The innovation of the doubling die, also known as the cube, brought a new level of enthusiasm to the already popular game of backgammon. This addition added an extra layer of excitement to the game, as a low-stakes match could quickly escalate to a more serious one with the possibility of the cube being raised to 8, 16, or even 32, potentially resulting in a gammon or even a backgammon win for the victor. However, the doubling die’s impact extends beyond just enhancing the excitement of backgammon. It has also introduced a new aspect of decision-making, making choices about the cube just as significant as any other skill in the game.
The Dilemma of Whether to Accept a Backgammon Double
When facing a backgammon double, your initial inclination might be to reject it if you are at a disadvantage. This is an easy mistake to make, but you need to think it through. Why should you increase the stakes when your opponent has the upper hand? The answer is that you should not agree to a double if you can keep the stakes the same. Unfortunately, you cannot. You must either drop the double and forfeit the original stake or agree to pay twice as much. In this case, you must choose the lesser of two evils.
The question that arises each time is: “Is it better to drop the current stake and decline the double or take the risk of doubling the stakes because there is still a reasonable chance of winning?” For now, we will only consider positions where there is no chance of a gammon.
Let’s consider a straightforward example where both players are bearing off, and you (White) believe that you are in the worst position, but you still have about a one in three chance of winning the game. Your opponent (Black) offers you a double. The easiest way to analyze the problem is to think about what would happen if you played the game three times and averaged the results, i.e., winning once and losing twice. If you declined the double, you would lose 1€ on each game, resulting in a total loss of 3€ over three games. If you accepted the double, the stake would be 2€. You would lose two games at 2€ each (a total of 4€ on the two games), and you would win one game at 2€. Thus, you would be £2 down overall. That is still better than the 3€ loss you would have incurred by refusing all the doubles. In other words, if you accept the double, you will lose an average of 0.67€ per game, compared to 1€ per game if you decline it. Therefore, it is better to accept the double in that position since you will lose less in the long run. The next question is, “How bad does the situation need to be before it is not worth accepting the double?” The borderline case, where accepting or refusing the double will make no difference in the long run, is when, after accepting the double, you have only a one in four chance of winning. You can work it out for yourself. If you play four games and refuse the double each time, you will lose 4€. If you accept the double each time, you will lose three games at 2€, which is 6€, and win one game at 2€, resulting in a net loss of 4€, which is the same as the net loss you incurred when you declined all the doubles. In either case, you will lose an average of 1€ per game. The conclusion, therefore, is that when there is no chance of a gammon, you should accept all doubles where you have a better than one in four chance of winning and decline all doubles where your chances are worse than that.
You might be wondering, “How can I determine whether my chances are better or worse than one in four?” Calculating your chances can be relatively straightforward in some situations, particularly towards the end of the game when you are bearing off. However, in many other scenarios, the possibilities are too numerous and complicated to accurately calculate the position. In such cases, you’ll need to rely on your experience or seek guidance from experts. To help you get started, I will present two positions: one where you can easily calculate your chances and another where you can get a good estimate.
If Black rolls a 1, they will fail to remove both their pieces from the board, and you will win. However, if they roll any other number, they will win the game. This situation is known in backgammon as a “must give, must take” scenario.
It may seem illogical at first that you are asked to give a double and then accept it, but the mathematics behind this decision are clear-cut. If Black knows anything about the game, they will offer you a double. You should remember that if you decline the double, you will lose the current stake anyway. Therefore, accepting the double and winning at the increased stake can result in a net gain for you.
Of the 36 possible rolls that Black can achieve, 11 of them will contain a 1, resulting in a win for you. This means that you will win 11 times out of 36 on average, which is better than one in four. As a result, you should accept the double.
The stake and value of the doubling cube should not influence your decision. Whether the cube goes from 1 to 2 or 16 to 32, you should make your decision based on the probability of winning.
In summary, this is a common situation in backgammon where Black offers a double, and you should accept it if you want to minimize your losses in the long run. While you will likely lose more often than you win, accepting the double can pay off over time
Let’s examine another backgammon position. In this particular situation, you (White) have a closed board and little chance of winning unless you hit one of Black’s men. Black, on the other hand, is highly likely to win in a straight runner, but he needs to move the man on his twelve point past your men on his two and six points. Black doubles, and although you could estimate your precise chances of winning with a pencil and paper and some time, there’s no time for that at the table. So let’s roughly evaluate the situation.
Your gut feeling is that you’ll likely get one shot, and that’s correct. If Black rolls double 6, double 5, double 4, double 3 or 6-5 (which are only 6 throws out of 36), you won’t get a shot at all. But any other roll by Black gives you a shot, and quite often, it will be a double shot. Sometimes, you may even get a second shot after missing the first because Black still won’t have moved past you.
If you do get a direct shot, you have a considerably better than three to one chance of winning, which means it’s worth taking the double. An exact calculation confirms this. In situations like this where Black has no choice but to move his blot whatever he rolls, you should accept the double, even if you only have one man threatening his blot, as long as that man is nine or more pips away from the blot.
However, you should be careful in similar situations. If you expect to have only one shot to save the game, make sure to calculate whether you’re sure to win if you hit it. If not, it’s not worth accepting a double for just one shot. But in this case, there’s no doubt about it. If you hit Black’s man, he won’t be able to accept the redouble which you’ll offer him immediately. With only two men borne off, his chances of winning are worse than one in four.
