Perpetual Redouble? by Bill Kennedy

Mathematics of Backgammon

Perpetual Redouble?

Bill Kennedy, 1982

Las Vegas Backgammon Magazine, April 1982

While thinking about the strange things that can happen in backgammon, I came up with the following position:

Position 1.
Should the player on roll redouble?

Although it is unusual to redouble with only one winning shot, I felt that it was correct here because:

The player who enters first loses a lot if he hasn't doubled, since he is very likely to gammon his opponent.
Possession of the cube has little importance; whoever comes in first will simply blast his opponent off the board. The cube is only likely to come to life if a shot is hit in the bearoff.

So the advantage of rolling first seemed to outweigh the value of owning the cube, and I offered to play this as a proposition, where I would redouble and my opponent wouldn't, alternating first rolls. However an analysis by Bob Floyd (see "Riding the Tiger") shows that, even though the redoubling side should win that proposition, it is not necessarily right to recube.

Bob is a professor of computer science at Stanford with an international reputation in the design and analysis of algorithms, and a strong interest in mathematical aspects of backgammon. After seeing Bob's analysis, I went back to the drawing board and came up with this:

Position 2.
Should the player on roll redouble?

This position is quite a bit different from the first one. The player on roll is now favored to roll the crusher — the key factor. Possession of the cube is important, for the player who is hit first is favored to get a decent ace-point game, where hitting a shot is a distinct possibility. The chance of an ace-point game makes this an easy take.

Nevertheless, the great advantage of the player on roll outweighs the value of owning the cube, and the roller must redouble (if he can afford it).

In this case, the infinite series discussed by Bob will add up, because each term decreases. Each successive term is multiplied by 2 and by 16/36 (the chance of a miss).

The idea of the cube rising to some astronomical level seems crazy — but consider that every time you play backgammon for money there is no limit on the cube. You need to be at least half crazy to play backgammon in the first place.