On the consistency of deferred acceptance when priorities are acceptant substitutable

In the context of resource allocation on the basis of responsive priorities, Ergin (2002) identifies a necessary and sufficient condition for the deferred acceptance rule to satisfy a consistency principle. In this note, we extend this result to the domain of substitutable priorities, complementing results of Kojima and Manea (2010) and Kumano (2009).Financial support from Plan Nacional I+D+i (ECO2008–04784), the Consolider-Ingenio 2010 (CSD2006–00016) program, the Barcelona Graduate School of Economics and the Government of Catalonia (SGR2009–01142) is gratefully acknowledged

Object allocation via deferred-acceptance: strategy-proofness and comparative statics

We study the problem of assigning indivisible and heterogenous objects (e.g., houses, jobs, offices, school or university admissions etc.) to agents. Each agent receives at most one object and monetary compensations are not possible. We consider mechanisms satisfying a set of basic properties (unavailable-type-invariance, individual-rationality, weak non-wastefulness, or truncation-invariance). In the house allocation problem, where at most one copy of each object is available, deferred-acceptance (DA)-mechanisms allocate objects based on exogenously fixed objects' priorities over agents and the agent-proposing deferred-acceptance-algorithm. For house allocation we show that DA-mechanisms are characterized by our basic properties and (i) strategy-proofness and population-monotonicity or (ii) strategy-proofness and resource-monotonicity. Once we allow for multiple identical copies of objects, on the one hand the first characterization breaks down and there are unstable mechanisms satisfying our basic properties and (i) strategy-proofness and population-monotonicity. On the other hand, our basic properties and (ii) strategy-proofness and resource-monotonicity characterize (the most general) class of DA-mechanisms based on objects' fixed choice functions that are acceptant, monotonic, substitutable, and consistent. These choice functions are used by objects to reject agents in the agent-proposing deferred-acceptance-algorithm. Therefore, in the general model resource-monotonicity is the «stronger» comparative statics requirement because it characterizes (together with our basic requirements and strategy-proofness) choice-based DA-mechanisms whereas population-monotonicity (together with our basic properties and strategy-proofness) does not

Alternative characterizations of Boston mechanism

Kojima and Ünver (2011) are the first to characterize the class of mechanisms coinciding with the Boston mechanism for some priority order. By mildly strengthening their central axiom, we are able to pin down the Boston mechanism outcome for every priority order. Our main result shows that a mechanism is outcome equivalent to the Boston mechanism at every priority if and only if it respects both preference rankings and priorities and satisfies individual rationality for schools. In environments where each student is acceptable to every school, respecting both preference rankings and priorities is enough to characterize the Boston mechanism

Efficiency and stability under substitutable priorities with ties

Many assignment mechanisms appeal to a priority structure to determine how over-subscribed indivisible goods are assigned to unit-demand individuals. We study substitutable priorities with ties which not only nest important classes of priorities and preferences studied in the literature, but also allow us to formalize plausible priority structures not captured in previous literature. Efficiency is typically in conflict with respecting priorities (i.e., stability), and therefore the natural welfare objective is constrained efficiency. A generalization of the deferred acceptance process yields a stable assignment, but this outcome is not necessarily constrained efficient. We identify an easily verifiable sufficient condition for a stable assignment to be constrained efficient, which then leads to an algorithm to compute a constrained efficient assignment. Finally we illustrate practical applications of our framework and algorithm, including a widely studied matching problem with distributional constraints

Pareto Dominance of Deferred Acceptance through Early Decision

An early decision market is governed by rules that allow each student to apply to (at most) one college and require the student to attend this college if admitted. This market is ubiquitous in college admissions in the United States. We model this market as an extensive-form game of perfect information and study a refinement of subgame perfect equilibrium (SPE) that induces undominated Nash equilibria in every subgame (SPUE). Our main result shows that this game can be used to define a decentralized matching mechanism that weakly Pareto dominates student-proposing deferred acceptance

