This article originally appeared in the August 2002 issue of GammOnLine.|
Thank you to Kit Woolsey for his kind permission to reproduce it here.
Editor's Note: I wrote the following article for Inside Backgammon back in 1991. It was the first time Woolsey's Law for Doubling appeared in print. This was pre-bot time, so things couldn't be verified by computer rollout. However, I believe that the concepts in this article have withstood the test of the bots, and that they are as accurate today as they were over 10 years ago. I thought that GammOnLine readers would appreciate seeing the article, so here it is in its entirety.
Several years ago when I first got interested in tournament backgammon, I kibitzed a 15-point match between Tony Goble and Lynn Goldsmith, two of the top players at that time. Soon the score was 13-12, after 25 games! That's right, 25 initial doubles which were passed. At the time, I was impressed -- this is the way good tight backgammon should be played. Since then, I have learned better. If they were playing correctly, what happened that match was virtually impossible. No doubt players of that era were too conservative in their takes, but, more important, they were waiting far too long to double. With correct cube play, I believe that at least 2/3's of initial doubles should be taken.
Today, most advanced and intermediate players have a pretty good idea about what doubles should be taken and what doubles should be passed. Unfortunately, they are unable to use this evaluation skill to determine when they should be doubling. As a result, most good players double too late, costing themselves opportunities to win two and four points. In this article, I will show how you can use your skill in evaluating whether or not a position is a take to determine whether or not you should double.
For any given position, if you are asked what is the correct cube strategy there are four possible answers. They are as follows:
1) Not good enough to double, take
If you have the advantage in a position and fail to double when you should, letting the position slide from category 2 to category 3, you have doubled too late. This is called losing your market. Even though you can lock up a sure point by doubling, if your opponent's pass is correct you have cost yourself considerable equity by failing to double earlier. This is the error we are trying to avoid.
Let's play through a few rolls of a game and see how the correct cube strategy changes as the position improves.
What's happening? Should Blue be doubling? This is always the most important question to ask whenever it is your roll if the cube is in the center or on your side. Every roll is a cube decision! The first part of any play, before rolling the dice, is making a conscious decision about whether or not to double. This cannot be emphasized enough. I have seen many missed doubling opportunities go flying by simply because it never occurred to the player that it might be right to double. A common scenario is for a player to roll a series of good rolls each of which improves his position, and for him to be so caught up in his good fortune that he completely forgets about the cube. It is a worthwhile habit to consider doubling before every roll of the dice, regardless of the position. The position in question here belongs in category (1) in my opinion -- not good enough to double, take. However, it is always worth looking at.
What's happening? Should Blue be doubling? I strongly believe he should. There are many strong sequences for him which will give him a substantial advantage, and not much bad figures to happen. However, I think White has enough play to justify a take, for if he is hit and hits back or isn't hit and makes the enemy bar point he will have almost an even game, and even if things go badly he has the ace point anchor to fall back on and Blue still has one man to escape. Consequently, I believe this position falls into category (2) -- double, take. Shortly I will show how I reach the conclusion to double.
Assuming Blue failed to double last time, what's happening now? In my opinion, White no longer has a take. We are in category (3) -- double, pass. Blue has erred by not doubling last time, and he has now lost his market. Even though he locks up a sure point by doubling now, he still wishes he had doubled earlier, for since White has a pass here Blue's equity in playing out the game with White owning a two-cube is higher than one point.
Assuming Blue still has not yet doubled, what is happening here? In this position, Blue is too good to double -- he should play on for a gammon. Granted a gammon isn't all that likely with White having an anchor, but Blue's position is so strong that nothing bad can happen next roll, so there is no harm in playing on. This is category (4) -- too good to double, pass.
Now, how do we determine when category (2) -- good enough to double -- has been reached? The procedure is as follows: When considering whether or not to double, walk around the table to the other side of the board (at least do it mentally, since you won't be able to do it physically), and ask yourself the question: If I were doubled in my opponent's position, is it a take?
