| Cube Handling in Races |
Thank you to Douglas Zare and Gammon Village
for their kind permission to reproduce it here.
We often have to estimate the chance to win a race with a 20–25% chance to win because that’s when we get doubled. A good rule of thumb is that in a race of medium length, the trailer can take for money (with about 22% chance to win) when the player on roll leads by 10% plus 2 pips. Each pip near the take/pass point is worth about 2%. This heuristic fails for more lopsided races where the trailer wins under 15%, which can be important for cube decisions in match play.
In this column, we’ll see a simple, computable racing formula due to Danny Kleinman. We’ll use it to analyze a checker play decision which depends on correctly evaluating two desperate races.
A Decision
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|
Double match point. Black to play 5-4. |
It’s double match point (DMP), so black is just trying to win. Should black run, or stay back for the shot? To understand this position, we need to understand two desperate races.
- If black runs, the race is pretty bad, 45-64, but how bad is that?
- If black stays, black gets an immediate shot on 6-5 2 times in 36, and then hits 11 times in 36, for about 1.7% hits. These hits are not gin, particularly with badly placed spares while white has 3 checkers off, but they are worth about 81% wins, for a total of about 1.4% wins. The cost of trying to stay back is not all of your racing chances. You waste about 5 effective pips by piling up checkers on the ace point, but how costly is that?
If we simplify the decision by saying that we will only stay back for one roll, we want to know if we give up about 1.4% racing chances. To analyze these races, we’ll use an invention of Danny Kleinman, the Kleinman count.
Kleinman Race Count
There are at least two different things people mean when they say the “Kleinman count” in racing. We’ll cover the one related to lopsided races. Suppose the player who is ahead in the race is on roll.
| (Difference + 4)2 |
| (Sum − 4) |
Difference and Sum refer to the difference between the pip counts and the sum of the pip counts. For example, suppose the pip counts are 60–80. The Difference + 4 is 24, which is the true lead in pips, since being on roll is worth half of the average pips rolled.
So the Kleinman count is 576/136, which is about 4.2.
The idea is that two races with the same Kleinman count are about as bad. The chance to win should be computable from this one number instead of a formula involving two numbers. The conversion from the Kleinman count to your winning percentage is not an easy calculation, but you can remember a few key values.
| K | Leader | Trailer |
| 1 | 76% | 24% |
| 2 | 84% | 26% |
| 3 | 89% | 11% |
| 4 | 92% | 8% |
| 5 | 95% | 5% |
| 6 | 96% | 4% |
| 7 | 97% | 3% |
| 8 | 98% | 2% |
At least the last half of the table is easy to memorize: For K = 5 or greater, the leader wins about (90 + K)%.
In case you want to evaluate a race with the trailer on roll, a reasonable approximation is to use
| (Difference − 4)2 |
| (Sum + 4) |
That’s just like the first formula, but you switch the signs in both the numerator and the denominator. The value squared in the numerator is still the true lead in pips, taking into account the advantage of the roll. This says that 8 pips is an average roll for the leader.
At a pip count of 60–80, the Kleinman count is 4.2, and we can interpolate between 92% at 4 and 95% at 5 to get about 92–93%.
If you calculate the Kleinman count, you’ll encounter the squares of numbers up to 40. There aren’t that many possibilities, and you may recognize or remember them after some practice.
If the wastage is uneven, then you may need to adjust the pip counts.
Back to the Position
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| Black to play 5-4. |
Suppose you run with the 5-4, going for the 45–64 race:
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| Black runs 20/11. |
The Kleinman count is
| (19 + 4)2 |
| 109 − 4 |
| 529 |
| 105 |
worth about 95% for the leader, and 5% for the trailer.
Suppose you stay back, and you don’t get or miss the shot. Burying two checkers on the ace point wastes about 5 effective pips, making the race about 45–69. What does that do to your racing chances?
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| Black buries 6/1, 5/1. |
The Kleinman count becomes
| (24 + 4)2 |
| 114 − 4 |
| 784 |
| 110 |
worth about 97% for the leader, or 3% for the trailer.
Giving up 2% racing chances to get 1.4% shots is not a good trade. Rollouts agree that staying back is a slight error, but perhaps slightly more of one, 1.2% instead of the 0.6% we calculated.
The discrepancy is small, but why were our calculations off by 0.6%? One reason for this is that the percentages we used for the Kleinman count are rounded, and you shouldn’t expect them to be more accurate than 0.5% each.
Another is that when you have a pile of checkers on the ace point, it is slightly harder to come from behind than the effective pip count would suggest. In addition, we assumed that white would leave black alone, but white gets to shift with 1-1. That’s just one number here, but in other situations where the race is not as desperate, white would gain from hitting with more numbers, and getting hit would cost more.
Summary
Absolute estimates of desperate races occur in checker play situations.
For races near the take/pass point, you can use a simple formula like 10% + 2 pips for 22% equity in medium races. That formula doesn’t help you to evaluate a desperate race.
The chance to win a desperate race can be approximated using the Kleinman count. When the leader is on roll, the Kleinman count is
| (Difference + 4)2 |
| (Sum − 4) |
When the trailer is on roll, use a Kleinman count of
| (Difference − 4)2 |
| (Sum + 4) |
When the Kleinman count is 5–8, the leader’s chance to win the race is about (90 + K)%.