The Blown Lead Paradox: Conditional Laws for the Running Maximum of Binary Doob Martingales

Abstract: Live win-probability forecasts are now ubiquitous in sports broadcasts, and retrospective commentary often cites the largest win probability attained by a team that ultimately loses as evidence of a "collapse." Interpreting such extrema requires a reference distribution under correct specification. Modeling the forecast sequence as the Doob martingale of conditional win probabilities for a binary terminal outcome, we derive sharp distributional laws for its path maximum, including the conditional law given an eventual loss. In discrete time, we quantify explicit correction terms (last-step crossings and overshoots); under continuous-path regularity these corrections disappear, yielding exact identities. We further obtain closed-form distributions for two extensions: the maximal win probability attained by the eventual loser in a two-player game and the minimal win probability attained by the eventual winner in an n-player game. The resulting formulas furnish practical benchmarks for diagnosing sequential forecast calibration.