Efficient and Essentially Stable Assignments

Efficient and Essentially Stable Assignments Andrew Kloosterman Department of Economics University of Virginia Peter Troyan Department of Economics University of Virginia August 11, 2016 Abstract Many institutions that must assign a group of objects to a group of agents on the basis of priorities (the canonical example of which is school choice) desire the assignment to be stable, ie, no agent should claim an object because she has higher priority than the agent to whom it is assigned However, it is well-known that stable matchings are in general Pareto inefficient, forcing upon designers a difficult trade-off This paper alleviates this tension between efficiency and stability by defining an assignment to be essentially stable if any claim an agent has to an object she prefers is vacuous, in the sense that it initiates a chain of reassignments that ultimately results in the initial claimant losing the object to a third student with even higher priority We then show that an essentially stable and Pareto efficient assignment always exists, and can be found using Kesten s efficiency adjusted deferred acceptance (EADA) algorithm A main practical advantage to our definition is its simplicity: explaining to agents why their claims are vacuous becomes straightforward, even to non-experts who may not fully understand the inner workings of the mechanism Our results give strong support for the use of EADA in practical applications as a way to improve upon standard mechanisms, achieve Pareto efficiency, and still adhere to the general principle that motivated stability in such markets in the first place PO Box 400182, Charlottesville, VA, 22904 Emails: ask5b@virginiaedu and troyan@virginiaedu 1

1 Introduction It is well known that there is a tradeoff between efficiency and equity in matching problems The celebrated deferred acceptance (DA) algorithm of Gale and Shapley (1962) always produces an equitable match, and in fact produces the most efficient equitable match among all equitable matches However, it does not always produce a Pareto efficient match: there may be non-equitable matches that Pareto dominate it Are all non-equitable matches equally non-equitable? Here, we argue that the answer is no, and in fact there is a non-equitable match that is essentially just as equitable as classically defined equitable matches There are many real-world examples of problems that fit into this framework, but perhaps the largest and most important is public school choice as instituted in many cities across the United States and around the world School choice has developed as a way to give students access to more schools beyond simply the school closest to their home However, when the demand for a school is greater than its capacity, school districts must decide which students are to be admitted To do so, students are given certain priorities at all of the schools How the priorities are determined varies across applications, but common characteristics used include proximity to the school, socioeconomic status, sibling attendance, test scores, languages spoken, and so forth After an assignment is implemented, a student is said to have a (justified) claim of a seat at a school x if she prefers x to her assigned school and she has higher priority than a student who is assigned to x An assignment is called equitable (or fair, or is said to eliminate justified envy) if no student has any claims Equity is a normative property, but has a direct connection to the concept of stability that is familiar from the two-sided matching literature, and indeed has lead the one-sided matching literature to directly adopt stability as a desideratum (albeit sometimes under a different name) This connection has also lead to deferred acceptance (DA), one of the most well-studied mechanisms in the two-sided literature, becoming one of the most commonly used mechanisms in one-sided matching problems When ensuring stability is an important concern, then DA is the clear choice, as it well-known to Pareto dominate all other stable assignments In fact, many school districts, often on the advice of economists, have begun using DA as their student placement algorithm, including Boston, New York City, Denver, and New Orleans, among others This suggests that stability constraints do indeed play an important role in many real-world markets 1 1 The first mechanism used in New Orleans was an unstable, yet Pareto efficient, mechanism (the top trading cycles mechanism of Shapley and Scarf, 1974) However, this was only in place for a short time, and was soon abandoned in favor of DA, providing further evidence that stability-type constraints are important to real-world practitioners, even at the cost of efficiency The goal of this paper is to provide a practically useful alternative definition of stability that will eliminate the need to make such a choice 2

At first glance, the classic definition of stability (and its implications for the use of DA) seems very reasonable However it misses an important point: if an agent were to have a claim on an object (eg, a school seat), granting their claim displaces another agent This agent will then have to be assigned a new match, and, using the same justification as the initial agent, she can claim her favorite school at which she has high enough priority This will displace yet another agent, and so on Eventually this chain of reassignments will end when some agent is reassigned to an object that is available Now, it is possible that the agent who made the initial claim will be displaced by some agent further down the chain, in which case the agent who made the initial claim does not end up with the object she claimed Since the initial agent will ultimately not receive the object she laid claim to, we call such a claim vacuous Imposing stability (eg, by using DA) eliminates all claims, but the price of doing so is a loss of efficiency We propose a new definition of stability that expands the set of stable matches by allowing some claims to remain: namely, those that are vacuous While this is a weaker definition of stability, we argue that it captures the essential feature of the standard definition, in the sense that if an agent were to try to assert their claim, we can clearly explain to them why such a claim will be for naught For example, if some student i were to claim a seat at a school, one could convince her to relinquish this claim by walking her through its ultimate effects in the following way: Yes, I agree that your claim at school A is valid because you have higher priority than student j But, if I assign you to that school, then student j will need to be reassigned to school B, where her claim is equally as valid as yours was at A This will displace student k, who will then claim school A from you, and, because she has higher priority, I will need to give it to her So, while you may claim school A, ultimately, you will not receive it For this reason, we call an assignment with only vacuous claims an essentially stable match The reason for imposing only essential stability is that it allows us to implement a larger set of assignments, including some that may Pareto dominate any (classically) stable assignment Our main contribution is to show that a Pareto efficient and essentially stable assignment always exists 2 Furthermore, whenever DA produces an inefficient assignment, there is an essentially stable and Pareto efficient assignment that Pareto dominates it Thus, by requiring only essential stability, we eliminate the tension with efficiency, while still ad- 2 Stable matches themselves are also essentially stable, so the existence of an essentially stable match is trivial Our contribution is to show that by requiring only essential stability, we are able to reach Pareto efficiency 3

