This article originally appeared in the May 2000 issue of GammOnLine.|
Thank you to Kit Woolsey for his kind permission to reproduce it here.
Cube decisions in blitz situations can be very difficult. It is far to easy
overestimate the danger of the blitz and pass a clear take, or overestimate
the defensive chances and take a clear pass. Unfortunately there is no
yardstick one can use to measure a blitz position and decide whether or
not it is a take. There are a lot of different variables which have to
be put together properly to come up with the right answer.
From the potential doubler's side, it is usually easy. If it even looks like it might be a double, it is almost always correct to double. Blitz positions are so volatile that doubling is often the proper action even if the equity isn't too great. One flunk by your opponent at a key moment and you will lose your market by a mile. In addition, these positions can be difficult to evaluate, so even if you happen to double a bit early there is always the chance that your opponent may make a huge blunder and pass what is a very clear take.
The pass/take decision is much more difficult. What I will do in this article is start with a typical blitz and then make small changes in the position, seeing how these changes effect the equity and cube decisions. I have rolled out each of these positions on Snowie, level 2 rollout, 360 trials with variance reduction. This is equivilant to several thousand regular rollouts without variance reduction, so the sample size should be sufficiently large to give us meaningful results. In addition I believe that Snowie plays this type of position quite well, so the rollout results can be trusted.
This is the base position we will be looking at:
This is a very typical blitz position. Blue has a good three-point board, and White has a man on the bar and another checker on the ace point. On White's side of the board he hasn't done much, and Blue has split his back men. In addition, Blue is well ahead in the race. Blue has the advantage on all fronts and has good gammon chances, but White will still have a lot of counterplay most of the time.
The rollout results for this position came out as follows:
Equity Blue BG Blue gammon Blue win White win White gammon White BG .605 1.6% 31.7% 67.2% 32.8% 6.7% 0.4%Note -- the Blue win column includes all of Blue's wins, not just his wins which aren't gammons. Thus, the Blue win and the White win will add up to 100%. Similarly, the gammon column includes both gammons and backgammons.
The total cubeless equity for this rollout comes to +.602, for which Snowie says White has a bare take.
Wait a minute! A take? Thought we needed 3 to 1 odds to have a take. That would be an equity of .500 or lower. How can this possible be a take? The answer is in the recube potential. To illustrate this, let's first look at a simple racing positions where we won't have to worry about gammons.
A cubeless rollout of this race had Blue winning 76.9% of the time, which indicate that it is a pass since White wins only 23.1% if we don't take the recube into account. However with White owning the cube, Blue won only 74.1% of the time. This was at a settlement level of .55, where if White achieved a position where his cubeless equity was .55 or higher he was immediately credited with the win. This gives White 25.9% wins, which is sufficient to take. Where do the extra 2.8% wins come from? Of the 25.9% White reaches the .55 settlement point, if the game goes to completiion occasionally Blue turns the game back around, and that 2.8% difference between the cubeless and cubeful results represents these turnarounds. Since in this case it is the difference between a pass and a take, it is clear that the recube can be very important.
The power of the recube is a function of the position. For some positions, the recube is of relatively little value. For example:
Here the rollouts have White winning 23.5% cubeless, but only 24.9% owning the cube. What is happening is that almost all of White's wins come from hitting a shot, and when he does hit that shot he has a near claim with the cube. Thus, it is rare that Blue turns the game around once White gets over the .55 settlement point, so the difference between the cubeless and cubeful wins for White is considerably less that in the previous example. White wins more often cubeless than he did in the previous position, but because his cube is usually inefficient has has a close pass while in the race he had a clear take.
For most positions with relative low gammon possibilities, a cubeless equity of about .560 or .570 is usually the break-even point between a pass and a take. The recube potential allows the underdog to take with equity worse that -.500. When there is a large gammon risk, the underdog can actually take with even worse cubeless equity. The reason is that for him to be even close to a take with a lot of gammon danger, he will have to win a larger percentage of the games to compensate. Win percentages in the mid-thirties are common for close pass/take decisions with high gammon risk. And the more games the taker wins, the more games he will reach a point where he has a game-winning recube. Since some of these games would have turned around again, there will be a larger difference in cubeless and cubefull wins than there would be for low gammon positions with similar equity. For this reason, it is not uncommon for a blitz position to be a take even though the cubeless equity is worse than -.600, such as in our actual position.
