| 146 135 | Woolsey 8 11 point match Readers 9 |
White on roll. Cube action?
Clearly White has a strong double. A good positional advantage along withplenty 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 edgeof White's blockade and is threatening to escape. White currently hasonly a Two-Point board with the five point slotted. Unless White rollsdoubles the best he can do is make the five point and hit loose, so Bluewill have some kind of shot. White might not be able to hit at all, whichgives Blue a chance to escape. If Blue does get away safely he is in fineshape despite his awkward structure up front, since White still has twoback checkers to extricate. For money, it would be quite reasonableto 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 bythe 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 himthose two points. We are talking about only a 2-cube, not somelarge cube. How can the match score turn this from a solid money taketo a huge pass?
In order to understand what is going on, it is necessary to examing thedynamics 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 andwin, 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) inorder to gain 3 points (from losing 1 to winning 2). Therefore, if youcan win the game 25% of the time or more, you have a take.
The above analysis does not include two factors -- gammons and recubepotential. If there is gammon danger you risk losing 4 points insteadof 2 points, so you need a higher winning percentage to take. Therecube potential means you need a lower winning percentage to take,since you will be able to win some games with the recube which wouldotherwise have been lost had they been played to conclusion.
For starters, let's look at a position with no gammon danger -- a straightrace.
| 109 97 | White money game Blue |
Blue doubles. Zero gammon danger. A rollout resulted in White winning21.6% 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 gamesbecause of the recube potential.
If White could be sure he could time his redouble perfectly, he could takewith 20% winning chances. Let's see why. In order for White to winplayed 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 win20% overall, he will get to that 80% mark 25% of the time -- and thenlose 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 getsto 80%, since at that point he redoubles and Blue has a borderlinetake/pass -- which is equivalent to a win for White.
Of course, White won't always get to time his redouble perfectly. Sometimeshe will boom out boxes and lose his market by a fair amount. Sometimes hewill 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 getthat extra 5%. It turns out that the break-even point in races is about21.5%, which comes to -.570 in cubeless equity. White will win approximatelyan additional 3.5% of the games because of his recube potential. This comesout to about an additional 1/6 wins over the cubeless result. This 1/6 figure is fairly accurate for most positions, and unlessthere is a strong indication otherwise this figure can be used.
When might this figure change? There are some positions where the recubeis quite likely to be inefficient. For example:
| 141 105 | White money game Blue |
A cubeless Snowie rollout has White winning 22.3% of the games, with both sideswinning 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 onthat 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 tohave to hit a shot. If he gets a shot, he won't be able to turn thecube then. He will have to wait until after he hits the shot. Hisboard figures to be perfect, so if he hits the shot and then doublesBlue will have a huge pass. Therefore, White won't get anything closeto an efficient cube. When he wins, he will almost always lose hismarket by a mile. This means that the 21.5% winning chances which wouldbe sufficient for a race shouldn't be sufficient here. White needsclose to 25% winning chances to justify the take, since the value of therecube is relatively small. If fact, a cubefuls rollout (with the sameseed, therefore the same win percentage), came out to a tiny pass.
On the other side of the coin, there are positions which figure to havebetter than average recube efficiency. For these positions, it may beright to take with lower winning chances. For example:
| 60 93 | White money game Blue |
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 chancethat 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 thatWhite has a pass.
Despite all this, White has a take. The possible losses in the variationswhere Blue misses are an illusion. White can redouble, and his redoubleis quite efficient -- Blue has to pass when he still has quite reasonablewinning chances. Thus, White wins immediately when Blue doesn't hit, soall he has to do is eke out 1 win in 28 from the bar -- and that he canjust about do with the help of his recube potential.
How do gammons fit into the picture? Obviously the greater danger ofbeing gammoned, the higher the winning percentage needs to be in orderto justify taking. What is not commonly realized is that in positionswith high gammon potential, one can take a cube with worse cubelessequity that the -.570 with is generally the break-even point forraces or normal positions.
Why can we take with lower equity in gammonish positions? It is becauseof the recube potential. In order for the equity to be such that we areclose 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 forthe 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 tobegin with, that makes our extra wins due to the recube higher still.To see this, let's look at an early blitz position:
| 158 147 | White money game Blue |
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 winningan additional 6% of the games. So, let's subtract 6% from Blue'ssingle 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. Thisillustrates 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 andgammons let's see what percentage of the time Blue would need to winin 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 towin 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 thevalue of the recube in gammonless positions is large enough so thatone 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 gammonsinto 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 doubleand a huge take for money becomes a pass at this match score.
Thirdly, we do have to take gammons into account. There is plenty ofgammon danger in the position, since Blue has no anchor and White maybe able to attack the lone back checker. I'm going to guess that 1/3 ofWhite's wins will be gammons (that is probably on the low side). Ifthat is the case, 2/3 of the time White wins he wins a single game andis ahead 2 away, 1 away (70% equity), while 1/3 of the time he winsa gammon and wins the match (100% equity). This comes out to an averageequity of 80% for White, or 20% for Blue. Thus, taking gammons intoaccount, we can see that Blue is risking 30% in order to gain 50%, so heneeds to win 30/80 or 37.5% of the time to justify a take. I think itis 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 thelow side, which is no surprise -- that probably means that Blue has towin 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 haveseen, for money one can take high risk gammon positions with equityover .600 due to the recube, so for money this would, in fact, be aquite solid take. At the match score, taking would be an error of sucha large magnitude you don't even want to think about it.
Hopefully this discussion helps to clarify how the leader must adjusthis normal cube decisions at this tricky match score. The match scorereally does make a large difference.