Performance rating in chess, tennis, and other contexts

Abstract: In this note, I introduce Estimated Performance Rating (PR$^e$), a novel system for evaluating player performance in sports and games. PR$^e$ addresses a key limitation of the Tournament Performance Rating (TPR) system, which is undefined for zero or perfect scores in a series of games. PR$^e$ is defined as the rating that solves an optimization problem related to scoring probability, making it applicable for any performance level. The main theorem establishes that the PR$^e$ of a player is equivalent to the TPR whenever the latter is defined. I then apply this system to historically significant win-streaks in association football, tennis, and chess. Beyond sports, PR$^e$ has broad applicability in domains where Elo ratings are used, from college rankings to the evaluation of LLMs.

• The paper introduces a novel Estimated Performance Rating (PR) system that generalizes traditional Tournament Performance Rating (TPR) by handling extreme score scenarios.
• It formulates PR as an optimization problem based on scoring probabilities and shows equivalence with TPR when TPR is applicable.
• Historical data from chess, tennis, and football illustrate PR's effectiveness in evaluating performance during win or loss streaks.

Abstract Overview

In competitive sports and games, player performance is commonly assessed using rating systems like the Elo system, established in various domains including chess, tennis, association football, and even fields outside of sports such as college rankings and AI evaluations. This paper presents a novel system known as Estimated Performance Rating (PR), which addresses limitations of the existing Tournament Performance Rating (TPR) by being applicable to all possible scores, including for zero or perfect score streaks.

Development of the PR System

The PR is defined by an optimization problem involving scoring probabilities and can provide a performance rating at any level of play. The study establishes a main theorem confirming that the PR equates to the TPR when the latter is well-defined—that is, other than in cases of zero or perfect scores. Through historical data analysis, the research applies PR to distinguish notable win-streaks in sports like tennis and association football, as well as chess performance. The findings suggest that PR provides an interpretation distinct from TPR and FIDE's Performance Rating (FPR) by emphasizing the maximum probability of a player scoring certain points in a series of games.

Illustrations and Applications

The paper includes examples and tables detailing how PR performs in different contexts compared to TPR and FPR. To illustrate, scenarios are examined where a chess player's TPR is undefined due to a perfect score series, and the equivalent PR is calculated instead. The study also applies this rating system to real-world scenarios, including tennis Grand Slam events and FIFA World Cup teams with perfect scores.

Implementation and Considerations

The implementation details and the code for PR calculations are provided, ensuring transparency and reproducibility. The paper concludes by reiterating the limitations of TPR and the necessity for a dynamic and adaptable system like PR in contexts where traditional ratings fall short. The PR's ability to incorporate the length of win- or loss-streaks into its evaluation demonstrates its potential broader applicability and significance in the current landscape of competitive sports and beyond.