When does an additional stage improve welfare in centralized assignment?

We study multistage centralized assignment systems to allocate scarce resources based on priorities in the context of school choice. We characterize schools’ capacity-priority profiles under which an additional stage of assignment may improve student welfare when the deferred acceptance algorithm is used at each stage. If the capacity-priority profile is acyclic, then no student prefers any subgame-perfect Nash equilibrium (SPNE) outcome of the 2-stage system to the truthful dominant-strategy equilibrium outcome of the 1-stage system. If the capacity-priority profile is not acyclic, then an SPNE outcome of the 2-stage system may Pareto dominate the truthful dominant-strategy equilibrium outcome of the 1-stage system. If students are restricted to playing truncation strategies, an additional stage unambiguously improves student welfare: no student prefers the truthful dominant-strategy equilibrium outcome of the 1-stage system to any SPNE outcome of the 2-stage system.


Introduction
Centralized clearinghouses are used to allocate scarce resources when money cannot be used as a medium of exchange.For example, pupils are assigned to schools, families to public housing, and college students to courses and dormitory rooms.In this paper, we study when an additional stage of assignment improves welfare in centralized clearinghouses.We answer this question in the context of assigning students to public schools, the so-called school choice problem.In this context, families submit preferences that are used, together with school priorities, to create an assignment.In the last decade, many school districts have adapted the celebrated deferred acceptance (DA) algorithm of Gale and Shapley (1962) for this purpose.
The DA algorithm has a number of attractive properties.First of all, it is strategyproof in the sense that families do not have any incentives to misreport their preferences.Furthermore, it produces the student-optimal fair matching-every student weakly prefers the DA outcome to any other fair matching.However, these important properties crucially depend on the assumption that there is no second stage of assignment to match students who are either unmatched in the first stage or who are not satisfied with their first-stage assignments.In practice, however, school districts have additional stages.For example, the newly adapted system in Chicago has two stages (Dogan et al. 2019).
To study the welfare effects of an additional stage, we assume that there are two stages of assignment, as in Chicago.At each stage, families submit preferences over schools and then DA is used to make an offer to students.Each student can accept the offer at the end of the first stage as their permanent assignment or reject it and participate at the second stage. 1 We compare the equilibrium outcomes of this 2-stage enrollment system with the benchmark 1-stage enrollment system, which produces the DA outcome, and characterize when the introduction of a second stage may improve student welfare.We assume that students are strategic both in submitting their preferences and in their acceptance/rejection decisions.Since DA is strategy-proof, we assume that at the second stage, which is the last stage, students submit their preferences truthfully and accept their offers.Therefore, we consider the environment in which students strategically submit their preferences and decide whether to accept or reject their offers at the first stage.
Our main results identify conditions on capacity-priority profiles under which a second stage may improve welfare and conditions under which a second stage cannot improve welfare.We focus on conditions on the capacity-priority profile because school capacities and priorities are typically observable and commonly known, while students' preferences are their private information.The acyclicity condition on capacity-priority profiles, due to Ergin (2002), turns out to be critical.
First, we show that if the capacity-priority profile is acyclic, then either there exists no pure-strategy SPNE of the 2-stage enrollment system or, for each student, every pure-strategy SPNE that it has is weakly less preferred to the truthful equilibrium outcome of the 1-stage enrollment system (Theorem 1).Note that this result states that the truthful equilibrium outcome of the 1-stage enrollment system cannot be improved, in a rather strong sense: even improving the welfare of some students at the expense of some other students is not possible.To show this, we first show that an SPNE outcome of the 2-stage enrollment system cannot have a blocking pair.The reason is that, if the priority of a student s is violated by another student s at a school c, then students s and s must be receiving their final assignments at different stages of the 2-stage enrollment system.Furthermore, student s, by a one-shot deviation, can instead participate at the stage where student s receives school c, and can guarantee himself at least school c, since a lower-priority student is receiving c before his arrival into the problem.Here, the acyclic capacity-priority profile assumption is crucial because, otherwise, student s could have been an interrupter in the new problem and could have ended up with a strictly less-preferred school than c, although without his presence a lower-priority student is able to receive school c. 2 Once we show that there is no blocking pair, the rest follows from the fact that the 1-stage enrollment outcome is the student-optimal stable matching.
Second, we show that if the capacity-priority profile is not acyclic, then there exists a preference profile such that an SPNE of the 2-stage enrollment system Pareto dominates the truthful dominany-strategy equilibrium of the 1-stage enrollment system (Theorem 2).The proof is constructive and the main complexity in the proof is to avoid profitable one-shot deviations at decision nodes resulting from non-truthful preference reports in the first stage.Several findings in the previous literature, in particular those in Ergin (2002) and Kesten (2010), suggest that if the capacity-priority profile is not acyclic, then there exists a problem with an interrupter; and, if the interrupter leaves with his assigned seat after the first stage and the remaining students participate in the second stage, the outcome can be Pareto improved.Yet, the literature is silent about whether such a Pareto improvement can be supported as an SPNE outcome. 3he critical nature of the acyclicity condition in how equilibrium outcomes of 1stage and 2-stage enrollment systems compare to each other is essentially due to the fact that it is closely related to consistency: DA is consistent under a given capacitypriority profile if whenever some students leave a problem with their DA assignments, the assignments of the remaining students at the DA outcome for the reduced problem does not change compared to their DA assignments at the original problem (Thomson 1990).DA is consistent under a capacity-priority structure if, and only if, the capacitypriority structure is acyclic (Ergin 2002).Given a problem, some students accepting their offers at the end of the first stage resembles a scenario where the same students leave the problem with their DA assignments and a new DA outcome is calculated for the remaining students and schools.In our setting, consistency is equivalent to acceptance and rejection strategies being outcome equivalent.Although consistency has been studied in the resource allocation literature mostly in reference to its normative appeal, in our setting consistency is important due to its relation to strategic behavior.
Another way to interpret our results on multistage centralized assignment systems is the following.Note that whenever the outcome of the 2-stage enrollment system Pareto dominates the 1-stage enrollment outcome, it must be that the 2-stage enrollment outcome is not fair, that is, the priority of at least one student is violated at a school.Even though a static enrollment system based on an unfair rule may be subject to legal action because student priorities may be violated, a multistage enrollment system based on a fair stage rule such as DA may not be susceptible to legal action since any outcome that is not fair can be justified by the preferences submitted by students and their decisions to accept or reject their offers.For example, at the 2-stage enrollment outcome, if Alice would rather go to Bob's school and Alice has a higher priority than Bob at that school, it must be that either Alice has not ranked that school over her assigned school at the stage she was assigned or Alice has accepted her offer at the first stage while Bob was offered a seat at the school in the second stage.Therefore, a multistage enrollment system based on DA brings a new type of justification for some unfair assignments.From this point of view, our results identify conditions under which this new type of justification can be invoked to achieve Pareto improvements over the 1-stage enrollment outcome (i.e., DA outcome).
Finally, we illustrate an environment where an additional stage unambiguously improves student welfare.We consider truncation strategies (Roth et al. 1999) that have attracted attention in the literature.These are strategies in which students restrict the number of schools they rank as acceptable schools, but truthfully report their preferences over schools.We show that, if students are restricted to playing truncation strategies, an additional stage unambiguously improves student welfare: No student prefers the truthful dominant-strategy equilibrium outcome of the 1-stage system to any subgame-perfect Nash equilibrium (SPNE) outcome of the 2-stage system (Proposition 3).

