Quantifying Distributional Model Risk in Marginal Problems via Optimal Transport

This paper studies distributional model risk in marginal problems, where each marginal measure is assumed to lie in a Wasserstein ball. We establish fundamental results including strong duality, finiteness of the proposed Wasserstein distributional model risk, and the existence of an optimizer at each radius. We also show continuity of the Wasserstein distributional model risk as a function of the radius. Using strong duality, we extend the well-known Makarov bounds for the distribution function of the sum of two random variables with given marginals to Wasserstein distributionally robust Makarov bounds. We illustrate our results on four distinct applications when the sample information comes from multiple data sources and only some marginal reference measures are identified: partial identification of treatment effects, externally valid treatment choice via robust welfare functions, Wasserstein distributionally robust estimation under data combination, and evaluation of the worst aggregate risk measures. Funding: This work was supported by the National Science Foundation Infrastructure Grant (PIHOT) DMS-2133244, acknowledged by Y. Fan. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2024.0557 .