In a game between Players A and B, there is a normal distribution centered at A’s rating and another centered at B’s rating. The standard deviation is 200. In the Elo system, the expected score for Player A is the probability that a random number from A’s distribution is higher than a random number from B’s. This seems to ignore the possibility of draws – there is a 0% chance that both random numbers are equal – but that will be addressed later. The expected score can be approximated with the logistic function:
Next, I model a tournament as n games against your
average opponent. This is an approximation (the expected score isn’t a linear
function, so a game against an 1800 followed by a game against a 2000 is slightly
different from playing two games against a 1900). With this assumption, your
score follows a binomial distribution. The mean is np and the variance
is np(1-p), where p is your expected score against the average
opponent. The issue with this binomial distribution is that there is no
accounting for draws. However, the binomial distribution converges to a normal
distribution, so I use that as an approximation. The normal distribution is
continuous, so scores such as 8.5 are possible. This means that we aren’t
ignoring draws.
