Compatibility and Exclusivity of Loss Functions under Statistical Optimality
In applications a loss is usually chosen from a family, like a quantile level, a robustness threshold or a surrogate margin loss. In this paper, we ask which of these choices change the optimal decision. A model, a class of decision rules and an optimality criterion define an optimality map $L\mapsto\cO(L)$. Two losses are compatible when they share an optimal rule, and exclusivity regions are unions of the connected components of this relation. When $\cO$ is single-valued these components are the fibres of the map, and the problem is to identify the statistically meaningful features of a loss that determine its fibre. Under explicit model assumptions, the answer is the quantile level for generalized piecewise linear quantile scores and the margin invariant (equivalently, the link) for regular margin losses read at the score level. For canonical Huber losses on strictly skewed location models, a class that includes every asymmetric Laplace model, it is the threshold. Two structural results hold in general. Convex interpolation between losses with distinct unique Bayes acts meets uncountably many compatibility classes, so the structure is conic rather than convex. Coarsening the optimality operator can only merge components, which is how the known common Bayes classifier of calibrated surrogates arises.