Cube Handling in Races

 The Effective Pip Count Douglas Zare, 2003

 This article originally appeared in GammonVillage, April 25, 2003. Thank you to Douglas Zare and GammonVillagefor their kind permission to reproduce it here.

 Contents Effective Pip Count A Few Reference Positions Estimating the EPC • Pips Positions • Rolls Positions • Stack and Stragglers • How Do I Determine the EPC? What about Penalizing Crossovers, Gaps, and Stacks? • Crossovers • Stacks • Gaps Cube Actions Examples Answers Summary

 The pip count is a simple way to assess most races, including races to save the gammon. In a long race (50–120 pips), you can double with a lead of 10% − 2 pips, redouble with a lead of 10% − 1 pip, and take with a deficit of 10% + 2 pips. In many situations, the raw pip count is unsatisfactory, and must be adjusted. Take a look at the diagram below. In almost all circumstances, you should prefer the position with 44 pips over the position with 15. (In a head-to-head contest, Black would be a favorite even if White were on roll.) Here, you can’t rely on the pip count.

 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
 White (15 pips) Which side do you like better? Black (44 pips)
There are several methods available for adjusting the pip count to penalize the candlestick, and the effective pip count (epc) is the best that I have encountered. It not only allows me to assess most races to within a pip, but it provides a framework for learning. If I misunderstand a position, I can improve. It is also simpler than many inaccurate systems.

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.

You can find some other introductions to the effective pip count in Walter Trice’s book Backgammon Boot Camp, Chapters 22, 23, and 24, and following my article “Burying vs. Bearing In.”

Effective Pip Count

The effective pip count is the average number of rolls needed to bear off times the average roll (8.167 pips). This assumes that the goal is to bear off in as few rolls as possible on average, which usually coincides with the correct strategy for beating your opponent.

 12 11 10 9 8 7 6 5 4 3 2 1
6 pips

This position takes an average of 1.25 rolls to bear off. The effective pip count is 1.25 × (49/6) = 10.208. Since the actual pip count is 6, the average wastage is 4.208 pips.

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.

 12 11 10 9 8 7 6 5 4 3 2 1
20 pips

This position takes an average of 3.97 rolls. The effective pip count is 32.425. The actual pip count is 20, so the average wastage is 12.425.

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.

A Few Reference Positions

 12 11 10 9 8 7 6 5 4 3 2 1
 79.00 +7.07 pips   wastage 86.07 epc
This is the 7-5-3 position. It is the most efficient bearoff position with 15 checkers. It is reasonable to aim for this position while bearing in. Anything close to it is almost as good, including positions with 8 or 9 checkers on the 6 point, but not those with even 2 checkers on the ace point, which will all waste at least 1.05 pips more.

 12 11 10 9 8 7 6 5 4 3 2 1
 57.00  +9.47 pips   wastage 66.47 epc
That’s 2.40 pips more wastage than the 7-5-3 position. Of course, that the nominal pip count is 22 pips less means that this is a better position, but not as much better as the pip count suggests.

 12 11 10 9 8 7 6 5 4 3 2 1
 15.00  +5.31 pips   wastage 20.31 epc
Short bearoffs with all checkers on high points tend to be more efficient than the 7-5-3 position. You can’t be sure that you will reach one of these positions from the 7-5-3 position, since you might roll awkward high doubles. 5-5 is a great roll, but you often don’t get full value from the nominal 20 pips.

 12 11 10 9 8 7 6 5 4 3 2 1
 12.00 +31.00 pips   wastage 43.00 epc
One of the most valuable reference positions: A pure n-roll position has an effective pip count of 7n + 1. Here we get 7 × 6 + 1 = 43. This is not perfectly accurate, but it is only off by 1/6 pip for a 1-roll position, and it is much closer for everything else. A triumph of the effective pip count is that it allows you to compare n-roll positions with normal, efficient positions.

