  An optimal randomized strategy for design of balanced, normalized mass
transport plans is developed. It replaces -- but specializes to -- the
deterministic, regularized optimal transport (OT) strategy, which yields only a
certainty-equivalent plan. The incompletely specified -- and therefore
uncertain -- transport plan is acknowledged to be a random process. Therefore,
hierarchical fully probabilistic design (HFPD) is adopted, yielding an optimal
hyperprior supported on the set of possible transport plans, and consistent
with prior mean constraints on the marginals of the uncertain plan. This
Bayesian resetting of the design problem for transport plans -- which we call
HFPD-OT -- confers new opportunities. These include (i) a strategy for the
generation of a random sample of joint transport plans; (ii) randomized
marginal contracts for individual source-target pairs; and (iii) consistent
measures of uncertainty in the plan and its contracts. An application in
algorithmic fairness is outlined, where HFPD-OT enables the recruitment of a
more diverse subset of contracts -- than is possible in classical OT -- into
the delivery of an expected plan. Also, it permits fairness proxies to be
endowed with uncertainty quantifiers.
