When might it be correct to double? In principle, it could be correctany time our winning chances are greater than 50%, since all turning thecube amounts to is doubling the stakes (I am assuming money play forthis article). The opponent can take if our winning chances are up to75%. These numbers are assumed to be taking gammons and recube potentialinto account. Thus, it might be a double and a take any time theperson on roll has greater than 50% winning chances but less than 75%winning chances. This area between 50% and 75% is called the doublingwindow.

In practice, it is generally not right to double when near the low endof the doubling window. It isn't because our winning chances aren'tgood -- 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 timewe will have an opportunity to double next roll and our opponent willhave a take even if things go well for us. If things go badly, thenof course we would rather not have doubled. In addition, if we own thecube then we want to be even more cautious about doubling, since that willgive 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 fromnot doubling is losing one's market. The more likely it is that we willlose our market, the more important it is to turn the cube. That iswhere 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 thedoubling window. However, if not much is likely to happen, then we willwant 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.

2 7

White money game Blue

Blue gets off 19 rolls out of 36, or about 52.8% of the time. Even thoughthis is way down in the doubling window, it is still correct for Blueto double. The reason is that the volatility couldn't be higher -- thisis the last roll of the game. There is no worry about a recube, and nopotential for Blue to turn the cube later. This is it! Therefore, Blueshould double simply because he is a favorite.

On the other side of the coin, we have:

41 93

White money game Blue

A rollout of this position has Blue winning 77.5% of the time, for a cubelessequity of .550. If White didn't have recube potential that would amount toa pass, but due to White's recube potential he has a bare take -- thatrecube will give him just enough extra wins to get above the 25% mark. Ifwe accept these results, then it is clear that Blue should not double eventhough he is right at the top of the doubling window. The reason is thatthis position is as involatile as can be. Nothing Blue can roll willhave any significant effect on Blue's winning chances, and of courseWhite's roll doesn't matter either. It is absolutely impossible forBlue to lose his market on the next exchange, therefore he cannot gainby doubling.

Players have often asked me how good their winning percentage needs to beto double. That question can't be answered. While obviously winning chancesare very important, volatility and the timing of the critical rolls isalso vital. Consider the following two positions:

4 10

White money game Blue

Blue's cubeless winning chances are only 59%. Despite this, it isquite correct for Blue to double. The reason is that the result of thegame is very likely to be decided on the next exchange. If Blue rollsa five or a six he will win, unless White rolls doubles in which casehe loses. If Blue misses, White will be able to double and Blue willhave to pass. Only in the variations where Blue rolls 2-1 or 1-1 andWhite 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 adecent favorite it is correct for him to turn the cube.

Contrast with:

10 8

White money game Blue

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, butthe main volatility is in White's direction. If Blue misses, White willhave a very powerful double of his own. Blue would have a take, butBlue's pass take decision would be so close that for all practical intentsand 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 willnow be left with two checkers on the two point and needing a non-ace towin the game. This common scenario results in a perfectly efficientcube for Blue, with maximum volatility and White having a borderlinetake (or possibly a close pass if White has rolled badly). The only timeBlue really regrets not having doubled is when he rolls doubles and getsall 4 men off this roll, and that is more than compensated for by thetimes 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)
Good: 1-1, 3-2, 4-2, 5-2, 6-2, 4-3, 5-3, 6-3, 5-4, 6-4, 6-5 (21 rolls)
Awful: 2-1, 3-1, 4-1, 5-1, 6-1 (10 rolls)

Suppose Blue doubles (as opposed to not doubling). If he rolls a greatnumber 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 tolosing (almost) 1 point. The reason I say (almost) is that White willdouble and Blue has a bare take, so his equity is slightly better thanthe value of the cube, but the take is so close that it isn't muchbetter.

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 himselfa point by doubling.
b) If White rolls a terrible number like 2-1, then Blue will have gainedby doubling since he will have lost his market. The market loss isn'thuge (and it is less if White's terrible number is 3-1). On other Whiterolls (even 4-1, 5-1, or 6-1) White will have a bare take because of hispowerful recube potential if Blue misses.
c) If White rolls an average number (not big doubles, but not an ace), Bluewill double and White will have a take. In that scenario, it won't matterwhether Blue has doubled the previous turn or not. The resulting positionis the same -- Blue is on roll needing a non-ace with White holding a 2-cube.

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 muchcancels 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 1296possible scenarios after we roll, he rolls. As we have seen, thereare three possibilities:

1) It is now double and pass (a market has been lost). If that happens,failing to double cost some equity.
2) It is now double and take. If that happens, it didn't matter whetheror not you doubled the previous roll.
3) It is now not a double. If that happens, doubling cost some equity.

