Across the Scores
Douglas Zare, 2011
Gammon Village Magazine, © 2011 Douglas Zare and GammonVillage

This article originally appeared in Gammon Village Magazine.
Thank you to Douglas Zare and Gammon Village
for their kind permission to reproduce it here.

Does the Score Matter?

When we play backgammon matches, we face a series of decisions at one match score, and then move on to the next score.

It is an enlightening exercise to do the reverse: Take one decision, and consider it at several match scores. This can help you to understand both the match scores and the position better.

Sometimes a play is right because it is simply strong. At other times, it is right because of specific features of the match score. At which other match scores is it also right?

Cube Decision

Here is a cube decision:

Black doubles at each match score.

White has a fairly well-timed deuce-point game, although early hits are far from gin. White will get a lot of late hits, but also risks losing a gammon. White might run off the gammon without hitting by rolling well or if black is forced to pile up checkers on the ace point. This is a close take/pass decision for money. How much is taking worth at each match score within a 5-point match?

Black White
2-away 3-away 4-away 5-away
2-away 1.101 .545 −.091 .366
3-away 1.353 1.176 .919 .963
4-away 1.434 1.346 1.340 1.306
5-away 1.099 1.074 1.049 1.008

Any surprises? I think several of the values make sense only after studying the 5-point match in detail.

If it wasn’t intuitive that there was a large difference between 5-away 5-away and 4-away 4-away, then you might benefit from studying 4-away 4-away more. The value of double/take reflects many factors, such as the racing take point, the value of the gammons, and the recube vigorish. Gammons on a 2-cube are more valuable at 4-away 4-away, and the racing take point is higher because it is an unusually small disadvantage to trail 3-away 4-away.

It is common advice to be more conservative taking when you lead in the match. However, white is much closer to being able to take while leading 4-away 5-away than while trailing. It is very efficient for the leader at 4-away 5-away to get 4 points. The advice to be conservative while leading is often correct, but there are some common exceptions within the 5 point match when there aren’t many gammons.

Although this position is a pass if black trails 2-away 5-away, it is closer than many people expect. Gammons on a 2-cube are valuable, but much less so than at 2-away 4-away, and the racing take point is low for the leader.

Checker Play

This is also a good exercise for checker play decisions.

Black to play 3-3 at each match score.

Two natural candidates are 24/21(2), 13/10(2) and 8/5(2), 6/3(2). I normally make the first play for money and when leading in the match, but when trailing in the match, the more gammonish play of making two inner-board points might be better. At which scores does the more aggressive play become correct? A third play, 24/21(2), 6/3(2), should also be considered, but in the following table, I gave the difference between the two main plays. A positive number means 24/21(2), 13/10(2) is better.

I used lower-level rollouts, so the individual values are slightly less accurate, but the pattern should be correct. In practice, you might not want to roll out all scores if the first few rollouts agree with the evaluations.

Black White
Crawford 2-away 3-away 4-away 5-away
Crawford .032 .120 .039 .073 .036
2-away −.031 .095 .093 .156 .111
3-away .023 .021 .060 .130 .099
4-away −.001 −.077 .015 .054 .077
5-away .025 .010 .027 .024 .063

It appears that the score has to be close to gammon go to make 8/5(2), 6/3(2) correct, although it is only a small error at many scores where black trails. At Crawford 4-away, where gammon losses don’t matter but gammon wins do, the decision is close.

As an additional variation, you can consider different cube values and locations, although that would make less sense in a position so close to the initial position.

The value in EMG is sometimes misleading. This is normalized so that winning a normal game is worth 1.000, and losing a normal game is worth −1.000. If the play affects gammons, or if there will be a double soon, this can magnify or shrink the size of the difference between plays in EMG. Misplaying an opening 3-1 13/10, 24/23 is a horrible blunder at all match scores, but it costs more EMG at some scores than others. Because of the rapid doubles at 2-away 2-away, errors with the cube centered are magnified by a factor of 3 compared with DMP.

I hope you have found this a useful exercise. It takes time, but it is a method I use when the inital answer of a bot leaves me asking why the play is better.

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