  This paper studies distributional model risk in marginal problems, where each
marginal measure is assumed to lie in a Wasserstein ball centered at a fixed
reference measure with a given radius. Theoretically, we establish several
fundamental results including strong duality, finiteness of the proposed
Wasserstein distributional model risk, and the existence of an optimizer at
each radius. In addition, we 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
Markarov bounds. Practically, 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. They are: 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.
