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Dr Ian Plummer

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The Croquet Grading System (CGS)

The following is an extract from articles by Stephen Mulliner published in Croquet, numbers 228 and 229, August 1993.

a). Every player has an index which is a number lying between 0 and 200 and is changed after every game a player plays. The index can be a volatile number and is exponentially smoothed to produce the grade which is much less volatile and is used to produce the rankings lists. The CGS uses the results of all level single games in C.A. Calendar events and some overseas events.

b). The CGS algorithm has a strong theoretical foundation supported by empirical research. The rating of competitive performance has been extensively studied (see bibliography in The Rating of Chessplayers, Elo A.E., Batsford 1978) and supports the basic tenet of both the CGS and the Elo Rating Syatem, namely that "the many performances of an individual will be normally distributed when evaluated on an appropriate scale". In practice, accuracy is improved by using the closely related Verhulst distribution which is represented by the logistic function. This function required that the increment added to the winner's index (and subtracted from the loser's index) can be calculated as follows:


IW and IL are the winner's and loser's indices immediately before the game. K is set at 4 for low significance events, 5 for normal events and 6 for high significance events.

The results of this formula can be expressed in tabular form.  

0.04 0.05 0.06 0.99  
2.00 2.50 3.00 0.50
0.36 0.45 0.55 0.91  
2.23 2.79 3.34 0.44
0.55 0.68 0.82 0.86  
2.45 3.07 3.68 0.39
0.80 1.00 1.20 0.80  
2.86 3.58 4.29 0.28
1.14 1.42 1.71 0.72  
3.20 4.00 4.80 0.20
1.55 1.93 2.32 0.61  
3.45 4.32 5.18 0.14
1.77 2.21 2.66 0.56  
3.64 4.55 5.45 0.09
3.96 4.95 5.94 0.01

The column headed Pw sets out the probabilities that the player with the higher index will win. Thus, if A, 130, plays B, 100, the theoretical probability that A will beat B is 80% (or 4 to 1 on). The probability that B will beat A is therefore 20% (or 4 to 1 against).

c). The CSG is an objective system. Initial indices are estimated by reference to handicaps and estimates of relative strength are given by reliable judges. Performance over the first 15 to 25 games is monitored to check the reasonableness of the initial estimate and adjustments are made in a few cases. Once a player has entered the system, his or her index is affected only by results. The principal administrative problem lies in ensuring that all the relevant data has been collected. The advent of automatic handicapping and the carrying of record cards may help in this regard. Players will be encouraged to send photocopies of their cards to the Grading Officer to ensure that all relevant games are included.

d). Some believe that the CGS is too volatile to provide reliable "championship" ranking lists or to determine with reliability who is ranked first. In fact , the CGS is adjustable and can produce ranking lists with different characteristics. The present ranking list is based on grades which are calculated after each games as follows:

Gn=(1 - Y) * Gn-1 + Y * In

Where Y s set to 0.1.

To produce a "championship" ranking list, Y could be reduced to 0.05 and certain weaker events (such as plate events) could be ignored completely.

e). It is worth recording that I have conducted extensive experiments with these and other ideas (such as increasing K in the first equation for certain prestigious events) and found that they produce lists which were very similar to those produced by the existing parameters. At the end of the day, all that really matters is how well you do against which opponents. The determination of the number one position will often be controversial. It is commonplace for strong subjective opinions to lead to selective treatment of the evidence, namely the remembering of data consistent with the opinion and the dismissal of inconsistent data. A merit of the objective system is that all data is used impartially.

f). The role of the CGS in selection of events and teams should be clearly understood. Its primary purpose is to ensure the selectors know which players are within the "selection envelope" for any given purpose. Such selection envelopes are large enough to ensure that every player with a remote but credible claim is included. The CGS then ensures that all relevant data is considered by providing individual records for each player within the envelope. Once that stage has been reached the selectors' subjective judgement takes over. Grade differences are noted if they exceed 5 but even these have only marginal influence on evaluation for the President's Cup and international selections. Selections for the lower Eights make greater use of grade differences to choose between players.

Transcribed by Ian Plummer August 1998

Author: Stephen Mulliner
All rights reserved © 1993

Updated 28.i.16
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