Market Design Under Constraints

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./nIn 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./nThe 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./nIn 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.

Advisor: Ergin, Haluk