Mastering Backgammon (39)

40. Playing for the Gammon


The following position came up in a recent chouette. Take a look.



Position 1: Money game. Black on roll. Black to play 4-4.


Black has closed out one checker and escaped all his men from behind White's prime. The win is nearly certain, and his has some substantial gammon chances. As the captain, I had a 4-4 to play. Three plays were possible.

• 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, and White has the opportunity to enter with a 6, eliminating all danger.

• 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 in (a), while the two extra checkers on the ace-point clearly slow down his bearoff and reduce his gammon chances.

• Play : 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 either (a) or (b).

This is a problem in balancing extra losing chances with extra gammon chances. The relationship between these two factors is easy to understand in theory. 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 games lost for the play to be profitable.

In Position 40, Black's losing chances are small after any play. After play (s), they're almost non-existent. Even after the risky play, , 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. (If this seems unclear, not that White's checker on the bar needs four crossovers to enter and reach White's home board, the checkers on the midpoint add another four, and the checkers on White's 7-point and 8-point require another four, plus one crossover to actually bear off a checker, for a total of 13.) This implies that our gammon chances must be pretty good, certainly in the 40% to 50% range, so extra checkers off should add to those chances substantially. (Or so it seemed at the time).

In the actual game, this reasoning seemed compelling, so we took three men off. We eventually won the game but not the gammon. Did we make the right play?

Let's start by calculating how often we get hit 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 a Snowie rollout, which will tell us how often we lose, after each play. The other is Hugh Sconyers' database, which can tell us how often we are hit, after each play. (Sconyers' database is calculated recursively from the simple 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 those 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 our rollouts and databases:

Probability that Black gets hit after each play.
(From Sconyers' database)

After Play (a), clearing the 6-point: 1.2%
After Play (b), stacking on the 1-point: 3.6%
After Play , taking three off: 7.7%

Probability that Black loses after each play.
(From Snowie rollout)

After Play (a), clearing the 6-point: 1.0%
After Play (b), stacking on the 1-point: 2.8%
After Play , taking three off: 5.0%

These results aren't particularly surprising. As expected, play (a) is safest by a wide margin, play 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 77% respectively) while play loses only 64% after being hit, showing that bearing lots of men off before being hit indeed has great 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 should win the most. But here we encounter a surprise.

Probability that Black wins a gammon after each play.
(From Snowie rollout)

After Play (a), clearing the 6-point: 45.4%
After Play (b), stacking on the 1-point: 53.7%
After Play , taking three off: 49.6%

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 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, and equally important, 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 when he throws the occasional ace or deuce. Instead, he'll keep bearing checkers off. Play (b) gives Black the best chance for never leaving a gap in his bearoff, which generates a surprising number of gammons when the gammon race is close.

The best play, and not by a close margin at all, is 9/1 5/1 4/off.

• 
• 
• Email to a friend
• Print