Coalitions and Cliques

Coalitions and Cliques in the School Choice Problem

S. Aksoy A. Azzam C. Coppersmith J. Glass G. Karaali X. Zhao  and  X. Zhu
July 25, 2011
Abstract.

The school choice mechanism design problem focuses on assignment mechanisms matching students to public schools in a given school district. The well-known Gale Shapley Student Optimal Stable Matching Mechanism (SOSM) is the most efficient stable mechanism proposed so far as a solution to this problem. However its inefficiency is well-documented; recently the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) was proposed as a remedy for this weakness. This note introduces two related adjustments to SOSM in order to address the same inefficiency issue. In one we create possibly artificial coalitions among students where some students modify their preference profiles in order to improve the outcome for some other students. Our second approach involves trading cliques among students where those involved improve their assignments by waiving some of their priorities. The coalition method yields the EADAM outcome as well as other Pareto dominations of the SOSM outcome, while the clique method yields all possible Pareto optimal Pareto dominations of SOSM. The clique method furthermore incorporates a natural solution to the problem of breaking possible ties within preference and priority profiles. We discuss the practical implications and limitations of our approach in the final section of the article.

Aksoy, Azzam, Coppersmith, and Karaali were partially supported by NSF Grant DMS-0755540. Karaali was partially supported by a Pomona College Hirsch Research Initiation Grant. Zhao was partially supported by the Hutchcroft Fund of the Department of Mathematics and Statistics at Mount Holyoke College. Zhu was partially supported by a Mount Holyoke College Ellen P. Reese Fellowship.

1. Introduction

Since the mid-eighties, in cities across the United States, public school assignment policies have shifted towards providing students the opportunity to influence their school assignment. The main objective of these school choice policies is to allow all students to attend more desirable schools. A standard theoretical framework for studying such policies is two-sided matching (cf. [14, 23]). Presented in this context, the goal of the School Choice Problem (SCP) is to devise a matching mechanism (designed by or for the school district) that allocates available resources (seats in schools) among players (students or parents) subject to district priorities and legal requirements.

The ideal way to solve the SCP would be to make all schools desirable to sufficiently many students so that all students could attend schools high on their preference profile. Short of a magic wand, we follow in the footsteps of other researchers in our attempt to create the most desirable matching possible given the seemingly intractable problem of too few seats in desirable schools. Current school choice mechanisms tolerate a large number of students receiving low preference schools (“inefficiency”) in order to respect school priority structures (“stability”). The ultimate purpose of these priorities is to benefit the students, but in many practical situations they are also the direct cause of the efficiency losses. This suggests that taking a stable solution as baseline (starting out with a balanced focus on school priorities and student preferences) and then making improvements for efficiency (emphasizing preferences over priorities at the expense of stability) may be a good compromise incorporating both preferences and priorities, resulting in more desirable matchings.111A relevant quote from [4]: “Pareto efficiency for the students is the primary welfare goal, but […] stability of the matching, and strategy-proofness in the elicitation of student preferences, are incentive constraints that likely have to be met for the system to produce substantial welfare gains over the [current] system.”

To this end we employ the language and methods of mechanism design as applied to the SCP a la [2]. In our context the designer/principal is the school district (or whoever is choosing the mechanism to be used). Students are the players; schools are merely items to be consumed, though the end result is influenced by the priorities defined by the district. The designer’s desired outcome is that all students get matched to a school high on their preference lists. Thus the underlying sentiment in our research coincides with that found in previous research in the SCP: the idea that the process should result in as many students as possible receiving placements at schools as high on their preference list as possible in a given matching. School districts may of course have additional motives when designing their policies, which might include, among other things, diversity and social justice considerations; these can to an extent be incorporated in the school priority structures.

In this article we examine several commonly applied mechanisms in two sided matching and school choice. We introduce two related approaches and examine how these new approaches measure up. In Section §§1.2 we introduce three standard mechanisms used in this area of investigation: SOSM222Since SOSM basically uses a particular type of deferred acceptance (DA) procedure applied to the SCP, in §§1.2, we also describe the DA algorithm briefly., EADAM, and TTC. In §2 we introduce our first new approach by studying the impact on outcomes if students were to form “coalitions” in order to affect their school assignments. This section closely follows [15] where it is shown that while the Gale-Shapley deferred acceptance algorithm (DA) disincentivizes strategic action by individuals, it is still feasible for groups to beat the system by coming together and strategizing. We adapt Huang’s methods to the SCP and along the way prove that SOSM (DA as applied to School Choice) is not coalition-strategyproof. We then show that coalitions created by the school district (or any designer/principal) could result in efficiency gains over the DA/SOSM outcome. We find that this approach produces many possible Pareto improvements to the DA/SOSM outcome and, in particular, will yield the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) outcome as one of its outcomes.333All acronyms in this paragraph will be explained in detail in §§1.2.

