Stick has admitted that he is either unwilling or unable to
write a dissertation explaining the "live cube take point"
and why it is useful. So let me step in and write that
dissertation for him. I'll explain some of the history behind
the term, what various people have meant by it, why it is
useful, how to use it, and also what its limitations are. At
the very end, I'll explain my objections, and how to address
those objections.
In what follows, I'll ignore gammons and backgammons. They
create complications that are best ignored initially.
At first glance, the concept of a "take point" seems simple
enough. Let P be my equity if I pass, let L be my equity if
I take and lose, and let W be my equity if I take and win.
Then the take point---or more precisely what people call the
"raw" or "dead cube" take point---is defined by
T = (P - L)/(W - L)
For example, for money with a centered cube, P = -1, L = -2,
and W = +2. So T = (-1 - (-2))/(+2 - (-2)) = 1/4, giving us
the familiar 25% take point value. This means that I should
take if I win more than 25% and pass if I win less than 25%.
To make sure we understand this, let's not just quote the
formula for T; let's derive it. Let's say that my winning
chances are w, so that my losing chances are 1-w. Then if
I take, my equity is
w*W + (1-w)*L
If I pass, then my equity is P. So to justify taking, I need
the following to be true:
w*W + (1-w)*L > P
But now we can solve for w:
w*(W - L) + L > P
Rearranging gives us w > (P - L)/(W - L) = T, the take point.
Note that this derivation applies equally well to a match as
to a money game.
Simple enough, but of course the big flaw in this argument is
that it ignores the fact that the taker's ability to recube
to 4, not to mention further recubes to even higher values.
How do we take this into account?
The earliest attempts to analyze the effect of recubes on the
take point was the so-called "continuous cube model." In order
to make the problem more mathematically tractable, we pretend
that the equity fluctuates randomly and continuously over the
course of a game, and that the cube can reach arbitrarily high
values. I'll skip the mathematical analysis here, but it turns
out that under these simplifying assumptions, you can justify
taking with 20% winning chances in a money game---significantly
lower than 25%. If you want more details, see this article:
https://bkgm.com/articles/KeelerSpencer/OptimalDoublingInBackgammon/
Clearly, redoubling is an important thing to take into account,
at least in some positions, and so we need some terminology to
make it clear whether we're talking about the 25% figure or the
20% figure. One natural possibility is to refer to 25% as the
"dead cube take point" and the 20% as the "live cube take point."
If life really were this simple, then we would simply use 20% as
the take point in almost all positions, with the exception that
in a last-roll position (where you really can't redouble because
the game will be over), we would use 25% as the take point.
Unfortunately, matters are not quite so simple. The continuous
model is an oversimplification, and it can diverge from practical
play significantly. Equities change in discrete jumps, sometimes
quite large, causing you to overshoot and "lose your market."
This means that in practice, you're often going to double well
below the 80% threshold predicted by the continuous model. This
divergence between reality and the continuous idealization is
sometimes referred to informally as "cube efficiency." The idea
is that if your (re)double has a D/T equity of 1.0 (EMG, i.e.,
normalized to the cube value), then your cube is "perfectly
efficient" and conforms closely to the continuous idealization.
The more the D/T equity differs from 1.0, the more "inefficient"
your (re)double is.
Because of these "inefficiencies," we can't simply say that in a
money game, we should use 20% as the take point (except in a
last-roll position). If we did do that, then in practice we
would be taking a significant number of cubes that we should be
passing. Some kind of further adjustment is needed. But what
adjustment should we make?
This is the point at which things get complicated. Opinions
diverge as to what sort of adjustments to make. If you set
up XG with a non-contact position with no gammons possible, then
you'll find it reports a "live cube take point" of 20% for an
unlimited game. This is nominally consistent with the documentation,
which says that the live cube take point assumes "perfect redouble
efficiency." (If you set up a position where gammons are possible,
then it will give you a different live cube take point, which is "gammon-adjusted." I won't explain further because I'm ignoring
gammons, but notice that this simply reinforces my point that the
term "live cube take point" by itself is ambiguous---do you mean "gammon-adjusted" or not?) In effect, XG is telling you to figure
out your own adjustment to make for "cube inefficiency." But it
doesn't say so explicitly, so you can be left scratching your head,
wondering why your take was deemed incorrect even though your
winning chances exceeded the "live cube take point," or even more
confusingly, how your pass was deemed incorrect even though your
winning chances were below the "live cube take point." How can
the cube efficiency be better than perfect? Well, if you unwind
all the definitions very carefully, you'll find no contradiction,
but it's certainly very confusing for the uninitiated. "Perfect"
recube efficiency just means that it conforms to some idealized
model, and doesn't necessarily mean "maximum possible" recube
efficiency. (Note, by the way, that the XG documentation never
gives a quantitative definition of efficiency. It also does not
explain how it computes "perfect recube efficiency" for matches,
when arbitrarily high cube values are not possible. GNU BG does
define efficiency, but it's not the same as what many others mean
by that term.)
