This article originally appeared in the December 2000 issue of GammOnLine.|
Thank you to Kit Woolsey for his kind permission to reproduce it here.
The question of whether or not to play on for a gammon can be a very
perplexing one. First of all, it must be 100% clear that your opponent
has a pass -- if there any chance that he might have a take or even that
he might think that he has a take, doubling is definitely correct. Since
if there is any question about the take the equity gain from playing on
will be small anyway even if playing on is theoretically correct, the
potential gain from an incorrect take (or, more seriously, when your opponent
actually has a take) makes doubling the proper action. So for this article
we will only be examining positions for which there is no question that it
is a pass.
For money, the mathematics is simple. If you double and he passes, you collect the value of the cube. If you play on and win the gammon, you get double the value of the cube. If you play on and lose the game, you lose the value of the cube. For example, let's suppose you are sitting on a 2-cube. If you play on and are right you gain 2 points (from +2 to +4). If you play on and are wrong you lose 4 points (from +2 to -2). Therefore, you are laying 2 to 1 odds when you choose to play on for a gammon.
In a match, these odds may be different due to the match score. Also in a match you can play on with the cube in the center, since undoubled gammons count. For money most players use the Jacoby rule which says that undoubled gammons don't count, so you would only consider playing on when you own the cube. For this article, unless other wise stated we will be assuming either money play or match play early in a long match (so the odds are about the same as they would be for money).
First, let's look at a true play-on problem where everything will hinge on the next exchange:
The issue is quite simple. If Blue clears the four point or if Blue leaves a shot and is missed he is certain to win and will have some chance to win a gammon. If Blue is hit, he is likely to lose the game. If Blue's gammon chances are twice as great as his losing chances he should play on, otherwise he should cash.
How often will Blue get hit? He leaves a shot on 8 numbers (3-1, 4-1, 5-1, and 6-1), and is hit a little less than 1/3 of the time he leaves a shot -- roughly 2.5 out of 36 rolls. If Blue is hit he will have somewhere from 3 to 5 men of depending on what he rolled to leave the shot and how he played it (how to play something like 3-1 is an interesting problem in itself, but we won't go into that here), so Blue can still win maybe 20% of these games, leaving him 2.0 losses out of 36 rolls. Note that if the cube were in the center White would be able to cash if he hits the shot, so Blue wouldn't get that extra 20%.
According to our estimates, Blue will lose about 1/18 of the time. Thus if he can win a gammon 1/9 of the time, he should play on. My guess is that he can (though it is close), so I would play on. If you estimated that his gammon winning chances are less than 1 in 9, you should cash. At any rate, this how the mathematics are calculated for this simple one roll position.
To see how things can get complex fast, let's back the position up one roll:
The previous position was reached from this position after both sides rolled 5-3. What is happening here? The gammon chances are probably a bit greater, since the gammon race is longer so Blue will have one more roll to roll big doubles and pull ahead. How about safety? In this position Blue leaves a shot only with 4 numbers (6-2 and 5-2), so it is much safer for one roll than the previous position. Given that, playing on from here looks to be clear.
But wait! In the previous position, Blue only had one point to clear -- if he survived that he was home free. That is not the case with this position. If Blue clears the five point, he still has to clear the four point safely. In addition to the shot numbers, 4-2 and 3-2 leave an ugly gap on the four point, and 6-1, 5-1, 4-1, and 3-1 force Blue to come down to three men on the four point where he will be facing a double-jeopardy situation unless he rolls doubles. Even if Blue rolls two big numbers, he will come down to the same danger as in the previous position. Only if Blue rolls doubles will he be safer. Thus Blue is much more likely to leave a shot here than from the previous position, which means that White has greater winning chances. Since Blue's gammon chances are only a small bit better than in the prvious position, it would appear that Blue should take the money and run.
But wait! In the previous position it was a now or never situation -- if Blue didn't double now the cube would be of no use to him if he weren't hit. That is not the case here. Blue may choose to play on for now, yet turn the cube next roll if he doesn't like the way the wind is blowing. If Blue were required to double now or be committed to play on for the gammon whatever happened, it would be correct to double now. But that is not the case. Suppose Blue chooses to play on and rolls one of his awkward numbers with an ace or a two. Regardless of how awkward his position is, he can still turn the cube next turn and White would have to pass. Also, suppose Blue rolls decently (5-3), but White comes back with, say 4-4. Blue would now be in just as much danger of losing as he was in the first position, but since White has gained an extra 8 pips Blue's gammon chances will have gone way down. In that scenario, Blue would again cash.
