at any stable matching. Thus, GS i (P k I , P S , q) = s for some school s . Because GS(P k I , P S , q) is the studentoptimal stable matching under (P k , P S , q), we have s = GS i P k I , P S , q P k i ϕ i P k I , P S , q = s.
Therefore, s P i s. Let P s i be a preference relation where school s is the only acceptable school for student i. As shown by Roth (1982), GS i (P s i , P k I\{i} , P S , q) = s . Since student i is matched at a stable matching, Lemma 1 implies that she is also matched at any stable matching and, in particular, under ϕ(P s i , P k I\{i} , P S , q). Since ϕ is individually rational and s is the only acceptable school under P s i , then ϕ i (P s i , P k I\{i} , P S , q) = s . Note now that because k ≥ 1, ϕ i (P s i , P k I\{i} , P S , q) = ϕ k i (P s i , P −i , q). This equation and (6) imply that This means that student i is a manipulating student of ϕ k at (P, q). Finally, we show that no student in I(s) is in M 1 ; that is, no student in I(s) is a manipulating student of GS k at (P, q). Let i ∈ I(s). Because student i is matched under ϕ(P k I , P S , q), at a stable matching, Lemma 1 implies that she is also matched under GS(P k I , P S , q). By Lemma 3, student i is not a manipulating student of GS k at (P, q) and, thus, i / ∈ M 1 .
Step 3: The mechanism ϕ k has weakly more manipulating agents than GS k at (P, q). First, for each s, s ∈M 2 such that s = s , we show I(s) ∩ I(s ) = ∅. Let i ∈ I(s) = ϕ s (P k I , P S , q) \ GS s (P k I , P S , q) and j ∈ I(s ) = ϕ s (P k I , P S , q) \ GS s (P k I , P S , q). Since ϕ(P k I , P S , q) is a matching and s = s , then we have i = j. That is, I(s) ∩ I(s ) = ∅. Second, because for each school s ∈M 2 , |I(s)| ≥ 1, we have That is, ϕ k has weakly more manipulating agents than GS k at (P, q).
Proof of Theorem 2. Statement (i). We divide the proof into two parts. In the first part, we show that every manipulating student of the constrained GS mechanism who is unmatched under the constrained Boston mechanism is also a manipulating student of the constrained Boston mechanism. In the second part, we show that every manipulating student of the constrained GS mechanism who is matched under the constrained Boston mechanism induces at least one new manipulating student under the constrained Boston mechanism.
Part 1: For every problem (P, q), every manipulating student of GS k at (P, q) who is unmatched under Boston k (P, q) is a manipulating student of Boston k at (P, q).
Let i ∈ I be a manipulating student of GS k at (P, q) and suppose that Boston k i (P, q) = ∅. By Lemma 3, there is a school s such that where s is the only acceptable school under P s i . First, student i did not rank school s first under P i . Otherwise, because she is matched to school s under GS k (P s i , P −i , q), then this matching would be stable at (P k I , P S , q). By Lemma 1, the same set of students is matched at all stable matchings. Therefore, student i is also matched under GS k (P, q). This result contradicts the assumption that GS k i (P, q) = ∅. Second, we claim that there are less than q s students who have ranked s first under P and have higher priority than i under s . Otherwise, GS k i (P s i , P −i , q) = s would imply that at least one of these students is not matched to school s under GS k (P s i , P −i , q).
This contradicts the stability of GS k (P s i , P −i , q) under (P s i , P k −i , q) because student i is matched to school s, while a student with a higher priority under s prefers this school to her assignment.
Therefore, by ranking s first, i is matched to it under the Boston mechanism. That is, That is, student i is a manipulating student of Boston k at (P, q). Part 2: Manipulating students of GS k at (P, q) who are matched under Boston k (P, q) can be associated in a one-to-one relation with a subset of manipulating students of Boston k at (P, q) who are not manipulating students of GS k at (P, q).
