Abstract

This thesis consists of three studies in matching theory where agents have dichotomous preferences, while institutions have strict rankings over agents, interpreted as preferences or priorities depending on the setting. Dichotomous preferences simply divide institutions according to whether they are acceptable or not. This preference domain is natural in applications where the primary concern is to match as many agents as possible to any acceptable
institution, such as centralized daycare assignments and other allocation problems with scarcity, as well as when preferences are naturally binary due to compatibility or eligibility criteria.

In Chapter 2, I study a two-sided matching problem with dichotomous agent preferences and institutions with strict rankings of agents. I design a new class of mechanisms called SAFE mechanisms (Sequential Allocation for Fairness and Efficiency), equivalently characterized as Rank-Maximal mechanisms, that produce a maximum-size matching while respecting individual rationality, institutional rankings, and satisfying all standard fairness
and incentive properties, including strategyproofness and nonbossiness. These mechanisms are computationally tractable and can be implemented in polynomial time.

Chapter 3 extends the analysis to a two-period overlapping dynamic setting motivated by daycare allocation. Children participate for two periods, report dichotomous acceptance
over daycares, and daycares have strict priority orderings over children. I study whether one can simultaneously achieve maximum size together with dynamic fairness, efficiency, and incentive properties. I propose two dynamic mechanisms, tailored to two distinct policy objectives: a history-dependent (HD) policy and a childcare-guarantee (CG) policy. Under HD, a dynamic SAFE mechanism attains maximum size and the entire set of targeted
dynamic properties. Under CG, I show that no reasonable mechanism can satisfy these dynamic axioms. However, the proposed SAFE mechanism retains strong period-by-period
properties (maximum and fair in each period), and impossibility results justify it as a best possible compromise under the CG policy.

In Chapter 4, I analyze matching under institutional constraints that limit how many schools an agent may list as acceptable, which is a typical practice in school choice and centralized university admissions. I ask whether dichotomous preferences mitigate the manipulation and fairness issues known from the strict-preference setting. I show that if the constraint binds, then any individually rational and maximum constrained mechanism is manipulable.
Nevertheless, for any constraint level, all Nash equilibria of the induced preference revelation game of individually rational and maximum mechanisms yield fair outcomes. Finally, I show that manipulability and fairness cannot be compared in an unambiguous way across SAFE mechanisms with different constraint levels.