The bottom line is that even though overall Blue will be leaving more shots from the second position if he plays to the end, in practice he will be hit less often because he will have doubled if he reaches one of the more dangerous positions or if White rolls big enough to make the gammon very unlikely. Thus, the second position is actually safer than the first position even in the long run. It is true that Blue won't get quite as many gammons since he won't see the whole thing through as often, but he will still get most of the gammons he would have gotten otherwise. Keep in mind that Blue's most favorable gammon possibilities occur when he rolls doubles, which also lead to his safest positions. Thus I believe that playing on for the gammon in the second position is clearer than it is in the first position.
The fact that the player who is playing on for a gammon has cube access means that immediate danger is often the number one consideration. Even if the gammon chances are small, if there is little or no danger that your opponent will have a take after the next exchange it has to be correct to play on. It is a no-lose proposition. For example, consider the classic prototype position which I have seen even the best players get wrong:
As we all know, Blue's gammon chances are very small while he could still lose the game on an unlucky sequence. In fact, a Snowie rollout indicates that Blue will lose about 5% of the time while he will win a gammon only about 2 1/2% of the time. Since Blue needs twice as many gammons as losses to justify playing on, it would appear that Blue should double. This is not correct. Blue should definitely play on. The reason is that absolutely nothing can happen on the next exchange which would allow White to take a double. Even if Blue rolls something very awkward such as 5-5, he can still turn the cube next turn and White will pass. Thus, Blue has nothing to lose by playing on for one roll.
Can playing on really gain? Most of the time it won't matter. It is likely that Blue will reach a position where he is in danger of leaving a shot on the next roll, and if Blue reaches such a position he will probably have to cash. However it is possible that Blue will be able to bear off in such a way so he never risks leaving a shot on the next roll. As long as he is safe for one roll, he can continue to play on. The moment he is in danger, he must double. This way, it is impossible for Blue to ever lose the game if he is careful. Blue will have to reconsider his decision to play on for a gammon before every roll, but that is as it should be.
When Blue has taken a lot of men off and White is still on the bar, Blue may decide to take a small risk of leaving a shot if he judges the gammon potential is great enough. If he does that and White hits the shot, Blue could lose the game. However that was a calculated risk which Blue chose to take. Blue never need risk losing the game if he chooses not to. In practice he will get fewer gammons than if he played the position to the end, since on some of those gammons he would have gotten he would have been forced to cash first. However since his losing chances are zero, it doesn't matter how small his gammon chances are. Playing on is definitely correct.
The ability to look down the road and determine what is likely to happen if often an important factor in deciding whether or not to play on for a gammon. If there is likely to be future danger which will force you to cash, you might as well cash now if there is any immediate danger at all. For example:
Blue's gammon chances aren't nearly good enough to compensate for his losing chances vs. this two-point game. However, not much is likely to go wrong next roll. The only roll which busts Blue's prime is 6-6. Even if he rolls that, White would still have to roll 5-5 or hit an indirect shot to be in the game -- otherwise Blue would be able to cash. Given that, it seems offhand worthwhile to take this small risk. After all Blue does have some gammon chances. He could play on, hoping to get a position something like this:
If Blue gets to here, playing on is clear. Blue could leave a shot if he rolls 6-4 or big doubles, but White would still have to hit the shot. Since White's board is crunched and Blue will have several men off Blue would still be the favorite -- in some variations Blue might even have a powerful recube. Blue's gammon chances aren't huge, but they are definitely significant. Thus in this position, Blue should be happy to play on.
