Hello,

recently I became interested in bear-off heuristics in a pure
race. Since greedy bear-off is even good enough for Kit Woolsey, see

https://bkgm.com/rgb/rgb.cgi?view+251

, there is no need for me to try to be clever here. So the interesting positions are those that miss.

On autopilot, I just fill the highest gap. Occasionally, though, awfully
high stacks in my position cause me to pause and then play such that the biggest stack difference (defined as checkers on "from" point minus
checkers on "to" point) is reduced. In case of ties, e.g., "play one of
4 checkers on point 4 to point 2, which before the move has 1 checkers"
versus "play one of 3 checkers on point 3 to point 1, which before the
move was empty", prefer playing to the "most empty" point. In case of
further ties, prefer playing to the highest "most empty" point. I can
imagine another heuristic, namely playing such that an adjusted pip
count, e.g., my Isight method, is minimized. This also requires some
rules for resolving ties. This last approach is mentioned here:

https://www.bkgm.com/rgb/rgb.cgi?view+138

Did anyone here compare in some more detail the above three approaches:

1. Fill the highest gap
2. Reduce the biggest stack difference
3. Minimize wastage according to some adjusted pip count

What techniques do you use over the board?

I expect the differences to be tiny, but it might be an interesting data
mining exercise.

Any comments as always welcome!

Best regards

Axel

--- SoupGate-Win32 v1.05
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On 4/7/2022 2:31 PM, Axel Reichert wrote:

Did anyone here compare in some more detail the above three approaches:

1. Fill the highest gap
2. Reduce the biggest stack difference
3. Minimize wastage according to some adjusted pip count

What techniques do you use over the board?

I have not studied these heuristics or compared them. Here are some of
the heuristics I use.

1. Playing 6/3 or 4/2 or 2/1 is usually an efficient use of a miss. So
if I have enough checkers on my 6pt then I will tend to play 4/2 rather
than 5/3 (if both the 2pt and 3pt are vacant), reasoning that the gap on
the 3pt is already "covered" by the 6pt. Here's an example.

XGID=-A--BBD------------dbb--a-:1:-1:1:62:0:0:3:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game, Jacoby Beaver
+13-14-15-16-17-18------19-20-21-22-23-24-+
| | | O O O O | +---+
| | | O O O | | 2 |
| | | O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | X |
| | | X |
| | | X X X |
| | | X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 43 O: 43 X-O: 0-0
Cube: 2, O own cube
X to play 62

1. Rollout¹ 6/Off 4/2 eq:+0.176
Player: 63.41% (G:0.00% B:0.00%)
Opponent: 36.59% (G:0.00% B:0.00%)
Confidence: ±0.001 (+0.175..+0.177) - [100.0%]

2. Rollout¹ 6/4 6/Off eq:+0.170 (-0.006)
Player: 63.21% (G:0.00% B:0.00%)
Opponent: 36.79% (G:0.00% B:0.00%)
Confidence: ±0.001 (+0.169..+0.172) - [0.0%]

3. Rollout¹ 6/Off 5/3 eq:+0.169 (-0.007)
Player: 63.15% (G:0.00% B:0.00%)
Opponent: 36.85% (G:0.00% B:0.00%)
Confidence: ±0.001 (+0.168..+0.170) - [0.0%]

¹ 1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves and cube decisions: XG Roller++
Search interval: Gigantic

eXtreme Gammon Version: 2.19.207.pre-release

2. When I'm down to 2 checkers, it's best to keep them e = 2.718... pips
apart, or as close to that as you can make them. So checkers on the
5 and 2 are better than checkers on the 4 and 3 or on the 6 and 1.

