Cube Handling

Cubeless Probability of Winning 
Number of Positions 
Fraction of Positions 
0.00 to 0.05 0.05 to 0.10 0.10 to 0.15 0.15 to 0.20 0.20 to 0.25 0.25 to 0.30 0.30 to 0.35 0.35 to 0.40 0.40 to 0.45 0.45 to 0.50 0.50 to 0.55 0.55 to 0.60 0.60 to 0.65 0.65 to 0.70 0.70 to 0.75 0.75 to 0.80 0.80 to 0.85 0.85 to 0.90 0.90 to 0.95 0.95 to 1.00 
86807 9053 6983 6063 5681 5650 5557 5088 5032 4676 4894 4904 5226 5244 5477 5907 6231 7341 9228 85679 
30.9% 3.2% 2.5% 2.2% 2.0% 2.0% 2.0% 1.8% 1.8% 1.7% 1.7% 1.7% 1.9% 1.9% 2.0% 2.1% 2.2% 2.6% 3.3% 30.5% 
Unfortunately, many positions in the database have one side or the other as an overwhelming favorite. These are not particularly useful for judging cube handling. But there are still thousands of "good" positions for us to work with.
Many equities were also confirmed by consulting Hugh Sconyers's Bearoff Database.
Equities are recorded from the point of view of the player whose turn it is to roll.
Here is the number of positions in the database of each type:
Correct initial doubles  109,638  (39.1%) 
Correct redoubles  106,777  (38.0%) 
Correct takes  181,227  (64.6%) 
Correct passes  99,494  (35.4%) 
The importance of the pip count has been recognized just about as long as the doubling cube has been around. Grosvenor Nicholas and Wheaton Vaughan give a method of counting pips in their 1930 book Winning Backgammon: Problems and Answers.
The straightforward method of counting pips is to multiply the number of checkers on each point by the point number, and then add up the products. The following diagram shows the process.
In this example, White's pip count is 54. He will have to roll a minimum of 54 pips (probably more) to bear off all his checkers.
The probabilities in the chart refer to the cubeless probability of winning the game (CPW). They tell you your chance of winning a game if it is played without a doubling cube. When a doubling cube is used, your cubeful equity is more extreme (higher if you are winning, lower if you are losing). Formulas for converting between cubeless and cubeful equities appear later in this article.
Cubeless Probability of Winning 
Each Pip is Worth 
10% to 20%  about 1.5% 
20% to 30%  about 2% 
30% to 70%  about 2.5% 
70% to 80%  about 2% 
80% to 90%  about 1.5% 
An extreme example of wastage is Red's position below:
Red has 15 checkers on his one point, giving him a pip count of 15; White has one checker on each of his points 4, 5, and 6, giving him a pip count of 4 + 5 + 6 = 15. Though both players have the same count, it is easy to see that Red will need many more rolls to bear of his checkers than White. That's because Red will waste many pips almost every roll.
Many checkers on lownumbered points lead to wastage. 
In a different illustration of wastage, look at the following position:
Any time White rolls an even number, he is forced to waste pips. For example, if White rolls a 4, he must move one of his checkers from the five point to the one point. We have already seen that many checkers on a lownumbered point will end up being wasteful. The same type of problem happens if White rolls a 2; he is forced to move a checker onto a point that already has many checkers.
Gaps and high stacks lead to wastage. 
Over the years many people have proposed formulas that attempt to do that. In this section we will take a look at some of these formulas.
For each player, start with the basic pip count, then:
The resulting adjusted pip count gives a more accurate picture of the relative merits of the players' positions. We will see later how to use the Thorp count to make cube decisions.
For each player, start with the basic pip count, then:
The net effect of Keeler's and Gillogly's alterations to the Thorp formula is to reduce the size of the numbers to be compared. This makes the arithmetic involved a little easier.
For each player, start with the basic pip count, then:
The Ward count is a little harder to use than Thorp or Keeler/Gillogly, but it seems to be more accurate and many experts prefer it.
For each player, start with the basic pip count, then:
With so many steps, it's natural to ask: "Is a formula of this length worth learning?" We will see later that Lamford/Gasquoine actually does no better than any of the other counts.
For each player, start with the basic pip count, then
We'll see how this formula compares to other counting methods in the next section.
Here is an example to illustrate. Let's take a look at all the positions in the database in which the playeronroll has a CPW of between 74% and 76%. For each such position we'll calculate the pip count of each player, giving an ordered pair (x, y) where x is the playeronroll's pip count and y is the opponent's pip count. Then we plot the ordered pairs.
The resulting graph looks like this:
This shows that a straight pip count does only a rough job of lining up points of equal equity.
We can do the same type of plot with the other formulas and see how their graphs compare.
We can see that any formula is an improvement over a straight pip count, and the Keith formula seems to do best. Of course this is only a subjective opinion, and we are looking only at positions with CPW around 75%. It would be nice to get some hard numbers, and to compare more positions than just CPW = 75%.
So here is what I did. For each counting method, I found a smooth function of the players' counts that approximated as closely as possible the average CPW of the positions in the database. This gives a way for each counting method to estimate the CPW of a position. Then, for all the positions in the database with a CPW between 50% and 95%, I found the average* error between the CPW estimated by a particular count and the actual CPW.
*  Average error was calculated as the square root of the sum of the squares of the differences between the estimated and actual CPW. 
The following graph shows the error rate of each method at different length races. The error rate is plotted according to the playeronroll's pip count.
Thorp and Keeler do significantly better than a straight pip count in all but very long races. Lamford does a little better than Thorp in midlength races but not as well at the extremes. Ward count does quite a bit better overall, especially at shorter and midlength races. And Keith count does better than Ward except at very short races.
