Overview
This work addresses the verification of multivariate stochastic dominance in operational decisions involving alternatives with multiple uncertain attributes. While exact stochastic dominance avoids reliance on a single multivariate utility function, its verification can be challenging due to the strictness of comparisons and the difficulty of identifying precise dominance relationships. The research develops two tractable relaxations: Compensated Stochastic Dominance (CSD) and Reference-Weighted Stochastic Dominance (RWSD).
Research Context
Many operational decisions necessitate the comparison of alternatives characterized by several uncertain attributes. Stochastic dominance offers a framework for such comparisons without requiring the explicit specification of a single multivariate utility function. However, the exact form of stochastic dominance is sensitive; a single lower-set or utility comparison with a differing rank can preclude exact dominance. Additionally, the verification of multivariate dominance often presents practical difficulties. The developed relaxations aim to overcome these challenges by providing more tractable methods for assessing stochastic orders.
Approach
The research develops two distinct relaxation approaches, each rooted in a standard representation of stochastic orders:
Compensated Stochastic Dominance (CSD)
- CSD represents a relaxation of Strassen's coupling characterization.
- First-order dominance, in its exact form, necessitates a coupling where the outcome from distribution $Q$ is at least the outcome from distribution $P$ in every coordinate, almost surely.
- CSD modifies this by permitting selected monotone scores of the outcome from $Q$ to be lower than those from $P$ in some coupled pairs. This is allowed provided that the expected weighted amount by which they are lower does not exceed a predefined fraction of the expected weighted amount by which they are higher.
- The entire CSD tolerance path is computable via a single optimal-transport problem.
- Through transport duality, the utility functions whose expected values generate the order are characterized exactly.
- This framework incorporates substitution multivariate almost stochastic dominance and extends multivariate orders between first- and second-order dominance.
Reference-Weighted Stochastic Dominance (RWSD)
- RWSD maintains the generating functions characteristic of an integral stochastic order.
- It introduces a restriction on the probability measures utilized to combine their expected-utility differences.
- Checking for RWSD reduces to solving a conditional value-at-risk (CVaR) problem.
- RWSD versions of first-order, lower-orthant, and increasing-concave dominance are constructed within this framework.
Findings
- Compensated Stochastic Dominance (CSD) allows for controlled deviations from strict dominance, where the aggregate expected weighted 'shortfall' is bounded relative to the 'excess'.
- The computation of the full CSD tolerance path is achievable through a single optimal-transport problem.
- Transport duality provides an exact characterization of the utility functions that align with the CSD order.
- CSD encompasses existing frameworks such as substitution multivariate almost stochastic dominance and multivariate extensions of orders between first- and second-order dominance.
- Reference-Weighted Stochastic Dominance (RWSD) modifies the aggregation of expected-utility differences by constraining probability measures.
- The verification of RWSD can be reframed as a conditional value-at-risk (CVaR) problem.
- Specific RWSD variants for first-order, lower-orthant, and increasing-concave dominance have been developed.
- The transport and CVaR characterizations offer a basis for deriving finite-sample statistical guarantees for both CSD and RWSD.
Why This Matters
The developed relaxations, CSD and RWSD, provide more practical tools for assessing stochastic dominance in situations with multiple uncertain attributes. By offering tractable computational methods (optimal transport for CSD and CVaR for RWSD), these approaches facilitate the analysis of decision-making under uncertainty without the strict limitations of exact dominance. The ability to characterize relevant utility functions and obtain statistical guarantees enhances their applicability in various operational contexts.