For the purposes of my own records, I call this a containment cube,
although I usually reserve that term for positions in which the
doubler is behind in the pip count. Alternatively, you could call
this a "blitz versus prime" cube, but X doesn't win as many gammons
as in a typical blitz position, and a broken four-prime barely
qualifies as a prime in my book.

XGID=aBBB-DB------A--c-bccB-ab-:1:1:1:00:0:0:0:0:10

X:Player 2 O:Player 1
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O O | | O O X O O |
| O O | | O O X O |
| O | | O O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | X |
| | | X | +---+
| | | X X X X X | | 2 |
| | | X X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 99 O: 103 X-O: 0-0
Cube: 2, X own cube
X on roll, cube action

---
Tim Chow

XGID=aBBB-DB------A--c-bccB-ab-:1:1:1:00:0:0:0:0:10

X:Player 2 O:Player 1
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O O | | O O X O O |
| O O | | O O X O |
| O | | O O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | X |
| | | X | +---+
| | | X X X X X | | 2 |
| | | X X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 99 O: 103 X-O: 0-0
Cube: 2, X own cube
X on roll, cube action

I was astounded by how differently the given position and its variant
below rolled out.

Analyzed in Rollout
No redouble
Player Winning Chances: 72.01% (G:11.99% B:0.09%)
Opponent Winning Chances: 27.99% (G:4.06% B:0.12%)
Redouble/Take
Player Winning Chances: 72.05% (G:12.13% B:0.09%)
Opponent Winning Chances: 27.95% (G:4.63% B:0.14%)

Cubeful Equities:
No redouble: +0.723 (-0.110)
Redouble/Take: +0.834
Redouble/Pass: +1.000 (+0.166)

Best Cube action: Redouble / Take

Rollout:
1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller
Confidence No Double: ± 0.007 (+0.717..+0.730)
Confidence Double: ± 0.010 (+0.823..+0.844)

eXtreme Gammon Version: 2.19.207.pre-release

-------
Variant
-------

XGID=aBBB-CC------A--c-bccB-ab-:1:1:1:00:0:0:0:0:10

X:Player 2 O:Player 1
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O O | | O O X O O |
| O O | | O O X O |
| O | | O O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | |
| | | X X | +---+
| | | X X X X X | | 2 |
| | | X X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 100 O: 103 X-O: 0-0
Cube: 2, X own cube
X on roll, cube action

Analyzed in Rollout
No redouble
Player Winning Chances: 75.57% (G:15.51% B:0.13%)
Opponent Winning Chances: 24.43% (G:3.86% B:0.13%)
Redouble/Take
Player Winning Chances: 75.61% (G:15.56% B:0.11%)
Opponent Winning Chances: 24.39% (G:3.89% B:0.14%)

Cubeful Equities:
No redouble: +0.819 (-0.181)
Redouble/Take: +1.072 (+0.072)
Redouble/Pass: +1.000

Best Cube action: Redouble / Pass

Rollout:
1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller
Confidence No Double: ± 0.006 (+0.813..+0.824)
Confidence Double: ± 0.011 (+1.061..+1.083)

