Dr Ian Plummer
by Louis Nel
Dec 2001
This system applies exclusively to 26 point Advanced Singles games. For entry to the system and to remain in it, a player needs to have played at least 10 games in the preceding 12 month period and be sufficiently competitive.
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.
Index Adjustments
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
D = (Winner's index) - (Loser's Index)
For example, if Peter, with Index 2179, beats John, with Index 2122, then
D = 2179 - 2122 = 57.
On the other hand, if John beats Peter then
D = 2122 - 2179 = -57 (a negative quantity).
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.
ADJUSTMENT TABLE
D A(D)
up to -998 50
-997 .. -755 49
-754 .. -640 48
-639 .. -562 47
-561 .. -503 46
-502 .. -454 45
-453 .. -413 44
-412 .. -377 43
-376 .. -345 42
-344 .. -315 41
-314 .. -288 40
-287 .. -263 39
-262 .. -239 38
-238 .. -216 37
-215 .. -195 36
-194 .. -174 35
-173 .. -154 34
-153 .. -135 33
-134 .. -116 32
-115 .. -98 31
-97 .. -80 30
-79 .. -62 29
-61 .. -44 28
-43 .. -27 27
-26 .. -9 26
-8 .. 8 25
9 .. 26 24
27 .. 43 23
44 .. 61 22
62 .. 79 21
80 .. 97 20
98 .. 115 19
116 .. 134 18
135 .. 153 17
154 .. 173 16
174 .. 194 15
195 .. 215 14
216 .. 238 13
239 .. 262 12
263 .. 287 11
288 .. 314 10
315 .. 344 9
345 .. 376 8
377 .. 412 7
413 .. 453 6
454 .. 502 5
503 .. 561 4
562 .. 639 3
640 .. 754 2
755 .. 997 1
998 and more 0
|
So after each game, we have
Winner's New Index = (Winner's Old Index) + A(D)
Loser's New Index = (Loser's Old Index) - A(D).
Example: Suppose Peter, with Index = 1881, beats John with Index = 1992. Then
D = (Peter's Index - John's Index) = 1881 - 1992 = - 111
In the table we find that - 111 lies in the range -115 .. -98, so A(D) = 31. Accordingly,
Peter's new index = 1881 + 31 = 1912
John's new index = 1992 - 31 = 1961
Example: Suppose in the preceding game John beats Peter. Then
D = (John's Index) - (Peter's Index) = 1992 - 1881 = 111
In the table we find that 111 lies in the range 98 .. 115, so A(D) = 19. Accordingly,
Peter's new index = 1881 - 19 = 1862
John's new index = 1992 + 19 = 2011
Grade Adjustments.
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:
New Grade = (90% of Old Grade) + (10% of New Index).
This effectively means that the New Grade equals
10% of most recent Index
+ 90% of 10% of second most recent Index
+ 90% of 90% of 10% of third most recent Index
and so on.
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
New Grade = (S% of Old Grade) + ((100 - S)% of New Index)
where S lies in the range 90 .. 97 as determined by the expression
S = 80 + (Old Grade - 1000)/100
or 97, whichever is smaller. For example, if the Grade is 2400, then the stabilization percentage S becomes
S = 80 +(2400 - 1000)/100 = 80 + 14 = 94
For a graphic illustration of how Grades relate to Indexes,
The purple dots depict the fluctuating Index of a player after each game over a season. The yellow dots depict the corresponding fluctuation of that player's Grade after each game.
New Players
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.
Further Remarks
(of peripheral intested only).
(a) The above Adjustment Table and actual computer calculations of adjustments are based on the formula
A(D) := 50/(1 + 10 ^ D/500))
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).
(b) A player whose Index is greater than his Grade, is gaining ground. Similarly, if his Index is less than his Grade, then he is losing ground. When the Index is close to the Grade, it indicates a fairly steady state.