XG's choice is not the only one. In Dirk Schiemann's book, "The
Theory of Backgammon," he carefully defines the "live cube take
point" by considering long races that are at the take/pass
borderline, and examining the winning chances of the underdog.
Of course, these winning chances vary slightly from one position
to another, but they don't vary too much; they hover close to
21.5%. So for Schiemann, the "live cube take point" for money
is 21.5%, in contrast to XG's 20%. There's certainly a plausible
rationale for this figure; rather than appealing to some fictional
idealized model of backgammon, he takes actual data from real
positions, and examines what the winning threshold has to be for
a take in a familiar reference position (i.e., a long race). But,
you need to be aware of the different definitions or you'll be
confused by apparent inconsistencies.
Whichever of the above definitions you use, there is an important
caveat you have to be aware of: now that we're wading into the
complexities of actual backgammon play, the "live cube take point"
usually *cannot* be simply taken to be a threshold above which
your winning chances permit you to take. You still have to take
into account how much "recube vig" you have in your current position.
In Schiemann's case, he's made long races his reference position,
so when judging recube vig, you need to be judging how much more
(or less) recube vig you have in your current position *relative to*
a long race. With XG's definition, the reference point is some kind
of idealized model, which typically yields a live cube take point
lower than Schiemann's, so you have to adjust accordingly.
We have by no means exhausted the possibilities. John O'Hagan has
advocated for something he calls the "true" take point.
https://www.facebook.com/groups/backgammonstrategy/posts/2872916849643635/
What he's doing here, I believe, is estimating how much recube vig
one "typically" has, by interpolating between XG's live cube take
point and the dead cube take point. I'm not sure how exactly his
"true" take point compares with Schiemann's live cube take point.
If things are still not confusing enough, the terms "live" and "dead"
cube take point are sometimes used even at scores where estimating
recube vig makes no sense. If you double while leading 2-away, or
redouble to 4 while leading 3-away or 4-away, and so on, then your
opponent has an automatic redouble, and so it really makes no sense
to talk about "live" versus "dead" (except in exceptionally strange
positions like the one Paul mentioned, where you refrain from your auto-redouble because you're too good). The cube action is trivial
so you can simply figure out whether your winning chances exceed
the relevant threshold. Talking about live versus dead in such
situations makes no sense (to me anyway; Stick still hasn't explicitly
agreed with me on this point, so maybe he has some arcane justification
for it).
Okay, you say, but what's the bottom line? Here are the conclusions
that I would draw.
- One certainly needs to understand that, at most match scores, the
dead cube take point is not sufficient for deciding whether to take,
even if you know your winning chances and no gammons are possible.
- At such match scores, where the cube action is not trivial, it is
useful to have some other reference point besides the dead cube take
point, to help you make take decisions---whether you use XG's "live
cube take point" or Schiemann's "live cube take point" or O'Hagan's
"true take point" or yet another flavor of the day. As long as you
know what reference point you're using and what it means, the choice
probably doesn't matter much.
- I think it is a bad idea to speak of the "live cube take point" as
if it is a definite thing. As we've seen above, there are different definitions floating around, and often it's not even clear what the
precise definition is. Furthermore, the terminology has a rather
strong connotation that it's a threshold above which your winning
chances permit you to take---yet that's not what the "live cube take
point" means, either according to XG or according to Schiemann
(except in long races for Schiemann). So the terminology is very
confusing and even actively misleading (especially if it's used at a
score where the cube action is trivial because of an auto-recube).
- These concerns could be largely addressed if a clear definition of
the term were agreed upon. Given XG's popularity, its definition is
the frontrunner, but the big disadvantage is that the definition is
not stated in the documentation. As things stand, it's just some kind
of mysterious black magic. I'd prefer Schiemann's definition
myself, but I don't know what the chances are for his definition to
prevail. And even if his definition were standardized, I'd still be
a bit unhappy that the term itself is so confusing and misleading, but
this wouldn't be the first time that we're stuck with inferior
terminology.
---
Tim Chow
--- SoupGate-Win32 v1.05
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