Market design with distributional objectives: Efficiency, incentives, and property rights

Given an initial matching and a policy objective on the distribution of agent types to institutions, we study the existence of a mechanism that weakly improves the distributional objective and satisfies constrained efficiency, individual rationality, and strategy-proofness. Such a mechanism need not exist in general. We introduce a new notion of discrete concavity, which we call pseudo M♮-concavity, and construct a mechanism with the desirable properties when the distributional objective satisfies this notion. We provide several practically relevant distributional objectives that are pseudo M♮-concave.