This article originally appeared in the June 1999 (Demo) issue of GammOnLine.|
Thank you to Kit Woolsey for his kind permission to reproduce it here.
Is the location of the cube an important factor for play selection?
We have all heard players say "I would have made a different move if
the cube had been in the center" or something like that. Do they have
any idea what they are talking about, or is it just rationalization.
In this article, I'm going to examine this very thorny problem.
First of all, let's see what our goal is. Ideally, we would like to turn the cube at a time when our opponent has a borderline take/pass decision. This is a maximally efficient double. If we reach this point we have for all practical intents and purposes won the game since, if he chooses to take, his equity is the same as if he had passed. If we overshoot this point to where he has a clear pass, it is overkill. We didn't need the extra advantage in the position to lock up the game. If we double before we have reached that point (as we often will, of course), we haven't quite achieved the equity of winning the game since our opponent will have a proper take.
It must be understood that try as we might our Utopian goal of doubling when our opponent has a borderline decision will often not be achieved. Sometimes a joker by us combined with a mediocre roll by our opponent turns a slight advantage into a huge advantage and we overshoot the mark by a lot. Other times the position is so volatile that we must turn the cube even though we are a good way from our opponent's dropping point, since there is too much danger that we will lose our market by a mile after the next exchange. Backgammon is by nature a volatile game. What we can do is attempt to control that volatility to suit our desires regarding the cube location.
The best known type of play alteration occurs when you have access to the cube and have just rolled a joker which will very likely give you a game-winning double on your next turn. Assuming you aren't going to be playing on for the gammon (either because your position isn't strong enough or because you are playing for money with the Jacoby rule in effect and the cube is in the center), it isn't necessary to make the strongest play. All that matters is that you guarantee that your opponent can't get to where he has a take if he rolls his best number. For example, consider the following position:
Without taking the cube into consideration, Blue's strongest play is almost certainly 13/7(2), 11/5(2)*. The bar point is powerful, and will help Blue bear in safely as well as contain White and increase Blue's chances of winning a gammon. True this leaves one checker back in White's board, but that checker is not likely to get into trouble. 21/9, 11/5(2)* is safer, but not nearly as strong.
If we take the cube into consideration, it is another story. Blue has access to the cube, so he certainly plans to double White out on his next turn unless something catastrophic happens. This would be 100% playing for money where undoubled gammons don't count, and would probably be correct if playing a match since this sort of position is dangerous to play on for a gammon. So, what catastrophic thing can happen? Obviously the danger is that White rolls 2-2. If Blue makes his bar point and White rolls 2-2 and makes the four point on Blue's head, it is a new ballgame. Suddenly White has a lot of play, since he could easily contain Blue's back checker and win a priming battle. Blue might still have a double, but it would be marginal, and White would have an easy take.
Let's suppose Blue instead plays 21/9, 11/5(2)*. What can go wrong? Whatever White rolls, Blue will have a powerful double which White will have to pass. In other words, by playing safe Blue guarantees winning the game. The resulting position probably won't be as strong as if Blue makes his bar point, but since Blue has access to the cube that isn't necessary. All that Blue needs is to have a position which is strong enough that White can't take a double. If Blue had previously doubled so White owned the cube then making the bar point would be best, but with Blue having cube access the super-safe play is superior.
On the other side of the coin, what about when we have been playing for a gammon but have just rolled a bad roll. After this bad roll we simply want to be able to claim with the cube next turn if possible. What we want to avoid is giving our opponent a good enough response so he will be able to take our double. Even if it cuts down on our chances of winning a gammon, this conservative approach is usually best. For example:
If White owned the cube, the best play is probably to go for the gusto with 6/2*, 4/2. This could blow up in Blue's face if White rolls a five, but the increased gammon chances make the risk worthwhile. If Blue conservatively plays 13/9, 5/3, his probability of winning a gammon goes down considerably. In addition, if White rolls a two, Blue now has to sweat out bearing in against the two-point anchor.
On the other hand, suppose Blue owns the cube as in the actual diagram. Clearly Blue was playing for a gammon, and it looked like a pretty safe play-on since if Blue covered the blot on the five point nothing could go wrong for quite a while. However Blue rolled badly, and must reassess the situation. If Blue makes the two point and White rolls a five, suddenly White is right back in the game. Blue will not have a cash; in fact, he won't even be strong enough to double. Suppose instead Blue plays safe with 13/9, 5/3. Now only if White rolls the 5-4 joker is Blue in jeopardy. If White enters without hitting, Blue has a strong double which White must pass. Consequently Blue doesn't have to worry about the problems of bearing in against the anchor, since he can claim with the cube before these problems occur. If White flunks, Blue can continue to play on for the gammon with reasonable safety. It seems paradoxical, but sometimes the best way to play on for a gammon is to make a play which is less likely to win a gammon than another play.
The above examples would seem to indicate that we should tend to play safe when we have access to the cube. That is not always the case. Sometimes cube access points to more aggressive play, when a likely result of the aggressive play leads to an immediate efficient double.
The natural play is B/21, 13/7, which looks somewhat better than the cowardly B/15. The interesting alternative is B/21, 7/1*. Offhand this looks a bit inferior, since most of the time White will enter and Blue's position will be too strung out.
