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Reference PositionsHow do we make our cube decisions? Most positions can't be calculated inany sensible manner, so we have to rely on our judgment and instincts.Our judgment is mostly based on experience, from positions we have seenbefore. However, every backgammon Position is different. Except for certainstandard types of positions, such as a closeout of one checker vs. some numberof 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 typesof 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 positionwhich we know. We make adjustments for the differences, and come to aconclusion. In order to make the proper adjustments, we need to know how various factorsaffect a type of position. Thus, just knowing a reference position isn'tsufficient. We also have to have an idea how much one side or the otherimproves as the position changes slightly from our reference position. What I have done in this article is examine one common type of position -- thefive-point holding game. I start with a prototype holding game, and then makeadjustments to the position and see how the equity and the doubling strategychanges with these adjustments. In order to get my equities, I have used Snowie rollouts. The setting I usedwere: 2-ply rollout with variance reduction 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 doingrollouts. For this type of position, I believe that the settings I usedare sufficient to get decent results. The rollouts generally were about what my guesses were, which is further confirmation that these settingsare adequate. First, let's look at a classic 5-point holding game position, which willbe our main reference position.
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. Inaddition Green may have to waste in order to bear in safely. Both sides have fairly ideal structure. White's board is perfect, and willbe improving for a while. White will not be able to hold his midpointmuch longer and still keep his board, but he will have plenty of shot-hittingpotential from his anchor on Green's five point. Green's position is alsowell-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 sortof position is relatively involatile -- not too much figures to happenimmediately. For this reason, Green needs better equity than he would fora more volatile position to justify a double because Green doesn't havetoo many market-losing sequences. However he does have a few, namely3-3 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.
Now White has a take on the race alone, which gives him an even easier takewith the shot-hitting potential. Granted he usually won't be able to take fulladvantage of both the shot-hitting and the racing potential, but he is stillin pretty good shape. Green's equity has dropped to .352; definitely notworth a double in this relatively involatile position. Let's move the other direction with the pip count:
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, hisshot-hitting chances have gone up. Now he may be able to stay on themidpoint long enough to get a direct Shot at Green as Green tries to clearhis own midpoint. This tradeoff of race vs. shot-hitting potential isquite common in holding games. Also, while Green does have a properdouble it is still fairly close. Once again, there are only a fewmarket-losing sequences. Let's extend Green's lead in the race even further and see what happens.
Interestingly enough, Green's cubeless equity has dropped back down to .445.This actually does make sense. The difference between a 30 pip lead anda 40 pip lead in the race isn't too significant -- White is pretty muchcooked 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 geta direct shot or otherwise be a nuisance as Green tries to clear hismidpoint. Once again, an easy take, and a close but correct double dueto the few market losers. Of interest is that the gammon chances forboth sides, while still small, have noticeably increased. We can beconfident that extending Green's lead even further isn't going to tell usanything new -- White will always have a comfortable take regardless ofthe racing lead, and Green will always have a double due to the potentialmarket losers if he rolls doubles or otherwise clears the midpoint safely. Let's try some other factors. How about backing up both positions in orderto give Green more time to clear the midpoint safely? We will retain Green'soriginal 20 pip lead in the race.
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 withthe pip count remaining the same. Green has more spares to work with, sohe has more time to either make his bar point or roll a lucky doubles andclear the midpoint safely. This extra leeway for Green is reflected inthe rollout results. Still an easy take, however, and still a close yetcorrect double for all the same reasons. Conversely, let's take a look at what is happening when Green is runningout of time.
Even though the race is closer to the finish and the pip lead remains thesame, Green's equity has gone down. He will be forced to leave at leastan indirect shot very soon. He no longer has the potential to make hisbar point with a 6-1, and he won't have many more turns to sit tight andhope 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 seewhat happens.
Very little equity change from the original position -- only slightly morefavorable to Green. The reason is that with Green having only a 20 piplead, White usually isn't going to be able to maintain his midpoint longenough to get a shot. White will have to play to keep his board in onepiece, and that will probably mean releasing the midpoint. Thus, nothaving it originally isn't too important here. Does owning the midpoint become more important for White as the racing leadgets wider? Let's try giving it up with a 30 pip lead.