Related literature
In an influential study, Ergin (2002) shows that DA is consistent (or Pareto efficient) if, and only if, the capacity-priority profile is acyclic.Acyclic capacity-priority profiles play a key role in our results.We show that acyclicity of the capacity-priority profile is essential to how the DA outcome compares to the equilibrium outcomes of the 2-stage enrollment system.
In a companion paper, Dogan et al. (2019), we study one aspect of the newly adapted enrollment system in Chicago that the set of schools is partitioned into selective schools and nonselective schools.There, we analyze when an alternative unified school choice system is better than the current divided enrollment system when there is only one stage of assignment.In contrast, in the current paper, we consider a unified enrollment system and analyze when having an additional stage can improve student welfare.In Dogan et al. (2019), we assume that there is no second stage of assignment. 4he distinguishing feature of our model is that the enrollment system may run in more than one stage. 5Manjunath and Turhan (2016) also consider a multistage enrollment system, where students submit preferences only once, at the beginning of the first stage without strategic considerations.Furthermore, they assume that students participate at all stages.In contrast, a first-stage acceptance by a student is permanent in our 2-stage enrollment system.In addition, students are strategic in our setting and submit preferences at every stage as in the enrollment system in Chicago.
Another paper that considers student assignment in multiple stages is Dur et al. (2018).Their main focus is understanding whether any sequential enrollment mechanism satisfying certain design goals such as non-wastefulness, strategy-proofness, fairness, and respecting improvements exists, while we take the mechanism to be used at each stage of the sequential process as DA, since all school districts with centralized clearinghouses use this mechanism, and compare the 2-stage and 1-stage enrollment system outcomes.In a recent work, Haeringer et al. (2021) study college admissions in France, which feature multiple stages.They introduce refitting rules, which govern the set of admissible preference lists that students can submit, and study when refitting rules improve student welfare.For example, they assume that a school that was deemed unacceptable at early stages cannot be acceptable at later stages.We do not make any such assumptions on preference lists that can be submitted across stages following the rules in school districts such as Chicago.More importantly, students are strategic in our setting and, therefore, we analyze the equilibria of the enrollment systems that we consider.Andersson et al. (2018) also study two-stage assignment systems of students and schools, which may be private and public, and show that when all schools have the same ranking of students, seats of schools who are assigned students at the second stage are not wasted for a particular mechanism.In contrast, we consider 1-stage and 2-stage enrollment systems where DA is used at each stage and compare equilibria of these systems from the student welfare perspective.Bó and Hakimov (2022) introduce a class of dynamic mechanisms and show that straightforward behavior by students is an equilibrium that produces the DA outcome.We do not focus on straightforward behavior and study when students can be better or worse off in the 2-stage enrollment system compared to the DA outcome.
Our paper also contributes to the literature that studies Pareto improvements over DA.In an influential paper, Kesten (2010) provides a modification of DA, which produces an efficient outcome by allowing some student priorities to be violated. 6thers consider alternative solution concepts to stability that allow for efficient matchings (e.g., Morrill 2015;Ehlers and Morrill 2019;Troyan et al. 2020).In a related approach, Alva and Manjunath (2019) study when a mechanism has strategy-proof Pareto improvement. 7In contrast to this literature, we consider a repeated implementation of DA and study the equilibria of the induced game and characterize when student welfare can be improved compared to the static DA outcome.

Model
In this section, we formally introduce our concepts and notation.

Preliminary definitions
Let S = {s 1 , . . ., s |S| } be a set of students and C = {c 1 , . . ., c |C| } be a set of schools.Each student s ∈ S has a preference relation R s over C ∪ {s}, 8 where s represents an outside option for the student, which can be a private school or homeschooling.Given c, c ∈ C ∪ {s}, we write c P s c if c = c and c R s c , i.e., student s strictly prefers c to c .A school c is acceptable to student s if the school is strictly more preferred than the outside option.
Each school c ∈ C has a capacity q c ∈ N, which represents the maximum number of students the school can admit and a priority ranking c over S ∪ {c}. 9 Here, c denotes the option of having an empty seat.The strict part of the priority ranking c is denoted as c , so, for any s, s ∈ S ∪ {c}, if s c s and s = s , then s c s .A student s is acceptable to school c if s c c.
The admissions policy of each school c ∈ C is represented by a choice function Ch c : 2 S × {1, . . ., q c } → 2 S , which maps each nonempty set S ⊆ S of students and each q ∈ {1, . . ., q c } to a subset Ch c (S; q) ⊆ S of chosen students such that |Ch c (S; q)| ≤ q.Here, q represents the remaining capacity, so the school cannot admit more than q students.In a 2-stage enrollment system, the admissions policy may depend on the number of seats available since a school needs to specify how to allocate the remaining seats at the second stage after some of the seats are allocated in the first stage.We assume that for each school c ∈ C, Ch c is responsive to the priority ranking c , i.e., for each S ⊆ S, Ch c (S; q) is obtained by choosing the highest-priority acceptable students in S until q students are chosen or no acceptable student is left.
A (school choice) problem is a tuple (S, C, (R s ) s∈S , (q c , c ) c∈C ).Sometimes, we fix all the primitives of a problem other than the student preference-relation profile; in that case, we refer to (S, C, (q c , c ) c∈C ) as the problem.
A matching μ is a function on the set of agents (students and schools) that specifies the assignment of each agent respecting school capacity constraints.More explicitly, • for each student s, μ(s) ∈ C or μ(s) = s, • for each school c, μ(c) ⊆ S and |μ(c)| ≤ q c , and • for each student s and school c, μ(s) = c if, and only if, s ∈ μ(c).
In words, every student is either matched with a school or matched to his outside option (which we also refer to as being unmatched), and every school is assigned a set of students with cardinality less than or equal to its capacity.In addition, there is a 8 More formally, a preference relation over C ∪ {s} is a complete, transitive, and anti-symmetric binary relation over 9 The priority ranking c is a complete, transitive, and anti-symmetric binary relation over S ∪ {c}.
feasibility constraint so that if a student gets matched with a school, then the student is in the set of students that is assigned to the school.
A matching μ Pareto dominates another matching ν, if, for every student s, μ(s) is weakly more preferred than ν(s), and, for one student, it is strictly more preferred.A matching that is not Pareto dominated is called Pareto efficient.
A matching μ is stable if it satisfies the following properties: • (individual rationality for students) for each student s, μ(s) R s s, • (individual rationality for schools) for each school c, Ch c (μ(c); q c ) = μ(c), and • (no blocking) there exists no student-school pair (s, c) such that c P s μ(s) and s ∈ Ch c (μ(c) ∪ {s}; q c ).
Individual rationality for a student means that she weakly prefers the outcome to the outside option.On the other hand, for a school, it means that the school would like to keep all students assigned to it.More explicitly, since choice functions are responsive, only acceptable students are matched with the school and the number of students assigned to the school is no more than its capacity.No blocking rules out the existence of a student-school pair such that the student strictly prefers the school to his match and the school would like to admit the student.In student-assignment settings, the stability of a matching is viewed as a fairness notion (Abdulkadiroglu and Sönmez 2003).
A stable matching is student optimal if every student weakly prefers the outcome in this matching to the outcome in any other stable matching.
