Title: Elo performance in sports
Abstract: For more than fifty years, an open problem has been how to measure Elo performance in tournaments with perfect scores. In this work, I introduce two new methods for evaluating player performance in sports at any score: the Estimated Performance Rating (EPR) and the Complete Performance Rating. I show that a player’s EPR coincides with their standard Tournament Performance Rating whenever the latter is well-defined. I then apply these methods to historically significant winning streaks in association football, tennis, and chess. Having defined a performance rating for any possible score, I then introduce a new performance rating system, the Performance Rating Equilibrium (PRE). A PRE is a vector of hypothetical ratings for each player, such that if these ratings were each player’s initial rating at the start of a tournament, scoring the same points against the same opponents would leave each player’s initial rating unchanged. In other words, all players’ initial ratings perfectly predict their actual scores in the tournament. This property does not hold for the Tournament Performance Rating. Formally, a PRE is defined as a fixed point of a multidimensional rating function. I show that such a fixed point, and hence a PRE, exists under mild conditions. I provide an implementation of PRE along with several empirical applications. PREs have broad applicability, from sports competitions to the evaluation of large language models.
School of Economics Sir Clive Granger Building University of Nottingham University Park Nottingham, NG7 2RD