Cubeless Equity

There is a huge element of luck in backgammon, and it visits good and bad players alike. What sets the very best players ahead of the rest, is their ability to calculate probabilities and to play accordingly.

In most real-time situations, it simply isn’t possible for you to look at a position and to know exactly what your chances of winning and losing are. Not unless you’re a bot, or are using one. The best that you can do is to estimate your chances based upon past experience, study, and strategies (devised from experience and study).

Equity is a backgammon term which is used to describe how good a position is. Actually, backgammon players refer to different kinds of equity. For the purpose of this article, we’ll restrict ourselves to a discussion of the equity in a money session where the stake is $1 a game and the doubling cube is not used. Hence the term, cubeless equity.

Let’s examine a position and what gnuBG has to say about it:

          GNU Backgammon  Position ID: 6M7EAQyY28gHAA
                          Match ID   : MAELAIAHmAQA

           +12-11-10--9--8--7-------6--5--4--3--2--1-+     O: P0
           | X  O        O  O |   | O     O  X       |     120 points
           | X           O  O |   | O     O  X       |     Rolled 62
           | X                |   | O                |
           |                  |   |                  |
           |                  |   |                  |
          ^|                  |BAR|                  |     (Cube: 1)
           | O                |   |                  |
           | O                |   |                  |
           | O                |   | X  X             |
           | O           X    |   | X  X             |
           | O     X     X    |   | X  X  X          |     147 points
           +13-14-15-16-17-18------19-20-21-22-23-24-+     X: P1

           1. Cubeless 2-ply   13/11 13/7   Eq.:  +0.497
              0.737 0.047 0.001 - 0.263 0.026 0.000

First of all, let’s explain the numbers.

gnuBG is saying that if this position were to be played out 1,000 times, it’s expectation is that the player who is on roll (0) would win 737 times (1 backgammon, 47 gammons, and 689 singles) and that the opposing player (X) would lose 263 times (0 backgammons, 26 gammons, and 237 singles).

When all is said and done, after the 1,000 games, Player O will have won:
(1 x 689) + (2 x 47) + (3 x 1) - (1 x 237) - (2 x 26) - (3 x 0) = $497.

Which, on a per-game basis, is $0.497.

And so the cubeless equity value of this position is 0.497.

What if Player O had made a different move? Would it have been a better play?

Here’s more of what gnuBG had to say about some of the 22 possible moves:

           2. Cubeless 2-ply   13/7 6/4   Eq.:  +0.431 ( -0.066)
              0.705 0.055 0.001 - 0.295 0.033 0.000

           3. Cubeless 2-ply   11/5 7/5   Eq.:  +0.354 ( -0.143)
              0.664 0.090 0.003 - 0.336 0.066 0.002

           4. Cubeless 2-ply   13/5       Eq.:  +0.326 ( -0.171)
              0.663 0.077 0.003 - 0.337 0.078 0.002

           5. Cubeless 2-ply   8/6 8/2    Eq.:  +0.297 ( -0.200)
              0.642 0.067 0.003 - 0.358 0.055 0.001

Player O would be said to have lost 0.2 in equity each time that the inferior 8/6 8/2 choice was made.

And while that might not seem like much at first glance, making that wrong choice 1,000 times would result in Player O winning $200 less in total!

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gnuBG is a wonderful FREE resource for anybody who is serious about improving their understanding of backgammon. To acquire the program, simply click on the “GNU Backgammon” link in the “Blogroll” sidebar section.