With the development of market design and the increasing number of applications from school choice to kidney exchange has come the need for flexibility. The wide variety of practical settings calls for a set of tools which can be used broadly to incorporate market specific details. This dissertation collects three efforts to this end. The first two chapters concern the development of incentive compatible and Pareto efficient mechanisms in a setting where constraints are taken very broadly. The third abandons Pareto efficiency in favor of stability.

In the first chapter, coauthored with David S. Ahn, we study private-good allocation mechanisms where an arbitrary constraint delimits the set of feasible joint allocations. This generality provides a unified perspective over several prominent examples that can be parameterized as constraints in this model, including house allocation, roommate assignment, and social choice. We first characterize the set of two-agent strategy-proof and Pareto efficient mechanisms, showing that every mechanism is a "local dictatorship.'' For more than two agents, we leverage this result to provide a new characterization of group strategy-proofness. In particular, an N-agent mechanism is group strategy-proof if and only if all its two-agent marginal mechanisms (defined by holding fixed all but two agents' preferences) are individually strategy-proof and Pareto efficient. To illustrate their usefulness, we apply these results to the roommates problem to discover the novel finding that all group strategy-proof and Pareto efficient mechanisms are generalized serial dictatorships, a new class of mechanisms. Our results also yield a simple new proof of the Gibbard--Satterthwaite Theorem.

The second chapter, coauthored with David S. Ahn, takes a more concrete approach. In the same setting as chapter 1, we introduce a large subclass of mechanisms which we dub "constraint-traversing" and explore their properties. In particular, we provide two weak conditions -- forward consistency and backward consistency -- which, if satisfied, guarantee that a mechanism is group strategy-proof and Pareto efficient. We illustrate the usefulness of this approach by deriving the set of 3-agent and 3-object house allocation mechanisms (already characterized by (Pycia and Unver 2017). In addition, we demonstrate that these conditions can be equally applied to many "nearby" problems that would otherwise be intractable. Constraint-traversing mechanisms have a number of convenient properties. First, group strategy-proofness implies Pareto efficiency. Second, the marginal mechanisms of any constraint-traversing mechanism is also constraint-traversing.

In the final chapter, I consider stability in a two-sided matching context rather than incentive compatibility and Pareto efficiency in allocation mechanisms. I introduce a unified framework for studying two-sided matching problems with constraints. I introduce a matching algorithm called the constrained cumulative deferred acceptance algorithm capable of accommodating a wide variety of constraints. Like the deferred acceptance algorithm, one side of the market makes proposals to another. A "constraint correspondence" dynamically limits the choices of the receiving side in order to enforce that the ultimate match satisfies the constraint. If the constraint correspondence satisfies a "generalized substitutes" condition, the ultimate match will be constrained stable in the sense that satisfying any blocking pair would lead to a violation of the constraint. I provide two further conditions, "aggregate monotonicity" and "constraint IIA," on the constraint correspondence which ensure the constrained cumulative deferred acceptance algorithm implements a strategy-proof mechanism. Finally, I study the comparative statics of constraint correspondences.

This dissertation studies the problem of allocating heterogeneous indivisible goods to agents without the use of monetary compensation when each agent receives at most one good. The three central concerns in designing allocation mechanisms are incentives of the agents, efficiency, and fairness. There are inherent trade-offs between competing notions of the three desiderata in such assignment mechanisms. This dissertation further explores these tradeoffs and constructs novel mechanisms for such settings.

Chapter 2 focuses on random assignment mechanisms for the canonical house allocation problem. It shows that any strategy-proof and envy-free mechanism must sacrifice efficiency in a very weak form captured by the notion of contention-free efficiency. The chapter also characterizes a large family of strategy-proof and envy-free mechanisms called Rank Exchange Mechanisms thereby showing the existence of mechanisms apart from the equal division mechanism that satisfy these two properties.

Chapter 3 studies the allocation problem in the presence of arbitrary linear constraints when agents exhibit indifferences in their preferences. It proposes the Constrained Serial Rule, a mechanism that is a generalization of the well-known Probabilistic Serial mechanism, and shows that this mechanism satisfies constrained ordinal efficiency and envy-freeness among agents of the same type.

Chapter 4 introduces Generalized Hierarchical Exchange mechanisms and shows that they satisfy strategy-proofness, Pareto efficiency, and admit many bossy mechanisms. Using this generalization, the chapter proposes novel mechanisms called priority trading mechanisms. Finally, it also provides a characterization of an important sub-class of Generalized Hierarchical Exchange mechanisms.

Since the 2000s, matching theory has seen an increasing number of applications, from school assignments to organ donation. This dissertation collects three papers contributing to the theory of one-sided matching with endowments.

