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Cube Theory
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Double-Beaver Revisited
Arthur Ramer, 1981
Las Vegas Backgammon Magazine, September 1981
In the July, 1979 issue of
Backgammon Magazine, M. Leifer suggested that there exist positions where the correct doubling strategy is to double for one opponent (say White) and to beaver for the adversary (say Black). The proposed illustrative position
was subsequently criticized by W. Robertie as not providing sufficient equity for the doubling side — in order to win gammon White has to risk about a five percent chance of losing the game altogether.
The following position meets all the requirements of a simultaneously correct double and beaver:
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White on roll
has borne off one checker.
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White is on roll and eleven out of thirty-six times will jump the prime and win gammon from Black. (Black does not have even a theoretical chance of saving a double game.) In the remaining twenty-five instances Black will double (or redouble) and White has to pass. To see that this is the case, let's note that White has a 90%+ chance of moving the prime (with all fifteen men in play) around the board and containing Black without ever giving him a chance of escaping (add to it a very high likelihood of White not escaping in the remaining cases when Black may, on occasion, keep a five-point prime only).
Black, having closed his home board with White's one man on the bar (and only one man borne off), has at least a ninety percent chance of winning.
The combined probability of success is 81%+, sufficient to force White to drop (under any assumptions, continuous game or sudden death).
In light of the above discussion we can now compute White's equity. If White does not double his equity is
when he jumps the prime and
when he fails, for a total of
.
If White does indeed double, he is beavered (optimum strategy for Black) and his equity becomes:
(gammon with the beavered cube) and
if he fails.
The total now becomes
,
a better result (that is a smaller average loss) for White.
The suggested position may appear a rather contrived one, but similar situations may arise as a result of a massive back game by Black, when White has proceeded to hit all his opponent's checkers.