This article originally appeared in the May 2001 issue of GammOnLine.
Thank you to Kit Woolsey for his kind permission to reproduce it here.

Reference Positions

By Kit Woolsey
How do we make our cube decisions? Most positions can't be calculated in any sensible manner, so we have to rely on our judgment and instincts. Our judgment is mostly based on experience, from positions we have seen before. However, every backgammon position is different. Except for certain standard types of positions, such as a closeout of one checker vs. some number of men off, there will always be something new about every position. Thus, it won't help to just memorize a bunch of positions and their equities.

What can we do? The best approach is to understand several common types of positions and know whether they are passes or takes and by how much. Then, when we see a position over the board, we can relate to a similar position which we know. We make adjustments for the differences, and come to a conclusion.

In order to make the proper adjustments, we need to know how various factors affect a type of position. Thus, just knowing a reference position isn't sufficient. We also have to have an idea how much one side or the other improves as the position changes slightly from our reference position.

What I have done in this article is examine one common type of position -- the five-point holding game. I start with a prototype holding game, and then make adjustments to the position and see how the equity and the doubling strategy changes with these adjustments.

In order to get my equities, I have used Snowie rollouts. The settings I used were:

2-ply rollout with variance reduction
Full rollout (no truncation)
72 trials
Super-tiny search space
20% speed
No live cube in play

Obviously better results could have been obtained with more powerful settings, but I was interested in getting the results of several positions quickly. The tradeoff beteween speed and accuracy is always a problem when doing rollouts. For this type of position, I believe that the settings I used are sufficient to get decent results. The rollouts generally were about what my guesses were, which is further confirmation that these settings are adequate.

First, let's look at a classic 5-point holding game position, which will be our main reference position.





 
Pip: 119
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.444
  0.1%   2.0%  72.1%    27.9%   2.0%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.444 0.028.
Rollout settings:
Full rollout,
72 games (equiv. 6511 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.684
2. No double 0.667 (-0.017)
3. Double, pass 1.000 (+0.316)
Proper cube action:   Double, take

Green is up 20 pips. If it were just a race this would be double and pass. White would need another 7 pips or so to have a borderline take. However, this is not just a race. White has plenty of potential to get a shot. In addition Green may have to waste in order to bear in safely.

Both sides have fairly ideal structure. White's board is perfect, and will be improving for a while. White will not be able to hold his midpoint much longer and still keep his board, but he will have plenty of shot-hitting potential from his anchor on Green's five point. Green's position is also well-balanced. His wastage is currently minimal, and he has some flexibility.

It is well-known that the proper cube action for this position is double/take, which is what the rollout indicates. In fact, with a cubeless equity of .444 for Green, the take is very clear and the double is close. This sort of position is relatively involatile -- not too much figures to happen immediately. For this reason, Green needs better equity than he would for a more volatile position to justify a double because Green doesn't have too many market-losing sequences. However he does have a few, namely 3-3 or higher, and these are sufficient to justify a cube turn.

Now let's start modifying the position and see what effect changes have. First, let's improve White's racing chances.






 
Pip: 109
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.352
  0.1%   2.5%  67.3%    32.7%   1.8%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.352 0.025.
Rollout settings:
Full rollout,
72 games (equiv. 7964 games),
played 2-ply (supertiny),
seed 2, without race database.
1. No double 0.517
2. Double, take 0.466 (-0.051)
3. Double, pass 1.000 (+0.483)
Proper cube action:   No double, take10%

Now White has a take on the race alone, which gives him an even easier take with the shot-hitting potential. Granted he usually won't be able to take full advantage of both the shot-hitting and the racing potential, but he is still in pretty good shape. Green's equity has dropped to .352; definitely not worth a double in this relatively involatile position.

