Exact characterization of optimal randomized site selection in a structured two-site case

Characterize the exact minimax‑regret optimal randomized site selection and associated treatment assignment rule in the stylized one‑dimensional setup with k = 1, degenerate signals (i.e., \hat{τ}_s = τ_s), covariates X_s = s, and experimental sites coinciding with policy sites {1,2}, under the Lipschitz constraint |τ_s − τ_t| ≤ C|s − t| on treatment effect heterogeneity across sites.

Background

Section 6.2 discusses extending the site selection framework from purposive (nonrandomized) to randomized sampling schemes and highlights how optimality can depend on timing assumptions in a minimax setting. In a first stylized example with S_E = {1,4}, S_P = {2,3}, X_s = s, and no sampling uncertainty (\hat{τ}_s = τ_s), the authors solve the exact minimax‑regret values, showing randomized sampling reduces worst‑case regret from 3C/4 (purposive) to C/2.

They then consider a closely related, still highly structured case where the experimental sites coincide with the policy sites at {1,2}. While they can find the minimax‑regret optimal combination for purposive sampling and verify that randomized sampling strictly reduces worst‑case regret, they explicitly state that they are unable to characterize the exact optimal randomized solution. This leaves a concrete gap in the theory: deriving the exact randomized minimax‑regret solution even in this simple two‑site configuration.