hering closely to the general priniciple behind imposing stability constraints in the first place We do this by showing that the efficiency-adjusted deferred acceptance (EADA) algorithm introduced by Kesten (2010) always produces an essentially stable match Intuitively, EADA as Kesten introduces it, works by asking students whether or not they will consent to having their priority violated, and only implements a Pareto improvement (relative to DA) if the student whose priority would be violated has consented Kesten (2010) shows that no student is ever harmed by consenting, and if all students do consent, then the EADA outcome is Pareto efficient Dur et al (2015) then strengthen the result of Kesten (2010) by showing that not only does consenting never harm a student in EADA, EADA is in fact the unique constrained efficient mechanism that Pareto dominates DA for which this is true 3 The main advantage of essential stability is that it is intuitively simple to describe to nonexperts, which we believe makes it a very appealing property for practical applications By imposing only essential stability, we can guarantee Pareto efficiency by directly implementing EADA under full consent, and if anyone protests ex-post, ensures that we can easily explain to them why their protest is vacuous Matchings are more concrete objects than mechanisms, which can be very abstract, especially to laypersons Given an explicit matching, it is easy to show a student that her claim is vacuous by walking her through the steps noted above for one specific preference profile It is much harder, if not nearly impossible, to prove to a student that consenting ex-ante will not hurt them, because of the sheer number of possible preferences that could be submitted to the mechanism While we can always just tell students that consenting is not harmful, they may not believe it 4 Additionally, if there is any degree of status quo bias, directly implementing an essentially stable match should lead to greater efficiency The reason is that, when asked ex-ante, not consenting is the default option, as there is no obvious reason to consent The status quo is thus not to consent, and, while not harmful, consenting is also not beneficial Directly implementing an essentially stable (and Pareto efficient) match effectively changes this status quo, and so will likely lead to higher welfare in practice Before closing, we highlight that while we show that EADA always produces an essentially 3 Constrained efficiency in their context essentially means that the final matching is not Pareto dominated by another matching that respects the priorities of any students who chose not to consent 4 While Kesten (2010) and Dur et al (2015) show that consenting is a dominant strategy under EADA, Li (2016) argues that not all dominant strategies are created equal: some dominant strategies are easy to understand as dominant, while others require a much higher degree of contingent reasoning and cognitive ability to fully understand that they are indeed dominant He introduces a formal definition of an obviously dominant strategy to capture those that fall into this latter class of easy strategies, and shows that in practice, participants are much more likely to play an obviously dominant strategy compared to one that is dominant, but not obviously so Under EADA, while consenting is a dominant strategy, it is not an obviously dominant strategy 4

stable match, essential stability itself is a property of a matching that is defined independently of any mechanism The aforementioned results of Kesten (2010) and Dur et al (2015) regarding consenting incentives for students are properties of mechanisms, and so there is no immediate correspondence between our results and theirs However, they are clearly closely related, and we view all of these results as complementary, in the sense that each gives a strong justification for Kesten s EADA as the mechanism of choice when trying to Pareto improve on the DA assignment 5 The rest of this paper is organized as follows Section 2 outlines the model and gives our new definition of essential stability Section 3 provides the main results Section 4 places our paper in the context of the related literature Section 5 concludes Proofs of minor lemmas not found in the main text can be found in the appendix 2 Preliminaries 21 Model There is a set of agents S who are to be assigned to a set of objects C Each i S has a strict preference relation P i over C and each c C has a strict priority relation c over S Let P = (P j ) j S denote a profile of preference relations, one for each agent, and = ( ) c C the profile of priority relations Each c C has a capacity q c which is the number of agents in S that can be assigned to it Let q = (q c ) c C denote a profile of capacities We assume that c C q c S A market is then a tuple (S, C,, P, q) We assume S, C,, and q are fixed throughout the paper and will associate a market with its preference profile P Probably the best-known application of this model is when S is a set of students and C is a set of schools (or colleges ) In this case, the priorities are determined by state/local laws and/or criteria determined by each school district While we will generally use the student/school terminology for concreteness, it should be noted that the model can be applied to many other real-world assignment problems Examples include the military assigning cadets to branches, business schools assigning students to projects, universities assigning students to dormitories, or cities assigning public housing units to tenants 6 5 Morrill (2016) provides another independent (but also related) justification for EADA, which is discussed in Section 4 6 For more detail on these markets, see Sönmez (2013), Sönmez and Switzer (2013), Fragiadakis and Troyan (2016), Chen and Sönmez (2002), Chen and Sönmez (2004), Sönmez and Ünver (2005), Sönmez and Ünver (2010), and Thakral (2015) Essential stability can also be useful in two-sided markets as well See Section 4 for a discussion of this point 5

