The Endogeneity of Miscalibration: Impossibility and Escape in Scored Reporting

Abstract

An agent's probability report is paid for twice: by a strictly proper scoring rule, and by an approval rule for the decision it triggers. In this classical decision-coupled setting, non-affine approval is known to defeat truthful reporting. We show the conflict is endogenous: when feasible, the welfare-maximizing approval rule is never affine. The distortion, however, is predictable and can be designed around. There is a reserve report at which pretending to be the marginal type costs exactly the approval prize. Approving at or above the reserve screens types perfectly under every strictly proper score, and the reserve does not depend on the type distribution. A Lipschitz rule with a single kink attains first-best exactly; under strict feasibility no continuously differentiable rule does. The binding constraint is steepness, not smoothness. First-best is attainable within a slope budget if and only if the budget is at least the critical slope: the steepest chord of the pretending cost up to the reserve. Below it the welfare loss is cubic in the shortfall. Where the pretending cost is convex up to the reserve, as for Brier, log and power scores, the critical slope is closed-form. The instances are AI-agent oversight and marketplace operation.