Cube Handling

 Cube concepts

 From: Peter Bell Address: usrobots@aol.com Date: 14 August 1995 Subject: The Doubling Cube Forum: rec.games.backgammon Google: 40mo4m\$noi@newsbf02.news.aol.com

```Hi all,

I'd like to share my thoughts on how to handle the doubling cube in money
games.  I'm hoping beginners and intermediates may find this information
useful. If you are a strong player, you already know everything I have
to say!

The information in my posting is almost certainly available in the book
"The Doubling Cube in Backgammon".  I don't have this book, however, so I
went ahead and figured out what I could on my own.  My sources are mostly
postings in this newsgroup and my own reasoning ability, which I hope is
sufficient!  I will not attempt to discuss cube handling in match play,
which is more complicated than money play - I'd recommend Kit Woolsey's
monograph on that subject to anyone serious about winning matches.

I will post this message in several sections, and then post the entire
text afterwards.  Comments, questions, and corrections can be sent to me
at usrobots@aol.com or posted to this newsgroup.  Enjoy!

Part I: The Basics
------------------

The simplest example of cube use is a last-roll situation in a money game.
Note that "last-roll" means "the game will be decided by this roll".  It
does not mean "this is the last theoretically possible roll in the game."
Last-roll situations may occur late in the bearoff.  For example, if you
are on roll, have two checkers left to bear off, and your opponent has one
checker left on her one-point, the game will be decided on this roll.
Either you will win, if you bear off both checkers, or you will lose if
you do not.  (Last-roll situations may also occur in the middlegame,
usually when one player has a game-deciding shot and the cube is still in
the middle.)

Cube decisions are straightforward in last-roll situations, assuming
gammon chances are negligible.  You should double if you are a favorite to
win, by even the slightest amount.  The game has become a one-roll
proposition, and you have a choice between playing at the current stakes
or at two times the current stakes.  If you are a favorite, you want to
play at two times the current stakes, so you should double.  Otherwise,
you should not.

Conversely, in a last-roll situation, you should take an opponent's double
if you have more than a 25% chance to win the game.  Consider the
possibilities after your opponent doubles: you can take and lose, you can
drop, or you can take and win.  If you take and lose, you lose twice the
stakes.  If you drop, you lose the stakes.  If you take and win, you win
twice the stakes.  These outcomes can be represented by -2, -1, and +2
respectively.  If you take, you are risking 1 (the difference between -1
and -2) to get 3 (the difference between -1 and +2).  This is a good bet
only if you win more than 1 out of 4 games, or more than 25%.

Easy, right?  However, last-roll situations are rare.  In non-last-roll
situations, cube access is often very important.  Say that you own the
doubling cube in a game because you took an earlier double.  In this
situation, you can always play the game to its natural conclusion (that
is, until one side bears off all checkers) by simply holding on to the
cube.  Your opponent can never "double you out" (offer a double that you
must pass), regardless of how good his position gets.  Therefore, you
always have a chance to come back and win.

Furthermore, if the game happens to swing around, you control the cube.
You can be patient, choosing to double when it is most effective for you.
In the simplest case, you could use the cube to double out your opponent
as soon as you have more than a 75% "cubeless" chance of winning
("Cubeless" means "assuming the game is played to its natural conclusion".
In most contexts, winning percentages are assumed to be cubeless.)  The
75% figure assumes no gammons, and is slightly inaccurate for a reason
I'll discuss below, but it's close enough for now.  In this situation, you
don't actually have to win the game outright: you only have to gain more
than 75% winning chances.  In games where you do so, your opponent will
never get the chance to come back.  You can instantly win the game with a
cube turn.

The importance of cube access discourages doubling in many backgammon
positions.  Think of a backgammon game as a tug of war, in which both
sides start at 50% and must get to 75% to win.  At any point during the
tug-of-war, one side may double the stakes ... but if the other side
accepts, the first side now has to get all the way to 100%, while the
other side still only has to get to 75%.  In other words, the winning
conditions become asymmetric.  Obviously, the decision of exactly when to
double could get quite subtle.  One point should be clear, however: in
most situations, you wouldn't want to double until you got close to 75%.
worse, since you would now have to pull almost twice as far to win, while
your opponent's winning conditions would not have changed.

Using this analogy, a last-roll situation is like a tug-of-war in which
you turn to the others on your team and say, "Lads, this is it.  One
all-out pull, we either get all the way to 100% or they do."  In such a
situation, the 75% mark is irrelevant.  You should double if you are even
a slight favorite.  In many situations, however, you should be in no
particular hurry to double, since your winning effort becomes so much
harder after doing so.

Cube access is the reason behind the often-stated rule that you should be
less eager to redouble (that is, double when you own the cube) than to
offer an initial double (that is, double when the cube is in the middle).
With the cube in the middle, both players have cube access - the
tug-of-war finish lines are at 75% for both.  If you offer an initial
double, you make your finish line 100%, which is a disadvantage.  However,
at least you don't change your opponent's finish line - it remains at 75%.
If, however, you redouble, you change both finish lines.  Yours moves
from 75% to 100%, while your opponent's moves from 100% to 75%.  Your
situation has gotten worse in two ways, not just one!
Therefore, when redoubling, you should be even closer to 75% then when you
are offering an initial double, in most cases.