A matching is a correspondence µ : S C S C such that, for all (i, c) S C, µ(i) C, µ(c) S, and µ(s) = c if and only if s µ(c) 7 A matching ν Pareto dominates a matching µ if ν(i)r i µ(i) for all i S, and ν(i)p i µ(i) for at least one i S 8 A matching µ is Pareto efficient if it is not Pareto dominated by any other ν µ Note that Pareto efficiency is evaluated only from the perspective of the students S, and not the schools C This is a standard view in the mechanism design approach to object assignment and school choice, beginning with the seminal papers of Balinski and Sönmez (1999) and Abdulkadiroğlu and Sönmez (2003) In addition to Pareto efficiency, in many applications (particularly in school choice), market designers also care about fairness (also called no justified envy) Typically, a matching is deemed unfair if student i desires an school c and has higher priority at c than some other student who is assigned there Fairness is the natural counterpart of the more familiar concept of stability from the two-sided matching literature Formally, given a matching µ, we say student i claims a seat at school c if (i) cp i µ(i) and (ii) either µ(c) < q c or i c j for some j µ(c) We will sometimes use (i, c) to denote i s claim of c If no student claims a seat at any school, then we say µ is stable 9 Let M denote the space of all possible matchings, and P denote the space of all possible preference relations A mechanism ψ : P S M is a function that assigns a matching to each possible preference profile that can be submitted by the students 10 For any mechanism ψ and preferences P, ψ(p ) is the matching output by ψ when the submitted preferences are P We denote student i s assignment at ψ(p ) by either ψ i (P ), or, if the preferences are understood, µ ψ (i) Mechanism ψ is said to be Pareto efficient if ψ(p ) is a Pareto efficient matching for all P Mechanism ψ is said to be stable if ψ(p ) is a stable matching for all P 7 We assume all students are assigned to a school and vice-versa While in practice some students may prefer taking an outside option to some schools, our model is without loss of generality, as we could simply model the outside option as a particular school o C with capacity q o S 8 R i denotes the weak part of i s preference relation P i Given the assumption that preferences are strict, ar i b if and only if a = b 9 Some papers break this definition into two parts, nonwastefulness and fairness We use stability because we think this terminology is more familiar, and will serve to highlight the connection of our results with the broader literature An additional important property is individual rationality, which says that no student is assigned to a school that she disprefers to taking an outside option However, as noted in footnote 7, we can model the outside option as a particular school o C that has enough capacity for all students Then, individual rationality is implied by nonwastefulness 10 A mechanism can also depend on S, C,, and q but we have assumed those are fixed for the remainder of the paper and so suppress the dependence of ψ on them 6

22 Motivating example In this section, we provide an example that exhibits a case where there are multiple unstable matchings that Pareto dominate DA, and use this as a motivation for our formal definition of essential stability, which is provided immediately after To do so, we first formally introduce the (student-proposing) deferred acceptance mechanism of Gale and Shapley (1962), which is one of the benchmark algorithms that forms the foundation for both the theory and practice of a myriad of matching markets DA operates as follows Deferred acceptance (DA) Step 1 Every student i applies to the first school on her preference list P i Each school tentatively accepts the highest priority students up to their capacity who have applied All other students are rejected Step k, k 2 Every student rejected in step k 1 applies to her favorite school among those that she has not yet applied to Each school keeps the highest priority students among the new applicants and the students tentatively held from step k 1 (if any) up to their capacity All other students are rejected Once there are no rejected students, the algorithm terminates and the tentative matches are made permanent For preferences P, the matching produced by the DA algorithm is denoted as either DA(P ), or, if the preferences are understood, as µ DA Student i s assigned school under the DA matching is denoted as either DA i (P ), or, if the preferences are understood, as µ DA (i) Deferred acceptance is an enormously successful mechanism in the field because it produces the so-called student-optimal stable match: that is, any other stable match ν is Pareto dominated by µ DA The prevalence of DA in the field suggests that such stability constraints do play an important role in many settings However, it is well-known that ensuring stability comes at the price of efficiency: DA is in general not a Pareto efficient mechanism The example below illustrates this point, and serves to motivate our new definition of stability Example 1 Let there be 5 students S = {i 1, i 2, i 3, i 4, i 5 } and 5 schools C = {A, B, C, D, E}, each with capacity 1 The priorities and preferences are given in the following tables A B C D E i 1 i 2 i 3 i 4 i 2 i 3 i 4 i 5 i 4 i 1 i 2 i 3 P i1 P i2 P i3 P i4 P i5 B C B A D A A D C E B C D 7