Essays on matching and preference aggregation

Cette thèse est une collection de trois articles dont deux portent sur le problème d’appariement et un sur le problème d’agrégation des préférences. Les deux premiers chapitres portent sur le problème d’affectation des élèves ou étudiants dans des écoles ou universités. Dans ce problème, le mécanisme d’acceptation différée de Gale et Shapley dans sa version où les étudiants proposent et le mécanisme connu sous le nom de mécanisme de Boston sont beaucoup utilisés dans plusieurs circonscriptions éducatives aux Etats-Unis et partout dans le monde. Le mécanisme de Boston est sujet à des manipulations. Le mécanisme d’acceptation différée pour sa part n’est pas manipulable mais il n’est pas efficace au sens de Pareto. L’objectif des deux premiers chapitres est de trouver des mécanismes pouvant améliorer le bien-être des étudiants par rapport au mécanisme d’acceptation différée ou réduire le dégré de vulnérabilité à la manipulation par rapport au mécanisme de Boston. Dans le Chapitre 1, nous étudions un jeux inspiré du système d’admission précoce aux Etats-Unis. C’est un système d’admission dans les collèges par lequel un étudiant peut recevoir une décision d’admission avant la phase générale. Mais il y a des exigences. Chaque étudiant est requis de soumettre son application à un seul collège et de s’engager à s’inscrire s’il était admis. Nous étudions un jeu séquentiel dans lequel chaque étudiant soumet une application et à la suite les collèges décident de leurs admissions dont les étudiants acceptent. Nous avons montré que selon une notion appropriée d’équilibre parfait en sous-jeux, les résultats de ce mécanisme sont plus efficaces que celui du mécanisme d’acceptation différée. vi Dans le Chapitre 2, nous étudions un mécanisme centralisé d’admissions dans les universités françaises que le gouvernement a mis en place en 2009 pour mieux orienter les étudiants dans les établissements universitaires. Pour faire face aux écoles dont les places sont insuffisantes par rapport à la demande, le système défini des priorités qui repartissent les étudiants en grandes classes d’équivalence. Mais le système repose sur les préférences exprimées pour départager les ex-aequos. Nous avons prouvé que l’application du mécanisme d’acceptation différée avec étudiant proposant aprés avoir briser les ex-aequos est raisonable. Nous appelons ce mécanisme mécanisme français. Nous avons montré que le mécanisme français réduit la vulnérabilité à la manipulation par rapport au mécanisme de Boston et améliore le bien-être des étudiants par rapport au mécanisme standard d’acceptation différée où les ex-aequos sont brisés de façon aléatoire. Dans le Chapitre 3, nous introduisons une classe de règles pour combiner les préférences individuelles en un ordre collectif. Le problème d’agrégation des préférences survient lorsque les membres d’une faculté cherchent une stratégie pour offrir une position sans savoir quel candidat va accepter l’offre. Il est courant de classer les candidats puis donner l’offre suivant cet ordre. Nous avons introuduit une classe de règles appélée règles de dictature sérielle augmentée dont chacune est paramétrée par une liste d’agents (avec répétition) et une règle de vote par comité. Pour chaque profile de préférences, le premier choix de l’agent en tête de la liste devient le premier choix collectif. Le choix du deuxème agent sur la liste, parmi les candidats restants, devient le deuxième choix collectif. Et ainsi de suite jusqu’à ce qu’il reste deux candidats auquel cas le comité vote pour classer ces derniers. Ces règles sont succinctement caractérisées par la non-manipulabilité et la