Following up on the coalition/cooperation theme, in §3 we focus on groups of students who form trading cycles (“cliques”) to improve their own assignments.444The term clique has a specific meaning in graph theory, unrelated to our work here. We examine the impact of these trading cliques when starting with the baseline assignment that results from the DA/SOSM mechanism. In particular we show that the coalition improvements of §2 can be integrated into this new framework, which proves to be a powerful construct to study cycle improvements of various kinds. We also note that indifferences in student preferences may be incorporated into this model. Although a considerable amount of research has been done regarding indifferences within school priority classes, indifference in student preferences has not been studied in as much depth. As far as we know, this characteristic of cycle improvement models has not been investigated before.

Interwoven throughout this work is our emphasis on student preferences as opposed to school priorities, that is, efficiency as opposed to stability. This is due to our wariness of accepting the cost of upholding priorities at expense to the students they are purported to help. We also note that to run the standard algorithms, strict priority rankings are needed. While priorities vary between districts, a single priority class will often have a large number of students. Thus to apply standard two-sided matching algorithms to the SCP, one must ultimately break ties randomly among students within a single priority class. This creates arbitrary rankings, introduces artificial conditions, and results in a sizable efficiency loss (see [11] for a study of tie-breaking and its efficiency cost). Both collaborative approaches presented (coalitions and cliques) make efficiency adjustments to a stable baseline solution which we see as a feasible way to partially address this tie-breaking conundrum as well as the more standard stability-efficiency tradeoff mentioned earlier.

1.1. Notation and basic terms used

Let denote a nonempty set of students, and a nonempty set of schools. A matching is a function that associates every student with exactly one school, or potentially no school at all. Write for the set of matchings. We will also use the related function and write if .

A preference profile for student is a tuple where the ’s form a partition of and every element of is preferred to every element of if and only if . Define the ranking function of a student by letting denote ’s ranking of . In other words if . When each is a singleton, we say that ’s preference profile is strict, (in which case we can view as an -vector). If for some , then we say that the student is indifferent between and . If prefers to , we write , or simply if is unambiguous. Note that the notation denotes a strict order; if we want to describe a weak order, we will write . We denote a set consisting of preference profiles for each student in by and the space of all such sets is denoted by .

A priority structure for school is a tuple where the ’s form a partition of and every element of is preferred to every element of if and only if . When each is a singleton, we say that ’s priority structure is strict, (in which case we can view as an -vector). If for some , then we say that the school is indifferent between and . If prefers to we write , or simply if is unambiguous. Once again, the notation denotes a strict order; if we want to describe a weak order, we will write . We denote a set consisting of priority structures for each school in by and the space of all such complete sets is denoted by .

A matching (Pareto) dominates if for all and is strict for some . A (Pareto) efficient matching is a matching that is not (Pareto) dominated.

A matching mechanism is a function that takes an ordered pair of preferences and priorities and produces a matching.

Let be a priority structure for school . We say that a matching violates the priority of for if there exist some and such that

  1. , : gets assigned under and gets assigned under .

  2. : prefers attending over and

  3. : prioritizes over .

We say that a matching is stable if

  1. does not violate any priorities.

  2. No student is matched to a lower-ranked school when a more preferred school is unfilled.

A stable mechanism is one that always produces stable matchings. A matching mechanism is strategyproof if there is no rational incentive for a student to misrepresent their preferences.

1.2. Background

In this section we introduce several well-known matching algorithms / mechanisms and give a few illustrative examples.555We should remark that all the mechanisms described in this section use strict preference lists for students. In the literature on two-sided matching the Gale-Shapley deferred acceptance algorithm [13] is highly touted, see [23] for an extensive review of the various applications of this algorithm and [21] for a more recent historical overview. Gale and Shapley first described their deferred acceptance method in the context of the stable-marriage problem: There are two distinct groups (men and women) each with an individual preference profile ranking the members of the opposite sex; the ultimate goal is to find a stable matching between the men and the women.666In this context stability means that no man will prefer a woman other than his own partner who also prefers him more than the man to whom she was matched. The deferred acceptance procedure (DA) is as follows:

In round each man proposes to his top choice. Each woman then tentatively accepts the man who is highest on her preference list among those who proposed to her that round (who is now her fiance) and rejects the rest. In step , each unengaged man proposes to his next choice, and each woman considers her new suitors along with her current fiance, tentatively accepting her top choice among them, and rejecting the rest. The algorithm ends when all men are engaged.

