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Maths isn't usually linear and it isn't usually easy either.
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peps...@gmail.com
Jul 18, 2023, 1:13:33 PM7/18/23
to
I understand (if I'm wrong, please correct me but refrain from
destroying my village) that, for a match to n points, the standard
time limit is to give each player 2 * n minutes for the match
(+ a per-move delay which I think is around ten seconds).
But isn't this too simplistic? Surely the relationship between
expected match length (in terms of total number of plays)
is non-linear in k where k is the minimum score needed to win
the match? Why don't the rules respect the complex (and perhaps
interesting) maths involved?
Paul
destroying my village) that, for a match to n points, the standard
time limit is to give each player 2 * n minutes for the match
(+ a per-move delay which I think is around ten seconds).
But isn't this too simplistic? Surely the relationship between
expected match length (in terms of total number of plays)
is non-linear in k where k is the minimum score needed to win
the match? Why don't the rules respect the complex (and perhaps
interesting) maths involved?
Paul
Timothy Chow
Jul 18, 2023, 9:02:07 PM7/18/23
to
---
Tim Chow
ah....Clem
Jul 19, 2023, 9:50:54 AM7/19/23
to
maximum number is 2n-1. I don't know what the "average" length is, but
it is probably of O(n) so the simple rule is probably fine.
I don't play with a clock, so it's not terribly relevant to me.
--
Ah....Clem
The future is fun, the future is fair.
Axel Reichert
Jul 19, 2023, 11:51:40 AM7/19/23
to
"ah....Clem" <ah_...@ymail.com> writes:
> For a match to n points, the minimum number of games is 1 and the
> maximum number is 2n-1. I don't know what the "average" length is,
> but it is probably of O(n)
Nice fundamental argument!
> For a match to n points, the minimum number of games is 1 and the
> maximum number is 2n-1. I don't know what the "average" length is,
> but it is probably of O(n)
> so the simple rule is probably fine.
After looking at
https://www.bkgm.com/rgb/rgb.cgi?view+712
I just ran 20 matches each with GNU Backgammon playing itself on
expert level to augment the table. Here is the average number of games
for different match lengths (my data starts with match length 13):
|--------------+-------------------------|
| Match length | Average Number of Games |
|--------------+-------------------------|
| 1 | 1.00 |
| 3 | 2.35 |
| 5 | 3.83 |
| 7 | 5.02 |
| 9 | 7.24 |
| 11 | 7.81 |
|--------------+-------------------------|
| 13 | 10.35 |
| 15 | 10.00 |
| 17 | 11.60 |
| 19 | 14.00 |
| 21 | 13.20 |
| 23 | 13.85 |
| 25 | 15.80 |
|--------------+-------------------------|
20 matches each is certainly to few, which possibly explains the
non-monotoneous growth. But I can easily imagine, as Paul put it, some
nonlinear effects, due to "overshooting" or "undershooting" the match
length with a particular (higher) cube level.
Which for me immediately raises the question which match lengths are
particularly interesting in the sense of having many match scores at
which the cube action deviates strongly from money sessions. Also, is
there any reason (theoretical or historical) for odd match lengths?
Perhaps this is one of the reasons why 3 matches to 7 might be more
attractive than one 21-pointer. After all, a very long match essentially
starts out as a money session, the tricky skills for match (which
distinguish the beginner from the expert) occur much later: Free drop at
even-away post-Crawford, free take at odd-away post-Crawford, automatic
redoubles, Gammon-Go, Gammon-Save, DMP, ...
By the way, some years back Chiva Tafazzoli told me that he once ran a
tournament with 2-point matches. When I looked at him in disbelief he
said that it was incredible fun watching the players (knowledgable or
not in theory) trying to outsmart each other. Of course often losing
their market by a mile ... (-:
Best regards
Axel
MK
Jul 19, 2023, 7:20:32 PM7/19/23
to
On July 18, 2023 at 11:13:33 AM UTC-6, peps...@gmail.com wrote:
> Surely the relationship between expected
> match length (in terms of total number of
> plays) is non-linear
How do you know this? Assuming that you
> Surely the relationship between expected
> match length (in terms of total number of
> plays) is non-linear
are referring to gamblegammon matches,
and assuming that what you are saying is
true, would you say that in backgammon
it is linear or at least relatively more linear
than in gamblegammon..?