Cube access is also the explanation for a concept known as "recube vig"
(where "vig" is short for "vigorish", which means additional winning
chances).   Say that you are doubled in a non-last-roll position in which
you judge your winning chances to be exactly 25%.  Using the reasoning
that you are getting 3 to 1 odds on your take, you will expect to win the
same amount whether you take or drop (in other words, you are
"indifferent" between taking and dropping).  However, using the tug-of-war
analogy, you realize that if you take, the finish lines change to be in
to the 3 to 1 odds you are already getting.  Therefore, you will do
slightly better to take in this situation.  In other situations, you might
have a take even if your winning chances were much lower than 25%.

When recube vig is present, it changes the strategy of both sides.  The
leader now has to get somewhat further than 75% before she can cash.  The
trailer, on the other hand, can sometimes eke out a take despite having
less than a 25% chance of winning.  The amount of recube vig varies quite
a bit from position to position.  I'll discuss why in more detail later in
this document.  For now, assume that you have been doubled, your winning
chances are a little under 25%, and you are searching for some recube vig.
In a last-roll situation, your recube vig is zero, and you must drop.  In
routine holding games, it is typically only one or two percent.  In racing
games and blitzes, it is somewhat higher.  In priming games, it is often
huge, often making takes possible with winning chances that are quite a
bit less than 25%.  Bearoff positions follow no general rule: recube vig
can range from zero to enormous, making proper cube handling deceptively
difficult in many of these simple-looking positions.

So, there you have the basics of cube handling.  In a last-roll situation,
double if you are a favorite, and take if you have at least a 25% chance
of winning.  Otherwise, before doubling, try to get as close to the
"finish line" (75%+recube vig) as you can, especially if you already own
the cube.  If your opponent doubles you when you have less than a 25%
chance, consider whether recube vig can give you a take.  That's it!  Of
course, the estimation of all of these winning chances and recube equities
is often very difficult.  Everyone gets better at it over time, however,
and you must know the right strategy to use once you've figured out your
winning chances.  The next section of this document will discuss equity, a
more general and useful concept than winning chances.

Part II: Equity
---------------

The concept of equity is crucial to a reasonably complete discussion of
the doubling cube, because of the possibility of gammons and backgammons.
Trying to talk in terms of percentages in situations involving gammons and
backgammons quickly gets cumbersome.  Equity, on the other hand, is one
number that represents the total value of a position.  Probably the
easiest way to think about equity is to assume that you are playing for
one dollar per point, and to figure out how much money you expect to win
or lose from your position, gammons and backgammons included.

In such a position, you may have a 25% chance to win a gammon, a 50%
chance to win a single game, and a 25% chance to lose a single game.  You
can figure out your equity using the following reasoning: "I will win two
dollars 25% of the time, win a dollar 50% of the time, and lose a dollar
25% of the time.  Therefore, my expected gain is .25(\$2.00) + .5(\$1.00) -
.25(\$1.00) = \$.75."  In such a situation, if your opponent offered to pay
you \$.75 to settle the game, you would do just as well by accepting the
settlement as you would by continuing.

Generalizing this concept, you can state that your equity in the position
is .75 of the stakes for which you are playing, or even simpler, just .75.
Your opponent's equity is always -1 times your equity.  In this case,
your opponent's equity is -.75.  A positive equity means that you are the
favorite in the position, while a negative equity means you are the
underdog.  Note that "favorite" means "favorite to win money (or whatever
stakes are involved)", not "favorite to win the game."  If you only win
40% of the games from a given position, but all of your wins are gammons
and all of your losses are single games, you are the favorite, since your
equity is .4(2) - .6(1) = .2.

Discussions about equity can sometimes get confusing.  For example,
someone might say "I have 60% equity."  This is an ambiguous statement.
Does he mean "My equity is .60" or "I have a 60% chance to win?"  In the
latter case, assuming no gammons, his actual equity is .20, much less than
.60.  Therefore, to avoid confusion, he should have said either "My equity
is .60" or "I have a 60% chance to win."  By the way, converting between
winning chances and equity is easy, assuming no gammons.  Let P=percentage
chance of winning and E=Equity.  Then, E=2P-1.  For example,
E=2*(60%)-1=.20.  Going the other way, P=(1+E)/2.  For example,
P=(1+.20)/2=60%.

Equity can also get confusing becuse plain old "equity" without a modifier
sometimes means equity taking the cube position into account, and
sometimes means cubeless equity.  In general, equity before a double mean
cubeless equity, while after a double means equity including the cube
position.  This probably sounds strange, but it actually makes the
discussion easier.  When considering a double, you have two options.
First, you can play on (at least for a while) without using the cube, in
which case cubeless equity is relevant.  Or, you can double, at which
point cube considerations become important, because you are changing the
finish lines.

Finally, equity sometimes includes the cube level (1, 2, 4, etc.) and
sometimes does not.  One again, the general rule is somewhat complicated.
When considering whether to double, the cube is usually assumed to be at 1
before the double and at 2 after the double, regardless of the actual
level of the cube.  This makes it easy to formulate rules like "Take if
your equity is greater than -.5."  Otherwise, you would have to say things
like "Take if your equity is greater than -.5x, where x is the level of
the cube before the double," which would soon get tiresome.