An assignment rule ϕ associates each problem (S, C, (q c , c ) c∈C ) with a matching ϕ(S, C, (q c , c ) c∈C ).We sometimes write ϕ s (S, C, (q c , c ) c∈C ) instead of ϕ(S, C, (q c , c ) c∈C )(s) to denote the assignment of a student.
The deferred acceptance (DA) rule due to Gale and Shapley (1962) is used in many school districts that have reformed their school choice systems.The DA rule associates each problem with the matching determined by the following deferred acceptance algorithm.
Step 1.Each student applies to her top-ranked acceptable school.If there is no such school, then she is unmatched.Each school c considers its applicants, say A 1 (c).Among these, it tentatively accepts Ch c (A 1 (c); q c ).It rejects all the other applicants.If there is no rejection by any school at this step, then stop and return the tentative matching.
Step ≥ 2. Each student who is rejected at Step t − 1 applies to her top-ranked acceptable school among the ones that have not rejected her.If there is no such school, she is unmatched.Each school c considers the union of the set of students that it tentatively accepted at Step t − 1 and the set of new applicants at Step t, say A t (c).Among these, it tentatively accepts Ch c (A t (c); q c ).It rejects all the other applicants.If there is no rejection by any school at this step, then stop and return the tentative matching.
The algorithm stops in finite time since there can only be a finite number of rejections.The DA algorithm produces the student-optimal stable matching (Gale and Shapley 1962).

Enrollment systems and games
The student enrollment systems in many districts have supplementary stages for students who are unmatched or matched with less-preferred schools.Thus, we consider two centralized enrollment systems depending on whether the enrollment system has two stages or only one stage.1-Stage Enrollment System: Each student reports a preference relation.Then, a matching is determined by the DA rule.Finally, each student either accepts her match or rejects it (in which case she is matched to her outside option).

2-Stage Enrollment System:
In the first stage, a matching is determined through the 1-stage enrollment system described above.In the second stage, if there are any remaining students (those who were unmatched or rejected their matches at the end of the first stage) and there are available seats, another matching is determined through the 1-stage enrollment system among the unmatched students and schools with reduced capacities.10Each remaining student submits a new preference relation in the second stage (as in Chicago).
We assume that students may be strategic when reporting their preferences or making their acceptance/rejection decisions, but schools are not strategic, and their priority rankings and capacities are common knowledge.Each enrollment system, at each preference profile, induces an extensive-form game.We study the Nash equilibria and subgame-perfect Nash equilibria (SPNE) of these games and compare the equilibria of the two enrollment systems.
In the extensive-form game at a given preference profile, a student's strategy should recommend, at each stage, what preference relation to report and whether to accept a matched school as a function of the history.Let us call a student's strategy truthful if it recommends reporting, at each stage and for each history, the true preference relation.Note that there can be many different truthful strategies, which differ in the acceptance-rejection recommendations but always report the preferences truthfully.
We say that a student's strategy weakly dominates another strategy if, for each strategy profile of the other students, the former strategy yields an outcome that is weakly preferred by the student to the outcome of the latter strategy. 11We say that two student strategies are outcome equivalent if, for each strategy profile of the other students, the student is indifferent between the outcomes of the two strategies.
It is a weakly dominant strategy for each student to truthfully report his preferences in DA (Dubins and Freedman 1981).In other words, the game induced by DA has a truthful Nash equilibrium with weakly-dominant strategies.As a result, most of the literature focuses on this particular equilibrium, even though the game induced by DA may have other Nash equilibria. 12nder the 1-stage enrollment system, it is a weakly dominant strategy to accept any offer that is weakly more preferred than the outside option.Therefore, there is a truthful SPNE in which students accept their offers.We also focus on this truthful equilibrium of the 1-stage enrollment system.

Consistency and acyclic capacity-priority profiles
An assignment rule is consistent under a given capacity-priority profile if, for each problem with that capacity-priority profile and for each reduced problem obtained by the departure of an arbitrary subgroup of students with their assigned seats at the original problem, the matching outcome of the assignment rule for the reduced problem is the restriction of the matching chosen for the original problem to that subgroup.
More formally, given a problem (S, C, (R s ) s∈S , (q c , c ) c∈C ), a set of students S ⊆ S, and a matching μ, let (S, C, (R s ) s∈S , (q c , c ) c∈C )| (S,μ) denote the reduced problem when students in S leave the problem with their assigned seats in μ.We say that an assignment rule ϕ is consistent under the capacity-priority profile (q c , c ) c∈C if, for each preference profile R, set of students S ⊆ S, and student ).We say that an assignment rule is consistent if it is consistent under any capacity-priority profile (Thomson 1990).
A capacity-priority profile (q c , c ) c∈C includes a cycle if there exist a pair of schools c and c , and three students s, s , and s such that i. s c s c s c s and ii.there exist (possibly empty) disjoint sets of students A capacity-priority profile (q c , c ) c∈C is acyclic if it does not include any cycle.Ergin (2002) shows that DA is consistent (or Pareto efficient) under the capacitypriority profile (q c , c ) c∈C if, and only if, (q c , c ) c∈C is acyclic.
We use the following interpretation of acyclicity.Given a problem (S, C, (R s ) s∈S , (q c , c ) c∈C ), we say that student s is an interrupter at school c if there is a step of the DA algorithm where student s is tentatively accepted by school c but another student is rejected, while at a later step student s is also rejected. 13 Kesten (2010) shows that a problem with the capacity-priority profile (q c , c ) c∈C does not have an interrupter if, and only if, (q c , c ) c∈C is acyclic.

Students are strategic only in acceptance-rejection decisions
To compare the outcomes of the 1-stage and 2-stage enrollment systems, we first consider the case when students use truthful strategies.This environment paves the way for our subsequent main analysis where students are more sophisticated and, in particular, they are also strategic in reporting their preferences.Suppose that each student reports her preferences truthfully in the first stage, and also in the second stage, if she rejects her offer at the end of the first stage.In addition, accepting the offer after the second stage is optimal for students, so the only strategic decision for a student is the acceptance-rejection decision at the end of the first stage.
Consider the game in which students observe their assigned schools at the first stage and decide whether to accept or reject an offer, and students who reject their offers move to the second stage, where they report truthfully and accept their offers.At each problem, this game can be represented by a one-stage simultaneous-move game where the strategy set of each student consists of accepting the first-stage offer or rejecting it.
Proposition 1 Suppose that students are strategic only in acceptance-rejection decisions.Then any strategy profile outcome, in particular any Nash equilibrium outcome, of the 2-stage enrollment system either Pareto dominates or is the same as the 1-stage enrollment outcome.
To show this result, we use the following property of DA.Given a problem, consider the DA outcome.Let a set of students leave the problem with their assigned seats in DA, reducing the capacities of the matched schools.Run DA again in the reduced problem.In this setting, it is known that the matching for the remaining students may change, i.e., DA may not be consistent, unless the capacity-priority profile is acyclic (Ergin 2002).Lemma 1 shows that each student remaining in the problem gets a weakly more preferred school (Chen 2017).This implies that rejecting the offer after the first stage is a weakly dominant strategy since at the second stage every student is matched with a weakly more preferred school, regardless of what other students decide.Hence, at any strategy profile of the 2-stage system, a student either accepts his offer at the first stage and is assigned to his 1-stage enrollment school, or some students leave with their 1-stage enrollment schools and in the second stage the student is assigned a school that he weakly prefers to his 1-stage enrollment school.
