One Proper Score for Many Decisions: Sharp Regret Transfer under Misspecification

Under model misspecification, minimizing different proper scoring rules can induce different downstream decisions. We characterize the smallest worst-case target-risk ratio achievable by minimizing one fixed proper score across populations and report classes, relative to the best predictor in each class. The target is the contextwise maximum of declared task regrets, augmented by a strictly proper baseline divergence. We show that the optimal risk ratio equals the smallest two-sided distortion between the target and a proper-score divergence. For finite-action tasks, decision boundaries force curvature charges that yield exact distortion $m$ for an orthogonal family of $m$ two-action tasks on $K$ outcomes, with $m\le K-1$. Independent shared components with common tie selections admit an exact covering formula that quantifies the benefit of shared geometry. Its optimal weights yield a baseline-augmented task-score sum that attains the optimum over all proper scores. We also derive exact binary and quadratic characterizations and a sharp local trade-off between smoothness and additive regret error at hard boundaries. Known-truth studies illustrate the distinction between worst-case guarantees and achieved projection risk.