Introduction to Doubling 
The doubling cube is used to double the stakes for which you are playing.
We don’t know who first added the doubling feature to backgammon, but we feel we owe him a great debt. Without the doubling feature, few people would be playing backgammon today, and we probably wouldn’t have written this book.
Doubling cube
We warn you that some of our discussion will be very tough going, but don’t get discouraged! Whatever you may not fully comprehend the first time will be clearer after reading later chapters. But now is the time to start.
Before the game begins the doubling cube is placed in the middle of the board on the bar, with the number 64 on top and facing left or right (i.e., not facing either player directly). This of course does not mean that you’re playing for sixtyfour units; rather, it reminds both players that you are starting at one unit.
If there is an automatic double, the number 2 is faced, again left or right; if there is a second automatic double, the number 4 is faced, and so on. The cube is left in the same position until the first doubling during play, to indicate that either player has the right to double when it is his turn to play.
Automatic Doubles
These have no official part in backgammon: they are not allowed in tournaments, and in most expert games they are either barred completely or else limited to one. But if both players agree, an automatic double occurs at the beginning of a game if both players, in tossing for first play, roll the same number. They may repeat tossing the same number several times, so even in a smallstake game it is best to limit automatic doubles to one, or perhaps two. Remember that three automatic doubles multiply the original stake by eight, not three!
Some people like to liven up the game by simply starting the cube at 2 — and after a few drinks possibly starting at 4 or even 8. This can be a dangerous procedure and can change a friendly game for small stakes into a heavy gambling game that can get quite out of hand.
You won’t get into this kind of trouble if you restrict yourself to one or two automatic doubles. Be very wary of anyone who suggests unlimited automatic doubles.
Regular Doubles
After the first roll either player may double at any time when it is his turn to roll. He should consider doubling when he gains an advantage, small or large, during the course of the game.
Normally, if he decides to double, he picks up the doubling cube, turns it to the 2 (or, if it’s a redouble, the next higher number — 4, 8, 16, etc.) and places it near his opponent’s side of the table, usually saying, “I double.”
His opponent must then decide whether the doubler’s position is too advantageous to allow him to risk accepting the double. If he decides to refuse the double, the game is then over and the doubler wins whatever the stake was before his double.
If he decides to accept the double, he usually says “Yes” and moves the doubling cube nearer to his side of the table. The game then continues at the new stake, which is double the old. Thus, if you have been playing for a dollar a game, the stakes become two dollars a game.
After the double is accepted, the person doubling cannot make the next double, no matter how much his advantage may grow as the game progresses. The man who has been doubled is then the only one who has the right to redouble, and the doubling block is now said to be “in his control .” He may double any time it is his turn, before he rolls his dice, if and when he finds himself ahead in the game.
He would then say “I double,” and if it is accepted he turns the cube to 4 (or whatever is then the next higher number) and moves it to his opponent’s side.
Later we will analyze in detail how and when to double, but first there are some general principles worth keeping in mind:
As we have often urged, you should learn how to figure odds, both in order to know when to double and so that when you are doubled you can figure your chances of winning the game. In most situations if you are at only a slight disadvantage, you ought to accept your opponent’s double. Even though the odds are slightly against you at that moment, the fact that the doubling cube then reverts to your control improves your odds. More on this later.
Before accepting any double, always make sure that you are not in danger of losing a gammon (which doubles the stake) or a backgammon (which triples it).
In deciding whether or not to accept your opponent’s double, bear in mind that many average players tend to accept doubles that are too risky. Especially in the running game, the average player will rise to the bait and chance it, hoping for a lucky roll (such as double 6), which would turn the advantage to him. Remember, your opponent has just as much chance as you of rolling that double 6.
There are both tangible and intangible factors to weigh in deciding whether to double, and whether to accept your opponent’s double. These include how good a player your opponent is; whether he usually accepts or turns down close doubles; how erratic a player he is; and how high the stakes are already.