Also note that in the 2-stage enrollment system, all students rejecting their firststage offers is a Nash equilibrium, which produces the same outcome as the 1-stage enrollment system.It is easy to show that a Nash equilibrium outcome of the 2-stage enrollment system may indeed Pareto dominate the 1-stage enrollment outcome when the capacity-priority profile has a cycle (see Example 1 in Appendix A).
We next show that a capacity-priority profile with a cycle is necessary and sufficient for the 2-stage enrollment system to feature a Nash equilibrium outcome which Pareto improves upon the 1-stage enrollment outcome.
Proposition 2 Suppose that students are strategic only in acceptance-rejection decisions.
i.If the capacity-priority profile is acyclic, then any strategy profile of the 2-stage enrollment system is a Nash equilibrim, which produces the same outcome as the 1-stage enrollment outcome.ii.If the capacity-priority profile has a cycle, then there exists a student preference profile such that a Nash equilibrium outcome of the 2-stage enrollment system Pareto dominates the 1-stage enrollment outcome.
The proof of Proposition 2 is provided in Appendix B. Part i follows from the fact that DA is consistent under an acyclic capacity-priority profile and therefore, for any student, accepting the first stage offer or rejecting it are outcome equivalent.The proof of Part ii is constructive.
Remark 1 The first statement in Proposition 2 holds for any assignment rule that is consistent under a capacity-priority profile.Kesten (2006) introduces a different cycle condition and characterizes capacity-priority profiles under which the top trading cycles rule is consistent.Kojima and Ünver (2014) and Dogan and Klaus (2018) characterize the Boston (or immediate acceptance) rule with some desirable properties which include consistency.Ergin (2000) studies consistent sequential rules for house allocation problems. 14Therefore, our result holds for the top trading cycles rule under the assumptions stated in Kesten (2006), the sequential rules, and the Boston rule.
Remark 2 When students are strategic only in acceptance-rejection decisions, Propositions 1 and 2 hold for the comparison between k-stage and 1-stage enrollment systems for every k ≥ 2. Furthermore, if k is sufficiently large, there exists a pure-strategy SPNE of the k-stage enrollment system that has the outcome of the efficiency-adjusted deferred acceptance algorithm, which is Pareto efficient (Kesten 2010). 15  4 Students are strategic also when reporting first-stage preferences Suppose now that students are strategic also when reporting their first-stage preferences, in addition to the acceptance-rejection decisions at the end of the first stage.
Consider the complete information game where students report preference relations, then observe the reported preferences at the first stage (and also the profile of offers), and decide whether to accept an offer or not, and the students who reject their offers move to the second stage, where they report truthfully and accept the new offer.At each problem (R s ) s∈S , this game can be represented by an extensive-form completeinformation game where a student strategy is a complete-contingent plan that specifies which preference relation to report and whether to accept or reject the offer following each possible reported preference profile.We compare the SPNE of this game with the truthful equilibrium of the 1-stage enrollment system. 14See also Sönmez and Ünver (2010) for house allocation problem with existing tenants. 15A proof of this result is available from the authors.
It turns out that an SPNE outcome of the 2-stage enrollment system may Pareto dominate the 1-stage enrollment system outcome (see Example 2 in Appendix A).Therefore, an SPNE outcome of the 2-stage enrollment system may Pareto dominate the truthful equilibrium outcome of the 1-stage enrollment system.We show that this is not possible under an acyclic capacity-priority profile.In fact, we prove a stronger result: Theorem 1 Consider any problem (S, C, (q c , c ) c∈C ) where (q c , c ) c∈C is acyclic.For every student preference profile, either the 2-stage enrollment system does not have a pure-strategy SPNE or, for each student, every pure-strategy SPNE that it has is weakly less preferred to the truthful equilibrium outcome of the 1-stage enrollment system.
When the capacity-priority profile is acyclic, there are two possibilities.First, the 2-stage enrollment system may not have a pure-strategy SPNE as demonstrated below by Example 5 in Appendix A. In this case, families that participate in such a system have to play a strategic game that does not have a pure-strategy SPNE.Second, when the 2-stage enrollment system has a pure-strategy SPNE, all pure-strategy SPNE are weakly dominated by the truthful equilibrium of the 1-stage enrollment system.In this case, students do not benefit from the 2-stage enrollment system and may indeed be hurt from it.Both cases are highly undesirable from the perspective of families.
In particular, when the capacity-priority profile is acyclic, a pure-strategy SPNE outcome of the 2-stage enrollment system is either the same as the truthful equilibrium outcome of the 1-stage enrollment system, or it is Pareto dominated by the latter.To show this result, we first show that an SPNE outcome of the 2-stage enrollment system cannot have a blocking pair.The intuition is that, if the priority of student s is violated by another student s at a school c, then students s and s must be receiving their final assignments at different stages of the 2-stage enrollment system.Moreover, student s, by a one-shot deviation, can instead participate at the stage where student s receives school c, and can guarantee himself at least school c, since a lower-priority student is receiving c before his arrival into the problem.Here, the acyclic capacitypriority profile assumption is crucial because, otherwise, student s could have been an interrupter in the new problem and could have ended up with a strictly less-preferred school than c, although without his presence a lower-priority student is able to receive school c.Once we show that there is no blocking pair, the rest follows from the fact that the 1-stage enrollment outcome is the student-optimal stable matching.
It turns out that the 1-stage enrollment system outcome may Pareto dominate an SPNE outcome of the 2-stage enrollment system when the capacity-priority profile is acyclic, which is essentially due to possible coordination failures among students (see Example 3 in Appendix A).The next result shows that the acyclic capacity-priority profile assumption is necessary for the previous result that students weakly prefer the 1-stage enrollment system outcome to any pure-strategy SPNE outcome of the 2-stage enrollment system.
Theorem 2 Consider any problem (S, C, (q c , c ) c∈C ) where (q c , c ) c∈C has a cycle.Then there exists a student preference profile such that there is an SPNE of the 2stage enrollment system that Pareto dominates the truthful equilibrium of the 1-stage enrollment system.
The proof of Theorem 2, which is provided in the Appendix, is constructive: For every school choice problem that has a capacity-priority profile with a cycle, we construct student preferences such that an SPNE of the 2-stage enrollment system Pareto dominates the truthful equilibrium of the 1-stage enrollment system.The construction builds upon the idea in Example 2 and resembles the construction in the proof of Proposition 2. Yet, the constructed strategy profile is more complex than the constructed strategy profile in the proof of Proposition 2 since profitable one-shot deviations at many more decisions nodes must be avoided.
Unlike the case when students are strategic only in acceptance-rejection decisions, the truthful equilibrium outcome of the 1-stage enrollment system may indeed Pareto dominate a pure-strategy SPNE outcome of the 2-stage enrollment system even when the capacity-priority profile has a cycle (see Example 4 in Appendix A).
