End Game Doubling:
A Science, Not an Art
"Dr. J." Jeff Ward, Ph.D., 1980
Las Vegas Backgammon Magazine, October 1980

Jeff Ward
Proper use of the doubling cube is probably the most important and most difficult part of backgammon beyond the beginner level. One reason it is so difficult is that it is normally hard to prove whether a particular doubling decision is correct or not. About the best that can usually be done is to play out a position (as a "proposition") to the end of the game over and over again, and keep track of the results. After a large number of repetitions the correct answer usually becomes clear.

This is a slow and laborious process, however, and one could do this night and day and still not investigate more than a tiny fraction of the situations that can come up. As a result, good cube play is more the product of much playing experience and the development of a feel for positional strengths and weaknesses. In this sense it is more an art than a science.

There is, however, one important part of backgammon where art gives way to science (mathematics), and doubling decisions can be determined precisely. This is the part of the end game when both sides are bearing off. Correct cube play has been determined for many such end game positions through mathematical analysis. Since mathematics is primarily of interest only to mathematicians, I leave a discussion of how this is done to the end of the article. It is suffice to say that the process is rather complicated and involves very expensive computer time even for simple positions.

One group of positions is worth some special attention because they occur fairly often and because they offer guidance as to how to handle more complicated positions. This is the group of positions characterized as "guaranteed, no-miss." A typical group of such positions are those in which both players have all their remaining checkers on their respective 1-points. With each normal roll a player will bear two checkers off, and with a double, four checkers off. These positions are best described by the number of normals rolls it takes to complete the bear-off. For example, three or four checkers on the 1-point is the "2-roll" positions, none or ten checkers is the "5-roll" positions, and fifteen checkers is the "8-roll" position.

Now, what do you do if you encounter one of these situations during a game?

Figure 1.
White has 6 checkers.
Black has 7 checkers.
In Figure 1, assume you are Black and it is your roll. Would you double? If so, should White take? Or assume it is White's roll. Should he double and should Black take? Mathematics provides the answer to each of these questions.

Let's first consider the situation in which one player needs more rolls to bear off than the other player. For example, in Figure 1 Black needs four rolls to complete the bear-off, and White needs three. If it is the turn of the player needing more rolls, he should not double. Of course, if he mistakenly doubles, it is a take by the other player. If it is the turn of the player needing less rolls, he should double and his opponent should drop. In Figure 1, if it is Black's turn, he should not double. If it is White's turn, he should double and Black should drop.

This takes care of most possibilities, but what if each player needs the same number of rolls to bear off? Here is where it gets interesting, because it depends on how many rolls are needed (Table 1). If Black is on roll, and needs three or less rolls, he should double and White should drop. If Black needs four rolls, he should double and White should take. When five or more rolls are needed, Black should double only if the cube is in the middle. In other words Black should not redouble; White should take in all cases.

Table 1

 No ofrolls No ofcheckers White's Position 1 2 3 4 5 6 7 8 No of rolls 1,2 3,4 5,6 7,8 9,10 11,12 13,14 15 No of checkers Black's  Position 1 1,2 D/D D/D D/D D/D D/D D/D D/D D/D 2 3,4 ND D/D D/D D/D D/D D/D D/D D/D 3 5,6 ND ND D/D D/D D/D D/D D/D D/D 4 7,8 ND ND ND RD/T D/D D/D D/D D/D 5 9,10 ND ND ND ND D/T D/D D/D D/D 6 11,12 ND ND ND ND ND D/T D/D D/D 7 13,14 ND ND ND ND ND ND D/T D/D 8 15 ND ND ND ND ND ND ND D/T

 ND   No Double RD/T  Double or Redouble/Take D/T   Double/Take D/D   Double/Drop

After studying Table 1, you may say to yourself, "Ok, but how often will both my opponent and I have 13 or 14 checkers stacked on our 1-points?" The answer is: probably not very often. But the information is still useful because it can be applied to a large number of closely related positions. For example, the rules still apply if a few of the checkers are on the 2-point instead of the 1-point.

Figure 2.
Should Black double?
Should White take?
In Figure 2, Table 1 applies because all doubles still remove four checkers and nondoubles remove two checkers.

Table 1 can be applied in another way. If a player has an odd number of checkers remaining, he can afford while bearing off to miss once and not have it hurt him.

Figure 3.
Should Black double?
Should White take?
In Figure 3, Black has five checkers left. If he misses by rolling, say, a 3-1 and takes only one checker off, he will be left with four checkers. But he is no worse off than if he rolls something like 6-1 and has three checkers left. In each case, he needs two more normal rolls to complete the bear-off. Therefore Black should consider this the equivalent of the 3-roll vs. 3-roll situation of Table 1. This means Black should double and White should drop.

With a little thought, Table 1 can be extended further.

Figure 4.
Should Black double?
Should White take?
In Figure 4 both players have four checkers, and the pip count in each case is 18. As a starting point, Table 1 says that 4 checkers vs. 4 checkers is a double and a drop. Because of the checker placement, however, both players are likely to miss should they roll a nondouble. This means that 3 rolls vs. 3 rolls is a better indicator. But most rolls that are doubles fail to take four checkers off, and so are not as effective as in the earlier positions we looked at. This necessitates making the judgment that 4 rolls vs. 4 rolls is probably an even better indicator.

If you are Black, you say to yourself, "Well, it's still a double (or redouble) in any case," and confidently thrust the cube at White. If you are White, you think it's a take but are not sure. You know, however, that at the very least, it is almost a take. (Incidentally, it is a take.)

One important word of caution needs to be made. Table 1 applies to stakes play (money games); it does not necessarily apply to match play (tournament games). When you are trying to be the first player to reach a certain number of points, as in a tournament match, the general rule is to be more conservative than usual with the cube when you are ahead, and less conservative when you are behind. The recommendations given in Table 1 should therefore be adjusted accordingly. Exact mathematical solutions to tournament doubling situations can be worked out, but space prohibits a discussion of the topic in this article.

The derivation of the information in Table 1 is based upon the mathematical concept of expectation. In this case, expectation is the expected return per one unit bet on the game. In order to know the correct doubling strategy for Black's position vs. White's position, it is necessary to know the expected return to Black should he double, should he not double owning the cube, and should he not double with the cube in the middle.

Table 2
Expected Return for Black in Selected Cases

 Case Number of RollsRemaining forBlack vs. White Black Doubles Black Does Not Double Owning Cube Cube in Middle 1 3-roll vs. 2-roll −1.444 −0.491 −0.722 2 3-roll vs. 3-roll +1.000 +0.768 +0.768 3 5-roll vs. 5-roll +0.727 +0.753 +0.704

Table 2 shows the expected return for Black with Black on roll in three different situations. Black always wants to maximize his expected return. In case 1, Black will probably lose money, but will lose more money (\$1.44 per dollar of original stake) if he doubles. So he doesn't double. Notice that he will lose less money if he owns the cube than if the cube is in the middle. (−0.491 means that if Black had to repeatedly face this same situation in many different games, he would lose on the average 49 cents out of each dollar bet in these particular games.)

In case 2, on the average Black will win no matter what he does, but will maximize his return if he doubles. This assumes White will drop and forfeit his original stake of one unit. If White takes, White will lose even more.

In case 3, with the cube in the middle, Black does slightly better if he doubles. However, if Black owns the cube, he does slightly better if he doesn't double. Notice that White does better if he takes. By taking White loses an average of 0.727 units per game versus one full unit if he dropped every time.