The most important tool of backgammon experts is the “reference position.” A reference position is a position that the expert knows the correct action in, and is close enough to positions that actually arise that they can use it in practical play.
Let’s start with a very simple reference position:
This is a pure “3-roll position.” Each side needs 3 rolls to all their checkers off, and all doubles save a full roll. The side on roll in this position is known to win exactly 78.8% of games, and the opposing side can never offer an effective redouble. Since the taking side needs 25% in a money game without cube use, it is a double and a pass.
There is a closely-related position. This is what happens if black fails to roll doubles. White is 74.5% to win here:
But consider this position:
What are black’s winning chances here? Yes, you can calculate it. Here’s what white can think:
“This position is identical on a lot of rolls. When he rolls 11, it’s as good as any double unless he rolls 21 next and I gets doubles. Let’s ignore that, it’s a small difference.
“If I roll 22, this doesn’t save him a roll. So about 3% of the time I am still in the game with 25% chances. So this gains me about 0.7% wins.
“The real difference is when he roll 21. Now, instead of being 25.5% to win, I will be 100%. I will double and he will have to drop. He rolls 2-1 5.6% of the time, so this gains me 75% of 5.6%, or 4.2%.
“Thus, in total and allowing for my owning the cube, I gain about 5% wins, from 21.2% to 26.2%. I can now take.”
Before we leave the subject of reference positions, let’s show just a few more racing positions.
This is a pure 2-roll position. The side on roll is 86.1% to win. In a cross-section of 1296 rolls (36 for the side on roll, 36 for the other) the side on roll will win on the first shake 216 times. Of the remaining 1080 rolls, the other side will roll doubles 180. So the side not on roll wins 180/1296, or 13.9%. You might ask “So what? I know the side not on roll is losing badly. He wouldn’t take a double, and he can’t control the dice, so why do I care?” There are two reasons. First, at some match scores, there might be cube actions. Imagine yourself on the white side, trailing 5-1 to 9, and your opponent redoubles to 4. Second, take a black checker and move it to the 3-point. Or the 4-point. As we did earlier, you can calculate the modified win chances if you know the chances in the reference position.
A four-roll position is 74.5% for the side on roll to win. In a money game this is a double and a take. You should double here because most of the time neither side will roll doubles and then your opponent will have a pass next turn. You get maximum value for the cube if you double here.
A 5-roll position is 71.7% for the side on roll to win. You should not double. Most of the time neither side will roll doubles, and then you can double next turn.
In some later articles, we will discuss a variety other types of reference positions.