3. The opponent's position can matter. Most obviously, if my only
chance of winning is to roll doublets then I will maximize my chance of
rolling doublets. Less obviously, if I'm losing then I will fill high
gaps and if I'm way ahead then I will fill low gaps. Example:

XGID=---BBBC------------dbbb---:1:-1:1:62:0:0:3:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game, Jacoby Beaver
+13-14-15-16-17-18------19-20-21-22-23-24-+
| | | O O O O | +---+
| | | O O O O | | 2 |
| | | O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | |
| | | X |
| | | X X X X |
| | | X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 42 O: 48 X-O: 0-0
Cube: 2, O own cube
X to play 62

1. Rollout¹ 6/Off 3/1 eq:+0.549
Player: 79.81% (G:0.00% B:0.00%)
Opponent: 20.19% (G:0.00% B:0.00%)
Confidence: ±0.001 (+0.548..+0.549) - [100.0%]

2. Rollout¹ 6/Off 4/2 eq:+0.545 (-0.003)
Player: 79.65% (G:0.00% B:0.00%)
Opponent: 20.35% (G:0.00% B:0.00%)
Confidence: ±0.001 (+0.544..+0.546) - [0.0%]

¹ 1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves and cube decisions: XG Roller+
Search interval: Gigantic

eXtreme Gammon Version: 2.19.207.pre-release

XGID=---BBBC------------abbb---:1:1:1:62:0:0:3:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game, Jacoby Beaver
+13-14-15-16-17-18------19-20-21-22-23-24-+
| | | O O O O |
| | | O O O |
| | | |
| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | X | +---+
| | | X X X X | | 2 |
| | | X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 42 O: 30 X-O: 0-0
Cube: 2, X own cube
X to play 62

1. Rollout¹ 6/Off 4/2 eq:-0.545
Player: 20.68% (G:0.00% B:0.00%)
Opponent: 79.32% (G:0.00% B:0.00%)
Confidence: ±0.000 (-0.546..-0.545) - [99.9%]

2. Rollout¹ 6/4 6/Off eq:-0.546 (-0.001)
Player: 20.58% (G:0.00% B:0.00%)
Opponent: 79.42% (G:0.00% B:0.00%)
Confidence: ±0.000 (-0.547..-0.546) - [0.1%]

3. Rollout¹ 6/Off 3/1 eq:-0.549 (-0.004)
Player: 20.70% (G:0.00% B:0.00%)
Opponent: 79.30% (G:0.00% B:0.00%)
Confidence: ±0.000 (-0.550..-0.549) - [0.0%]

¹ 5184 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves and cube decisions: XG Roller+
Search interval: Gigantic

eXtreme Gammon Version: 2.19.207.pre-release

---
Tim Chow

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

Timothy Chow <tchow12000@yahoo.com> writes:

1. Playing 6/3 or 4/2 or 2/1 is usually an efficient use of a miss.

Yes, this is also mentioned in

https://www.bkgm.com/rgb/rgb.cgi?view+1520

2. When I'm down to 2 checkers, it's best to keep them e =
2.718... pips apart

Yes, I knew this and especially like the mnemonics with Euler.

Less obviously, if I'm losing then I will fill high gaps and if I'm
way ahead then I will fill low gaps.

Indeed less obvious! Thanks for this one, I will keep an eye on it.

Of course the equity losses per decision are usually tiny, but these
positions occur again and again, so, like opening moves, it pays to
learn a thing or two about them.

Are things like this covered in Michi's "Endgame Technique"? Any other
comments on this book?

Best regards

Axel

--- SoupGate-Win32 v1.05
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On 4/13/2022 3:04 PM, Axel Reichert wrote:

Are things like this covered in Michi's "Endgame Technique"? Any other comments on this book?

Here are a couple of positions from that book.

XGID=-BAC-B-------------b-ca---:1:1:1:22:0:0:0:0:10

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 |
| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | X | +---+
| | | X X X | | 2 |
| | | X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 23 O: 27 X-O: 0-0
Cube: 2, X own cube
X to play 22

1. XG Roller++ 5/1 3/1 2/Off eq:+0.135
Player: 49.75% (G:0.00% B:0.00%)
Opponent: 50.25% (G:0.00% B:0.00%)

2. XG Roller++ 5/3 5/1 2/Off eq:+0.091 (-0.044)
Player: 47.96% (G:0.00% B:0.00%)
Opponent: 52.04% (G:0.00% B:0.00%)

3. XG Roller++ 5/1(2) eq:+0.091 (-0.044)
Player: 47.96% (G:0.00% B:0.00%)
Opponent: 52.04% (G:0.00% B:0.00%)

eXtreme Gammon Version: 2.19.207.pre-release

XGID=-AB--AB------------baaa---:1:1:1:41:0:0:0:0:10

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 |
| | | |
| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | | +---+
| | | X X | | 2 |
| | | X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 22 O: 24 X-O: 0-0
Cube: 2, X own cube
X to play 41

