This article originally appeared in the October 2003 issue of GammOnLine.|
Thank you to Kit Woolsey for his kind permission to reproduce it here.
There is much discussion about the importance of volatility when making
the decision of whether or not to turn the cube. But exactly what does
this mean? Is there some way we can quantify it? How do we use this
concept at the table when making cube decisions?
When might it be correct to double? In principle, it could be correct any time our winning chances are greater than 50%, since all turning the cube amounts to is doubling the stakes (I am assuming money play for this article). The opponent can take if our winning chances are up to 75%. These numbers are assumed to be taking gammons and recube potential into account. Thus, it might be a double and a take any time the person on roll has greater than 50% winning chances but less than 75% winning chances. This area between 50% and 75% is called the doubling window.
In practice, it is generally not right to double when near the low end of the doubling window. It isn't because our winning chances aren't good -- as far as that goes we are a favorite and are doubling the stakes. It is because if we are at the low end of the window, most of the time we will have an opportunity to double next roll and our opponent will have a take even if things go well for us. If things go badly, then of course we would rather not have doubled. In addition, if we own the cube then we want to be even more cautious about doubling, since that will give our opponent the potential to recube.
So, how far into the doubling window is it necessary to be to have a cube? That depends upon the position. As all readers know, the big danger from not doubling is losing one's market. The more likely it is that we will lose our market, the more important it is to turn the cube. That is where volatility kicks in. If a lot is likely to happen on the next exchange (we roll, he rolls), then it may not be necessary to be too far in the doubling window. However, if not much is likely to happen, then we will want to be high up in the window near where the opponent has a pass.
As an extreme example, let's look at the most volatile position of all.
Blue gets off 19 rolls out of 36, or about 52.8% of the time. Even though this is way down in the doubling window, it is still correct for Blue to double. The reason is that the volatility couldn't be higher -- this is the last roll of the game. There is no worry about a recube, and no potential for Blue to turn the cube later. This is it! Therefore, Blue should double simply because he is a favorite.
On the other side of the coin, we have:
A rollout of this position has Blue winning 77.5% of the time, for a cubeless equity of .550. If White didn't have recube potential that would amount to a pass, but due to White's recube potential he has a bare take -- that recube will give him just enough extra wins to get above the 25% mark. If we accept these results, then it is clear that Blue should not double even though he is right at the top of the doubling window. The reason is that this position is as involatile as can be. Nothing Blue can roll will have any significant effect on Blue's winning chances, and of course White's roll doesn't matter either. It is absolutely impossible for Blue to lose his market on the next exchange, therefore he cannot gain by doubling.
Players have often asked me how good their winning percentage needs to be to double. That question can't be answered. While obviously winning chances are very important, volatility and the timing of the critical rolls is also vital. Consider the following two positions:
Blue's cubeless winning chances are only 59%. Despite this, it is quite correct for Blue to double. The reason is that the result of the game is very likely to be decided on the next exchange. If Blue rolls a five or a six he will win, unless White rolls doubles in which case he loses. If Blue misses, White will be able to double and Blue will have to pass. Only in the variations where Blue rolls 2-1 or 1-1 and White doesn't roll doubles will there be another roll to be taken. It is on this roll that the volatility is huge, so since Blue is a decent favorite it is correct for him to turn the cube.
This time, Blue's cubeless winning chances are over 61%. Despite this, it would be a blunder for Blue to double. The position is volatile, but the main volatility is in White's direction. If Blue misses, White will have a very powerful double of his own. Blue would have a take, but Blue's pass take decision would be so close that for all practical intents and purposes Blue loses the game if he misses. What if Blue doesn't miss, but doesn't roll doubles? Assuming White does nothing special, Blue will now be left with two checkers on the two point and needing a non-ace to win the game. This common scenario results in a perfectly efficient cube for Blue, with maximum volatility and White having a borderline take (or possibly a close pass if White has rolled badly). The only time Blue really regrets not having doubled is when he rolls doubles and gets all 4 men off this roll, and that is more than compensated for by the times Blue misses.
Let's examine this position further from a mathematical point of view. We can group Blue's possible numbers into three categories:
Great: 2-2, 3-3, 4-4, 5-5, 6-6 (5 rolls)
Suppose Blue doubles (as opposed to not doubling). If he rolls a great number he gains a point, winning 2 points as opposed to winning 1 point.
If he rolls an awful number, he loses (almost) 2 points as opposed to losing (almost) 1 point. The reason I say (almost) is that White will double and Blue has a bare take, so his equity is slightly better than the value of the cube, but the take is so close that it isn't much better.
If he rolls a good number, then it depends on what White rolls:
a) If White rolls 4-4, 5-5, or 6-6, then Blue will have cost himself
a point by doubling.
Putting these together, it is quite clear that Blue should not double now. He gains big on 5 rolls, loses big on 10 rolls, and the rest pretty much cancels out.
In theory, what one should do when considering whether or not to double (assuming it is a trivial take) is to look at the entire range of 1296 possible scenarios after we roll, he rolls. As we have seen, there are three possibilities:
1) It is now double and pass (a market has been lost). If that happens,
failing to double cost some equity.
All you have to do as add up the equity costs and gains from doubling over the 1296 rolls, and out pops the answer. Simple for the bots which do this at lightning speed (in fact, this is exactly what they do when making a 3-ply analysis), but for mere mortals it is pretty impossible. For most positions we pretty much have to rely on our judgment and experience. Still, it is sometimes possible to break down the rolls into various categories and come up with a reasonable estimate. For example:
It is easy to break Blue's rolls into categories:
Great rolls: 6-6, 6-5, 6-4, 6-3, 6-2, 6-1 (11)
If Blue rolls a great roll, he loses his market by a pretty large margin. White's chances from a crushed ace-point game are bleak.
