Performance Rating Equilibrium

TL;DR

This paper introduces Performance Rating Equilibrium (PRE), a new rating system where initial ratings predict tournament scores perfectly, with broad applications from sports to AI model evaluation.

Contribution

The paper proposes PRE as a novel fixed-point rating system that improves prediction accuracy over existing methods like Tournament Performance Rating.

Findings

PRE exists under mild conditions

PRE accurately predicts tournament outcomes

PRE can be applied to diverse domains like sports and AI

Abstract

In this note, we introduce a novel performance rating system called Performance Rating Equilibrium (PRE). A PRE is a vector of 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, however, does not hold for the well-known Tournament Performance Rating. PRE is defined as a fixed point of a multidimensional rating function. We show that such a fixed point, and hence a PRE, exists under mild conditions. We provide an implementation of PRE along with several empirical applications. PREs have broad applicability, from sports competitions to the evaluation of large language models.