Dr Ian Plummer
Technical
Simple Guide to the Croquet
Grading System
Stephen Mulliner writes
As the original author, I will try and oblige with a "plain man's guide" to
the CGS. It recognises that there is more merit in scoring 5/10 (say) against
10 very strong players than 5/10 against 10 very weak players. That is why
a simple ranking system based on % wins is unsatisfactory.
The system needs a consistent basis for deciding how much more credit to
give X for beating a relatively strong player than a relatively weak players
and, the other side of the coin, how much less discredit to give for losing
to a relatively strong player than a relatively weak player. All mentions of "relative" are
to the strength of X.
The basis chosen is a relationship called the Verhulst or logistic distribution
which is a close relative of the normal distribution but easier to use in a
simple computer (which is all I had in the early 80s). This is in turn based
on research done by Arpad E. Elo who created the modern chess ranking system.
Elo's central principle is that "the many performances of a player in pairwise
competition will be normally distributed [about an average level]". He observed
this as a phenomenon and was able to prove that it worked mathematically because
his ranking lists agreed with surveys of players' opinion and, more importantly,
his rankings had reasonable predictive power. The Normal Distribution is the
formal name for the bellshaped curve that appears to describe many naturallyoccurring
distributions (heights of children in a class etc).
After a game has been entered into the system the primary output from the
formula is the "increment" which is added to the winner's index and subtracted
from the loser's index to produce the new indices. If the winner had a much
higher index than the loser, this implies that he was highly likely to win
and the increment will be small, ie close to zero. Hence the winner's and loser's
indices will be little changed which is consistent with the low degree of "surprise" in
the result. If the winner had a much lower index than the winner, this is a
surprise result and the increment will be large, tending to 50 for most games.
This increases the change of the two indices.
The index is necessarily fairly volatile to match the amateur characteristic
of the game which means that shortterm form can and does vary. However, in
order to avoid the ranking order changing every day, this effect is damped
by also calculating an average index or "grade" which does not change as quickly
or as much as the index. The ranking list is composed of grades rather than
indices. In any average calculation, even the "exponential moving average" used
by the CGS, there is a lag effect so the impact of the most recent games is
really their comparison with game splayed some time ago. Hence the slightly
odd effect where Robert Fulford's comparatively weak performance in the President's
Cup (7/14) takes a little time to work through the system.
I hope this helps a bit. If you want to understand exactly how it works,
the articles already published are the best source. A degree in maths is not
necessary but some familiarity with probability theory would help.
Chris Williams adds
>Is there any approximate relationship between index in
the CGS and index on a handicap card?
Yes. It was done deliberately.
When the rankings were first published the numbers used to be in what I call
base 100. In fact internally in the ranking system (CGS) they are still calculated
in this base and converted to the 2000 base when the list is produced. Originally
the AHS (Automatic Handicap System) was also in base 100, i.e., a 0 handicap
was 100 points rather than the 2000 it is today.
When the AHS was expanded to include level games the numbers in the AHS were
changed from base 100 to the current base 2000. Prior to that 1 point was gained
for winning a handicap game in the AHS, rather than the current 10 points.
At the same time I changed the CGS (Croquet Grading System, i.e., the Rankings)
so that the values listed were in base 2000.
I have found that for most players with a UK handicap of 2 or below, their
AHS index is fairly close to their CGS grade. When a new player is entered
into the system they are usually given a CGS starting grade and index which
reflects their handicap. I have found that this rarely has to be altered, apart
from rapid improvers. For example last year I altered [..]'s starting grade
at least once if not twice. Now that he has played enough games I should not
have to alter it again.
Author: Stephen Mulliner
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