Cube Handling in Match Play

Woolsey’s Rule

In my December 2002 column, “Giving Gifts,” we looked at how many terrible takes or passes are needed to justify any money redouble. If your opponent will err 50% of the time, and then play perfectly, you should redouble any position, whether you are about to lose, have a reasonable double, or are too good. This may be viewed as a starting point for a quantitative understanding of Woolsey's rule (by Kit Woolsey):

Woolsey’s rule is very powerful in money play, but it doesn’t work as well in match play. Consider the following position:

Is it a take? Is it a pass?

This is a strange decision, and there is a good chance of drawing an error if you double. Whether your opponent takes or passes, you are much worse off if you double than if you hold the cube, since by doubling you eliminate your gammon threat.

That can’t happen in money play, as we saw in December. Here, even if your opponent promises to make the wrong decision, you still shouldn’t double here because of the match score. (By the way, rollouts say it is a 1.150 pass.)

A priori, my feeling was that Woolsey’s rule applies at even match scores and when one trails, but is dangerous to apply when one leads. It is much easier to encounter positions while leading in which one is better off after not doubling than double/take and double/pass.

Let’s see if the numbers bear that out. It’s also worth getting a feel for how many bad passes are needed to justify a bad double in a variety of situations in match play. The sections of most direct practical value are on doubling in gammonish positions and when your opponent has borne off one too many checkers. The other sections are more useful as rough guidelines, and for understanding the differences between match scores.

Woolsey’s rule is also supposed to encourage you to double when doubling is correct, but here we’ll concentrate on doubling when it is wrong — bluffing.

I don’t play poker, but I’ve been told that you should tend to bluff when you have a hand slightly weaker than what you need, rather than complete garbage. That way you don’t require your opponent to be bluffed in order to win. You only need to recover a small amount of lost equity. Here, we are trying to understand which bluffs in backgammon require only a small number of mistakes by your opponent to justify, and which are the equivalent of bluffing with garbage.

Gin Lost

How many passes do you need to make it correct to double with a dead-lost position? For money play, you risk 1 point to try to gain 2 points from a bad pass, so you need 33% bad passes in order to double from a gin lost position. Let’s look at initial doubles within a short match. The columns show the number of points you need, while the rows show the number of points your opponent needs.

For example, if you lead 3-away 4-away, you only need 36% passes to double a gin-lost position, but if you trail 3-away 4-away, you need 49% passes to make up for the error of doubling.

That seems backwards! It looks like by this measure, Woolsey’s rule is more powerful when you lead than when you trail. What’s wrong?

In match play, the trailer benefits greatly by being able to win 4 points at a time. The leader would prefer to see cubeless games with no gammons. The leader benefits more from putting one extra point at stake, while the trailer typically benefits more from putting 3 extra points at stake. We’ve seen earlier that the racing take point on an initial double may be lower than for money for the leader, while it may be higher than for money for the trailer, but that this is reversed for redoubles.

How many passes are needed to make a double correct in a gin lost position only depends on your opponent’s gammon price on a 1-cube. The (completely useless) formula for it is g/(1+g), where g is your opponent’s gammon price. So, what this table means is that the trailer values gammons on a 1-cube less than the leader values gammons on a 1-cube.

This doesn’t have enough in common with the positions in which one might apply Woolsey’s rule in a real match. In those positions, there would be real chances to win after a double is accepted, while there might also be recubes. We’ll need a better test than this. It should serve as a caution, though, not to double too recklessly while behind.

Even

I took a bearoff position that was roughly even, and evaluated the doubling decision using Snowie on 1-ply. An even position is a more plausible use of Woolsey’s rule than with no chance of winning. Although it is unlikely that someone would pass an even race, this serves as an abstract test case.

For example, you need to get 30% passes in order to justify doubling with an even position trailing 3-away 5-away, but if you lead 3-away 5-away you need 35% passes.

In money play, you need to get about 27% passes in order to double from an even position, assuming no beavers. The shaded entries in the table are those that are lower than 27%.

Here we start to see that Woolsey’s rule is more powerful for the trailer, but it looks less clear than I expected. One complication is that the racing take points are generally lower for the leader on initial doubles than for the trailer. Perhaps because 50% is closer to the take point, it means that doubling is a smaller blunder for the leader. Again, we see that the trailer should not double for no reason.

I would have expected that when the leader needs 2 points to win, a high number of passes would be needed. You can compare this column to the one in the next section. It could be that these values are not accurate because they do not properly take into account the automatic recube, unlike the values in the next table.