neutralité sous l’extension lexicographique des préférences. Nous avons montré aussi que ces règles sont non-manipulables sous une variété d’extensions raisonable des préférences. Mots-clés : Appariement, mécanisme d’acceptation différée, mécanisme de Boston, mécanisme français, agrégation des préférences, règle non-manipulable, règle de dictature sérielle augmentée.This thesis is a collection of two papers on matching and one paper on preference aggregation. The first two chapters are concerned with the problem of assigning students to schools. For this problem, the student proposing version of Gale and Shapley’s deferred acceptance mechanism and a mechanism known as Boston mechanism are widely used in many school districts in U.S and around the world. The Boston mechanism is prone to manipulation. The deferred acceptance mechanism is not manipulable ; however, it is not Pareto efficient. The first two chapters of this thesis deal with the problem of either improving the welfare of students over deferred acceptance or reducing the degree of manipulation under Boston. In Chapter 1, we study a decentralized matching game inspired from the early decision system in the U.S : It is a college admission system in which students can receive admission decisions before the general application period. But there are two requirements. First, each student is required to apply to one college. Second, each student commits to attend the college upon admitted. We propose a game in which students sequentially make one application each and colleges ultimately make admission decisions to which students commit to accept. We show that up to a relevant refinement of subgame perfect equilibrium notion, the expected outcomes of this mechanism are more efficient than that of deferred acceptance mechanism. In Chapter 2, we study a centralized university admission mechanism that the French government has implemented in 2009 to better match students to viii university schools. To deal with oversubscribed schools, the system defined priorities that partition students into very coarse equivalence classes but relies on student reported preferences to further resolve ties.We show that applying student-proposing deferred acceptance mechanism after breaking ties is a reasonable procedure. We refer to this mechanism as French mechanism. We show that this mechanism is less manipulable than the Boston mechanism and more efficient than the standard deferred acceptance in which ties are broken randomly. In Chapter 3, we introduce a class of rules called augmented serial rules for combining individual preferences into a collective ordering. The aggregation problem appears when faculty members want to devise a strategy for offering an open position without knowing whether any given applicant will ultimately accept an offer. It is a commonplace to order the applicants and make offers accordingly. Each of these augmented serial rules is parametrized by a list of agents (with possible repetition) and a committee voting rule. For a given preference profile, the collective ordering is determined as follows : The first agent’s most preferred alternative becomes the top-ranked alternative in the collective ordering, the second agent’s most preferred alternative (among those remaining) becomes the second-ranked alternative and so on until two alternatives remain, which are ranked by the committee voting rule. The main result establishes that these rules are succinctly characterized by neutrality and strategy-proofness under the lexicographic extension. Additional results show that these rules are strategy-proof under a variety of other reasonable preference extensions