The table on the right indicates three different potential matchings, a matching µ (denoted by boxes ), and two Pareto efficient matchings µ (denoted by stars ) and µ (denoted by daggers ) The DA matching µ DA in this example is µ which therefore can readily be shown to be stable It is, however, not Pareto efficient: it is easy to see that it is Pareto dominated by both µ and µ This of course implies that µ and µ are both unstable For instance, under µ, student i 4 claims the seat at C (because i 4 C i 2 = µ (C)) and under µ, student i 3 claims the seat at B (because i 3 B i 1 = µ (B)) and student i 5 claims the seat at D (because i 5 B i 3 = µ (D)) There are simpler examples to show that the DA match may not be Pareto efficient We present this one to illustrate the main point of our paper, which is that not all instability is the same We argue that µ is truly unstable while µ is not To understand our argument, consider µ first Suppose student i 3 protests and claims the seat at school B If we grant i 3 s claim and assign her to B, then student i 1 becomes unmatched Student i 1 must be assigned somewhere, and (using the same logic as i 3 ), she will ask to be assigned to A, her next most-preferred school where she has higher priority than i 4 who is matched to A Granting i 1 s claim just as we did i 3 s, she is assigned to A and now student i 4 is unmatched Student i 4 then asks for C, which is her most preferred school where she has high enough priority to be assigned Student i 2 is now unmatched, and asks for B, 11 which means student i 3 is removed from B In summary, student i 3 starts by claiming B If her request is granted based on the fact that i 3 B i 1, then we should also grant the next request of i 1, since he has the same justification for claiming A as i 3 did for claiming B Continuing, we see that ultimately another student with higher priority than i 3 at B (in this case i 2 ) ends up claiming it, and so i 3 s initial claim is vacuous A protest from i 5 in which i 5 claims the seat at D also begins a chain of reassignments where eventually i 4 takes D away, so i 5 s claim is also vacuous Now let us contrast this with the instability found in matching µ Similarly, assume that student i 4 protests and claims the seat at C, and this request is granted Following similar logic to the above, i 2 then asks for B, and i 3 asks for D This is the end of our reassignments because D is the school that i 4 gave up to claim C In this case, the original claimant s (student i 4 ) request does not result in her ultimately losing the school she claimed to a higher priority student, and therefore this claim is not vacuous in the manner that the claims the seat at under µ were Thus, both µ and µ are unstable according to the classical definition, but they are 11 Note that her next most preferred school is A, but school A is now assigned to i 1 and i 1 A i 2, so she cannot get A and must go to B 8

unstable in different ways (which will be made more precise in a moment), and this difference is relevant for the set of matchings that should be deemed feasible While student i 3 can protest µ and request B, if she does so, she will ultimately be rejected from B What is more, while in practice students may not fully understand the workings of the mechanism, if a student were to protest, it would be very easy to walk her through the above chain of reassignments to show her that granting her request would ultimately not be beneficial to her, and therefore convince her to relinquish her claim Thus, to the extent that the goal of imposing stability is to prevent students from protesting an assignment, matchings like µ, while not fully stable in the classical sense, are essentially stable On the other hand, the instability of matching µ is a much stronger type of instability, because i 4 claiming C will ultimately benefit her, and so we would not be able to convince her to relinquish her claim Our new definition of stability is designed to capture this idea and, in the process, recover inefficiencies by expanding the set of feasible matchings to include those like µ, but exclude those like µ 23 Essentially stable matchings We now formalize the intuition from the previous example Recall that, fixing a matching µ, we use the notation (i, c) to denote i s claim of a seat at c Definition 1 Consider a matching µ and a claim (i, c) The reassignment chain initiated by claim (i, c) is the list i 0 c 0 i 1 c 1 i K c K where, i 0 = i, µ 0 = µ, c 0 = c and for each k 1: i k is the lowest priority student in µ k 1 (c k 1 ) µ k is the matching defined as: µ k (j) = µ k 1 (j) for all j i k 1, i k, µ k (i k 1 ) = c k 1, and student i k is unassigned c k is student i k s most preferred school at which she can claim a seat under µ k and terminates at the first K such that µ K (c K ) < q c K We should note a few features of assignment chains First, a student i may claim a seat at a school c either because c is not filled to capacity, or because she has higher priority 9

than someone currently assigned there Whenever the former occurs, that is the end of the reassignment chain (if this occurs immediately, then the original matching µ was not nonwasteful) Second, the definition of a reassignment is closely related, but slightly different from, the common definition of a rejection chain The main difference is that in a rejection chain, when a student is rejected from a school, she applies to the next school on her preference list In a reassignment chain, she may in principle apply to a school that is more preferred than the school she is rejected from Note that a school or a student can appear multiple times in a reassignment chain 12 If the initiating student i appears later in the reassignment chain then the reassignment chain returns to i If the reassignment chain returns to i, then i will ultimately be removed from the school c that she claimed initially by some student with higher priority When this is the case, we say that claim (i, c) is vacuous Definition 2 At matching µ, if all claims (i, c) are vacuous, then µ is essentially stable If there exists at least one claim that is not vacuous, we say µ is strongly unstable Returning to Example 1, we can check that µ is not essentially stable, because the reassignment chain initiated by (i 4, C), i 4 C i 2 B i 3 D, does not return to student i 4, and hence the claim (i 4, C) is not vacuous In matching µ, on the other hand, the claims (i 3, B) and (i 5, D) are vacuous, because the reassignment chains are respectively i 3 B i 1 A i 4 C i 2 B i 3 D, and i 5 D i 3 B i 1 A i 4 C i 2 B i 3 C i 4 D i 5 E, both of which return to the initial claimant Similarly to classical stability, a mechanism ψ is said to be essentially stable if ψ(p ) is an essentially stable matching for all P If ψ is not essentially stable, then we say it is strongly unstable 12 Note also that reassignment chains are well-defined (ie, they must end in finite steps) This is because, if a student i k is rejected from a school c k 1, then µ k (c k 1 ) must be filled to capacity with higher priority students For all k > k, the lowest priority student in µ k (c k 1 ) only increases, and thus, if i k is ever rejected again later in the chain, the best school at which she can claim a seat is ranked lower than c k 1 Since students only apply to worse and worse schools as the chain progresses and preference lists are finite in length, the chain must end 10

