Tractable Relaxations of Multivariate Stochastic Dominance via Optimal Transport and CVaR

Abstract

Many operational decisions involve alternatives with several uncertain attributes. When comparing such alternatives, stochastic dominance requires every decision maker in a prescribed class to prefer one alternative to the other. These orders have two known limitations: a single extreme decision maker can rule out dominance, and multivariate dominance can be difficult to verify. To address these limitations, we develop two relaxations of their standard representations: compensated stochastic dominance (CSD) relaxes Strassen's coupling representation, while reference-weighted stochastic dominance (RWSD) relaxes the integral representation. We characterize the utility class generated by CSD, recovering known almost stochastic dominance orders as special cases, and characterize it for RWSD in several cases. Importantly, we show that CSD and RWSD can each be checked by computing a single value, an optimal-transport value for CSD and a CVaR value for RWSD, and verifying that this value is nonpositive. This inequality characterization allows us to derive finite-sample guarantees for both orders, even when the outcome dimension is large. An application to Olist e-commerce data shows that, even when empirical FOSD fails, CSD and RWSD quantify how small this failure is and what is needed to overcome it.