> in k where k is the minimum score needed
> to win the match?
As one side reaching the minimum points,
i.e. the match length, while the other side is
still at zero..?
If so, I can't see how do you relate minimum
score with expected total number of plays in
a match but you may be onto something and
I would appreciate if you expand/explain it.
MK
MK
Jul 19, 2023, 7:27:16 PM7/19/23
to
On July 19, 2023 at 7:50:54 AM UTC-6, ah....Clem wrote:
>> On 7/18/2023 1:13 PM, peps...@gmail.com wrote:
>>> expected match length (in terms of
>>> total number of plays)
> For a match to n points ..... I don't know
>> On 7/18/2023 1:13 PM, peps...@gmail.com wrote:
>>> expected match length (in terms of
>>> total number of plays)
> what the "average" length is, but .....
He is talking about "number of plays", not
"number of games". Duh!
MK
MK
Jul 19, 2023, 8:01:30 PM7/19/23
to
On July 19, 2023 at 9:51:40 AM UTC-6, Axel Reichert wrote:
> "ah....Clem" <ah_...@ymail.com> writes:
>> For a match to n points, the minimum number
>> of games is 1 and the maximum number is 2n-1.
>> I don't know what the "average" length is, but it
>> is probably of O(n)
> Nice fundamental argument!
Except that it's unrelated to what Paul was talking
> "ah....Clem" <ah_...@ymail.com> writes:
>> For a match to n points, the minimum number
>> of games is 1 and the maximum number is 2n-1.
>> I don't know what the "average" length is, but it
>> is probably of O(n)
> Nice fundamental argument!
about, i.e. the "number of plays", not the "number of
games" and even then it's at best a *probably* "nice
fundamental argument"... :)
What is remotely related to Paul's comments there
is the "average and maximum sumber of moves in
games", which may or may not apply to matches.
> I just ran 20 matches each with GNU Backgammon
> playing itself on expert level to augment the table.
> Here is the average number of games for different
The average number of games for different match
lengths are the "effective match lengths" between
players with zero ELO difference. :)
How is your other experiment coming along...? ;)
> is there any reason (theoretical or historical) for odd
> match lengths?
played. Maybe initially gamblegammon matches
were also the same lengths but using the cube..?
> Perhaps this is one of the reasons why 3 matches
> to 7 might be more attractive than one 21-pointer.
what I have been explaining all along how the cube
magifies luck, i.e. longer matches = higher cubes =
"effective match lengths" increasing at slower rates.
MK
Timothy Chow
Jul 20, 2023, 8:02:47 AM7/20/23
to
On 7/19/2023 11:51 AM, Axel Reichert wrote:
> Which for me immediately raises the question which match lengths are
> particularly interesting in the sense of having many match scores at
> which the cube action deviates strongly from money sessions. Also, is
> there any reason (theoretical or historical) for odd match lengths?
I learned from Douglas Zare that the cube action for 7-point matches is
> Which for me immediately raises the question which match lengths are
> particularly interesting in the sense of having many match scores at
> which the cube action deviates strongly from money sessions. Also, is
> there any reason (theoretical or historical) for odd match lengths?
very similar to the cube action for money (more so than for 9-point
or 11-point matches, I believe), including the recubes (though certainly
not the re-recubes). I never confirmed this calculation myself, but
Zare is usually right about that sort of thing.
As for odd match lengths, a surprising number of people will suggest
that it's to avoid ties. This of course makes no logical sense, but
there are of course many contests in sports that have a "best-of-n"
form, where n needs to be odd to avoid ties. So perhaps people just
got used to match lengths being an odd number in other sports, and
carried over this tradition to backgammon even though there's no
mathematical reason for it.
