MATHEMATICAL BASIS FOR GOLF CROQUET RANKINGS

The Mathematics of the System

The system chosen is based upon that used for the Association Croquet rankings. It was originally developed by Stephen Mulliner in the mid 80's and is now used by Chris Williams for the world rankings in Association Croquet.

The calculations for golf croquet are based upon each player having an index that for the start of the calculations will range from 95 to 130 dependant on the players conceived ability. The allocation of an initial index is made in accordance with a rather crude table with 130 being given to the very top players. As players within the system play competitive games, the winner will receive additional index points whilst the looser will give away the same number of points. Naturally if a very good player beats a weak player, he does not deserve the same credit as if a supposedly weak player beats a supposedly much stronger player. Thus the index points that are interchanged between the two players have to vary dependant on the estimated strengths of the two players at the start of the game. In practice the interchange of points is governed by the difference in index points of the two players at the start of the game.

The actual points changed are calculated from the following formula

Index Change = Class Factor/( 1+ 10^(idiff/50))

The variable 'idiff' is defined as difference in index points at the start of a game between the winner's and loser's index. It will be positive if the higher index player wins, but negative if the lower indexed player wins. The Class Factor is the basic number of points that change hands if the players are of equal index at the start of a game. This will be 5 for normal tournaments, inter club matches or heats of the national final but 6 for the finals of the National Championship, The Opens, the European or World Championships. Thus if two players of equal index play, the winner receives 2.5 index points for a normal tournament event but 3 for a major event. If, however, the players start with an index difference of say 50, the higher indexed player will only receive .45 points if he wins, but should he loose he will give away 4.55 points in a minor event or .55 / 5.45 in a major event. The attached graph gives the index changes for different index differences at the start of a game.

It should be noted that the phrase 'at the start of a game ' is used regularly throughout this text. It is therefore important that when results are reported they should be reported in order of play as this can make minor differences to the calculations if the results are not entered in the correct order.

The final placings in the ranking system are decided by a grading rather than by index points. This is because the ranking should not respond fully to day to day changes in a players form but must follow the general trends of a players index points. A smoothing calculation has therefore been added and the actual grading is calculated as follows.

New Grade = (· *Old Grade)+(1- · )*New Index)

Where · is the damping factor. In the Association Croquet rankings the damping factor varies with the ability of the players, but this has been simplified for Golf Croquet and a constant factor will initially be used for all calculations. Nominally a value of .9 will be used, however this value can be changed. One other difference from the Association rankings is that the starting index for Golf Croquet is nominally 100 whereas 1000 is used in Association.The graph below shows the difference between index and grading. The black line shows the rapid change of index from game to game. The mauve line is the resulting grading after the smoothing has been employed.

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Operating the System

It is simply not practical for any form of ranking system to produce good results from day one. Any system needs masses of data on which it can operate and experience has already shown that three seasons results have been necessary before the rankings truly reflect most players form.

In the first section it was stated that players were started at an index ranging from 95 to 130 dependant on perceived ability. Theoretically it is possible to start all players from the same index and after sufficient results have been input to the system the player finds his true index. Unfortunately we do not have sufficient results to make this practical and it also means that new players coming into the system have a false position for some considerable time. However as the system operates by recalculating from the first data every time it is run; a badly chosen start index can be detected, changed and the calculations rerun.

Since collection of results was started three years ago, all results from tournaments in which UK players have been involved have been put into the system. As such this includes matches involving many overseas players. All overseas players results are used in the calculations each and every time the system is run but their names are filtered out before any ranking lists are published. Similarly English players who do not have a total of at least 10 results in the system since it started are also excluded from any listings.

The calculation system has been developed using a Visual Fox Pro Database with a series of screens which allow input by use of mouse only for players already in the system.

Bill Arliss

Golf Croquet Ranking Officer.

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Updated 26 April 2003