Almost stochastic dominance via optimal transport

Abstract

We study parametric classes of almost stochastic dominance on general Polish spaces as order relations for probability distributions with a parameter $\gamma \in [0,1]$. Larger values of $\gamma$ correspond to weaker order relations: $\gamma=0$ gives classical stochastic dominance $\le_{st}$, whereas $\gamma=1$ gives a complete preorder based on comparison of expectations of a fixed increasing function $g$. It is well known that $X \le_{st} Y$ can be characterized by the existence of a solution to an optimal transport problem with $\mathrm{OT}_c(X,Y)=0$ for a suitable cost function $c$. We generalize this idea so that the best possible parameter $\gamma$ for almost stochastic dominance can be determined from the solution of an optimal transport problem. Using a generalization of the classical Kantorovich--Rubinstein duality theorem to quasi-pseudo-metrics, we derive a dual characterization of the order in terms of expectation comparisons for a parametric class of test functions. Consequently, our relations are always transitive, in contrast to some other recent approaches to almost stochastic dominance based on optimal transport. A natural multivariate approach to almost stochastic dominance, based on classes of test functions with bounds on partial derivatives, was recently introduced by M\"uller et al. (2025). We show that this approach is a special case of our framework and derive the best possible parameters $\gamma$ for examples considered there, as well as for other examples from the literature. We also prove a robustness result showing that, under small perturbations of the distributions in a Wasserstein-type metric related to the optimal transport problem, the best possible $\gamma$ increases only slightly.