  Stochastic dominance is an important concept in probability theory,
econometrics and social choice theory for robustly modeling agents' preferences
between random outcomes. While many works have been dedicated to the univariate
case, little has been done in the multivariate scenario, wherein an agent has
to decide between different multivariate outcomes. By exploiting a
characterization of multivariate first stochastic dominance in terms of
couplings, we introduce a statistic that assesses multivariate almost
stochastic dominance under the framework of Optimal Transport with a smooth
cost. Further, we introduce an entropic regularization of this statistic, and
establish a central limit theorem (CLT) and consistency of the bootstrap
procedure for the empirical statistic. Armed with this CLT, we propose a
hypothesis testing framework as well as an efficient implementation using the
Sinkhorn algorithm. We showcase our method in comparing and benchmarking Large
Language Models that are evaluated on multiple metrics. Our multivariate
stochastic dominance test allows us to capture the dependencies between the
metrics in order to make an informed and statistically significant decision on
the relative performance of the models.