Quite a difference! Green's cubeless equity has shot up to .579, and now Whitehas a borderline pass/take decision. This does make sense. With a 30 pip leadWhite's main winning chances lie in hitting a shot, and he has a greater chanceof maintaining the midpoint for getting that shot than he had when Green'slead was 20 pips. Thus, the loss of the midpoint is more costly to Whitewhen Green's racing lead increases. So far, we have been giving White a near-perfect board. It isn't alwayslike that in real life. White still has to contain a checker once hehits it, and if his board isn't ready that may be a problem. Let's startmessing White's board up a bit and see what happens.
White's five point is open, and while White may be able to make it later itwon't be too convenient to do so. Greens equity is up to .498, but Whitestill has an easy take. White's board is structurally okay, so Whiteremains 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.
Now Green's equity has shot up to .551, and while White still has a takeit is getting close. It is clear that as the racing lead increases theimportance of winning by hitting a shot becomes greater, which means thathaving a solid board is crucial for White. Let's try doing a different kind of damage to White's board, advancingit a bit far by making the ace point. We'll cut the racing lead back downto 20 pips again.
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 winningchances to justify a take. He may not need to build a full prime infront of a hit checker -- just holding the checker on the bar for a rollor two might be sufficient. We'll now look at the same structure with the 30 pip lead.
Not much difference. Even though White is more dependent upon hitting ashot to win, having made the ace point isn't as much of a liability as wemight have thought. Part of the reason is that White can concentrate onmaking inner board points and perhaps hold his midpoint longer than hewould have been able to were he working on keeping a pure board. Let's do some serious damage to White's board.
This does make a significant difference. Now only will White find it verydifficult to put together much of a board with all those dead checkers onthe two point, but if White does get into a race those checkers representconsiderable potential wastage. Now White's take is quite slim. Moving over to the other side of the board, let's start playing with Green'sstructure. One possible improvement for him is to give him his bar point.
This is a significant improvement for Green. The made bar point means thatonce Green gets down to two checkers on the midpoint he will be able toclear the midpoint safely with a 6-5 as well as doubles. Also, once themidpoint is cleared it will be easier to clear the eight point safely.These changes are almost enough to give White a pass. Any slightdeterioration in White's position and he could no longer justify takingthe double. We can check this hypothesis out by keeping the same structure butimproving Green's racing lead by 10 pips.
Not surprisingly, Green's cubeless equity is up to .637, and now White hasa clear pass. White probably needs to hit a shot, and with the help ofhis 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 whathappens.
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 pointare costly for several reasons. The big reason is the race. If Whiteturns the game into a race, those checkers represent potential wastage whichcould cost Green at least one roll in the bearoff. Also, it may be importantin the future for Green to have a board so he can pounce on White if Whitegets 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 inagainst a five-point anchor. Let's take a look at some of the otherpoints.
It appears that the 11 point is just about as difficult to clear as the midpointagainst the defensive five point. Green's equity is close to that of ouroriginal reference position. Once again, easy take and close but correctdouble. Let's move in a bit, maintaining our 20 pip differential, and see how muchdifference that makes.
Moving the outer point closer makes a significant difference. These checkershave three safety numbers (6's, 4's, and 2's), while from the 11 point theyhad only 2 safety numbers (5's and 3's). This makes it much more likelythat Green will be able to bring the position home safely, upping hisequity to .536. White still has a take with his combination of racingand shot-hitting potential, but it is getting close. Will moving the outer point one closer be enough to swing things to a pass?
The result is no surprise. Clearing the nine point safely is relativelyeasy with only one gap to hurdle over. White's shot-hitting chances aresmall, 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 eachof them has from our experience in looking at other positions which wereclose to our reference position. From this, we can get a good idea ofthe proper cube action for our actual position. For example:
Obviously White has a trivial take. Does Green have a double? What canhappen next roll? If Green gets the back checker safe, the position willbe approximately like our original reference position -- clear take. Greenmight make a new outer board point with 3-2, 4-3, or 5-4, but even if thathappens our analysis has shown that White will still have a take. Greenwould have to roll 6-6 in order to lose his market. On the other side ofthe 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 losinghis market, so the conclusion is that doubling must be wrong. And, thatis exactly what the rollout tells us.
Green has made his bar point, which as we know gives White a close takewith 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 therace, 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 closedecision if all other things were equal. And all other things are notequal -- White's board is not in perfect shape. Thus, the conclusion isthat this is definitely a pass -- again, exactly what the rollout says. This article has covered one particular type of position. There are manyposition types. The better equipped a player is with reference positionsand their variants, the better he will be able to make accurate cube assessementsin actual play.
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