Graph Causal Optimal Transport and Wasserstein Distances

Abstract

We study the graph causal optimal transport problem, a generalisation of the classical optimal transport problem in which the allowed couplings satisfy causal restrictions prescribed by a directed graph. We characterise fully the directed acyclic graphs for which the associated graph causal Wasserstein discrepancy is a metric and show that the induced topology agrees with other natural adapted topologies. We characterise the gluing properties of graph causal couplings, prove denseness of Monge couplings, and obtain a dynamic programming principle which allows us to deduce when the graph causal Wasserstein and the adapted Wasserstein distances are equal. Our results link fundamental properties of graph causal optimal transport to structural properties of its underlying graph. Complementing Cheridito and Eckstein (2025), who first introduced such distances and established Lipschitz continuity for the average treatment effect in structural causal models, we obtain Lipschitz continuity of the value function in stochastic team problems.