In the first chapter, I study the testable implications of the core in an exchange economy with unit demand when agents' preferences are unobserved. To do so, I develop a model of aggregate matchings in which the core is testable; the identifying assumption is that agents' preferences are solely determined by observable characteristics. I give conditions that characterize when observed economies are compatible with the core. These conditions are meaningful, intuitive, and tractable; they provide a nonparametric test for the core in the style of revealed preferences. I also develop a parametric method to estimate preference parameters from multiple observations of exchange economies. An allocation being in the core implies necessary moment inequalities, which I leverage to obtain partial identification.

The second chapter is coauthored with Will Sandholtz. We study the classic house-swapping problem of Shapley and Scarf (1974) in a setting where agents may have “objective” indifferences, i.e., indifferences that are shared by all agents. In other words, if any one agent is indifferent between two houses, then all agents are indifferent between those two houses. The most direct interpretation is the presence of multiple copies of the same object. Our setting is a special case of the house-swapping problem with general indifferences. We derive a simple, easily interpretable algorithm that produces the unique strict core allocation of the house-swapping market, if it exists. Our algorithm runs in O(n^{2}) time, where n is the number of agents and houses. This is an improvement over the O(n^{3}) time methods for the more general problem.

The third chapter is also coauthored with Will Sandholtz. We note that the proof of Bird (1984), the first to show group strategy-proofness of top trading cycles (TTC), requires a correction. We provide a counter-example to a critical claim, then present a corrected proof in the spirit of the original.

In this dissertation, I study the properties of and propose the use of a family of random mechanisms for a large class of problems where agents need to be matched to objects or to each other without the use of monetary transfers.

In the first chapter, I study the problem of kidney exchange under strict ordinal preferences and with constraints on the length of the trading cycles. The requirement of individual rationality in this setting incentivizes patient-donor pairs who are compatible with each other to participate in the kidney exchange, thus increasing the match rate for incompatible pairs. I show that deterministic mechanisms have poor properties in this environment. Instead, I explicitly define an individually rational, efficient and fair random mechanism for the case of pairwise kidney exchange. Finally, I show that individual rationality, efficiency and weak strategyproofness are incompatible for the cycle-constrained case making the proposed mechanism, called the 2-Cycle Probabilistic Serial (2CPS) mechanism, a second-best mechanism.

In the second chapter, I extend the idea behind the 2CPS mechanism to arrive at a constrained efficient mechanism for a general matching environment and regardless of what the ex-post constraints on the outcome are, including individual rationality, limits on the cycle lengths, maximizing the number of proposed matches etc. Several mechanisms from the existing literature are special cases of this mechanism, called the Generalized Constrained Probabilistic Serial (GCPS) mechanism.

In the final chapter, I consider two natural notions of strategyproofness in random matching mechanisms based on ordinal preferences. The two notions are stronger than weak strategyproofness but weaker than strategyproofness. I demonstrate that the two notions are equivalent and provide a geometric characterization of the new intermediate property which I call convex strategyproofness. I then show that the probabilistic serial mechanism (a special case of the GCPS mechanism) is, in fact, convexly strategyproof. I finish by showing that the property of weak envy-freeness of the random serial dictatorship can be strengthened in an analogous manner.

This thesis studies dynamic markets with endogenous entry, where access hurdles are either an explicit feature of the market or an implicit result of productive externalities.

The first chapter concerns markets of influence, where formal gatekeepers are in charge of selecting and promoting the most promising ideas. I model this as a two-sided matching market between a continuum of experts and a finite number of gatekeepers under sequential directed search. Real-world examples include academic publishing, venture capitalism or political agenda setting. Uniqueness of the resulting equilibrium allows for clear-cut pre- dictions: First, sorting may fail in equilibrium. Second, gatekeepers may have an incentive to add artificial delay. Such red tape occurs in equilibrium – and only at the top – when the impact of two gatekeepers is very different. Third, artificial delay may improve equilib- rium sorting and thereby enhance welfare. Finally, the bottom gatekeeper may endogenously specialize on a quality-irrelevant attribute.

In other applications, there are no formal gatekeepers who regulate market access, yet endogenous access hurdles play a similarly selective role. In this vein of research, the second chapter studies the evolution of labor market composition under mentoring externalities. This chapter is joint work with Aniko Oery. We provide a continuous time, overlapping generations framework to analyze the costs and benefits of affirmative action policies if mentoring complementarities are present. Senior workers reduce the young population’s educational cost through mentoring, and thus act as ‘trailblazers’ for future generations. In such a framework, the main trade off is between using the most able workers and reducing mentorship misallocation. We identify conditions under which persistent market intervention is warranted to improve long-term surplus. We also contrast different channels through which the planner can affect market outcomes, and highlight the benefits of educational fellowships over hiring restrictions.