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A Guide to the Ranking System

Stephen Mulliner was the originator of the ranking system, which is currently run by Chris Williams. He wrote the following as a simple guide to the ranking system aimed at the non-mathematician.

The latest Association Croquet Rankings and Golf Croquet Rankings can be found on this web-site.

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 bell-shaped curve that appears to describe many naturally-occurring 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, i.e. 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 short-term 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.