The better the game, the fewer doubles and redoubles. If you see a game with the cube bouncing back and forth like a pingpong ball, you’ll be watching an exciting game — but you won’t be watching good backgammon.
A player who knows when to double and when not to double is probably an expert. If he also knows when to refuse a double, there is no question about it: he is an expert.
The point is not that doubles aren’t made and accepted in expert games, but rather that doubles don’t come so frequently that the stakes end up at 32, 64, or higher. Quite the contrary: few games between experts go beyond 4, and a 16 game is almost as rare as a baseball nohitter.
An early book on backgammon contained the absurd statement: “In a perfect game no double will ever be accepted.”
You might be inclined to doubt that such a statement is absurd, arguing as follows: Clearly, no expert would ever offer to double unless the odds were in favor of his winning, and equally clearly, no expert would ever accept a double under conditions which favored his opponent.
The real point at issue, however, is: How much do the odds favor the doubler? Let’s consider a simple situation. It’s my roll. I have two men left on the board; one is on the two point and one is on the four point. My opponent has one man left on the one point. Thus, unless I win on this roll, he is certain to win on his next.

A quick count will show that of the thirtysix possible outcomes of my next roll, I will win the game if I get one of the twentythree rolls that are favorable to me and I will lose it if I get one of the thirteen unfavorable rolls (any 1 and 23).
11  12  13  14  15  16 
21  22  23  24  25  26 
31  32  33  34  35  36 
41  42  43  44  45  46 
51  52  53  54  55  56 
61  62  63  64  65  66 
 If I roll without offering to double I have a probability of 23/36 of winning the game at the present stakes (which we’ll call one unit) and 13/36 of losing that one unit, giving me a net “positive (or plus) expectation” of 23/36 less 13/36 or 10/36 of one unit.
 If I double and the double is accepted, my “plus expectation” also doubles and becomes 10/36 of two units, or, what is the same thing here, 20/36 of one unit.
Hence, I double. You may still say that no expert would accept such a double since it is so crystal clear that the double is to my advantage. So let us examine the situation from my opponent’s point of view.
If he accepts the double, he has a net “negative expectation” of 20/36 of one unit (simply, the reverse of my plus expectation); but if he refuses the double, he has a certainty of minus one whole unit. He would be very foolish to refuse to accept the double — he would have preferred that I hadn’t doubled but, once I did, he had no intelligent alternative but to accept!
It was fairly easy to work out your plus expectation in the preceding situation. You were either going to win or lose the game depending on your next roll, and since you were more likely to win the game than to lose it, you had a good double. Specifically, your plus expectation if you failed to double was 10/36. The double increased this to 20/36 and hence the right to double in this position was worth 10/36 of a unit.
It was also correct for your opponent to accept. Refusal would have cost him one unit; acceptance gave him a “minus expectation” of 20/36 of that unit, so that his right to accept your double was worth 16/36 or 4/9 of that unit.
Just what is meant by a “plus expectation”? All this means is that the odds are in your favor, and the better the odds in your favor, the higher your plus expectation.
Suppose we translate your plus expectation in the previous example into money — American money, for the sake of convenience. The stake is a dollar a point; if you arrived at this particular position exactly thirtysix times, you would expect to win twentythree dollars and lose thirteen dollars, for a new profit of ten dollars in thirtysix games, or an average profit per game of 10/36 dollars or just under twentyeight cents.
By doubling the stakes you double your plus expectation to 20/36 dollars, or almost fiftysix cents.
Thus, your position is “worth” fiftysix cents to you if you double. It represents an expected loss or minus expectation of fiftysix cents to your opponent, once he accepts your double. If he refuses your double he is all through with “expectation.” He can’t possibly lose two dollars; but he has certainly lost one dollar, which is considerably more than the expected fiftysixcent loss for taking the double.
It should therefore be easy to see that in all oneroll situations you should double any time that the odds are in your favor (i.e., when you have a plus expectation).
Suppose that your two men in the previous example were on the five and two points.

