  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. We show that such a mechanism
need not exist in general. We introduce a new notion of discrete concavity,
which we call pseudo M$^{\natural}$-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$^{\natural}$-concave.
