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The Croquet World E-Mail Interview:
Chris Williams, Co-Proprietor,
World Croquet Federation Ranking List

Only the Croquet God knows the true, utterly incontestable ranking of the world's croquet players. All players, however, whether they admit it or not, burn to know their place in the croquet universe. Lacking a clear channel to the divine being's font of knowledge, we must settle for the WCF World Ranking List produced by Chris Williams and Stephen Mulliner and distributed to the Nottingham croquet internet board.

The World Ranking List, sanctioned by the World Croquet Federation, ranks the top 350 international tournament croquet players, based on the results of "major" tournaments held around the world. The list contains players from thirteen countries (Australia, Canada, Switzerland, England, France, Ireland, Italy, Jersey, New Zealand, Scotland, South Africa, the USA, and Wales). The official list appears at the beginning and the end of the year, with periodic updates appearing on the Nottingham board. Since 1985, when the list began, 59,000 games have been included in the database.

The listing itself is not complicated. Click here to see an example of a recent ranking list. Along with the names of the players only three numbers appear to the right of each name: the ranking (a number equal to or greater than 1700), the number of games played, and the number of games won. For example: "Robert Fulford 2884 103 86." But to the uninitiated the ranking list is mysterious. Its surface simplicity deepens the puzzlement players feel when they scrutinize it.

How is the ranking derived? How are the games chosen for inclusion? How does a player get included in the list?

For an explanation of the World Ranking List, Croquet World Online Magazine turned to Chris Williams for enlightenment.

-- Mike Orgill

Croquet World Online Magazine: What is the World Ranking List?

Chris Williams: The World Ranking List is a ranking list of advanced level singles international rules 26-point croquet games. An official list is produced twice a year - at the end of March and the end of October. For a player to be included in the published list they must have played at least 10 games in the previous 12 months. Intermediate unofficial lists are put onto the net during the year. The World Ranking List has been officially sanctioned by the World Croquet Federation (WCF).

CW: How does it relate to other countries' ranking lists?

Chris Williams: The United Kingdom ranking list is the same as the World Ranking List, but with only the UK based players included. The Croquet Grading System (CGS), which is the name of the computer program used to produce the list, started in the UK and has been expanded to include events played around the world. Other countries' ranking lists are produced using completely different methods. Some are computer based, some are 'Grand Prix' based, i.e., based on how far a player has got in certain key events.

CW: When did the list begin, and who started it?

Chris Williams: The list began in the early 1980s. Initially Stephen Mulliner did the calculations by hand. A number of people had input into the early systems. The current system has been running since 1985 and includes some 59,000 games.

CW: How do you split responsibilities with Stephen Mulliner for producing the Ranking List?

Chris Williams: Ideally we should both be entering the data independently and checking that we have the same results. In practice, I enter the data and Stephen checks it. Ideally, ranking data and any comments should be sent to both Stephen and myself to make certain of its receipt, but we keep closely in touch with each other to reduce the risk of going astray.

CW: What is the purpose of the list? What role does it play in Britain in selecting players for international teams?

Chris Williams: The list gives an indication of which players fit into a selection criteria. It is useful for ensuring that no players fail to be considered for selection.

CW: How does a player get listed if one isn't on the list already?

Chris Williams: A player can get into the database by playing in an event which is included in the system. To get into the published list a player must play at least 10 tournament games in the relevant year. Usually the lists includes only players with a minimum grade and up.

CW: Which events are included in the system?

Chris Williams: At first only games played in the UK were included, but in the late 1980s when the WCF World Championship started, data from the major non-UK events such as the New Zealand and Australian Opens were entered. Over the next few years more and more non-UK data was added, but the problem has been obtaining it, especially for the minor events. To get as accurate a ranking as possible we need as much data as possible. For example, every game played by the leading players in all countries should be in. This then means that games played by their opponents should be in, because their grade would then be correct.

CW: How are results collected for input into the system?

Chris Williams: Data is supplied by a number of correspondents worldwide. It is hoped that, as the Internet grows in size and more overseas players get on-line, then more and more results will be sent electronically. I am gradually building up contacts in all the major countries, but would welcome copies of national Fixture Lists, so I can tell which events are missing and hence what I need to chase up. Don Gaunt is currently producing a world-wide calendar, and this will help us make the list complete.

CW: What criteria do you use in including tournament results from the United States?