With all that said, we can now rephrase the situations described in Part I
in terms of equity.  Winning chances of 50% with no gammons is equivalent
to a cubeless equity of .5 - .5 = 0.  Winning chances of 25% with no
gammons is equivalent to a cubeless equity of .25 - .75 = -.5.  Therefore,
in a last-roll situation, a player should double with equity greater than
zero, and that player's opponent should take with equity greater than -.5.
Another way to see this is to realize that an accepted double exactly
doubles the equity of both players in a last-roll situation.  Therefore,
if a player considering a double has positive equity, no matter how small,
she will want to double that equity.  Her opponent will want to take if
her own equity after taking is greater than -1, since by dropping, her
equity becomes exactly -1.  In a last-roll situation, equity after taking
of -1 corresponds to equity before taking of -.5, which agrees with our
previous analysis.

In the tug-of-war analogy, introduced in Part I, each player starts with a
cubeless equity of zero.  The first to get to a cubeless equity greater
than .5 (assuming no recube vig) can win the game with a cube turn.  If
either player chooses to double before that point, he then has to get to a
cubeless equity of 1 to win.  If a player redoubles, his own finish line
increases from .5 to 1, while his opponent's drops from 1 to .5.
Unfortunately, gammons complicate this picture somewhat - in a gammonish
position, one player can have an equity greater than 1 but his opponent
can still have winning chances.  This analogy is also limited in other
ways.  Therefore, we'll leave it behind at this point, and just talk in
terms of equity.

As a final point, thinking in terms of equity is somewhat unnatural.  It's
easier for most people to think in terms of percentages of gammons won,
single games won, single games lost, and gammons lost.  However, equity is
a more useful number, and converting from percentages into equity is quite
easy.  For example, say that your opponent doubles, and you have a 10%
chance to win a gammon, a 20% chance to win a single game, a 45% chance to
lose a single game, and a 25% chance to lose a gammon.  Should you take?
"Eyeballing it" might work, but since the equity calculation is so easy,
you should do it.  2(.1) + .2 - .45 - 2(.25) = -.55 = marginal take/drop,
depending on recube vig.  In the next section, I will use equity to
consider the concept of cube leverage.

Part III: Cube Leverage
-----------------------

Most of the subtlety in cube use involves the decision about when to
double.  The taking decision is usually straightforward, although by no
means easy.  "Take if your cubeless equity is greater than -.5: if it is
less, try to find recube vig that will allow you to take: if you do not
find it, drop."  Even simpler:  "Take if your equity after taking is
greater than -1."  There is no similar rule about when to double, however.
Sometimes is it correct to double with a cubeless equity of .06 and
incorrect to double with a cubeless equity of .6.  In fact, given three
positions with a cubeless equity of .6, the first might be not good enough
to double, the second might be a correct double, and the third might be
too good to double! ("Too good to double" means that the player should go
for a gammon instead of cashing the game with a cube turn, for at least
one more roll.)

To begin to get a handle on the doubling decision, let's consider the
potential gains in various positions from a double.

POSITION 1

24  23  22  21  20  19      18  17  16  15  14  13
-----------------------------------------------------
| X                     |   |                       |
| X                     |   |                       |
|                       |   |                       |
|                       |   |                       |
|                       |   |                       |
|                       |BAR|                       |
|                       |   |                       |
|                       |   |                       |
|                       |   |                       |
|                       |   |                       |
|     O           O     |   |                       |
|_______________________|___|_______________________|
1   2   3   4   5   6       7   8   9  10  11  12

OFF: X-13  O-13
CUBE:  1

In this last-roll position, O has 19 rolls that win and 17 that lose.
Therefore, her equity is 19/36 - 17/36 = 2/36 = .06.  By doubling, she
increases her equity to .12 (since her opponent will take), for a gain of
.06.  Nice, but nothing to get excited about.  If she neglected to double,
it would be no great tragedy.

POSITION 2

24  23  22  21  20  19      18  17  16  15  14  13
-----------------------------------------------------
| X                     |   |                       |
| X                     |   |                       |
|                       |   |                       |
|                       |   |                       |
|                       |   |                       |
|                       |BAR|                       |
|                       |   |                       |
|                       |   |                       |
|                       |   |                       |
|                       |   |                       |
|                     O |   |                       |
|_______________________|___|_______________________|
1   2   3   4   5   6       7   8   9  10  11  12

OFF: X-13  O-14
CUBE:  1

In this position, on the other hand, O would make a huge mistake by
neglecting to double.  She has 27 out of 36 rolls to win, for an equity of
exactly .5.  It doesn't matter whether her opponent takes or drops a
double:  in either case, she will increase her equity to 1 by doubling,
for a gain of .5.  This is the best equity gain she can get by doubling in
a last-roll situation, assuming her opponent always makes correct cube
decisions.

POSITION 3

24  23  22  21  20  19      18  17  16  15  14  13
-----------------------------------------------------
|                 X   X |   |                       |
|                 X   X |   |                       |
|                       |   |                       |
|                       |   |                       |
|                       |   |                       |
|                       |BAR|                       |
|                       |   |                       |
|                       |   |                       |
|                       |   |                       |
|                       |   |                       |
|     O   O             |   |                       |
|_______________________|___|_______________________|
1   2   3   4   5   6       7   8   9  10  11  12

OFF: X-11  O-13
CUBE:  1

This position represents the opposite extreme from Position 1.  Here, O's
position is so good that she gains little equity by doubling.  To lose, O
must roll a 1 (11/36) and X must roll a 6-6 (1/36).  O's total equity
works out to be .983.  After she doubles, X will drop instantly, and she
will have gained .017.  Once again, doubling is correct, but doesn't gain
much equity.

