The ATP finals is the concluding tournament of the tennis season since its initiation over 50 years ago. It features the 8 best players of that year and is often considered to be the most prestigious event in the sport other than the 4 grand slams. Unlike any other professional tennis tournament, it includes a round-robin stage where all players in a group compete against each other, making it a unique testbed for examining performance under forgiving conditions, where losing does not immediately result in elimination. Analysis of the distribution of final group standings in the ATP Finals for singles from 1972 to 2021 reveals a surprising pattern, where one of the possible and seemingly likely outcomes almost never materializes. The present study uses a model-free, optimization approach to account for this distinctive phenomenon by calculating what match winning probabilities between players in a group can lead to the observed distribution. Results show that the only way to explain the empirical findings is through a “paradoxical” balance of power where the best player in a group shows a vulnerability against the weakest player. We discuss the possible mechanisms underlying this result and their implications for match prediction, bettors, and tournament organization.

中文翻译：

---

ATP网球巡回赛单打决赛中的一个奇特现象及其潜在解释

ATP 总决赛是网球赛季自 50 多年前创立以来的收官赛事。它汇集了当年 8 位最佳球员，通常被认为是这项运动中除 4 大满贯之外最负盛名的赛事。与任何其他职业网球锦标赛不同，它包括一个循环赛阶段，一组中的所有球员相互竞争，使其成为在宽容条件下检查表现的独特试验台，输球不会立即导致淘汰。对 1972 年至 2021 年 ATP 总决赛单打决赛小组排名分布的分析揭示了一个令人惊讶的模式，其中一种可能的和看似可能的结果几乎从未实现。本研究使用无模型，优化方法通过计算一组玩家之间的比赛获胜概率可以导致观察到的分布来解释这种独特的现象。结果表明，解释实证研究结果的唯一方法是通过“矛盾的”力量平衡，在这种平衡中，团队中最好的球员对最弱的球员表现出弱点。我们讨论了这一结果背后的可能机制及其对比赛预测、投注者和锦标赛组织的影响。