Chris Williams: I include all U.S. international rules tournaments that are submitted. This year the following U.S. tournaments were included:

U.S. Open
USCA N.Carolina Championship
USCA N.Carolina Plate
Sonoma Cutrer World Croquet Championship
USCA Delaware Invitational
USCA National Championship
USCA California State Championship
USCA Chattooga Challenge
CW: How is the World Ranking System calculated?

Chris Williams: The World Ranking system uses a system similar to the Elo system in chess. A player gains and loses points after each game depending on who has won and the current indices of the two players. If a player beats a player on a much higher index then more points change hands than if the higher ranked player wins. Each event is given a weighting depending on its importance. In a top class event (e.g., the World Championship) up to 6 points can change hands, whilst a lower class event such as a consolation one only 4 can do. Most events are 5-point events. The formula used is

where "idiff" is the winner's index minus the looser's index (keeping track of the sign) before the game and "Class Factor" is 6, 5 or 4 depending on the event.

This means that for a Class Factor 5 game a difference of 15 points in the grades leads to 1.67 points changing hands if the player with the higher index wins and 3.33 if the player with the lower index wins. Each player has an "Index" which is changed after every game according to the result and the current indices of the two players concerned.

The winner adds an amount to his existing "Index", while an equal amount is subtracted from the loser's "Index". The following procedures are used to calculate new "Grades" after each game.

CW: Hold on! What's the difference between a "Grade" and an Index"?

Chris Williams: The "Grade" is simply a damped value of the "Index". After each game, the new "Index" often varies enough to move a player 10 places, and this would result in a very volatile ranking list. The change in "Index" is therefore converted in a change in "Grade" by using the damping formula, which is:

New Grade = (0.9 * old Grade) + (0.1 * New Index)
CW: OK, go on. How is the new index calculated?

Chris Williams: First, a new "Index" is calculated for each player using the formula above. This formula assumes that a player's index is in the range 0 to about 200, and that each game has a weighting factor of 4, 5 or 6. In practice, a factor of 6 is used for major events, such as world championships, international matches, and some national championships; a factor of 5 for national and regional championships and some major national events; and a factor of 4 for other "A" class events that are reported.

The new Indices are used to calculate intermediate grades, using the damping formula given, after which the intermediate grade figure is multiplied by 10. The new "Grade" to be used in ranking each player is then obtained by adding 1000 to this result. Since the changes in the index can be quite volatile we apply the smoothing function to arrive at the grade used for ranking purposes. All this means that the system is objective and depends on the results achieved by a player and the standard of opponents. Lots of wins against opponents with low grades will not give many points.

For an example of the ranking calculation, click here.

CW: How is the initial grade set when a player enters the Ranking List for the first time?

Chris Williams: When they first enter the system, players are given an initial grade based on their handicap. Internally the grades of players vary from about 40 for the worst to 190 for the best. In the UK we have introduced an Automatic System for handicaps which uses numbers in the thousands. In order to make the systems appear similar 100 in the ranking system equates to 2000 in the published rankings. (110 = 2100, 120 = 2200, etc.) A review of this initial grade is carried out after the first few game results have been processed for the new player, and if necessary, the initial grade will be altered and the calculation re-run to give a result that appears correct. New players are usually given a grade between 1700 and 1900.

CW: Have any statistical studies been done on the list's utility in predicting the outcome of matches?

Chris Williams: Yes. The calculations do have a statistical basis. The CGS is based upon work done in Chess, namely the Elo Chess ranking system. Players gain points based on the difference in the grades of the two players. If the player with a higher grade wins then fewer points change hands than if the lower graded player wins. As a player's grade rises it gets harder and harder for them to gain points. If a player only plays 'weak' opponents then he or she will gain very few points.

CW: Is the list more valid for British players because of the greater number of matches and tournaments used in the ranking? Are the rankings of players from other countries less valid for this reason?

Chris Williams: Yes. The more games played the more accurate the position. Though above a certain number of games there is very little difference. A reasonably good ranking can be done on as few as 10 games.

CW: Do you have any data giving the probability of differently ranked players beating each other?

Chris Williams: In our grading system, a difference of 150 points means there is a two to one probability of the higher- ranked player winning, that is, the better player should win a best-of-3 match by 2 games to 1. A difference of 300 points corresponds to a probability of 4 to 1.

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