Playing against a friend this afternoon, and this cube decision was noteworthy enough to take a picture of the board and run an eval later in GNU.

I had to think hard about whether or not to take, so I was very surprised that it was not even a worthy double.

He has 24/36 rolls to cover his 5 point. Even if he doesn't, I have to roll a 5 (11/36), so his chances of geting hit are only 10%. Assuming he doesn't get hit, I will be closed out with only 6 checkers off. Since 8 off with one on the bar is 50/50 (at least, I think I read that somewhere...) I will be far behind by the time I'm able to come in.

So, why a bad double?

This from Stick:

The table below is a summary of one checker closed out at double match point according to XG 4-ply rollouts, with x checkers off for your opponent, and assuming you have the perfect spare distribution of 654.

Number Off––––––Win Percentage
––––––––––––––––––––––––––––
–––0 off––––––––––––––98
–––1 off––––––––––––––96
–––2 off––––––––––––––94
–––3 off––––––––––––––90
–––4 off––––––––––––––85
–––5 off––––––––––––––78
–––6 off––––––––––––––70
–––7 off––––––––––––––61
–––8 off––––––––––––––51
–––9 off––––––––––––––42
––10 off––––––––––––––32
––11 off––––––––––––––24
––12 off––––––––––––––17
––13 off––––––––––––––22
––14 off–––––––––––––– 8

Source: http://www.bkgm.com/rgb/rgb.cgi?view+1543

Stick does not say which points hold the remaining checkers of your opponent, but it is a fair assumption that with 6 borne off, the rest are likely to be found on the lower points.

These semi-short rollouts are easy to do, so if you like, you can verify Stick’s results for yourself.

According the numbers developed by Stick (see preceding post), your opponent’s chance of winning after he closes your checker out is 70%. That number depends on his spares being ideally situated, with one each on the 6pt, 5pt and 4pt.

But your opponent can never achieve an ideal arrangement of spares. He already has an extra checker on the 2pt. In a quick 3-ply rollout, XG found that the difference will cost him about 2% of the games, so his real chance of winning is only about 68.5%, provided, that is, that he can complete the closeout. If he cannot manage to position his last two spares on the 5pt and 6pt, his odds of winning go down further.

After a closeout, you will not be winning any gammons, but you will be winning 31% or 32% of the games. Do you think that your opponent should double even at that stage? (See diagram above.)

There is another point worth discussing about your original position. Did you see the hint I gave above, where I mentioned that after your opponent gets the closeout, you cannot expect to win any gammons?

Well, what about before then? Although the chance seems small, it turns out you are still winning a gammon in about 4% of the games. You can see this in the rollout you posted.

Now, I know Bill Robertie does not like to post rollouts in his messages. All the numbers can be overwhelming to the beginners and intermediates he is targeting here at TwoPlusTwo. But advanced players, and Backgammon Giants, as well, rely on them. Bill makes one or two rollouts for every Problem of the Week, and Bill is not just looking at the “equity” result. He delves deeply into the these difficult and disorienting numbers.

In terms of marketing TwoPlusTwo to the widest audience, there is no doubt that Bill is right to do without rollouts. But do you see the irony? Bill himself depends on rollouts as part of the data that go into every one of his POTW analyses (or, at least, the confirmation of said analyses), but he doesn’t want his readers to think they have to trouble themselves with the confusing results! Well, Bill is right. If you are trying to teach people how to add 2 + 2, you cannot begin your lesson with a treatise on the Fundamental Theorem of Calculus.

That said, I am foolish enough to take the plunge anyway. The next section describes how to read a rollout. It is intended for the uninitiated. If you already know how to find the rate of gammon wins and losses in a rollout, you can skip ahead to the following section.

How To Read the Gammon Rate from a Rollout
For your position (as the player on the bottom), a key statistic is the rate at which you win a gammon. In the rollout, it is displayed as one of the cubeless figures. It is called cubeless because it gives the percent of games in which you will win a gammon when the doubling cube is not is use.

Actually, there are 6 cubeless statistics that are displayed in a GNU Backgammon rollout. You can see them in two different places in your original screen shot. First are the 6 numbers on the line right below the “No Double” equity of +0.138. The second appear under the line giving the “Double, take” equity of +0.068.

In each line, the first three numbers give winning percentages for the player who is on roll or who is offering a double. The second set of three numbers gives winning percentages for his opponent. In order, the three numbers are:

1. Percentage of games a player won in the rollout, including single wins, gammons and backgammons.
2. Percentage of games in which a player won a gammon or backgammon in the rollout.
3. Percentage of games in which a player won a backgammon in the rollout.

In the position of this thread, the first three numbers in each line give the winning percentages for the player on top, i.e, the player who is contemplating a double. Looking at the line beneath “No Double,” the first number is 0.590. That means that the player on top won 59.0% of the games in the rollout (after choosing not to double). The second and third numbers are both 0.000. That means that the player on top won no gammons and no backgammons.

The next three numbers give the winning percentages for the player who is not on roll, i.e, the player on the bottom. He won 0.410, or 41.0%, of the time. That makes sense, because 59.0% + 41.0% add up to 100%. The fifth number on the line is 0.041, or 4.1%. This is the rate at which the player on the bottom won a gammon in the rollout. Somewhat surprisingly, the player on the bottom can still win a backgammon. The third number 0.001, indicates that this happened about 1 time in a thousand in the rollout.

Putting It All Together
Woe to your friend who doubled. Although he is the favorite, it is way too early to double. He is winning only about 59% of the games. But it is worse. The fact that you win a gammon in 4% of the games reduces his effective win rate even further.