eXtreme Gammon Version: 2.19.207.pre-release

---
Tim Chow

On Tuesday, February 15, 2022 at 8:53:51 AM UTC-5, Tim Chow wrote:
> XGID=aBBB-DB------A--c-bccB-ab-:1:1:1:00:0:0:0:0:10
>
> X:Player 2 O:Player 1
> Score is X:0 O:0. Unlimited Game
> +13-14-15-16-17-18------19-20-21-22-23-24-+
> | X O O | | O O X O O |
> | O O | | O O X O |
> | O | | O O |
> | | | |
> | | | |
> | |BAR| |
> | | O | |
> | | | X |
> | | | X | +---+
> | | | X X X X X | | 2 |
> | | | X X X X X | +---+
> +12-11-10--9--8--7-------6--5--4--3--2--1-+
> Pip count X: 99 O: 103 X-O: 0-0
> Cube: 2, X own cube
> X on roll, cube action
> I was astounded by how differently the given position and its variant
> below rolled out.
>
> Analyzed in Rollout
> No redouble
> Player Winning Chances: 72.01% (G:11.99% B:0.09%)
> Opponent Winning Chances: 27.99% (G:4.06% B:0.12%)
> Redouble/Take
> Player Winning Chances: 72.05% (G:12.13% B:0.09%)
> Opponent Winning Chances: 27.95% (G:4.63% B:0.14%)
>
> Cubeful Equities:
> No redouble: +0.723 (-0.110)
> Redouble/Take: +0.834
> Redouble/Pass: +1.000 (+0.166)
>
> Best Cube action: Redouble / Take
>
> Rollout:
> 1296 Games rolled with Variance Reduction.
> Dice Seed: 271828
> Moves: 3-ply, cube decisions: XG Roller
> Confidence No Double: ± 0.007 (+0.717..+0.730)
> Confidence Double: ± 0.010 (+0.823..+0.844)
>
> eXtreme Gammon Version: 2.19.207.pre-release
>
> -------
> Variant
> -------
>
> XGID=aBBB-CC------A--c-bccB-ab-:1:1:1:00:0:0:0:0:10
> X:Player 2 O:Player 1
> Score is X:0 O:0. Unlimited Game
> +13-14-15-16-17-18------19-20-21-22-23-24-+
> | X O O | | O O X O O |
> | O O | | O O X O |
> | O | | O O |
> | | | |
> | | | |
> | |BAR| |
> | | O | |
> | | | |
> | | | X X | +---+
> | | | X X X X X | | 2 |
> | | | X X X X X | +---+
> +12-11-10--9--8--7-------6--5--4--3--2--1-+
> Pip count X: 100 O: 103 X-O: 0-0
> Cube: 2, X own cube
> X on roll, cube action
> Analyzed in Rollout
> No redouble
> Player Winning Chances: 75.57% (G:15.51% B:0.13%)
> Opponent Winning Chances: 24.43% (G:3.86% B:0.13%)
> Redouble/Take
> Player Winning Chances: 75.61% (G:15.56% B:0.11%)
> Opponent Winning Chances: 24.39% (G:3.89% B:0.14%)
>
> Cubeful Equities:
> No redouble: +0.819 (-0.181)
> Redouble/Take: +1.072 (+0.072)
> Redouble/Pass: +1.000
>
> Best Cube action: Redouble / Pass
>
> Rollout:
> 1296 Games rolled with Variance Reduction.
> Dice Seed: 271828
> Moves: 3-ply, cube decisions: XG Roller
> Confidence No Double: ± 0.006 (+0.813..+0.824)
> Confidence Double: ± 0.011 (+1.061..+1.083)
>
> eXtreme Gammon Version: 2.19.207.pre-release
>
> ---
> Tim Chow

Why? If you only take the numbers that make the full board immediately from the original to the variant you go from one [11] to three [21 11]. So you have two more rolls or ~2.78% on the first turn to have the nuts. That explains the bulk of the win % right there. You can add on top of that of course when the opponent enters with [42 41] or when they simply fan again and again you still have extra numbers to close your board that you wouldn't have had the other way.

Stick

On Wednesday, February 16, 2022 at 6:42:41 AM UTC, Stick Rice wrote:
> On Tuesday, February 15, 2022 at 8:53:51 AM UTC-5, Tim Chow wrote:
> > XGID=aBBB-DB------A--c-bccB-ab-:1:1:1:00:0:0:0:0:10
> >
> > X:Player 2 O:Player 1
> > Score is X:0 O:0. Unlimited Game
> > +13-14-15-16-17-18------19-20-21-22-23-24-+
> > | X O O | | O O X O O |
> > | O O | | O O X O |
> > | O | | O O |
> > | | | |
> > | | | |
> > | |BAR| |
> > | | O | |
> > | | | X |
> > | | | X | +---+
> > | | | X X X X X | | 2 |
> > | | | X X X X X | +---+
> > +12-11-10--9--8--7-------6--5--4--3--2--1-+
> > Pip count X: 99 O: 103 X-O: 0-0
> > Cube: 2, X own cube
> > X on roll, cube action
> > I was astounded by how differently the given position and its variant
> > below rolled out.
> >
> > Analyzed in Rollout
> > No redouble
> > Player Winning Chances: 72.01% (G:11.99% B:0.09%)
> > Opponent Winning Chances: 27.99% (G:4.06% B:0.12%)
> > Redouble/Take
> > Player Winning Chances: 72.05% (G:12.13% B:0.09%)
> > Opponent Winning Chances: 27.95% (G:4.63% B:0.14%)
> >
> > Cubeful Equities:
> > No redouble: +0.723 (-0.110)
> > Redouble/Take: +0.834
> > Redouble/Pass: +1.000 (+0.166)
> >
> > Best Cube action: Redouble / Take
> >
> > Rollout:
> > 1296 Games rolled with Variance Reduction.
> > Dice Seed: 271828
> > Moves: 3-ply, cube decisions: XG Roller
> > Confidence No Double: ± 0.007 (+0.717..+0.730)
> > Confidence Double: ± 0.010 (+0.823..+0.844)
> >
> > eXtreme Gammon Version: 2.19.207.pre-release
> >
> > -------
> > Variant
> > -------
> >
> > XGID=aBBB-CC------A--c-bccB-ab-:1:1:1:00:0:0:0:0:10
> > X:Player 2 O:Player 1
> > Score is X:0 O:0. Unlimited Game
> > +13-14-15-16-17-18------19-20-21-22-23-24-+
> > | X O O | | O O X O O |
> > | O O | | O O X O |
> > | O | | O O |
> > | | | |
> > | | | |
> > | |BAR| |
> > | | O | |
> > | | | |
> > | | | X X | +---+
> > | | | X X X X X | | 2 |
> > | | | X X X X X | +---+
> > +12-11-10--9--8--7-------6--5--4--3--2--1-+
> > Pip count X: 100 O: 103 X-O: 0-0
> > Cube: 2, X own cube
> > X on roll, cube action
> > Analyzed in Rollout
> > No redouble
> > Player Winning Chances: 75.57% (G:15.51% B:0.13%)
> > Opponent Winning Chances: 24.43% (G:3.86% B:0.13%)
> > Redouble/Take
> > Player Winning Chances: 75.61% (G:15.56% B:0.11%)
> > Opponent Winning Chances: 24.39% (G:3.89% B:0.14%)
> >
> > Cubeful Equities:
> > No redouble: +0.819 (-0.181)
> > Redouble/Take: +1.072 (+0.072)
> > Redouble/Pass: +1.000
> >
> > Best Cube action: Redouble / Pass
> >
> > Rollout:
> > 1296 Games rolled with Variance Reduction.
> > Dice Seed: 271828
> > Moves: 3-ply, cube decisions: XG Roller
> > Confidence No Double: ± 0.006 (+0.813..+0.824)
> > Confidence Double: ± 0.011 (+1.061..+1.083)
> >
> > eXtreme Gammon Version: 2.19.207.pre-release
> >
> > ---
> > Tim Chow
> Why? If you only take the numbers that make the full board immediately from the original to the variant you go from one [11] to three [21 11]. So you have two more rolls or ~2.78% on the first turn to have the nuts. That explains the bulk of the win % right there. You can add on top of that of course when the opponent enters with [42 41] or when they simply fan again and again you still have extra numbers to close your board that you wouldn't have had the other way.

