The Elo rating system, commonly but not exclusively used in chess and various other competitive activities, has a mathematical basis designed to predict the outcome of a match and adjust players' rankings accordingly. Let's break down the maths involved.
The core concept is the expected score , which is calculated using the following formula:
Here and are the current Elo ratings of player A and player B, respectively. and are the expected scores for both players. The expected score essentially estimates the likelihood of each player winning the match. The values of will range between 0 and 1, representing probabilities. If both players have equal ratings, the expected score will be 0.5 for each, meaning each has an equal chance to win. If one player has a higher rating, their expected score will be greater than 0.5, and for the lower-rated player, it will be less than 0.5.
After a match, the Elo ratings are updated using the actual and expected scores. The formula for this is:
In these equations, and are the new Elo ratings for players A and B. and represent the actual score of the match for each player. In the case of a win, the score is 1; for a draw, is 0.5; for a loss, is 0.
The variable is the 'K-factor', which determines how much a match should affect each of the players' ratings. Higher K-factors increase the size of adjustments and so the volatility of the rankings. This makes the system more responsive to match outcomes. Lower K-factors make the system more conservative. We set the K factor based on the number of matches played, so for players with lots of data in the system a single loss will affect their ranking less.
The initial rating of 1500 in the Elo system is largely a convention, chosen as a midpoint value to allow ample room for players to move either upward or downward based on their performance. It doesn't hold any special mathematical property; instead, it serves as a reasonable starting point for competitors. By starting everyone at the same value, the system gets a baseline that provides a degree of equality. As players engage in matches, their ratings will diverge from this starting point in a way that reflects their relative skill levels.
So, the Elo system incorporates both your expected performance based on your current rating and your actual performance to update your rating in a manner that aims to be both fair and reflective of your skill level.