One Checker Too Many

Let’s suppose you are considering doubling after closing out one checker. Perhaps you have a good feel for how much these positions win, but you aren’t sure that your opponent knows. Let’s suppose you would have a borderline double if you had the same distribution of spares, but your opponent had one fewer checker borne off. How many passes do you need to justify doubling?

It’s messy to try to answer this. I set up different positions for each match score. This required me to use a lot of judgement, hence the entries in this table are not necessarily reproducible. To control the volatility, which greatly affects decisions to double or not, all of the positions that I chose had closed boards, no spares on the 6 point, one spare on the 5 point, and usually one checker borne off. This means that there was the possibility of opening the 6 point with 6-x followed by a dance, usually a market loser. The opposing side had a crashed board with many checkers on the ace point, and I took one of the checkers on the ace point off to produce the position that had one too many checkers off. For most match scores I looked at multiple such positions.

Unfortunately, Snowie does not understand the absolute equities of these positions well, although it plays well enough on 3-ply. At the start of the bearoff, you can use Snowie evaluations to determine the aggression index of a decision, since that depends on relative, not absolute evaluations. However, what Snowie calls a borderline money take is often a pass by 0.050. This makes it hard to trust rollouts of double/no double decisions, which is a shame because rollouts of closed out positions are rapid.

The tactic I tried is simply to accept the evaluations. Although they are wrong, they are reasonably consistent, and they tell a reasonably consistent story about the match scores. That’s what we are interested in here. Based on the flexibility in choosing the positions and the errors in evaluations, many of the smaller percentages in the following table may be off by 2% in either direction. The larger values in the first column may be off by 5%.

I think this is a particularly good time to use Woolsey’s rule due to the misapplications of Robertie's rule of 5. This table should also apply to other nongammonish positions. For example, in a race of medium length, you might double early if you think your opponent may make an arithmetic mistake counting pips, or might not know the take point, or might misjudge how much wastage there is.

The shock to me was that it took only 16–19% passes to justify a bluff double leading 3-away versus 5-away and more. It was very different from the high number of passes needed to justify an early double when leading with just 2 points to go. It could be that at 3-away, the doubling point is lower, and the recube to 4 is not automatic.

Except for the caution necessary when 2 points suffice to win the match, this experiment finds only a small difference between the passes needed to justify an early double when leading (the bottom left) versus trailing (the top right).

Highly Gammonish Positions

Gammonish positions are an ideal context for using Woolsey’s rule in money play. They are tough for people to judge, and yet take/pass errors are often huge blunders. Since the swing from losing to winning is greater, the positions are often volatile, which means that doubling is often correct anyway.

It is worth considering positions with gammon threats for both sides, double-edged positions. However, I felt that it was too difficult to try to standardize double-edged positions. Instead, I’ll focus on partially primed positions with one checker closed out. These are tough to judge, and I’m still working on them. Kit Woolsey’s The Backgammon Encyclopedia, Volume 1 has a good section on similar types of positions.

A crucial factor is how many pips black has to spare before black must break. The exact locations of the two spares matter, but the number of pips is the most important consideration. These positions are volatile enough that the window of correct double/take counts is pretty wide, but let’s suppose that black is doubling with 3 spare pips too few (varying the location of the two spares based on the match score). How many incorrect passes are needed to make up for this mistake?

This is a borderline take.

This is a borderline double without the Jacoby rule.

Black has too few spare pips. This is a borderline beaver.

Snowie 4’s evaluations are farther off near the take/pass point than in the bearoff positions. It evaluates the borderline decision as a big pass, worth 1.175 if white takes.

The evaluations are more accurate near the doubling point for money, in the sense of estimating the number of backgammons, gammons, and normal wins for each side. However, the cube adjustment is wrong, since there are few ways to improve slightly, leading Snowie to wait when doubling is correct.

Since we are studying match scores rather than these positions, what matters is that the evaluations are sufficiently consistent, and I believe that is the case. The following table was constructed using evaluations only, doubling with 3 pips fewer than is needed for a correct double according to the evaluations.

For money without the Jacoby rule, roughly 15% passes are needed to make up for the error of doubling, according to live-cube rollouts. By the way, since the rollouts suggest that the third position above is a borderline beaver (I think it is a small error to beaver), good luck getting people to pass! You may also view these as abstract test positions.

I’m going to ignore 2-away 2-away.

