return to indexFor money, the mathematics is simple. If you double and he passes, youcollect the value of the cube. If you play on and win the gammon, youget 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 ona 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 amatch you can play on with the cube in the center, since undoubled gammonscount. 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 ownthe cube. For this article, unless other wise stated we will be assuming either money play ormatch play early in a long match (so the odds are about the same as theywould be for money).
First, let's look at a true play-on problem where everything will hingeon the next exchange:
| 84 23 | White money game Blue |
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 thecenter 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 ifhe 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 hisgammon 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:
| 92 31 | White money game Blue |
The previous position was reached from this position after both sides rolled5-3. 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 rollthan the previous position. Given that, playing on from here looks to beclear.
But wait! In the previous position, Blue only had one point to clear -- ifhe 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 fourpoint, and 6-1, 5-1, 4-1, and 3-1 force Blue to come down to three men onthe four point where he will be facing a double-jeopardy situation unless he rolls doubles. Even if Blue rolls two big numbers, he will comedown to the same danger as in the previous position. Only if Blue rollsdoubles 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 betterthan in the previous 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 Bluewere 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 canstill 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 wouldnow 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 goneway down. In that scenario, Blue would again cash.
The bottom line is that even though overall Blue will be leaving more shotsfrom the second position if he plays to the end, in practice he will be hitless often because he will have doubled if he reaches one of the more dangerouspositions or if White rolls big enough to make the gammon very unlikely. Thus, the second position is actually safer than the first position evenin 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 getmost of the gammons he would have gotten otherwise. Keep in mind that Blue's most favorable gammon possibilities occur when he rolls doubles, which alsolead 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 accessmeans that immediate danger is often the number one consideration. Even ifthe gammon chances are small, if there is little or no danger that youro pponent will have a take after the next exchange it has to be correct toplay on. It is a no-lose proposition. For example, consider the classic prototype position which I have seen even the best players get wrong:
| 78 72 | White money game Blue |
As we all know, Blue's gammon chances are very small while he could stilllose 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 lossesto justify playing on, it would appear that Blue should double. This isnot correct. Blue should definitely play on. The reason is that absolutelynothing 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 canstill turn the cube next turn and White will pass. Thus, Blue has nothingto 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 probablyhave 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 decisionto 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 maydecide to take a small risk of leaving a shot if he judges the gammonpotential 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. Inpractice he will get fewer gammons than if he played the position to theend, since on some of those gammons he would have gotten he would have beenforced to cash first. However since his losing chances are zero, it doesn'tmatter how small his gammon chances are. Playing on is definitely correct.
The ability to look down the road and determine what is likely to happenif 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 tocash, you might as well cash now if there is any immediate danger at all.For example:
| 94 98 | White money game Blue |
Blue's gammon chances aren't nearly good enough to compensate for his losingchances 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 herolls that, White would still have to roll 5-5 or hit an indirect shotto be in the game -- otherwise Blue would be able to cash. Given that, its eems offhand worthwhile to take this small risk. After all Blue does have some gammon chances. He could play on, hoping to get a position somethinglike this:
| 71 50 | White money game Blue |
If Blue gets to here, playing on is clear. Blue could leave a shot if herolls 6-4 or big doubles, but White would still have to hit the shot. SinceWhite'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. Thusin 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 doingso he may be at risk. For example, suppose Blue reaches this position:
| 80 81 | White money game Blue |
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. Hecould roll 6-5 or 5-5 and be forced to leave a shot. In addition, themoment he breaks his eight point or Bar-Point White may barrel out with5-5 or 6-6 and Blue will find himself behind in the race. These dangersare one-roll dangers which may occur on the next exchange, so Blue doesn'thave any safety margin. Even if Blue survives these dangers and sucessfullyclears 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 leavea shot and get into trouble. It is true that if Blue can clear the eightand bar points without White doing anything Blue will probably reach aposition 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 thisis about as good as Blue can expect, that means he might as well cash inthe original position. The problem is that for Blue to get from theoriginal position to where he is happy to play for a gammon he has to passthrough a high risk position.
