Additive Distributionally Robust Ranking and Selection

Ranking and selection (R&S) aims to identify the alternative with the best mean performance among $k$ simulated alternatives. The practical value of R&S depends on accurate simulation input modeling, which often suffers from the curse of input uncertainty due to limited data. Distributionally robust ranking and selection (DRR&S) addresses this challenge by modeling input uncertainty via an ambiguity set of $m > 1$ plausible input distributions, resulting in $km$ scenarios in total. Recent DRR&S studies suggest a key structural insight: additivity in budget allocation is essential for efficiency. However, existing justifications are heuristic, and fundamental properties such as consistency and the precise allocation pattern induced by additivity remain poorly understood. In this paper, we propose a simple additive allocation (AA) procedure that aims to exclusively sample the $k + m - 1$ previously hypothesized critical scenarios. Leveraging boundary-crossing arguments, we establish a lower bound on the probability of correct selection and characterize the procedure's budget allocation behavior. We then prove that AA is consistent and, surprisingly, achieves additivity in the strongest sense: as the total budget increases, only $k + m - 1$ scenarios are sampled infinitely often. Notably, the worst-case scenarios of non-best alternatives may not be among them, challenging prior beliefs about their criticality. These results offer new and counterintuitive insights into the additive structure of DRR&S. To improve practical performance while preserving this structure, we introduce a general additive allocation (GAA) framework that flexibly incorporates sampling rules from traditional R&S procedures in a modular fashion. Numerical experiments support our theoretical findings and demonstrate the competitive performance of the proposed GAA procedures.