Cube Handling in Races
We’ll look at wastage, reference positions, techniques for computing the effective pip count of a position, and how to translate effective pip counts to cube actions. These are techniques I use over the board. They require less computation than tallying up the pip count in the first place.
Some of the potential wastage comes on the first roll, if it is more than 6 pips. If the first roll is less than 6 pips, it is likely that most of the second roll will be wasted.
The effective pip count comes from both the pip count and the wastage. Advancing checkers awkwardly will generally decrease the effective pip count. The pip count will decrease, and the wastage will increase, but not as much as the pip count decreases.
In the bear-in and the early bearoff, efficient positions waste a bit more than 7 pips.
What happens when there are checkers on the lower points? In the early bearoff, it doesn’t make a large difference to have one checker on the ace point. Two checkers starts to be a problem. It doesn’t cost much (perhaps a third of a pip) to have two checkers on the deuce point, but 3 checkers starts to be a problem.
The adjustment I use for pips positions is to penalize the second checker on the ace point by 1 pip, and the third and subsequent checkers on the ace point by 2 rising to 2.5 pips. For the third and subsequent checkers on the deuce point, I add 1 rising to 1.5 pips. Having extra checkers on the 3 point isn’t great, but it usually doesn’t warrant an adjustment by itself.
In addition to adjustments for checkers on the ace and deuce points, I add a synergy (combination) factor for the general ugliness of the position, having high gaps, too few checkers on the 6 point, or too many checkers on the lower points. How severely to penalize a gap depends on how well those numbers will play.
This synergy factor requires judgement, but the factor is usually small. I can usually estimate the synergy, hence the effective pip count, within a pip.
A pure n-roll position has an effective pip count of about 7n + 1. In real life, I hope you don’t end up with all of your checkers stacked up on the ace point. You have to deal with misses and non-working doubles.
The 7n + 1 formula says that it costs 7 pips either to miss or roll a non-working double. 1/36 is close to 1/35, and 1/35 of 7 pips is 0.2 pips. That means that if a particular double does not work for one roll, it costs 0.2 pips. So, if you can estimate the missing doubles, you can estimate the effective pip count.
It is typically more costly if there are an even number of checkers so that any roll taking off an odd number of checkers counts as a miss. Taking a checker off the 4 point saves a roll, but the effective pip count only goes down to 51.77, 1.77 more than a 7-roll position.
If the straggler is far back, then it is more accurate to use Walter Trice’s stack-and-straggler formulas. A single checker coming home wastes about 4.7 pips. Any position with that many pips will waste as least as much. If there is a stack of checkers on the low points and a straggler far from home, that has an effective pip count of roughly 3.5 times the number of checkers to bear off including the straggler plus the straggler’s position.
I recommend spending a short amount of time studying racing positions with one of the programs able to report an effective pip count. It doesn’t take long to build up your intuition about the epc with numerical feedback.
The reason is that, in general, these adjustments are wrong. They are often redundant, but sometimes they are even in the wrong direction. The only reason the methods get the right answer is that the adjustments are small. Nevertheless, the added complication is bad.
Stacks are bad if they are on low points, but on high points they are not particularly inefficient. It is wrong to penalize a position a fixed amount for a stack, ignoring the location of the stack and the other checkers.
Of course, I chose these positions because the other 5 checkers are placed well. Stacks make it more important to avoid having checkers on low points. Far more dangerous than having 10 checkers on the 6 point would be having 5 checkers on the bar point that might have to be sent deep as you bear in, particularly if you already have a checker on the ace point.
Gaps Gaps can be bad, but their level of importance is not linear. A gap on the 3 point may be no problem at all, if there are a lot of checkers on the 6 point. If there are few checkers on the 6 point, but many on the 1, 2, 4, and 5 points, then in addition to the wastage you typically expect from the checkers on the ace and deuce points, there will be additional checkers sent deep with every 3. Before the bearoff, the importance of a high gap depends on how easy it may be to fill the gap as you bear in.
A gap on the 4 point may be very serious, while gaps on the lower points are typically not a problem. Recall that the 7-5-3 position has no checkers on the lower points, and is the optimal bearoff position with 15 checkers.
So, you need to pay attention to gaps, but the effect depends strongly on the locations of the other checkers, not just where the gap is. I estimate the importance of gaps in what I called synergy in the adjustments for pips positions above.
Most players can compare pips versus pips positions. A good rule of thumb is that in races of
Most players can compare relatively pure rolls versus rolls positions. A 3-roll versus 3-roll position is a pass, a 4-roll versus 4-roll position a redouble and a take, and a 5-roll versus 5-roll position not a redouble, but still an initial double.
Most players don’t know how to compare pips positions and rolls positions. They also make too large or too small adjustments for awkwardness and missing numbers. A good way to do both is to use the effective pip count.