In [13], Gale and Shapley proposed applying their deferred acceptance algorithm to the college admissions problem, with the men replaced by students applying to colleges and the women replaced by the colleges to which they applied. In 2003 Abdulkadiroǧlu and Sömnez adapted the Gale-Shapley algorithm to the SCP [2] calling it the Student Optimal Stable Mechanism (SOSM). The Gale-Shapley deferred acceptance algorithm, as adapted to the SCP in the form of SOSM, is widely held to be a practical mechanism for implementation. In particular, several large districts such as New York City and Boston [3, 4, 5, 6] have adopted SOSM as their mechanism of choice. Pareto efficiency, stability, and strategyproofness are the main criteria used to evaluate a school choice matching mechanism,777We will add a fourth criterion to our consideration, see §§1.3 and within these measures, SOSM performs well. Indeed, SOSM offers a stable strategyproof mechanism whose outcomes Pareto dominate all other stable matchings.888Note that a stable mechanism can never really be strategy-proof in the complete sense. More specifically, no stable matching mechanism exists for which stating the true preferences is always a best response for every agent where all other agents state their true preferences (see for instance [23, Cor. 4.5]). However the DA/SOSM is practically strategy-proof as we only view the students as strategic players and the student optimality implies that there is no incentive for the students to misrepresent their preferences (cf. [20]). This perspective does not take into account manipulation by schools in capacity (cf. [25]) or preferences, see [10] for recent work addressing these issues.

In this article we use DA/SOSM as a baseline to improve upon. Indeed certain improvements are possible, feasible, and desirable because SOSM suffers from documented efficiency and welfare losses [4, 17]. More precisely, although the SOSM outcome in a given setting dominates any other stable matching, it can often be dominated by an unstable matching. We now illustrate this potential trade-off between stability and efficiency with an example due to Roth, which we will label as . Assume there are three schools, and three students . The priorities of the schools and the preferences of the students are given by:

where stands for “ is preferable to ” (more about notation and terminology can be found in §§1.1). Here, the only stable matching is:

but this matching is (Pareto) dominated by:

We see that (Pareto) dominates because it assigns and schools they prefer over their assignment. Furthermore is (Pareto) efficient. However the matching is no longer stable because is in the position of violating ’s priority for .

In order, in part, to address the weakness illustrated by the example above, Kesten in [17] proposes a new mechanism, and calls it the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM). In order to understand EADAM we must first define an interrupter. Let student be one who is tentatively placed in a school at some step while running the SOSM, and rejected from it at some later step . If there exists at least one other student who is rejected from school after step and before step , then we call student an interrupter for school and the pair is an interrupting pair of step . An interrupter is consenting if she allows the mechanism to violate her priorities at no expense to her. The EADAM then runs as follows:

  • Round 0: Run the SOSM.

  • Round 1: Find the last step (of the SOSM run in Round 0) at which a consenting interrupter is rejected from the school for which he/she is an interrupter. Identify all interrupting pairs in that step which contain a consenting interrupter. If there are no such pairs, then stop. Otherwise for each identified interrupting pair , remove school from the preference list of student without changing the relative order of the remaining schools. Rerun the SOSM with the new preference profile for until all students have been assigned.

  • And in general,

  • Round k, k 1: Find the last step (of the SOSM run in the previous round) at which a consenting interrupter is rejected from the school for which he/she is an interrupter. Identify all interrupting pairs in that step which contain a consenting interrupter. If there is no such pair, stop. Otherwise for each identified interrupting pair , remove school from the preference list of student without changing the relative order of the remaining schools. Rerun the SOSM with the new preference profile until all students have been assigned.

In , is an interrupting pair and the EADAM with the consent of outputs the Pareto efficient matching .

Even though the end result of consenting for interrupters is that they allow the mechanism to violate their priorities for schools they would not be assigned anyway, the step-by-step description above points toward a different route of obtaining the same outcome. The practical outcome would be the same if the consenting interrupters were to modify their preference lists in such a way as to drop schools that they’d not have been assigned to anyway. Thus instead of asking students to sign consent forms to waive priorities, as would be required to run the EADAM, we could in theory ask them to reconsider their preference lists.999In reality this is not desirable; we emphatically want students to be truthful in declaring their preferences.

