I would never dream of Axelising the below because it just seems like
a rolls vs rolls position where you need EPC and all that.
I also never suspected I was even close to a double. After all, I have 7
checkers against 6 checkers and we're all on low points.

But I think that Axelisation does actually give the right answer here,
and since I can't really do EPC, it may have been a smart approach.

Raw pip counts are 11 to 18
I have three extra on my ace point so add 2 * 3 to my score to get 17.
I have 4,5,6 all empty so add 3 to get 20. I need 1 more crossover
than XG so add 1 to get 21.
Add 1/6 of that to get 24.5
For the opponent we 1 for the stack on the 3 point and 2 more
for the open 5 and 6 points. So we get 21.
24.5 - 21 = 3.5 which is squarely within the D/T range.
Shame on me for missing such a great Axelisation opportunity.

Paul

XGID=-E-B-----------------ada--:2:1:1:00:0:0:3:0:10
X:Daniel O:eXtremeGammon

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

Analyzed in XG Roller+
Player Winning Chances: 68.65% (G:0.00% B:0.00%)
Opponent Winning Chances: 31.35% (G:0.00% B:0.00%)

Cubeless Equities: No Double=+0.373, Double=+0.746

Cubeful Equities:
No redouble: +0.575 (-0.072)
Redouble/Take: +0.647
Redouble/Pass: +1.000 (+0.353)

Best Cube action: Redouble / Take

eXtreme Gammon Version: 2.10

On 8/26/2022 3:45 PM, peps...@gmail.com wrote:

> Raw pip counts are 11 to 18
> I have three extra on my ace point so add 2 * 3 to my score to get 17.

Yes.

> I have 4,5,6 all empty so add 3 to get 20.

No, you only get penalized for this if your opponent has a checker on
the corresponding point. So add 1 to get 18

>I need 1 more crossover than XG so add 1 to get 21.

Hmmmm. You add one pip for each extra checker on the board. I think you
only count crossovers to bear in, not to bear off, otherwise you're
double counting. So, offsetting errors here, but since when is
18+1==21? 19 it is.

> Add 1/6 of that to get 24.5
> For the opponent we 1 for the stack on the 3 point and 2 more
> for the open 5 and 6 points. So we get 21.

Correct to add one for the extra spare on the three point. Incorrect to
add two for the open points. So the adjusted pipcount is 19-19.

You're on roll, so a clear D/T using Axel's formula or the simpler Trice
formula. And it should be a R/T for O if she were on roll - which it is
- see below.

> 24.5 - 21 = 3.5 which is squarely within the D/T range.
> Shame on me for missing such a great Axelisation opportunity.

It's always worth doing a race cube calculation in these situations.

>
> XGID=-E-B-----------------ada--:2:1:1:00:0:0:3:0:10
> X:Daniel O:eXtremeGammon
>
> 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 |
> | | | |
> | |BAR| |
> | | | X |
> | | | X |
> | | | X | +---+
> | | | X X | | 4 |
> | | | X X | +---+
> +12-11-10--9--8--7-------6--5--4--3--2--1-+
> Pip count X: 11 O: 18 X-O: 0-0
> Cube: 4, X own cube
> X on roll, cube action
>
> Analyzed in XG Roller+
> Player Winning Chances: 68.65% (G:0.00% B:0.00%)
> Opponent Winning Chances: 31.35% (G:0.00% B:0.00%)
>
> Cubeless Equities: No Double=+0.373, Double=+0.746
>
> Cubeful Equities:
> No redouble: +0.575 (-0.072)
> Redouble/Take: +0.647
> Redouble/Pass: +1.000 (+0.353)
>
> Best Cube action: Redouble / Take
>
> eXtreme Gammon Version: 2.10

Invert active player and cube:

XGID=-E-B-----------------ada--:2:-1:-1:00:0:0:3:0:10

X:Player 2 O:Player 1
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 | | 4 |
| | | X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 18 O: 11 X-O: 0-0
Cube: 4, X own cube
X on roll, cube action

Analyzed in Rollout
No redouble
Player Winning Chances: 74.94% (G:0.00% B:0.00%)
Opponent Winning Chances: 25.06% (G:0.00% B:0.00%)
Redouble/Take
Player Winning Chances: 74.93% (G:0.00% B:0.00%)
Opponent Winning Chances: 25.07% (G:0.00% B:0.00%)

Cubeless Equities: No Double=+0.499, Double=+0.997

Cubeful Equities:
No redouble: +0.786 (-0.167)
Redouble/Take: +0.954
Redouble/Pass: +1.000 (+0.046)

Best Cube action: Redouble / Take

Rollout:
1296 Games rolled with Variance Reduction.
Moves: 3-ply, cube decisions: XG Roller
Confidence No Double: ± 0.002 (+0.784..+0.788)
Confidence Double: ± 0.003 (+0.951..+0.956)

