Score   Minimum Doubling Point  Opponent's takepoint

0-0       50.0%                   75.0%
0-1       50.0%                   73.5%
0-2       52.6%                   74.3%
0-3       65.2%                   80.0%
1-0       50.0%                   73.5%
1-1       50.0%                   75.0%
1-2       60.0%                   76.7%
1-3       63.0%                   80.0%
2-0       47.4%                   71.4%
2-1       40.0%                   65.1%
2-2       50.0%                   70.0%
2-3       56.0%                   71.4%
3-0*      71.7%                   78.6%
3-1*      77.9%                   83.0%
3-2*      66.7%                   75.0%

3-0       34.8%
3-1       37.0%
3-2       50.0%

No double:

Hit shot, claim with cube (55.6%)—match equity 85%
Miss shot, drop cube (44.4%)—match equity 68%
Match equity if you do not double: 77.5%

Double:

Hit shot (55.6%) and win game (95%)—match equity 100%
Hit shot (55.6%) and lose game (5%)—match equity 0%
Miss shot (44.4%) and drop cube on 2—match equity 60%

Multiplying these outcomes gives you match equity of 79.5%

Thus, even though your win chances are well below the 71.7% needed when the cube is "live," you should double here.

It is important to remember why the opponent's takepoint is key. The minimum doubling point for you is the point at which you should double on the last roll of the game. At this point you will either win or lose; the cube will have no further value.

However, when you are above your minimum doubling point but not near your opponent's droppoint, there is no need to double. The cube is likely to have value in the future, so there is no need to give it up now. For example, at a score of 0-0, even with 65% win chances you are well into your doubling window, but you would have to shoot from 65% to 75% win chances by your next shake of the dice. (Or since most of our readers play online, perhaps I should say "click.") This is possible, but unlikely, and even if you go past 75%, you are unlikely to go past it by much. So in most cases, it is right to hold the cube.

Also notice that the takepoints above do not account for the value of owning the cube. Therefore, in general you can take a double even with winning chances a few percent lower.

Special Doubling Situations

The Free Drop

A special situation arises after the Crawford game when the trailer needs an even number of points to win the match.

Consider for simplicity a score of 1-away/2-away. If doubled, the leader can take and play the game for the match. If he drops, then the score will be tied, and the next game will be for the match. Therefore, he can drop in such a game with no loss of match-winning chances. Thus, it follows that he should take if he is at least 50% to win the current game, and drop otherwise.

This is a small, but perceptable advantage, and like most little things in backgammon, adds up over time. So let's consider a few possible cases:

Leader wins the opening roll

This is a source of some dispute. In general, the opening move is felt to confer about a 52.5% chance to win the game, but of course that depends on the roll itself. Some of the worse rolls, like 2-1, 4-1, 5-1, and 5-2, may actually make the opening roller an underdog. Nonetheless, most experts always take the cube if doubled if they moved first.

Trailer wins the opening roll.

Now, the trailer will have to wait for the opening roller's reply before doubling. At this point he will of course double.

Simply put, the leader should take if he is a favorite to win the current game, and drop otherwise. There are some theoretical considerations where he might want to drop if he is a slight favorite to win but the position is more gammonish than usual, but as a practical matter at this early stage, you are unlikely to be a favorite to win the game yet in greater-than-average gammon danger.

1-away/4-away and higher

When the score is 1-away/2-away the free drop must be used now or lost. However, at a score of 1-away/4-away or higher (6-away, 8-away, etc.) the free drop may have value later. Thus, in a marginal position it may pay to take the cube. For example, even if you might drop after a poor opening roll, you should definitely take if you got the opening roll. You are surely no worse than 49.5% to win the game, and you'll feel quite foolish if you drop, and in the next game your opponent opens with 3-1 and you reply with something feeble like 5-2. The cube will come over and you'll wish you still had the free drop available.

Checker play in free drop situations

As the trailer, when you get the opening roll in a free drop situation, you should be more cautious than usual. Slotting the 5-point with an ace is particularly pointless. If your opponent hits, he will happily grab the cube; while if he misses he will be pleased to drop. Having a strong initiative on your second roll is not your goal --minimizing your downside is.

Trick Plays

Occasionally you can hustle a little extra equity out of an opponent. Consider this sequence.

At a score of 1-away/3-away, you do not double right away. Your opponent, wondering if you're out of your mind, quietly plays on hoping you won't noticed the centered cube.

The game develops into a race, or perhaps a mutual holding game, a non-gammonish position. Suddenly you get a good roll and shoot up to 85% winning chances, say a 75-60 pip lead in the race. Acting like you just remembered the cube, you give a little shrug and turn it. Your opponent happily drops, thinking how glad he was to lose only one point in the game, not two.

The joke, however, is on him. Whether he takes or drops, he needs one more win to win the match. By winning just one point you have given him back a free drop, but you turned an 85% win into a 100% win.

There are three limitations to this trick play though.

1. You cannot use it when your opponent has a free drop available. If you do, he will gladly use his free drop when you are a huge favorite.
   
2. You must double before the position becomes gammonish. To win only one point rather than two is almost no loss at all when you need an odd number of points to win the match; to win one point rather than four is a disaster.
   
