  We employ scoring functions, used in statistics for eliciting risk
functionals, as cost functions in the Monge-Kantorovich (MK) optimal transport
problem. This gives raise to a rich variety of novel asymmetric MK divergences,
which subsume the family of Bregman-Wasserstein divergences. We show that for
distributions on the real line, the comonotonic coupling is optimal for the
majority of the new divergences. Specifically, we derive the optimal coupling
of the MK divergences induced by functionals including the mean, generalised
quantiles, expectiles, and shortfall measures. Furthermore, we show that while
any elicitable law-invariant coherent risk measure gives raise to infinitely
many MK divergences, the comonotonic coupling is simultaneously optimal.
  The novel MK divergences, which can be efficiently calculated, open an array
of applications in robust stochastic optimisation. We derive sharp bounds on
distortion risk measures under a Bregman-Wasserstein divergence constraint, and
solve for cost-efficient payoffs under benchmark constraints.
