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The Comparison
We will assess the accuracy of four different methods of bridge distribution evaluation via some standard common mathematical approaches.
The first one is the already mentioned Charles Goren’s system, known as the “3-2-1” system, named after the points assigned for short-suits holdings.
The second method is the Marty Bergen’s “Rule of 20” method from his famous book-series “Points Schmoints”. The approach of Bergen is to assign points equal to the sum of the lengths of the 2 longest suits of a hand, i.e. (a+b), using our notation.
We will also compare with the newest method from the late 90-ies, the Drabble rule of “adding the 2 longest suits, divide by 3, and subtract the length of the shortest suit, rounding downwards. Since Drabble’s scale starts with -1 for the 4-3-3-3, we have adjusted it by shifting the entire table with (+1) to eliminate the negative numbers.
In all cases we consider the initial base points, before the “fine tuning” in one way or another. The fourth method is the Zar distribution Points method you are already familiar with – assigning the value of (a+b) + (a-d), i.e. the sum of your 2 longest suits, plus the difference between your longest and your shortest suit (effectively representing the SUM of all the 3 suit-differences of the hand).
As we mentioned, there are 39 different suit-distributions in a bridge hand. The table below covers them:
Como ya mencionamos, hay 39 distribuciones diferentes en una mano de bridge. La tabla de abajo las cubre.
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Distribuciones de las manos por el palo mas largo |
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4-3-3-3 |
6-4-2-1 |
8-2-2-1 |
10-1-1-1 |
It is interesting to know what the probabilities for holding these distributions are, so here they go:
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Distribuciones de las manos y sus Probabilidades |
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4-3-3-3 = 10.5% |
6-4-2-1 = 4.7% |
8-2-2-1 = 0.19% |
10-1-1-1 = ~0 |
The numbers marked as ~0 are numbers less than 0.01%. It is worth noticing that the 4-3-3-3 distribution is not among the top 3 most probable distributions and that by far the most probable one is 4-4-3-2 – 6% above the second-most -probable 5-3-3-2.
The distributive part of the Zar Points varies from 8 for flat hand to 26 for the “wildest” hand with 3 voids . This means that it classifies the hands in 17 categories. Here they go:
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Puntos Zar para todas las Distribuciones |
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4-3-3-3 = 8 |
6-4-2-1 = 15 |
8-2-2-1 = 17 |
10-1-1-1 = 20 |
We are going to compare the 4 methods by 3 criteria:
1) span of base, given by the number of the groups the method classifies the hands in;
2) separation power, given by the maximum number of distributions which can fall in a single group;
3) standard deviation, which is explained below in the article.
To prepare for this exercise, we will present the following table with the points assigned by all four
evaluation methods:
| Zar Points | Bergen Points | Goren 3-2-1 Points | Drabble Points |
| 4-3-3-3 = 8 | 7 | 0 | 0 |
| 4-4-3-2 = 10 | 8 | 1 | 1 |
| 4-4-4-1 = 11 | 8 | 2 | 2 |
| 5-3-3-2 = 11 | 8 | 1 | 1 |
| 5-4-2-2 = 12 | 9 | 2 | 2 |
| 5-4-3-1 = 13 | 9 | 2 | 3 |
| 6-3-2-2 = 13 | 9 | 2 | 3 |
| 5-4-4-0 = 14 | 9 | 3 | 4 |
| 6-3-3-1 = 14 | 9 | 2 | 3 |
| 7-2- 2-2= 14 | 9 | 2 | 2 |
| 5-5-2-1 = 14 | 10 | 3 | 3 |
| 5-5-3-0 = 15 | 10 | 3 | 4 |
| 6-4-2-1 = 15 | 10 | 3 | 3 |
| 6-4-3-0 = 16 | 10 | 3 | 4 |
| 7-3-2-1 = 16 | 10 | 3 | 3 |
