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 describe the sharp upper bound for variance of treatment effect Y−X given marginals when monotone, or Y≥X. thus focus on supermodular costs (or submodular rewards) and coupling P∗ that all such yields bound. This admits manifold characterizations—geometric, order-theoretic, as transport, through cdf, via kernel—that explain its structure imply useful bounds. When first marginal atomless, concentrated graphs two maps which described in terms marginals, second map arising due binding constraint.