Let M denote the set of the manipulating students of GS k at (P, q) who are matched under Boston k (P, q). For the rest of the proof, the strategy is to pair each student in M with a manipulating student of Boston k at (P, q) who is not a manipulating student of GS k at (P, q). Let μ = GS k (P, q) and ν = Boston k (P, q).
We label the seats of each school s into q s different copies s 1 , , s q s . Let denote the collection of these copies with a generic element x. We call them seats. We assume that each student who is matched to the same school under both μ and ν is matched to the same copy of this school. That is, for each student i and each school s such that μ(i) = ν(i) = s, then μ(i) = ν(i) = s . To do our pairing, define a directed graph with vertices I as follows. For each students i, j ∈ I, define an edge from i to j if there is a seat x ∈Ŝ such that ν(i) = x and μ(j ) = x. We label the edge from i to j as x. The edge i x − → j means that, under ν, student i has taken the seat x that was allotted to student j under μ. A chain in this graph is a sequence of κ > 1 different vertices (i 1 , , i κ ) and κ − 1 different edges (x 1 , , x κ−1 ) such that (a) for each = 1, there is no outgoing edge from i κ ; that is, there is no vertex i and a seat x such that i κ We call the vertex i 1 the tail of the chain and i κ the head of the chain. We establish five steps to proving the theorem.
Step 1: No loop. Suppose that there is a sequence of κ > 1 different vertices (i 1 , , i κ ) and κ − 1 different edges (x 1 , , x κ−1 ) such that i 1 ∈ M and for each Then there is no outgoing edge i κ x − → j such that j ∈ {i 1 , , i κ−1 }. Suppose that there is an outgoing edge i κ x − → j from i κ . First, j = i 1 because μ(i 1 ) = ∅ and, under ν, i κ could not have taken a seat that was allotted to student i 1 under μ. Suppose, to the contrary, that j = i for some ∈ {2, , κ − 1}. Thus, i κ x − → i and i −1 By assumption, i −1 and i κ are different vertices. Since ν is a matching, students i κ and i −1 are allotted (if at all) different seats under ν. Then, under ν, student i −1 and student i κ have taken seats that were allotted to student i under μ. This conclusion contradicts the fact that μ is a matching and that i was allotted only one seat under μ.
Step 2: Every vertex in M is the tail of a chain. Let i ∈ M. First, there is an outgoing edge from i. To see this, recall that, by assumption, student i is matched under ν. That is, ν(i) = x for some seat x ∈Ŝ, while μ(i) = ∅. Suppose that x is a seat at school s. Since the GS mechanism is individually rational, s is one of the top k acceptable schools under P i . Thus, we have s P k i μ(i) = ∅. Since μ = GS(P k I , P S , q) is stable at (P k I , P S , q), we have |μ −1 (s)| = q s . Therefore, there is a student j such that μ(j ) = x and, thus, i 1 x − → j. Next, there is κ ≥ 1 and a sequence (i 1 , , i κ+1 ) of different vertices and different edges (x 1 , , x κ ) such that i 1 = i, and for each ∈ {1, , κ}, i x − → i +1 . The sequence (i, j ) and x is one of these sequences. Since there is a finite number of students, there is a finite number of these sequences. By Step 1, the sequence with the greatest number of vertices is a chain.
Step 3: The head of each chain with a tail in M is a manipulating student of Boston k at (P, q). Let j be the head of a chain with a tail in M. There is an edge i Since there is no outgoing edge from j, either ν(j ) = ∅ or ν(j ) = x such that there is no student j with μ(j ) = x . We claim that μ(j ) P j ν(j ). Otherwise, ν(j ) P j μ(j ) = x and, thus, ν(j ) P k j μ(j ) = x. Because μ is individually rational under P k , we have ν(j ) = x . Suppose that x is a copy of school s. Then s P k j μ(j ). Since μ is stable at (P k I , P S , q), we have |μ −1 (s)| = q s . Therefore, there is a student j such that μ(j ) = x and j x − → j . This contradicts the fact that there is no outgoing edge from j. Therefore, s = μ(j )P j ν(j ).
