# Value of the cube?

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### midas

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Dec 12, 1997, 3:00:00 AM12/12/97
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Having read the article posted by Gary Wong which showed the value of
the cube, a part of which I have quoted below. I decided to do some
rollouts and it seems the figures dont agree although I may have
overlooked something. Here's part of the original article and
following are the rollout results. Could someone explain the
discrepancies ? Here's part of the original article:

GARY WONG WROTE:
=================================================================

cube owning equity

0.5 +
|
0.4 + - - - - - - - - - - - - - - - - - - -,* - - - - -
| _,-`~ |\
0.3 + _,-`~ `,
| _,-`~ | \
0.2 + _,-`~ \
| _,-`~ | \
0.1 + _,*`~ `,
| _,-`~ | \
0.0 *----+----+----+----+----+----+----+----+----+----*
| | | | |
0 10 20 30 40 50 60 70 80 90 100% Winchances
| | | | |
-1 -0.6 0 0.6 1

cubeless equity

NB: below your take point, the cube is worth nothing to you if your
opponent also has cube access, because you can expect to be cubed out
immediately.

You want to maximise the sum of the cubeless equity + cube owning
equity.
As examples of how to calculate these: at the start of the game, you
have 50/50 winning chances for 0 cubeless equity; and the centred cube
is worth 0.25 points to you minus 0.25 points to your opponent (since
you both have access) for a total sum of 0 (as you would expect). If
you own a 2-cube with 30% winning chances, your cubeless equity in the
game is 0.3 - 0.7 = -0.4;
your opponent has no cube owning equity at all and you have 0.15, for
a total equity of -0.25 -- which, at a 2-cube, leaves you expecting to
lose on average half a point per game.
======================================================================

Here's some rollout results(14400 settlement limit .60 )I got from
race positions:

Pip Count MWC C/less Center Owned NotOwned
Leader/Trailer

104/100 50.7 .013 .022 .155 -.121
100/101 60.1 .202 .292 .378 .110
96/99 64.0 .28 .4 .488 .182
93/99 69.4 .388 .547 .602 .321
121/132 74.6 .496 .721 .762 .43

So unless I've calculated something wrong here the following seems to
be true:
When cube is in center Gary Wong's model seems to be correct

However owning the cube for both sides seems to be worth LESS than
expected. Can someone explain this please?

### Chuck Bower

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Dec 15, 1997, 3:00:00 AM12/15/97
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Good observation. This is a VERY OLD (OK, 20 year old) problem
which was well understood back then (see articles by Keeler and Spencer,
and also by Yadeh and Kobliska). What Gary illustrates is known as the
"continuous model" of backgammon. Here you NEVER lose your market.
You never cross over your opponent's drop/take point. If you get there
at all, you must land exactly on it. Then, with proper cube strategy,
you double and it doesn't matter whether your opp takes or passes, in
the long run the result is the same--you receive the value of the cube
before it was turned.

Real backgammon is DIScontinuous. Equity changes in chunks, and
sometimes the chunks are huge. In races they tend to be less than in
contact positions, but they still exist. (For example, one player
rolls big doubles, the other doesn't.) All of this has led (again,
many years ago) to the concept of "losing one's market". It is often
technically correct to double your opponent "IN", that is offer
the cube when the opponent has a take. If you wait a roll and things
go really well for you, you've lost some of your previously calculated
equity. (This argument can be made mathematically rigorous, but who
besides Bob Koca has the time to do that!)

There is a second thing which your rollouts illustrate which further
makes the cube ownership equity less for JF level-5 cubeless rollouts:
here JF NEVER doubles the opponent in, thus JF ALWAYS loses its market.
In real backgammon, playing optimally, you lose a bit of cube equity
by doubling your opponent in, but you lose even more cube equity if you
decide NEVER to lose your market (as JF does for the level-5 limited cube
rollouts which you used).

Hopefully the next version of JF will have FULL CUBE rollouts. Then
you can rerun your study positions and see that "REAL BG" is somewhere
between your results and Gary's.

Chuck
bo...@bigbang.astro.indiana.edu
c_ray on FIBS

### mi...@depreli.demon.co.uk

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Dec 16, 1997, 3:00:00 AM12/16/97
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On 15 Dec 1997 17:40:41 GMT, bo...@bigbang.astro.indiana.edu (Chuck
Bower) wrote:

Im reading the JellyFish manual and it says......
The cube is only used for cashing or settling games, so cubes are
never accepted. This is done to reduce the random error of the
results. The game is cashed when Jellyfish' estimated equity exceeds a
predefined settlement limit ,set by the user. The default value of the
settlement limits is 0.55. In most positions cubes can be taken with a
cubeless equity highet than this, because the cube has value in
itself. For this reason the player with the advantage gets to cash
some positions which are takeable. But on the other hand he will
sometimes lose equity by not doubling when the equity is close to 0.5,
which is often correct. The default value is chosen in the hope that
these effects will cancel each other out.