Last, we describe the Top Trading Cycles (TTC) mechanism, first introduced in [24] (also see [1, 18]) and adapted to the school choice context in [2] as an alternative to SOSM. TTC is a strategyproof mechanism that compromises on stability to achieve efficiency, and proceeds as follows:

  • Round 1: Each student points to his or her first choice school. Similarly each school points to its first choice applicant. Since there are finitely many students and schools, there is at least one cycle. For each such cycle do the following: Assign each student in the cycle to the school he or she is pointing to and remove the student and the school from the market. All unassigned students and unfilled schools move on to the next round.

  • And in general,

  • Round k, k 1: Each unassigned student points to his or her top choice school among the unfilled ones. Each unfilled school points to the student whom it ranks highest among the unassigned students. There should be at least one cycle. For each such cycle do the following: Assign each student in the cycle to the school he or she is pointing to and remove the student and the school from the market. All unassigned students and unfilled schools move on to the next round. The algorithm runs until all students have been assigned.

Thus in essence, once in a trading cycle, students are allowed to trade schools among themselves.

1.3. A new evaluation criterion

The three main criteria most commonly used to evaluate school choice mechanisms are stability, strategyproofness and (Pareto) efficiency. In [7] we introduced a new student-optimal criterion, a “preference reverence index”, and showed that it incorporates a measure of student optimality that is not fully captured by these three previously emphasized criteria. When evaluating matching outcomes in the later parts of this article we make use of this index, so we provide a brief exposition about it in this section.

With the notation from §§1.1 we define by

For any given we will call the preference reverence index of or simply the preference index. We summarize some results regarding the preference index in the following:

Proposition 1.1.

The following are some properties and implications of the preference reverence index as applied to the SCP:

  1. There can exist two stable matchings with the same preference index.

  2. The stable matching with the smallest preference index is the SOSM outcome and it is the unique stable matching with that preference index.

  3. If a matching (Pareto) dominates , then has a lower preference index.

See [7] for more on the preference reverence index.

2. Coalitions in the school choice problem

In [15] Huang discusses a weakness found in the Gale-Shapley stable matching in the context of the stable marriage problem and introduces the idea of coalition cheating in the marriage problem. More specifically he shows that a coalition can be formed where some men, without forgoing their own Gale-Shapley stable matching assignment, can cheat (misrepresent their preferences) so that some other men marry women who are higher on their preference list.

In this section we apply these ideas to the school choice problem. In §§2.1 we give the details of Huang’s construction. Then in §§2.2 we introduce the elements of cheating coalitions in the context of the SCP, and discuss some implementation issues. In §§2.3 we focus on some interesting theoretical consequences of coalitions in the context of the SCP. In §§2.4 and §§2.5, we compare the possible outcomes of coalitions to that of EADAM, and to that of TTC, respectively.

2.1. Huang’s Construction and Coalitions

Originally proved in [8], the following theorem establishes that in the stable marriage problem, there exists no coalition of men that may falsify their preferences such that every member of the coalition receives a strictly better assignment:

Theorem 2.1 (Dubins-Freedman 1981).

In the Gale-Shapley men-optimal algorithm, no subset of men can improve their assignment by falsifying their preference lists.

Therefore in order to study coalitions which falsify preferences to improve their assignments, Huang introduces a nuanced notion of coalitions, which incorporates a separation between two main components: those who falsify their preferences, and those that benefit from these falsifications. In the following we provide a detailed exposition of his construction.

Let and be the set of men and women respectively in a given stable marriage problem. Let be the Gale-Shapley stable matching assignment for this problem when all members of and submit their true preferences. A coalition is defined in terms of a pair of subsets of the set . The first subset, the cabal of the coalition , is a list of men such that each man , , prefers to his own partner , indices taken module . In other words, we have for , what we will call a cabal loop, written , a closed chain of men each of whom would prefer the stable partner of the person before him to his own partner. The second subset, the accomplice set of cabal , is a set of men such that if for some , and . In other words, an accomplice is a man who in his truthful preference list ranks the stable partner of someone in the cabal higher than his own stable partner, while he himself is ranked higher by that woman than the next member of the cabal who would prefer her to his own partner. Note that and may or may not be disjoint.

For any given man we can write the preference profile of as a disjoint union of three sets: . Here the set (respectively ) is simply the list of women on ’s preference profile to the left (respectively to the right) of his stable partner .