1. XG Roller++ 6/2 1/Off eq:+0.108
Player: 48.68% (G:0.00% B:0.00%)
Opponent: 51.32% (G:0.00% B:0.00%)

2. XG Roller++ 5/Off eq:+0.066 (-0.042)
Player: 46.78% (G:0.00% B:0.00%)
Opponent: 53.22% (G:0.00% B:0.00%)

eXtreme Gammon Version: 2.19.207.pre-release

---
Tim Chow

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

Here are some positions from my files. I have lots more if there
are specific things you're looking for. For brevity, I have omitted
much of the rollout information that XG normally displays.

XGID=---AAAA----------------bd-:1:-1:1:32:0:0:0:0:10

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 | | 2 |
| | | O | +---+
| | | O |
| | | |
| |BAR| |
| | | |
| | | |
| | | |
| | | |
| | | X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 18 O: 8 X-O: 0-0
Cube: 2, O own cube
X to play 32

1. Rollout¹ 5/Off eq:-0.117
2. Rollout¹ 6/4 3/Off eq:-0.153 (-0.035)
3. Rollout¹ 4/2 3/Off eq:-0.173 (-0.055)

XGID=----BBB------------aa-aa--:1:1:1:52:0:0:0:0:10

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 |
| | | |
| | | |
| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | | +---+
| | | X X X | | 2 |
| | | X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 30 O: 16 X-O: 0-0
Cube: 2, X own cube
X to play 52

1. Rollout¹ 6/4 5/Off eq:-0.782
2. Rollout¹ 5/Off 4/2 eq:-0.838 (-0.056)
3. Rollout¹ 5/3 5/Off eq:-0.843 (-0.061)

XGID=--AAABB--------------b-ce-:1:-1:1:11:0:0:0:0:10

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 | | 2 |
| | | O O | +---+
| | | O |
| | | O |
| |BAR| |
| | | |
| | | |
| | | |
| | | X X |
| | | X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 31 O: 19 X-O: 0-0
Cube: 2, O own cube
X to play 11

1. Rollout¹ 5/4 3/Off eq:+0.019
2. Rollout¹ 6/4 2/Off eq:+0.018 (-0.001)
3. Rollout¹ 5/3 2/Off eq:+0.003 (-0.016)
4. Rollout¹ 3/Off 2/1 eq:-0.004 (-0.023)
5. Rollout¹ 4/Off eq:-0.007 (-0.026)

XGID=--ABAAA-------------a-aab-:1:1:1:21:0:0:0:0:10

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 |
| | | |
| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | | +---+
| | | X | | 2 |
| | | X X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 23 O: 12 X-O: 0-0
Cube: 2, X own cube
X to play 21

1. Rollout¹ 5/4 2/Off eq:-0.754
2. Rollout¹ 4/3 2/Off eq:-0.785 (-0.031)
3. Rollout¹ 6/5 2/Off eq:-0.795 (-0.041)
4. Rollout¹ 3/Off eq:-0.796 (-0.042)

XGID=--BCCBD------------cba-dc-:0:0:1:44:0:0:0:0:10

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 O O O |
| | | O O O |
| | | O |
| | | |
| |BAR| |
| | | |
| | | X |
| | | X X X |
| | | X X X X X |
| | | X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 59 O: 43 X-O: 0-0
Cube: 1
X to play 44

1. Rollout¹ 6/2 4/Off(3) eq:-0.047
2. Rollout¹ 5/1 4/Off(3) eq:-0.077 (-0.030)

XGID=-A-BB--------------a-aaa--:1:1:1:21:0:0:0:0:10

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 |
| | | |
| | | |
| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | | +---+
| | | X X | | 2 |
| | | X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 15 O: 15 X-O: 0-0
Cube: 2, X own cube
X to play 21

