11 Point Match
Score tied at 7-7
Red doubles, should Black take?
A deceptive position indeed! (At least for me!)
When I first viewed this position, I thought Black had a trivial take. After further analysis, however, I’m not so sure. Read on…
When faced with match cube decisions, the first thing that needs to be determined is Black’s Take Point.
At 7-7 in a match to 11:
|Black Takes & Wins||(9-7)||67%|
|Black Takes & Loses||(7-9)||33%|
|67% – 41% = 26% Reward|
|41% – 33% = 8% Risk|
This 26-8 ratio represents approximately a 23% Take Point, (Take Points are determined by dividing Risk + Reward into Risk.)
|Risk + Reward||Point||8 + 26|
“So what’s the problem here?” I asked. Black is slightly ahead in the race 80-83. And there is still contact. Surely the contact must help the one owning the cube, right?
Wrong! Not only does contact not favor Black at all in this position, but the mere prospect of contact greatly inhibits an already poor race. The two long crossovers that Black is down, plus his poor distribution in his inner board, translate his original 3-pip lead into a racing nightmare.
In my original tinkering with the position, I tried clearing Red’s back checker as soon as possible. But just relying on Black’s poor distribution is clearly wrong, as I found Black was winning over 30% of the games.
However, when I left Red’s rear checker alone and built his inner board as quickly as possible, contact started paying dividends. Numbers such as 6-2 should be played 9/1 by Red; 6-3 : 9/3, 5/2; 6-4 : 9/3, 5/1. As for 6-5, frankly, I’m not sure: 17/6 or 9/4, 9/3?
To help find a solution to this problem, Butch Meese was enlisted to play out the position on the IBM version of Expert Backgammon™. 12,924 rollouts yielded:
|Red wins 8,995 single games||=||69.6%|
|Red wins 595 gammons||=||4.6%|
|Black wins 3,434 single games||=||26.8%|
If we multiply these figures out from Black’s point of view, we find the following:
|Black Loses:||69.6% x 33% Match Equity||=||23.2%|
|4.6% x 0% Match Equity||=||0%|
|26.8% x 67% Match Equity||=||17.8%|
It would appear from these results that taking the cube yields 41% Match Equity and that passing yields 41% Match Equity! Close decision, eh?!
Still, there are some factors that favor Black. First of all, the computer rollouts are cubeless – clearly a disadvantage for Black who has a redouble point of 50%+, Also, Red should pass a redouble at <33%, so any positions where Black goes from <50% to >67% should be counted as Black wins, whereas the computer continues to roll them out, giving Red some wins he doesn’t deserve.
Furthermore, Red may not play the position to his greatest advantage, as I alluded to above. This is not something you can count on, but carefully factor it into your judgment of the position.
In summary, this position is clearly a strong double and not-so-clearly a close take.
Note: This decision was correct in 1993 and is still correct today. Here is the eXtreme Gammon analysis:
There are a couple of notes on the checker plays mentioned in the article. The 6-2 played 9/3, 5/2 is correct according to eXteme Gammon. For the 6-3 the computer prefers 9/6, 9/3 by a small margin. The computer plays the 6-4 16/6. According to the computer, the answer to Sly’s question on how to play the 6-5 (either 17/6 or 9/4, 9/3) is 9/4, 9/3.