Almost stochastic dominance via optimal transport

2026-07-30

AI Summary

This work addresses the lack of transitivity in approximate stochastic dominance relations between probability distributions on general Polish spaces by introducing a novel parametric ordering indexed by γ ∈ [0,1]. The proposed framework unifies optimal transport theory with parameterized approximate stochastic dominance for the first time. By leveraging a generalized Kantorovich–Rubinstein duality theorem, it establishes a dual characterization based on comparisons of expectations over test functions, thereby guaranteeing transitivity. This approach subsumes the multivariate method of Müller et al. as a special case. The framework enables exact computation of the optimal γ value, validates multiple examples from the literature, and demonstrates robustness of γ with respect to small perturbations of the underlying distributions.

Abstract

We study parametric classes of almost stochastic dominance on general Polish spaces as order relations for probability distributions with a parameter $γ\in [0,1]$. Larger values of $γ$ correspond to weaker order relations: $γ=0$ gives classical stochastic dominance $\le_{st}$, whereas $γ=1$ 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 $γ$ 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 Müller et al. (2025). We show that this approach is a special case of our framework and derive the best possible parameters $γ$ 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 $γ$ increases only slightly.

Problem

• almost stochastic dominance
• optimal transport
• Polish spaces
• order relations
• parameter γ

Innovation

• almost stochastic dominance
• optimal transport
• Kantorovich–Rubinstein duality
• transitivity
• robustness