Next we claim that there are less than q s students who have ranked school s first and have higher priority than student i under P. Otherwise, the fact that μ(j ) = s would imply that one of such students is not matched to school s under μ. This conclusion contradicts the fact that μ is stable at (P k I , P S , q) because student i is matched to school s and a student with higher than i under s prefers s to her assignment.
Finally, we claim that student j did not rank school s first under P j . Otherwise, she would be matched to school s under ν = Boston k (P, q) because there are less than q s students who have ranked it first under P and have higher priority than j under s . Let P s j be a preference relation where student j has ranked only school s as acceptable. Then Boston j (P s j , P k −j , q) = s. Since s = μ(j ) P j ν(j ), we have s = Boston k j P s j , P −j , q P j Boston k j (P, q) = ν(j ).
This means that j is a manipulating student of Boston k at (P, q).
Step 4: The head of each chain with a tail in M is not a manipulating student of GS k at (P, q). Let i be the head of a chain with a tail in M. Then there is an edge j x − → i. Thus, μ(i) = x. That is, student i is matched under GS k (P, q). By Lemma 3, student i is not a manipulating student of GS k at (P, q).
Step 5: No two chains with different tails in M have the same head. This follows from the fact that no two chains with tails in M have a vertex in common. Otherwise, since such chains have different tails, there are different edges j x − → i and j x − → i,where i is one of the common vertices. Since ν is a matching, students j and j are allotted different seats under ν. This means that both students j and j have taken seats x and x that were allotted to student i under μ. This conclusion contradicts the fact that μ is a matching and student i was allotted one seat under μ.
We are ready to complete the proof of the theorem (see Figure 1 for an illustration). Let (P, q) be a problem. Let M ∅ denote the set of manipulating students of GS k at (P, q) who are unmatched under Boston k (P, q). By Part 1, every student in M ∅ is a manipulating student of Boston k at (P, q). The set M ∪ M ∅ is the set of all manipulating students of GS k at (P, q). Let h(M ) denote the collection of students such that each of them is the head of a chain with a tail in M. By Step 3, each student in h(M ) is a manipulating student of Boston k at (P, q). By Step 4, M ∅ ∩ h(M ) = ∅. By Step 5, there are as many students in M as there are in h(M ). Therefore, each student in M ∅ ∪ h(M ) is a manipulating student of Boston k at (P, q) and |M ∅ ∪ M| = |M ∅ ∪ h(M )|. There are weakly more manipulating students of Boston k than GS k at (P, q).
Next, we formulate and prove Lemma 4, which is the main part for proving Theorem 2(ii). Recall the notation used to formulate Lemma 3 above. Lemma 4. Let N I and i / ∈ N. For each problem (P, q), the mechanism GS N∪{i} has weakly more manipulating students than GS N at (P, q).
Proof. LetP = (P N , P k −N , P S ). Then GS N (P, q) = GS(P, q) and GS N∪{i} (P, q) = GS(P i ,P −i , q). We compare the number of manipulating students of GS N at (P, q) to the number of manipulating students of GS N∪{i} at (P, q). We consider two cases depending on the matching status of student i.
Case 1: Student i is unmatched under GS N (P, q) or matched under GS N∪{i} (P, q). For this case, we will show that every manipulating student of GS N at (P, q) is also a manipulating student of GS N∪{i} at (P, q).