A set of decision criteria tells us when to double, when to redouble, when to take, and when to pass. We will use the term double to refer to the first double in a game, and redouble to refer to a subsequent double in the same game. Generally you want to have a slightly stronger position to redouble than to double.
For example, suppose your count is 80 and your opponent's count is 90. The difference is 10 pips. Eight percent of 80 is 6.4 pips, so a 10pip lead is easily enough to double. Nine percent of 80 is 7.2 pips, so 10 pips is also enough to make a redouble. Twelve percent of 80 is 9.6 pips, but since the opponent trails by 10 pips, he doesn't quite have enough to take if a double is offered.
Thorp's decision criteria are also used with the Keeler/Gillogly count and the Ward count.
where  CPW  =  cubeless probability of winning the game expressed in percent, 
PC  =  pipcount for the player on roll,  
L  =  number of pips lead he has in the race. 
You can then use the calculated CPW to make your cube decision. For example, you could say:
One advantage of this approach is that in match play you have an actual CPW that you can use with a match equity table to adjust your cube decisions according to the score. Of course there is a huge downside—the Lamford/Gasquoine approach is considerably more complicated to apply than any of the other methods presented here.
ND  No double. This is the player's cubeful equity if the cube is in the center and he does not double this turn. 
NRD  No redouble. This is the player's cubeful equity if he owns a 1cube and does not double this turn. 
TAKE  Take. This is the player's cubeful equity if he doubles this turn and his opponent takes. 
These equities give all the information we need to make "perfect" cube decisions:
So we can go through the positions in the database and see how often each method makes the correct cube play. And when a method makes an error, we can calculate the size of the error, measured in lost equity.
Cube Method  Doubling Errors  Redoubling Errors  Take/Pass Errors 
8912 Rule  2063.64  2380.06  7493.83 
Lamford/Gasquoine  794.13  840.84  1475.70 
Thorp  752.82  808.13  2097.53 
Keeler/Gillogly  490.17  593.30  1466.11 
Ward  435.98  526.23  1202.67 
Keith  348.66  353.63  633.89 
High numbers in the chart are bad; they indicate either that a particular cubing method makes a lot of errors, or that it makes costly errors, or a combination of the two.
The 8912 rule fairs worst. That's because it is using a straight pip count. Any formula that attempts to account for wastage does significantly better. This just shows how important it is to do some kind of wastage adjustment before using a pip count to make cube decisions.
The Lamford/Gasquoine count does a little better than Thorp. However, we will see that the Thorp count is not yet being shown in its best light.
It is interesting to see how much better Keeler/Gillogly does compared to Thorp considering how similar the formulas are.
The Ward count does a little better than Keeler/Gillogly. And the Keith count does the best overall.
These are the revised error numbers.
Cube Method  Doubling Errors  Redoubling Errors  Take/Pass Errors 
8912 Rule  2063.64  2380.06  7493.83 
Lamford/Gasquoine  794.13  840.84  1475.70 
Thorp*  574.14  579.54  1712.00 
Keeler/Gillogly*  401.24  467.17  1239.32 
Ward*  378.29  448.97  1032.81 
Keith  348.66  353.63  633.89 
As you can see, the 30pip condition helps quite a bit with all three formulas that use Thorp's criteria.
The cubeless equity of a position refers to the likelihood of a player winning the game if it is played to conclusion (without a doubling cube). We have been expressing cubeless equity as CPW (cubeless probability of winning).
Cubeful equity refers to the number of points you expect to win or lose when the doubling cube is in play. This depends not only on the position of the checkers, but also on the value of the doubling cube and who owns it.
From the database of noncontact positions, we can find the average cubeful equities at each possible CPW.
Each colored line represents a different cube action. The red line shows your equity if you don't double. The green line shows your equity if you double and the opponent takes. The blue line shows your equity if you double and the opponent passes.
You can see that when your CPW is less than about 68% the proper cube action is "no double." When your CPW is between about 68% and about 78%, the proper cube action is "double/take." And above 78%, the proper cube action is "double/pass."
Each colored line represents a different cube action. The red line shows your equity if you don't redouble. The green line shows your equity if you redouble and the opponent takes. The blue line shows your equity if you redouble and the opponent passes.
This looks similar to the centeredcube chart, but there are some important differences. The NRD line has no curve on its left side, so a player has a higher equity before he doubles than in the previous chart. This means you need slightly higher CPW before you can redouble because it takes longer for TAKE to overtake NRD.
When your CPW is less than about 71% the proper cube action is "no redouble." When your CPW is between about 71% and about 78%, the proper cube action is "redouble/take." And above 78%, the proper cube action is "redouble/pass." You can see that the double/take region of this chart is narrower than in the centeredcube chart.
I tried varying the decision thresholds and testing them on the positions in the database to see what numbers gave the lowest errors. Here are the results:
These are not necessarily the best universal numbers; they are just what worked best on the positions in the database. But the best universal numbers should be close.
Cube Method  Doubling Errors  Redoubling Errors  Take/Pass Errors 
CPW  80.31  102.99  18.42 
The CPW method of cube handling has a very low error rate compared to the other methods we looked at. Unfortunately, it is usually harder to accurately estimate CPW than it is to make correct cube plays, so this method is really more of academic interest.
For each player, start with the basic pip count and:
Increase the count of the player on roll by oneseventh (rounding down).

July 2004: Followup correspondence archived here.
April 2009: Stein Kulseth looks at how the length of the race affects the size of the doubling window. See: Keith count doubling window
Page last updated: June 22, 2011