XGID=aBBBBBB-B--aa-a--bbcbb--A-:0:0:1:00:2:2:3:0:10
X:Daniel O:eXtremeGammon

Score is X:2 O:2. Unlimited Game, Jacoby Beaver
+13-14-15-16-17-18------19-20-21-22-23-24-+
| O O O | | O O O X |
| O O | | O O O |
| | | O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | |
| | | |
| X | | X X X X X X |
| O O X | | X X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 82 O: 129 X-O: 2-2
Cube: 1
X on roll, cube action

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 7/16/2022 5:35 PM, peps...@gmail.com wrote:

XGID=aBBBBBB-B--aa-a--bbcbb--A-:0:0:1:00:2:2:3:0:10
X:Daniel O:eXtremeGammon

Score is X:2 O:2. Unlimited Game, Jacoby Beaver
+13-14-15-16-17-18------19-20-21-22-23-24-+
| O O O | | O O O X |
| O O | | O O O |
| | | O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | |
| | | |
| X | | X X X X X X |
| O O X | | X X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 82 O: 129 X-O: 2-2
Cube: 1
X on roll, cube action

I don't think X has enough spare pips to double here. O's outfield
blots give me pause, because they increase X's gammon wins, but I
still don't think it's enough for a double.

---
Tim Chow

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Sunday, July 17, 2022 at 3:36:51 AM UTC+1, Tim Chow wrote:

On 7/16/2022 5:35 PM, peps...@gmail.com wrote:

XGID=aBBBBBB-B--aa-a--bbcbb--A-:0:0:1:00:2:2:3:0:10
X:Daniel O:eXtremeGammon

Score is X:2 O:2. Unlimited Game, Jacoby Beaver +13-14-15-16-17-18------19-20-21-22-23-24-+
| O O O | | O O O X |
| O O | | O O O |
| | | O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | |
| | | |
| X | | X X X X X X |
| O O X | | X X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 82 O: 129 X-O: 2-2
Cube: 1
X on roll, cube action

I don't think X has enough spare pips to double here. O's outfield
blots give me pause, because they increase X's gammon wins, but I
still don't think it's enough for a double.

This could be a (somewhat) rare case where Jacoby is relevant.
I haven't rolled out the non-Jacoby case.

Paul

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Sunday, July 17, 2022 at 3:36:51 AM UTC+1, Tim Chow wrote:

On 7/16/2022 5:35 PM, peps...@gmail.com wrote:

XGID=aBBBBBB-B--aa-a--bbcbb--A-:0:0:1:00:2:2:3:0:10
X:Daniel O:eXtremeGammon

Score is X:2 O:2. Unlimited Game, Jacoby Beaver +13-14-15-16-17-18------19-20-21-22-23-24-+
| O O O | | O O O X |
| O O | | O O O |
| | | O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | |
| | | |
| X | | X X X X X X |
| O O X | | X X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 82 O: 129 X-O: 2-2
Cube: 1
X on roll, cube action

I don't think X has enough spare pips to double here. O's outfield
blots give me pause, because they increase X's gammon wins, but I
still don't think it's enough for a double.

---
Tim Chow

Do you see it as an ND/beaver or an ND/T?
(Try and pretend that you didn't see my follow-up post.)

Paul

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 7/17/2022 7:14 AM, peps...@gmail.com wrote:

Do you see it as an ND/beaver or an ND/T?
(Try and pretend that you didn't see my follow-up post.)

ND/T. This doesn't look like a Kauder paradox to me.

---
Tim Chow

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Monday, July 18, 2022 at 8:22:45 AM UTC+1, peps...@gmail.com wrote:

On Monday, July 18, 2022 at 12:53:53 AM UTC+1, Tim Chow wrote:

On 7/17/2022 7:14 AM, peps...@gmail.com wrote:

Do you see it as an ND/beaver or an ND/T?
(Try and pretend that you didn't see my follow-up post.)

ND/T. This doesn't look like a Kauder paradox to me.

Holding is a big blunder. I get only just over 50% wins but approx 60% of the wins are gammons.
This position has the (somewhat) unusual trait in that you have to double, even
though you don't have that big an edge.
I was (wrongly) quite confident in the hold.
I will post the rollout later -- it's on my other computer.

