Goldilocks and the Three Bearoffs

Fifteen men in each home board

Yo ho ho and a bottle of rum

“Today”, said Goldilocks, “we are going to look at three positions which are useful in calculating cube actions in bearoffs.”

“The golden triangle, which is the most efficient distribution of 70 pips, is below. Interestingly, moving any checkers, while keeping to the 70 pips, reduces the winning chances for that side.”

The following position should be memorised:

Money Game. Blue on Roll, Cube action?

“I am sure none of you three would double this one.” Goldilocks began. “The reason that I ask you to remember it is that it is close to 60% wins for Blue, and can be used as a benchmark.”

• No double
  • Player Winning Chances: 60.05% (G:0.00% B:0.00%)
  • Opponent Winning Chances: 39.95% (G:0.00% B:0.00%)
• Double/Take
  • Player Winning Chances: 60.02% (G:0.00% B:0.00%)
  • Opponent Winning Chances: 39.98% (G:0.00% B:0.00%)
• Cubeless Equities: No Double=+0.201, Double=+0.401
• Cubeful Equities:
  • No double: +0.302
  • Double/Take: +0.138 (-0.164)
  • Double/Pass: +1.000 (+0.698)
• Best Cube action: No double / Take

So, not a beaver, but a big error to double as some people “feeling lucky” might do. They think that they are on roll in an equal race, and were it not for redoubles, they would be right. But cube ownership plays a big part. And the fact that it is so close to 60% wins make it very useful. As a general rule (and only a general one as distribution plays a part) you can add 2% to the winning chances for each extra pip Black has from the golden triangle, with Blue’s position remaining the same. Let us start with 5 pips, as that is a very important position which should be known.

Money Game. Blue on Roll. Cube Action?

On our golden rule, we expect to add 5 x 2 or 10% to the winning chances for Blue at 70-70, and we find that this is pretty close to the mark. A whisker more than 10% because Black has slightly worse distribution. 70% in a 15-a-side competition is a borderline double/no double. Any improvement, say a checker off, and it would be a big double. I recall when Wales last won the Six Nations at rugby, in their first two matches they won with 15 v 14 men when the opponents each had a red card. In the third, some astonishing errors by the referee led to them beating England, and the commentator stated that they had 16 v 15 this time. But in a race at backgammon, it is an advantage to have men off. Let us see the rollout:

• No double
  • Player Winning Chances: 70.57% (G:0.00% B:0.00%)
  • Opponent Winning Chances: 29.43% (G:0.00% B:0.00%)
• Double/Take
  • Player Winning Chances: 70.54% (G:0.00% B:0.00%)
  • Opponent Winning Chances: 29.46% (G:0.00% B:0.00%)
• Cubeless Equities: No Double=+0.411, Double=+0.821
• Cubeful Equities:
  • No double: +0.638
  • Double/Take: +0.636 (-0.002)
  • Double/Pass: +1.000 (+0.362)
• Best Cube action: No double / Take
• Percentage of wrong pass needed to make the double decision right: 0.6%

The last figure is often ignored, but is important. It must be right to double this if there is any chance your opponent would pass. And a miscount of the pips is always on the cards.

So far we have 70-70 producing 60% wins and 70-75 producing 70% wins. We would expect 70-80 to produce 80% wins. Of course it cannot be linear or we would be winning more than 100% at some point. It is a gentle curve which is tangential to the x-axis so that at some point it reaches 100%. Indeed if Black had 15 checkers on his six-point, Blue would be winning close to 95%, so the curve is above the straight line there. With all Black’s checkers on his mid-point, the race is 70-180 and Blue wins close to 100% of the time. 99.98% to be precise. But back to a more relevant position:

Money Game. Blue on Roll. Cube Action?

The rollouts show that this is a small pass. Generally you need about 21-22% wins to take these positions where you are quite likely to get a redouble in. Black does not have this here:

• No double
  • Player Winning Chances: 79.60% (G:0.00% B:0.00%)
  • Opponent Winning Chances: 20.40% (G:0.00% B:0.00%)
• Double/Take
  • Player Winning Chances: 79.52% (G:0.00% B:0.00%)
  • Opponent Winning Chances: 20.48% (G:0.00% B:0.00%)
• Cubeless Equities: No Double=+0.592, Double=+1.181
• Cubeful Equities:
  • No double: +0.879 (-0.121)
  • Double/Take: +1.060 (+0.060)
  • Double/Pass: +1.000
• Best Cube action: Double / Pass

So, concluded Goldilocks, there you have some Golden Rules. 70-70 is 60% wins and no double but no beaver either. 70-75 is borderline double/no double. 70-80 is a borderline pass. If you compare any position to those, you may get quite close to the right action.