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

Duration: 0.3 second

eXtreme Gammon Version: 2.10

On Friday, August 26, 2022 at 10:45:33 PM UTC+1, ah...Clem wrote:
> On 8/26/2022 3:45 PM, peps...@gmail.com wrote:
>
> > Raw pip counts are 11 to 18
> > I have three extra on my ace point so add 2 * 3 to my score to get 17.
> Yes.
> > I have 4,5,6 all empty so add 3 to get 20.
> No, you only get penalized for this if your opponent has a checker on
> the corresponding point. So add 1 to get 18
> >I need 1 more crossover than XG so add 1 to get 21.
> Hmmmm. You add one pip for each extra checker on the board. I think you
> only count crossovers to bear in, not to bear off, otherwise you're
> double counting. So, offsetting errors here, but since when is
> 18+1==21? 19 it is.
> > Add 1/6 of that to get 24.5
> > For the opponent we 1 for the stack on the 3 point and 2 more
> > for the open 5 and 6 points. So we get 21.
> Correct to add one for the extra spare on the three point. Incorrect to
> add two for the open points. So the adjusted pipcount is 19-19.
>
> You're on roll, so a clear D/T using Axel's formula or the simpler Trice
> formula. And it should be a R/T for O if she were on roll - which it is
> - see below.
> > 24.5 - 21 = 3.5 which is squarely within the D/T range.
> > Shame on me for missing such a great Axelisation opportunity.
> It's always worth doing a race cube calculation in these situations.
> >
> > XGID=-E-B-----------------ada--:2:1:1:00:0:0:3:0:10
> > X:Daniel O:eXtremeGammon
> >
> > 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 |
> > | | | |
> > | |BAR| |
> > | | | X |
> > | | | X |
> > | | | X | +---+
> > | | | X X | | 4 |
> > | | | X X | +---+
> > +12-11-10--9--8--7-------6--5--4--3--2--1-+
> > Pip count X: 11 O: 18 X-O: 0-0
> > Cube: 4, X own cube
> > X on roll, cube action
> >
> > Analyzed in XG Roller+
> > Player Winning Chances: 68.65% (G:0.00% B:0.00%)
> > Opponent Winning Chances: 31.35% (G:0.00% B:0.00%)
> >
> > Cubeless Equities: No Double=+0.373, Double=+0.746
> >
> > Cubeful Equities:
> > No redouble: +0.575 (-0.072)
> > Redouble/Take: +0.647
> > Redouble/Pass: +1.000 (+0.353)
> >
> > Best Cube action: Redouble / Take
> >
> > eXtreme Gammon Version: 2.10
> Invert active player and cube:
>
>
> XGID=-E-B-----------------ada--:2:-1:-1:00:0:0:3:0:10
>
> X:Player 2 O:Player 1
> 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 | | 4 |
> | | | X X X | +---+
> +12-11-10--9--8--7-------6--5--4--3--2--1-+
> Pip count X: 18 O: 11 X-O: 0-0
> Cube: 4, X own cube
> X on roll, cube action
> Analyzed in Rollout
> No redouble
> Player Winning Chances: 74.94% (G:0.00% B:0.00%)
> Opponent Winning Chances: 25.06% (G:0.00% B:0.00%)
> Redouble/Take
> Player Winning Chances: 74.93% (G:0.00% B:0.00%)
> Opponent Winning Chances: 25.07% (G:0.00% B:0.00%)
>
> Cubeless Equities: No Double=+0.499, Double=+0.997
>
> Cubeful Equities:
> No redouble: +0.786 (-0.167)
> Redouble/Take: +0.954
> Redouble/Pass: +1.000 (+0.046)
> Best Cube action: Redouble / Take
> Rollout:
> 1296 Games rolled with Variance Reduction.
> Moves: 3-ply, cube decisions: XG Roller
> Confidence No Double: ± 0.002 (+0.784..+0.788)
> Confidence Double: ± 0.003 (+0.951..+0.956)
>
> Double Decision confidence: 100.0%
> Take Decision confidence: 100.0%
>
> Duration: 0.3 second
>
> eXtreme Gammon Version: 2.10

Thanks for this analysis.
The only error I made was my misremembering the high-open-points
penalty method. But my variant of it is extremely similar.
Other wrong numbers given are a consequence of this previous error.
And I present the algo in a unified way by counting all crossovers whether they
are from needing to bear in or checkers left. This isn't a difference.
I don't know the Trice method but your post might inspire me to learn it.

Thanks again,

Paul

On 8/27/2022 3:45 AM, peps...@gmail.com wrote:

>
> Thanks for this analysis.
> The only error I made was my misremembering the high-open-points
> penalty method. But my variant of it is extremely similar.

