An Explanation of the World Ranking System
by Louis Nel
Relative skill level is expressed for every player in terms of two 4-digit integers: the Grade and the Index. For example, a player may have a Grade of 2179 and an Index of 2201. As explained below, the Index is a rather volatile indicator while the Grade is a more stable indicator, derived from the sequence of preceding indices. Ranking of players is based on their Grades.
After every game, the Index of the winner is increased by a certain amount, the Adjustment, while the Index of the loser is decreased by the same amount. The size of the Adjustment depends on the difference
For example, if Peter, with Index 2179, beats John, with Index 2122, then
On the other hand, if John beats Peter then
The Adjustment A(D) is proportional to the probability of winning the game that the losing player had just before the game. The weaker player will have a small winning probability. So if the weaker player loses, the Adjustment is small compared to the case where he wins. In the latter case, the Adjustment is relatively large. For a given value of this difference D, the corresponding Adjustment A(D) for a normal game can be found from the following table.
So after each game, we have
Example: Suppose Peter, with Index = 1881, beats John with Index = 1992. Then
In the table we find that - 111 lies in the range -115 .. -98, so A(D) = 31. Accordingly,
Example: Suppose in the preceding game John beats Peter. Then
In the table we find that 111 lies in the range 98 .. 115, so A(D) = 19. Accordingly,
A small number of games can cause a large change in the Index of a player. This volatility motivates introduction of the Grade, a more stable indicator of relative skill level. The Grade is effectively a weighted average of the preceding Indices. The most recent Index carries the greatest weight, the second most recent Index carries the second largest weight and so on. The actual weights depend on the present Grade. For players with Grade 2000 or less, the Grade adjustment after a game is defined as follows:
This effectively means that the New Grade equals
Only 1.2% of the 20th most recent Index is added. Thus the influence of earlier games gradually fades away.
For players with a Grade more than 2000, the influence of earlier games fade away somewhat slower, so their Grades are even more stable. The definition is now
where S lies in the range 90 .. 97 as determined by the expression
or 97, whichever is smaller. For example, if the Grade is 2400, then the stabilization percentage S becomes
For a graphic illustration of how Grades relate to Indexes,
When a player enters the system for the first time, an initial Index and Grade is determined by the Ranking Officer after observing the newcomer's performance against players who are already Graded. Thereafter all adjustments are automatic, as described. The system is self correcting: any imperfection in initial ranking will gradually disappear as game after game is played -- the more games played the faster the correction.
Classification of Events
Events are classified as Class 1 or Class 2 or Class 3. Most events are of Class 2, which we temporarily called "normal' events: the above table of Adjustments applies to such events directly.
Class 1 events are prestigious events (e.g. the British Open). For such events the normal adjustments are multiplied by 1.2 before rounding.
Class 3 events are typically consolation (plate) competitions. In this case the normal adjustments are multiplied by 0.8 before rounding.
(of peripheral intested only).
where 10 ^ x means "10 to the power x". When published, the calculated value is rounded to the nearest integer. (The official calculations are first done in terms of a related number, the Internal Index, via a different but mathematically equivalent formula. The published Index is then 1000 + 10 * (Internal Index). Similarly for the Grade).
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