Let's move the other direction with the pip count:





 
Pip: 129
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.511
  0.0%   2.9%  74.9%    25.1%   1.6%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.511 0.025.
Rollout settings:
Full rollout,
72 games (equiv. 5781 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.838
2. No double 0.796 (-0.042)
3. Double, pass 1.000 (+0.162)
Proper cube action:   Double, take

Green's cubeless equity is now up to .511. Still a very easy take, of course, due to the potential recube. While White's racing equity is down, his shot-hitting chances have gone up. Now he may be able to stay on the midpoint long enough to get a direct shot at Green as Green tries to clear his own midpoint. This tradeoff of race vs. shot-hitting potential is quite common in holding games. Also, while Green does have a proper double it is still fairly close. Once again, there are only a few market-losing sequences.

Let's extend Green's lead in the race even further and see what happens.





 
Pip: 129
Game 1
Money session

Green-White:
Score 0-0
Pip: 89
 


Cube action equity
Rollout Money equity: 0.445
  0.2%   4.9%  72.0%    28.0%   4.3%   0.2%
  95% confidence interval:
- money cubeless eq.: 0.445 0.033.
Rollout settings:
Full rollout,
72 games (equiv. 4869 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.686
2. No double 0.646 (-0.040)
3. Double, pass 1.000 (+0.314)
Proper cube action:   Double, take

Interestingly enough, Green's cubeless equity has dropped back down to .445. This actually does make sense. The difference between a 30 pip lead and a 40 pip lead in the race isn't too significant -- White is pretty much cooked in the race regardless. However by being back an additional 10 pips, White is more likely to be able to hold his midpoing long enough to get a direct shot or otherwise be a nuisance as Green tries to clear his midpoint. Once again, an easy take, and a close but correct double due to the few market losers. Of interest is that the gammon chances for both sides, while still small, have noticeably increased. We can be confident that extending Green's lead even further isn't going to tell us anything new -- White will always have a comfortable take regardless of the racing lead, and Green will always have a double due to the potential market losers if he rolls doubles or otherwise clears the midpoint safely.

Let's try some other factors. How about backing up both positions in order to give Green more time to clear the midpoint safely? We will retain Green's original 20 pip lead in the race.





 
Pip: 136
Game 1
Money session

Green-White:
Score 0-0
Pip: 116
 


Cube action equity
Rollout Money equity: 0.496
  0.0%   2.1%  74.2%    25.8%   0.9%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.496 0.021.
Rollout settings:
Full rollout,
72 games (equiv. 7651 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.803
2. No double 0.774 (-0.029)
3. Double, pass 1.000 (+0.197)
Proper cube action:   Double, take

Green's cubeless equity has gone to .496 (from .444 in our original position), despite the fact that making the race longer is better for the trailer with the pip count remaining the same. Green has more spares to work with, so he has more time to either make his bar point or roll a lucky doubles and clear the midpoint safely. This extra leeway for Green is reflected in the rollout results. Still an easy take, however, and still a close yet correct double for all the same reasons.

Conversely, let's take a look at what is happening when Green is running out of time.





 
Pip: 106
Game 1
Money session

Green-White:
Score 0-0
Pip: 86
 


Cube action equity
Rollout Money equity: 0.428
  0.0%   1.7%  71.1%    28.9%   1.1%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.428 0.023.
Rollout settings:
Full rollout,
72 games (equiv. 7218 games),
played 2-ply (supertiny),
seed 2, without race database.
1. No double 0.673
2. Double, take 0.666 (-0.007)
3. Double, pass 1.000 (+0.327)
Proper cube action:   No double, take2%

Even though the race is closer to the finish and the pip lead remains the same, Green's equity has gone down. He will be forced to leave at least an indirect shot very soon. He no longer has the potential to make his bar point with a 6-1, and he won't have many more turns to sit tight and hope to roll doubles. Now the double is borderline.

How important is the contact created by White's checkers on the midpoint? Let's move them away, keeping the pip count difference at 20, and see what happens.





