You may know the name of the famous chess player Garry Kasparov who hit the highest rating in the history of FIDE at 2849. You may also know that the FIDE rating is calculated by the Elo system. But do you know how?

The Elo System

The system is named after Dr. Arpad Elo who improved the original one developed by Kenneth Harkness. It has been in use in the USA since 1960 and was taken on by FIDE in 1970. The system is twofold:
1. It shows how strong the player is: Player A rated 2400 is stronger than Player B rated 2300.
2. The system also calculates the results of a game, tournament, or chess event as numerical Elo results.
Originally designed as a chess rating system, nowadays it is also being used in a number of other sports.

The Main Elo Idea

Each chess player has chance to win a game. The stronger player, the more chances to win. FIDE uses a special winning probability table for a game which is based on the rating difference between the two opponents.
If the rating difference between the two is 0, each player has equal chances to win, and his or her winning probability is 0.50. If the rating difference is 100, the stronger player has the winning probability 0.64 while the weaker 0.36. Please remember 100, 0.64, and 0.36.

Let's imagine that Player A rated 2400 and Player B rated 2300 are to officially play 100 games. The rating difference being 100, the expected result for Player A is therefore 0.64 and for Player B 0.36. And now the main Elo idea follows...
If Player A is really playing as strong as 2400 and Player B as 2300, at the end of the event Player A will score 64 and Player B 36 for sure. If Player A scores only 55 (but not expected 64) and Player B 45 (more than expected 36), the Elo system will change their new ratings.

The K-factor

The Elo rating system uses the K-factor which is necessary for Elo calculation. The K-factor is assigned to the chess player, and its possible values in FIDE are 10, 15, and 25 as follows:
* 25 for players new to the rating list, until they have completed events with a total of at least 30 games.
* 15 for players with a rating under 2400.
* 10 once the player has reached 2400 and been registered for at least 30 games. Thereafter it remains permanently at 10, even if the player's rating is under 2400 at a later stage.

Calculating the Rating Change

The current rating of the chess player changes after each game. The one-game Rating Change depends on:
* The player's K-factor.
* The player's score (1, 0.5, or 0).
* The player's Expected Result for a game.

Example 1. With the K-factor 10, Player A rated 2400 defeated Player B rated 2300. The Rating Change for Player A is therefore calculated as this:
Rating Change = K-factor x (Result - Expected Result)
Rating Change = 10 x (1 - 0.64) = 10 x 0.36 = 3.6

Example 2. With the K-factor 10, Player A rated 2400 lost to Player B rated 2300. In this case, the Rating Change for Player A is calculated as this:
Rating Change = K-factor x (Result - Expected Result)
Rating Change = 10 x ( 0 - 0.64) = 10 x (- 0.64) = - 6.4

Example 3. With the K-factor 10, Player A rated 2400 made a draw with Player B rated 2300. The Rating Change for Player A is now calculated as this:
Rating Change = K-factor x (Result - Expected Result)
Rating Change = 10 x (0.5 - 0.64) = 10 x (- 0.14) = - 1.4

Conclusion

The new rating of the chess player is calculated based on the rating change. Updated, the FIDE rating list is available online on 1 January, 1 April, 1 July, and 1 October.