1. Get the base pip count
2. Add ? point for each extra crossover
3. Adjust for "wastage" as follows:
a. Add two pips for each checker over two on the ace-point
b. Add one pip for each checker over two on the 2-point
That's reasonable, though imperfect. If one has 3 checkers on the 2-point, this is not great, but not terrible.
However, it is a horrible addition to 4 checkers on the ace-point, worth much more than an extra pip. There are some nonlinear effects.
c. Add 0.5 pip for each checker over five on any point
I disagree quite strongly. It's not very bad to have a huge stack of checkers on the 6-point. See this. For example, 12 checkers on the 6-point and 3 checkers on the 5-point wastes less than one pip more than the optimal 7-5-3 position (7.99 vs 7.07 pips wasted on average).
Part c) alone would say that it should be 2.5 pips worse, and then there is the matter of the gaps on the 4-point, 3-point, and 2-point.
On the other hand, having 4 checkers on the bar-point might warrant a serious penalty if combined with other checkers on the ace-point
(and sometimes mid-point).
4. Adjust for gaps as follows if there are at least six checkers above the gap:
a. Add four pips for a gap on the 4-point
That's often way too much. Some gaps can be fixed while bearing in. Further, having no checkers on the 4-point may be a minor problem if there are plenty of checkers on the 6-point and none on the 2-point, for example.
b. Add three pips for a gap on the 5-point
The seriousness depends on the numbers of checkers on the 6-point and ace-point.
c. Add two pips for a gap on the 3-point
d. Add one point for a gap on the 2-point
Gaps on the 3-point and 2-point are often not problems at all. Three pips is a very large penalty. One should penalize an unfixable gap or lightness on the 6-point.
5. Make your decision:
a. Double if you lead in the adjusted pip count by at least 10% (of *your* pip count)
b. Redouble if you lead in the adjusted pip count by at least 11%
c. Take if you trail by no more than 12.5% (of opponent's pip count)
It would be interesting to hear opinions about how this system compares to the other race formulae out there, both for accuracy and ease of use. It seems a bit too complicated for the level of accuracy it will produce.
On the other hand, it does point out some of the features to watch, and the adjustments are usually in the right direction.
When reviewing these counts, I usually ask what the formulas think of the 7-5-3 bearoff, which wastes 7 pips, and the 12-3-0-0-0-0 bearoff, which wastes 8 pips, and the 3-3-3-2-2-2 bearoff, which wastes 9.5 pips.
--Douglas Zare [posted probably early-2002]
I usually add one pip for the second checker on the ace-point, and two pips for the third and each additional checker.
I don't worry about additional crossovers except when they indicate that there will be wastage in the future, e.g., multiple checkers on your bar-point mean that you will bury checkers with high numbers as you bear in.
The problems from a gap are often overestimated. You need to look at whether there are opportunities to fill the gap, so a gap on the 4-point is very bad if you have a lot of checkers on the 6, 5, 2, and ace-points.
A gap on the 4-point is not so bad if you have no checkers on the deuce-point, and a lot on the 6-point.
In many positions I think it is better to estimate the effective pip count (epc) by comparison with reference positions.
For example, if one side has a no-miss position, the epc is 7*rolls + 1. If you almost have a no-miss position, you might penalize this by a pip
or two, depending on the chance of missing, and the doubles that don't reduce the rolls by two.
Other good positions to know include that checkers on the 654 waste about 5 pips, and will be off in 2 rolls about half of the time, and that
a good closed board may waste 9-10 pips (compared with the optimal 7 for a 15 checker bearoff position, 7-5-3).
It's helpful if you have a database of epc values of positions. One comes with Trice's Bearoff Quizmaster.
For a while I tried to estimate how bad it was to have gaps on high points, extra crossovers, and extra checkers on the low points, but I used to misestimate positions by several pips regularly, often because the modifications were redundant.
Then I started trying to estimate the effective pip count, using Trice's formulas and building up a few reference positions, and in most positions I can estimate the effective pip count within a pip. I still have problems determining how much a gap is worth when there are a lot of lower checkers. It's probably worth taking a bit of time to figure that one out. I also spent a while with Trice's Bearoff Quizmaster on positions with fewer than 6 checkers.
I've recently started trying to understand the shape of the variations of the number of rolls needed to bear off. In total, I think I've spent at most 8 hours studying bearoffs, outside of playing. If a position looks like a rolls position, I count the rolls, and determine how many working doubles there are, and how many misses, and whether these will persist, and I adjust the 7n+1 formula to account for these.
