This article originally appeared in the August 1999 issue of GammOnLine.|
Thank you to Kit Woolsey for his kind permission to reproduce it here.
I am sure that a lot of our beginning and intermediate readers will be turned
off just by the title of this article. It contains two scary words,
"Doubling" and "Theory."
The new player thinks of the doubling cube this way:
"The cube is an annoyance. The idea of backgammon is to race your checkers around the board, hit your opponent without getting hit, trap him where he doesn't want to be. If I get lucky maybe I'll win two points for a gammon. But backgammon is about the checkers and the dice. Anyway, I don't understand the cube and I really don't want to."
The advanced player thinks of the doubling cube this way:
"The cube is the best part of the game. It's one part of the game where it's all skill and no luck. When I handle the cube right, I can win a game without sitting helplessly while my opponent gets lucky at the end of the game. Sometimes I can even steal a game because he drops incorrectly. I can confuse him and psych him out. When I double and he takes and I win a gammon, that four-point loss that he could have gotten out for one may destroy him psychologically. A really bad checker play decision can gain or lose me maybe an average of a tenth of a point on the scoresheet; a really bad cube decision can cost a whole point! I'm glad I understand the cube."Readers of this series are beginning players hoping to become advanced players. That's why they care about the doubling cube, and why they care about the theory behind it. Doubling theory is not particularly abstract. Mostly it comes down to a few mathematical principles, and as we've seen in earlier articles, backgammon is all about basic math.
Many beginning players take a seriously wrong view of the cube. They hate to double when there is any significant chance they could lose, and they also hate to drop when they have more than a long-shot chance of winning - they feel like a quitter. It is very important therefore to understand the idea of the doubling cube:
The goal of doubling and taking or dropping is not to "cash" a game that is already all but won, or to show your resolve not give up easily. It is not to end the game, nor to prolong it. The goal of doubling is to increase your long-run expected number of points won per game.It will take several articles to cover doubling theory. There are entire books about it. In this installment, we will just scratch the surface.
We need to start with some terminology. Not only will this be useful in reading this series, but it will be helpful in reading articles by other authors on this subject.
The Basics of Doubling TheoryLet's begin with the simplest of examples: Throughout this article, we will assume that there are no gammons, unless specifically mentioned otherwise.
First, start with a simple principle:
When you drop the cube, you lose the current value of the cube. If you take, your starting point is twice your current equity in the game. Therefore, you should take when your equity is greater than negative .50.
How do we apply this?
Should Blue double? Should White Take?
It is easy to see that on the next roll, Blue has 27 rolls that will allow him to take his last checker off. If he succeeds, he will win the game; if he fails, he will lose. The doubling decision isn't hard on a common-sense basis. With a 75% chance to win no matter what the value of the cube, Blue should want to play for the highest stakes available. Blue should clearly double.
What is White's equity? He has 25% winning chances and 75% losing chances; his equity is .25 minus .75, or minus .50. This is a borderline decision. What we learn from this is that you should take a double when your game-winning chances are 25% or greater. This is the equivalent of an equity of minus .50.
Like most "rules" we will modify this later.
The Value of the CubeLet's look at another position:
Should Blue double? Should White take?
This is a relatively easy position to calculate, although a bit harder than the last one. Blue has 25 rolls that win the game immediately (any roll that doesn't contain an ace). If he fails to do so, then White has 29 rolls that will win him the game (everything but 1-1, 2-1, 3-1, or 3-2). Blue's chances of winning are 25/36 plus 11/36 times 7/36, for a total of 77.6%. Clearly he should double, and White should pass, right?
Not quite. Remember that White can double also. Whenever White takes, he has the opportunity to redouble anytime Blue fails to get both checkers off, and Blue should pass, since he will lose 29/36 of the time. Because of the cube, Blue gets no advantage from White's possibility of not getting both checkers off. Allowing for the possibility of a correct redouble, White's actual losing chances are only 72.2%. He should take. In the long run, he will lose 2 * (72.2% − 27.8%), or about nine-tenths of a point, by taking, compared to a full point by dropping.