1. Rollout¹ 3/Off eq:-0.126
2. Rollout¹ 4/2 1/Off eq:-0.163 (-0.037)

XGID=--A-A-B-------------aaa---:1:1:1:61:0:0:0:0:10

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 |
| | | |
| | | |
| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | | +---+
| | | X | | 2 |
| | | X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 18 O: 12 X-O: 0-0
Cube: 2, X own cube
X to play 61

1. Rollout¹ 6/Off 4/3 eq:-0.429
2. Rollout¹ 6/5 6/Off eq:-0.447 (-0.019)
3. Rollout¹ 6/Off 2/1 eq:-0.463 (-0.034)

XGID=-BCEB-B-------------a-acg-:1:1:1:22:0:0:0:0:10

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 |
| | | O O |
| | | O |
| | | 7 |
| |BAR| |
| | | X |
| | | X |
| | | 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: 43 O: 21 X-O: 0-0
Cube: 2, X own cube
X to play 22

1. Rollout¹ 6/4 2/Off(3) eq:-0.494
2. Rollout¹ 4/Off 2/Off(2) eq:-0.520 (-0.026)
3. Rollout¹ 6/4(2) 2/Off(2) eq:-0.535 (-0.042)

XGID=---AADA---------------abb-:1:1:1:62:0:0:0:0:10

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 |
| | | |
| | | |
| | | |
| |BAR| |
| | | |
| | | X |
| | | X | +---+
| | | X | | 2 |
| | | X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 33 O: 9 X-O: 0-0
Cube: 2, X own cube
X to play 62

1. Rollout¹ 6/Off 3/1 eq:-0.895
2. Rollout¹ 6/Off 4/2 eq:-0.903 (-0.009)
3. Rollout¹ 6/4 5/Off eq:-0.932 (-0.037)
4. Rollout¹ 6/Off 5/3 eq:-0.934 (-0.039)

---
Tim Chow

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

Timothy Chow <tchow12000@yahoo.com> writes:

Here are some positions from my files. I have lots more if there are specific things you're looking for.

Thanks, also for the excerpts from Michi's book. If there is more need,
I think I would rather repurpose Tom Keith's database of endgame
positions. It should be easy to create tons of checker play problems
from it using GNU Backgammon's scripting/automation capabilities. Then I
could test and compare various heuristics, similar to the approach for
finding my Isight method.

My gut feeling for such a project is that minimizing EPCs will be the
best strategy (with rare exceptions), which makes the heuristic of
minimizing an adjusted pipcount the favourite. And since during
researching the Isight count I also found a nice EPC approximation
(which I ignored henceforth), it might be time to reactivate it. A bonus
is that the adjustments for this EPC approximation and for the Isight
count are very similar, see page 28 of

https://bkgm.com/articles/Reichert/insights-with-isight.pdf

Best regards

Axel

--- SoupGate-Win32 v1.05
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On 4/14/2022 5:10 PM, Axel Reichert wrote:

I think I would rather repurpose Tom Keith's database of endgame
positions. It should be easy to create tons of checker play problems
from it using GNU Backgammon's scripting/automation capabilities.

This is true; however, there is an advantage to examining hand-picked positions: those are the positions where one is in most need of a
heuristic. A machine-generated list may contain a lot of positions
which are "easy" for a human in the sense that the correct move is
what a human would naturally play anyway. Heuristic 1 might solve
more positions in this list than Heuristic 2, but Heuristic 2 might
solve more of the "difficult" positions.

One way to mitigate this problem would be to begin with a simple
heuristic H, such as the one you began this thread with. Run your
script and keep only the positions that H gets wrong. Then study
just these "hard" positions, and try to modify H to yield a more
sophisticated, but still humanly usable, heuristic H+. Run H+ on
the whole corpus again, isolate the positions that H+ gets wrong,
and see if you can come up with H++, etc.

My gut feeling for such a project is that minimizing EPCs will be the
best strategy (with rare exceptions), which makes the heuristic of
minimizing an adjusted pipcount the favourite.

This is a pretty good guess, but it could also be that the positions
that are hard for humans are the positions for which it is difficult
to assess the EPC, and for which EPC heuristics give the wrong answer.

