## Doubling Cube Strategy

Many beginning and intermediate players play like they're afraid of the doubling cube. They will almost never double, and when doubled they will take or drop almost at random. In my experience, beginning players tend to take much too often, and intermediate players tend to drop much too often.

You will never be successful at backgammon if you don't know proper cube play. If you let your opponent win two points every game he wins while winning only one yourself, it's like giving 2-to-1 odds in every game.

### The Basics

Should you double? Should you take? Should you drop?

Consider a simple case. You are playing for money (not in a match), and you are on roll. You have two checkers on the board, on your 5 and 2 points; your opponent has one checker left on his one-point. Either you will win the game on this roll, or you will lose it.

A bit of math will show that you have 19 good dice rolls and 17 bad ones. You have a 52.8% chance of winning the game. So you should double.

Should your opponent take? This is easy. He should take when he is 25% or more likely to win the game. Say you have 4 games where your opponent doubles when you are 25% to win. If you drop all four, you'll lose four points. If you take all four and win one, losing three, you'll break even compared to dropping all four - losing four points. So your opponent should take anytime you have 26 or fewer rolls that win the game, and drop when you have 28 or more. If the number is exactly 27, well, does he feel lucky today?

### A Little Bit More Advanced Playing

Well, a lot of times, players have an advantage and don't double. Say you open with a 6-1 (making your bar point) and your opponent rolls 6-2. Surely you have the advantage. Why not double?

Let's say your game-winning chances are 52% at that point. What do you think they'll be the next time it's your turn to roll? Well, it depends on how your opponent played his 6-2 and what the next two rolls are, but let's say the range is plus or minus 15% - say, 37% to 67%. And that's a pretty wide range, kind of a best-case to worst-case, since this position isn't very volatile.

Well, if next two rolls turn out badly, you'll be glad you didn't double. If they turn out well, you can always double next time. If your opponent has a 33% chance to win, he'll still take. (And if he drops, great!) So you don't need to be in a hurry to double this turn.

I'm going to introduce the concept of market loser here. A "market loser" is a sequence of rolls (one for you and one for your opponent) that would cause your position to get so strong that your opponent would drop. Anytime you have no market losers, you should wait to double. If things go well, you haven't lost anything, and if things go badly, you will be glad you did.

The next statement is probably the single most important piece of advice you can ever get in backgammon:

Every roll is a new cube decision!

No matter what your position was last roll, you should always be at least thinking about the cube. Not necessarily long and hard, but if you got a good roll, or your opponent a bad one—think. "How good is my position now? Should I at least be thinking about doubling?"

### To Double or Not to Double?

Consider the four basic game possibilities:

1. Not good enough to double. Obviously, if you are trailing, you don't want to double (except in certain match situations, which we'll discuss later). Or if you have only a small lead and no (or very few) market losers.

2. Double/Take. Many players are afraid to double if their position is not overwhelming. They are worried that their opponent might take. That shouldn't be something to worry about, though. Suppose you have bet on a football game, and after the team you bet on scores a touchdown, you have a chance to double the bet. That's great—you're in the lead and get to double the bet! You might lose double sometimes, but you're going to win double a lot more often. And anytime you're in the lead in a game, there's usually some gammon possibilities—wouldn't you much rather win four points than two?

3. Double/Drop. Obviously, if your opponent doubles and your position is worse than about a 25% chance to win, you should drop. Give up a point and get on to the next. If you have significant chances of being gammoned, that's another good reason to drop and get on to the next.

4. Too good to double. Let's start with a little math here. Suppose you have 90% chances to win the game, or which 30% are gammons, and 10% to lose. On average you will win 1.1 points per game (0.9 plus 0.3 times 1 [the extra point for a gammon], minus 0.1). Why settle for just one point? Hold the cube for now.

Think also about the idea of market losers in reverse. We said earlier that if, no matter what sequence of rolls occurred, your opponent would still take, then there's no hurry to double. Well, the same thing applies here. Suppose that in the above example, the worst set of rolls you can get would leave you with no gammon chances and only 80% odds to win the game. Well, your opponent still has only 20% chances to win the game, and would still have to drop. So you can hold the cube for now. If things go well, you'll be glad you did, if things go badly, you can always double next roll.

### The Cube Value

Consider the following position. You are on roll, with 4 checkers on your ace-point. Your opponent has one on his six-point and one on his 4-point. What are your chances of winning the game?

If you have doubled, that means you can't double again. So you have to play the game to the end. 5/6 of the time you will not roll doubles. If you don't, your opponent can win with the following rolls: 6-4, 6-5, and any doubles of 3 or larger. That's a total of 8 of the 36 possible rolls on two dice (allowing for the fact that 6-4 and 6-5 are really two rolls out of 36, since they can come up either of two ways). So his winning chances are 5/6 times 8/36, or 40/216—about 18%.

But if you haven't doubled, your chances of winning the game are 100%! You should double, and your opponent should drop. So having doubled has cost you something. Now—you're not sorry you doubled, of course. When you're in this strong a position, you're happy to be playing for two points rather than one. But the point is that you shouldn't double with a weak advantage (if you doubled with a weak advantage, you were lucky to get to this strong a position), because giving up control of the cube costs something.

### Game Implementation of the Cube Strategy

We've talked mostly in principles so far. How do you decide whether to double or not?