On the other hand, it may be correct to take an immediate risk if survivalmeans that you will have a comfortable playon from then on.
| 158 73 | White money game Blue |
Blue has several shot numbers. 6-1, 6-2, 5-1, 5-2, 6-6, and 5-5. If Whitehits a shot Blue will no longer be able to cash -- in fact, White may soonprove to be the favorite. Despite this large immediate risk, I believe thatBlue is correct to play on. If Blue does clear his nine point successfully he will have an easy play on from then on. Sure he could lose -- we haveall 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 veryclear. 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 chancesmay not be as great.
| 118 83 | White money game Blue |
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, whengetting hit back may not be fatal.
With Blue owning the cube, however, I believe it is another story. Theproblem 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 mustpass. If Blue does have a redouble at all (which is questionable), Whitecan 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 theanchor, Blue can turn the cube. White will be forced to pass, and Bluewill not have to face the two-point game which he would have had toface if White owned the cube. I believe this cube leverage is sufficientto 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 aggressiveplays:
| 140 74 | White money game Blue |
If White owned the cube, I would be inclined to go conservatively with9/3. 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 believethe 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 verypowerful redouble -- one that White would probably have to pass. If my assessment is correct then the loose hit doesn't risk losing the game exceptfor those double-aces, so it is worth the risk because of the significant increased gammon chances.
Match score considerations can lead to interesting play on decisions. For example:
| 128 67 | White 8 11 point match Blue 3 |
Once again Blue is coming in against a two-point game. This time Blue hascleared the eight point and White is far enough back so Blue has some moderate gammon chances. However Blue has very real immediate danger clearinghis bar point, and will of course have problems down the road also. At aneven 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 anotherstory. For example, suppose the following position is reached down the road:
| 101 34 | White money game Blue |
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 recubevig. 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 hugecube leverage which Blue will have for the rest of the game is in my opinionsufficient 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 majortournament 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-pointmatch I got to behind 10-9, post-Crawford. I got an opening 4-3 andplayed 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. Thiscompletely stunned me -- I had never seen this before. A few seconds thoughtand it was clear what was going on, and the concept of the free drop wasborn 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 onbackgammon at the time) whether there were any opening roll and response sequences after which it was correct for the person behind 2 away, 1 awayto 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 rolloutscould be trusted, I was able to reassess this problem. By looking atall 1296 possible exchanges and the resulting equities, I was able to formsome kind of conclusion. Let's take a look at an opening 3-1 and a 6-2 response.
| 159 163 | White 10 11 point match Blue 9 |
What makes this different from a normal play on problem is that White'stake point isn't the usual 25% -- it is 50%. If White is any kind ofunderdog he is better off dropping a double and playing the next game forthe match than he is playing the current game. This means that as longas Blue is likely to be the favorite whatever happens on the next exchange, he can safely playon 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 turnsthe game around. However the odds are that Blue will be the favorite, andif he isn't the favorite he won't be much of an underdog. Blue's philosophyis 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 safefor Blue to play on, but if there is a reasonable chance that White can turnthe game around in one roll then Blue should cash unless he has some decentgammon chances.
There is no way that I know of one can calculate whether or not to play onin this sort of position, just as it is often impossible to determine whether or not to play on in a normal play on decision. As we have seen itisn't just the possibility of a turnaround on the next roll or just thegammon chances which matter -- it is the way the game is likely to go andthe resulting types of positions. However as best as I could determineit 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 play on, 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.
| 87 26 | White 5 11 point match Blue 7 |
I had been playing for a gammon, and had been hit by a last ditch shot. Nowwhat? Let's see how things look from his point of view if I double:
He passes: Behind 9-5, 19% equity.
He takes and wins: Wins match (since he recubes to 8).
He takes and loses: Loses match.
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 thinkso. Therefore, the correct cube action is double and pass if my assessmentis 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 withthe 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 costinghim a gammon. Then I doubled. As it happened he took, but my luck was inand 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 thebar against a closed board and my opponents checkers aren't blocked. Yetthis 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, thatgammon threat would have no longer been there. Admittedly this is a retherobscure 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 dubiousvalue 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 ussolve these problems.