 
Pip: 119
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.477
  0.0%   2.1%  73.5%    26.5%   1.4%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.477 0.018.
Rollout settings:
Full rollout,
72 games (equiv. 8309 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.762
2. No double 0.736 (-0.025)
3. Double, pass 1.000 (+0.238)
Proper cube action:   Double, take

Very little equity change from the original position -- only slightly more favorable to Green. The reason is that with Green having only a 20 pip lead, White usually isn't going to be able to maintain his midpoint long enough to get a shot. White will have to play to keep his board in one piece, and that will probably mean releasing the midpoint. Thus, not having it originally isn't too important here.

Does owning the midpoint become more important for White as the racing lead gets wider? Let's try giving it up with a 30 pip lead.





 
Pip: 129
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.579
  0.3%   3.5%  78.0%    22.0%   1.8%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.579 0.057.
Rollout settings:
Full rollout,
72 games (equiv. 1760 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.998
2. No double 0.939 (-0.060)
3. Double, pass 1.000 (+0.002)
Proper cube action:   Double, take

Quite a difference! Green's cubeless equity has shot up to .579, and now White has a borderline pass/take decision. This does make sense. With a 30 pip lead White's main winning chances lie in hitting a shot, and he has a greater chance of maintaining the midpoint for getting that shot than he had when Green's lead was 20 pips. Thus, the loss of the midpoint is more costly to White when Green's racing lead increases.

So far, we have been giving White a near-perfect board. It isn't always like that in real life. White still has to contain a checker once he hits it, and if his board isn't ready that may be a problem. Let's start messing White's board up a bit and see what happens.





 
Pip: 119
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.498
  0.1%   2.8%  74.2%    25.8%   1.5%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.498 0.019.
Rollout settings:
Full rollout,
72 games (equiv. 7828 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.808
2. No double 0.777 (-0.031)
3. Double, pass 1.000 (+0.192)
Proper cube action:   Double, take

White's five point is open, and while White may be able to make it later it won't be too convenient to do so. Greens equity is up to .498, but White still has an easy take. White's board is structurally okay, so White remains with decent shot-hitting chances and racing chances.

Does the weaker board become more significant as the racing lead increases? Let's take a look.





 
Pip: 129
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.551
  0.1%   4.1%  76.1%    23.9%   1.2%   0.1%
  95% confidence interval:
- money cubeless eq.: 0.551 0.027.
Rollout settings:
Full rollout,
72 games (equiv. 4791 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.927
2. No double 0.883 (-0.044)
3. Double, pass 1.000 (+0.073)
Proper cube action:   Double, take

Now Green's equity has shot up to .551, and while White still has a take it is getting close. It is clear that as the racing lead increases the importance of winning by hitting a shot becomes greater, which means that having a solid board is crucial for White.

Let's try doing a different kind of damage to White's board, advancing it a bit far by making the ace point. We'll cut the racing lead back down to 20 pips again.





 
Pip: 119
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.496
  0.0%   1.9%  74.4%    25.6%   1.1%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.496 0.022.
Rollout settings:
Full rollout,
72 games (equiv. 5987 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.806
2. No double 0.770 (-0.036)
3. Double, pass 1.000 (+0.194)
Proper cube action:   Double, take

Green's cubeless equity is up to .496, but White is still in the game. As long as White has some decent structure, he has sufficient winning chances to justify a take. He may not need to build a full prime in front of a hit checker -- just holding the checker on the bar for a roll or two might be sufficient.

We'll now look at the same structure with the 30 pip lead.





 
Pip: 129
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.510
  0.1%   4.5%  74.4%    25.6%   2.3%   0.1%
  95% confidence interval:
- money cubeless eq.: 0.510 0.029.
Rollout settings:
Full rollout,
72 games (equiv. 7835 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.833
2. No double 0.795 (-0.038)
3. Double, pass 1.000 (+0.167)
Proper cube action:   Double, take

Not much difference. Even though White is more dependent upon hitting a shot to win, having made the ace point isn't as much of a liability as we might have thought. Part of the reason is that White can concentrate on making inner board points and perhaps hold his midpoint longer than he would have been able to were he working on keeping a pure board.

Let's do some serious damage to White's board.