11  12  13  14  15  16 
21  22  23  24  25  26 
31  32  33  34  35  36 
41  42  43  44  45  46 
51  52  53  54  55  56 
61  62  63  64  65  66 
You would have nineteen winning and seventeen losing rolls and thus a plus expectation of just 2/36 of a unit, or just over five cents (translating it into money at a dollar a point. A double would increase your plus expectation to eleven cents).
It should be even clearer that your opponent should accept this double. Why should he pay out a dollar for sure instead of his “expected” eleven cents?
When is a double so overwhelming that it should not be taken? The answer to this is that threetoone odds represent the breaking point. When the odds are less than three to one in your favor, your opponent does better in the long run if he accepts your double. He doesn’t win, but he loses less, on average.
When the odds are more than three to one in your favor, he should refuse. His right to accept in such situations represents a liability, not an asset.
To see why this is so, let’s go back to our dollar game. If the odds are three to one in your favor, your plus expectation is seventyfive cents minus twentyfive cents, or half a dollar. If you double the stakes, you double your plus expectation (and your opponent’s minus expectation) to one dollar, which represents the exact cost to your opponent if he refuses the double.
When to Make the First Double
Is it possible for you to have a position that is too good to double? Of course! Your opponent always has the right to refuse when you double. If you have a good chance to win a gammon, you may prefer to play on for it.
The following position was reached after three rolls by each player. You started with 65 and ran one back man to the black twelve point. Black rolled 64 and brought a back man to your eleven point. You rolled double 2 and made your eleven and four points, hitting his blot on your eleven point in the process. He rolled 41 and decided to bring his man on the bar up to your five point. We don’t approve of that play, and you might have doubled right then, but you rolled once more and got double 6. This enabled you to make your bar and five points and to put one black man back on the bar. Black proceeded to roll 65 and could not come in from the bar.

Should White 
If you double, you will win a point, unless you are playing against someone who takes almost any double offered him. If you play on, your plus expectation is greater than that: your onepoint profit is in no danger, since you can double the stake at any subsequent turn to play, and if you get a few more good rolls, you will win two points with a gammon.
For instance, suppose that black’s next roll is 32. He comes in from the bar with the 3 and makes your three point by moving from your one point with the 2. Your chance to gammon has gone down to practically nothing, but your advantage is still overwhelming; you should now double him.
Let’s consider another situation. Assume that your position is such that you estimate that you have one chance in three of winning a gammon, one chance in three of winning an ordinary game, and one in three of losing. It is your roll.
Should you double? Should your opponent accept?
If you don’t double and play on, your plus expectation is twothirds of a unit. The unit you win when you win a single game cancels the unit you lose when you lose a single game, so the two units you win for gammon produce your net plus expectation.
No Double  Prob.  Game Value  Cube  Expectation  
Lose single  1/3  ×  −1  ×  1  =  −0.333 
Win single  1/3  ×  +1  ×  1  =  +0.333 
Win gammon  1/3  ×  +2  ×  1  =  +0.667 
+0.667 
If you double and your double is accepted, your plus expectation doubles also and thus grows from twothirds of a unit to one and onethird units, while if your double is refused you win one unit.
Double/Take  Prob.  Game Value  Cube  Expectation  
Lose single  1/3  ×  −1  ×  2  =  −0.667 
Win single  1/3  ×  +1  ×  2  =  +0.667 
Win gammon  1/3  ×  +2  ×  2  =  +1.333 
+1.333 
Hence you should double, and your opponent should refuse.
A Paradox
Look at the following position.

Should white redouble? 
It is your roll, and there are nineteen out of thirtysix rolls that will end the game immediately (double 6, double 5, double 4, double 3, double 2, 65, 64, 63, 62, 54, 53, 52).
11  12  13  14  15  16 
21  22  23  24  25  26 
31  32  33  34  35  36 
41  42  43  44  45  46 
51  52  53  54  55  56 
61  62  63  64  65  66 
game this turn
If black’s two men were on his one point, you should double or redouble, and thereby double your plus expectation. But here is black with a poorer position than that, and this time you should not double if the cube is on your side. The reason is that if you keep the cube on your side and fail to get both your men off, you may get a second turn. Black has seven possible rolls that won’t take both his men off.
