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.

翻译：我们引入了一种有限的最佳运输问题,即原产地为x美元只能运往目的地$y\geqx$。我们的统计动机是描述处理效果差异的急剧上限。当效果为单质或Y\geqx$时,当效果为一元或一元为一元时,我们只关注超模式成本(或亚模式回报 ), 并引入对所有这些成本最有利的组合美元, 并产生锐质约束。这种组合接受多种特征 -- -- 几何、秩序理论、最佳运输方式、通过 cdf 和通过运输内核 -- -- 解释其结构并暗示了有用的界限。当第一个边缘无孔时, 美元集中在两幅地图的图表上, 这两张地图可以用边边框描述, 第二张地图是因约束性限制而生成的。