 
Pip: 119
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.561
  0.2%   3.2%  76.8%    23.2%   0.8%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.561 0.027.
Rollout settings:
Full rollout,
72 games (equiv. 4537 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.954
2. No double 0.905 (-0.049)
3. Double, pass 1.000 (+0.046)
Proper cube action:   Double, take

This does make a significant difference. Now only will White find it very difficult to put together much of a board with all those dead checkers on the two point, but if White does get into a race those checkers represent considerable potential wastage. Now White's take is quite slim.

Moving over to the other side of the board, let's start playing with Green's structure. One possible improvement for him is to give him his bar point.





 
Pip: 133
Game 1
Money session

Green-White:
Score 0-0
Pip: 113
 


Cube action equity
Rollout Money equity: 0.558
  0.0%   1.5%  77.4%    22.6%   0.5%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.558 0.020.
Rollout settings:
Full rollout,
72 games (equiv. 7552 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.951
2. No double 0.906 (-0.045)
3. Double, pass 1.000 (+0.049)
Proper cube action:   Double, take

This is a significant improvement for Green. The made bar point means that once Green gets down to two checkers on the midpoint he will be able to clear the midpoint safely with a 6-5 as well as doubles. Also, once the midpoint is cleared it will be easier to clear the eight point safely. These changes are almost enough to give White a pass. Any slight deterioration in White's position and he could no longer justify taking the double.

We can check this hypothesis out by keeping the same structure but improving Green's racing lead by 10 pips.





 
Pip: 133
Game 1
Money session

Green-White:
Score 0-0
Pip: 103
 


Cube action equity
Rollout Money equity: 0.637
  0.0%   2.9%  80.7%    19.3%   0.6%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.637 0.021.
Rollout settings:
Full rollout,
72 games (equiv. 6077 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, pass 1.000
2. No double 0.991 (-0.009)
3. Double, take 1.132 (+0.132)
Proper cube action:   Double, pass

Not surprisingly, Green's cubeless equity is up to .637, and now White has a clear pass. White probably needs to hit a shot, and with the help of his bar point Green has a good chance of coming home safely.

On the other side of the coin, let's damage Green's structure and see what happens.





 
Pip: 119
Game 1
Money session

Green-White:
Score 0-0
Pip: 99
 


Cube action equity
Rollout Money equity: 0.409
  0.0%   1.9%  70.6%    29.4%   2.1%   0.1%
  95% confidence interval:
- money cubeless eq.: 0.409 0.028.
Rollout settings:
Full rollout,
72 games (equiv. 4798 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.604
2. No double 0.604 (-0.000)
3. Double, pass 1.000 (+0.396)
Proper cube action:   Optional double, take

Even though the pip count is the same, Green's equity has dropped down to .409 and now his double is borderline. Those dead checkers on the ace point are costly for several reasons. The big reason is the race. If White turns the game into a race, those checkers represent potential wastage which could cost Green at least one roll in the bearoff. Also, it may be important in the future for Green to have a board so he can pounce on White if White gets even in the race and then tries to run with one checker.

The midpoint is not the only point which might have to be cleared when coming in against a five-point anchor. Let's take a look at some of the other points.





 
Pip: 113
Game 1
Money session

Green-White:
Score 0-0
Pip: 93
 


Cube action equity
Rollout Money equity: 0.462
  0.0%   0.9%  73.6%    26.4%   1.6%   0.3%
  95% confidence interval:
- money cubeless eq.: 0.462 0.026.
Rollout settings:
Full rollout,
72 games (equiv. 7136 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.730
2. No double 0.709 (-0.021)
3. Double, pass 1.000 (+0.270)
Proper cube action:   Double, take

It appears that the 11 point is just about as difficult to clear as the midpoint against the defensive five point. Green's equity is close to that of our original reference position. Once again, easy take and close but correct double.

Let's move in a bit, maintaining our 20 pip differential, and see how much difference that makes.