Even though the 2-stage enrollment system has a pure-strategy Nash equilibrium when students are only strategic in acceptance-rejection decisions, it does not necessarily have a pure-strategy SPNE when students are also strategic in submitting their first-stage preferences.Note that each subgame that follows a preference-revelation profile resembles an allocation problem where each student has an endowment school (the school that she is assigned to at the first stage) and has to decide on whether to stay out of the second stage and enjoy her endowment or stay in together with her endowment.In the second stage, a student can possibly be assigned a strictly more preferred school than her endowment, which is an incentive to reject the first-stage offer.However, a school being your endowment does not provide any additional priority at that school in the second stage, which is an incentive to accept the first-stage offer.In Example 5 in Appendix A, we show that there exists a problem at which the 2-stage enrollment system does not have a pure-strategy SPNE even when all schools have the same priority ranking of students.
Note that when students are strategic also when reporting first-stage preferences, we allow for any possible manipulation strategy.Next, we restrict attention to truncation strategies (Roth et al. 1999) that have attracted attention in the literature, which allows us to illustrate an environment where an additional stage unambiguously improves student welfare.These are strategies in which students restrict the number of schools they rank as acceptable schools, but truthfully report their preferences over the schools.Formally, given the true preferences R s of a student s, a preference report R s is a truncated preference relation if for each pair of schools c, c ∈ C, c R s c if and only if c R s c , and s R s c implies s R s c.In the 2-stage enrollment system, we say that a student plays a truncation strategy if the student submits a truncated preference relation in the first stage.
Proposition 3 Consider any problem.For each student, every pure-strategy SPNE outcome of the 2-stage enrollment system where students are restricted to playing truncation strategies is weakly preferred to the truthful equilibrium outcome of the 1stage enrollment system.Moreoever, there exists a problem for which a pure-strategy SPNE outcome of the 2-stage enrollment system where students are restricted to playing truncation strategies Pareto dominates the truthful equilibrium outcome of the 1-stage enrollment system.
The key to the proof of Proposition 3, which is provided in the Appendix, is the fact that when a student is restricted to reporting truncation strategies, the only other assignment he can obtain by misreporting is his outside option, in which case all other students become weakly better off.In particular, at the end of the first stage, any student who truthfully reports his preference relation becomes weakly better off after some other students truncate their preferences.This is the sense in which truncation strategies provide a sufficiently narrow set of possible deviations so that Proposition 3 holds.

Conclusion
When designing a centralized clearinghouse to assign scarce resources based on priorities, such as in school choice, fairness is a critical desideratum.As a result, school districts that have reformed their admissions systems have employed the celebrated deferred acceptance (DA) algorithm.Even though the DA algorithm produces the student-optimal fair matching in one stage, in practice school districts have a second stage of assignment.Our study provides an explanation of why an additional stage of assignment may be preferable.In particular, an additional stage of assignment may improve student welfare if, and only if, the capacity-priority structure is not acyclic.Furthermore, in an environment where students are only strategic in acceptance-rejection decisions, having a second stage of assignment can never hurt any student.
Our paper provides a first look at student welfare implications of having an additional stage in centralized clearinghouses.One limitation of our study is that, although we show that an additional stage of assignment may improve student welfare provided that the capacity-priority structure is not acyclic, we do not characterize the domain of problems (preference profiles together with capacity-priority structures) where such welfare improvement is indeed possible.It would be interesting to know more about such domain of problems.
In our analysis, we restrict ourselves to the DA rule as the second-stage assignment rule.It would be interesting to know which of our results are robust to variations in the second-stage assignment rule, in particular, whether they would hold if DA is replaced with another rule that Pareto improves over the DA rule.Also, we have considered a setting without "frictions."An additional stage can naturally improve welfare when there are frictions such as information asymmetries, application costs, limited choice for students"16 and so on.17It would be interesting to know which of our results are robust to the introduction of such frictions.
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Appendix A: Examples
Example 1 This example illustrates that when students are strategic only in acceptancerejection decisions, the 2-stage enrollment system equilibrium outcome may Pareto dominate the 1-stage enrollment outcome.
Let S = {s 1 , s 2 , s 3 } and C = {c 1 , c 2 , c 3 }.Each school has capacity one.Let the student preference profile (R s ) s∈S and the school priority profile ( c ) c∈C be as below.
In the first stage, student s 1 gets an offer from school c 3 , student s 2 gets an offer from school c 2 , and student s 3 gets an offer from school c 1 .Consider the strategy profile where student s 1 accepts her offer and the other two students reject their offers.The outcome of this strategy profile is that student s 1 gets school c 3 , student s 2 gets school c 1 , and student s 3 gets school c 2 .Since students s 2 and s 3 get their best schools, they do not have any profitable unilateral deviation.If student s 1 instead rejects, he still receives school c 3 , so he also does not have a profitable unilateral deviation.Therefore, we have a Nash equilibrium outcome that Pareto dominates the 1-stage enrollment outcome in which student s 1 gets school c 3 , student s 2 gets school c 2 , and student s 3 gets school c 1 .
Example 2 This example illustrates that if students are strategic also when reporting their first-stage preferences, an SPNE outcome of the 2-stage enrollment system may Pareto dominate the 1-stage enrollment system outcome.
Consider the school choice problem in Example 1 again.The 1-stage enrollment outcome is that student s 1 gets school c 3 , student s 2 gets school c 2 , and student s 3 gets school c 1 .Consider the following strategy profile.Student s 1 reports all school unacceptable (e.g., truncates his preferences over c 1 ) in the first stage, and accepts an offer if, and only if, it is strictly more preferred than her 1-stage enrollment outcome school c 3 .Each other student reports truthfully in the first stage and at each possible preference profile realization, and they accept an offer if, and only if, it is strictly more preferred than their 1-stage enrollment outcome.
Consider student s 1 .Consider her preference-revelation decision node.By changing her preference report, she either causes an outcome where student s 2 gets school c 2 and student s 3 gets school c 1 , or an outcome where student s 2 gets school c 1 and student s 3 gets school c 2 .In the former case, all students reject their offers, and student s 1 in the end receives school c 3 .In the latter case, only student s 1 rejects her offer, and in the end she still receives school c 3 .So there is no one-shot profitable deviation at her preference-revelation decision node.Consider any of her acceptance-rejection decision nodes.If she is offered school c 1 , she clearly cannot be strictly better off by rejecting it. 18If she is offered school c 3 and she rejects it, in the worst case she receives school c 3 in the second stage, so again there is no one-shot profitable deviation.If she receives school c 2 and rejects it, then she receives a school at the second stage that cannot be strictly less preferred than school c 2 .Again, there is no profitable one-shot deviation.
Consider student s 2 .Consider his preference-revelation decision node.Clearly, there is no one-shot profitable deviation since he gets his best school.Consider any of his acceptance-rejection decision nodes.If he is offered school c 1 , he can clearly not be better off by rejecting the offer.If he is offered school c 2 and he rejects it, in the worst case he receives school c 2 in the second stage, so there is no one-shot profitable deviation either.If he is offered school c 3 in the first stage, by rejecting this offer he gets a school at the second stage that cannot be worse than school c 3 .Again there is no profitable one-shot deviation.
The case for student s 3 is symmetrical to the case for student s 2 .Thus, student s 3 does not have a profitable one-shot deviation either.

Example 3
This example shows that the 1-stage enrollment system outcome may Pareto dominate an SPNE outcome of the 2-stage enrollment system when the capacity-priority profile is acyclic.
Let S = {s 1 , s 2 } and C = {c 1 , c 2 }.Each school has capacity one.Let the student preference profile (R s ) s∈S and the school priority profile ( c ) c∈C be as below.Note that the capacity-priority profile is acyclic.