So is the above statement about these effects canceling each other out
not valid then or have I misunderstood somewhere?

### Kit Woolsey

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Dec 16, 1997, 3:00:00 AM12/16/97
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mi...@depreli.demon.co.uk wrote:
: Im reading the JellyFish manual and it says......

: The cube is only used for cashing or settling games, so cubes are
: never accepted. This is done to reduce the random error of the
: results. The game is cashed when Jellyfish' estimated equity exceeds a
: predefined settlement limit ,set by the user. The default value of the
: settlement limits is 0.55. In most positions cubes can be taken with a
: cubeless equity highet than this, because the cube has value in
: itself. For this reason the player with the advantage gets to cash
: some positions which are takeable. But on the other hand he will
: sometimes lose equity by not doubling when the equity is close to 0.5,
: which is often correct. The default value is chosen in the hope that
: these effects will cancel each other out.

: So is the above statement about these effects canceling each other out
: not valid then or have I misunderstood somewhere?

In pre-Jellyfish days, suppose you had a position which you were very
interested in what the equity was -- say, whether it was a pass or a take.
How would you go about determining it? You would roll the position out a
couple of hundred times by hand, and see what the total results were.

Since the cube is part of the game, you would have to take it into
account. So if during the rollout you thought the guy who owned the cube
had a double you would have him redouble to 4 (he owned the cube at 2
originally, since it was a pass/take question), and if you thought the
other guy had a take you would have him take and continue the rollout
with the cube on 4.

The problem with this approach is that there might be a couple of 8 or 16
cubes in the rollout. While if you are playing the position as a
proposition for money these high cubes are definitely a part of the
action, for the rollout purposes one or two high cubes can distort the
results. So, what can we do about this?

The generally accepted practice is to decide whether it is the double or
the take which is the question (on the first redouble). If you judge
that is is a marginal redouble, you just continue the rollout with the
redouble having not been made. On the other hand, if you judge that it
is a clear redouble but a marginal take, you assume that it was a pass
and score it up as a 2-point win for the redoubler. This way, you avoid
large cubes. The idea is that sometimes you will fail to redouble when
you should, but other times you will pass a redouble you should take.
Hopefully, the equity swings from these decisions will cancel out in the
long run.

When Jellyfish does rollouts with the cube in play, it is doing exactly
the same thing. No doubles up to a point, and above that point it is
scored as double and pass.

Where does the settlement figure of .55 come from? Good old trial and
error. In fact when Fredrik first came up with the idea of rollouts with
the cube he guessed at a .50 settlement figure, and sent me a
beta-version to test out. Rollouts of "known" positions convinced us
that the .50 figure was too low, and after a lot of experimentation it
appeared that .55 gave the most accurate results for most positions. Of
course the user can make the settlement number what he wishes.

I feel that since for Jellyfish rollouts we can do a few thousand if
necessary (as opposed to the couple of hundred if doing it by hand) we can
afford some moderately high cubes, since the sample size should be large
enough to cut down on the luck factor. Perhaps the user should be able
to set the cube limit -- then in the rollout the program allows the cube
to go up to that limit, and uses the settlement approach to make sure it
doesn't get any higher. But that is a matter for the next version of
Jellyfish.

Kit

### Chuck Bower

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Dec 17, 1997, 3:00:00 AM12/17/97
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In article <3496c9d...@news.demon.co.uk>,
<mi...@depreli.demon.co.uk> wrote:

(snip)

>Im reading the JellyFish manual and it says......
>The cube is only used for cashing or settling games, so cubes are
>never accepted. This is done to reduce the random error of the
>results. The game is cashed when Jellyfish' estimated equity exceeds a
>predefined settlement limit ,set by the user. The default value of the
>settlement limits is 0.55. In most positions cubes can be taken with a
>cubeless equity highet than this, because the cube has value in
>itself. For this reason the player with the advantage gets to cash
>some positions which are takeable. But on the other hand he will
>sometimes lose equity by not doubling when the equity is close to 0.5,
>which is often correct. The default value is chosen in the hope that
>these effects will cancel each other out.
>
>So is the above statement about these effects canceling each other out
>not valid then or have I misunderstood somewhere?