3 Results It is obvious that any stable match is also essentially stable and so the existence of an essentially stable match is trivial The more interesting question is whether any other essentially stable matches exist The first theorem shows that, not only is the answer to this question yes, but further, we can always find an essentially stable match that is also Pareto efficient Theorem 1 An essentially stable and Pareto efficient matching exists for every market (S, C, P,, q) The proof of this theorem will immediately follow by showing the efficiency-adjusted deferred acceptance (EADA) mechanism of Kesten (2010) always results in an essentially stable match 13 To do this, we actually use an alternative mechanism of Tang and Yu (2014) that is outcome-equivalent to EADA: the simplified efficiency adjusted deferred acceptance (SEADA) mechanism To describe the algorithm, we first need the following definition Definition 3 School c is underdemanded at matching µ if µ(i)r i c for all i Intuitively, school c is underdemanded at µ if nobody desires c relative to their assignment µ(i) In the running of DA, the set of underdemanded schools are the schools that never reject a student With this definition, we can formally define the SEADA algorithm Recall that DA(P ) denotes the DA outcome under preferences P, and DA i (P ) denotes student i s DA assignment The simplified efficiency-adjusted deferred acceptance (SEADA) mechanism Round 0 Compute DA(P ) Identify the schools that are underdemanded, and for each student to these schools, make their assignments permanent Remove these students and their assigned schools from the market Round r 1 Compute the DA outcome on the submarket consisting of those students who still remain at the beginning of round r Identify the schools that are underdemanded, and for each student at these schools, make their assignments permanent Remove these students and their assigned schools from the market Let µ 0 = DA(P ), and, for r 1, let µ r denote the matching at the end of round r, defined as follows: if i was removed from the market prior to the beginning of round r, then µ r (i) = µ r 1 (i); if i remains in the market at the beginning of round r, then µ r (i) is the 13 More precisely, we directly implement the Pareto efficient version of the EADA mechanism in which all students consent 11

school she is assigned at the end of DA on the round r submarket We define µ SEADA = µ R, where round R is the final round of the above algorithm The formal theorem we will prove is a slightly stronger statement than what is needed to show Theorem 1, which is that, not only is the SEADA outcome essentially stable, every intermediate matching of SEADA is also essentially stable While both the formal statement of Theorem 2 below and its proof rely extensively on the specific structure of the SEADA algorithm as defined by Tang and Yu (2014) (as opposed to the description of the EADA algorithm as it is defined by Kesten, 2010), 14 SEADA is outcome-equivalent to EADA, and so, by Theorem 2, the final outcome of EADA is also essentially stable (and Pareto efficient) Theorem 2 Every matching µ r for r = 0,, R is essentially stable Proof Fix r Consider the matching at the end of round r of SEADA, µ r, and choose an arbitrary claim (i, c) 15 Let Γ denote the reassignment chain initiated by this claim We will show that student i must be rejected from c at some point in Γ, and hence the claim (i, c) is vacuous, and µ r is essentially stable We will need the following lemma, part (i) of which is due to Kojima and Manea (2010) To state it, say that a preference relation P i is a monotonic transformation of P i at c C if br ic = br i c Preference profile P is a monotonic transformation of P at a matching µ if P i is a monotonic transformation of P i at µ(i) for all i In words, P is a monotonic transformation of another preference profile P at a matching µ if, for all i, the ranking of µ(i) only increases in moving from P i to P i Lemma 1 (i) If P is a monotonic transformation of P at DA(P ), then DA i (P )R ida i (P ) for all students i S (ii) If P is a monotonic transformation of P at DA(P ), then DA i (P )R i DA i (P ) for all students i S Now, consider again the claim (i, c) under µ r For student i to have a claim, then she must have been removed in a round strictly earlier than r To understand why, consider some student j who is still in the market at the beginning of round r We argue that she cannot claim a seat at any school a C First, note that she cannot claim a seat at any school still in the market at the beginning of round r, because DA on the round r submarket is stable Thus, consider a school a that is not still in the market at round r, and suppose a was removed in some round r < r By construction, a was underdemanded at the end of 14 This is especially true for the proofs of the key lemmas found in the appendix 15 If there are no claims, then the matching is classically stable, and so is also essentially stable trivially Also, all matchings µ r are nonwasteful, and so any claim (i, c) must be because there exists some j µ r (c) such that i c j 12

