This work reconciles two perspectives on the Elo ranking that coexist in the literature: the practitioner's view as a heuristic feedback rule, and the statistician's view as online maximum likelihood estimation via stochastic gradient ascent. Both perspectives coincide exactly in the binary case (iff the expected score is the logistic function). However, estimation noise forces a principled decoupling between the model used for ranking and the model used for prediction: the effective scale and home-field advantage parameter must be adjusted to account for the noise. We provide both closed-form corrections and a data-driven identification procedure. For multilevel outcomes, an exact relationship exists when outcome scores are uniformly spaced, but approximations are preferred in general: they account for estimation noise and better fit the data. The decoupled approach substantially outperforms the conventional one that reuses the ranking model for prediction, and serves as a diagnostic of convergence status. Applied to six years of FIFA men's ranking, we find that the ranking had not converged for the vast majority of national teams. The paper is written in a semi-tutorial style accessible to practitioners, with all key results accompanied by closed-form expressions and numerical examples.

翻译：本研究调和了文献中共存的两种关于Elo排名的视角：实践者将其视为一种启发式反馈规则，而统计学者则将其视为通过随机梯度上升进行的在线最大似然估计。在二元情形下（当且仅当期望得分服从逻辑函数时），两种视角完全一致。然而，估计噪声迫使排名模型与预测模型在原理上必须解耦：有效标度和主场优势参数需要针对噪声进行调整。我们同时提供了闭式修正方法和数据驱动的识别流程。对于多级结果，当结果得分均匀分布时存在精确关系，但一般情况下更倾向于使用近似方法：此类方法能够处理估计噪声并更好地拟合数据。这种解耦方法显著优于复用排名模型进行预测的传统方法，并可作为收敛状态的诊断工具。应用于六年国际足联男子排名数据时，我们发现绝大多数国家队的排名尚未收敛。本文采用半教程式写作风格以方便实践者理解，所有关键结论均附有闭式表达式和数值示例。