This article originally appeared in the November 2003 issue of GammOnLine.
Thank you to Kit Woolsey for his kind permission to reproduce it here.

Two Away Three Away

By Kit Woolsey
The cube action by several readers in the online match indicated that many players have serious difficulties making the necessary adjustments to their cube decisions at this match score. This was the position:




Woolsey 8

11 point match

Readers 9

White on roll. Cube action?

Clearly White has a strong double. A good positional advantage along with plenty of threats. It is the take which is the issue.

Blue has play. He is ahead in the race. His back checker is at the edge of White's blockade and is threatening to escape. White currently has only a two point board with the five point slotted. Unless White rolls doubles the best he can do is make the five point and hit loose, so Blue will have some kind of shot. White might not be able to hit at all, which gives Blue a chance to escape. If Blue does get away safely he is in fine shape despite his awkward structure up front, since White still has two back checkers to extricate. For money, it would be quite reasonable to take this cube.

At the match score, it is a pass. Not a small pass. A monstrous pass. Taking this cube would have been by far the largest blunder made by the readers the entire match.

It may seem surprising that the match score can make that much difference. Yes, Blue needs only two points to go, but winning the game gets him those two points. We are talking about only a 2-cube, not some large cube. How can the match score turn this from a solid money take to a huge pass?

In order to understand what is going on, it is necessary to examing the dynamics of the cube. For starters, let's look at money games. If you are doubled and you pass, you lose one point. If you take and win, you win two points. If you take and lose, you lose two points. Thus, you are risking one additional point (from losing 1 to losing 2) in order to gain 3 points (from losing 1 to winning 2). Therefore, if you can win the game 25% of the time or more, you have a take.

The above analysis does not include two factors -- gammons and recube potential. If there is gammon danger you risk losing 4 points instead of 2 points, so you need a higher winning percentage to take. The recube potential means you need a lower winning percentage to take, since you will be able to win some games with the recube which would otherwise have been lost had they been played to conclusion.

For starters, let's look at a position with no gammon danger -- a straight race.





money game


Blue doubles. Zero gammon danger. A rollout resulted in White winning 21.6% of the games. Yet, Snowie evaluated this as a borderline take. That means that Snowie thinks White will win an extra 3.4% of the games because of the recube potential.

If White could be sure he could time his redouble perfectly, he could take with 20% winning chances. Let's see why. In order for White to win played to conclusion, he has to get up to a point where he is 80% to win. Once he gets there, he will win 4/5 of the time. Thus, in order to win 20% overall, he will get to that 80% mark 25% of the time -- and then lose 1/5 of the games from there because he is only 80% to win. However, if he has recube potential he will always win when he gets to 80%, since at that point he redoubles and Blue has a borderline take/pass -- which is equivalent to a win for White.

Of course, White won't always get to time his redouble perfectly. Sometimes he will boom out boxes and lose his market by a fair amount. Sometimes he will roll well enough so he has a bare cube, but Blue still has a clear take. In these cases, White's redouble is not perfectly efficient. Therefore, White can't take with 20% cubeless winning chances, since he won't get that extra 5%. It turns out that the break-even point in races is about 21.5%, which comes to -.570 in cubeless equity. White will win approximately an additional 3.5% of the games because of his recube potential. This comes out to about an additional 1/6 wins over the cubeless result. This 1/6 figure is fairly accurate for most positions, and unless there is a strong indication otherwise this figure can be used.

When might this figure change? There are some positions where the recube is quite likely to be inefficient. For example:





money game


A cubeless Snowie rollout has White winning 22.3% of the games, with both sides winning under 2% gammons as would be expected. Given the above analysis, it might appear that this is a take, and Snowie thinks it is based on that rollout. I believe that is wrong. How does White win this game? The race is virtually out of sight, so he is almost certainly going to have to hit a shot. If he gets a shot, he won't be able to turn the cube then. He will have to wait until after he hits the shot. His board figures to be perfect, so if he hits the shot and then doubles Blue will have a huge pass. Therefore, White won't get anything close to an efficient cube. When he wins, he will almost always lose his market by a mile. This means that the 21.5% winning chances which would be sufficient for a race shouldn't be sufficient here. White needs close to 25% winning chances to justify the take, since the value of the recube is relatively small. If fact, a cubefuls rollout (with the same seed, therefore the same win percentage), came out to a tiny pass.