Anytime you have the ability to double, you are able to win some games you might otherwise lose. When you don't have the ability to double, you will lose some games you might otherwise win. Therefore we're going to introduce a very useful rule of thumb: When you can double and your opponent cannot, increase your expected winning chances by 10%. If you would win 23% of the time with the cube in the middle, expect to win about 25.3% of the time with the cube on your side. Thus, you can take most doubles when your winning chances are as good as about 22.5%, or an equity of as low as minus .55.
If the cube had no value, it would usually be right to double anytime you had greater than a 50% chance of winning. Why shouldn't you? And actually, there are many situations in backgammon where the cube does have no value. For example, when playing a match, once the value of the cube is such that whomever wins the game will win the match, the cube has no further value. Or, consider this position:
Blue has 19 rolls that bear both checkers off on his next turn. If he fails, to do so, he will lose regardless of the position of the doubling cube. Therefore, playing for money, he should double. His expected win in this game is 2/36 of a point (19/36 wins minus 17/36 losses); if he doubles he increases his winnings to 4/36 of a point. White of course should take, since his expected loss of 4/36 of a point is far less than the full point he loses if he drops.
There is not really any precise mathematical formula for using "value of the cube" in doubling decisions. The point is that there is a cost to doubling, and you need a solid enough advantage in the game to make it worthwhile to give up access to the cube.
Market Losers(For simplicity, in this section we will assume that the correct borderline between taking and dropping is minus .50 in equity, even though in the last section we showed that, because of the value of the cube, you can really take when your equity is a little worse.)
Strangely enough, the key factor in the decision to double or not to double is not so much your chances nowbut your chances next roll. This is not an easy concept to understand, but it is crucial to using the cube effectively, which in turn is key to being a successful backgammon player. So take the time to try to understand this section.
We defined a market loser above as a sequence of rolls, one for each player, that would turn a correct take into a correct drop. Why is this important? Well, you really don't want to double any earlier than you have to, thus giving up control of the cube. Nor do you want to double so late that you lose the chance to play a strong position for one point rather than two.
Let's introduce another important concept:
When you are leading in the game, your effective current equity can never exceed .50 until the cube is turned. Whether your equity is .51 or .60 or .90, you can only increase it to 1.00 when you turn the cube.
Therefore, if you turn the cube when your equity in the game is .51, you gain .49 points. If you turn it when your equity is .80, you gain only .20. Therefore, you want to double when your equity is as close to .50 as possible.
Looking at it another way, let's say your current equity is .49. On your next roll, things can get worse for you, but they can't get much better unless you turn the cube.
At the same time, let's say your current equity is .45, but no matter how well you roll and how badly your opponent rolls, you know it will not be greater than .50 by your next turn. You have no need to double. If things get better for you, you can still double, your opponent will still take, and you'll be in the same position you were before. If things go badly for you, you'll be glad you didn't double. Even if your equity could go as high as .52 and your opponent would drop, you haven't lost muchyour cube turn has gained .48 in equity rather than .50.
This leads us to the most important rule of doubling:
The Market Loser Principle: You should double when you have a solid advantage, and when you have a significant risk that at your next turn, you will be well above the borderline takepoint for your opponent. In this situation, you must double now to unlock the value of your favorable sequences. At the same time, no matter how strong your advantage, you need never double if your opponent will always have a take on the next turn, or if when he drops you have lost only a small amount of equity by doubling too late.
It is not intuitive to think that in one case you can double with a smaller advantage and in another case you might hold the cube with a larger one. In a future article we will discuss the way in which volatility affects the doubling decision.
It is difficult to assemble the pieces of doubling theory. No one said that
backgammon is an easy game. But a proper understanding of doubling theory is
one of the most significant differences between the expert and the beginning
backgammon player. Hopefully our readers, by the time this series is over,
will be that much closer to being able to play on a par with the big boys.
Next Article: Match Equity and Doubling Windows.
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.