Access to the cube may tell another story. If Blue hits loose on the ace point, 25% of the time White will flunk which gives Blue a strong position. Naturally the 75% of the time White enters, Blue will wish he had made a more conservative move. The key is that if White flunks, it appears that Blue will have a very efficient cube -- one which gives White a close pass/take decision. Thus, 25% of the time Blue will have essentially won the game without any wastage. I believe this is sufficient to make the loose hit the superior play. If White owned the cube, B/21, 13/7 looks better.
It should be noted that the feature which makes this sort of play correct is that the 25% of the time White flunks Blue has a very efficient double. If Blue's position were stronger, then the loose hit would be overkill. Blue would be better off making a quieter play and slowly building on his advantage. True he would win the game outright 25% of the time with the loose hit, but that wouldn't compensate for the costs when the hit goes badly. Similarly, if Blue's position were weaker then he should avoid the over-aggressive play. Now even if White flunks Blue wouldn't have a particularly powerful cube, so Blue should just concentrate on making the best overall play.
The same sort of reasoning can apply when you are playing for a gammon. If your position is so strong that even if your opponent rolls well you still have a very strong double, it may be worth taking an aggressive play which increases your gammon chances if you can claim (or nearly claim) with the cube when things go badly.
If White owned the cube, the solid 13/7, 6/5 would be correct. While 8/7, 8/2* generates more gammons, there are too many bad things which can happen if White rolls a two. Blue will be forced to play the game to the end with White owning the cube, and White could easily win.
With Blue owning the cube (having been playing for a gammon, of course), it isn't so dangerous to hit loose with 8/7, 8/2*. The reason is that if White hits back he isn't yet a real contender. Unless White's hit is 2-2 or 2-6 Blue will still have a very powerful cube from the bar, and White will have a bare take at best. This means that on 8 of White's 11 hitting numbers Blue effectively wins the game anyway with the cube, and this is sufficient justification for making the bigger play. Of course if White's board were stronger Blue would not take this risk, since now if White hits Blue would not have a super-efficient double.
Even in positions where a cube turn is not imminent, cube position may dictate one's choice of plays. The player who owns the cube wants to achieve a position where he has an efficient double, while the player who doesn't own the cube wants to avoid such positions.
Clearly Blue will come out with two of the fours; the question is whether he should play 20/12(2) or 20/16(2), 13/9(2). If he were behind in the race, he would maximize contact and stay back on the 16 point. If he were ahead in the race, he would try to make it a pure race and move to the 12 point. Here, however, he is four pips ahead after his play. Since the average roll is a little over eight pips, for all practical intents and purposes the race is dead even.
Suppose Blue stops on the 16 point. If nobody rolls doubles in the next few rolls, one of the players may be squeezed off the anchor and be forced to leave a shot. Since the player squeezed off will probably be ahead in the race, the outcome of the next roll is likely to decide the game. If this scenario occurs, cube ownership is of little value. You won't be able to double before you hit the shot, and after you hit the shot with a powerful board you will be losing your market by a mile.
Suppose Blue runs all the way. Now it is basically a race, since both players will have plenty of rolls to roll doubles or two numbers greater than three. In a race one often creeps up on an efficient cube by slowly gaining in the race. Therefore, cube ownership figures to be quite valuable.
Now we know the answer. Since the race is so close that both plays figure to be equally good, the position of the cube decides the issue. If Blue owns the cube, as in the diagram, he should run all the way with 20/12 and play for the race where his cube ownership figures to be valuable. However if White owned the cube, Blue should hang back on the 16 point. He would then want to make it a one-shot proposition, where ownership of the cube was less valuable to his opponent since it would be less likely that his opponent could get off an efficient double.
What about when your opponent has cube access and is nearing a double. Just as your goal when you have cube access is to maneuver into an efficient double, your goal when your opponent has cube access is to attempt to make his doubles less efficient. Oddly enough, often this means making a play which provokes a double by raising the volatility of the position, when if you had made an alternative play he would not have doubled. For example:
Blue must choose between breaking the anchor with 21/12 or sitting tight but starting the collapse with 7/1, 6/3. The race is fairly close and White's board is not complete, so running is quite reasonable. White will have a clear advantage after either play, but if White doubles it looks like Blue should have a very easy take. In other words, White's double will not be efficient.
If Blue runs, White has no choice but to turn the cube. The volatility just hit the ceiling, and after the next exchange White might be crushing Blue or Blue might be the favorite.
If Blue stays, White definitely will not double. His winning chances might be just as good or even better, but the position will be relatively involatile. Not too much is likely to happen on the next exchange, so White will wait until there is more excitement or until his advantage is greater before turning the cube.
The most likely scenario if Blue stays is that his board will crunch while White's position slowly improves. If this occurs, White will creep up to having a very efficient double. This is not what Blue wants to happen. Blue would rather provoke White into doubling now, when the double is not efficient. Therefore if running and staying are otherwise about equally good plays, which they might well be, Blue should lean towards running in order to prevent White from having an efficient double.
As we have seen from the above examples, there are indeed several common types of positions where it might be correct to alter ones checker play depending upon the location of the cube. You should not overdo this. If one play is clearly superior, it will generally be best regardless of cube location. However if the choice of plays appears close, it makes a lot of sense to let the final deciding factor be the cube.