First, suppose that student i is unmatched under μ = GS N (P, q). Note that because i / ∈ N,P = (P k i ,P −i ) and GS(P k i ,P −i , q) is stable at (P k i ,P −i , q). Since student i is unmatched under GS(P k i ,P −i , q) and < k, GS(P k i ,P −i , q) is also stable at (P i ,P −i , q). By Lemma 1, the same set of students is matched in every stable matching. Therefore, the same set of students is matched under GS(P k i ,P −i , q) and at GS(P i ,P −i , q). Second, suppose that student i is matched under GS(P i ,P −i , q). Since k > , GS(P i ,P −i , q) is also stable at (P k i ,P −i , q). By Lemma 1, the same set of students is matched under GS(P i ,P −i , q) and GS(P k i ,P −i , q). In either case, the same set of students is matched under GS N (P, q) = GS(P k i ,P −i , q) and GS N∪{i} (P, q) = GS(P i ,P −i , q).
Let j ∈ I be a manipulating student of GS N at (P, q). By Lemma 3, j is unmatched under GS N (P, q) and there is a school s such that where P s j is a preference relation where j has ranked only s as an acceptable school. Because the same set of students is matched under GS N (P, q) and GS N∪{i} (P, q), student j is also unmatched under GS N∪{i} (P, q). That is, First, suppose that j = i. Since ≥ 1, the truncation of P s i after the th acceptable school is nothing but P s i . Therefore, GS N∪{i} (P s i , P −i , q) = GS N (P s i , P −i , q) and we have This means that student i is also a manipulating student of GS N∪{i} at (P, q). Second, suppose that j = i. Note that since k > , student i has extended her list of acceptable schools under P k i compared to P i . Gale and Sotomayor (1985) showed that, after such an extension, no student other than i is better off in GS. In particular, where the equality in the last part follows from the fact that GS N j (P s j , P −j , q) = GS j (P s j , P k i ,P −{i,j} , q) = s. Since GS is individually rational, we have GS j P s j , P i ,P −{i,j} , q = s = GS N∪{i} j P s j , P −j , q .
This equation and (8) yield the relation s = GS N∪{i} j P s j , P −j , q P j GS N∪{i} j (P, q) = ∅.
This means that student j is a manipulating student of GS N∪{i} at (P, q). As a conclusion of Case 1, for each problem (P, q), each manipulating student of GS N at (P, q) is also a manipulating student of GS N∪{i} at (P, q). Therefore, GS N∪{i} has weakly more manipulating students than GS N at (P, q).
Case 2: Student i is matched under GS N (P, q) and unmatched under GS N∪{i} (P, q). Let μ = GS(P, q) and ν = GS(P i ,P −i , q). Let us summarize our proof strategy in the following diagram. We divide the set of students into matched and unmatched at μ. The manipulating students of GS N at (P, q) are unmatched under GS N (P, q). We would like to construct the set of manipulating students of GS N∪{i} at (P, q) from the set of manipulating students of GS N at (P, q).
First, we will show that student i joined the set of manipulating students of GS N∪{i} at (P, q). Second, we will show that all manipulating students of GS N at (P, q), but at most one, remain manipulating students of GS N∪{i} at (P, q). Figure 2. The flow of students across matched, unmatched, and manipulating students at (P, q) when moving between GS N and GS N∪{i} . Notes: The arrow that roots at ∅ shows an impossible flow and the remaining arrows show possible flows. (i) At most one student can leave the set of manipulating students of GS N at (P, q); (ii) student i, who is not a manipulating student of GS N at (P, q) became a manipulating student of GS N∪{i} at (P, q), and no student can leave the set of manipulating students of GS N at (P, q) and remain unmatched under μ. There can be new manipulating students of GS N∪{i} that were unmatched under GS N (P, q).
Step 1: Student i is a manipulating student of GS N∪{i} at (P, q), but not a manipulating student of GS N at (P, q). Because student i is matched under μ = GS N (P, q), by Lemma 3, she is not a manipulating student of GS N at (P, q). Let s = μ(i) and let P s i be a preference relation where she has ranked only school s as an acceptable school. As shown by Roth (1985), GS i (P, q) = s ⇒ GS i P s i ,P −i , q = s.