In terms of the take point, i.e, the minimum percentage of wins a player needs to take a double, the gammons won by the taker allow him to adjust his take point downward. Normally, the live cube take point in unlimited (cash) games is around 22%. Because of the net gammon difference of 4% in this position, that number can be adjusted downward to 20%. That’s because the gammon price in a cash game is 0.5, and 0.5 * 4% = 2%.

In a smoothly changing, non-volatile position, the doubler should be trying to get as close as possible to his opponent’s take point without exceeding it. Thus, in this position, he should not double unless his winning chances approach 80%. In a volatile game, where a significant number of rolls can cause the doubler to lose his market, he should double earlier, possibly much earlier.

At first glance, it is tempting to say that is the case here. In fact, Teddosan made that argument in his initial post. With 24 rolls that close the board, it is easy to think that the player on top should double now, fearing market loss when he covers his blot. But that is where this discussion began. We saw from the data in Stick’s table, that this is not even a double, let alone a pass, even after the board is closed.

Hope that helps. Let me know if you think my explanation is too technical.

Great post, learned a lot here. Can you explain why the live cube in cash games is around 22%? Why not 25%? And does that differ in tournament games?

The answer is something called recube vig or recube vigorish. Here is how it works:

A dead cube is one that can never be effectively offerred. Dead cubes occur on the 2nd-to-last roll of money games, and also in positions where the game will be decided on the next roll. In match play, the cube dies when it reaches a level that will give the match to its owner should he win the game. Leading 5-4 in a match to 7, for instance, and holding a cube at 2, there is no profit to be had in doubling. The cube is dead.

In unlimited games, the take point for a cube that will be dead after the take is 25%. When your winning chances are 25%, you can take or drop as you like. You expected loss will be the same either way.

A live cube is one that can still be profitably offerred in the event it becomes favorable to do so. In unlimited games, for instance, the taker will often reach a point where he can force a recube on his opponent. Instead of winning those games on a 2-cube, he will win most of them on a 4-cube. He will also lose some of them, but less often than he wins. On average, the taker’s ability to offer a recube translates into increased winnings (or, rather, a net smaller expected loss). The difference between a taker’s equity on a live cube, and his equity on a dead cube is called recube vigorish.

Here is the definition from Backgammon Galore!

Recube Vigorish (Vig)
The value of cube ownership to the player being offered a double; the additional equity that comes from being the only player who may redouble.

Take points in match play are significantly different than in unlimited games, both for cubes that are alive, and those that are dead. In general, every score has a different take point.

In XG, you can use the command Analyze -> Cube Information -> Market Window to play around with scores, gammons and take points.

Stick discusses the 20% take point of the 5-away, 5-away match in the free video lesson now up on his web site. If you plan to watch it, don’t tarry. It won’t be up for very much longer.

Hope this helps.

Thanks, Mike. That's a great answer. By the way, I don't think that it's too technical; I comprehend these numbers. My problem is usually the why.

For this particular problem, I was missing two important pieces of information:
1) I assumed that the winning probability dropped off much more quickly with less checkers off (Stick's chart). I knew that 8 checkers off is around 50%, but I assumed that 6 checkers off would put me at more like 10-15% rather than 30%.
2) I failed to consider that, even given a closeout, it's not a given that his home board won't get awkward resulting in a non-optimal bear in and, potentially, gaps and/or blots.

I also didn't figure in my own gammon chances, but that was just me being lazy...

Thanks again for your detailed explanation; great info.

Ted

As ever, great stuff Mike.

This position makes me feel nostalgic to the days I used to charge into these spots cube flying - psychologically completely convinced I was near 100% to win - only to feel the grim shudders of terror and the hopeless unfairness of it all creep up my spine as I watched it all dissapear.

In spite of tmike's prozaic talents, i didn't read previous posts. Why should I. When white has got 6 off, and black has got an ideal closed board, it's no double, according to 501, problem 386, p.296. Minus the small probability that the board is not closed, blot is hit, followed by some double throw, which will compensate for the 2cube.

So, rather than learn the thought processes that tmike has presented and apply them to future games, you are recommending that I simply memorize each of the 501 backgammon problems in the book?

I don't think he meant that. (I certainly wouldn't recommend trying to memorize the book.) The point of '501' was to give the reader a sense of the different kinds of backgammon positions and what you're trying to do in each type of position. Backgammon is a complicated game with a lot of different types of positions, and '501' is really just an overview of the sort of things that you should be thinking about in each type.

Having said that, there are certain positions that arise very frequently and where it's worth memorizing some simple fact or formula. The best example is a straight race with a pip count in the 50 to 110 range. These come up all the time, and the 8-9-12 rule will almost always lead you to the correct doubling decision. (Give an initial double if you're up at least 8% in the pip count, give a redouble if you're up at least 9%, and take if you're down no more than 12%.)

Another example is the 'Rule of 5', which is what Mike is talking about. It applies to positions of this sort:

> Your opponent is bearing off against contact (usually an ace-point game).

> He bears off some number of checkers.

> You finally hit him, close your board, and get ready for your own bearoff.

> When do you redouble?

The 'Rule of 5' states that, in a normal position, you can double and he can take if he bore off 5 men before getting hit. If he bore off 6 or more, you don't have a double and he has an easy take. If he only got 4 men off, you can double and he has a pass.

The Rule of 5 has a lot more exceptions than the 8-9-12 rule, depending on how well-distributed your spare checkers are when you are finished bearing in, and whether he has gaps in his inner board. But it covers a lot of positions and it's very handy.