Limited Arbitrage Is Necessary and Sufficient for the Existence of a Competitive Equilibrium

A condition of limited arbitrage is defined on the endowments and the preferences of the traders in an Arrow-Debreu economy. It bounds the diversity of the traders in the economy, and the gains from trade which they can afford from initial endowments. Theorem 1 shows that limited arbitrage is necessary and sufficient for the existence of a competitive equilibrium, when consumption sets are either positive orthants or the whole euclidean space. The results apply therefore to market economies with or without bounds on short sales. Theorem 6 establishes that an Arrow-Debreu economy has a competitive equilibrium if and only if every subeconomy with N + 1 traders does, where N is the number of commodities. Limited arbitrage has been shown elsewhere to be equivalent to the contractibility of spaces of preferences, and therefore, by the results of Chichilnisky and Heal [12], to be necessary and sufficient for the existence of social choice rules defined on individual preferences over allocations, rules which are continuous, anonymous and respect unanimity

Markets, Arbitrage and Social Choices

The paper establishes a clear connection between equilibrium theory and social choice theory by showing that, for a well defined social choice problem, the conditions which are necessary and sufficient to establish existence of a competitive equilibrium. We define a condition of limited arbitrage on the preferences and the endowments of an Arrow-Debreu economy. This bounds the utility gains that the traders can afford from their initial endowments. Theorem 2 proves that limited arbitrage is necessary and sufficient for the existence of a social choice rule which allocates society's resources among individuals in a manner which depends continuously and anonymously on their preferences over allocations, and which respects unanimity. Limited arbitrage is also necessary and sufficient for the existence of a competitive equilibrium in the Arrow-Debreu economy, with or without bounds on short sales, Theorem 7. Theorem 4 proves that any market allocation can be achieved as a social choice allocation, i.e. an allocation which is maximal among all feasible allocations according to a social preference defined via a social choice rule which is continuous, anonymous and respects unanimity

Essays on Matching Markets

The thesis "Essays on Matching Markets" contributes to the theory and applications of matching theory. The first chapter analyzes the German university admissions system and proposes an alternative admissions procedure that outperforms the currently used mechanism. In particular, the new mechanism provides strong (i.e. dominant strategy) incentives for applicants to reveal their true preferences and achieves a notion of stability that is adapted to the German system. In the second chapter we analyze the school choice problem with indifferences in priority orders. In this context, stability (with respect to student preferences and school priorities) can be understood as a fairness criterion which ensures that no student ever envies another student for a school at which she has higher priority. Since school seats are objects to be allocated among students, it is important to ensure that a constrained efficient allocation is selected, i.e. an allocation that is stable and not (Pareto-) dominated by any other stable matching. A counterexample of Erdil and Ergin (American Economic Review, 2008) shows that there may not exist a non-manipulable and constrained efficient mechanism. We consider the case where students either all have the same priority or all have distinct priorities for a given school. For this important special case we investigate whether the negative result of Erdil and Ergin is the rule or an exception and derive sufficient conditions for the existence of a constrained efficient and (dominant strategy) incentive compatible mechanism. The proof is constructive and shows how preferences of students can (sometimes) be used to prevent any welfare loss from tie-breaking decisions. The third chapter deals with a more general matching model recently introduced by Ostrovsky (American Economic Review, 2008). For this model we analyze the relation between Ostrovsky's chain stability concept, efficiency, and several competing stability concepts. We characterize the largest class of matching models for which chain stable outcomes are guaranteed to be stable and robust to all possible coalitional deviations. Furthermore, we provide two rationales, one based on efficiency and the other based on robustness considerations, for chain stability in the general supply chain model

Top trading with fixed tie-breaking in markets with indivisible goods

We study markets with indivisible goods where monetary compensations are not possible. Each individual is endowed with an object and a preference relation over all objects. When preferences are strict, Gale's top trading cycle algorithm finds the unique core allocation. When preferences are not necessarily strict, we use an exogenous profile of tie-breakers to resolve any ties in individuals' preferences and apply Gale's top trading cycle algorithm for the resulting profile of strict preferences. We provide a foundation of these simple extensions of Gale's top trading cycle algorithm from strict preferences to weak preferences. We show that Gale's top trading cycle algorithm with fixed tie-breaking is characterized by individual rationality, strategy-proofness, weak efficiency, non-bossiness, and consistency. Our result supports the common practice in applications to break ties in weak preferences using some fixed exogenous criteria and then to use a 'good and simple' rule for the resulting strict preferences. This reinforces the market-based approach even in the presence of indifferences because always competitive allocations are chosen

Essays in Behavioral Industrial Organization

How do consumers make consumption decisions if the acquisition of product information is difficult? What determines their search behavior and ultimately their choices? In three essays I examine the way in which the answers to these questions affect market outcomes and the incentives of firms to obfuscate product information

Pareto Optimal Matchings in Many-to-Many Markets with Ties

We consider Pareto-optimal matchings (POMs) in a many-to-many market of applicants and courses where applicants have preferences, which may include ties, over individual courses and lexicographic preferences over sets of courses. Since this is the most general setting examined so far in the literature, our work unifies and generalizes several known results. Specifically, we characterize POMs and introduce the \emph{Generalized Serial Dictatorship Mechanism with Ties (GSDT)} that effectively handles ties via properties of network flows. We show that GSDT can generate all POMs using different priority orderings over the applicants, but it satisfies truthfulness only for certain such orderings. This shortcoming is not specific to our mechanism; we show that any mechanism generating all POMs in our setting is prone to strategic manipulation. This is in contrast to the one-to-one case (with or without ties), for which truthful mechanisms generating all POMs do exist

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

Pareto optimal matchings in many-to-many markets with ties

We consider Pareto optimal matchings (POMs) in a many-to-many market of applicants and courses where applicants have preferences, which may include ties, over individual courses and lexicographic preferences over sets of courses. Since this is the most general setting examined so far in the literature, our work unifies and generalizes several known results. Specifically, we characterize POMs and introduce the Generalized Serial Dictatorship Mechanism with Ties (GSDT) that effectively handles ties via properties of network flows. We show that GSDT can generate all POMs using different priority orderings over the applicants, but it satisfies truthfulness only for certain such orderings. This shortcoming is not specific to our mechanism; we show that any mechanism generating all POMs in our setting is prone to strategic manipulation. This is in contrast to the one-to-one case (with or without ties), for which truthful mechanisms generating all POMs do exist