The Endogeneity of Miscalibration: Impossibility and Escape in Scored Reporting

A theoretical paper demonstrates that principals using standard scoring rules to oversee strategic AI agents face an inherent impossibility: achieving both honest reporting and accurate calibration simultaneously. The research identifies step-function approval thresholds as the only mechanism that preserves calibration while maintaining incentive compatibility, with specific equivalence properties under the Brier score.

This academic research addresses a fundamental challenge in AI alignment and autonomous agent oversight. When principals score agent reports using strictly proper scoring rules but agents derive additional benefits from approval or resource allocation, a structural conflict emerges. The paper proves that optimal screening requires non-affine approval functions, yet these same functions create incentives for strategic misreporting whenever undetectable—a paradox the authors call endogeneity of miscalibration. This impossibility extends universally across all strictly proper scoring rules, with mathematically derived perturbation formulas quantifying the distortion. The practical significance lies in AI deployment: smooth, continuous oversight mechanisms cannot reliably extract truthful reports from economically rational agents, undermining the assumption that scoring rules alone ensure calibrated predictions. The constructive solution—binary step-function thresholds—reframes the agent's decision space as a binary choice (inflate or not), creating type-dependent screening independent of the underlying scoring rule's mathematical properties. Under the Brier score, step functions achieve welfare equivalence between second-best and first-best outcomes; the authors prove this unique property distinguishes Brier from all other C¹ scoring rules, with measured welfare gaps for alternatives. The framework bridges AI alignment and classical mechanism design, suggesting that sharp decision boundaries rather than continuous approval functions represent the calibration-preserving design principle. For practitioners deploying autonomous systems, this implies that gradual, continuous incentive structures risk systematic truth distortion regardless of formal scoring properties.

• →Strictly proper scoring rules cannot simultaneously achieve honest reporting and calibration when agents have external approval incentives.
• →Step-function approval thresholds uniquely solve the screening problem while preserving calibration across all scoring rules.
• →The Brier score exhibits special welfare properties that eliminate the gap between optimal and second-best solutions under threshold mechanisms.
• →AI oversight systems using smooth, continuous approval functions inherently incentivize strategic misreporting even when undetectable.
• →Binary decision thresholds represent the calibration-preserving design principle for autonomous agent supervision.