It's almost the same since it increases the adjusted count for each
player equally, but results in a larger adjusted count. Probably
doesn't matter for most positions.

> And I present the algo in a unified way by counting all crossovers whether they
> are from needing to bear in or checkers left. This isn't a difference.

You are correct, and the mathematician in me favors this as it reduces
the number of rules by one. Whether it would be clear to many readers
in this form is debatable. Then again, as currently written the rules
could be interpreted to double count extra checkers in the homeboard.

> I don't know the Trice method but your post might inspire me to learn it.

I've been using iSight's adjusted pipcount along with the Trice metric
(rule of 62) since iSight's adjusted count is easier to do in my head
than Trice's, and the Trice metric is easier than iSight.

Using the iSight adjusted counts, apply the metric as follows:

If leaders adjusted count is >= 62, POLT = count/10 + 1 round up
If leaders adjusted count < 62, POLT = (count-5)/7 round down

The POLT is the point of last take; if the trailer is behind by
more than the POLT it's a pass.
If the leader is within 3 of the POLT it's an initial double, if
within 2 it's a redouble.

I find that computing 10% and adding one is much easier than increasing
by 1/6th. And subtracting 5 and dividing by 7 is somewhat easier. YMMV>

On Saturday, August 27, 2022 at 3:33:09 PM UTC+1, ah...Clem wrote:
> On 8/27/2022 3:45 AM, peps...@gmail.com wrote:
>
> >
> > Thanks for this analysis.
> > The only error I made was my misremembering the high-open-points
> > penalty method. But my variant of it is extremely similar.
> It's almost the same since it increases the adjusted count for each
> player equally, but results in a larger adjusted count. Probably
> doesn't matter for most positions.
> > And I present the algo in a unified way by counting all crossovers whether they
> > are from needing to bear in or checkers left. This isn't a difference.
> You are correct, and the mathematician in me favors this as it reduces
> the number of rules by one. Whether it would be clear to many readers
> in this form is debatable. Then again, as currently written the rules
> could be interpreted to double count extra checkers in the homeboard.
> > I don't know the Trice method but your post might inspire me to learn it.
> I've been using iSight's adjusted pipcount along with the Trice metric
> (rule of 62) since iSight's adjusted count is easier to do in my head
> than Trice's, and the Trice metric is easier than iSight.
>
> Using the iSight adjusted counts, apply the metric as follows:
>
> If leaders adjusted count is >= 62, POLT = count/10 + 1 round up
> If leaders adjusted count < 62, POLT = (count-5)/7 round down
>
> The POLT is the point of last take; if the trailer is behind by
> more than the POLT it's a pass.
> If the leader is within 3 of the POLT it's an initial double, if
> within 2 it's a redouble.
>
>
> I find that computing 10% and adding one is much easier than increasing
> by 1/6th. And subtracting 5 and dividing by 7 is somewhat easier. YMMV>

Thanks for your very interesting thoughts on this.
Personally speaking, I'm extremely good at mental arithmetic.
As far as I know, I've never met anybody better than me at it. I
am probably weaker at it then Art Benjamin, but only just.
Therefore, ease of calculation is absolutely not a factor in my choice of algo.
I did get the count in this thread wrong (as you said) but this was a rules-remembering
issue, not a calculation issue.
If I understand him correctly, Axel chose an enormous set of possible rules, and optimised
by picking the one that worked best with the Fibs database of positions.
In this particular position, with me on roll, my variant actually works better than the genuine rule.
Both methods give D/T but my method gives larger counts and therefore makes the double
more marginal and the take easier (which is right).
However, if you give the roll to the other side, the real rule works better. Here, it's the take that
is marginal so we want a smaller count that gives a stronger advantage to the on-roll player.
But both rules work here in both positions. It feels neat to have four distinct hypotheses which
are all correct -- real validation at work.

Paul

"ah...Clem" <ah_clem@ymail.com> writes:

> I've been using iSight's adjusted pipcount along with the Trice metric
> (rule of 62)

As I have written repeatedly here, I do not think this is a good idea,
see table 9 in my article. Your combination is about a third less
accurate (1424 total error versus 1064 for "pure" Isight).

> I find that computing 10% and adding one is much easier than
> increasing by 1/6th. And subtracting 5 and dividing by 7 is somewhat
> easier.

Since you obviously do not mind the additional mental effort of
distinguishing between long and short races, please see the very last
paragraph of my article. For race lenghts >= 50 this boils down to the
time-tested "10 % +/- 2 pips", for race lengths <= 50 you get a point of
last take of 20 % - 3 pips, with the (re)doubling windows 4 or 3 pips
wide, respectively.

Best regards

Axel