On the other side of the coin, there are positions which figure to have better than average recube efficiency. For these positions, it may be right to take with lower winning chances. For example:





money game


White has a lot of ways to lose this game. Blue has 28 hitting numbers, and if Blue hits White needs a miracle win from the bar. If Blue misses, White still has to get past Blue's anchor, and there is always the chance that Blue could win a freak race. A rollout had Blue winning 78.5% of the time. That along with 2.2% gammons would seem to indicate that White has a pass.

Despite all this, White has a take. The possible losses in the variations where Blue misses are an illusion. White can redouble, and his redouble is quite efficient -- Blue has to pass when he still has quite reasonable winning chances. Thus, White wins immediately when Blue doesn't hit, so all he has to do is eke out 1 win in 28 from the bar -- and that he can just about do with the help of his recube potential.

How do gammons fit into the picture? Obviously the greater danger of being gammoned, the higher the winning percentage needs to be in order to justify taking. What is not commonly realized is that in positions with high gammon potential, one can take a cube with worse cubeless equity that the -.570 with is generally the break-even point for races or normal positions.

Why can we take with lower equity in gammonish positions? It is because of the recube potential. In order for the equity to be such that we are close to a take in the first place when we are getting gammoned a lot, we will have to win a much higher percentage of games to compensate for the lost gammons. And this is where the recube formula comes into play. Remember that we will win an additional 1/6 of the games, roughly, because of the recube potential. If our win percentage is higher to begin with, that makes our extra wins due to the recube higher still. To see this, let's look at an early blitz position:





money game


A rollout produced the following cubeless results:

Blue wins backgammon 1.6%
Blue wins gammon 36.3%
Blue wins single 28.1%
White wins backgammon 0.3%
White wins gammon 7.5%
White wins single 26.2%

This comes to a cubeless equity of .635. However, according to Snowie, this is a borderline pass/take.

Let's see if we can understand why. White won the game 34% of the time. According to our 1/6 value for the recube, that means he will be winning an additional 6% of the games. So, let's subtract 6% from Blue's single wins, add them to White's, and now we have:

Blue's equity = (1.6% X 3) + (36.3% X 2) + (22.1% X 1) - (0.3% X 3) - (7.5% X 2) - (32.2% X 1)

Which comes to .514

If this adjusted equity were less than .500, White would have a take. This illustrates how the power of the recube figures into this sort of position.

Now, back to the original position where the match score is 3 away, 2 away. There are several differences. First of all, ignoring recubes and gammons let's see what percentage of the time Blue would need to win in order to justify taking.

Blue passes: 2 away, 2 away, 50% equity
Blue takes and wins: Wins match, 100% equity
Blue takes and loses: Behind 1 away, 2 away, 30% equity

Blue would be risking 20% in order to gain 50%, so he would have to win 20/70 or about 28.5% of the time. That's a lot worse than the 25% he would need for money.

Secondly, Blue has no recube potential. We have already seen that the value of the recube in gammonless positions is large enough so that one needs to win about 21.5% of the time cubeless in order to have a take. Here, Blue has to win 28.5% of the time cubeless (not taking gammons into account) in order to have a take. That is a very signifant difference, enough so that in a straight race what might have been a marginal double and a huge take for money becomes a pass at this match score.

Thirdly, we do have to take gammons into account. There is plenty of gammon danger in the position, since Blue has no anchor and White may be able to attack the lone back checker. I'm going to guess that 1/3 of White's wins will be gammons (that is probably on the low side). If that is the case, 2/3 of the time White wins he wins a single game and is ahead 2 away, 1 away (70% equity), while 1/3 of the time he wins a gammon and wins the match (100% equity). This comes out to an average equity of 80% for White, or 20% for Blue. Thus, taking gammons into account, we can see that Blue is risking 30% in order to gain 50%, so he needs to win 30/80 or 37.5% of the time to justify a take. I think it is pretty clear that Blue isn't going to be able to do this.

A Snowie rollout had White winning 67.0% of the games, with 29.1% gammons. It looks like my 1/3 of White's wins being gammons was a little on the low side, which is no surprise -- that probably means that Blue has to win 38% of the time to have a take. Blue won only 33% of the games, which isn't even close.

It is to be noted that White's cubeless equity came to .558. As we have seen, for money one can take high risk gammon positions with equity over .600 due to the recube, so for money this would, in fact, be a quite solid take. At the match score, taking would be an error of such a large magnitude you don't even want to think about it.

Hopefully this discussion helps to clarify how the leader must adjust his normal cube decisions at this tricky match score. The match score really does make a large difference.

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