The 1-stage enrollment outcome is that student s 1 gets school c 2 and student s 2 gets school c 1 .Consider the following strategy profile.Student s 1 reports only school c 1 as acceptable in the first stage, and accepts an offer following a preference profile realization R if, and only if, both students receive offers from schools (i.e., no student is assigned to his outside option at the DA outcome at R). Student s 2 reports only school c 2 as acceptable in the first stage, and accepts an offer following a preference profile realization R if, and only if, both students receive offers from schools.It is easy to verify that this strategy profile is a SPNE with the outcome where student s 1 is assigned to c 1 and student s 2 is assigned to c 2 , which is Pareto dominated by the 1-stage enrollment system outcome.

Example 4
This example shows that the truthful equilibrium outcome of the 1-stage enrollment system may indeed Pareto dominate a pure-strategy SPNE outcome of the 2-stage enrollment system even when the capacity-priority profile has a cycle.
Let S = {s 1 , s 2 , s 3 } and C = {c 1 , c 2 , c 3 }.Each school has capacity one.The student preference profile and school priority profile can be depicted as below.
In the 1-stage enrollment system, each student is matched with his top-ranked school: student s 1 is matched with school c 1 , student s 2 is matched with school c 2 , and student s 3 is matched with school c 3 .
Let R be a preference profile such that student s 1 finds only school c 2 acceptable at R s 1 , student s 2 finds only school c 3 acceptable at R s 2 , and student s 3 finds only school c 1 acceptable at R s 3 .Consider the following strategy profile for the 2-stage enrollment system: Each student s reports R s in the first stage and accepts his offer following a preference profile realization R if, and only if, R s = R s and there is at least one other student s ∈ S \ {s} such that R s = R s .
Note that the outcome of this strategy profile is that student s 1 is matched with school c 2 , student s 2 is matched with school c 3 , and student s 3 is matched with school c 1 , which is Pareto dominated by the 1-stage enrollment outcome.We show that this strategy profile is an SPNE.Consider student s 1 .Since student s 2 has the top priority at school c 3 and student s 3 has the top priority at school c 1 , by a one-shot deviation at his preference-revelation decision node student s 1 cannot affect the assignments of students s 2 or s 3 , but can only change his own assignment from school c 2 to his outside option s 1 .At the given strategy profile, students s 2 and s 3 accept their offers at the first stage, student s 1 rejects his offer and in the second stage he receives school c 2 .Thus, student s 1 does not have a profitable one-shot deviation at his preference-revelation decision node.
Consider the acceptance-rejection decision node of student s 1 following an arbitrary preference profile realization R. Suppose that R is such that at the given strategy profile, it recommends student s 1 to reject his offer at the acceptance-rejection decision node following R.There are two possible cases.The first case is that R s and R s 3 = R s 3 .In this case, the first-stage offer of student s 1 is either school c 2 or his outside option s 1 , and the strategy recommends student s 1 to reject his offer, student s 2 to accept school c 3 , and student s 3 to accept school c 1 .Clearly, student s 1 does not have a profitable one-shot deviation at this decision node.The second case is that all students reject their offers at their decision nodes following R.In this case, student s 1 is assigned to his top-ranked school c 1 and there is no profitable one-shot deviation.
Suppose that R is such that at the given strategy profile, it recommends student s 1 to accept his offer at the acceptance-rejection decision node following R.There are three possible cases.The first case is that for each student s ∈ S, R s = R s , in which case student s 1 clearly does not have a profitable one-shot deviation at this decision node.The second case is that R s 1 = R s 1 , R s 2 = R s 2 , and R s 3 = R s 3 .In this case, if student s 1 rejects his offer by a one-shot deviation at this decision node, in the second stage he will still be assigned to school c 2 since his more preferred school c 1 will be assigned to student s 3 , who has a higher priority at school c 1 , which is therefore not a profitable deviation.The third case is that R s 1 = R s 1 , R s 2 = R s 2 , and R s 3 = R s 3 .In this case, if student s 1 rejects his offer by a one-shot deviation at this decision node, he will still be assigned to school c 2 in the second stage since his more preferred school c 1 will be taken by student s 3 in the first stage, which is therefore not a profitable deviation.
Hence, student s 1 has no profitable one-shot deviation.By symmetrical arguments, no student has a profitable one-shot deviation.
Example 5 This example illustrates that the 2-stage enrollment system may have no pure-strategy SPNE, even with an acyclic capacity-priority profile.
Let S = {s 1 , s 2 , s 3 , s 4 } and C = {a, ω(s 1 ), ω(s 2 ), ω(s 3 ), ω(s 4 )}.Each school has capacity one.Let the student preference profile R and the school priority profile be depicted as below.The part of the preference profile denoting the unacceptable schools are not relevant and not depicted.School have a common priority ordering over students.
Consider the subgame that follows after student s i reports school ω(s i ) as the only acceptable school, for each i ∈ {1, . . ., 4}.Note that student s i gets an offer from school ω(s i ) at the first stage, for each i ∈ {1, . . ., 5}.Suppose that this subgame has a pure-strategy Nash equilibrium, say σ .Since a P s 2 ω(s 2 ) and student s 2 has the top priority from among the students who find a acceptable, it must be that student s 2 rejects his offer at σ since he can guarantee himself at least a by participating in the second stage.Now, we claim that student s 4 must accept his offer, ω(s 4 ), at σ .Suppose not, that is, suppose that student s 4 rejects his offer at σ and participates in the second stage.Now, if student s 3 also rejects his offer and participates in the second stage, irrespective of the strategy that student s 1 chooses, s 2 will receive ω(s 4 ) since s 2 top ranks ω(s 4 ) and has the highest priority among all students who find ω(s 4 ) acceptable, student s 3 will receive a, and student s 4 will receive his outside option s 4 .Since a P s 3 ω(s 3 ), student s 3 must also reject his offer in the equilibrium.But then, student s 4 accepting his offer and not participating in the second stage is a unilateral profitable deviation, contradicting that σ is a Nash equilibrium.Hence, student s 4 accepts his offer at σ .Now, if student s 3 also accepts his offer at σ , irrespective of the strategy that student s 1 chooses, student s 4 will receive a if he rejects his offer and participates in the second stage (note that s 2 will receive ω(s 4 ) since s 2 top ranks ω(s 4 ) and has the highest priority among all students who find ω(s 4 ) acceptable), which would be a unilateral profitable deviation.Hence, student s 3 rejects his offer at σ .Since ω(s 3 ) P s 1 ω(s 1 ) and student s 1 has the top priority, student s 1 rejects his offer at σ since he receives ω(s 3 ) by participating at the second stage.Therefore, students s 1 , s 2 , and s 3 reject their offers and student s 4 accepts his offer at σ .But then, student s 3 receives his outside option s 3 .Since ω(s 3 ) P s 3 s 3 , student s 3 has a profitable unilateral deviation, contradicting that σ is a Nash equilibrium.Hence, the subgame does not have a pure-strategy Nash equilibrium.
Since there is a subgame without a pure-strategy Nash equilibrium, there is no pure-strategy SPNE.