This is the point where most players make their big mistake. They may do all this, then if they think it is a pass they will double, while if they think it is a take they will not double. This will lead to doubling way too late in many games. The important thing to realize is that there are not two possible answers to the question of is it a take -- there are three possible answers. They are as follows:
1) Yes, I'm absolutely sure it is a take
We now get to Woolsey's Law of Doubling, which may be the most important backgammon principle you will ever learn. It is as follows: If the answer to the "is it a take" question falls into category (3) "I'm not 100% sure", then it is ALWAYS correct to double. This applies to money play and to match play. It is true that there are certain exceptions in match play when you are ahead in the match and the turn of the cube will put you out or nearly out, but in all other situations the law applies.
Many readers will assume that I am exaggerating. Let me assure you that I am not exaggerating in the slightest! I repeat: If there is ANY doubt in your mind as to whether a position is a take or a pass, then it is ALWAYS correct to double. To put it another way: Suppose Magriel or Sylvester or even God came up to you and said: I think this position is a pass, and I am willing to pay a point and the cube for the highest stakes you can afford. If you would not instantly say: "Sit down", then you are not 100% sure, which means you must automatically double.
Let's see why this rule works. If there is some doubt in your mind as to whether it is a take or not, there are several possibilities:
a) Maybe it actually is a pass. I hope most of the readers are sophisticated enough to realize that if you fail to turn the cube when your opponent's correct action is to pass you are costing yourself a ton of equity, regardless of what your opponent will actually do.
b) Maybe your opponent will think it is a pass. Differences of opinion make horse races and backgammon decisions. If there is ANY doubt in your mind, then there is always a possibility that your opponent may view his position more pessimistically than you and choose to pass the double. If you fail to double in such a position you have given your opponent a free roll to get back in the game when you could have collected a sure point, which is extremely expensive.
c) Maybe it is a correct double, even if it is a take and your opponent correctly takes. This is not an unlikely scenario. The fact that you thought there was some possibility that it might be a pass indicates that your position is strong and there are undoubtedly a few market-losing sequences. If this is the case, it could easily be correct to double even if your opponent is correctly taking.
d) Worst case -- It is not theoretically correct to double, and your opponent correctly takes. How bad is this? Not very bad. Unless your assessment of the position is completely off base you still have a clear advantage. True, you have forfeited future use of the cube to your opponent, but you have doubled the stakes with an advantage -- how bad can that be? You will only regret your decision to double if he turns the game around -- then you will lose twice as much as if you hadn't doubled. Since you are the favorite in the game, things are not all that bad.
When I first discovered this principle and started to employ it, I was astounded at how my results improved, both at tournament and money play. I found myself winning doubled games and gammons that I never would have won before, because I previously would have waited too long. At the same time, I collected unexpected points when my opponent passed a double that I thought he should take (but I was not sure, of course). I was outperforming players whose checker play was superior to mine simply because my doubles were more timely. These pressure doubles caused opponents to make mistakes -- dropping positions that they should take and taking positions which led them to larger losses.
Does this law apply only to initial doubles, or does it apply to redoubles as well? Surprisingly enough, it applies just as strongly to redoubles, for all the same reasons. It is true that it is more costly to sacrifice a cube that you own than to sacrifice a center cube when you err and make a theoretically incorrect redouble (and your opponent correctly takes, which he might not), but in the long run you will still gain by using the law for redoubles as well as initial doubles.
Let's replay our sample game asking the correct questions:
Blue: 3-1 8/5, 6/5
Let's walk over to White's side and ask the question: Is it a take? Personally, I am 100% convinced that it is. Blue certainly has the advantage, but if Magriel, Sylvester, or God declared that it was a pass I would unhesitatingly tell them to sit down. As we shall see later just because a position is a 100% clear take doesn't necessarily mean that it is not a double, but that is not what we are considering here. The key to the rule is that if it is not a 100% clear take, then it is definitely a double.
Is it a take? I certainly think so. Despite Blue's lead in the race and in development, White still has all his men in play, is threatening to make an advanced anchor, and is in no immediate danger of being blitzed or primed. However, the key is that I think it is a take, as opposed to being 100% sure. If Magriel, Sylvester, or God declared it was a pass and pulled out his wallet, I would certainly pause and give the position a second look. That is all I need. If there is the slightest doubt in my mind, then it is automatically correct to double, and I need look no further. Readers who have watched me play may notice that I often double a complicated position with apparently no thought. What has happened is that I was not able to instantly assess the position as a take for my opponent, which makes it automatic to double. Maybe he'll take, maybe not, but that is his problem. It is odd but true that the less sure you are about the position (i.e. whether or not it is a take), the more sure you can be that it is correct to double.
