An Analysis of Elo Rating Systems via Markov Chains

We present a theoretical analysis of the Elo rating system, a popular method for ranking skills of players in an online setting. In particular, we study Elo under the Bradley–Terry–Luce model and, using techniques from Markov chain theory, show that Elo learns the model parameters at a rate competitive with the state of the art. We apply our results to the problem of efficient tournament design and discuss a connection with the fastest-mixing Markov chain problem.