Thus if you don’t double, your chance of winning is 19/36 plus the product of 17/36 (your chance of not getting off in one roll) times 7/36 (black’s chance of not getting off). This represents a total chance of winning of 803/1296 (19/36, or 684/1296, plus 17/36 times 7/36, or 119/1296); your chance of not winning is 1296 less 803, or 493 out of 1296.
Probability  
Win immediately  19/36  =  684/1296 
Win next turn  17/36 × 7/36  =  119/1296 
Total winning chances  803/1296 
Thus your net plus expectation is 803/1296 minus 493/1296, which is 310/1296; to express it as a decimal, divide 310 by 1296, which comes to .239 times the stake.
If you do double, your opponent will accept, and now your whole chance depends on your first roll, because if you don’t get off you are going to be redoubled, and you will have to refuse. Your net plus expectation if you double is therefore 2/36 (19/36 less 17/36) times 2 (the doubled stake), which is 4/36 or .111 times the undoubled stake.
In other words, you do better than twice as well if you leave the cube alone.
The Position of the Cube
The preceding paradox illustrates a point that is generally overlooked: It is a twofold advantage to have the doubling cube on your side of the table. The first advantage is that you can double and force your opponent to give up or to play at a doubled stake any time that it is your turn to roll. The second is that, no matter how good his position becomes, you are going to get to play the game to the finish since he can’t double you.
When to Double, Redouble, Refuse, and Accept
So far, this has been a tough chapter. We have shown a position so good that you don’t want to double. We have discussed the advantage that you gain by having the doubling cube on your side of the table. We have pointed out that in the end game you should accept a double any time the odds against you are less than three to one. We have given an example of a proper double when your chance of winning is just better than even money (nineteen out of thirtysix) — and in general we’ve kept things at a level that has probably confused you.
If it hasn’t, you’re a genius!
Now let’s see if we can’t give you a few rules of thumb. We’ll start by considering only cases in which there is no possibility of a gammon.
In such situations you should consider a first double (moving the cube from the center to your opponent’s side of the bar) at any time that you estimate that the odds in your favor are seven to five, or better. You should definitely make a first double any time you think the odds in your favor are nine to five or better.
When there has already been at least one double, you should be a trifle more conservative about redoubling. You should consider a redouble when the odds in your favor are three to two, and you should definitely redouble when the odds are two to one in your favor or better.
Of course these doubles will all be accepted, or should be accepted. But there is no guarantee — we have seen incredible refusals and acceptances!
When you have a chance to win a gammon, you double a trifle quicker and your opponent is more likely to refuse.
So much for rules of thumb. How do you figure out what your chances are? This comes only with experience. You start by counting the position (see the previous chapter). Once you have done that, Table 5 gives a rough estimate of your chances, but only if all else if even.
Table 5
The Approximate Odds in Your Favor When It Is Your Roll
In a Running Game and There Is Little or No Positional Advantage
Your lead over your opponent*  
Your count  0  4  8  12  16  20 
60  6 to 5  8 to 5  11 to 5  17 to 5  5 to 1  8 to 1 
70  11 to 10  7 to 5  2 to 1  3 to 1  4 to 1  6 to 1 
80  21 to 20  13 to 10  9 to 5  13 to 5  18 to 5  5 to 1 
90  31 to 30  12 to 10  8 to 5  12 to 5  16 to 5  9 to 2 
100  41 to 40  11.5 to 10  3 to 2  11 to 5  3 to 1  4 to 1 
110  51 to 50  11 to 10  7 to 5  2 to 1  14 to 5  19 to 5 
* The number of additional points in your opponent’s count. 
We make no claim that this table is completely accurate, but it does represent the best experience of over thirty years of backgammon.