Let be a coalition as described above and let denote a random permutation of . The coalition cheating procedure proceeds as follows ([15, Thm.2]): Each accomplice submits a falsified list of the form

Here is the set

if , and

if . In other words, accomplices modify their preference profiles by moving women on the left of their stable partner to the right of their stable partner if they are desirable to other men in the cabal. In particular if is an accomplice, then the set of women moves to the right of his stable partner will consist of all the stable partners of members of the cabal who rank higher than the man following their stable partners in the cabal loop. Huang then shows that in the resulting matching , for and for . Note that in the above the falsified preference lists incorporate a random permutation of the preferences to the left and the right of the stable partner. The cheating coalition procedure is quite robust, in that such a random permutation will not affect the outcome. In other words, the resulting matching creates a cyclical reassignment of those within the cabal loop while leaving all other assignments as they were.

2.2. Coalitions and school choice

Here is an analogue of Theorem 2.1 in the SCP context:

Theorem 2.2.

In the SOSM algorithm, no subset of students can improve their assignment by falsifying their preference lists.

This is particularly easy to see in our previous example. Assume there are three schools, and three students . The school priorities and the student preferences are given by:

Recall that the SOSM yields the (unique stable) matching

Each non-singleton, nonempty subset of the set of students corresponds to a valid coalition:

For coalition , the only way for both students to improve their lot is for to receive and to receive . However, unless receives a better assignment with this coalition, the resultant matching will be unstable since violates ’s priority at . So must receive a better school if or have a chance of receiving a better assignment, so must receive or . Thus clearly and cannot both strictly benefit. The arguments for the remaining coalitions are similar.

The underlying intuition for the above result is much clearer than brute force computation. For a coalition of students to benefit from their falsification under SOSM, either their interests conflict, or the constraint of stability requires appeasing other students not in the coalition, and this conflicts with the interests of those in the coalition. For a deeper and more enlightening discussion, see [8]. With this example and motivating theorem in mind, we now proceed to formalize our coalition model for the SCP, which we will call the Coalitional Improvement Mechanism (CIM).

Let and be the set of students and schools respectively in a given SCP. Let be the SOSM stable matching assignment for the case where all students submit their true preferences. A coalition is defined in terms of a pair of subsets of the set of students. The first subset, the cabal of a coalition , is a list of students such that each student , , prefers to , indices taken modulo . In other words, we have for , and a cabal loop, written , a closed chain of students each of whom would prefer the stable assignment of the person before him to his own stable assignment. The second subset, the accomplice set of cabal , is a set of students such that if for some , and . In other words, an accomplice is a student who in his truthful preference list ranks the stable assignment of someone in the cabal higher than his own stable assignment, while he himself is ranked higher by that school than the next member of the cabal who would prefer it to his own school. Note that and may or may not be disjoint.

For any student we can write the preference profile of as a disjoint union of three sets: . Here the set (respectively ) is simply the list of schools on ’s preference profile to the left (respectively to the right) of his stable assignment . Let denote a random permutation of . The coalition cheating procedure (CIM) is described in:

Theorem 2.3 (cf. Huang 2006 [15]).

Let be the SOSM matching for a given SCP when students submit their true preferences. Consider a coalition , and suppose that each accomplice submits a falsified list of the form , where

  • if , then , and

  • if , .

Then in the resulting matching , for and for .

The proof of Theorem 2.3 is an easy adaptation from the analogous result of Huang [15].

Accomplices modify their preference profiles by moving schools on the left of their stable assignment to the right of their stable assignment if they are desirable to other students in the cabal. In particular if is an accomplice, then the set of schools moves to the right of his stable assignment will consist of all the stable assignments of the members of the cabal that rank higher than the student following their stable assignment in the cabal loop. Again, this procedure is robust, i.e., a random permutation of the two sides of the stable assignment will not affect the outcome.

Let us now consider an example which we will label . Let and be the set of students and schools, respectively, and let their respective preference and priority profiles be given as follows:

Note that the matching output by the SOSM for is:

and has preference reverence index 10 (§§1.3).

We now consider the following coalition : Let with the cabal loop . The accomplice set is and the set for is . In other words the only student who modifies his preference profile is . We display his old and new profiles:

[We underlined ’s stable assignment .] The outcome matching when we rerun SOSM is:

which improves the outcome for all members of the cabal, does not affect the remaining students, and has preference index 6.101010We note that this is also the EADAM outcome if consents. We will discuss this example further in §§3.1.