3. It will not, of course, work against an opponent who understands the trick.

2away/2-away

The situation where both players need two points to win the match is one of the more interesting ones in backgammon. Experts disagree on the appropriate strategy, and they are probably all correct!

First, consider a game between two perfect players. Say that after the opening roll, the player on roll is 49% to win the game. Should he even think of doubling?

Observe that the gammonless droppoint at this score is 30% win chances. If you drop the cube you are 30% to win the match, so you must win the current game (and the match) at least that often.

Now, the player on roll considers the following sequences:

1. A normal but moderately bad sequence. Say that after his roll he is 48% to win. His opponent now has a choice. If he holds the cube, his match-winning chances are:

   52% * 70% + 48% * 30% = 51.2%

   If he doubles his match-winning chances are 52%. So it seems he should double.

   What about the loss from giving up control of the cube though? Well, once the cube is turned at this score, it is dead. He no longer can double, but neither can his opponent. Only if he perceives the cube of being of more value to him than to his opponent is there any value in holding it. So it seems that in this instance the cube will be turned.

2. A very bad sequence. But how bad can you be after you roll? Can you really be below your 30% dropoint? Surely not. In this case, your opponent will presumably double, and you will take. So it costs nothing if you have doubled on the previous roll.

3. A very good sequence. Say you roll 6's, or perhaps 5's after your opponent has split with 24-22. You might already have lost your market! Now you wish you had doubled.

4. A normal but moderately good sequence. Say you are now 51% to win. This just transposes to the same situation from the other side of the board.

In cases 1 and 2, you will be doubled and take anyway. In case 3, you will wish you had doubled. In case 4, the situation just repeats until case 1, 2, or 3 arises.

So you will either wind up doubled, or doubling, or wishing you had. So the theoretically correct action is to double.

Theory is a wonderful thing, but try getting Mr. Theory to pay your losses after a session where you play perfectly and still lose 20 points. He'll be nowhere to be found. Let's talk about practice.

First, your opponents are not perfect. You are perfect, because you read GammOnLine and understand cube theory. But sometimes your opponents will not double when it is theoretically correct. Why do their thinking for them? If they wait too long and lose their market, let them.

Second, if you are playing a weaker player, your theoretical doubling window may not be 50% to 70%. Suppose you are 60% to win each game, but 20% of each side's wins are still gammons. Now, if you lead 4-3, your match equity is 77.6%, not 70%, so that is your opponent's droppoint, and your doubling window opens at 65.6%, not 50%. Now, only a huge mismatch of ability will lead to 60% odds of winning a single game, so don't take these actual figures too seriously. But the point is still valid, that as the stronger player you might be a little more reluctant to double.

There is another reason perhaps to hold the cube as the stronger player, which is, to give your opponent the opportunity to make a cube error. But - at least in my opinion—the value of this is far overestimated. The equity gained when you can estimate the value of the position better than your opponent is generally far less than the equity lost when you suddenly zoom past your market

Of course the reverse is true—as the weaker player you should be quicker to double, and since it is theoretically correct to double anyway, the weaker player should surely double at the first opportunity. So maybe the stronger player should double anyway, since the weaker one will, unless it is to his advantage not to? Well, that depends on how much weaker the weaker player is. In most cases, he will not know the correct 2-away/2-away strategy! Of course you do now, since you've read this article. So anytime you are playing a stronger player at this score, double at your first turn.

Match Cubes

A not particularly uncommon situation is this: The score is tied, and you are holding the cube. If you double, the cube level will be such that the match is on the line.

There is nothing unique about this situation that cannot be solved by general doubling principles, but there is an easy shortcut for this particular situation.

Assume, for simplicity, that the score is 3-3 in a match to 9 and you are holding a 4-cube. If you win the game, you will be leading 7-3, with match equity of 81%. If you lose you will be trailing 7-3—obviously with match equity of 19%. Regardless of the actual match equity, your gain when you double and win will always be the same as your loss when you double and lose. So your gain/loss ratio is always 50%, and your doubling window opens at 50%.

However, your opponent's droppoint is always equal to his equity if he drops, since when he takes the match is on the line. In this example, if he drops, he is 19% to win the match, so he needs odds of 19% to win this game (and the match) to take.

Thus, your doubling window is always 50% to your match equity if you double and your opponent drops. In theory, you should then double anytime you are at least 50% to win, and you also have any market losers—sequences of two rolls that will make your winning chances greater than your opponent's droppoint. Remember that you are offering a cube that will be dead if taken, so there is no value to your opponent in owning the cube.

In effect this is a special case of the 2-away/2-away situation. The only difference is that one side owns the cube, so only one side can double. Therefore there is no need to offer a 'preemptive" double, doubling in case things go well next roll, since you will be doubled anyway if things go badly. But the same principle applies—that you can double with 50% win chances, and should double anytime you have any market losers.

In our next article, we are going to take a break from doubling theory and discuss reference positions.
 

Next Article: Reference Positions.

Practical Backgammon is a column for beginning and intermediate players. Its goal is to offer specific solutions to common backgammon situations, and to provide the tools for advancing players to make use of more advanced material.