The problem is getting from here to there. Blue isn't allowed to magically lift up those checkers on the bar and eight points and bring them in. He will have to roll the numbers to clear these points, and while he is doing so he may be at risk. For example, suppose Blue reaches this position:
This is about as good as Blue could expect. His distribution is ideal, and White has crunched. But Blue is in plenty of immediate danger. He could roll 6-5 or 5-5 and be forced to leave a shot. In addition, the moment he breaks his eight point or bar point White may barrel out with 5-5 or 6-6 and Blue will find himself behind in the race. These dangers are one-roll dangers which may occur on the next exchange, so Blue doesn't have any safety margin. Even if Blue survives these dangers and sucessfully clears hit eight and bar points, he is a long ways from winning a gammon. Not only will White probably be able to scramble off, but Blue could leave a shot and get into trouble. It is true that if Blue can clear the eight and bar points without White doing anything Blue will probably reach a position where he is happy to be playing on for a gammon as we have seen. However that is quite different from actually winning the gammon. Thus, I'm pretty sure Blue is supposed to cash in this position. And, since this is about as good as Blue can expect, that means he might as well cash in the original position. The problem is that for Blue to get from the original position to where he is happy to play for a gammon he has to pass through a high risk position.
On the other hand, it may be correct to take an immediate risk if survival means that you will have a comfortable playon from then on.
Blue has several shot numbers. 6-1, 6-2, 5-1, 5-2, 6-6, and 5-5. If White hits a shot Blue will no longer be able to cash -- in fact, White may soon prove to be the favorite. Despite this large immediate risk, I believe that Blue is correct to play on. If Blue does clear his nine point successfully he will have an easy playon from then on. Sure he could lose -- we have all lost to ace-point games before. However Blue is going to win a ton of gammons in this position, and these gammons will make future playons very clear. Given all this, I believe Blue is correct to take a roll here.
Checker play problems when playing on for a gammon can be very interesting. Sometimes it is correct to go conservative, even though your gammon chances may not be as great.
Blue has two obvious choices, the safe 13/10, 13/7 or the daring 13/7, 5/2*. The loose hit will lose more often, of course, but it will also bring in more gammons. If White owned the cube, my judgment would be to go for the loose hit. The problem with the safe play is that it isn't a claimer. If White rolls a two, Blue will have to struggle to come in against a well-timed two-point game. It looks better to try to end it now, when getting hit back may not be fatal.
With Blue owning the cube, however, I believe it is another story. The problem with the loose hit is that if White hits back suddenly Blue loses his big advantage. No longer will Blue have a redouble which White must pass. If Blue does have a redouble at all (which is questionable), White can certainly shoot out a take and hope that he is the one who enters first. On the other hand, if Blue plays the quiet 13/10, 13/7 he does not have to worry about losing the game. If White enters and makes the anchor, Blue can turn the cube. White will be forced to pass, and Blue will not have to face the two-point game which he would have had to face if White owned the cube. I believe this cube leverage is sufficient to make the safe play better, even though it will get the gammon which Blue is going after less often.
On the other hand, cube ownership may be good reason for more aggressive plays:
If White owned the cube, I would be inclined to go conservatively with 9/3. This gives Blue excellent builder distribution if White doesn't enter, and if White does enter Blue will be quite happy he didn't hit loose. While Blue is a clear favorite to escape if he hits loose and is hit back, he could easily lose the game if he is unable to escape. I don't believe the extra gammon chances justify this risk. If Blue owns the cube, however, I think he should hit loose. The reason is that if White hits back with something other than double-aces, I believe that Blue would have a very powerful redouble -- one that White would probably have to pass. If my assessment is correct then the loose hit doesn't risk losing the game except for those double-aces, so it is worth the risk because of the significant increased gammon chances.
Match score considerations can lead to interesting playon decisions. For example:
Once again Blue is coming in against a two-point game. This time Blue has cleared the eight point and White is far enough back so Blue has some moderate gammon chances. However Blue has very real immediate danger clearing his bar point, and will of course have problems down the road also. At an even match score I would be inclined to simply redouble and take the point. At this match score, however, I believe Blue can afford to risk playing on. He is still in big trouble if he is hit now, but a later hit may be another story. For example, suppose the following position is reached down the road:
For money or early in a match Blue would be nuts to redouble -- relinquishing cube ownership might turn him from the favorite (or close to the favorite) to the underdog. At this match score, it is another story. White can't use the four points which would be at stake, and White would have no recube vig. In addition Blue could win a gammon and scoop the whole match. Even without going into the match equities it is easy to see that Blue has a very powerful redouble and White might not even have a take. This huge cube leverage which Blue will have for the rest of the game is in my opinion sufficient for him to justify playing on for the gammon in the original position.
