The Information Cost of Asymmetric Risk Control

When prediction errors have unequal consequences, overall accuracy may conceal failures that warrant separate risk limits. We ask how the granularity of these limits changes the information a representation must retain. We study finite-message representations fixed before numerical risk budgets are specified, with frontier error defined as the largest increase in optimal prediction error across all budgets. For a regular class of classification problems with a common default that avoids severe errors and one competing action, we prove that grouping the same severe error events into $m$ groups yields optimal $M$-message frontier error $\Theta (M^{-2/m})$. With the source and error events held fixed, finer control thus changes the rate at which additional messages recover decision performance, even when each grouping receives its own optimal representation. Separate limits make distinctions between error types relevant to action; we identify cases where proportional risks or a default action that is optimal at every input eliminate the added cost. Preserving the frontier further bounds the loss in optimal objective value when risk preferences are specified after encoding. Synthetic experiments illustrate these effects, while experiments on frozen, calibrated WiCE verifier outputs show a corresponding cost of separate overclaim control over the tested penalty family.