When the leader is 2-away, no number of spare pips makes it right to double. Not doubling is better than double/take (not good enough), and might or not be better than double/pass (too good). In positions that the evaluations say are borderline money takes, black is not too good, and 63–77% passes are needed to justify the bad double. The 63% was for 2-away 6-away, and the 77% occurred both at 2-away 4-away and 2-away 5-away.

Here we see a significant trend favouring early doubles when trailing. It takes particularly many bad passes in a position like this when the leader is 3 points away from victory, and this is most evident at 3-away 4-away, since the leader will probably face a nasty recube soon after a premature double.

Gin Gammons

Let’s try another abstract test case: Suppose you will definitely win a gammon if you don’t double. How many bad takes do you need to justify doubling anyway? Of course, you need 100% if a gammon would win the match for you, so I’ll leave out that column. For money, you need to get 33% bad takes in order to redouble when you are certain to win a gammon.

A gin gammon is extreme. For money, a 1.2 pass is a big pass, and a 1.5 pass is a nightmare. A gin gammon is a 4.0 pass: If you take a 2 cube, you lose 4 points. These values are therefore not directly useful, but they might serve as upper bounds. If you think the chance your opponent will make a bad take is comparable to the entry in this table, you can double.

Particularly interesting to me is the 46% for 3-away 4-away, which is higher than the surrounding entries. This match score is a strange one. The racing take point for the trailer is much higher than for money, even taking into account the considerable recube power. The reason is that 2-away 4-away is much better for the trailer than Crawford 4-away. This also means that gammons on a 1-cube are quite valuable to the leader. The combination means that it is easy to become too good to double with a small improvement from a position that is not good enough to double. It turns out that one often becomes not only too good to double, but too good to double with a very high fraction of bad takes needed to justify doubling.

Analysis

These tables are a first step, but they are only a small part of the story, and for the most part they are not directly applicable. They may tell you which way to lean compared to money play. There still many other aspects that come into play when considering a technically incorrect double.

Woolsey’s rule applies in money play when the doubler is unsure of how much a position is worth. In this article, we have studied what happens when you know that you are making an incorrect double. Over the board, you may be weighing the costs of doubling early not just against the benefits of provoking a bad take/pass, but against the costs of failing to make a correct double. We’ll look at those costs in a future column.

How likely is it for your opponent to misjudge the position? If you are about to get gammoned while trailing by a lot, you might need only a 5% chance for your opponent to pass in order to make a double right, but in reality there is a 0% chance that your opponent will pass. In other situations, you might need a 15% chance of a pass, and you may be confident of getting more like 30%.

It may be that the above tables underestimate the effectiveness of bluffing while ahead in the match. The reason is that we tried to fix the distance away from a correct double. When the trailer is considering doubling, borderline doubles tend to be farther away from the borderline passes. This may mean it is less plausible that the leader would pass a marginal double than that the trailer would pass.

Another issue to consider is the future set of errors by both sides. Of course considering projected checker play errors will complicate matters tremendously, but you can also consider the possible cube errors on the next roll. This is usually an argument against an early double. The current position may be too clearly a take. However, next turn if your position improves, you may have a better chance to get your opponent to err by erroneously taking or passing, perhaps even making a larger mistake than could happen this turn. This means that even if you will get more bad passes or bad takes than you need to “justify” the double, you may be better off waiting. We’ll consider this in a future column.

I’m still studying Woolsey’s rule for redoubles. Recubes are more complicated than initial doubles, since the take points vary more. I would have liked to add parallel sections on recubes to 4 and 8 at a variety of match scores for gin, racing, and gammonish position, but I will leave it to you to try these techniques if you like. I believe that you will find that all recubes behave more like gammonish decisions on initial doubles, and that this allows the trailer to be more aggressive with early recubes than the leader, but I haven’t tested this.

Summary

Doubling for no reason is a serious blunder for the trailer, just as it is for the leader.

When a position without many gammons starts to get close to a reasonable double, only slightly fewer bad passes are needed to justify a double by the trailer than by the leader, even if the leader needs 3 points and the trailer many more.

If the leader needs 2 points to win, this is an exception, and the trailer must pass a much higher fraction of the time in order to justify a double that is even slightly early.

In gammonish positions, the leader must be much more confident of bluffing successfully in order to make an incorrect double, particularly at 3-away.

Be very cautious applying Woolsey’s rule when leading with 2 points to go or at 3-away 4-away.

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