Backgammon

The key difference between doubling and redoubling is that when you redouble you give your opponent access to something- the cube - which he did not previously have. This is not true before the initial double where both players have equal access to the cube. As an example, what could be simpler than this bear-off problem, where after the next roll by each side the game will be decided? But beware. Make the wrong decision with the cube here and you will more than halve your expected winnings!

First let's look at Black's raw probability of winning. He will win if he does not throw a 1 or a 2 on his first roll (except 2-2 which wins) or if White throws a 1 on his roll. This happens just over 63 per cent of the time, so Black is a clear favourite to win. Notice one very important point: if Black does not win on his first roll then White will have a very strong redouble which Black will have to take. But White can redouble only if he has access to the cube.

This points the way to the solution. If the cube is in the middle, Black should double and White should take. This is because if Black fails to bear off his two men, White will double anyway. By doubling initially Black doubles his expected equity to 0.12. (Equity - your anticipated profit - is calculated by multiplying your probability of winning by the level of the cube.) If, however, Black owns the cube, he should not redouble, thus killing White's potentially very strong redouble to 4. If Black holds the cube on 2 his equity is 0.53; if he redoubles to 4 his equity drops to 0.24.

This type of dramatic swing shows how careful you must be in giving the cube to an opponent who may have the chance to get in an efficient redouble. Thus in races it quite often correct to try to double an opponent out, (when he drops your double) rather than trying to double him in (when he takes).