Paul

There are two obvious variantizations -- non-Jacoby or owning the cube.
These are probably holds but I have difficulties setting up positions (as opposed
to just exporting positions that I've already met) so I won't be
doing rollouts for these.

Paul

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Monday, July 18, 2022 at 12:53:53 AM UTC+1, Tim Chow wrote:

On 7/17/2022 7:14 AM, peps...@gmail.com wrote:

Do you see it as an ND/beaver or an ND/T?
(Try and pretend that you didn't see my follow-up post.)

ND/T. This doesn't look like a Kauder paradox to me.

Holding is a big blunder. I get only just over 50% wins but approx 60% of the wins are gammons.
This position has the (somewhat) unusual trait in that you have to double, even though you don't have that big an edge.
I was (wrongly) quite confident in the hold.
I will post the rollout later -- it's on my other computer.

Paul

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 7/18/2022 3:25 AM, peps...@gmail.com wrote:

There are two obvious variantizations -- non-Jacoby or owning the cube.
These are probably holds but I have difficulties setting up positions (as opposed
to just exporting positions that I've already met) so I won't be
doing rollouts for these.

According to the rollouts below, the non-Jacoby is a borderline D/ND
and X owning the cube is a clear hold. Very nice position for
illustrating the differences among these three scenarios!

------------------------
No Jacoby, centered cube
------------------------

XGID=aBBBBBB-B--aa-a--bbcbb--A-:0:0:1:00:0:0:0:0:10

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-+
| O O O | | O O O X |
| O O | | O O O |
| | | O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | |
| | | |
| X | | X X X X X X |
| O O X | | X X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 82 O: 129 X-O: 0-0
Cube: 1
X on roll, cube action

Analyzed in Rollout
No double
Player Winning Chances: 51.91% (G:37.28% B:0.71%)
Opponent Winning Chances: 48.09% (G:4.52% B:0.14%)
Double/Take
Player Winning Chances: 52.08% (G:37.19% B:0.63%)
Opponent Winning Chances: 47.92% (G:4.53% B:0.14%)

Cubeless Equities: No Double=+0.372, Double=+0.746

Cubeful Equities:
No double: +0.288 (-0.014)
Double/Take: +0.303
Double/Pass: +1.000 (+0.697)

Best Cube action: Double / Take

Rollout:
1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 4-ply, cube decisions: XG Roller+
Search interval: Large
Confidence No Double: ± 0.008 (+0.280..+0.296)
Confidence Double: ± 0.015 (+0.288..+0.317)

eXtreme Gammon Version: 2.19.207.pre-release

-----------
X owns cube
-----------

XGID=aBBBBBB-B--aa-a--bbcbb--A-: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-+
| O O O | | O O O X |
| O O | | O O O |
| | | O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | |
| | | | +---+
| X | | X X X X X X | | 2 |
| O O X | | X X X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 82 O: 129 X-O: 0-0
Cube: 2, X own cube
X on roll, cube action

Analyzed in Rollout
No redouble
Player Winning Chances: 51.65% (G:37.42% B:0.71%)
Opponent Winning Chances: 48.35% (G:4.60% B:0.13%)
Redouble/Take
Player Winning Chances: 52.08% (G:37.19% B:0.63%)
Opponent Winning Chances: 47.92% (G:4.53% B:0.14%)

Cubeful Equities:
No redouble: +0.468
Redouble/Take: +0.303 (-0.165)
Redouble/Pass: +1.000 (+0.532)

Best Cube action: No redouble / Take
Percentage of wrong pass needed to make the double decision right: 19.2%

Rollout:
1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 4-ply, cube decisions: XG Roller+
Search interval: Large
Confidence No Double: ± 0.005 (+0.463..+0.473)
Confidence Double: ± 0.015 (+0.288..+0.317)

eXtreme Gammon Version: 2.19.207.pre-release

---
Tim Chow

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Tuesday, July 19, 2022 at 5:36:01 AM UTC+1, Tim Chow wrote:

On 7/18/2022 3:25 AM, peps...@gmail.com wrote:

There are two obvious variantizations -- non-Jacoby or owning the cube. These are probably holds but I have difficulties setting up positions (as opposed
to just exporting positions that I've already met) so I won't be
doing rollouts for these.