Since ≥ 1, the truncation of P s i after the th acceptable school is nothing but P s i . Therefore, GS N i P s i , P −i , q = s ⇒ GS N∪{i} i P s i , P −i , q = s.
Since GS N∪{i} i (P, q) = ∅ and school s is an acceptable school under P i , we have GS N∪{i} i P s i , P −i , q = s P i GS N∪{i} i (P, q) = ∅.
This means that student i is a manipulating student of GS N∪{i} at (P, q).
Step 2: Every manipulating student of GS N at (P, q) who is unmatched under ν is a manipulating student of GS N∪{i} at (P, q). Let j be a manipulating student of GS N at (P, q) and suppose that she is unmatched under ν = GS N∪{i} (P, q). Since she is a manipulating student of GS N at (P, q), by Lemma 3, we have GS N j (P, q) = ∅ and there is a school s such that s P i GS N j (P, q) and GS N j (P s j , P −j , q) = s. Student i has extended her list of acceptable schools under P k i compared to P i . As shown by Gale and Sotomayor (1985), no other student is better off under GS after such an extension. In particular, we have GS j P s j , P i ,P −{i,j} , q R s j GS j P s j ,P −j , q = s.
Since GS is individually rational, GS j (P s j , P i ,P −{i,j} , q) = s. Let x be a natural number such that x = if j ∈ N and x = k if j ∈ I \ N. Since x ≥ 1, the truncation of P s j after the xth choice is nothing but P s j . Therefore, GS N∪{i} j P s j , P −j , q = s.
Since by assumption GS N∪{i} j (P, q) = ∅, we have s = GS N∪{i} j P s j , P −j , q P j GS N∪{i} j (P, q) = ∅.
This means that student j is a manipulating student of GS N∪{i} at (P, q).
Step 3: Every student but i who is matched under GS N (P, q) is also matched under GS N∪{i} (P, q). Note that student i has extended her list of acceptable schools under P k i compared to P i . As shown by Gale and Sotomayor (1985), no other student is better off in GS after such an extension. Thus, for each student j = i, ν(j ) = GS j P i ,P −i , q R j GS j P k i ,P −i , q = μ(j ).
Let j = i be a student other than i and suppose that μ(j ) = s for some school s. Since μ is individually rational underP, then ν(j ) = ∅.
Step 4: There is at most one student who is a manipulating student of GS N at (P, q) but not a manipulating student of GS N∪{i} at (P, q). By Step 2, any manipulating student of GS N at (P, q) who is not a manipulating of GS N∪{i} at (P, q) is matched under GS N∪{i} (P, q). We prove, more generally, that there is at most one student who is unmatched under μ = GS N (P, q) but matched under ν = GS N∪{i} (P, q). To do that, we compare the number of students who are matched to each school under μ and ν.
Because school s has an empty seat under μ, by assumption, this contradicts the fact that μ = GS(P, q) is stable at (P, q). Thus, there is no student who is matched to school s under ν but not under μ. Therefore, |ν −1 (s)| ≤ |μ −1 (s)|.
Recall that by Step 3, all students but student i, who are matched under μ are also matched under ν. Then inequality (10) implies that there is at most one student who is unmatched under μ but matched under ν.
To sum up, among the manipulating students of GS N at (P, q), at most one of them is not a manipulating student of GS N∪{i} at (P, q). By including student i, who is a manipulating student of GS N∪{i} at (P, q), but not a manipulating student of GS N at (P, q), there are weakly more manipulating students of GS N∪{i} at (P, q) than GS N at (P, q).
Statement (ii). Let (P, q) be a problem. For simplicity, let I = {1, , |I|}. Let m(ϕ) denote the number of manipulating students of ϕ at (P, q). Then where each inequality follows from Lemma 4. Note now that GS ∅ = GS k and GS I = GS . Thus, GS has weakly more manipulating students than GS k at (P, q).