The Blown Lead Paradox: A Pathwise Calibration Benchmark for Win Probability Forecasts

Abstract

Live win probabilities have given sports collapses a common numerical language: the highest win probability attained by the team that eventually lost. Yet this familiar number is a selected pathwise extreme. The losing path is identified by the terminal outcome and then searched for its most favorable point. Under ideal sequential calibration, we derive an exact continuous-path benchmark for this statistic, a conservative bound for discretely reported paths, and a probability-integral-transform diagnostic for collections of games. Unlike fixed-time calibration summaries and proper scores, the diagnostic asks whether a published feed produces severe losing paths as often as a calibrated sequential model should. We apply the benchmark separately to public regular-season NFL and NBA feeds from 2018-2024, using a season-stratified dyadic bootstrap to account for recurring teams. We detect no global departure in the NFL and a systematic excess of extreme losing-team peaks in the NBA. In the upper 5% benchmark tail, the NBA excess is 1.3% (95% interval 0.4% to 2.3%); first crossings of a published win probability of 0.95 show the same positive gap between observed and implied loss rates.

27 pages, 6 figures. Revised version with dependence-robust inference, expanded NFL and NBA applications, and supplementary material