Problem of the Week #73: Solution


Cash game, Black owns the cube.




Should Black double? If he does, Should White take?


Note: All ‘cash game’ problems assume the Jacoby Rule is in effect. That is, you can’t win a gammon unless the cube has been turned.


In Problem 72 we started to look at the class of positions I like to call “One Man on the Bar”. White is on the bar with a single checker, while Black has a five-point board with a high point open. Black is in the process of shepherding his remaining checkers home, while White is hoping to hit a last-ditch shot, or win the race.

Here in Problem 73 we’re going to start to look at the doubling rules that govern these positions. This position is a good place to start, because we’ve stripped the position down to its essentials: White has no extra checkers in the outfield, and Black has just two men left to get home. White also has one open point high in his board, which is often typical of these positions.

Position 73 is what we call a reference position for these situations. In backgammon, reference positions are very useful; by memorizing one or two simple situations, we can hopefully solve a very large class of possible positions at the board just by comparing the real situation to our reference position and making some common-sense deductions.

Not all positions make good reference positions. The characteristics of a useful reference position are these:

(1) It’s an example of a class of positions that arise frequently.

(2) It’s simple – no extraneous tactical issues going on.

(3) It’s governed by a clear rule, often based on nothing more than the pip count.

Position 73 meets all these criteria, so let’s see what’s going on here.

First of all, Black is trailing by 8 pips: his pip count is 74, White’s is 66. It should be pretty clear that Black is some sort of a favorite now. If he can roll as much as 8 pips this turn, and White then dances (a very possible result), Black will be even in the pip count with White still on the bar against a 5-point board. At that point Black will obviously be a big favorite, and we’ll be wondering if White even has a take. On the other hand, White clearly has some chances now, so if White is doubled, he’ll have to seriously consider taking.

Positions like Problem 73 hinge on the exact pip count. If Black is close enough in the race, he’ll double and White will have to pass. White will be a big underdog in the race, and he won’t get many shots from the bar under the best of circumstances. If Black is too far behind in the race, however, he won’t even have a double. So what we want to know are those endpoints: at what point is Black close enough in the race that he can double (and White has an easy take), and when does the race become so close that White has to pass? Once we know those points, we’ve solved this exact problem, and we can use the knowledge to extrapolate to a wide class of related problems.

Let’s start by looking at Problem 73a:




Problem 73a: Black on roll, trails by 12 pips in the race.

Here we’ve moved Black’s checkers back a bit, so now he trails by 12 pips. A Snowie rollout shows this is an easy take for White, and doubling is in fact a small mistake. Black’s equity owning the cube is about +0.75, but if he doubles and White takes, it drops to +0.72. So Black needs to be closer than 12 pips in the race to double.

Now take a look at 73b:




Problem 73b: Black on roll, trails by 4 pips in the race.

Black’s checkers have moved up a bit, and now the pip count is Black: 70, White: 66. Snowie rollouts show that Black’s equity after a double and a take has risen to +1.07, so White in fact has a clear pass.

These two examples show the rough parameters for positions of this type. If Black is within about 5 pips in the race, it’s a strong double and a marginal take/pass. If Black is trailing by more than about 10 pips, he can’t double yet. This “5-10” rule is simple, useful, and easy to remember.

Knowing this rule, our original position becomes a matter of simply knowing the pip count. Trailing by 8 pips, it should be a solid redouble and a clear take, and in fact that’s right. Black’s equity if he doesn’t redouble is about +0.80, and it rises to +0.90 if he doubles and White takes. White should take, of course, since losing 9/10 of a point (on average) is better than dropping and losing a full point.

Keep in mind that this “5-10” rule is very specific: both sides have a 5-point board with a high point open, White has one man on the bar with all his other men home, and Black is moving through the outfield. That covers a lot of positions that arise in practical play, but not all; some minor variations will require common-sense adjustments. Next week we’ll look at some variations on these positions and see how we can reason our way to good cube decisions.


Solution: Black should double, and White should take.

The fact that gammons are activated in that kind of position doesn't matter since gammon threats are virtually inexistent here.

However, I have a question for you Bill. You explained that 5-10 rule in a context where Black was already owning the cube. Does it mean that if the cube was in the center, Black could double a little bit "looser" and that the 5-10 rule would shift a bit, like to 6-11 or 7-12 for instance, or it doesn't matter that much?

(Actually, for White's take it doesn't really change, so the 5 would probably remain, but the rule could shift to 5-11 or 5-12 maybe.)

Basically, does it make a difference if it's an initial double or a redouble?

Rollouts from GNUBG

Cubeful equities:

1. +0.910 Double, take
2. +1.000 Double, pass
3. +0.802 No double

Rollout details:

Player Black owns 2-cube:
0.754 0.009 0.000 - 0.246 0.005 0.000 CL +0.512 CF +0.802
[0.000 0.000 0.000 - 0.000 0.000 0.000 CL 0.001 CF 0.001]

Player White owns 4-cube:
0.759 0.009 0.000 - 0.241 0.004 0.000 CL +1.047 CF +0.910
[0.001 0.000 0.000 - 0.001 0.000 0.000 CL 0.003 CF 0.004]

Full cubeful rollout with var.redn.
648 games, Mersenne Twister dice gen. with seed 894982135 and quasi-random dice
Play: supremo 2-ply cubeful prune [world class]
keep the first 0 0-ply moves and up to 16 more moves within equity 0.32
Skip pruning for 1-ply moves.
Cube: 2-ply cubeful prune [world class]

Yes, it makes a very small difference. In Position 73a, for instance, the double is a mistake by .03 when Black owns the cube. With a centered cube, it's a mistake by .001. On average, Black can double about 1 pip sooner with a centered cube. This is analogous to the 8-9-12 rule in straight races, where you can double with an 8% lead with a center cube, but you need a 9% lead if you own the cube. In most races, that means you can double 1 pip sooner with the cube in the middle.

Thanks Bill. I will therefore think of it as the 5-10-11 rule !