Probably both. First off, in Midas's study (which showed that Gary
Wong's continuous model and JF level-5 limited cube rollouts were
inconsistent FOR THE CONDITIONS HE CHOSE), he used a settlement limit
of 0.6, which is the correct value for CONTINUOUS backgammon.

The 0.55 settlement limit default is the proper setting for "typical"
positions, but it's not always optimal. And even when it is optimized,
it's still not going to give as accurate a picture as real backgammon
played with a real cube. Full cube rollouts are the answer, but the
only commercially available software which does rollouts (Jellyfish)
doesn't as yet do full cube rollouts. Hopefully v4.0....

### Gary Wong

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Dec 22, 1997, 3:00:00 AM12/22/97
to

(Sorry for taking so long to respond; I've been out of the country and beyond
the reach of Usenet for two weeks.)

midas writes:
> Having read the article posted by Gary Wong which showed the value of
> the cube, a part of which I have quoted below. I decided to do some
> rollouts and it seems the figures dont agree although I may have
> overlooked something. Here's part of the original article and
> following are the rollout results. Could someone explain the
> discrepancies ? Here's part of the original article:
>
> GARY WONG WROTE:
> =================================================================
>
> cube owning equity
>
> 0.5 +
> |
> 0.4 + - - - - - - - - - - - - - - - - - - -,* - - - - -
> | _,-`~ |\
> 0.3 + _,-`~ `,
> | _,-`~ | \
> 0.2 + _,-`~ \
> | _,-`~ | \
> 0.1 + _,*`~ `,
> | _,-`~ | \
> 0.0 *----+----+----+----+----+----+----+----+----+----*
> | | | | |
> 0 10 20 30 40 50 60 70 80 90 100% Winchances
> | | | | |
> -1 -0.6 0 0.6 1
>
> cubeless equity
>

> ======================================================================
>
> Here's some rollout results(14400 settlement limit .60 )I got from
> race positions:
>
> Pip Count MWC C/less Center Owned NotOwned
> Leader/Trailer
>
> 104/100 50.7 .013 .022 .155 -.121
> 100/101 60.1 .202 .292 .378 .110
> 96/99 64.0 .28 .4 .488 .182
> 93/99 69.4 .388 .547 .602 .321
> 121/132 74.6 .496 .721 .762 .43
>
> So unless I've calculated something wrong here the following seems to
> be true:
> When cube is in center Gary Wong's model seems to be correct
>
>
> However owning the cube for both sides seems to be worth LESS than
> expected. Can someone explain this please?

My guess is that there's two reasons: Jellyfish is giving you results which
are somewhat lower than the true (cube owning) equities; and using the model
above (maximum cube equity 0.4 at 80%) really represents an upper bound on
the cube equity and not its true value. I would argue that the `real'
equity is somewhere in between.

Firstly, I think using a settlement value of 0.6 is a bit on the high side,
especially in a non-volatile position like a long race. I would be inclined
to use a value like 0.55 or even lower. Although it's difficult to justify
a particular value from a theoretical basis, using a lower value would
increase the benefit of owning the cube (since you can claim more easily) and
raise your results a little bit.

Secondly, the model I gave makes one or two assumptions which mean it is not
directly appliciable to these positions. You snipped part of the answer to
your question out of your quote; what that graph really said was:

> VALUE OF ACCESS TO THE CUBE FOR VARIOUS WINNING CHANCES
> assuming a perfectly live cube in a gammonless game
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

By "perfectly live" I mean that whoever has access to the cube will be able
to make perfectly efficient doubles, ie. right on their opponent's drop
point. Since it is impossible to get more leverage from the cube than this,
it represents an upper bound on the cube ownership benefit (the lower bound
is no benefit at all, holding a totally dead cube -- eg. the last roll of
the game). More volatile positions tend to be less efficient. It's
difficult (or at least I don't know how) to quantitatively define the
cube efficiencies into the future of the game, but you could take a stab in
the dark with an estimate like "if X owns the cube in this position, he is
likely to be able to double with a winning probablility in the range 60-85%,
for an average efficiency of 87.5%" and so lop off 12.5% of the full
theoretical value to compensate for the inefficiency.

If you lower the settlement value in Jellyfish, and scale a fudge factor for
cube inefficiency into the cube equity value, then I expect you could
certainly get the two equities to agree.

Cheers,
Gary (GaryW on FIBS).
--
Gary Wong, Computer Science Department, University of Auckland, New Zealand
ga...@cs.auckland.ac.nz http://www.cs.auckland.ac.nz/~gary/

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