Another consideration involving blitz positions is as follows: When you survive a blitz, you usually get an anchor and have hit a couple of the enemy checkers. He is often over-extended, while your timing becomes very good. As a result, when things go your way you will slowly move into a winning position rather than getting there by one lucky roll. This means that your recubes are likely to be very efficient, which makes the potential value of the recube even higher than normal.
It is for these reasons that our base position is a take despite the cubeless equity figure. However it is a bare take. Now we will look at the main factors involved in a blitz, make small changes to our base position, and see what happens.
The number of builders in firing range is extremely important in a blitz position. Just one extra builder can easily be the difference between a take and a big pass. Let's modify our base position by moving the third checker on Blue's midpoint down to the 11 point.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .676 1.4% 36.2% 68.6% 31.4% 6.8% 0.5%Quite a difference. Just putting that checker in direct range of the five point is sufficient to turn a close take into a big pass. This is an illustration of something we will be seeing throughout this article -- that every factor counts.
It isn't just a builder on a new point which makes a difference. When you are running a blitz, every checker in the firing range can be the checker which decides the fate of the blitz. For example, let's move that floating checker to the six point. No new numbers to hit with, but more wood in place with which to attack.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .677 1.5% 38.5% 67.5% 32.5% 7.0% 0.4%Again, a big pass for White. Just having that extra bit of firepower on the six point makes Blue's attack that much stronger. Builders make all the difference in the world in a blitz.
As a final example, let's take the spare on Blue's eight point back to the midpoint and see how things look.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .546 1.7% 29.5% 65.6% 34.4% 7.2% 0.4%Just moving that checker out of the firing range was sufficient to turn the position into a fairly easy take. Once again, every builder in the blitzing zone makes a big difference.
INNER BOARD POINTS
When a blitz or a potential blitz is on, every inner board point the attacker owns makes a big difference. Each point gives the opponent several more flunking numbers, which greatly increases the chances for the attack to succeed. To see this, let's give Blue a fourth inner board point (merely the two point), and as compensation let's take one of his builders back to the midpoint so he only has one builder in range.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .692 1.3% 39.2% 68.0% 32.0% 7.0% 0.5%Despite the lack of builders, the fourth inner board point is sufficient to give White a big pass. The big difference is the jump in gammons. Having that fourth point locked up is a big improvement for the attacker. In fact, just the threat of making that fourth point is big.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .654 1.4% 37.9% 66.6% 33.4% 6.7% 0.4%Blue has the two point slotted, but there is no guarantee that he will be able to cover it. Even so, just the threat of making that fourth inner board point is sufficient to push White to a clear pass.
On the other hand, there is a big difference between having a point made and having it slotted. Even if you are a big favorite to make it, sometimes you might not. Also you have to use half of your roll to make it. For example:
Equity Blue BG Blue gammon Blue win White win White gammon White BG .515 1.5% 30.6% 63.9% 36.1% 7.9% 0.5%I gave Blue the builder on the 11 point to compensate for the fact that the four point is made. It isn't nearly enough compensation. Even though Blue has 28 numbers out of 36 which cover the blot on the four point, and three of the non-cover numbers point on White on the ace point, White now has a very easy take as opposed to a borderline take. Those few horror rolls for Blue plus the fact that he has to use this turn to make the four point rather than do something else are sufficient to improve White's chances considerably.
When a blitz is on, the vulnerability of the defender's blots is very important. The danger of getting another checker being put on the bar is a serious consideration. Also important is if he has or is able to make an anchor.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .676 2.4% 35.1% 68.9% 31.1% 7.1% 0.6%Normally White would be quite happy having that builder on his nine point. Not when he is in danger of being blitzed, however. The builder is much more of a target, and is sufficient to turn what was a close take into a clear pass.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .642 1.1% 34.9% 67.7% 32.3% 6.8% 0.4%Normally White would like to be up on Blue's five point, since he would love to make that advanced anchor. Here, however, he is better off hiding on the ace point out of harms way. Blue will be fast to attack the blot on the five point, and White will be under immediate pressure to roll a good number or be wiped off the board. It is not so convenient for Blue to attack deep in his board, so White will have a better chance of making and anchor and staying in the game.
Speaking of anchors, how important are they? Very important! An anchor retains winning chances no matter how bad things look, and it is also good gammon protection. For example:
Equity Blue BG Blue gammon Blue win White win White gammon White BG .606 3.0% 28.7% 67.9% 32.1% 6.4% 0.4%Normally it would be quite costly to have a checker moved back from the midpoint to the enemy ace point. Here the building of the anchor makes it a worthwhile tradeoff. White is still in trouble, but he can't be blitzed by brute force. He still has a close take.
