Using the Logistic Function to Compute Elo

When designing ordered rankings (sports, gaming, even dating), the Elo Rating System in chess has proven very useful.

FIDE has tweaked the original rating scheme over the years, but in its current form it is:

\[E_A = \frac{1}{1 + 10^\frac{R_B - R_A}{400}}\]

\(E_A\) represents the probability (expected outcome) of Player A beating Player B. The base of 10 and standard deviation of 400 have been chosen such that when players have a 400 point difference, there is a 90% chance that the stronger player will win:

\[\frac{1}{1 + 10^\frac{-400}{400}} = \frac{1}{1 + 10^{-1}} = \frac{1}{\frac{11}{10}} = 0.90\]

And a 50/50 chance for either player winning when they have equal ratings:

\[\frac{1}{1 + 10^\frac{0}{400}} = \frac{1}{1 + 1} = \frac{1}{2} = 0.50\]

After a given outcome has occurred, the formula below is used to update each player’s rating is:

\[R' = R + K(S - E)\]

This updates a player’s rating given: current rating \(R\), expected probability of winning \(E\), and outcome \(S\).

Some new variables that require definitions: \(K\) is the K-factor constant controlling the sensitivity of update: larger for more sensitive and smaller for less. \(S\) is the score which takes value 1 for win, 0 for loss, and 0.5 for tie.