2.3. Selfless and hopeless students

In the example above, student modified his preference profile in order to change the group outcome to from , but in the end, he did not improve his own assignment. In fact, in any coalition the accomplice set will include some students who do not benefit from the coalition. To understand why, we go back once again to the stable marriage problem. In that context, Dubins and Freedman [8] showed that it is impossible for every man in a coalition to improve his assignment. In other words a subset of men cannot falsify their preference lists so that all of them get better partners. Thus in Huang’s coalition cheating framework, in order to improve the assignments of some members of the cabal, there have to be some selfless men who are willing to make adjustments to their preference profiles despite the fact that they themselves cannot benefit from the arrangement. Analogously in the SCP context, for cheating coalitions to work, there have to be selfless students who are willing to adjust their preferences even though this will not improve their assignment. It should be noted that in these cases, the selfless participant does not end up with a less desirable assignment.

This situation raises the question of the feasibility of coalition cheating, as we now see that some members of the accomplice set have no obvious incentive to make the required adjustments to their preference profiles. As a possible resolution to this issue, Huang [15] proposes a randomized strategy in which every man can expect to marry women ranked higher on their preference lists. This strategy requires men to take risks, since some men can end up with less desirable partners. In the School Choice Problem however, students and parents are relatively risk-averse. Thus a coalition agreeing to a randomized arrangement like the one mentioned by Huang in the Stable Marriage Problem is not rationally practical in the SCP. Moreover, in school districts where there may be many students competing against each other for a limited number of seats at desirable schools, cooperation among parents and students is not plausible.

Nonetheless we propose that our work does not merely present a theoretical framework to investigate collaboration and cooperation issues in the context of the SCP, but in fact it can have implementable outcomes in this context. More specifically, we propose that a school district, given perfect information of all student preferences, can create “artificial coalitions” that would result in a more efficient outcome than that of SOSM. Using the SOSM matching as a baseline, a computer program could identify all matchings that would result from all possible coalitions.111111However, arguably, the computer power needed for this could be quite large. It would then be up to the district to decide which coalitions would be most appropriate to further its own district goals. For instance one district could select the coalition(s) which results in the matching with the most Pareto improvements on the SOSM outcome, while another might choose the coalition(s) based on minimizing the preference index, and yet another could count the number of priority violations (the matching with fewer priority violations being more “fair”) or weight the magnitude of the various priority violations (in terms of the priority level of the violator(s)).

We have already pointed out that in order for coalitions to work, there have to be some accomplices willing to modify their preferences despite the fact that they themselves will not benefit. While some of these accomplices might benefit from different coalitions, others, it turns out, have no hope of ever benefiting from any coalition. Huang calls these hopeless men in the context of the stable marriage problem, and proves that there always exists at least one hopeless man for any given set of preference profiles. We now consider the analogous construct for the School Choice Problem:

Definition 2.4.

Given a specific SCP, a student who cannot benefit from any Pareto improvement upon SOSM is said to be hopeless.

It turns out that there is always such a hopeless student. In other words, we have:

Theorem 2.5.

There is always at least one hopeless student when Pareto improvements are made on SOSM. More specifically, in EADAM or CIM, there is always at least one hopeless student.

In fact this follows directly from a stronger result:

Theorem 2.6.

The students who propose in the last round of SOSM are hopeless.

Proof.

Let be the set of all students and be the set of all schools. Label the set of schools that get proposals in the last step of SOSM as and the set of students that propose in the last step as . If a student proposes to in the last step of SOSM, then we can conclude that prior to that step, had not yet filled its capacity. Otherwise would be replacing another student at and thus forcing that student to propose in a next round, which would in turn mean that the algorithm could not stop.

Assume now that is not hopeless. This means that he can improve his lot via some Pareto improvement. Let us denote the set of ALL students who are moved with this improvement . If any of these students preferred vacant seats to their assigned seats at the end of SOSM we would have a contradiction; they should have included or ranked those schools higher on their preference lists. Thus the Pareto improvement must move all these students to seats that have been assigned under SOSM. Thus the improvement in fact should correspond to a permutation of , and by the pigeonhole principle there exists another student that prefers to her original assignment school . (It doesn’t matter if is from or not. Similarly may or may not belong to ). Since in SOSM students propose to schools in the order that they rank them on their preference lists, must have applied to , been rejected, then applied to . This would imply that school must have filled its capacity before the last step of SOSM, which contradicts our assumption that was accepted by in the last step of SOSM: no filled school can receive and accept a proposal in the final step of SOSM because that would result in a student being displaced which would require another step in the mechanism. ∎

Looking now back at we see that is a hopeless student. In other words, will not benefit from any Pareto improvement to the SOSM matching. Similarly in , the last proposer in the SOSM is who is thus a hopeless student.

