Backgammon

This position is an example of the Jacoby Paradox, first explained by Oswald Jacoby in The Backgammon Book, published in 1970. Besides this position, there are four others in all, of which Black has two men, one on his 2-point and one on his 5-point. White's positions are: one man on the 19-point; one man on the 21-point and one man on the 24-point; two men on the 23-point; one man on the 22-point and one man on the 23- point.

In each position, if the cube were in the centre, Black would have a perfectly correct initial double. This is because Black is the favourite and he doubles his equity by turning the cube. With Black owning the cube, the situation is different. In the cases where Black does not throw one of his immediate 19 winning numbers, White will have a very powerful redouble. In the diagram position, say Black rolls 41 and plays 5-off. He would have to drop a redouble as White has 29 winning numbers. But if he has not redoubled the original position, then he will still win in those cases where White rolls one of his seven losing numbers. By redoubling, Black reduces his equity from 0.48 to 0.22, a considerable difference.

The paradox is that if White had a stronger position, for example two men on the 24-point, then Black would have a clear redouble! This is because his next roll would then be the last of the game, and White would get no benefit from owning the cube. In this pioneering piece of analysis, Jacoby was one of the first players to demonstrate clearly the power of cube ownership. Even this close to the end of a game, the difference in equity generated by the right doubling decision is huge. It gives one just some idea of the complexity of doubling cube theory.