Post-Crawford scores combined with the free drop can lead to some strange situations. I learned about the free drop firsthand in the first major tournament I played in back in 1975. In the fourth round I met Chuck Papazian, one of the top players in the world at the time. In our 11-point match I got to behind 10-9, post-Crawford. I got an opening 4-3 and played 13/10, 13/9. He responded with a 4-3 and also played 13/10, 13/9. I routinely doubled, and he couldn't set the checkers up fast enough. This completely stunned me -- I had never seen this before. A few seconds thought and it was clear what was going on, and the concept of the free drop was born in my mind. Instinctively I then mentally asked myself the question: Since he had a clear pass, perhaps I should have been playing for the gammon.
A few days later I asked Paul Magriel (who was the number one authority on backgammon at the time) whether there were any opening roll and response sequences after which it was correct for the person behind 2 away, 1 away to play for the gammon. Paul said that he didn't think there were. I was not convinced, and kept the topic in my mind.
When we finally had computer programs which played well enough that rollouts could be trusted, I was able to reassess this problem. By looking at all 1296 possible exchanges and the resulting equities, I was able to form some kind of conclusion. Let's take a look at an opening 3-1 and a 6-2 response.
What makes this different from a normal playon problem is that White's take point isn't the usual 25% -- it is 50%. If White is any kind of underdog he is better off dropping a double and playing the next game for the match than he is playing the current game. This means that as long as Blue is likely to be the favorite whatever happens on the next exchange, he can safely play on for the gammon. Of course Blue can't be sure that he will be the favorite after the next exchange -- White might roll some good doubles which turns the game around. However the odds are that Blue will be the favorite, and if he isn't the favorite he won't be much of an underdog. Blue's philosophy is pretty much the same as a normal playon problem. If White isn't going to have a take (which means be the favorite at this match score) it is safe for Blue to play on, but if there is a reasonable chance that White can turn the game around in one roll then Blue should cash unless he has some decent gammon chances.
There is no way that I know of one can calculate whether or not to play on in this sort of position, just as it is often impossible to determine whether or not to play on in a normal playon decision. As we have seen it isn't just the possibility of a turnaround on the next roll or just the gammon chances which matter -- it is the way the game is likely to go and the resulting types of positions. However as best as I could determine it is correct to play on from this position, and probably from other almost as good starts. For what it's worth Snowie agrees with me that this is a playon, although I have no idea how Snowie comes to that conclusion.
As a final illustration of the oddities of playing on for a gammon, here is a position I had several years ago.
I had been playing for a gammon, and had been hit by a last ditch shot. Now what? Let's see how things look from his point of view if I double:
He passes: Behind 9-5, 19% equity.
Thus, he can justify taking if he has 19% winning chances.
Can White win this game 19% of the time? I had no idea -- I didn't think so. Therefore, the correct cube action is double and pass if my assessment is correct.
But what if I am wrong, or what if he thinks I am wrong and takes the double? Then he will be playing all out to win. This may mean that if he rolls something like 2-1 he might play 2/1 with the ace in the hopes of picking up my other blot. Is this his correct play? Again I have no idea, but it might be. However if I don't double now he certainly can't afford to do this, since one thing he cannot risk is losing a doubled gammon with the cube on 2.
Therefore, I waited until he had brought all his men home and was about to take a checker off, so there was no longer the issue of the trap play costing him a gammon. Then I doubled. As it happened he took, but my luck was in and I won the game. Afterwards Kent Goulding who had been watching said to me: What were you doing -- playing for the gammon? When I said yes he was startled -- after all, how can I play for a gammon when I am on the bar against a closed board and my opponents checkers aren't blocked. Yet this really was what I was doing. The gammon threat when I didn't double prevented him from going after my other checker. Had I doubled immediately, that gammon threat would have no longer been there. Admittedly this is a rether obscure example of playing for a gammon, but it shows the extent one can go to if willing to use a little imagination.
The question of when to play for a gammon remains a sticky one. The bots can help some, but their opinions (and even their rollouts) are of dubious value since while they can look two rolls ahead very accurately they cannot forecast the type of path the game is likely to take when making their decisions. We simply have to use our judgment and experience to help us solve these problems.