Staying Ahead
This text deals with a common tournament backgammon: How to protect a lead in a long match, e.g. when the match mightstill have a long way to go. For now, the main focus is on taking doubles when leading in the match. First we'll have a look at accepting initial doubles when leading in the match; then we turn to accepting redoubles.
The approach I'm going to take is a general one. I won't be explaining how to calculate take points, and I won't be presenting lots of numbers, formulas and algebra. Plenty of excellent sources for that sort of thing are already available, many right here on GammOnLine. Rather, I'll be presenting charts showing the patterns ofhow the match score influences take points, and develop some general guidelines, along withsome practical examples. The reason for this approach is that I'm having a really hard time doing match equity calculations over the boardunder tournament conditions. Back home at my desk, with pen,paper and spreadsheet hey sure, no problem. At the quarterfinals in the average regional tournament, after eight hours of play, with a crowd of kibitzers watchingand mumbling, possibly under time pressure, it's a different story. If you feel the same way please read on. Since the approach I take is kind of an experiment, I'd be more than happy to receive some feedback as to whether or not it is a useful way to present the information.
Accepting initial double
Generally speaking, a match lead should not make too big a difference in your taking policy when we're taking initial doubles. What might make difference is that gammons usually are more costly than normal, and that you don't get as efficient redoubles as usually. How big are those factors?
The chart shows the winning chances you need to accept a 2-cube whenyour opponents still has 17 point to go. The blue curves indicate no-gammon situations; the red ones that 21% of either players wins will be gammons. Of course this is seldom going to be exactly the case, but it gives an idea of the effect of gammon, in many opening and middle game situations. The lines with markers indicate the cube equity is taken into account, thus producing alower take point (measured in cubeless probability of winning, CPW sometimes also referred to as cubeless game winning chances, cgwc).
We're looking at the big picture here, so the chart includes all scores right from 2-away, 17-away to27-away, 17-away. The opponent score is held constant at 17-away, while the player, whose take strategy we're examining, is treated as a variable).
In the simple case, where there are no gammons an dno recube equity, as in, say, a last roll bear off situations, the chart indicates that you will generally need around 25% to accept acube, just as in money game. No big surprise here. It should be noted, however, the matchlead increases it becomes slightly more attractive to acceptan initial double under these conditions. It's not a big deal, however, but at 6-away, 17-away, the leader should be able to accept an initial double in a last roll bear off situation with about 24% winning chances rather than the usua l25%. Few people will be able to judge winning chances that accurately, so in practice this is not tooimportant.
When we grant the leader cube access it becomes a bit easier to accept the initial double; the take point is just shy of 22% cubeless winning chances, just as in money game. (See the bluecurve with markers). The value of owning the cube diminishes, however,as the end of the match approaches, since the leader would be reluctant toredouble to four: Notice how the distance between the blue lines gets smaller asthe lead gets bigger. Obviously, when the leader needs only two points to win, he gets no cube value at all. The main point to notice is that the take points are still pretty much as in money game, with a small exception when the leader can't take advantage of cube ownership.
When gammons are included (at the rate of 21% for both sides) we still have pretty much a money game situation, except when very close to the end of the match. The worst situation for the leader is at 2-away, 17-away where the lead is really big and where he gets norecube equity at all. In this case he needs about 33% winning chances to justify a take, and that's quite different from the roughly 26% he'd need at money game or further away from the end of the match, he could put the cube to some use. Before we turn to some actual examples, let's try to summarize what we've learned so far about accepting initial double with a match lead:
It's not the size of the lead but the number of points still needed to go that has thegreatest effect on the leader's take point. With lots of point to go, even a big lead shouldn't cause the leader to be more cautious in accepting doubles when gammons are unlikely.
With lots of points to go, a big lead should cause the leader to be only slightly more cautious,in accepting gammonish doubles; the takepoint is typically about 2 percentage points higher compared to even scores.
Near the end of the match, when the leader is within about five point of victory, his take points increases a little 1-2 percentage points) for non-gammonish positions, and a good deal percentage points) for gammonish positions. With a big lead you can take a last roll bear off position with slightly less than 25% winning chances.
Let's take a look at a couple of examples of this:
ption of being 2 or 3 points within victory, since the take might depend on being able to redouble, should the game turn around. That is indeed an accurate assessment; White should take with any kind of lead in the match, except when he's 2-away or 3-away; in that case he has small pass.