Part ii.Let (q c , c ) c∈C be a capacity-priority profile that has a cycle.Then, there exist a pair of schools c and c , and three students s, s , and s such that i. s c s c s c s and ii.there exist (possibly empty) disjoint sets of students S c , S c ⊂ S \ {s, s , s } with Consider a preference profile (R s ) s∈S such that i. c P s c P s s P s c for every c ∈ C \ {c, c }, ii.c P s s P s c for every c ∈ C \ {c}, iii.c P s c P s s P s c for every c ∈ C \ {c, c }, iv.c P i i P i c for every i ∈ S c and every c ∈ C \ {c}, v. c P i i P i c for every i ∈ S c and every c ∈ C \ {c }, and vi.i P i c for every i ∈ S\{{s, s , s } ∪ S c ∪ S c } and every c ∈ C.
The relevant part of the restriction of the preference relation profile and the priority profile to {s, s , s , c, c } looks as follows.
We construct a Nash equilibrium of the 2-stage enrollment system for the problem (R s ) s∈S that Pareto dominates the truthful equilibrium of the 1-stage enrollment system.In the truthful outcome of the 1-stage enrollment system, students in {s} ∪ S c get matched with c, and students in {s } ∪ S c get matched with c , whereas the remaining students (including s ) get their outside options.
Consider the following strategy profile σ for the 2-stage enrollment system in which students s and s reject and every other student accepts their offers.Therefore, only students s and s participate in the second stage with schools c and c , both of which have a remaining capacity of only one.At the second stage, s applies to c and s applies to c, both of which are accepted.In the outcome under the strategy profile σ , students in {s, s } ∪ S c ∪ S c get their most preferred schools, whereas the other students get their outisde options.Therefore, the outcome of the 2-stage enrollment system under σ Pareto dominates the truthful outcome of the 1-stage enrollment system.
We now show that σ is a Nash equilibrium of the 2-stage enrollment system.Every student other than s gets their most preferred outcome.Furthermore, if student s deviates and rejects her offer, then students s, s , and s participate in the second stage with schools c and c with remaining capacity of one.Then the outcome of DA at the second stage matches s with her outside option, so student s does not have a unilateral deviation in which she is matched with a more preferred outcome.Therefore, σ is a Nash equilibrium of the 2-stage enrollment system.when the schools have their original capacities, which is not possible since this new problem has the same capacity-priority profile with the original problem, which is an acyclic capacity-priority profile.Hence, there is no interrupter in the subproblem.
Next, we will show that, since initially student s receives school c and student s has higher priority than student s at school c, student s gets either school c or something better in the subproblem after his one-shot deviation.Suppose, for contradiction, that student s gets a school that is worse than school c.For any step of DA at which student s is tentatively accepted by a school that he prefers to school c, no other student should be rejected from that school at the same step, since there is no interrupter.Also, for any step of DA at which student s is tentatively accepted by school c, no other student should be rejected from c at the same step.Now, consider the step at which s is rejected by c.Note that until student s gets rejected by school c, DA if student s had not deviated to participate in the second stage and DA if he deviated to participate in the second stage run exactly the same.Also note that, since student s gets a school that is worse than school c and the allocation must be stable, student s must be rejected from school c at some step.If student s is tentatively accepted at school c in the same step at which student s gets rejected by school c, then student s is an interrupter at school c, which is a contradiction.Suppose that student s is not one of the tentatively accepted students at school c in the step that student s gets rejected by school c.Then, there exists a student s who has lower priority than student s at school c and gets tentatively accepted by school c at the step student s gets rejected by school c, since in DA when student s had not deviated to participate in the second stage, student s applies to school c at a later step and gets tentatively accepted.But then, student s gets rejected by school c at a later step, by the latest at the first step student s applies to school c.Hence, student s is an interrupter at school c, which is a contradiction.Thus, student s gets either school c or something better after his one-shot deviation in the subproblem without the added seat.By resource monotonicity of DA (Kelso and Crawford 1982;Chambers and Yenmez 2017), we conclude that student s gets either school c or something better after his one-shot deviation, which is a profitable one-shot deviation.Thus, we get a contradiction.
Case 4: Students s and s accept their offers at the second stage.Since students submit their true preferences at the second stage and DA is stable, student s cannot be matched with school c while student s does not receive an offer from school c or a better school with respect to his preference ranking R s because student s has a higher priority than student s at school c.This is a contradiction.
In all possible cases, we have established a contradiction.Hence, matching μ has no blocking pairs.In addition, it must be individually rational since in equilibrium a student does not accept an unacceptable school and a school never chooses unacceptable students.Yet this does not mean that matching μ is stable because it can have waste: a student may prefer an empty seat to his assigned seat.Consider the reduced problem obtained by decreasing the capacities of the schools (by the amount of empty seats).Then, matching μ is stable at this new problem.Let matching μ * be the student-optimal stable matching at this problem.Note that each student is weakly better off at matching μ * compared to matching μ.Also, by resource monotonicity of the DA, when the school capacities are increased to the initial levels, which is the 1-stage enrollment outcome, each student must be weakly better off, compared to μ * and also compared to μ.Hence, there is no student at μ who is better off compared to the outcome of the 1-stage enrollment system.
Proof of Theorem 2 Let (q c , c ) c∈C be a capacity-priority profile that is not acyclic.Then, there exist a pair of schools c and c , and three students s, s , and s such that  We will construct an SPNE of the 2-stage enrollment system for the problem (R s ) s∈S that Pareto dominates the truthful equilibrium of the 1-stage enrollment system.The construction generalizes the idea in Example 2, and, essentially, students s, s , and s play the roles of students s 3 , s 1 , and s 2 in Example 2, respectively, schools c and c play the roles of schools c 2 and c 1 in Example 2, respectively, and the outside option of student s 2 plays the role of school c 3 in Example 2. As will be seen shortly, the strategy profile in Example 2 will need to be modified due to the additional students S c and S c .First note that in the 1-stage enrollment outcome, students in {s} ∪ S c get school c, students in {s } ∪ S c get school c , and every other student gets his outside option.Consider the following strategy profile for the 2-stage enrollment system: Consider student s .By following the above strategy, he gets his outside option.His only acceptable school is c.By unilaterally changing his preference report, even if he applies to school c, he will not get an offer from c and he will make students s and s get their best offers from their 1-stage enrollment schools, in which case both s and s reject their offers.Then, in the second stage, student s will get his outside option, while s and those students in S c who did not get an offer from c and accepted in the first stage will occupy the remaining seats of c.So there is no one-shot profitable deviation at his preference-revelation decision node.Consider any of his acceptance-rejection decision nodes.If he is offered school c, clearly he can not be strictly better off by rejecting it.If he is offered his outside option or any school worse than that, and he rejects, in the worst case he receives his outside option in the second stage, so again there is no one-shot profitable deviation.
Consider student s .By following the above strategy, he gets school c, which is his top choice.In the preference-revelation decision node, there is no one-shot profitable deviation since he gets his best choice by following the prescribed strategy.Consider any of his acceptance-rejection decision nodes.If he is offered a school strictly better than c or strictly worse than c , clearly he cannot be better off by a one-shot deviation at that decision node.Suppose that he is offered school c .
Case 1: s is offered c and s is offered c.The prescribed strategy recommends s accept c .If he instead rejects, he cannot get c in the second stage since s accepts c in the first stage, and, in the second stage, the remaining seats of c are taken by those students in S c who did not get an offer from c in the first stage.Therefore there is no one-shot profitable deviation.
