When bearing off against contact, you often have a choice between playing aggressively to win more gammons, and playing safely to win the most games. Some of these plays are tricky. Here’s a good example.
Black on roll, money game, White owns the cube.
Black – Pips 60 (-58)
Black to Play 4-4
In this position, Black has closed out one checker and escaped all his men from behind White’s prime. The win is nearly certain, and he has some substantial gammon chances. With a 4-4 to play, Black has three choices:
Play A: 9/5 6/2(2) 4/off. This is clearly the safest play. Black is even on the end, with two spares on the highest point; White also has the opportunity to enter with a six, eliminating all danger of losing.
Play B: 9/1 5/1 4/off. Legal, but this seems at first glance the worst of the three. Black is stripped on the end, and so more likely to leave a shot than with play (A), while the two extra checkers on the ace-point might later force him to open high points more quickly, reducing his gammon chances.
Play C: 9/5 4/off (3). This is the gusto play, going full blast for the gammon, at the cost of obviously leaving more shots over time than (A) or (B).
This is a problem in balancing extra losing chances against extra gammon chances. As we’ve discussed in previous posts, losses trade against gammons won at a 2:1 ratio. Changing a simple win to a gammon gains two points (the difference between +2 and +4). But changing a simple win to a loss costs four points (the difference between +2 and -2). So if you’re contemplating a play that wins some extra gammons but loses some extra games as well, you have to pick up at least twice as many extra gammons as you do games lost for the play to be profitable.
In this position, Black’s losing chances are very small after any play. After play A, they’re almost non-existent. Even after the risky play C, we might easily have six or seven men off when and if we get hit, so a hit won’t necessarily win for White. On the other hand, White needs 13 crossovers to get off the gammon. This implies that our gammon chances must already be pretty good, certainly in the 40% to 50% range, so it seems extra checkers off should add to these chances substantially.
Let’s start by calculating how often we get hit and lose after each play. While we know from a casual examination of the position that clearing the 6-point is going to be the safest play, and clearing the 4-point will be the riskiest, what really interests us is the relative riskiness of each play. How many extra losses does our aggressiveness cost us?
To solve this problem, we’ll use two tools. One is an Extreme Gammon rollout, which will tell us how often we lose after each play. The other is Hugh Sconyer’s database, which can tell us how often we are hit after each play. (Sconyer’s database is calculate recursively from the simplest bearoff positions, so it’s completely accurate, given the assumption that the player trying to hit will maximize his hitting chances by maintaining contact. In this position, that’s a valid assumption.) Both these pieces of information are useful, and by comparing them, we get a third piece of information, namely how often we can save the game even after being hit.
Here’s the information from the rollouts and database:
Probability that Black gets hit after each play.
(From Sconyer’s database)
After play (A), clearing the 6-point: 1.2%
After play (B), stacking on the 1-point: 3.6%
After play (C), taking three off: 7.7%
Probability that Black loses after each play.
(From Extreme Gammon rollout)
After play (A), clearing the 6-point: 1.0%
After play (B), stacking on the 1-point: 2.9%
After play (C), taking three off: 5.2%
These results aren’t particularly surprising. As expected, play A is safest by a wide margin, play C is the most likely to be hit, and play B sits in the middle. The numbers show yet another expected result. If we divide the loss numbers by the hit numbers, we find that A and B lose most of the time after being hit (83% and 81% respectively), while play C loses only 67% after being hit, showing that bearing lots of men off before being hit indeed has value.
Next we need to consider the probabilities of winning a gammon. We would expect the two safe plays, A and B, to win the fewest gammons, while play C should win the most. But here we encounter a surprise.
Probability that Black wins a gammon after each play.
(From Extreme Gammon rollout)
After play (A), clearing the 6-point: 45.3%
After play (B), stacking on the 1-point: 54.4%
After play (C), taking three off: 48.7%
Taking three checkers off did indeed win more gammons than clearing the 6-point. No surprise there. But the play that won the most gammons by far was play B, taking one checker off while moving two checkers to the ace-point! How can we account for this?
There are two reasons that play B wins so many gammons. The first reason is that it keeps a closed board. The longer Black can keep White from moving, the better his chances of eventually winning a gammon. The second reason is that by putting two more checkers on the ace-point, Black creates a speed board. Once White does enter, Black won’t lose time if he throws the occasional ace in the bearoff. Instead, he’ll keep bearing checkers off. Play B gives Black the best chance for never leaving a gap in his bearoff, as well as the best chance for using a small double late in the game. Both these factors help generate gammons when the race is close.
In a play-to-play comparison, B easily beats A by generating 9.1% more gammons at a cost of just 1.8% fewer losses, far above the 2:1 ratio needed. But B crushes C by generating both more gammons and fewer losses! Can’t do better than that. Moving two checkers to the ace-point while keeping a closed board is the right play.