round r which means µ r (j)r j a Since j was removed later, µ r (j) a and so µ r (j)p j a Tang and Yu (2014, Lemma 2) show that j s assignment only gets better as rounds progress so µ r (j)r j µ r (j) This implies that µ r (j)p j a, and so j will not claim a So, i must have been removed in a round strictly earlier than r; denote this round by ˆr Define an alternative preference profile P ˆr as follows: for any student j removed before round ˆr, Pj ˆr ranks their assignment µˆr (j) first, and the remaining schools in the same order as the true P j ; for all j not removed before round ˆr, Pj ˆr = P j Note that this is a simple way to describe preferences so that DA(P ˆr ) = µˆr 16 Define a second preference profile P j as follows: for each j i, Pj ranks µ r (j) first, and every other school is listed in the same order as the true P j, while for student i, Pi ranks c first and the remaining schools in the order of the true P i Lemma 2 Student i s DA assignment at the end of round ˆr is the same as her DA assignment under P, which is the same as her assignment at the end of round r: DA i ( P ) = DA i (P ˆr ) = µ r (i) The lemma is formally proved in the appendix, but the main step is that P is a monotonic transformation of P ˆr at DA(P ˆr ) Now, it is well-known that the following is an alternative description of the DA mechanism (McVitie and Wilson, 1971; Dubins and Freedman, 1981): DA At each step t, arbitrarily choose one student among those who are currently unmatched, and allow him to apply to his most preferred a school that has not yet rejected him All schools other than a tentatively hold the same students as the last step School a holds the highest priority students up to their capacity among those held from last step combined with the new applicant and reject the (at most one) other In this new method, the choice of the applicant at each step is arbitrary, in the sense that the order in which they are chosen does not affect the final outcome So, for any fixed preference profile, one way to find the DA outcome is to have i be the last student chosen to enter the market That is, as long as there is some other student besides i who is tentatively unmatched, we always choose one of these students to make the next application Once all of these students have been (tentatively) assigned to a school, we allow i to enter by applying to the first school on his preference list Student i s application then initiates a rejection chain, where i applies to some school a, a rejects its lowest priority student i 1, i 1 applies to his most preferred school that has not yet rejected him, and so on, until we reach a school 16 Raising school µˆr (j) for all j removed prior to round ˆr to the top of her preferences is a way to effectively remove student j from the market, because no student who has not been removed prior to round ˆr will ever apply to such a school because it is underdemanded Tang and Yu (2014) show that this is an equivalent way to define the SEADA algorithm 13

a K with an empty seat, at which point the rejection chain (and the entire DA algorithm) end, and all tentative matchings are made final Now, let us consider running DA on the preference profile P in this manner where i enters the market last All the students j other than i are tentatively matched to µ r (j), and then i starts a rejection chain which, as the next lemma proves, turns out be identical to the reassignment chain Γ Lemma 3 The final matching at the end of Γ is DA( P ) By Lemma 3, the outcome of the reassignment chain Γ is the same as the outcome of DA( P ) By Lemma 2, DA i ( P ) = µ r (i), and so i s assignment at the end Γ is also µ r (i) The only way this is possible is if i is rejected from c at some point in Γ; that is, the claim (i, c) is vacuous Every round of SEADA produces an essentially stable match, including the final output of the mechanism, µ SEADA Tang and Yu (2014) show that µ SEADA is Pareto efficient, which completes the proof of Theorem 1 Furthermore, they show that, when DA is not efficient, the SEADA outcome Pareto dominates DA This leads to the following immediate corollary Corollary 1 If µ DA is not Pareto efficient, then µ SEADA is Pareto efficient, essentially stable, and Pareto dominates µ DA One additional remark is worth making We defined a claim (i, c) as vacuous if the induced reassignment chain returns to i and ultimately rejects her from c However, one might argue that maybe i claims a seat at a very good school and, while she is rejected from c in the reassignment chain, perhaps i s final match at the end of the chain is still better than i s initial match However, the proof Theorem 2 actually showed that not only does the reassignment chain return to reject i from c, but also i ends up with their initial match Hence, we equally could have defined a vacuous claim as the property that i s match at the end of the reassignment chain is the same as i s initial match and still obtained the same result We return to Example 1 to exhibit how SEADA runs and check directly that it s outcomes are essentially stable Example 1, continued Recall that there are five students S = {i 1, i 2, i 3, i 4, i 5 } and five schools C = {A, B, C, D, E} with 1 seat each The preferences and priorities are reprinted below for convenience: 14

A B C D E i 1 i 2 i 3 i 4 i 2 i 3 i 4 i 5 i 4 i 1 i 2 i 3 P i1 P i2 P i3 P i4 P i5 B C B A D A A D C E B C D To start SEADA, we first run the DA algorithm on this market using the true preferences P The round 0 match is µ 0 = µ DA = ( A B C D E i 1 i 2 i 3 i 4 i 5 ) Of course, there are no claims and so the match is essentially stable School E is the only underdemanded school, so student i 5 and school E are removed Running DA on the remaining students and schools, the round 1 outcome is ( ) µ 1 A B C D E = i 1 i 2 i 4 i 3 i 5 The only claim is (i 5, D) with reassignment chain i 5 D i 3 C i 4 D i 5 E which returns to i 5 and is therefore vacuous Now, school D is the only underdemanded school, so school D and student i 3 are removed Running DA on the remaining students and schools, the round 2 outcome is µ 2 = ( A B C D E i 4 i 1 i 2 i 3 i 5 ) All schools are underdemanded, and so the SEADA algorithm ends here, and the final output is µ SEADA = µ 2 This is the matching µ that we found before to be essentially stable Other mechanisms The major shortcoming of DA is its lack of Pareto efficiency Using SEADA instead is one way to recover these inefficiencies and reach a Pareto efficient outcome, but it is not the only way In this section, we consider other mechanisms that have been proposed as alternatives 15