 
Pip: 110
Game 1
Money session

Green-White:
Score 0-0
Pip: 90
 


Cube action equity
Rollout Money equity: 0.536
  0.0%   0.8%  76.8%    23.2%   0.8%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.536 0.018.
Rollout settings:
Full rollout,
72 games (equiv. 8326 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, take 0.903
2. No double 0.874 (-0.028)
3. Double, pass 1.000 (+0.097)
Proper cube action:   Double, take

Moving the outer point closer makes a significant difference. These checkers have three safety numbers (6's, 4's, and 2's), while from the 11 point they had only 2 safety numbers (5's and 3's). This makes it much more likely that Green will be able to bring the position home safely, upping his equity to .536. White still has a take with his combination of racing and shot-hitting potential, but it is getting close.

Will moving the outer point one closer be enough to swing things to a pass?





 
Pip: 107
Game 1
Money session

Green-White:
Score 0-0
Pip: 87
 


Cube action equity
Rollout Money equity: 0.629
  0.0%   1.1%  81.0%    19.0%   0.3%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.629 0.016.
Rollout settings:
Full rollout,
72 games (equiv. 9386 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, pass 1.000
2. No double 0.967 (-0.033)
3. Double, take 1.119 (+0.119)
Proper cube action:   Double, pass

The result is no surprise. Clearing the nine point safely is relatively easy with only one gap to hurdle over. White's shot-hitting chances are small, and with Green's equity up to .629 White has a clear pass.

How can we use all this information? When we see a five-point holding game, we compare it to our original reference position. There will be differences, of course. We examine those differences, and judge how much effect each of them has from our experience in looking at other positions which were close to our reference position. From this, we can get a good idea of the proper cube action for our actual position. For example:





 
Pip: 131
Game 1
Money session

Green-White:
Score 0-0
Pip: 111
 


Cube action equity
Rollout Money equity: 0.435
  0.0%   2.4%  71.4%    28.6%   1.7%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.435 0.031.
Rollout settings:
Full rollout,
72 games (equiv. 5413 games),
played 2-ply (supertiny),
seed 2, without race database.
1. No double 0.674
2. Double, take 0.661 (-0.013)
3. Double, pass 1.000 (+0.326)
Proper cube action:   No double, take4%

Obviously White has a trivial take. Does Green have a double? What can happen next roll? If Green gets the back checker safe, the position will be approximately like our original reference position -- clear take. Green might make a new outer board point with 3-2, 4-3, or 5-4, but even if that happens our analysis has shown that White will still have a take. Green would have to roll 6-6 in order to lose his market. On the other side of the coin Green might not get the outfield blot safe and White might hit. Or, maybe it will be White who rolls 6-6 and pulls even in the race. The chance of a big downside is greater than the chance of Green losing his market, so the conclusion is that doubling must be wrong. And, that is exactly what the rollout tells us.





 
Pip: 133
Game 1
Money session

Green-White:
Score 0-0
Pip: 109
 


Cube action equity
Rollout Money equity: 0.620
  0.1%   2.7%  80.1%    19.9%   0.9%   0.0%
  95% confidence interval:
- money cubeless eq.: 0.620 0.018.
Rollout settings:
Full rollout,
72 games (equiv. 12296 games),
played 2-ply (supertiny),
seed 2, without race database.
1. Double, pass 1.000
2. No double 0.980 (-0.020)
3. Double, take 1.094 (+0.094)
Proper cube action:   Double, pass

Green has made his bar point, which as we know gives White a close take with the other factors in our main reference position remaining the same. Here White is behind 24 pips instead of 20 pips, which is worse for the race, and we know that if White were behind 30 pips it would be a pass. So the race combined with the made bar point would make it a close decision if all other things were equal. And all other things are not equal -- White's board is not in perfect shape. Thus, the conclusion is that this is definitely a pass -- again, exactly what the rollout says.

This article has covered one particular type of position. There are many position types. The better equipped a player is with reference positions and their variants, the better he will be able to make accurate cube assessements in actual play.

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