The Directional Optimal Transport

We introduce a constrained optimal transport problem where origins $x$ can only be transported to destinations $y\geq x$. 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 $Y\geq X$. 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.