How about when you should be playing for a gammon. Simple -- ask the same question. Obviously you must think it is a pass, otherwise you would never consider playing for a gammon. However, you must be 100% sure. If there is the slightest doubt in your mind, then it is never correct to play on -- always double. The reasons are largely the same:
a) Maybe it actually is a take. If there is any doubt in your mind, then this is a possibility. Naturally it would be foolish to play on for a gammon when your opponent has a take, regardless of what actual decision he makes.
b) Maybe he will take even though it is a pass. Opponents are not infallible. I have several times doubled positions which I had considered playing on for the gammon, only to have my opponent shock me and take the double. Needless to say, you literally cost yourself double the value of the cube when you play for the gammon when your opponent would have taken, since doubling would have given you the opportunity to play for the gammon at twice the level of the cube.
c) Maybe it is correct to cash anyway. A position has to be very strong to justify playing on for the gammon. If there was some doubt in your mind as to whether or not it is a take, it is almost certainly correct to cash.
d) Even if it correct to play on and you mistakenly double and your opponent correctly passes, you usually haven't cost yourself much in equity. If you thought it might be a take, there must be a few rolls which will get your opponent back into the game, so the gain from playing on can't be much even if it is correct to do so.
I have often heard players attempt to justify not doubling by saying that they weren't sure if they were not good enough or too good. This is obviously nonsense most of the time. If they aren't sure if they are good enough they obviously are not sure it is a pass, while if they think they might be too good they obviously aren't sure it is a take. Therefore, by Woolsey's law, they have an automatic double. Following the law will prevent you from falling into this trap -- if you aren't sure about the position, double.
Let's continue with our sample game:
Is it a take? I don't think so. Blue has a strong threat to form a four-prime with three White men trapped behind it. Blue has only one man back, and White hasn't started to develop his position. However, White has an anchor and still has all his men in play, so there is some doubt in my mind -- I can't be 100% sure it is a pass. Therefore, any thoughts of playing for a gammon are out the window -- Blue must double. Who knows, White might even take.
Is it a take? Absolutely not! I am 100% sure about this one, and would not hesitate to pay a point to anyone who claimed it was a take. Consequently it is not automatic to double -- Blue can consider playing on for the gammon. As we shall see, playing on in this position is, in fact, correct.
Now, let's examine the more difficult question of whether or not to double even though we are 100% sure it is a take. In most positions it is correct to wait a roll. However, there are certain positions which are so volatile that it is imperative to double now. In other words, so much is likely to happen on the next roll that even though your opponent has a clear take now there is a good chance that he will have a huge pass on the next roll. In this case, it may well be correct to double now. The most basic example:
Obviously this is a take, since White will win 17/36 of the time. However, the position couldn't be more volatile, since the whole game will be decided by Blue's next roll. Consequently, it is clearly correct for Blue to double. Note the famous Jacoby paradox here -- if White's men were on the one and four points then it would still be correct to make an initial double but not to redouble. The difference is that here the next roll doesn't decide everything -- Blue still has some life after death if he holds the cube.
Look at the following position reached after this opening sequence:
White: 6-4 24/14
Is it a take? This position has been rolled out by many different players, and take my word for it -- it is a very clear take. However, the position is so volatile that it is still correct for Blue to double. If Blue hits White's blot and White fans or enters with only one man, or if Blue continues to develop and White again fans, White will suddenly have a big pass. On the other hand, if Blue does nothing special and White enters and brings the man on the 14 point to his outer board it will be anybody's game, with Blue's lead in the race and development compensated for by his early commitment to the ace point and having two men back. No matter. The position is so volatile , and Blue has a reasonable chance of losing his market by quite a bit on the next sequence of rolls, so it is correct for Blue to double.