The borderline takes in pips versus pips positions are at about 22% in a medium length race (and 24% in a short race). That corresponds to a nominal lead of 10% + 2 pips in a race of at least 50 pips. Smaller leads are needed to induce a pass in a shorter or longer race. Many symmetric positions of about 12 pips are borderline takes, a lead of 10% - 1 pip.
If one side has a rolls position, the variance is lower. If the trailer has a rolls position, high doubles are not as valuable as in a pips position, since they only take 2 rolls off, 14 effective pips, rather than the 24 possible with 6-6 in a pips position. On the other hand, if the leader has a rolls position, it is harder for the leader to roll much below average, since 1-2 takes off 7 effective pips. The combination makes a 3-roll versus 3-roll position a pass at 22 effective pips, and a 4-roll versus 4-roll position a small take at 29 effective pips. Pips versus rolls positions have variances between those of rolls versus rolls and pips versus pips positions. So, in a pips versus rolls position, you can’t take quite as deeply as in a pips versus pips position of the same effective pip count. You may have to be a pip more conservative in a short race. This is discussed in a bit more detail by Walter Trice in Backgammon Boot Camp, Chapter 23.
Try to estimate the winning chances for each side, and the correct cube action. My proposed answers appear in the next section.
If you start with the linear rules penalizing the second checker on the ace point 1 pip and each additional checker 2 pips, you only get a total of 5 pips added,
Black’s checkers are also not ideally placed for the pip count. High numbers will force checkers deep, and it is likely that there will be a combination of a thin 4 point and too many checkers on the deuce point. Zbot estimates that Black’s position wastes about 9.5 pips, slightly more than in the second position in the section on gaps.
The combination says that the nominal pip count is misleading: White is not ahead by 6 pips, but behind by about a half pip. Black is on roll, worth 4 pips, but Black is not close to a redouble (or an initial double). On the other hand, it is also far from a beaver. In order to beaver in an even race, the pip count must be about 100, and here the race is shorter and favors Black slightly.
White’s position is pretty efficient, although not ideal for a position with 36 pips, since it would be better to have no checkers on lower points and more checkers off. It wastes 7.11 pips, and it is easy to get this estimate almost exactly right over the board.
Black’s position is ugly. The three checkers on the ace point get penalized 1 + 2 = 3 pips. In addition, there are too many checkers on low points, as well as a shortage of checkers on even points. That means that there will be a tendency to miss on 2s later in the bearoff. Black’s position wastes 12.74 pips, so the synergy is worth about 3 pips. (Over the board, my estimate for the synergy was off by about a pip.)
Instead of leading by 7 pips plus the roll, the lead is more accurately described as a lead of about a pip and a half plus the roll. That makes it a small redouble. Redoubling is correct by 0.030 EMG: Not redoubling is worth 0.703 (times 4) while Redouble/Take is worth 0.733.
Is Black the favorite? It is easy to estimate the effective pip count of White. An efficient position with 25 pips has an effective pip count of about 25 + 6 = 31. (31.14, here.) Black’s position is slightly better than a 5 roll position. It is very likely that Black will roll 1 or 2 aces in the next few rolls, in which case the position will behave as a 5 roll position. It’s more common that Black rolls no aces than that Black misses twice by rolling 3 or more aces. A true 5-roll position would have an effective pip count of 7 × 5 + 1 = 36, and this position has an epc of 34.83.
Black trails by just under 4 pips and is on roll, so Black is the cubeless favorite. (Black wins 51.0% of the time.) That is enough to redouble to 8 at this match score, according to live-cube rollouts using Snowie’s match equity table. Of course, it is a huge take.
Redouble/Take is worth 0.281 EMG. No Redouble is worth ~0.257 EMG.
In a pips position, the second checker on the ace point wastes a pip, and each additional checker wastes two pips. The third and subsequent checkers on the deuce point waste 1 pip each. These adjustments can be added to 6–7 plus the pip count to estimate the effective pip count.
An n-roll position has an effective pip count of 7n + 1. Misses and nonworking doubles cost 0.2 pips each.
Gaps, stacks, and crossovers are typically less important than described by other systems for estimating the race. Most of their effect is already described by the above penalties for having too many checkers on low points, and the remainder depends on the other checkers.
If the player on roll trails by 4 effective pips, the race is very close to even. Cube actions in pips versus pips positions are close to those for efficient positions of the same effective pip counts. Slight adjustments need to be made to determine proper cube actions in pips versus rolls positions, but the effective pip count is a good guide.
In some subsequent columns, we will consider more aspects of the effective pip count, such as estimating winning percentages, gammon wins, and cube actions after a closeout. We’ll see checker play decisions that are clarified by effective pip count methods.
Sets & Boards
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