用得分梯度更新评分，让体育比赛评级更精准公平。
Score-Driven Rating System for Sports
- 以得分梯度为更新依据，替代传统胜负判断。
- 理论证明评分均值为零且随实力上升而下降。
- 适用于多种比赛结果，适合构建新动态模型。
本文提出一种基于得分的评分系统，作为经典Elo系统的推广，其更新机制采用对数似然梯度（即得分）来调整球员或队伍的评分。该框架不仅处理胜负结果，还能兼容分差、平局/胜/负、完整排名等多种比赛输出。理论上证明了得分具有零期望值，在所有参赛者中总和为零，并随个体评分升高而降低，确保系统内部一致性和公平性。此外，该评分系统表现出均值回归特性，即评分长期趋向于真实隐含实力。该框架为现有体育表现动态模型提供理论支持，并为构建新模型提供系统化方法。
原文摘要 · Abstract (English)
This paper introduces a score-driven rating system, a generalization of the classical Elo rating system that employs the score, i.e. the gradient of the log-likelihood, as the updating mechanism for player and team ratings. The proposed framework extends beyond simple win/loss game outcomes and accommodates a wide range of game results, such as point differences, win/draw/loss outcomes, or complete rankings. Theoretical properties of the score are derived, showing that it has zero expected value, sums to zero across all players, and decreases with increasing value of a player's rating, thereby ensuring internal consistency and fairness. Furthermore, the score-driven rating system exhibits a reversion property, meaning that ratings tend to follow the underlying unobserved true skills over time. The proposed framework provides a theoretical rationale for existing dynamic models of sports performance and offers a systematic approach for constructing new ones.
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