Mathematics of Elo Rating

Details

Supervisors

Absil, Pierre-Antoine

Faculty

Ecole polytechnique de Louvain

Degree label

Master [120] : ingénieur civil en mathématiques appliquées

Abstract

This master thesis aims to study the Elo rating system in three parts. The first one describes the system itself and the simulations of its convergence to an equilibrium rating as a function of different parameters. We will then conduct an analysis of the system in the form of a dynamical system in which the equilibria are shown to be stable for the continuous-time representation. This will result in an infinity of equilibria, but which are invariant by shift of ratings. The conclusion of that chapter will be that if we fix one of the ratings, the optimal rating is unique. The second chapter deals with the efficiency of Elo’s formula to estimate real probabilities inside a competition. The notion of transitivity between the relative strengths between players will be important in this part. Indeed, if probabilities are perfectly transitive, we will show that the optimal rating to describe a competition is easy to find and that Elo’s formula gives a correct estimation. On the other side, when probabilities are not transitive, the optimized rating is more complicated to determine but nevertheless exists. The last chapter will aim to understand behaviors of optimization algorithms and to find which one we should use to reach the rating with the best estimation of probabilities. The comparison between the steepest descent algorithm for different step sizes and the BFGS method will allow us to conclude that the BFGS seems to be more appropriate to obtain the best rating according to Elo’s formula.