To summarize these three positions, we can say that O's "cube leverage"
(her potential equity gain by turning the cube) is small in Positions 1
and 3 and large in Position 2.  Generalizing this concept, your cube
leverage in any position is typically highest when your equity is near .5
and lowest when it is near 0 or near 1.  In the latter two cases, it might
be correct to double, but your gains from doing so will typically be
small, unless your opponent makes a mistake and either drops in a
relatively good position or takes in a hopeless position.

Cube leverage provides another way of looking at the concept of waiting to
double until you are close to an equity of .5.  If you double with an
equity of .1, you will, at best, gain .1 by doing so.  While this is nice,
it is nowhere near the .5 you could potentially gain by waiting until
later to turn the cube.  If it is the last roll, there's no sense waiting,
but in the opening or middlegame, you just don't have that much to gain by

Furthermore, in non-last-roll positions, recube vig becomes an important
factor.  In last-roll positions, you can double your equity by doubling.
In non-last-roll positions, you will never be able to do quite as well,
because of recube vig.  In many positions, in fact, you can end up worse
off after doubling, despite having positive equity, if the recube vig is
greater than your equity.  As further motivation for waiting to double,
consider the fact that your opponent's recube vig is highest when you have
larger and larger.  Remember that recube vig represents the additional
winning chances your opponent gains from accepting a double.  If you are
close to .5 equity, you are fairly close to winning, and can often change
the finish lines without giving up too much.  If you have, say, a .1
equity, you are far from winning, and there is a large probability that
changing the finish lines will make a difference in the game.  Therefore,
doubling is likely to actually make your situation worse.

Of course, you don't want to wait too long to double!  Say your opponent's
"take point" (that point at which he is indifferent between taking and
dropping) in a given position is .55.  If your equity is above .55, you
should "cash" (offer a double that your opponent must drop) immediately,
except in special situations involving gammons that I will consider later.
By doing so, you raise your equity to 1, which is the best you are going
to be able to do.  It is no longer correct to think in terms of cube
leverage: just cash the game and move on to the next one, no matter what
your equity happens to be.  By playing on, you are merely giving your
opponent a chance to get back into the game.

Overall, the lesson should be clear: double when your equity is as close
to your opponent's take point as possible.  That point is when your cube
leverage is highest and your opponent's recube vig is lowest.  If you are
playing a good player, you offer a double, and she takes without a
moment's thought, you may have doubled too soon.  Conversely, if she drops
instantly, you may have waited too long.  (Of course, you may also have
used correct cube strategy -- it depends upon the position.  You can
always ask the other player's opinion!)

The best doubles are those where your opponent stares at the position for
a long time, grits his teeth, furrows his brow, and then finally makes a
decision.  In such a situation, you don't really care whether he takes or
drops -- in either case, you have offered a good double, and raised your
equity to somewhere close to 1.  This fact is behind the aphorism "An
efficient double is as good as a win."  If you have offered an "efficient
double" -- one offered very close to your opponent's take point -- you
will end up very close to an equity of 1, which is the same equity as an
outright win.

If the equity of a backgammon position always changed slowly, this would
now be the end of the discussion about the doubling cube (at least with
respect to money play, and assuming that one's opponent always makes
correct cube decisions).  Unfortunately, equity can often change
dramatically from roll to roll.  This fact means that some additional
advanced concepts often become important to proper cube handling.
However, these advanced concepts can also be misused, as we will see.
Therefore, if you still feel like you don't quite understand one of the
advanced concepts, you will probably do better by ignoring it and sticking
to the basics.  Later, after you feel that you grasp it completely, you
can incorporate it into your game.

--------------------------

Thrity-six different dice rolls are possible in any backgammon position.
Therefore, one way you can derive the equity of a position is by averaging
the equities of the positions that result from each of those 36 rolls.
This process is recursive -- looking two rolls into the future, you can
derive the equity of the current position by averaging the equities of the
36 x 36 = 1296 resultant positions.  This process is quite interesting in
and of itself, and has led to some detailed discussions in this newsgroup
in the past.  For the purposes of this discussion, we'll assume that
nothing out of the ordinary occurs during this type of process, and that
it can be continued indefinitely, for as many rolls as you would like.

Thinking of the equity of a position in this way, you can classify certain
rolls by the effect they will have on equity if they occur.  For example,
your equity in two different positions might be the same, say, .45.
However, the distribution of resultant equities might be quite different
in the two positions.  Let's assume you are on roll throughout this
section, since you can only double before your roll.  In the first
position, your best roll might increase your equity to .47, while your
worst might decrease it to .40.  In other words, none of your rolls are
going to make much difference in the game.  In the second position, your
best roll might increase your equity to .95, while your worst decreases it
to -.60.  Obviously, some rolls in this position would be extremely good,
while others would be extremely bad.  It turns out that the nature of the
possible resultant equities in a position is often an important input into
the doubling strategy for that position.  In other words, it is not only
the current equity of a position, but also the distribution of the
resultant equities, that affects doubling decisions.