All you have to do as add up the equity costs and gains from doublingover the 1296 rolls, and out pops the answer. Simple for the bots whichdo this at lightning speed (in fact, this is exactly what they do whenmaking a 3-ply analysis), but for mere mortals it is pretty impossible.For most positions we pretty much have to rely on our judgment andexperience. Still, it is sometimes possible to break down the rollsinto various categories and come up with a reasonable estimate. Forexample:

101 108

White money game Blue

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)
Good rolls: 1-1, 2-1, 3-1 (5)
Average rolls: 5-5, 5-4, 5-3, 5-2, 5-1, 4-3, 4-2, 4-1, 3-2, 2-2 (18)
Terrible rolls: 4-4, 3-3 (2)

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 powerfulcube, which Blue probably has to pass -- if Blue does have a take, itmust 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 lookat White's rolls. They are:

Good rolls: 6-6, 6-5, 6-4, 6-3, 6-2, 6-1, 5-5, 5-1 (14)
Fair rolls: 5-2, 5-3, 5-4, 4-2, 4-1, 3-2, 3-1, 2-1, 1-1 (17)
Bad rolls: 4-4, 3-3, 2-2, 4-3 (5)

This breakdown isn't absolute, since obviously 5-2 is considerablybetter than 4-2. Also, the rolls with 6's are much better in thevariations where Blue has to break the bar point, although theyare 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 willstill 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.
Blue wishes he hadn't doubled 2 X 36 plus 23 X 14.
The rest it doesn't matter whether Blue doubled or not.

Of course the size of the market loss and the amount by which Bluewishes he hadn't doubled is important. However, it does appear thatBlue loses his market more often than he wishes he hadn't doubled.Also, the size of the market loss is pretty large when Blue rollsa six, while in many of the variations where Blue wishes he hadn't doubledhe is still in decent shape -- just not as good shape as he was before.Therefore, the indications are that doubling is correct. And thatis the proper conclusion, at least according to a Snowie rollout. Thecubeless equity of the rollout was only .319, but the high volatilityin the position makes doubling worthwhile.

58 69

White money game Blue

Blue has 15 hitting numbers. 6-6 and 5-5 also put him in good shapein the race, and after 3-3 he is in decent shape. With any other ofthe 18 rolls, he is an underdog. He will either be forced to leavea fatal direct shot, or, if he can play safe, leave both checkers onthe 11 point. With White then on roll and owning the cube, Whitewould be a favorite.

Despite all this, it is correct for Blue to double. The reason is thathis 15 hitting numbers result in a monstrous market loss, whilewhen he misses he is only a small underdog. Thus, Blue will be throwingaway almost a full point if he fails to double and hits, while if hedoubles and misses he won't have cost himself anywhere near a full pointsince he will still have plenty of chances in a close race. Eventhough Blue's cubeless equity rolls out to only .286 (63.6% winning chances),the large volatility in this position makes doubling mandatory.

166 90

White money game Blue

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 withsome 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. Thereason is that the position is very involatile. Virtually nothing canhappen on the next exchange which will change things. Even if Blue rollsdoubles and clear his midpoint, that has the downside of improvingWhite's timing for the back game. Also, White has enough time now thathe can swallow rolling one set of large doubles and still probablybe able to hold on. Thus, Blue can't lose his market on the nextexchange, or if he does lose his market it will be by a very smallamount. Therefore, doubling is incorrect. There simply isn't enoughvolatility.

118 111

White money game Blue

Blue's cubeless equity here comes to about .500. White has combinedracing and hitting chances, along with his grip on the four pointmaking it potentially awkward for Blue to bear in effeciently.The combination of these chances are sufficient to give White aneasy take. It might seem as though it is wrong for Blue to double,since Blue has very few market losing sequences. However, I believeit is correct for Blue to double despite the relative involatility ofthe position. The key is that nothing really bad can happen on thenext exchange. White can't boom out with boxes, because Blue willstill 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 cannever get to a position where he is really sorry that he turned thecube after the next exchange. Therefore the volatility is in hisdirection, so it looks right to double.

Contrast with the following position:

130 77

White money game Blue

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, becauseBlue's structure is awkward and Blue's four point is open. Whitehas an easy take. There is some volatility in the position, butmost of it goes White's way. After Blue clears the 11 point he willstill 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 marketby much if at all on the next exchange, but he can lose the gameimmediately on a bad sequence. Therefore it is correct for Blue towait until he comes down to one checker on the 11 point and Whitemisses the indirect shot before Blue turns the cube.

158 147

White money game Blue

This is a very well-known position. It is the result of 6-4 played24/14, 5-5, flunk. Rollouts have shown that, despite the blitzthreat, White has a very clear take. Blue's equity rolls out toaround .510, but a ton of that come from gammons, so White's winningchances are quite reasonable -- definitely over 35% cubeless. Owningthe cube, White is going to win even more, because he tends to getvery efficient recubes when Blue's Blitz stalls. Despite this,it would be a blunder for Blue not to double. There is a lot ofvolatility here. Whether or not Blue rolls a two on his next roll andpicks up the blot is very important. Even more important is whatWhite does. If White flunks next roll, Blue will have lost his marketby quite a bit regardless of what Blue rolls -- one tempo in a blitzis huge. On the other hand, if White enters decently he will beright in the game and can become the favorite very quickly.

144 124

White money game Blue

Blue's good-sized lead in the race gives him winning chances of closeto 75%, although the gammon threat is obviously very low. Despitethis, Blue should hold off doubling. The position is relativelyinvolatile -- not a whole lot is likely to happen next turn. Bluecan't clear his midpoint if he rolls boxes, but White can catch up inthe race with boxes, so what volatility there is seems to be inWhite's favor now.

Volatility in backgammon is a difficult thing for us to measure. Yetit is so important when deciding whether or not to double that itmust be considered, even if it has to be judged subjectively.Failing to turn the cube in those volatile positions where a lot islikely to happen on the next exchange is very costly.