Estimating the EPC

In efficient positions, with almost all checkers on high points, the epc is easy to estimate. In speed board positions, with no misses and all checkers on low points, the epc is also easy to estimate. When there is a mixture of checkers on high points and low points, it takes some practice to estimate the epc accurately.

Pips Positions

A pips position is one in which there a lot of checkers on high points. A 6-5 is a better roll than 2-2. In pips positions, I estimate the wastage, and add this to the actual pip count.

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.

 12 11 10 9 8 7 6 5 4 3 2 1
 59.00 +7.59 pips   wastage 66.59 epc

 12 11 10 9 8 7 6 5 4 3 2 1
 56.00 +8.37 pips   wastage 64.37 epc
Black is better off in the second position by 3 nominal pips, but only 2.22 effective pips, because Black wastes almost a pip more. I would estimate that the third checker on the deuce point costs one pip. It looks like the correct value in this case is 0.78 pips.

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.

Rolls Positions

A rolls position is one with few misses likely. There may be little or no difference between rolling 6-5 and 4-3. In rolls positions, I estimate the effective pip count directly.

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.

 12 11 10 9 8 7 6 5 4 3 2 1
A 2-roll position would have an effective pip count of 7 × 2 + 1 = 15. Here, 1-1 doesn’t work on the first roll, and the effective pip count is 15.20.

 12 11 10 9 8 7 6 5 4 3 2 1
Eleven numbers don’t work as well as in a 2-roll position on the first roll, and 21/36 of the time, there will be 10 missing aces next roll. That comes out to about 17 misses, each costing 0.2, for an estimated effective pip count of 15 + 3.4 = 18.4. The actual epc is 18.476.

 12 11 10 9 8 7 6 5 4 3 2 1
It is tough to count potential misses in a “no miss” position directly. However, in my experience, these positions take roughly an extra effective pip beyond the value given by the closest n-roll position. An 8-roll position would be 7 × 8 + 1 = 57 effective pips, and this position has an epc of 57.73.

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.

Stack and Stragglers

After a late shot is hit, there may be one or more checkers to bring home, and a pile of checkers on low points. Sometimes I view such positions as modifications on n-roll positions: A position with 2n − 1 or 2n crossovers needed to bear off is at best an n-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.

How Do I Determine the EPC?

The effective pip count can be extracted from Snowie’s databases, albeit with difficulty. Gnu backgammon has tools that allow you to construct a database that includes the effective pip counts of some racing positions. Walter Trice’s Bearoff Quizmaster comes with a database that includes the epc of bearoff positions. An upcoming commercial bot, Zbot, will provide estimates of the epc in both races and contact positions, among other data.

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.

What about Penalizing Crossovers, Gaps, and Stacks?

There are other systems for estimating the race. In addition to the pip count, these sometimes add in factors that penalize a side for crossovers, for having gaps, and for stacking checkers too high. The method I described above makes no direct adjustments for extra crossovers or stacks on points higher than the deuce point. There are no direct adjustments for gaps.

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.

Crossovers

Are extra crossovers bad? It sounds like they ought to be. However, extra crossovers generally mean more pips, and we are already counting pips. For a fixed pip count, extra crossovers mean piling up checkers on lower points of quadrants. If you properly penalize a position for stacks on the ace and deuce points, you don’t need to worry about crossovers.

 12 11 10 9 8 7 6 5 4 3 2 1
7.7 wastage

 12 11 10 9 8 7 6 5 4 3 2 1
7.11 wastage

These positions have the same pip counts, 78, and not too many buried checkers. The first position wastes about 7.7 pips. The second wastes 7.11 pips. Two crossovers don’t mean much (0.6 pips here) if your structure is still efficient.

Stacks

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.