For example, the possibility of large equity swings on a given roll means
that you might suddenly move from a position with high cube leverage to
one with low cube leverage.  In both positions above, you have high cube
leverage, with an equity of .45.  Let's assume no recube equity and no
gammons, for simplicity.  Then, you can gain .45 in equity by doubling in
either position.  This is close to your maximum doubling equity gain of
.5.  If, however, you roll your best roll in position 2, your equity has
increased all the way to .95.  Therefore, your cube leverage is now low.
before your next roll, he will drop, and you will have gained .05 in
equity.  Nice, but not nearly as nice as the .45 you could have gained by
doubling one roll earlier.  Therefore, it *may* have been correct to
double one roll earlier, despite the fact that you weren't quite at the .5
take point.

Now, let's consider the first position.  Your best roll gets you to .48.
will get you to .5, your point of maximum cube leverage.  Then, in the
initial position, it *cannot* be correct to double.  If things go your
way, you will definitely end up with higher cube leverage, unlike position
2.  (If things do not go your way, you will be happy you didn't double, of
course.)  Therefore, you should wait at least one roll to double.

The term "market loser" is often useful in these types of discussions.  A
market loser is a two-roll sequence that raises your equity above your
opponent's take point.  Using this term, the result of the previous
paragraph can be summarized in the following rule: "If you have no market
losers, do not double."

Unfortunately, some players incorrectly interpret this rule to be, "If you
have a market loser, double."  Following this incorrect rule, these
players often double in nearly equal positions, just because they have a
few market losers.  As we have discussed, such doubles increase equity
only a little, at best, and often decrease equity.  The overriding
criterion for offering a sound double is an equity that is close to your
opponent's take point.  The mere presence of market losers does *not*
imply that a double is correct.

I think the term "market loser" itself is misleading to many over-eager
doublers, because it sounds like something to be avoided at all costs.  If
such players double and their opponent drops, their first reaction is, "OH
NO!  I lost my market!  What an idiot!"  Remember, if you double and your
opponent drops, you lost your market, but YOU WON THE GAME.  Take your
money or points and start the next game!

You should definitely not worry about losing your market if you are going
to *barely* lose it.  For example, say your equity starts at .35, then
double, your opponent drops.  Should you be upset that you didn't double
in the initial position?  Not at all - in fact, doubling would have been a
mistake!  In that position, the best you could have done was increase your
equity by .35.  In the final position, you increased your equity by .4.
Therefore, you did *better* with the double that was dropped, despite the
fact that you lost your market!  By this reasoning, we can add a corollary
to the above rule:  "If your equity gain by doubling after losing your
market is always greater than your equity gain by doubling now, do not
double now."

As long as you keep the above caveats in mind, the concept of market
losers can be useful.  The trick is to compare your current equity with
your equity after the market losers.  If you are winning, but aren't that
close to your opponent's take point, you must have many "big" market
losers (those that raise your equity close to 1) before considering a
double.  If you are very close to your opponent's take point, just a few
big market losers can make a double correct.  In general, you shouldn't
even worry about "small" market losers -- rolls that barely lose your
market -- since you will still have high cube leverage after these rolls.
With many "medium" market losers, and an equity that is close, but not
that close, to your opponent's take point, you are in a gray area in which
judgement and experience take over!

2, after all, you have a roll that reduces your equity all the way to
-.60!  These types of rolls are known as "market crashers".  More
specifically, any roll that makes your cubeless equity negative (in other
words, that makes you an underdog) is a market crasher.  If you double,
your opponent accepts, and you roll a market crasher, you immediately wish
you could take back your double, since your negative equity is now twice
what it would have been if you did not double.  Even worse, your opponent
now controls the cube, which is often a big advantage, as we discussed
earlier.  Therefore, your double has turned into something of a disaster.
Obviously, market crashers should make you more reluctant to double.

Furthermore, market crashers can consist of sequences that are longer than
one roll.  In fact, any sequence of rolls that makes your equity negative
is a market crasher.  Therefore, you have market crashers in every
position in which your opponent has some chance to win, no matter how
small.  However, some market crashers are more important than others,
because of the importance of your opponent's cube leverage.

Let's think about a situation in which you have just had a double accepted
and have rolled a market crasher.  This is always bad, but is particularly
In this case, your opponent's cube leverage has suddenly reached its
maximum.  He will offer an efficient redouble, and your equity will fall
all the way to the neighborhood of -1.  Therefore, your double has become
a total disaster.  This type of market crasher is what I call a "big"
market crasher.  It should discourage you from doubling more than other
market crashers.  It should especially discourage you from redoubling,
since you could have prevented your opponent's efficient cube turn merely
by holding on to the cube for one more roll.

What if your market crasher reduces your equity to somewhere near -1 to
you had rolled a big market crasher, because your opponent's cube leverage
will be low.  He will double, and you will drop, but he will not have
gained much extra by doubling.  Meanwhile, this terrible roll was already
factored into your equity in the initial position, which means that your
non-market-crashers were that much better.  In other words, you lose big
in this case, but since you don't face an efficient redouble, at least you
don't lose more than the equity loss of the roll itself.  Therefore,
paradoxical though it may seem, this kind of disaster roll is actually a
"small" market crasher.  The other small market crasher is one that
reduces your equity to just a little less than zero.  Once again, you're
not happy about such a roll, but at least your opponent still has some
work to do before offering a double.