2.4. Coalitions and EADAM

The SOSM has already been implemented in several school districts due to its desirable properties of strategyproofness and stability and the fact that it generates the most (Pareto) efficient assignment among all stable matchings. However, as has already been mentioned (and documented in [4, 17] and elsewhere) its strict adherence to a stable outcome (emphasizing priorities) can result in substantial inefficiency. In this section we compare two efficiency-oriented modifications to SOSM, namely the EADAM from [17] (described in §§1.2) and the coalition cheating SOSM improvement.

The main result of this section is:

Theorem 2.7.

There exists a coalition improvement on SOSM yielding the EADAM outcome with full consent.

More generally we will prove

Theorem 2.8.

For any possible combination of consenters, the associated EADAM outcome may be obtained by forming an appropriately designed coalition that improves on SOSM.

The intuition behind this is that “accomplices” can be viewed as interrupters who consent to waive their priority so that they do not start a rejection chain. By each accomplice waiving his priority - those on the rejection chain are given the opportunity to be accepted into better schools on their lists without violating the priorities of the accomplices.

Proof of Theorem 2.8.

Let and be the sets of students and schools, respectively. Let be a given school choice problem for the pair , and let be the set of students who consent to waiving their priorities under the EADAM mechanism. Denote by and the SOSM and the EADAM outcome matchings of this problem, respectively. We will now construct a coalition which will result in the same outcome . First define the cabal set to be the set of all students whose assignments are different under and :

These are the students who benefit from the EADAM; they will also be the students who will benefit from the coalition . Since every student whose assignment changes under EADAM is in , we can partition into cabal loops. This is equivalent to the basic algebraic fact that any finite permutation can be written as the product of disjoint cycles. Hence an elementary algorithm to decompose into its individual cabal loops can be described as follows:

  • Step 0: Define a permutation of by setting ( points to ) if . In words, points to if EADAM matches to the school that SOSM matches .

  • Step 1: Pick a student and label her . Then let be the student and more generally label . This process will stop at some with as is a finite permutation. Then is a cabal loop.

  • And in general,

  • Step k, k 1: Pick a student who has not yet been assigned to a cabal loop and label her . If none exists then the algorithm stops. Otherwise, label as and more generally label . This process stops at some with as is finite. Then is a cabal loop.

Note that the algorithm has to stop because is finite. Furthermore each student in shows up in exactly one round and hence in exactly one cabal loop, because is invertible.

Next we describe how to form the accomplice set . A student will be in if and only if the following two conditions are both satisfied:

  • , or equivalently, consents to waive her priorities in the EADAM; and

  • There is a school such that is a last interrupter pair at some round of EADAM.

The new preference profile for an accomplice will be of the form

where

  • if , then , and

  • if , .

Here we are using the notation of §§2.2 where (respectively ) is the list of schools on ’s preference profile to the left (respectively to the right) of his stable assignment .

Finally Theorem 2.3 allows us to conclude that the outcome matching of will be as follows: for all , and for in some cabal loop in . But then and we are done. ∎

In order to see the analogy between accomplices and interruptors, we analyze a minor modification of from §§2.2 which we label : Let and be given with the following preference and priority structures, respectively:

We now run the SOSM algorithm for (this is also Round 1 for EADAM assuming full consent):

Round 1
Step 1 ,
Step 2 ,
Step 3 ,
Step 4 ,
Step 5 ,
Step 6 ,
Step 7 ,
Step 8 ,
Step 9 ,
Step 10 ,
Step 11 ,
Step 12

The SOSM outcome is:

Note that student is an interrupter for (causes to reject in step 4 and is rejected herself by when comes along, in step 8.) Student is an interrupter for (causes to reject in step 2 and is rejected herself by when comes along, in step 4.) Student is an interrupter for (causes to reject in step 6 and is rejected herself by when comes along, in step 10.) Student is an interrupter for (causes to reject in step 1 and is rejected herself by when comes along, in step 5.) Student is an interrupter for (causes to reject in step 5 and is rejected herself by when comes along, in step 9.) Thus is the last interrupter pair. We remove from ’s preference list.121212In each round of EADAM, there may be multiple interrupter pairs. However, only the LATEST interrupter pair is used for the next round. For example, in Round 1, there are 5 interrupter pairs, but we only remove the latest interrupter pair .