---
Tim Chow

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

Timothy Chow <tchow12000@yahoo.com> writes:

There is an advantage to examining hand-picked positions: those are
the positions where one is in most need of a heuristic. A
machine-generated list may contain a lot of positions which are "easy"
for a human in the sense that the correct move is what a human would naturally play anyway. Heuristic 1 might solve more positions in this
list than Heuristic 2, but Heuristic 2 might solve more of the
"difficult" positions.

Yes. This is something that came up a couple of times as feedback to my
Isight method: That in position A it fails (see Paul's recent example),
but method X gets position A right. And in position B, both method X and
mine fail, but method Y gets it right. Throw in the "horses for courses"
adage, and you have a convincing argument, don't you?

My answer was always that in this case you need another heuristic that
tells you when to apply method X and when Y. This reminds me on

https://en.wikipedia.org/wiki/Fredkin%27s_paradox

Some people apply several methods and then go by "majority vote". In my
opinion this is too much overhead for often tiny differences in
equity. I tend to stick to my guns and use my Isight method even for
cub-offs, see page 18 ("Comments on Cub-offs"), since I am very sure
that I will not be able to do better than 12 % wrong decisions over the
board under time pressure. And using the "universal weapon" throughout
will also avoid mistakes in mental calculations that might creep in if
you are juggling with a zoo of different methods: "Was it 76 or 78 as a threshold value?"

Run your script and keep only the positions that H gets wrong. Then
study just these "hard" positions, and try to modify H to yield a more sophisticated, but still humanly usable, heuristic H+. Run H+ on the
whole corpus again

I had this idea some years back, when Mochy reported a rather large
error resulting from my Isight method "in a friendly chouette for not so friendly stakes" (whatever that means ...). The thing is, though, that
using H+ might of course give different results for the positions that H
got right, even to the point that the whole corpus is "solved" in an
inferior manner. So it is either again back to switching between
different heuristics or back to optimizing one heuristics over the whole corpus.

I readily admit, though, that playing around with H+, H++ etc. might
give you ideas about the correct parameter space to use for this one optimization.

If it were possible (to use an analogy from linear, structural dynamics)
to augment the "orthogonal base" of eigenvectors for a real world
transient dynamic process by a "residual mode" that improves precisely
on the stuff not captured by the orginal "orthogonal base", that would
be great. However, I do not see how to ensure this orthogonality, be it
for cubing or moving in non-contact positions.

Best regards

Axel

--- SoupGate-Win32 v1.05
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I think my ending post has been completely misunderstood.
But the truth is maybe even sadder than what Axel thought.
The reality is that I've so far been too lazy to learn the Isight method, despite taking an intererest in it.
At the time of my post, I didn't know that the Isight method gave the wrong answer.
I just thought it was an interesting bearoff position, so I tagged Axel knowing that he is interested in racing algos.

There are two reasons my position doesn't make a good criticism of Isight. Firstly, there is the too often overlooked philosophical point that, for a judgment
to be valid, the comparators must be clear and relevant. It isn't valid to say "I know X is a terrible umpire. She once called the ball in when Hawkeye showed
it was at least an inch out!" No, this doesn't show she's a "terrible umpire", only
that she's a suboptimal umpire. This one bad call is totally consistent with her being
the greatest human umpire in tennis history. The criticism would only be valid if
most human umpires can call the lines with 100% accuracy (and they can't).

Secondly, there is the point that my position is difficult and marginal. Getting that
one wrong isn't so bad.

Paul

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On Saturday, April 16, 2022 at 8:30:52 AM UTC+1, Axel Reichert wrote:

Mochy reported a rather large
error resulting from my Isight method "in a friendly chouette for not so friendly stakes" (whatever that means ...)...

To me, the meaning seems actually quite clear.
There are two aspects:
1) The chouette was friendly.
2) The stakes were not friendly.

First note that "not friendly" and "unfriendly" have different meanings.
For example, the majority of transactions at a grocery till (I'm talking about a human cashier here) are not friendly -- there generally isn't much chatting or conversation.
But they also aren't usually unfriendly -- there aren't usually arguments or rudeness.