It should be used carefully. Thus, if you have six men on your five point and your opponent has two men each on his six, five, and four points, the count of each side is 30 — but his position is decidedly better.
 Black's position is better than White's 
Table 6 is based upon Table 5; it presents the same information in a more usable form.
Table 6
Should You Double?
Your lead  Your count lies between these limits  Should you double?  Should you redouble?  Should your opponent accept? 
0  55–115  no  no  — 
4  55–75 75–115  yes no  no no  yes — 
8  55–75 75–115  yes yes  yes no  yes yes 
12  55–75 75–115  yes yes  yes yes  no yes 
16  55–105 105–115  yes yes  yes yes  no yes 
Let’s go back to a position we discussed in the last chapter.

Should White 
Your count is 109 and your opponent’s 118. The odds in your favor are better than seven to five and worse than two to one. You should double, but if you already have been doubled you should not redouble.
Table 5 shows that the odds in your favor are between seven to five and two to one: your count of 109 is closest to the 110 in the first column; your lead of 9 lies between 8 (which has odds of seven to five) and 12 (which has odds of two to one). Table 6 shows that you should double if the game has not yet been doubled, but you should not redouble: your lead is nearest the 8 group in the table, and your count lies between 75 and 115.
If you don’t want to learn these tables (and we see no reason for you to do so unless you want to spend a large part of your life learning backgammon), here is a simple rule for doubling in a running game:
 Count your position and your lead.
 If your lead is as much as 7½ per cent of your count, make a first double.
 If it is as much as 10 per cent of your count, make a first double or redouble.
 If your opponent doubles in a running game, accept any time you are less than 15 percent of your opponent’s count behind (i.e., his lead is less than 15 per cent).
Thus, in the position above your count is 109 and your lead over your opponent’s count (118) is 9; 9 is about 8¼ per cent of your count, so you should make a first double, but you should not make a redouble.
In the last chapter we raised the question of what action, if any, you should take with the doubling cube when you have two men on your six point while your opponent has three men on his one point.

Should White 
Let’s begin by calculating your chance to win if the game is played to the end. You will win if you get off in one roll, or if you get a second roll and get off then.
Table 4 shows that your total chance of getting off in two rolls is 78 per cent, and of getting off in one roll 11 per cent. Hence, your chance of getting off on your second roll is the difference, or 67 per cent. However, you may not get a second roll. Your opponent has one chance in six (six possible rolls out of thirtysix) of rolling a double and winning before you get that second roll. Therefore, your chance to win is 11 per cent plus 5/6 of 67 per cent. This adds up to 67 per cent.
A probability of 67 per cent is equivalent to odds of about two to one in your favor, and so you should double or redouble, as the case may be. You may wish you hadn’t redoubled if you proceed to roll 21 and get redoubled right back, but you should not let this possibility dissuade you.
The Psychology of the Double
The more you play backgammon the more you’ll see the importance of learning your opponent’s particular habits of play — especially his treatment of the doubling cube.
Is he a quick doubler? Always hesitate before taking a double from this kind of player. He always wants you to accept. Make him worry before you do accept; he may start to double even more prematurely. You might even refuse one unsound double in order to get him to double even more hastily in the future.
Is he a slow doubler? Such a man really hopes that you will refuse his double. If you do refuse him, do so slowly. If you accept, do so quickly and cheerfully. He will tend to be even slower with his future doubles.
Is he inclined to refuse close doubles? You can afford some unsound doubles against him; if he refuses one out of three, you will gain.
Is he inclined to take sound doubles? Be even more sound in your doubles. They will still be accepted.
Watch how the score affects his doubling and his acceptance and refusal of your doubles. Losers tend to double unsoundly and to take doubles when they should refuse; they can’t get even winning 1point games or losing any games at all.
Winners go in the other direction. They are quick to refuse doubles and slow to turn the cube themselves. They want to stay a winner and don’t want to let the action speed up.
Don’t be discouraged if there are things you don’t understand the first time you read this chapter. Go on to the next chapters, but plan to refer here again and again.