Well, that's not a question with a simple answer. It's certainly not about pip count (except in racing positions). The important thing is to recognize that when you have a solid lead, you should be thinking about doubling. Most players don't realize that.

One good rule is to put yourself in your opponent's shoes. If he doubled, would you take? Would you even think of dropping? Many times, players will look at a position and say "I would drop, my position is pretty bad"—yet, on the other side, they think "I'm not good enough to double." Someone has to be wrong!

And if your opponent takes your double, don't think you've necessarily made a mistake. You might have doubled too early—when your position isn't as strong as you might like—but how bad can doubling the stakes when you're in the lead be? And also, maybe your opponent's take was a mistake!

### Doubling in Non-Contact Game Positions

The simplest kinds of doubles are in non-contact (racing) positions.

A general rule of thumb is that if you are ahead by 10% in pip count, you should double. You can take with up to a 12-13% deficit in pip count. The needed percentages get larger as the race gets shorter. For example, a lead of 88-80 is enough to double, but a lead of 44-40 probably isn't.

You also need to pay attention to how smooth your board is for bearing off, and how many checkers (if any) each side has off. Particularly bad are "useless gaps." If you have no checkers on your 3-point but have checkers on your 6-point, if you roll a 3, you can't take any checkers off, but at least you can move 6-3 and have useful 3's next time. But if you have no checkers on the 6-point—or a gap on the 5-point—or a gap on the 2-point and no checkers on the 4-point—be wary. You can waste an entire die movement in a position where every roll is crucial.

And remember—every roll is a new cube decision. Don't think that just because it's the last or next-to-last roll of the game, you can't double. Suppose you are trailing badly, and then roll a big set of doubles. Suddenly—let's say you are on roll with one checker on the ace-point and one on the three, and your opponent has just one checker left. Double! Why let yourself get unlucky and roll 2-1 and win when your opponent would surely have dropped?

### Doubling in Match Situations

Doubling in match situations is slightly different than in money play, where every point is equal. It is a complex subject and very mathematical. But I will try to simplify it a little.

If you are leading in a match, you should be a little more conservative about doubling, and about taking. But only a little. Don't let yourself get stolen blind.

When one player is close to winning the match, you should be aware of some special situations.

• Post-Crawford. When the Crawford game has already been played, the trailer should double at his first opportunity. Period. You can't lose more than one meaningful point in the game, so why not play to win as many as you can?

• Overage. Suppose you are trailing 3-1 in a match to 5 and your opponent doubles. You decide to take. Redouble on your next roll, no matter how bad your position! What can you lose? If you lose this game you lose the match—why not put it all on the line? The same would go if you were trailing 3-0, or any time you will lose the match if you lose this game.

Similarly, if you are leading, be more cautious about doubling in such a situation. Remember, if your opponent takes, he will immediately redouble. Instead of betting one additional point to win one, you're effectively betting three.

• Free Drop. This is a special situation after the Crawford game. Say you are leading a match to 5 by 4-2 and your opponent wins the Crawford game, so it's now 4-3. He rolls an opening 3-1, playing 8/5 6/5, and you roll 5-2. Regardless of what you play, you are clearly trailing in this game. Your opponent now doubles.

Normally, you would take. You're the underdog, but by no means as bad as 25% to win the game. But if you take, the match is on the line on this game. If you drop, it will be 4-4 and the match will be on the line next game. So you might as well drop, and take the chance to start the game over. Basically, you should drop anytime you are the underdog at the time your opponent doubles.

Note that this only applies when the difference between the two scores is an odd number. Say that the score, instead of being 4-3, is 4-2. Now, you should take. If you drop, your opponent will double in the next game and the match will be on the line in that one game. If you take and lose, the match will be tied 4-4. In short, you always take when a drop would mean your opponent needs one less game to win the match, and you at least consider dropping when dropping will not decrease the number of games he needs to win.

• 2-away/2-away. This refers to a particular situation where both players need exactly two points to win the match. In theory, you should double anytime you have an advantage. This is because once the cube is turned by either player, it has no more value. So you are not giving away control of the cube by doubling, because once you do, neither player controls it. The theory that shows you should double with any advantage is complicated. In fact, the same theory suggests you should double even when behind! (Because if you're behind after you roll, your opponent will double anyway, and if you're ahead, you'll wish you had doubled.) In practice, most players double with a smaller lead at this match score, but it's still pretty rare to see the cube turned immediately.

### Doubling in a Game and Match Equity

From a given score, it is possible to know your chances of winning the match. Consider a simple example. You are trailing 2-0 in a 3-point match. You need to win two games to win the match, period. (If you win the first game without a gammon, you will of course double on the first roll of the next game.) Since you are 50% to win each game, you are 50% times 50% or 25% to win the match.

Suppose instead you are trailing 2-1 in the Crawford game. Now your chances are a little better. You can win a gammon in this game to win the match, and on average you'll do that about 10% of the time. The other 40% of the time that you win (but don't win a gammon) you'll win the match 50% of those times. 10% plus 40% times 50% makes you 30% to win the match.

It is possible to know all the match-winning odds for all scores. That discussion is too complex for this article, but a good table and shortcut for memorizing it is contained in the sample issue of GammOnLine.com, Kit Woolsey's excellent online backgammon magazine.

So, suppose you are leading 1-0 in a match to 5. When should you double? When should you take?