1) is probably clear in meaning. Besides the competitive nature of the chouette, the players
also experienced friendship. Between games, they probably chatted. Some of the players
like to view surreal numbers as functions from initial segments of the ordinals. On the other hand,
other players in the group prefer to view them as being derived from a left set and a right set
Before all the chouetteers were ready to start, they gently mocked those who had a different
viewpoint: "You're a left-right deriver. Don't you know there's no real canonicity that way? Ha Ha Ha!"
"You use ordinals to define them! Don't you think that concept's a bit unnecessary here? Ha Ha Ha!"

2) The stakes were high enough that the money really mattered. If a player saw something as a 1.02 drop,
they would drop, and wouldn't deliberately do an even slightly bad take "for the fun of it".
They weren't gambling irresponsibly. However, they were professionals or semi-professionals and need
to monitor the financial aspect carefully. Of course, it's a zero-sum situation so some of the players
(perhaps only one) are expected to lose money overall. For the expected losers, the hope is that the
expected loss is compensated for by the improvement in backgammon from the experience of meeting
better players. "You need to spend money to learn something" might be part of the thinking there.

Paul

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"peps...@gmail.com" <pepstein5@gmail.com> writes:

On Saturday, April 16, 2022 at 8:30:52 AM UTC+1, Axel Reichert wrote:

Mochy reported a rather large
error resulting from my Isight method "in a friendly chouette for not so
friendly stakes" (whatever that means ...)...

To me, the meaning seems actually quite clear.
There are two aspects:
1) The chouette was friendly.
2) The stakes were not friendly.

So far, so clear. But I would have been curious about the number
quantifying aspect 2 ...

Best regards

Axel

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On Saturday, April 16, 2022 at 8:30:52 AM UTC+1, Axel Reichert wrote: This reminds me on

https://en.wikipedia.org/wiki/Fredkin%27s_paradox

I had a Fredkin paradox situation at work once. Our team needed to do an estimate of when
we could finish a software development project. So we met in the kitchen to decide between ourselves
how long it would take us to do the estimate.
However, this meeting in the kitchen was sudden and unplanned. No one gave any thought to estimating
how long the meeting should be which decides how much time to provide to the estimation process.

Paul

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On Saturday, April 16, 2022 at 10:45:10 AM UTC+1, Axel Reichert wrote:

"peps...@gmail.com" <peps...@gmail.com> writes:

On Saturday, April 16, 2022 at 8:30:52 AM UTC+1, Axel Reichert wrote:

Mochy reported a rather large
error resulting from my Isight method "in a friendly chouette for not so >> friendly stakes" (whatever that means ...)...

To me, the meaning seems actually quite clear.
There are two aspects:
1) The chouette was friendly.
2) The stakes were not friendly.

So far, so clear. But I would have been curious about the number
quantifying aspect 2 ...

Oh, I understand you better now.
Yes, it's like the phrase "a lot of money".
A lot of money to one person is spare change to a richer person.
So what is the "not friendly stakes" bracket for Mochy's chouette circle?

Paul

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"peps...@gmail.com" <pepstein5@gmail.com> writes:

I think my ending post has been completely misunderstood.

No, Paul, no worries. I know that you have not learned my method and
thus assumed that you did not check your position with it. But your
example was one for the method's failure nevertheless. I can live with
this. (-:

Best regards

Axel

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On 4/16/2022 3:30 AM, Axel Reichert wrote:

If it were possible (to use an analogy from linear, structural dynamics)
to augment the "orthogonal base" of eigenvectors for a real world
transient dynamic process by a "residual mode" that improves precisely
on the stuff not captured by the orginal "orthogonal base", that would
be great. However, I do not see how to ensure this orthogonality, be it
for cubing or moving in non-contact positions.

I agree. Crafting heuristics that are useful for humans is more of an
art than a science. While automated tools can be helpful, they cannot
entirely replace the human element. There's always a temptation to
design heuristics that require mathematical computations, but even a
small amount of calculation tends to make the heuristic unusable for
most people in practice.

---
Tim Chow

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