to reach Pareto efficient outcomes It is no surprise that these mechanisms will not be stable; our question is whether they are essentially stable or strongly unstable Top Trading Cycles One of the most commonly proposed alternative mechanisms to DA is the top trading cycles (TTC) mechanism of Shapley and Scarf (1974) This mechanism was originally defined for so-called house allocation problems in which there are no priorities, but each agent owns one house It was extended to school choice with priorities by Abdulkadiroğlu and Sönmez (2003) Briefly, the mechanism works as follows Every student starts by pointing to her favorite school Every school starts by pointing to its highest priority student In the resulting graph, there is at least one cycle i 1 c 1 i 2 c 2 i K where i K = i 1 For each cycle in the graph, implement all trades (ie, assign i 1 to c 1, i 2 to c 2, etc) Remove any students involved in a trade from the market, along with the schools they are assigned In the next round, all students point to their favorite school still remaining, and all schools point to the highest priority student left in the market Again remove any cycles and make assignments as in round 1 In each round, there must be at least one cycle, and the algorithm stops when all students have been assigned a school The intuition behind this mechanism is that priority at a school is interpreted as an ownership right to a seat at that school, which can be traded away: if i has high priority at j s first choice, and j has high priority at i s first choice, then TTC allows i and j to trade, even though this may violate the priority of some third student k who is not involved in the trade By continually making all mutually beneficial trades, we eventually end up at a final assignment that is Pareto efficient However, as TTC ignores the priorities of students not involved in a cycle, it will generally be unstable It turns out that TTC is not just unstable, but is in fact strongly unstable Theorem 3 The top trading cycles mechanism is strongly unstable Proof The proof is by example Let there be 5 students S = {i 1, i 2, i 3, i 4, i 5 } and 5 schools C = {A, B, C, D, E} with one seat each The preferences and priorities are given in the table below 16

A B C D E i 2 i 3 i 1 i 4 i 1 i 4 i 2 i 3 i 5 i 1 i 3 i 4 P i1 P i2 P i3 P i4 P i5 B C C A A A A D B E D B D i 3 The TTC outcome for this market is ( µ T T C A B C D E = i 2 i 1 i 3 i 4 i 5 ) Now, consider student i 2, who claims a seat at school C The reassignment chain initiated by this claim is: i 2 C i 3 B i 1 A We see that the i 2 s claim is not vacuous, and so µ T T C is strongly unstable For completeness, we also calculate the SEADA outcome for the above market matching produced by SEADA is ( ) µ SEADA A B C D E = i 4 i 1 i 2 i 3 i 5 The Under µ SEADA, students i 1, i 2, and i 4 are getting their top choice, and so will not claim any seat Student i 3 desires school C, but i 2 C i 3, and so he has no claim Student i 5 does have a claim at A The reassignment chain initiated by the claim (i 5, A) i 5 A i 4 B i 1 A i 5 E, which we see is vacuous, and so µ SEADA is essentially stable In sum, the main argument for the use of TTC is that it results in a Pareto efficient outcome SEADA also results in a Pareto efficient outcome, and so on this dimension the two perform equivalently However, on the stability dimension, SEADA outperforms TTC, because SEADA is essentially stable while TTC is strongly unstable DA + TTC One drawback of using TTC is that it completely ignores the priorities of students not involved in a trade, and so can produce very unstable assignments Another alternative is 17

to first calculate DA, and then allow students to trade by running TTC, using the initial DA assignments as the ownership rights in the TTC mechanism 17 The advantage of this is that it will guarantee that students always receive an assignment that is no worse than their DA school, which is not guaranteed by using TTC alone However, the next theorem shows that this, too, is a strongly unstable mechanism Theorem 4 The DA+TTC mechanism is strongly unstable Proof Consider again Example 1 The DA outcome is ( ) µ DA A B C D E = i 1 i 2 i 3 i 4 i 5 Now, TTC is applied by first giving each student her DA assignment as her initial endowment, and then creating a graph in which each student points to her favorite school, and the school points to the student who is endowed with it Carrying this out, we see that there is only one cycle: i 2 C i 3 B i 2, and so we implement this trade between i 2 and i 3 and remove them from the market with their assignments After this, all remaining cycles are self-cycles, and so the final allocation is ( ) µ DA+T T C A B C D E = i 1 i 3 i 2 i 4 i 5 This was the matching µ introduced in Example 1 which we found had a non-vacuous claim and so we have that µ DA+T T C is strongly unstable 4 Related literature The tension between stability and efficiency in priority-based allocation problems has been pointed out in many papers, starting with Balinski and Sönmez (1999) and Abdulkadiroğlu and Sönmez (2003) The first papers to point out this tension showed that Pareto efficiency and stability are not compatible for every possible instance that may arise; of course, this does not mean that they are never compatible If the class of problems for which this tension is present is small, it may not be relevant in practice Ergin (2002) and Kesten 17 Similar improvement cycles algorithms are proposed by Erdil and Ergin (2008) to recover inefficiencies in school choice caused by ties in priority among students 18

