Not surprisingly, I was playing some backgammon against the computer today. What was surprising, to me at least, was the following position came up that got marked as “doubtful” by GnuBG:
Now, I didn’t even think about this one, clearly the right move is 11/7,10/7 and making the 6 prime. But GnuBG said no, 10/6, 9/6 is better. Huh?
GnuBG says 10/6, 9/6 has equity +0.788, 11/7, 10/7 has equity +0.715; a difference of -0.073 and so a blunder! Extreme gammon rates it similarly, 10/6, 9/6 gets +0.797 while 11/7, 10/7 gets +0.736, a difference of -0.061 and again a blunder!
It gets worse for me. After the next exchange of rolls, I was at the following position:
D’uh – clearly 9/2 is the right move. But no, GnuBG says 7/2, 5/2!!! 9/2 has +0.728 equity, versus +0.792 for 7/2, 5/2, a difference of -0.064 and so another blunder! Extreme gammon agrees again.
The key here is the cube. With the cube in the middle, the moves I actually made are the correct moves. But the cube has already been offered and taken, and that makes a difference. I no longer have the option to double red out after a rough roll, I have to take what I get – and if I get something bad red has the redouble.
As it happens, what I got was a 5-6 on the first roll bearing in against the men still on my one point, forcing me to leave two blots. Red rolled a 5, putting me on the bar, I danced, and then got redoubled out. So it really could have worked out better for me. But the main thing was these back-to-back positions where the difference in correct play depends so much on the cube position.
This table shows the winning chances for each move for the first position, per GnuBG:
| Win | W g | W bg | Lose | L g | L bg | |
| 11/7, 10/7 | 0.811 | 0.129 | 0.005 | 0.189 | 0.005 | 0.000 |
| 10/6, 9/6 | 0.835 | 0.146 | 0.005 | 0.165 | 0.005 | 0.000 |
And this is the equity, depending on who owns the cube:
| Equity Cube in Middle | Equity, Red’s cube | Equity, Black’s cube | |
| 11/7, 10/7 | +1.000 | +0.715 | +1.000 |
| 10/6, 9/6 | +0.934 | +0.788 | +0.978 |
| Difference | +0.066 | -0.073 | +0.022 |
Very similar charts for the second position:
| Win | W g | W bg | Lose | L g | L bg | |
| 9/2 | 0.824 | 0.115 | 0.004 | 0.176 | 0.005 | 0.000 |
| 7/4, 7/3 | 0.833 | 0.145 | 0.006 | 0.167 | 0.005 | 0.000 |
| Equity Cube in Middle | Equity, Red’s cube | Equity, Black’s cube | |
| 9/2 | +1.000 | +0.728 | +1.000 |
| 7/4, 7/3 | +0.925 | +0.781 | +0.970 |
| Difference | +0.075 | -0.053 | +0.030 |
I find it also interesting that the drop in equity for leaving an indirect shot is a lot lower when black owns the cube in both positions as well. I think either of these would make a good “quiz” question for Phil Simborg. Having them back to back like that in a game definitely made me stop and think about the value of owning the cube!
-Mark