When gammons are a real possibility, things are notquite as happy for the leader, even at initial doubles:
For money, White should have a reasonable take here, although not a particularly happy one. It's easy to see people passing this one. Gammons are very possible, but with the anchor and not too many blots around for Orange to scoop up, it's not as if White is grave gammon danger. So White takes in a money game and at any even match score with lots of points to go. With a biglead, however, White might have a pass, based on the gammon risk. If you think the take is really borderline in the first place, then a pretty small leadlike, say, 12-away, 17-away could turn it into a pass. It you think the takeis pretty clear, you would either need a really big lead, like 6-away, 17-away to justify a pass, or to be close to the end of the match, like 4-away, 8-away. At the double-edged 2- and 3-awayscores, you would most likely not want totake this one.
The point here is not so much exactly what White's winning chances are, and what the precise take point at all conceivable scores would be. We're trying to build a general feel for just howmuch more cautious the leader should be.
I wouldn't claim to know the theoretical correct cube Play at various scores, but an educated guess would be that can take the double unless he's within six points of victory
Supposewe weaken Orange's position, by giving him four checkers on the 20-point, while doing damage to his racing lead:
In this position White should havea pretty clear take for money and at almost any match score, with the exception of the notorious 2-or 3-points away. When White is 2- or 3-awayand enjoying only a small lead, like[2-away, 4-away]; [2-away,5-away]; [3-away,5-away] or something like that, he has a rather big pass, since Orange is now threatening to win the match or take the lead by winning a gammon, which is still not too unlikely. 2-away,10-away, for instance, would probably also be a pass, but not nearly as big asat 2-away 4-away where Orange's gammons operate at maximum efficiency.
Next is a simple position illustrating one of the finerpoints of taking with a match lead:
As most players are aware, for money this last roll situation is a true borderline take/pass decision. What fewerknow, however, is that it doesn't take much of a match lead to turn it into a take. With a 5-point lead or more, White has a pretty clear, although still small,take. Don't over estimate this effect, though. Pure 3-roll positions are still passes, with any kind of lead, forexample. (In fact, initial doubles in 3-roll positions can't be taken at anymatch score, unless there's an automatic redouble available). Also, note that this really only works in last roll positions; in longer bearoffs diminishing for the leader would balance the slight incentive to take more aggressively.
Now it's time take a closer look at the scores where the leader is near the end of the match. In the next chart, we'll fix the leaders score at two, three, four, and five points awaywhile treating the opponents score as a variable. This may be abit confusing at first, since the x-axis is now the other guy's score, ratherthan ours, so take your time to familiarize yourself with the chart.
From chart 1 we know that when you need four or five points to win, your take point is somewhat higher when you enjoy a big lead and face a gammonish initialdouble. Chart 2 verifies our suspicion that the bigger the lead, the more true this is. The red and green curves clearly indicate higher takepoint when the opponent needs lots of points to win. It's not a dramatic effect, though; each extra point you're leading raises you takepoint by only about onesixth of a percentage point. For example a 6 point lead, 4-away, 10-away suggest atake point of about 27%, compared to the roughly 26% you'd need at 4-away 4-away (taking into account gammons andrecube potential).
What's really interesting about chart 2, though, is the cyclic pattern the curves depict, especially at when the leader is two points from victory. This suggests that it's actually easierto accept an initial double, even a moderately gammonish one, when the opponent has an even number of points to go. That's quite counterintuitive, since after the Crawford-game the leader has a free
Let's se a couple of examples of this phenomenon:
In the above diagram, is trailingby four pips, 26 to 22, which is pretty serious in a race this short. For money,and at most match scores, White would probably have a small pass, winning 21.1% cubeless according to one database. Being 2-away with a big lead changes things, however. White should passif Orange needs 7, 9, 11, 13, 15, 17, 19 or 21 points towin, but take if Orange needs 8, 10, 12, 14, 16, 18, 20 or 22 points. That's kind of funny, but it seems to hold up to further analysis.
It should be noted, however, that this principle phenomenon only occurswhen White is holding a sizeable lead. If Orange needs 4, 5or 6 point to go, White has a pretty low take point, around 20%, as long asgammons are not possible. If Orange on the other hand needs 2 or 3 point, White< would be quick topass, with takepoints of 30% and 28% respectively.
The same pattern can, perhaps surprisingly, be seen in positions with some gammons chances:
).
A summary of taking redouble before we turn toexamples:
With a matchlead, even a smallone, be As a rule ofthumb, your take point increases with about one percentage point for each pointyou lead, when there's still a long way to go.
Typically ittakes around 30% or more to accept a gammonish redouble when leading. And make that 40% when you're leading substantially
Whengammons are an issue, a smaller lead might make 
Orange
