  The Elo system for rating chess players, also used in other games and sports,
was adopted by the World Chess Federation over four decades ago. Although not
without controversy, it is accepted as generally reliable and provides a method
for assessing players' strengths and ranking them in official tournaments.
  It is generally accepted that the distribution of players' rating data is
approximately normal but, to date, no stochastic model of how the distribution
might have arisen has been proposed. We propose such an evolutionary stochastic
model, which models the arrival of players into the rating pool, the games they
play against each other, and how the results of these games affect their
ratings. Using a continuous approximation to the discrete model, we derive the
distribution for players' ratings at time $t$ as a normal distribution, where
the variance increases in time as a logarithmic function of $t$. We validate
the model using published rating data from 2007 to 2010, showing that the
parameters obtained from the data can be recovered through simulations of the
stochastic model.
  The distribution of players' ratings is only approximately normal and has
been shown to have a small negative skew. We show how to modify our
evolutionary stochastic model to take this skewness into account, and we
validate the modified model using the published official rating data.