Now, in addition to focusing on big market crashers in a position, you
should be most concerned about 1-, 3-, and 5-roll market crashers.  These
"short-term" market crashers are the ones you can avoid by waiting to
double for a while.  Every position will have long-term, or residual,
market crashers.  In many positions, you should just accept these as
unavoidable, and worry more about the short-term market crashers.  The
reason the sequences with an odd number of rolls are important is because
your opponent can double after such sequences.  Looking at it another way,
if the first two rolls in a position (your roll, then your opponent's
roll) are a big market crasher, you will still get another roll to try to
salvage the position before your opponent can offer an efficient double.
Overall, the focus on short-term market crashers is so important that when
"market crasher" is used by itself, it usually means short-term market
crasher.  I will follow this convention from now on.

At this point, we can investigate doubling strategy and recube equity for
various types of positions, in terms of the advanced concepts in this
section.  In priming games, each player has few market losers but many
market crashers.  When playing such a game, you are trying to build and
roll forward a prime, which is a difficult task.  Any large doubles, or a
couple of large non-doubles in a row, are likely to crunch your prime or
give you a big timing disadvantage (remember, in a priming game, you want
to be behind in the race.)  Therefore, your position is always on the
brink of disaster.  Even worse, your market crashers are likely to be big.
You're cruising along, suddenly you roll double fours, and you partially
crunch.  You are by no means lost ... but here comes an efficient double,
which is as bad as a loss.  Meanwhile, if your opponent rolls one of her
market crashers, it is *not* likely to be a big market loser for you.  You
will probably have an efficient double.  The net result of all this is
that cube access is extremely important in priming games.  You therefore
want to be reluctant to give up that access by doubling, and willing to
accept a double based on recube vig, even if your cubeless equity is
significantly less than -.5.

Holding games represent the opposite extreme.  If you have escaped both of
7-point, and you have a healthy racing lead, you are likely to have an
excellent double.  Typically, you will have several market losers but no
(short-term) market crashers.  Even better, your long-term market crashers
are usually small.  You may leave a shot, but doing so is usually not a
market crasher, since you will still have positive equity.  Therefore,
your opponent cannot double.  If he hits the shot, he can then double, but
if he has managed to build a prime, his cube leverage will be low.  He
will be so much of a favorite that he won't gain much by doubling.  He
doubles, you drop, and you get on with the next game.  All in all, recube
equity is typically very low in holding games.

In racing games, the leader typically has many market losers but many
market crashers as well.  These tend to balance each other out, making it
best to use the default strategy of trying to sneak up on your opponent's
take point.  That take point in many positions, according to Robertie in
Robertie also states that you should double with a racing lead of 8% and
redouble with a lead of 9%.  At those points, your market losers have
increased and your market crashers have decreased enough to make a double
correct.  Since the window between the double point and the take point is
so small, many racing games end without any doubles being accepted.
Instead, a two-roll sequence often vaults a player past both the double
and take points, and that player doubles out his opponent.

Blitzes are a special case because, if you win, you typically win a
gammon.  If the blitz fails, however, your opponent will often become the
favorite in the game.  Blitzes are among the most difficult positions in
which to make rational cube decisions.  As an attacker, you might
intuitively feel that you are not good enough to double, and then one roll
later, feel that you are too good to double.  Although such a situation
happens occasionally, more often you will have a good double somewhere in
the middle of your blitz.  Therefore, you should be particularly careful
to estimate your equity.  "OK, let's see, I think I have 40% gammon wins,
10% single wins, 35% single losses, and 10% gammon lossses.  That works
out to be .35 equity.  Not nearly good enough to double."  As a defender,
you must also be careful to calculate equity.  Using this example, some
defenders get as far as the 40% gammon wins for their opponent, and
instantly drop if they are doubled, in a position in which they have a
very easy take.  Furthermore, recube equity is fairly high for the
defender, since if the blitz fails, she will be a cubeless favorite in the
game.  Therefore, the defender must strive to make an objective decision
about taking, despite the large chance of losing 4 points by being
gammoned on a 2 cube.

In bearoffs, volatility is typically huge, with market losers and market
crashers all over the place, in various combinations.  This fact makes it
difficult to figure out correct doubling strategy by reasoning from
general principles.  Various formulas exist to help, but fortunately,
close doubling decisions do not occur that often.  Of course, every once
in a while the cube can get turned three or four times in these types of
positions.  Therefore, a true expert must invest the time to understand
them.  The rest of us have too many other things to learn first!

Finally, when playing against a well-timed backgame, you should be in no
hurry to double.  Until you have brought all of your checkers around the
board and started to clear points, you have no market losers.  At that
point, you will have some market losers but also some market crashers.
Each category will often include small, medium, and big types.  The mix
within each category and the balance between the categories will often
change significantly from roll to roll. Furthermore, gammon and backgammon
possibilites change from roll to roll.  Meanwhile, if you are playing
against a questionably-timed backgame, you may have market losers, or
reach a position in which you can cash the game, before bringing all of
your men around or before clearing any points. For all of these reasons,
backgames are another area in which expert cube handlers can distinguish
themselves from all others!

To conclude this section, let's consider positions in which it may be
correct to play on for a gammon rather than cashing.  The most obvious
type of position in which you should play on for a gammon is one in which
your equity is already greater than 1.  In such a position, you are giving
away equity by doubling, assuming your opponent is bright enough to drop.
Even if you still have significant losing chances, you should play on
until your equity falls below 1.