If Blue rolls a terrible roll, he loses the game. White has a powerful cube, which Blue probably has to pass -- if Blue does have a take, it must be close enough to a pass to make it basically a loss.
If Blue rolls a good roll or an average roll, now we have to look at White's rolls. They are:
Good rolls: 6-6, 6-5, 6-4, 6-3, 6-2, 6-1, 5-5, 5-1 (14)
This breakdown isn't absolute, since obviously 5-2 is considerably better than 4-2. Also, the rolls with 6's are much better in the variations where Blue has to break the bar point, although they are good anyway.
If Blue rolls a good or average roll and White rolls a bad roll, Blue will lose his market. If White rolls a fair roll, it will still probably be double and take (if not a double, close to one). If White rolls a good roll, Blue will wish he hadn't doubled.
So, putting this all together, we have something like:
Blue loses his market 11 X 36 plus 23 X 5.
Of course the size of the market loss and the amount by which Blue wishes he hadn't doubled is important. However, it does appear that Blue loses his market more often than he wishes he hadn't doubled. Also, the size of the market loss is pretty large when Blue rolls a six, while in many of the variations where Blue wishes he hadn't doubled he is still in decent shape -- just not as good shape as he was before. Therefore, the indications are that doubling is correct. And that is the proper conclusion, at least according to a Snowie rollout. The cubeless equity of the rollout was only .319, but the high volatility in the position makes doubling worthwhile.
Blue has 15 hitting numbers. 6-6 and 5-5 also put him in good shape in the race, and after 3-3 he is in decent shape. With any other of the 18 rolls, he is an underdog. He will either be forced to leave a fatal direct shot, or, if he can play safe, leave both checkers on the 11 point. With White then on roll and owning the cube, White would be a favorite.
Despite all this, it is correct for Blue to double. The reason is that his 15 hitting numbers result in a monstrous market loss, while when he misses he is only a small underdog. Thus, Blue will be throwing away almost a full point if he fails to double and hits, while if he doubles and misses he won't have cost himself anywhere near a full point since he will still have plenty of chances in a close race. Even though Blue's cubeless equity rolls out to only .286 (63.6% winning chances), the large volatility in this position makes doubling mandatory.
Snowie rolled the above position out to a cubelss equity of .472 for Blue. This is probably in the ball park, since while Snowie has difficulty with some kinds of timing and back games, this one looks fairly straightforward. Even with this high equity, it is wrong for Blue to turn the cube. The reason is that the position is very involatile. Virtually nothing can happen on the next exchange which will change things. Even if Blue rolls doubles and clear his midpoint, that has the downside of improving White's timing for the back game. Also, White has enough time now that he can swallow rolling one set of large doubles and still probably be able to hold on. Thus, Blue can't lose his market on the next exchange, or if he does lose his market it will be by a very small amount. Therefore, doubling is incorrect. There simply isn't enough volatility.
Blue's cubeless equity here comes to about .500. White has combined racing and hitting chances, along with his grip on the four point making it potentially awkward for Blue to bear in effeciently. The combination of these chances are sufficient to give White an easy take. It might seem as though it is wrong for Blue to double, since Blue has very few market losing sequences. However, I believe it is correct for Blue to double despite the relative involatility of the position. The key is that nothing really bad can happen on the next exchange. White can't boom out with boxes, because Blue will still have his full prime if Blue doesn't roll one of his good doubles. Thus, Blue can lose his market if he rolls good doubles, but he can never get to a position where he is really sorry that he turned the cube after the next exchange. Therefore the volatility is in his direction, so it looks right to double.
Contrast with the following position:
This one rolled out to a cubeless equity of .488, which seems reasonable. White's position is far stronger than a normal two-point game, because Blue's structure is awkward and Blue's four point is open. White has an easy take. There is some volatility in the position, but most of it goes White's way. After Blue clears the 11 point he will still be facing the same problems unless he rolls a perfect 3-3 joker. In the meantime Blue may have to leave one checker on the 11 point, and White could hit that with a 6-3. Thus Blue can't lose his market by much if at all on the next exchange, but he can lose the game immediately on a bad sequence. Therefore it is correct for Blue to wait until he comes down to one checker on the 11 point and White misses the indirect shot before Blue turns the cube.
This is a very well-known position. It is the result of 6-4 played 24/14, 5-5, flunk. Rollouts have shown that, despite the blitz threat, White has a very clear take. Blue's equity rolls out to around .510, but a ton of that come from gammons, so White's winning chances are quite reasonable -- definitely over 35% cubeless. Owning the cube, White is going to win even more, because he tends to get very efficient recubes when Blue's blitz stalls. Despite this, it would be a blunder for Blue not to double. There is a lot of volatility here. Whether or not Blue rolls a two on his next roll and picks up the blot is very important. Even more important is what White does. If White flunks next roll, Blue will have lost his market by quite a bit regardless of what Blue rolls -- one tempo in a blitz is huge. On the other hand, if White enters decently he will be right in the game and can become the favorite very quickly.
Blue's good-sized lead in the race gives him winning chances of close to 75%, although the gammon threat is obviously very low. Despite this, Blue should hold off doubling. The position is relatively involatile -- not a whole lot is likely to happen next turn. Blue can't clear his midpoint if he rolls boxes, but White can catch up in the race with boxes, so what volatility there is seems to be in White's favor now.
Volatility in backgammon is a difficult thing for us to measure. Yet it is so important when deciding whether or not to double that it must be considered, even if it has to be judged subjectively. Failing to turn the cube in those volatile positions where a lot is likely to happen on the next exchange is very costly.