Round 2
Step 1 ,
Step 2 ,
Step 3 ,
Step 4 ,
Step 5 ,
Step 6 ,
Step 7 ,
Step 8 ,
Step 9 ,
Step 10 ,
Step 11

The outcome is the same in Round 2. The pair is the last interrupter pair. We remove from ’s preference list.

Round 3
Step 1 ,
Step 2 ,
Step 3 ,
Step 4 ,
Step 5 ,
Step 6

The outcome is different in Round 3; , and ’s assignments have changed. This time is the last interrupter pair. We remove from ’s preference list.

Round 3
Step 1

The last round (Round 4) takes only one step; everybody gets matched with his or her first choice. (Of course has had to make adjustments to his profile multiple times; his new first choice was in fact his fourth truthful choice.) Thus EADAM with full consent131313In fact we only used the consent of . returns the following matching:

Can we find a coalition that will output this same outcome? Indeed yes! The cabal will be the set and the singleton accomplice will be . The set for will be . Note that there are two cabal loops: and .

Note that the coalition we create (and equivalently the EADAM with full consent) in this problem corresponds to a drastic improvement in the preference index. The preference index of this outcome is 3 while the preference index of the original SOSM outcome was 11.

There are indeed other coalitions that could be used for the same SCP. Take for instance the cabal to be and let to be the singleton accomplice set. Then and we get:

The preference index for this matching is 7. Thus it is still an improvement by this measure on the SOSM, but the larger cabal of the previous example (which is equivalent to EADAM with ’s consent) is more optimal with respect to reducing the preference index. On the other hand it may be interesting to observe that this outcome cannot be obtained via EADAM no matter which students consent. This is due to the fact that once consents to waive his priorities, he has to consent fully. In other words, any other Pareto improvement involving the interrupter pairs he was a part of will also be made. This in particular implies that the converse of Theorem 2.8 is not true.

Looking more closely at , we notice that in practice what we have done amounts to changing the preferences for to ; this way we were able to give their first choice without making worse off. However, we can alternatively change the preferences for and give their first choice and her 4th choice which is essentially the same matching (also with preference index 3!) except that is worse off now than in the SOSM matching.

Student seems to be set to lose out from the beginning, even in the SOSM matching. We may ask whether this is due to the fact that , unlike , is not highly prioritized at any school, and how the fact that is a hopeless student relates to this situation. The crucial point is that the matching obtained via the coalition formed by changing ’s preferences, and including everyone else in the cabal, or equivalently the EADAM outcome, (Pareto) dominates the original SOSM, while the matching obtained when is made to change his preference list does not. Changing ’s preferences would be more objectionable than changing ’s preferences because the associated matching indeed harms somebody when compared to their stable assignment under SOSM, which is viewed as a baseline assigning an initial endowment to each student.

2.5. Another efficiency oriented mechanism: TTC and coalitions

We now look for a relationship between the Top Trading Cycles mechanism and SOSM with the coalition efficiency improvement. When we incorporate coalitions into the SOSM, we alter preferences so that some students have improved placements and no students have worse placements (relative to their standard assignments under SOSM). We now seek to understand if we can mimic the outcome of TTC via some coalition/preference manipulation of SOSM.

We begin with an example in which the TTC and SOSM find two different matchings, yet a coalition can be formed so that SOSM with this coalition results in the same matching as the TTC. Let us look once again at from §1: There are three schools, , three students , and only one seat at each school. Student preferences and school priorities are given by:

Here, the SOSM finds the matching (of preference index ):

The TTC finds a matching that (Pareto) dominates (and has preference index ):141414Recall that this is the same as the EADAM outcome in the case of full consent: .

If modifies her preference list such that is her first choice, then each student has a distinct first choice. Then SOSM assigns all their first choice, which yields the same matching as the TTC output for the original setup. The relevant coalition is given by and .

Similarly if we run the TTC algorithm on , we see that the outcome is the same as that obtained when we use the coalition with the cabal (the cabal loop is ), the accomplice set , and the set for . In other words the TTC outcome is equivalent to the coalition-adjusted outcome of SOSM with being the only person who needs to modify her preference profile (§§2.2, also see §§3.1).

With these examples as background we now prove:

Theorem 2.9.

One cannot always obtain the TTC outcome by coalition adjustments to SOSM.

Proof.

We construct a counterexample. We modify slightly and call the new setup . Assume there are three schools, , three students , and there are two seats at and one seat each at and . School priorities and student preferences are as in :

Here the SOSM matching is:

and it has preference index .

If we run the TTC for , we find one cycle in the first round: