The Directional Optimal Transport

We introduce a constrained optimal transport problem where origins x can only be transported to destinations y≥ x. Our statistical motivation is to describe the sharp upper bound for the variance of the treatment effect Y-X given marginals when the effect is monotone, or Y≥ X. We thus focus on supermodular costs (or submodular rewards) and introduce a coupling P* that is optimal for all such costs and yields the sharp bound. This coupling admits manifold characterizations -- geometric, order-theoretic, as optimal transport, through the cdf, and via the transport kernel -- that explain its structure and imply useful bounds. When the first marginal is atomless, P* is concentrated on the graphs of two maps which can be described in terms of the marginals, the second map arising due to the binding constraint.