  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.
