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.

翻译：本文介绍了一种基于评分的评级系统，它是经典埃洛评级系统的推广。该系统利用评分（即对数似然的梯度）作为运动员和队伍评分的更新机制。所提出的框架超越了简单的胜负比赛结果，能够适应多种比赛结果，如分差、胜/平/负结果或完整排名。本文推导了评分的理论性质，表明其期望值为零、所有运动员评分之和为零，且随运动员评分值的增加而减小，从而确保了内部一致性与公平性。此外，基于评分的评级系统具有回归特性，意味着评分会随时间趋近于潜在的真实技能水平。该框架为现有的运动表现动态模型提供了理论依据，并为构建新模型提供了系统性方法。