Good problem and variant, Tim, and good commentary by Stick.
I think Tim's intuition behind his fainting with shock and wonder comes from the feeling that:
"This inflexibility in our position makes us worse than the variant, but not much worse."
However, this inflexibility is worse than usual because it's _permanent_ --- that's the important point.
We can't legally move back from our 5 point to our 6 point.
However, the intuition from our memory bank comes from remembering positions where (for example) an inflexibility
like 4 on the six and 2 on the 5 wasn't hugely worse than 3 on the six and 3 on the 5. That inflexibility
gets removed when we roll an ace, so it isn't so serious.
This type of permanent inflexibility is much rarer than a removable inflexibility so it skews the intuition of non-experts.

Paul

On 2/16/2022 1:42 AM, Stick Rice wrote:
> Why? If you only take the numbers that make the full board immediately from the original to the variant you go from one [11] to three [21 11]. So you have two more rolls or ~2.78% on the first turn to have the nuts. That explains the bulk of the win % right there.

I don't buy this explanation. In the original position, if X rolls 21,
he doesn't lose immediately.

---
Tim Chow

On 2/16/2022 8:21 AM, Timothy Chow wrote:
> On 2/16/2022 1:42 AM, Stick Rice wrote:
>> Why?  If you only take the numbers that make the full board
>> immediately from the original to the variant you go from one [11] to
>> three [21 11].  So you have two more rolls or ~2.78% on the first turn
>> to have the nuts.  That explains the bulk of the win % right there.
>
> I don't buy this explanation.  In the original position, if X rolls 21,
> he doesn't lose immediately.

Maybe a better way to make my point is the variant position below.
I don't believe that the swing on 21 is the primary explanation.

XGID=aBBB-BD------A--c-bccB-ab-:1:1:1:00:0:0:0:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O O | | O O X O O |
| O O | | O O X O |
| O | | O O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | X |
| | | X | +---+
| | | X X X X X | | 2 |
| | | X X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 101 O: 103 X-O: 0-0
Cube: 2, X own cube
X on roll, cube action

Analyzed in XG Roller+
Player Winning Chances: 74.14% (G:14.87% B:0.09%)
Opponent Winning Chances: 25.86% (G:3.72% B:0.10%)

Cubeless Equities: No Double=+0.594, Double=+1.199

Cubeful Equities:
No redouble: +0.801 (-0.199)
Redouble/Take: +1.028 (+0.028)
Redouble/Pass: +1.000