How about playing with only one man back? That is strong. Even though the checker doesn't have anybody to anchor with, it may slip out on its own.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .504 0.8% 28.8% 63.7% 36.3% 6.4% 0.3%With only one man back, the gammon danger goes down and the winning chances go up. Now it is a very easy take.
In our base position, Blue is not committed to a blitz. He hasn't made his deep points, so he can shift to a priming game if that is what the dice dictate. It is another story if Blue has made deep points. Now all he can do is blitz, so he needs a considerably stronger position before White has a pass.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .581 1.1% 37.7% 64.1% 35.9% 8.3% 0.5%White is much more vulnerable to being attacked here than in our base position, since his blot on the two point is threatened by Blue's checkers on the six and eight points. However this doesn't compensate for Blue's over-extension of making the ace point. White has a take.
MEN ON BAR
In a blitz situation, it is not surprising that the number of men the defender has on the bar is a major factor. If he isn't stuck on the bar, he will be able to regroup and work on winning the game. If he is on the bar, however, he can't do anything. Keep in mind that being in danger of being put on the bar isn't the same thing as actually being on the bar. Sometimes your opponent doesn't roll the perfect attacking number.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .532 1.5% 29.4% 65.2% 34.8% 7.5% 0.4%Here White has both his men in, but to compensate we gave Blue another builder on the six point which as we have seen is very important. It isn't nearly as important as having that checker off the bar. White has a clear take.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .654 1.4% 37.9% 66.6% 33.4% 6.7% 0.4%Here White only has his four point slotted with no extra builders, which is far worse than having the four point made as we have seen. However Blue having both back checkers on the bar is worse still. Blue has a solid pass.
So far we have been looking mainly at the attacker's side of the board. This is not the only thing that matters. It is very important to consider what is going on over on the other side of the board. The reason is that if the player being doubled survives the blitz, he needs to have something to work with for offense. Most players recognize that this matters some, but they don't realize just how important it is.
If the person being blitzed has some offense, he will be able to snap back into the game if he survives the blitz. If he has nothing, he still will have a lot of work to do. A slight weakening or strengthening of his position can make a surprising difference.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .730 1.7% 36.5% 70.8% 29.2% 6.3% 0.5%Here we have taken away White's ten point, putting the checkers on the eight point and the midpoint. That might seem like a little thing, but it means a lot. Now White doesn't have any strucutre to get started with even if he survives the blitz. He will be floundering for quite a while. That change in White's position is sufficient to make it a huge pass.
On the other hand, let's strengthen White's position from the original by giving him the five point instead of the ten point. I'll also move a checker back to the midpoint from the six point, which is inferior.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .472 1.5% 28.6% 63.2% 36.8% 8.8% 0.5%The five point makes a huge difference. Now White has an offense ready to go. He can be a contender in a blot-hitting contest, and if he survives the blitz he is in position to take the advantage. White has a huge take. In fact, it is the double which is borderline.
ATTACKER'S BACK MEN
Another important factor is the location of the attacker's back men. Why should this matter when a blitz is being run? The answer is what will happen when the blitz fails. If the attacker has his men out or has an advanced anchor, he will still hold his advantage even when the blitz backfires. However if his back men aren't well-placed, the tide can turn very quickly.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .690 1.3% 34.1% 69.9% 30.1% 6.0% 0.3%Possession of the golden point means that nothing bad will happen to Blue even if his blitz fails. This makes White's winning task a very difficult one. All the dangers are still there, so White has a big pass.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .719 1.5% 33.5% 71.3% 28.7% 5.4% 0.3%Escaping one checker to the midpoint is doubly good for Blue. Not only does this put him farther ahead in the race and he only has one back checker to worry about, but this also gives him another potential attacker for his blitz. That escape is sufficient to turn the position into a huge pass.
Equity Blue BG Blue gammon Blue win White win White gammon White BG .566 1.6% 31.4% 66.0% 34.0% 7.9% 0.5%With Blue's back checkers not being split, he will be just a little bit slower when he has to extricate them. This probably won't matter if the blitz succeeds, but if it fails having the checkers split could mean the difference between Blue getting an advanced anchor or not getting one. This is sufficient to turn the original very close take into a fairly clear take.
The above examples illustrate the effect of the relevant factors for evaluating a blitz position. It doesn't help to memorize positions, since every position is different. However when you have to make a cube decision involving a blitz, you can compare the position to one you are familiar with. Then you can filter the plusses and minuses for each side which have been shown in this article, and come up with a very good evaluation.