Position 46a, page 84
From Better Backgammon, by Tim Holland

Should White
double to 2?

Although you will certainly win fewer times than in Position 46, it will still be more than your opponent.

 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66
19 rolls for Black
fail to take both off

Out of 36 games you will win 5 times on your first roll (double 6s, 5s, 4s, 3s, and 2s). This leaves 31 games. Of these 31 games, Black will win slightly less than 15 times. (He will fail to bear off both men with any roll that contains a 1 or a 2, with the exception of double 2s. This will occur 19 out of 36 times.)

Therefore, if you were to play this identical game 36 times, you would win slightly more than 21 times (the 5 where you win on the first roll and the 16+ where he fails to bear off both of his men). I’m sure that it is not necessary to say that whenever you will win more than 50 percent of the time you should want to increase the stakes to their maximum.

Now that I have shown you that White must double in the above situation I am going to explain the paradox that exists in backgammon where it is correct for one side to double and yet correct for the other side to accept.

Using the figures from 46a we came to the conclusion that out of 36 games White would win a little more than 21 times, and Black would win a little less than 15 games. At the end of those 36 games White has doubled Black from 1 to 2.

 21 times Black will lose 2 points −42 15 times Black will win 2 points +30 Black’s net −12

If Black declined all 36 times, he would have a minus of 36. This is proof positive that Black must accept the double.

Rollout

Tom Keith 2013
Money play
Centered cube
White on roll

1296 games with VR
Checker play: 3-ply
Cube play: XG Roller
XGID=-BB------------------aa---:0:0:1:00:0:0:0:0

 Cube Action Game G BG Equity No double W L .5934 .4066 .0000 .0000 .0000 .0000 +0.1867 (0.1868) Double Take W L .5934 .4066 .0000 .0000 .0000 .0000 +0.3735 +0.3735 Drop +1.0000

White should double and Black should take

 Previous Position No. 46, page 82
 Next Position No. 47, page 85

List of Positions from Better Backgammon

Better Backgammon (1974), by Tim Holland