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

TL;DR

This work addresses how to interpret extreme excursions of binary Doob martingales that track conditional win probabilities by deriving sharp, closed-form distribution laws for the running maximum $M_N$ and its continuous-path analogue $M$, including conditional laws given eventual loss. Discrete-time results incorporate last-step and overshoot corrections, while continuous-path results yield exact identities; the study extends to two-player and $n$-player settings, delivering distributions for the eventual-loser's maximum $M_\lambda$ and the eventual-winner's minimum $M_\omega$. These laws function as principled benchmarks for calibration of sequential forecasts in domains such as sports and finance, enabling objective assessment of narratives about “blown leads.” Overall, the paper provides a rigorous probabilistic foundation for interpreting large extrema under correct model specification and outlines practical extensions and empirical validations.

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.

Figures (3)

• Figure 1: Theoretical cumulative distribution function $F_{M_\lambda}(x)$ of the maximum win probability of the eventual loser for different pre-game win probabilities $p_0$. The symmetric case ($p_0 = 0.5$) shows a smooth curve, while asymmetric cases exhibit a kink at $x = p_0$ where both conditional distributions begin to contribute.
• Figure 2: Theoretical cumulative distribution function $F_{M_\omega}(x)$ of the minimum win probability of the eventual winner for symmetric $n$-player games. As $n$ increases, winners are more likely to have had very low win probabilities at some point during the game.
• Figure 3: Theoretical cumulative distribution function $F_{M_{\mathrm{loss}} \mid Y=0}(x)$ of the maximum win probability of losing trades for different inception probabilities $p_0$. As $p_0$ increases, the distribution shifts rightward, reflecting that trades starting with higher success probability are more likely to reach higher peak probabilities before losing.

Theorems & Definitions (16)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5: Discrete-Time Unconditional
• proof