Best Cube action: Redouble / Pass

eXtreme Gammon Version: 2.19.207.pre-release

---
Tim Chow

On Wednesday, February 16, 2022 at 3:02:53 PM UTC, Tim Chow wrote:
> On 2/16/2022 8:21 AM, Timothy Chow wrote:
> > On 2/16/2022 1:42 AM, Stick Rice wrote:
> >> Why? If you only take the numbers that make the full board
> >> immediately from the original to the variant you go from one [11] to
> >> three [21 11]. So you have two more rolls or ~2.78% on the first turn
> >> to have the nuts. That explains the bulk of the win % right there.
> >
> > I don't buy this explanation. In the original position, if X rolls 21,
> > he doesn't lose immediately.
> Maybe a better way to make my point is the variant position below.
> I don't believe that the swing on 21 is the primary explanation.
>
> XGID=aBBB-BD------A--c-bccB-ab-:1:1:1:00:0:0:0:0:10
>
> X:Player 1 O:Player 2
> Score is X:0 O:0. Unlimited Game
> +13-14-15-16-17-18------19-20-21-22-23-24-+
> | X O O | | O O X O O |
> | O O | | O O X O |
> | O | | O O |
> | | | |
> | | | |
> | |BAR| |
> | | O | |
> | | | X |
> | | | X | +---+
> | | | X X X X X | | 2 |
> | | | X X X X X | +---+
> +12-11-10--9--8--7-------6--5--4--3--2--1-+
> Pip count X: 101 O: 103 X-O: 0-0
> Cube: 2, X own cube
> X on roll, cube action
> Analyzed in XG Roller+
> Player Winning Chances: 74.14% (G:14.87% B:0.09%)
> Opponent Winning Chances: 25.86% (G:3.72% B:0.10%)
>
> Cubeless Equities: No Double=+0.594, Double=+1.199
>
> Cubeful Equities:
> No redouble: +0.801 (-0.199)
> Redouble/Take: +1.028 (+0.028)
> Redouble/Pass: +1.000
>
> Best Cube action: Redouble / Pass
> eXtreme Gammon Version: 2.19.207.pre-release

But I explained this, didn't I??
My analysis is totally consistent with this result.
The point (that I've made before even while describing exactly this position)
is that the new variant allows you to restore flexibility with your next ace.

I also don't agree that you are rebutting Stick in any sensible way.
If you change a position so that 21 switches from being a loser to a winner.
Then the difference is 2/18 = 1/9 = 11%.
That situation doesn't pertain here (as you say) and that is why Stick attributed
the 21 swing to 2.78% rather than 11%.
In your new variant, we don't make the point by 21 but we do have an extra pointmaker
-- 22 and 11 instead of just 11.
Further, as Stick and I said, the new variant allows us to roll an ace to prepare for 21 later.

Paul

On 2/16/2022 11:48 AM, peps...@gmail.com wrote:
> But I explained this, didn't I??
> My analysis is totally consistent with this result.
> The point (that I've made before even while describing exactly this position)
> is that the new variant allows you to restore flexibility with your next ace.
>
> I also don't agree that you are rebutting Stick in any sensible way.
> If you change a position so that 21 switches from being a loser to a winner.
> Then the difference is 2/18 = 1/9 = 11%.
> That situation doesn't pertain here (as you say) and that is why Stick attributed
> the 21 swing to 2.78% rather than 11%.

You're right that I initially missed that Stick said 2.78%.
Nevertheless, note that the equity difference between my original
position and my original variant is about 1.07 - 0.83 = 0.24.
Now suppose we turn the cube and ask XG for the equity after X
rolls 21 in each case. We get approximately 1.00 in the original
position and 0.22 in the original variant, but we have to multiply
these by 2 because XG divides by the cube value. So the swing is
about 2.00 - 0.44 = 1.56, and dividing by 18 gives about .087, which
does not account for the "bulk of" the equity difference.

Stick did say the win % rather than the equity difference; I'm not
sure what calculation he did, but in any case, my point is that the
swing on an immediate 21 isn't the bulk of the explanation.

> In your new variant, we don't make the point by 21 but we do have an extra pointmaker
> -- 22 and 11 instead of just 11.
> Further, as Stick and I said, the new variant allows us to roll an ace to prepare for 21 later.

Fair enough; I hadn't thought about how the new variant allows a
good use of an ace. Nevertheless I am surprised by the size of
the equity difference. I passed the cube, and if I had been told
that my original variant was approximately a 1.07 pass, I would
have guessed that my original position was at best a 0.95 take.
If it *had* been a 0.95 take then I would have accepted that the
huge immediate swing on 21 was the main point and left it at that.

---
Tim Chow

On 2022-02-16, peps...@gmail.com <pepstein5@gmail.com> wrote:

> But I explained this, didn't I??
> My analysis is totally consistent with this result.
> The point (that I've made before even while describing exactly this position)
> is that the new variant allows you to restore flexibility with your next ace.

One can add another variant and get a finer gradation. Looking at the
pattern for points 5 to 7:

420 Initial position: 1 joker, inflexible, unrepairable
222 1 joker, inflexible, very repairable
240 2 jokers, inflexible, repairable
330 3 jokers, flexible