This deceptive position came up several years ago:
Should Blue double? Obviously it is a trivial take, since if Blue fails to roll a six the game is up for grabs, and if Blue rolls a horror White becomes instant favorite. However, it is still correct for Blue to double! The point is that if Blue rolls one of his 11 sixes he loses his market by a mile, while if he rolls any other number except 3-3 or 4-4 it is still anybody's ball game. This is easily enough to justify a double. I played this as a proposition (double vs. no double) for several hours and slaughtered my opponents.
However, most positions are not as volatile as the above example. In these less volatile positions, it is usually not correct to double if your opponent has a clear take, even if you have a reasonable advantage. For example:
A typical five-point holding game. The race is 119 to 107, so White clearly has a take on the race alone. In addition, the contact appears to favor White. He will have some chance of getting a shot as Blue attempts to clear his midpoint and his eight point, and Blue will be forced to bear in with an empty five point until White leaves. These factors should more than compensate for any difficulty White has leaving the enemy five point. Consequently, it is a trivial take. Should Blue double anyway? No. There is not enough volatility in the position. Blue needs to roll a set to make great headway on the next roll, and only 5-5 or 6-6 really give White a big pass -- after 3-3 or 4-4 White will probably have a pass, but it will be close so Blue won't lose his market by much if at all. On the downside, if White rolls the big set the game will be a tossup. Since Blue is very unlikely to lose his market by a lot, he should wait.
Finally, let's examine playing for the gammon. As we have seen, it must be 100% that your opponent has a pass -- if not, it is always correct to doublt. Who knows, he might even take! Assuming that the pass is 100%, it is quite often correct to play for the gammon. Here, many players make a serious error. The first question they ask is: How good are my gammon chances? While this can be an important factor, it is not the most important one. More important is the following question: Is there a reasonable possibility that, on the next sequence (that is I roll, he rolls), the position can swing so much that my opponent now has a clear take? If the answer is no, then playing for the gammon is mandatory. Here is a common example which I have seen experts misplay:
Should Blue double? Obviously White has a clear pass, so playing for the gammon can be considered. Admittedly Blue's gammon chances are slim, but that is irelevant. The key point is that there is nothing that Blue can roll on his next roll that will give White a take, so going for the gammon is clearcut. If the spare man on the six point were on the five point then there would be some risk in playing of (6-6 by Blue, any 6 by White), so whether or not to play on would become a moot point. In the actual position, it is clear. It should be emphasized that Blue does not commit himself to play for the gammon to the end just because he take a roll here. Every roll is a cube decision! After each roll, Blue must reevaluate the position, and if there is a chance of leaving a shot it might well be correct to cash. However, as long as Blue cannot leave a shot on the next roll it is clear to play on. This is another common error by many players. They decide to play for a gammon, and continue to do so regardless of what happens. Each individual position must be analyzed before every roll and a new decision about the cube must be made. This is very difficult to do, which may explain why many players fail to play for the gammon when they should, but it must be done if you are to make the most out of the cube.
For another example, consider the resulting position after the fourth roll of our sample game, where Blue has completed his five-prime. Clearly, White must pass if Blue should double. Also, nothing really bad can happen to Blue on the next sequence. Give White his best sequence, and he will have at best a marginal take. Consequently Blue should play on for a gammon, even though he probably won't get it since White already has an anchor and has a chance to advance his anchor. However, if on the next roll Blue does nothing special and White either makes Blue's three point or starts to develop his own board by making his five or four point, then Blue should five serious consideration to cashing, for if White makes one additional improvement he could well move into take land. As always, before every roll Blue must make the big decision of whether or not to double.
Clearly White has a pass if Blue doubles. Should Blue play on? I think not. There is an immediate disaster sequence (6-4 by Blue, hit by White), and there are likely to be problems later. Here it is of importance that Blue's gammon chances aren't all that good, since White has most of his men in and Blue is facing a possible disaster immediately. If White had more men out in the outfield it might well be worth playing on. Note that if Blue's fourth man on the five point were on the six point instead, it would be correct to play on. Now, nothing bad can happen immediately. Admittedly something like double-fours would leave a very awkward position, but that wouldn't matter. Blue would then double, and White could still not afford to take. The point is that if Blue were to roll double-sixes and White rolled nothing special Blue might now decide that his gammon chances are worth risking losing the game and play on, while if anything else happens Blue can just cash, so there would be nothing to lose by playing on for a roll.