The Elo chess rating system

The Elo chess rating system is a mathematical process developed by Arpad E. Elo, after whom it is named. Players are assigned numerical ratings according to their playing strength; higher numbers indicate stronger players, and a difference in ratings denotes a ratio of playing strengths.

The original basis, or the reference value, was the average playing strength of the competitors in the U.S. National Chess Championship competitions; this level was assigned a value of 2,000 points. A statistical ‘standard deviation’ was set at 200 points, and so a difference of 200 points between players now indicates a ratio of about 76:24 in terms of their relative playing strengths. Since the scale is essentially logarithmic, this matter remains the same at any level (i.e. the scores in a long match between players rated 1,600 and 1,400 would be expected to be the same as those from a match between players rated 2,800 and 2,600).

During a competition, the average rating of each player’s opponents is calculated and compared with the player’s actual rating at the time. This is used to calculate the player’s expected score, and this is in turn compared with their actual score (with ‘scores’ in tournament terms being one point for a win, half a point for a draw and nothing for a loss). The difference of (actual score – expected score) is multiplied by a standardized figure, dependent upon the player’s current status, and applied to the player’s rating to determine their performance during that event. This gives a ‘partial rating’ for a newcomer or other unrated player; unrated players are usually assumed (for their opponents’ purposes) to have ‘artificial’ ratings of 2,000 points. This process is applied to all games over a playing season, usually six or twelve months in length, to determine the player’s rating for the following season.

In the event of a player not having played (during a single season) a number of games deemed to be statistically sufficient for ratings purposes, two or more seasons’ games may be added together to produce a sufficient number. If this total – normally 20 or 30 games – cannot be achieved by considering the results of games played over a period of three years, the player is deemed to be unrated.

The British grading system uses a different method entirely; however the two sets of numbers can be translated from one to the other with reasonable accuracy; a player’s BCF (British Chess Federation) grade may be multiplied by 8 and then 600 points added to the result: This will yield a close approximation to that player’s FIDE (International Chess Federation) rating based on the Elo system. This is particularly so at the higher levels of play.

The statistical Nature of the Elo system is tricky to handle, but a non-statistical mathematical analysis is available; it forecasts the same basic results in match and tournament play as the statistical method over differences up to about 700 Elo ratings points. Here’s my method:

Take the difference in points between a player’s own rating and the average of their opponents’ ratings and call this ‘P’. Then take 0.337046 – (P x 1.063x10^-4) and raise this to the power of (P/200). This will give you the lower rated player’s score as a fraction of the higher rated player’s score. It is then a very simple process to calculate the individual player’s expected score; just remember to take the higher percentage share if the player is rated higher than their opposition, or the lower percentage if the opposition has the higher rating.

As an indication of what a rating means in its own right, grandmasters must – amongst other things – reach a rating of 2,500 points, and the top players are often in the 2,700+ range. By contrast, the average rating of the 20,000 or so players on the BCF grading list (after applying the conversion mentioned above) is about 1,500 points. The international lists published by FIDE start at 2,000 points.