How far behind can you be in a pure race and still (barely) have a take? Most backgammon textbooks attempt to answer this question by giving tables based on pip counts. The classic Jacoby and Crawford book simplifies this to a “15% criterion”: If your deficit is no more than 15% of your opponent’s pip count, you can take his double.
All criteria in terms of such a percentage deficit must be inaccurate. They incorporate two different errors — fortunately, errors in opposite directions.
The first error is to view the difference in pip count as measuring the lead in the race. This ignores the shaker’s extra advantage in having the roll. The mean shake contains 81⁄6 pips. Since there is some inefficiency, some wastage of pips in bearing off, it is simpler and more accurate to count the average shake as 8 pips.
In a race with equal pip counts, therefore, the difference of a roll between being on shake and not being on shake is 8 pips. We may think of having the shake as being 4 pips ahead, and of not having the shake as being 4 pips behind. Before using pip counts in any computations, therefore, we will always subtract 4 from the shaker’s pips.
Let’s call the shaker’s true pip count, as adjusted downward by 4, T, for “true shaker’s pips.” We still call the nonshaker’s count N. We may use S for the sum of N and T to represent the length of the race, and D for the difference N − T to represent the shaker’s lead.
The nonshaker’s chances of winning the race are simply his chances of gaining D pips or more when both sides are trying to roll a total of S pips. The average stride in this race is 8 pips. If everyone rolls the average, the shaker who leads will keep his lead until the end and win. The nonshaker must hope for deviations from the mean.
Statisticians use standard deviation as a measure of variation from the mean. I calculate the standard deviation of pips rolled in a shake as about 4.3.
Again, this measure of variance is an overestimate. Very large numbers such as double sixes will usually be used wastefully, for we seldom have four or more men stacked up on the six point when we roll them. And very small numbers, in contrast, are used efficiently: It is impossible not to be able to play an ace smoothly, either to bear a man off from the one point or to fill gaps and even out your distribution. It is simpler and more accurate to regard the standard deviation as 4 than as 4.3 pips.
It is a theorem of statistics that the standard deviation over a number of trials is proportional not to the number itself but to its square root. This is the source of the second error in any percentage-deficit criterion for takes. The longer the race, the smaller percentage deficit that figures to be made up, provided you consider the true deficit D, which is 4 more than the apparent difference in pip counts.
If the race lasts a total of about S pips, then there will be about S/8 shakes. The standard deviation during the race will then be 4 times the square root of S/8, which is the same as the square root of 2S. We can use the normal distribution, popular in statistics as an approximation in order to estimate the probability of making up a deficit of D pips.
Standard tables show this probability as a function of the ratio of the deviation in question to the standard deviation. In our case, this ratio is of D to the square root of 2S. Interpolating in the tables, we get these results:
We can eliminate the troublesome square roots by substituting the square of the ratio: D2/2S. Then we can simplify further by doubling this square, so that our critical ratio becomes D2/S. Finally, we may use only two decimal places:
I have shown the critical ratios for a range of winning probabilities from 20% to 25% because 20% and 25% represent the liberal and conservative extremes for the chances you need to warrant a take. 25% take points err by ignoring the equity in owning the cube after taking. 20% take points err by assuming that the cube can be used with perfect efficiency by its owner when the game turns around. The true minimum probability justifying a take lies somewhere between these extremes.
A truly conservative taker, wanting at least 24% raw winning chances in order to take, can use a very simple critical ratio: 1.00. Likewise, a truly liberal taker, asking only 21% raw winning chances, can use a critical ratio almost as simple: 1.30.
But I think that, on the average, the cube can be used with slightly more than 60% efficiency. That is to say, in most races, a raw winning probability of 22% does justify a take. Examining the chart above, we see that this corresponds to a critical ratio of 1.19. For ease of calculation, therefore, we may adopt 1.2 as our critical ratio.
Thus whenever D2 fails to exceed S, it’s an easy take. When D2 exceeds S, however, we can simplify the calculation by taking the excess, D2 − S, and comparing it to S. If D2 − S is as great as 20% of S, it’s a pass; otherwise, a take.
Let’s try some examples.
Suppose the race is long and we trail 108 pips to 96 when doubled. N = 108, T = 92, S = 200, and D = 16. So D2/S = 256/200, a critical ratio of 1.28. We must pass.
Alter the race by giving our opponent just one more pip, so that we trail 108 to 97. Now N = 108, T = 93, S = 201, and D = 15. So D2/S = 225/201. This about 1.12, and we have a clear take.
Suppose that the race isn’t so long and we trail 82 pips to 72. N = 82, T = 68, S = 150, D = 14, and D2/S = 196/150. The excess D2 − S = 46; 20% of S would be 29.8, and the excess is less than 29.8, so we may take.
These results illustrate that a pip-count deficit of 14% can be a pass in a medium-length race, and a pip-count deficit of 121⁄2% can be a pass in a long race. This suggests that the traditional Jacoby-and-Crawford rule that you take if your pip-count deficit is less than 15% is a bit too liberal, especially in longer races.
Can we formulate comparable criteria for offering doubles and redoubles in pure races? Hardly. For unlike your pass-or-take decisions, your cube turning should depend more on your opponent’s liberal or conservative taking propensities than on mathematics.
Nonetheless, calculating the critical ratio may still prove helpful by letting you anticipate your opponent’s response to a cube turn. When it reaches 1, you may expect a very conservative opponent to pass. When it reaches 1.2, you may expect a middle-of-the-road opponent to pass. And against a very loose taker, you should require that D2/S reach at least 1.2 before turning the cube.