Essays on matching markets

Tesis llevada a cabo para conseguir el grado de Doctor por la Universidad Autónoma de Barcelona.--2016-07-04Essays on matching markets This thesis is on matching theory and its applications to the analysis and design of matching markets. A predominant concept in this thesis and in the matching literature is stability. Stable matchings are desirable both from a practical and a theoretical point of view and have played a critical role in the success of clearinghouses for medical residency matching and school choice. In what follows we provide a brief summary of each chapter. In the first chapter, entitled Stability of Equilibrium Outcomes under Deferred Acceptance: Acyclicity and Dropping Strategies, we study the preference revelation game induced by the deferred acceptance (DA) mechanism. Under DA no student can improve by misrepresenting his preferences. However, hospitals typically can. Furthermore, Nash equilibrium outcomes may be unstable with respect to agents' true preferences. We focus on Nash equilibria in which each hospital plays a so-called dropping strategy. A dropping strategy for a hospital is a list obtained by deleting some students from its true preference list. Dropping strategies are exhaustive in the following sense: the outcome that any hospital can obtain by playing some strategy can be replicated or improved upon by a dropping strategy. Given their simplicity and their exhaustiveness property, dropping strategies are appealing to hospitals. The main result in this chapter is that acyclicity of the hospitals' preference profile (Romero-Medina and Triossi, 2013a) is a necessary and sufficient condition to ensure that the outcome of any Nash equilibrium in which each hospital plays a dropping strategy is stable with respect to agents' true preferences. This result gives theoretical support as to why markets that employ the DA mechanism perform well in real-life applications even though hospitals may have incentives to misrepresent their preferences. In the second chapter, entitled Matching with Contracts, Substitutes and Two-Unit Demand, we study matching with contracts markets where agents on one side of the market can sign at most two contracts (two-unit demand). This model covers several important matching problems. For example, the problem of allocating couples of medical students to hospitals and the problem of assigning students to pairs of studies. The main result in this chapter establishes that for any choice function that satisfies irrelevance of rejected contracts and two-unit demand, bilateral substitutability and weak substitutability are equivalent. As a corollary, we obtain a new maximal domain for the existence of stable matchings in couples problems. At the end of Chapter 2 we show with an example that the equivalence depends crucially on the two-unit demand property. In the third chapter, entitled Investing before Stable Matching, we study matching markets where students make human capital investments before participating in the market for residency positions. We consider a stylized model where students have the same preferences over hospitals, and focus on the equilibria of the induced investment game. The main result provides necessary and sufficient conditions under which the equilibrium outcome of a market where students invest and match sequentially coincides with the equilibrium of a market where students invest and match simultaneously. We also give a sufficient condition on preference profiles such that, in equilibrium, students make efficient investments.Financial support from the Consejo Nacional de Ciencia y Tecnología (CONACyT), AGAUR-Generalitat de Catalunya (Project 2014 SGR 1064), Universitat Autònoma de Barcelona through PIF grant 412-01-9/2010 and the Spanish Ministry of Economy and Competitiveness through FPI grant BES-2012-055341 (Project ECO2011-29847- C02) is gratefully acknowledged.Peer reviewe