Efficiency in Matching Markets with Regional Caps: The Case of the Japan Residency Matching Program

In an attempt to increase the placement of medical residents to rural hospitals, the Japanese government recently introduced "regional caps" which restrict the total number of residents matched within each region of the country. The government modified the deferred acceptance mechanism incorporating the regional caps. This paper shows that the current mechanism may result in avoidable ineffciency and instability and proposes a better mechanism that improves upon it in terms of effciency and stability while meeting the regional caps. More broadly, the paper contributes to the general research agenda of matching and market design to address practical problems.medical residency matching, regional caps, the rural hospital theorem, sta- bility, strategy-proofness, matching with contracts

Two-Sided Matching Markets: Models, Structures, and Algorithms

Two-sided matching markets are a cornerstone of modern economics. They model a wide range of applications such as ride-sharing, online dating, job positioning, school admissions, and many more. In many of those markets, monetary exchange does not play a role. For instance, the New York City public high school system is free of charge. Thus, the decision on how eighth-graders are assigned to public high schools must be made using concepts of fairness rather than price. There has been therefore a huge amount of literature, mostly in the economics community, defining various concepts of fairness in different settings and showing the existence of matchings that satisfy these fairness conditions. Those concepts have enjoyed wide-spread success, inside and outside academia. However, finding such matchings is as important as showing their existence. Moreover, it is crucial to have fast (i.e., polynomial-time) algorithms as the size of the markets grows. In many cases, modern algorithmic tools must be employed to tackle the intractability issues arising from the big data era. The aim of my research is to provide mathematically rigorous and provably fast algorithms to find solutions that extend and improve over a well-studied concept of fairness in two-sided markets known as stability. This concept was initially employed by the National Resident Matching Program in assigning medical doctors to hospitals, and is now widely used, for instance, by cities in the US for assigning students to public high schools and by certain refugee agencies to relocate asylum seekers. In the classical model, a stable matching can be found efficiently using the renowned deferred acceptance algorithm by Gale and Shapley. However, stability by itself does not take care of important concerns that arose recently, some of which were featured in national newspapers. Some examples are: how can we make sure students get admitted to the best school they deserve, and how can we enforce diversity in a cohort of students? By building on known and new tools from Mathematical Programming, Combinatorial Optimization, and Order Theory, my goal is to provide fast algorithms to answer questions like those above, and test them on real-world data. In Chapter 1, I introduce the stable matching problem and related concepts, as well as its applications in different markets. In Chapter 2, we investigate two extensions introduced in the framework of school choice that aim at finding an assignment that is more favorable to students -- legal assignments and the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) -- through the lens of classical theory of stable matchings. We prove that the set of legal assignments is exactly the set of stable assignments in another instance. Our result implies that essentially all optimization problems over the set of legal assignments can be solved within the same time bound needed for solving it over the set of stable assignments. We also give an algorithm that obtains the assignment output of EADAM. Our algorithm has the same running time as that of the deferred acceptance algorithm, hence largely improving in both theory and practice over known algorithms. In Chapter 3, we introduce a property of distributive lattices, which we term as affine representability, and show its role in efficiently solving linear optimization problems over the elements of a distributive lattice, as well as describing the convex hull of the characteristic vectors of the lattice elements. We apply this concept to the stable matching model with path-independent quota-filling choice functions, thus giving efficient algorithms and a compact polyhedral description for this model. Such choice functions can be used to model many complex real-world decision rules that are not captured by the classical model, such as those with diversity concerns. To the best of our knowledge, this model generalizes all those for which similar results were known, and our paper is the first that proposes efficient algorithms for stable matchings with choice functions, beyond classical extensions of the Deferred Acceptance algorithm. In Chapter 4, we study the discovery program (DISC), which is an affirmative action policy used by the New York City Department of Education (NYC DOE) for specialized high schools; and explore two other affirmative action policies that can be used to minimally modify and improve the discovery program: the minority reserve (MR) and the joint-seat allocation (JSA) mechanism. Although the discovery program is beneficial in increasing the number of admissions for disadvantaged students, our empirical analysis of the student-school matches from the 12 recent academic years (2005-06 to 2016-17) shows that about 950 in-group blocking pairs were created each year amongst disadvantaged group of students, impacting about 650 disadvantaged students every year. Moreover, we find that this program usually benefits lower-performing disadvantaged students more than top-performing disadvantaged students (in terms of the ranking of their assigned schools), thus unintentionally creating an incentive to under-perform. On the contrary, we show, theoretically by employing choice functions, that (i) both MR and JSA result in no in-group blocking pairs, and (ii) JSA is weakly group strategy-proof, ensures that at least one disadvantaged is not worse off, and when reservation quotas are carefully chosen then no disadvantaged student is worse-off. We show that each of these properties is not satisfied by DISC. In the general setting, we show that there is no clear winner in terms of the matchings provided by DISC, JSA, and MR, from the perspective of disadvantaged students. We however characterize a condition for markets, that we term high competitiveness, where JSA dominates MR for disadvantaged students. This condition is verified, in particular, in certain markets when there is a higher demand for seats than supply, and the performances of disadvantaged students are significantly lower than that of advantaged students. Data from NYC DOE satisfy the high competitiveness condition, and for this dataset our empirical results corroborate our theoretical predictions, showing the superiority of JSA. We believe that the discovery program, and more generally affirmative action mechanisms, can be changed for the better by implementing the JSA mechanism, leading to incentives for the top-performing disadvantaged students while providing many benefits of the affirmative action program