Case 2: s is offered c and s is not offered c.The prescribed strategy recommends s to accept c .If he instead rejects, he cannot get c in the second stage since s accepts c and s rejects his best offer in the first stage, and in the second stage seats of c are taken by s and those students in S c who did not get an offer from c in the first stage.Therefore there is no one-shot profitable deviation.
Case 3: s is not offered c and s is offered c.The prescribed strategy recommends s to accept c .If s is offered c, both s and s accept their offers, and s cannot get c in the second stage since the remaining seats of c are taken by those students in S c who did not receive an offer from c in the first stage.If s is offered a school different than c and c , then s accepts c, s rejects his best offer, and s cannot get c in the second stage since the remaining seats of c are taken by those students in S c ∪ {s} who did not receive an offer from c in the first stage, except for the lowest-priority one (who may or may not be student s).Therefore, there is no one-shot profitable deviation.
Case 4: s is not offered c and s is not offered c.The prescribed strategy recommends s reject c , in which case s eventually gets c , which is equivalent to accepting.Therefore there is no one-shot profitable deviation.
Consider student s.By following the above strategy, he gets school c , which is his top choice.In the preference-revelation decision node, there is no one-shot profitable deviation since he gets his best choice by following the prescribed strategy.Consider any of his acceptance-rejection decision nodes.If he is offered a school strictly better than c or strictly worse than c, clearly he cannot be better off by a one-shot deviation at that decision node.Suppose that he is offered school c.Case 1: s is offered c and s is not offered c.The prescribed strategy recommends s accept c.Suppose that s is offered offered c .If s instead rejects, he cannot get c in the second stage since s accepts c in the first stage and in the second stage, the remaining seats of c are taken by those students in S c who did not get an offer from c in the first stage.Suppose that s is offered a school worse than c .If s instead rejects, he still cannot get c in the second stage since s rejects his offer and s accepts c in the first stage, and, in the second stage in the DA algorithm, s will be rejected by c in the first step, and in turn student s will be rejected by c in the second step.Therefore, there is no one-shot profitable deviation.
Case 2: s is offered c and s is also offered c.The prescribed strategy recommends s reject c.Since both s and s accept their offers from c, in the second stage student s gets c after rejecting in the first stage.Therefore, there is no one-shot profitable deviation.
Case 3: s is not offered c.The prescribed strategy recommends s reject c.If s is offered c, then s accepts the offer, and, in the second stage, s gets c , which he prefers to c, together with those students in S c who did not get an offer from c in the first stage.If s is offered c , s accepts c and in the second stage s gets c.If s is offered a school different than c and c , then both s and s reject their offers, and, in the second stage, s gets c together with those students in S c who did not get an offer from c in the first stage.Therefore, there is no one-shot profitable deviation.
Consider any student s * ∈ S \{s, s , s }.By following the above strategy, he gets his top choice.In the preference-revelation decision node, there is no one-shot profitable deviation since he gets his best choice by following the prescribed strategy.Consider any of his acceptance-rejection decision nodes.If he is offered his top choice, the prescribed strategy recommends s * accept it, in which case, clearly there is no oneshot profitable deviation.If he is not offered his top choice, then it must be that s * ∈ S c ∪ S c , and by rejecting s * can ensure an outcome at least as good as his outside option, which is his second-ranked school and therefore at least as good as his first-stage offer.Therefore, there is no one-shot profitable deviation.

Proof of Proposition 3
Consider any pure-strategy SPNE profile, say σ , of the 2-stage enrollment system where students are restricted playing truncation strategies.Let us start with the following observation: the only case where a student can change the DA outcome with a truncation strategy is that the student truncates above his DA assignment, he is then assigned his outside option, and all other students become weakly better off in the new DA outcome.Therefore, when students follow σ , at the end of the first stage, each student either gets something he weakly prefers to his 1-stage enrollment outcome, or he is assigned his outside option that is less preferred to his 1-stage enrollment outcome.In particular, at the end of the first stage, any student who truthfully reports his preference relation (without strict truncation) gets something he weakly prefers to his 1-stage enrollment outcome.
Consider any student s.Suppose that, when students follow σ , at the end of the first stage, s gets something, say c, that he weakly prefers to his 1-stage enrollment outcome.In the following subgame, his strategy either recommends he accept school c or recommends he reject school c, in which case he receives a school at least as good as school c at the end of the game, since σ induces a Nash equilibrium in every subgame.In any case, s weakly prefers the outcome of σ to the truthful equilibrium outcome of the 1-stage enrollment system.
Suppose that, when students follow σ , at the end of the first stage, s is assigned his outside option that is less preferred to his 1-stage enrollment outcome.Note that s must be playing a truncation strategy where he truncates above his 1-stage enrollment assignment.Consider a one-shot deviation of student s at the preference-revelation decision node where he instead reports his true preference relation.By the above observation, now at the end of first stage, s gets something, say c , that he weakly prefers to his 1-stage enrollment outcome.In the following subgame, his strategy either recommends he accept school c or recommends he reject school c , in which case he receives a school at least as good as school c at the end of the game, since σ induces a Nash equilibrium in every subgame.For this one-shot deviation not to be profitable, it must be that s weakly prefers the outcome of σ to c , and in particular, to the truthful equilibrium outcome of the 1-stage enrollment system.
To see that there exists a problem for which a pure-strategy SPNE outcome of the 2stage enrollment system where students are restricted to playing truncation strategies Pareto dominates the truthful equilibrium outcome of the 1-stage enrollment system, consider Example 2. Note that in the constructed strategy profile, all students are playing truncation strategies.Moreover, since students do not have profitable oneshot deviations in their preference-revelation decision nodes considering all possible preference reports, they do not have profitable one-shot deviations also when they are restricted to truncation strategies.
i. s c s c s c s and ii.there exist (possibly empty) disjoint sets of students S c , S c ⊂ S \ {s, s , s } such that S c ⊂ {i ∈ S : i c s }, S c ⊂ {i ∈ S : i c s}, |S c | = q c − 1, and|S c | = q c − 1.Consider a preference profile (R s ) s∈S such that i. c P s c P s s P s c for every c ∈ C \ {c, c }, ii.c P s s P s c for every c ∈ C \ {c}, iii.c P s c P s s P s c for every c ∈ C \ {c, c }, iv.c P i i P i c for every i ∈ S c and every c ∈ C \ {c}, v. c P i i P i c for every i ∈ S c and every c ∈ C \ {c }, and vi.i P i c for every i ∈ S\{{s, s , s } ∪ S c ∪ S c } and every c ∈ C.The relevant part of the restriction of the preference relation profile and the priority profile to {s, s , s , c, c } looks as follows.
i. Student s reports no school acceptable in the first stage and accepts his offer if, and only if, it is strictly preferred to his 1-stage enrollment school, which is his outside option s .ii.Student s reports truthfully in the first stage and accepts his offer if, and only if,• either it is strictly preferred to his 1-stage enrollment school, • or it is his 1-stage enrollment school and "student s has an offer from c student s has an offer from c." Student s reports truthfully in the first stage and accepts his offer if, and only if,• either it is strictly preferred to his 1-stage enrollment school,• or it is his 1-stage enrollment school and "student s has an offer from c and student s does not have an offer from c." iv.Each other student in S {s, s , s } reports truthfully in the first stage and accepts his offer if, and only if, it is weakly preferred to his 1-stage enrollment school.