| 7-3-3-0 = 17 | 10 | 3 | 4 |
| 8-2-2-1 = 17 | 10 | 3 | 3 |
| 6-5-1-1 = 16 | 11 | 4 | 3 |
| 6-5-2-0 = 17 | 11 | 4 | 4 |
| 7-4-1-1 = 17 | 11 | 4 | 3 |
| 7-4-2-0 = 18 | 11 | 4 | 4 |
| 8-3-1-1 = 18 | 11 | 4 | 3 |
| 8-3-2-0 = 19 | 11 | 4 | 4 |
| 9-2-1-1 = 19 | 11 | 4 | 3 |
| 9-2-2-0 = 20 | 11 | 5 | 4 |
| 10-1-1-1 = 20 | 11 | 6 | 3 |
| 6-6-1-0 = 18 | 12 | 5 | 5 |
| 7-5-1-0 = 19 | 12 | 5 | 5 |
| 8-4-1-0 = 20 | 12 | 5 | 5 |
| 9-3-1-0 = 21 | 12 | 5 | 5 |
| 10-2-1-0 = 22 | 12 | 5 | 5 |
| 11-1-1-0 = 23 | 12 | 7 | 5 |
| 7-6-0-0 = 20 | 13 | 6 | 5 |
| 8-5-0-0 = 21 | 13 | 6 | 5 |
| 9-4-0-0 = 22 | 13 | 6 | 5 |
| 10-3-0-0 = 23 | 13 | 6 | 5 |
| 11-2-0-0 = 24 | 13 | 7 | 5 |
| 12-1-0-0 = 25 | 13 | 8 | 5 |
| 13-0-0-0 = 26 | 13 | 9 | 5 |
The table is ordered by the amount of Zar Points assigned, in ascending order. As might be expected, ALL methods basically follow the same ascending line, giving the least amount of points for the balanced distributions and the biggest amount of points for the “wildest” distributions. Since for everyone the 4-3-3-3 case is the “base” to which everybody assigns the minimum points we are going to consider only the rest of the groups in the evaluation methods (taking 4333 distribution as base).
In the table below, the columns of the table are the displacements from the “base”, (e.g. +1 means the first group after the base of 4-3-3-3) while the actual number in the body of the table represent the number of distributions the corresponding group.
| Método | +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | +10 | +11 | +12 | +13 | +14 | +15 | +16 | +17 |
| M.Bergen | 1 | 2 | 1 | 2 | 4 | 2 | 3 | 4 | 3 | 3 | 4 | 2 | 2 | 2 | 1 | 1 | 1 |
| Goren | 3 | 6 | 7 | 9 | 6 | 7 | – | – | – | – | – | – | – | – | – | – | – |
| Drabble | 2 | 3 | 12 | 8 | 13 | – | – | – | – | – | — | – | – | – | – | – | – |
Marty Bergen’s Points classifies the hands in 6 groups , the 3-2-1 in 9, Drabble in 5, and Zar Points in 17. This means by the criteria of span of base (number of classification groups) Zar points are between 2 to 3.4 times better than the rest of the methods.
The separation power of the methods is given by the max number of distributions in a group. In Zar Points this number is 4, while Bergen has 9, Goren – 8, and Drubble – 13. Again between 2 and 3.2 times better results.
When we take into account the number of elements (hands) in each group, we can now find the Standard Deviation for each method and see the difference there. Here is what is meant by that.
The root-mean-square (RMS) of a variant x, sometimes called the quadratic mean, is the square root of the mean squared value of x: (http://mathworld.wolfram.com/Root-Mean-Square.html).
Scientists often use the term root-mean-square as a synonym for standard deviation when they refer to the square root of the mean squared deviation of a signal from a given baseline or fit.
Applying the standard deviation from the basis (the x coordinate) measure to the three hand-evaluation methods (using the number of hands in each group) yields the following:
Recursive Zar Points: root-square ( 91/17) = rs( 5.35)= 2.31
Marty Bergen Points root-square ( 260/ 6) = rs(43.33)= 6.58
Goren 3-2-1 for void-x-xx root-square ( 220/ 9) = rs(24.44)= 4.94
Drabble’s method root-square( 390/ 5) = rs(78.00)= 8.87
So by this 3rdcriteria, the standard deviation of the evaluation method, Zar Points demonstrate between 2.2 and 3.6 times better results.
The interesting part is that by ANY of the applied three criteria:
1) Span of base
2) Separation power
3) Standard Deviation
Zar Points manifests roughly three times better results than any of the three competitors.