Cube Theory

I recently read an article that stated, contrary to common belief, 25% is not the minimum necessary chance to win that justifies taking a double (in no-gammon positions). The article demonstrated that, in at least one case, a 20% chance of winning was sufficient to justify accepting the double.

The following example shows the minimum acceptable chance of winning (and to justify a take), is actually 18.75%! Suppose a particular game has developed into a race and Black doubles. Furthermore, suppose White has some way of determining that he can reach the following position 25% of the time, and lose 75%.

Now, since there is no equity gained or lost by Black in either accepting or declining the recube, then let us assume he takes. It should be easy to see that after the original acceptance, White will win 25% of the games. Multiply this 25% by his chance of winning after Black accepts the redouble, or 25% × 75%. Therefore, in this original position, White has a legitimate take, even though his winning chance is only 18.75%.

It is fairly simple to prove that (in no-gammon positions) this is the minimum winning chance to justify a take (steaming is not a justifiable reason). There are certain one-way gammon positions in which a smaller winning percentage allows an acceptance of the cube. In fact, if every win were to lead to a backgammon for the person accepting the cube and all his losses were single games, then a 12.5% chance of winning would be the minimum. My God, is everything a take?!

No, of course not. The gammon chances on either side influence the drop/take decisions a great deal. However, there are many positions that appear to be very bad, which in fact are actually good takes.

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