Whether or Where? Tail Calibration Signals Finite-Budget Reward Hacking in Best-of-N Selection

Cheng Gui, Yihua Shao, Lei Li, Jingcai Guo, Maolin Wang

Best-of-N (BoN) inference returns the candidate with the highest proxy reward, so a larger $N$ tests the proxy's extreme tail more than its average ranking quality. We show that the exact BoN value is a Beta-weighted integral of conditional true reward over within-prompt score quantiles. It follows that BoN depends on scores only through their ranks. It also follows that BoN can decline only if the calibration curve in the tail turns downward, and the budget where the decline appears depends on where that turn lies. Under a bounded additive model, the error tail sets the limit: Gaussian errors keep improving, Laplace errors reach a plateau, and heavy-tailed errors return to the prior mean. When error variance is largest at low true reward, even Gaussian noise can push BoN below random selection. Synthetic phase maps agree with this taxonomy. On real candidate pools spanning several verifiable domains, base policies, and learned reward models, tail failures are rare and all occur in code generation. A held-out tail calibration statistic reliably predicts where a cross-fitted budget cap helps. It does about as well as rank accuracy, and the cap is competitive with a Soft-BoN…