The standard formula for match-pointing pairs events results in percentages whose range tends to be wider as the number of tables gets smaller. The formula contains a bias, but it can be removed by a simple modification which is proposed in this article.
Consider a pairs event with 10 tables. The top score, if it is unique, will get 18 match-points out of a possible 18, which is 100%. Now suppose that there was another section of 10 tables which happened to produce the same unique top score. If the sections are match-pointed together, the shared top will get 37 out of 38 match-points, which is only 97.4%. In both cases, the top score was achieved by 10% of the field, so it is arguably equally good.
Let us examine the match-pointing process. First, we divide the range of numbers from 0% to 100% into segments, each of which corresponds to one of the scores obtained, and whose width is proportional to the number of pairs sharing that score. Then, for each segment, we assign the average of the numbers in that segment to all pairs who obtained the corresponding score.
In a section of 10 tables, the top score represents the top 10% of the field. The top 10% of the numbers are those between 90% and 100%, so the top score should be assigned the average of those numbers, which is 95%.
Currently the top score is given 100%. This is supposed to be the average of a range of numbers whose width is 10%, so that range must be 95% to 105%. Similarly, the range of numbers whose average is assigned to the bottom score must be Š5% to 5%. We conclude that the process actually starts with a range of numbers from Š5% to 105%, which is wider than it should be. It therefore results in percentages whose range is upwardly biased.
If the size of the bias were constant, it would not matter, but in fact it is inversely proportional to the number of tables being match-pointed together. Clearly this is important if the field is divided into sections of different sizes. The smaller sections will have a greater bias in the range of their percentages, so the leading pairs in those sections will tend to get higher scores than their counterparts in the larger sections.
The bias may also be important in a multiple-section event run over several sessions. In each session you should ideally match-point all sections together, because it means that all pairs who obtained the same score on a board will get the same number of match-points. However, in some sessions the sections may have been match-pointed separately for some reason. Those sessions will have a greater influence on the outcome of the event because they will have a greater upward bias in the range of their percentages.
The bias can be eliminated by a simple modification to the match-pointing process, which would require a change to Law 78A. Each score would be compared with all of the scores obtained on that board, including itself. Each score is obviously equal to itself, so the bottom score would get 1 match-point and the top score (of 10 tables) 19 out of 20 match-points, which is 95%. If the two identical sections considered at the start were match-pointed together, the two top scores would each get 38 out of 40 match-points, which is also 95%.
Travelling scoresheets are typically printed with a list of even numbers starting at 0, to assist players with the match-pointing process. Players are taught to find the top score by subtracting 1 from the number of tables and doubling the result. The modification I propose would require printing a list of odd numbers instead; the top would be found by doubling the number of tables and then subtracting 1. The process is otherwise identical.
Editor's note: This method for printing travelling score-sheets is not common outside New Zealand.
To test this proposal, I examined the results of a club duplicate session of 26 boards, with 50 tables playing in 4 sections. The standard deviations of the percentages were as follows:
|How match-pointed||Standard formula||Modified formula|
|Each section separately||7.223%||6.645%|
|All sections together||6.716%||6.582%|
Note that 7.223 / 6.645 = 1.087 and 6.716 / 6.582 = 1.020, confirming that the upward bias is equal to the reciprocal of the number of tables minus 1. In a section of 13 tables, a bias of this magnitude is enough to raise the leading pairÕs score by one percentage point, eg from 62% to 63%.
Also, 7.223 / 6.716 = 1.075 and 6.645 / 6.582 = 1.010, so the modification has greatly reduced the difference in the standard deviation of the percentages between separate and collective match-pointing. The difference cannot be eliminated completely as it is exacerbated by the presence of an additional random factor (for example, whether the other pairs who bid the slam were sitting in the same section as you).