 12 11 10 9 8 7 6 5 4 3 2 1
7.35 wastage

 12 11 10 9 8 7 6 5 4 3 2 1
8.39 wastage

 12 11 10 9 8 7 6 5 4 3 2 1
10.50 wastage

The first position, with a stack of 10 checkers on the 6 point, wastes 7.35 pips (as compared to 7.07 for the 7-5-3 position). Some systems of adjustments would penalize this structure by 2 or 3 pips, 10 times too much. The second position, with a stack of 10 checkers on the 5 point, wastes 8.39 pips. The third position wastes 10.50 pips. The stack on the 4 point isn’t the problem. The reason it wastes more than 3 pips more than the 7-5-3 position is that there aren’t enough checkers on the 5 and 6 points.

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.

 12 11 10 9 8 7 6 5 4 3 2 1
7.49 wastage

 12 11 10 9 8 7 6 5 4 3 2 1
9.31 wastage

In the first position, the gap on the 4 point is not too serious. The average wastage is 7.49, less than a half pip greater than the 7-5-3 position. In the second position, the gap on the 4 point is a major problem, since no number will fill the gap, and 4s will have to be played to the deuce or ace points. The wastage is 9.31, more than 2 pips greater than optimal.

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.

Cube Actions

Your opponents probably won’t be impressed if you announce that you have estimated the effective pip counts for each side. To impress them, confidently recube to 8 in races they don’t understand. In fact, in asymmetric positions, beavers are real possibilities; a player on roll trailing by more than 4 effective pips is a cubeless underdog, and even more of an underdog without cube access.

Most players can compare pips versus pips positions. A good rule of thumb is that in races of 50–120 pips, a lead of 10% plus being on roll means you will win about 75%. You can double with a couple of pips less than this, you can redouble with one pip less, and you can take if your opponent’s lead is up to 2 pips more. So, leading 75–70 double, leading 76–70 redouble, and take up to 79–70. In a shorter or longer race, the trailer can’t trail by as many pips and still take.

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.

Examples

The following are examples from actual play.

Position A:
 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
 69 pips Money play. Should Black redouble?Should White take? 75 pips

Position B:
 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
 36 pips Money play. Should Black redouble?Should White take? 29 pips

Position C:
 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
 25 pips White leads 4–3 to 9. Should Black redouble?Should White take? 20 pips

Try to estimate the winning chances for each side, and the correct cube action. My proposed answers appear in the next section.

 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
 69 pips Money play. Should Black redouble?Should White take? 75 pips
In Position A, the question is how much wastage there is from the 10 checkers on the ace, deuce, and trey points. There is more wastage than in the reference position with a closed board and spares on the lower points, 13.71 pips, or 7 pips more than optimal. How much does this position waste? Zbot estimates that it wastes about 16 pips, about 9 pips more than optimal. I think the best possible 15 checker bearoff forward from White’s position would have the shape 2-2-1-3-4-3, wasting about 15.5 pips. (In other words, from the reference position, moving a checker from the 4 point to the deuce point only saves 0.2 effective pips! 1.8 pips are wasted.)

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, 1 + 2 for the second and third checkers on the ace point and 1 + 1 for the third and fourth checkers on the deuce point. Having a large number of checkers on low points calls for a synergy adjustment, here about 4 pips.

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.

 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
 36 pips Money play. Should Black redouble?Should White take? 29 pips
Position B is actually from the same game as Position A, after the player holding the 4 cube rolled well including a great 2-2 in the bearoff. An efficient race of 36–29 would be a huge pass; to take in a race of that length you must be within about 10% + 0.5 pip.

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.

 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
 25 pips White leads 4–3 to 9. Should Black redouble?Should White take? 20 pips
Position C is interesting because of the match score. Black gains much more than Black risks by redoubling, so Black should redouble as a substantial underdog in a last-roll position. Of course, the game will not end on the next exchange, so there are more things to think about than the doubling window.

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.

Summary

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.

© 2003 by Douglas Zare and GammonVillage.

Douglas Zare is a mathematician and backgammon theorist. He writes a monthly column at GammonVillage on the theoretical aspects of backgammon. His web site is douglaszare.com.

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