The other type of position in which it is correct to play on for the
gammon is one in which your equity is greater than your opponent's take
point and you have no "cash crashers", that is, two-roll sequences that
positions, if you have any gammon chances whatsoever, you should roll
instead of doubling.  You can always cash next roll if a cash crasher
rears its ugly head at that point.

In most gammonish positions, the issue is not as clear-cut as in the above
two cases, and you must be careful to make a rational assessment of the
potential equity gains and losses of various actions.  Cash crashers, for
example, may be small, medium, or large.  A small cash crasher reduces
your equity to just below your opponent's take point.  Since you will
still have a good double, raising your equity almost back to 1, it is not
particularly troublesome.  A medium cash crasher reduces your equity to
quite a bit below your opponent's take point.  These sequences are much
worse, since you will not have a good double.  Big cash crashers, which
reduce your equity close to zero, and full-blown market crashers which
make you an underdog, are the worst of all.

For example, if your equity is .95, you are only gaining .05 by cashing.
You may therefore decide to play on, even with a few medium and big cash
crashers.  If your equity is .6, however, even one such cash crasher might
be enough to induce you to cash, since you can gain .4 by doing so.  There
are many gray areas here, but in general, if you have several cash
crashers, you have to be quite close to an equity of 1 and also have a
large chance of winning a gammon before you can justify playing on rather
than cashing.

Let's summarize this section.  With no market losers, do not double.  With
an equity close to your opponent's take point, several big market losers,
and no market crashers, a double is probably an excellent choice.  In the
many positions with both market losers and market crashers, consider the
number and type (small, medium, and large) of each one before making a
decision.  Finally, play on for the gammon rather than cashing if you have
no cash crashers, or your equity is greater than 1, or your equity is
close to 1 and you have few medium and big cash crashers.  All of this
applies strictly only if both you and your opponent are perfect players,
which, of course, is never true.  The next section will discuss practical
doubling strategy in a world in which both you and your opponent will
sometimes make mistakes.

Part V: Game Theory
-------------------

Game theory deals with decision-making in an imperfect world.  In the case
of a backgammon position, there are several imperfections that influence
doubling cube decisions.  Both you and your opponent may estimate the
equity of a position incorrectly.  Often, neither of you knows what the
other thinks the equity is, a very important gap in knowledge.
Imperfections in the play of both you and your opponent may affect the
equity of a position.  Once again, often neither of you will be aware of
your opponent's weaknesses, or your own.  Finally, in most cases, both of
you are human, and will occasionally make non-rational decisions.  For all
of these reasons, practical cube decisions in an imperfect world sometimes
differ from theoretically correct cube decisions.

The most common practical application of game theory is the bluff double.
This is a double in a position that, theoretically, is not good enough to
double.  If you double in such a situation, you hope to steal some equity
by inducing your opponent to drop incorrectly.  For example, say you have
an equity of .35, and after your opponent takes a double, your equity
would be .6 (because of recube vig).  If your opponent instead drops, you
have stolen .4 of equity, a big gain.  Meanwhile, if your opponent takes,
you still have a healthy lead.  The lower your equity, the bigger the
bluff - you have more to gain, but more to lose as well.  In any position
where you do not yet have a theoretically correct double, but you think
there is a good chance your opponent will drop, you should consider
turning the cube.

Entire books have been written on bluffing, but the basics are obvious.
When you bluff, you want to project an image of strength.  Therefore, if
you offer a bluff double, you should do so instantaneously and firmly.  If
you think about it for a while, you will probably ruin the bluff.  If you
are playing in person, you might even want to reach for the scoresheet, as
if there is no conceivable way your opponent could take.  You also want to
bluff when your opponent is feeling disappointed.  This situation often
occurs after he dances.  He's probably thinking, "Uh-oh.  Here comes the
gammon."  Whipping the cube around instantly can induce many incorrect
drops.  Again, in person, you might want to reach halfway towards the cube
after you hit your opponent, indicating that you intend to double even if
he enters.  If he dances, and you then double, you will have set him up
beautifully for an incorrect drop!

The opposite strategy from the bluff double is the delayed double.  You
should use this strategy if you have a theoretical cash, but you think
For example, if your have .58 equity and your opponent's theoretical take
point is .55, but you suspect she will take all the way up to .70, you
should consider waiting a while to double.  The arguments for waiting are
the same as those for sneaking up on the theoretically correct take point.
If things go your way, you will be able to gain more equity with a later
cube turn than if you double now.  If things don't go your way, you will

When you finally offer a delayed double, you should use the opposite
approach from the bluff double, that is, project weakness.  Think about
the double for a while.  Reach for the cube, then pull back.  Finally,
turn it like it is breaking your heart to do so ... and cringe (while
smiling inside) when your opponent snaps up the cube!

Another game theory double is the gammon double. You can use this when you
theoretically should play on for the gammon, but you double instead,
hoping that your opponent will take incorrectly.  The potential equity
gain from a gammon double is enormous, making it a good practical double
in many cases.  If your equity is only slightly greater than 1, you don't
give up much if your opponent drops, and you gain around a full point of
equity if he makes a mistake and takes.  Therefore, if you have any reason
to expect your opponent to take in such a position, a double is probably
wise.  It should, of course, be offered just like a delayed double,
tentatively rather than authoritatively.

