Score-Driven Rating System for Sports

TL;DR

This paper presents a generalized score-driven rating system for sports that extends traditional methods like Elo by using score gradients for updates, ensuring fairness and consistency.

Contribution

It introduces a flexible, theoretically grounded framework that accommodates various game outcomes and provides insights into rating dynamics and properties.

Findings

The score has zero expected value and sums to zero across players.

Ratings tend to revert to true skills over time.

The framework unifies and extends existing dynamic sports performance models.

Abstract

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.