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Double or Take

The backgammon position below is a money game, but this situation also applies to match scores where neither player is close to winning the match. White is holding a 2-cube and is on roll and has a decision to make. Should he redouble? If White does redouble, should black take the cube or drop it?

The correct answer is that White should double, and Black should take. Do not feel bad if you didn’t get this right as I showed this position to several very excellent players and several got it wrong, and even the ones that got it right were not completely sure.

The White Side of the Board

Let’s talk about White’s situation first. It’s easy to see how things can go wrong for White. If he does not roll a 6 and Black covers with a 2 or if Black rolls a 3 very soon and does not have to break his board, White is in trouble. There are even times when White rolls the 6 and still loses the game later by leaving a shot. But for these things to happen, the bottom line is that Black had better roll a couple of 3’s pretty soon or his board breaks down and White is home free.

Backgammon is all about odds, and while there is a chance that bad things can happen to White, the chances are not that strong against him. In fact, if you look at the statistics in the Snowie evaluation below in Figure 2, you will see that statistically, White will win this game about 69 percent of the time. It further shows that things go really well for White 16 percent of the time and he wins gammons (double). With that kind of odds in your favor, you would certainly be happy to play for twice as many points (or twice as much money).

There is another reason to double that is most important to consider. Yes, things can go bad for White, but they can’t go bad if he doubles and Black drops. It is possible that the game will end right here with the cube if Black drops, and that ends all of White’s worries.

Back to Black

Now, let’s talk about Black’s decision. According to the Snowie evaluation, Black should take. In fact, Snowie says it is a .229 error to drop, and that means it would be a very large mistake to drop. Why should Black take the cube when he is going to lose 30 percent of the time? One might ask why anyone should ever take a double when he is going to lose more than 50 percent of the time.

The answer is all in the math. If you drop, you lose money or points, but if you take, in the long run, you lose less money or points in total. The “break-even” point in money games is 25 percent. So forgetting about gammons for a second, anytime you will win more than one out of four games (25 percent), you lose less in the long run by taking the cube instead of dropping it.

Do the Math

To prove this to you with math, let’s say you have a position, like the one above, where you will win 30 percent of the time. Let’s say you play 100 games like this. If you drop the 2-cube every time, you will lose a total of 200 points. But if you take the 2 cube every time and now have a 4 cube, you will lose 4 points 70 games, or a total of 280 points, but you will win 4 points 30 games, or 120 points. So your net loss after 100 games is 280-120, or 160. So instead of losing 200 points, you only lose 160 points, so you have saved 40 points by taking the cube.

Now, you might argue that you are not playing 100 games, but the point is, on average, every time you are in a situation where the odds are over 25 percent, you will lose less if you take.

Of course, you have to consider gammons into your thinking. When you take the 4 cube above and get gammoned 16 percent of the time, that means you will lose 8 points each of the 16 games out of 100 that you lose. But you have to look at the other side of this too. As the Snowie evaluation points out, if things go well for Black, he wins gammons too, about 6 percent of the time. So that brings the “net” gammons losses down to 10 percent, and that’s not enough to make Black drop.

One of the reasons Black has a take is that the Snowie numbers only show what the odds are at the given moment, with White holding the cube. The instant Black takes the cube, his odds of winning go up. They reason they go up is that if things go well for Black he can redouble again and either win twice as much or play for twice as much in a position where he has a big advantage.

Since White no longer has access to the cube once he doubles, he cannot end the game and “cash in” on his advantage. The player holding the cube has power, and has some additional advantage in holding the cube. The more game there is left to play, the greater advantage there is, of course, depending on the volatility of the position. This is the kind of position that could turn around with a couple of good rolls for Black, and holding the cube, he has an easy take.