Essays in Market Design

Thesis advisor: Utku UnverThis dissertation consists of two chapters. The first chapter: Dynamic reserves in matching markets with contracts. In this paper we study a matching problem where agents care not only about the institution they are assigned to but also about the contractual terms of their assignment so that they have preferences over institution-contractual term pairs. Each institution has a target distribution of its slots reserved for different contractual terms. If there is less demand for some groups of slots, then the institution is given opportunity to redistribute unassigned slots over other groups. The choice function we construct takes the capacity of each group of seats to be a function of number of vacant seats of groups considered earlier. We advocate the use of a cumulative offer mechanism (COM) with overall choice functions designed for institutions that allow capacity transfer across different groups of seats as an allocation rule. In applications such as engineering school admissions in India, cadet-branch matching problems at the USMA and ROTC where students are ranked according to test scores (and for each group of seats, corresponding choice functions are induced by them), we show that the COM with a monotonic capacity transfer scheme produces stable outcomes, is strategy proof, and respect improvements in test scores. Allowing capacity redistribution increases efficiency. The outcome of the COM with monotone capacity transfer scheme Pareto dominates the outcome of the COM with no capacity transfer. The second chapter: On relationships between substitutes conditions. In the matching with contracts literature, three well-known conditions on choice functions (from stronger to weaker)- substitutability, unilateral substitutability (US) and bilateral substitutability (BS) have proven to be critical. This paper aims to deepen our understanding of them by separately axiomatizing the gap between the BS and the other two. We first introduce a new “doctor separability” (DS) condition and show that BS, DS and irrelevance of rejected contracts (IRC) are equivalent to IRC and US. Due to Hatfield and Kojima (2010) and Aygün and Sönmez (2012), it is known that US, “Pareto separability” (PS), and IRC are equivalent to substitutability and IRC. This, along with our result, implies that BS, DS, PS, and IRC are equivalent to substitutability and IRC. All of these results are given without IRC whenever hospital choices are induced from preferences.Thesis (PhD) — Boston College, 2015.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Economics

Three Essays on Matching with Contracts

Thesis advisor: Tayfun SonmezThis dissertation consists of three theoretical essays. In all essays matching with contracts is a key factor. The first essay tries to explain effects of choosing primitives of the model and irrelevance of rejected contracts condition on some key existence theorems and results in matching with contracts literature. The second essay analyzes the properties of cumulative offer algorithm and presents an application of matching with contracts. It studies the achievability of responsive choices under a constrained setup. The last essay presents a new market design application of program-student matching where affirmative action policies are effective. The first essay develops a hospital-doctor many-to-one matching with contracts model. Doctor preferences over contracts are part of primitive of the model. The other primitive of the model, our first essay suggests, hospital choice functions on sets of contracts. The first essay shows that if choice functions of hospitals are primitives of the model, then existence theorems used in many papers do not hold even when they satisfy strongest conditions. As a remedy, we introduced Irrelevance of Rejected Contracts (IRC) which guarantees stability if it is satisfied along with one substitutes condition. Next, we show the relation between IRC and law of aggregate demand (LAD) conditions. Since LAD is satisfied by many application naturally, many models satisfying LAD and the strongest substitutes conditions are immune to our criticism. On the other hand, many of the new and exiting applications satisfy only weakened substitutes condition. Therefore, assuming IRC explicitly does not only make their proofs accurate and also close the gap between theory and application. The second chapter studies properties of cumulative offer algorithm under weakened substitutes condition. In this part we showed that in many-to-one matching with contracts problems order of proposals of COA does not change the outcome, under bilateral substitutes and IRC conditions. Also, bilateral substitutes and IRC conditions make COA equivalent to generalized deferred acceptance algorithm which produces the outcome in fewer steps. This chapter also presents a new application area of matching with contracts. We used cadet-branch matching problem in USMA. In this application our main objective is, for a given branch, increasing cadet quality without giving up useful properties of allocation mechanism, such as stability and strategy-proofness. The third essay studies a college admission with affirmative action problem. With this application, for the first time in the literature, we presented an affirmative action problem where students need to claim privilege if they want to be subject to affirmative action. We analyzed the current system and showed that current guideline is unfair and causes incentive compatibility issues. Also we showed that it fails to satisfy affirmative action requirements described in affirmative action law. To solve these problems with the current system, we introduced a new choice function which is fair, respects affirmative action requirements and makes student optimal stable allocation stable and incentive compatible when used in conjunction with generalized deferred acceptance algorithm.Thesis (PhD) — Boston College, 2014.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Economics