According to the rollouts below, the non-Jacoby is a borderline D/ND
and X owning the cube is a clear hold. Very nice position for
illustrating the differences among these three scenarios!

------------------------
No Jacoby, centered cube
------------------------

XGID=aBBBBBB-B--aa-a--bbcbb--A-:0:0:1:00:0:0:0:0:10

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-+
| O O O | | O O O X |
| O O | | O O O |
| | | O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | |
| | | |
| X | | X X X X X X |
| O O X | | X X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 82 O: 129 X-O: 0-0
Cube: 1
X on roll, cube action
Analyzed in Rollout
No double
Player Winning Chances: 51.91% (G:37.28% B:0.71%)
Opponent Winning Chances: 48.09% (G:4.52% B:0.14%)
Double/Take
Player Winning Chances: 52.08% (G:37.19% B:0.63%)
Opponent Winning Chances: 47.92% (G:4.53% B:0.14%)

Cubeless Equities: No Double=+0.372, Double=+0.746

Cubeful Equities:
No double: +0.288 (-0.014)
Double/Take: +0.303
Double/Pass: +1.000 (+0.697)

Best Cube action: Double / Take

Rollout:
1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 4-ply, cube decisions: XG Roller+
Search interval: Large
Confidence No Double: ± 0.008 (+0.280..+0.296)
Confidence Double: ± 0.015 (+0.288..+0.317)

eXtreme Gammon Version: 2.19.207.pre-release

-----------
X owns cube
-----------

XGID=aBBBBBB-B--aa-a--bbcbb--A-: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-+
| O O O | | O O O X |
| O O | | O O O |
| | | O |
| | | |
| | | |
| |BAR| |
| | O | |
| | | |
| | | | +---+
| X | | X X X X X X | | 2 |
| O O X | | X X X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 82 O: 129 X-O: 0-0
Cube: 2, X own cube
X on roll, cube action
Analyzed in Rollout
No redouble
Player Winning Chances: 51.65% (G:37.42% B:0.71%)
Opponent Winning Chances: 48.35% (G:4.60% B:0.13%)
Redouble/Take
Player Winning Chances: 52.08% (G:37.19% B:0.63%)
Opponent Winning Chances: 47.92% (G:4.53% B:0.14%)

Cubeful Equities:
No redouble: +0.468
Redouble/Take: +0.303 (-0.165)
Redouble/Pass: +1.000 (+0.532)

Best Cube action: No redouble / Take
Percentage of wrong pass needed to make the double decision right: 19.2%

Rollout:
1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 4-ply, cube decisions: XG Roller+
Search interval: Large
Confidence No Double: ± 0.005 (+0.463..+0.473)
Confidence Double: ± 0.015 (+0.288..+0.317)

eXtreme Gammon Version: 2.19.207.pre-release

---
Tim Chow

Glad you liked it. Time for me to follow on with the original rollout. Analyzed in Rollout
No double
Player Winning Chances: 51.56% (G:35.79% B:0.66%)
Opponent Winning Chances: 48.44% (G:4.44% B:0.12%)
Double/Take
Player Winning Chances: 52.13% (G:37.44% B:0.66%)
Opponent Winning Chances: 47.87% (G:4.52% B:0.12%)

Cubeless Equities: No Double=+0.350, Double=+0.754

Cubeful Equities:
No double: +0.185 (-0.123)
Double/Take: +0.308
Double/Pass: +1.000 (+0.692)

Best Cube action: Double / Take

Rollout:
1296 Games rolled with Variance Reduction.
Moves: 3-ply, cube decisions: XG Roller
Confidence No Double: ± 0.008 (+0.177..+0.194)
Confidence Double: ± 0.019 (+0.289..+0.326)

Double Decision confidence: 100.0%
Take Decision confidence: 100.0%

Duration: 20 minutes 09 seconds

eXtreme Gammon Version: 2.10

Now, let's check for consistency between all three rollouts.
First look at the hold situation. Non-Jacoby is clearly better for
the favourite than mine but clearly not as good as actually holding
the cube. So, for holding, we expect original equity < non-Jacoby equity < own cube equity.
And indeed we get 0.185 < 0.288 < 0.468.
For doubling, we expect all equities to be the same.
Tim gets 0.303 and I get 0.308. This is a small discrepancy which is easily explained
by the different settings and the confidence bounds.

Paul

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)