If a position is very efficient, usually it's trivial to estimate the epc: pip count plus 6 if there are few checkers, pip count plus 7 if there are close to 15 checkers.
In bear-ins without high gaps, I count 1 pip of wastage for the second checker on the ace-point, and 2 increasing to 2.5 for additional checkers on the ace-point. I count 1 for every checker on the deuce-point beyond the first two.
(I increase the count by some to account for the nonlinearity of having lots of checkers on low points, but I don't have a set method for this.) The best linear approximation has an average error under one pip over all bearoff positions.
The biggest mistake in racing formulas is to penalize a position for not having checkers on low points. The pip count is sufficient penalty for this, and sometimes is even excessive.
654 (one checker on each of the high points) is _much_ better than 54321. A thin 4-point is not too serious if there are plenty of checkers on the 6-point and none on the 2-point, but if there are checkers on the ace and deuce-points a gap on the 4-point can be quite bad.
Another mistake in racing formulas is to penalize excessively for extra crossovers. While it is great to have extra checkers off in positions with few checkers, as long as you have an efficient structure and won't be forced to bury checkers, it doesn't matter much if you are still bearing in. The higher pip count is sufficient penalty.
--Douglas Zare [posted probably mid-2002]
You add an estimate of the wastage to the nominal pip count. The wastage is a bit more than 7 for an efficient position with 15 checkers. Actually, in most positions, I just add the excess of the wastage over that of a typical efficient position to the pip count, e.g., I might count 70 pips plus 1 pip of extra wastage.
I believe Walter Trice discovered that a pure n-roll position has an epc of very close to 7n+1, e.g., an 8-roll position has an epc of 57.
This is comparable with an efficient position with a nominal pip count of 50. Very efficient positions with only a few checkers on low points are easy to understand. So, too, are pure n-roll positions.
Between the two, it is tougher to estimate the epc accurately. I estimate the "synergy," the nonlinear part of the wastage, using reference positions and experience. The synergy is typically small because the basic adjustments are good.
I believe the best tool at the moment for these types of situations is the effective pip count.
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Pip counts; White 33, Red 24 Money Game Red on roll, cube decision? • 0-ply/expert, cube:2-ply/33%, Full, 12960 games Pip counts; White 26, Red 15 Money Game Red on roll, cube decision? • 0-ply/expert, cube:2-ply/33%, Full, 12960 games
White has a pip count of 33, and a fairly nice distribution. I figured the wastage was 6.5 pips, for an epc of 39.5.
A database says 6.774, so 39.774 effective pips. Red has 10 checkers, so is at best a 5-roll position, which would be 7*5+1 = 36 effective pips. Each miss and nonworking double (out of 36) on each roll costs 0.2 effective pips, and there are a lot of those: 3-3 is a double miss,
while 2-2 and 4-4 are not working doubles this turn, and may not be for a few turns. 1-1 would take off 4 checkers, but would set up many future misses. Repeated 3's would miss.
I figured that this was about 2.5 pips worse than a 5-roll position, but a database says it is 3.213 pips worse, for a total of 39.213 effective pips. White trails by the roll and about a half pip, and wins 30.3% of the time.
An exact cubeful database gives the following equities:
ND: 0.622 NR: 0.661 DT: 0.656
This is an initial double (ND: -.034), but not quite a redouble, although it is very close (- .005).
Cube decision Rollout cubeless equity +0.394 Cubeful equities: 1. No redouble +0.611 2. Redouble, pass +1.000 (+0.339) 3. Redouble, take +0.658 (- 0.003) Proper cube action: No redouble
Now, Red is much closer to a 4-roll position. 2-2 and 3-3 don't work, but they might work next turn, and it is hard to miss without rolling doubles. I guessed Red is 0.8 pips worse than a 4-roll position, and a database says 0.936 pips, for an epc of 29.936.
White's position is efficient as before, wasting 6.527 pips, for an epc of 32.527.
Now that Red leads by 2.5 pips, the pass is clear, and White only wins 19.5% of the time.
ND: 0.830 NR: 0.851 DT: 1.159
Cube decision Rollout cubeless equity +0.609 Cubeful equities: 1. Redouble, pass +1.000 2. Redouble, take +1.160 (+0.160) 3. No redouble +0.851 (- 0.149) Proper cube action: Redouble, pass
I presented these positions together because I didn't find it intuitive that more than a third of White's winning chances had evaporated
on the 5-4 5-2 exchange. Red only gained 2 nominal pips, and 2 effective pips, but this was the distance between a borderline redouble and
a solid 1.159 pass.
--Douglas Zare [posted probably early-2003]