(2006) investigate this question and provide characterizations of the priority structures under which efficiency and stability are compatible Unfortunately, these conditions turn out to be quite restrictive and are rarely satisfied in practice These negative results motivate the approach that we take, which is to find a new definition of stability that includes Pareto efficient assignments without giving up the reason practitioners impose stability in the first place, which is to prevent students from protesting their assignment based on their priority The papers most related to ours are Dur et al (2015) and Morrill (2016) Dur et al (2015) generalizes EADA to a wider class of partially stable mechanisms that takes the set of allowable priority violations as a primitive and show that EADA is the unique (constrained) efficient mechanism that Pareto dominates DA and is no-consent-proof While clearly related, as mentioned in the Introduction, the difference between our results and those of Dur et al (2015) is that essential stability is a property of a matching, and so can be checked individually for any given matching, while no-consent-proofness is a property of a mechanism that takes students consent decisions as an input Dur et al (2015) (and Kesten (2010)) argue that Pareto efficiency should be achieved, because consenting never harms any student However, it also does not benefit them (they are indifferent), and for students who do not understand the details of the mechanism or may not fully trust it, they may likely default to no as a safe answer to this question (or simply because it is the default) The main advantage of essential stability for practical application is that it simplifies the EADA mechanism by not asking the students about priority violations ex-ante, and for anyone who objects to the final outcome, it ensures that they can be walked through a very clear, concrete example of why their claim is vacuous, rather than having to consent to an abstract idea beforehand Morrill (2016) introduces the concept of a legal assignment, where a student i s claim at a school c is not redressable (and thus disregarded) unless i can actually be assigned to c under some other legal assignment He shows that the EADA assignment is always legal, and so the concepts seem quite related However, as we show in Appendix B, they are formally distinct Moreover, Morrill s notion of legality is defined on a set of matchings F ; that is, one cannot say whether an individual matching is legal independently, but only whether an entire set of matchings F is legal, which depends on the way the matchings in F relate both to each other and to matchings outside of F Again, we think from a practical standpoint, essential stability is easier for the participants in the market to understand and therefore is advantageous for the reasons outlined above Still, we view the results of Dur et al (2015) and Morrill (2016) as complementary to ours, as each gives support for the EADA mechanism as a way to reach Pareto efficiency while still adhering to the general 19

spirit of classical stability 18 Two other closely related papers that introduce alternative stability notions are Alcalde and Romero-Medina (2015), who introduces the concept of τ fairness, and Cantala and Pápai (2014), who introduces the concepts of reasonable stability and secure stability These concepts are (loosely) based on the idea that a matching µ should be defined as stable, if, whenever a student i objects to her assignment µ(i) by claiming some other school c, then, for any other matching ν such that ν(i) = c, some other student will object to ν 19 In other words, a matching is stable if, for every possible objection, there will be a counterobjection While the original motivation is similar, these stability concepts are distinct from ours This can be seen immediately by noting that both Alcalde and Romero-Medina (2015) and Cantala and Pápai (2014) show that the DA+TTC mechanism satisfies their respective definitions of stability, while we showed in Section 3 that DA+TTC is not essentially stable (for example, the matching µ from Example 1 is τ-fair, reasonably stable, and securely stable according to their respective definitions, but is strongly unstable according to the definition used in this paper) Thus, not only are these concepts distinct from ours, they also select entirely different mechanisms as the way to improve on DA: their results suggest DA+TTC, while ours, along with Dur et al (2015) and Morrill (2016), suggest EADA 20 Beyond one-sided matching applications like school choice, we also believe our concept of essential stability can be a useful concept in the two-sided matching literature, where both sides of the market have preferences and are strategic players The canonical example of this is the National Resident Matching Program (NRMP), which assigns newly graduated doctors to residency programs in the United States, and similar procedures used worldwide In the late 1990s, Alvin Roth and Eliot Peranson were enlisted to redesign the algorithm that was then in use (see Roth and Peranson (2002)) One of the main changes made to the algorithm was to benefit the residents over the hospitals by switching to the doctor-optimal stable match instead of the hospital optimal stable match If instead only essential stability was imposed, it would be possible to implement a different assignment that benefited all of the applicants even further Stability in two-sided markets is generally imposed as a positive (rather than normative) criterion, because if a matching was not stable, then two agents 18 Other papers that study EADA in different contexts are Bando (2014), who shows that the EADA outcome is equivalent to a strictly strong Nash equilibrium outcome in the preference revelation game induced by standard DA, and Dogan (2014), who proposes a mechanism similar to EADA in the context of affirmative action 19 This idea is similar in spirit to the idea of bargaining sets introduced by Zhou (1994) for cooperative games 20 Afacan et al (2015) take an entirely different approach to weakening stability, called sticky-stability Sticky-stability is motivated by the fact that appealing a priority violation may be costly, and so students will only invest in appealing if they can greatly improve their assignment They then construct a variation of EADA that always produces a sticky-stable assignment 20