From the above discussion, it's obvious that knowledge about your
opponent's tendencies can give you a large advantage.  If you know your
opponent is a reluctant taker in certain positions, you can offer early
doubles in those positions, gaining extra equity when he drops
incorrectly.  If you know he is a reluctant dropper in certain positions,
you can offer delayed doubles or gammon doubles, hoping to induce
incorrect takes.  If he is a "steamer" (someone who routinely accepts
hopeless doubles), squeeze the cube until it hurts ... then double, and
watch your equity skyrocket when he takes!

If you are playing an opponent for the first time, it might be worthwhile
to try various combinations of these game theory doubles to try to discern
her tendencies.  If she is verbose or has animated body language, you can
also guess her state of mind when considering a double.  Of course, she
also might be a hustler, sighing in despair to get you to offer an early
double!  Always remember, game theory strategies are very subtle and can
sometimes backfire.

Another game theory double is the money pressure double.  Objectively
speaking, the level of the cube and the status of the scoresheet should
have no effect on a cube decision.  In reality, someone who is already
down quite a bit of money, then faces a double for large stakes, might
drop incorrectly because they are afraid - or cannot afford - to lose at
the higher stakes.  On the other hand, they might take a hopeless double,
trying to recoup their losses in one fell swoop.  Similarly, someone well
ahead on the evening might drop a high cube to guarantee a winning
evening, or might take a hopeless one because they figure they will still
be even on the night if they lose.  Of all of these situations, the first
is almost certainly the most common.  Therefore, you might stand to gain
quite a bit by offering a theoretically unsound double to 16 or 32 to
someone who is behind on the evening or playing for stakes they cannot
afford.  (By the way, before you sit down to play a chouette, you might
want to ask yourself if you could take four 32-cubes in a gammonish
position without worrying about the money involved.  If not, consider
looking for a game with smaller stakes!)

Finally, you yourself are imperfect, and will often be uncertain about the
equity of a position.  Woolsey's Rule is often useful in such cases.  It
states:  "If you are unsure whether a position is theoretically a take or
a drop, you should double."  There are four relevant cases after such a
double.  In the first two, your opponent makes a mistake, either dropping
a take or taking a drop.  In both cases, you gain extra equity.  In the
third case, where your opponent drops a drop, your double was correct, and
all is well.  The fourth case, where your opponent takes a take, is the
only situation in which you might lose out.  In such cases, you might have
doubled too soon.  Therefore, Neil Kazaross has amended Woolsey's Rule
with what I call the Kazaross Corollary: "Follow Woolsey's Rule only if
you have market losers."  Personally, I like to add one more corollary:
"Follow Woolsey's Rule only if you feel that you  understand the
position."  As a beginner and low intermediate, I often reached positions
in which I didn't have a clue about my equity, even though I was pretty
sure I was ahead.  (I still reach such positions frequently today, alas.)
Applying Woolsey's Rule to such positions would have led to many premature
doubles, costing me large amounts of equity.  I think is it better to hold
off from turning the cube in such positions and wait for further
developments that might clarify your equity.

Well, I guess that's enough for this series on the doubling cube.  I hope
you enjoyed this document!  I'd like to hear your comments - send me an
email if you get a minute.  Good dice, and happy doubling!

Thanks,
USRobots
usrobots@aol.com
```

### Cube Handling

Against a weaker opponent  (Kit Woolsey, July 1994)
Closed board cube decisions  (Dan Pelton+, Jan 2009)
Cube concepts  (Peter Bell, Aug 1995)
Early game blitzes  (kruidenbuiltje, Jan 2011)
Early-late ratio  (Tom Keith, Sept 2003)
Endgame close out: Michael's 432 rule  (Michael Bo Hansen+, Feb 1998)
Endgame close out: Spleischft formula  (Simon Larsen, Sept 1999)
Endgame closeout: win percentages  (David Rubin+, Oct 2010)
Evaluating the position  (Daniel Murphy, Feb 2001)
Evaluating the position  (Daniel Murphy, Mar 2000)
How does rake affect cube actions?  (Paul Epstein+, Sept 2005)
How to use the doubling cube  (Michael J. Zehr, Nov 1993)
Liveliness of the cube  (Kit Woolsey, Apr 1997)
PRAT--Position, Race, and Threats  (Alan Webb, Feb 2001)
Playing your opponent  (Morris Pearl+, Jan 2002)
References  (Chuck Bower, Nov 1997)
Robertie's rule  (Chuck Bower, Sept 2006)
Rough guidelines  (Michael J. Zehr, Dec 1993)
The take/pass decision  (Otis+, Aug 2007)
Too good to double  (Michael J. Zehr, May 1997)
Too good to double--Janowski's formula  (Chuck Bower, Jan 1997)
Value of an ace-point game  (Raccoon+, June 2006)
Value of an ace-point game  (Øystein Johansen, Aug 2000)
Volatility  (Chuck Bower, Oct 1998)
Volatility  (Kit Woolsey, Sept 1996)
When to accept a double  (Daniel Murphy+, Feb 2001)
When to beaver  (Walter Trice, Aug 1999)
When to double  (Kit Woolsey, Nov 1994)
With the Jacoby rule  (KL Gerber+, Nov 2002)
With the Jacoby rule  (Gary Wong, Dec 1997)
Woolsey's law  (PersianLord+, Mar 2008)
Woolsey's law  (Kit Woolsey, Sept 1996)
Words of wisdom  (Chris C., Dec 2003)