---
Tim Chow
Timothy Chow
Jul 20, 2023, 8:03:42 AM7/20/23
to
On 7/20/2023 8:02 AM, I wrote:
> I learned from Douglas Zare that the cube action for 7-point matches is
> very similar to the cube action for money (more so than for 9-point
> or 11-point matches, I believe), including the recubes (though certainly
> not the re-recubes).
Just to clarify, I meant the *initial game* of a 7-point match.
> I learned from Douglas Zare that the cube action for 7-point matches is
> very similar to the cube action for money (more so than for 9-point
> or 11-point matches, I believe), including the recubes (though certainly
> not the re-recubes).
---
Tim Chow
Message has been deleted
peps...@gmail.com
Jul 20, 2023, 10:45:42 AM7/20/23
to
On Thursday, July 20, 2023 at 1:02:47 PM UTC+1, Timothy Chow wrote:
...
I don't quite follow what you're saying here. For example, in mens pro tennis,
best of 3 sets and best of 5 sets are both common formats. And yes, both 3 and 5
are odd, and best of (for example) 6 sets wouldn't make any sense.
But what (some) people are puzzling over is why the winner of a backgammon match
is (almost) always first to 2 * n + 1 rather than first to 2n.
But first to 2n means best of 4n - 1 which is of course odd.
The thinking process you describe does nothing to explain why backgammon matches,
in contrast to other sports, are (almost) always best of 4n + 1 (for some n) rather than
best of 4n + 3. This can't be explained by pointing out that "best of k" needs k to be odd.
Paul
...
>
> As for odd match lengths, a surprising number of people will suggest
> that it's to avoid ties. This of course makes no logical sense, but
> there are of course many contests in sports that have a "best-of-n"
> form, where n needs to be odd to avoid ties. So perhaps people just
> got used to match lengths being an odd number in other sports, and
> carried over this tradition to backgammon even though there's no
> mathematical reason for it.
...
> As for odd match lengths, a surprising number of people will suggest
> that it's to avoid ties. This of course makes no logical sense, but
> there are of course many contests in sports that have a "best-of-n"
> form, where n needs to be odd to avoid ties. So perhaps people just
> got used to match lengths being an odd number in other sports, and
> carried over this tradition to backgammon even though there's no
> mathematical reason for it.
I don't quite follow what you're saying here. For example, in mens pro tennis,
best of 3 sets and best of 5 sets are both common formats. And yes, both 3 and 5
are odd, and best of (for example) 6 sets wouldn't make any sense.
But what (some) people are puzzling over is why the winner of a backgammon match
is (almost) always first to 2 * n + 1 rather than first to 2n.
But first to 2n means best of 4n - 1 which is of course odd.
The thinking process you describe does nothing to explain why backgammon matches,
in contrast to other sports, are (almost) always best of 4n + 1 (for some n) rather than
best of 4n + 3. This can't be explained by pointing out that "best of k" needs k to be odd.
Paul
Timothy Chow
Jul 21, 2023, 8:07:09 AM7/21/23
to
*logical* sense. My hypothesis is that people just got used to
odd numbers showing up in match lengths, and blindly chose to use
odd numbers in backgammon match lengths. Or in other words, my
hypothesis is that people are illogical. Evidence for this
hypothesis is that a surprising number of people, when asked this
question, will offer an illogical answer ("it's to avoid ties").
A former director of the lab where I work has a line which I love:
"Your problem, Tim, is that you're trying to use logic." I believe
that that applies here.
---
Tim Chow
Bradley K. Sherman
Jul 21, 2023, 8:14:13 AM7/21/23
to
Timothy Chow <tchow...@yahoo.com> wrote:
> ...
Logic is not concerned to discover premises that persuade
an audience to accept, or to believe, the conclusion.
This is the subject of rhetoric.
--bks
> ...
>A former director of the lab where I work has a line which I love:
>"Your problem, Tim, is that you're trying to use logic." I believe
>that that applies here.
From the Encyclopedia Britannica article on logical fallacies:
>"Your problem, Tim, is that you're trying to use logic." I believe
>that that applies here.
Logic is not concerned to discover premises that persuade
an audience to accept, or to believe, the conclusion.
This is the subject of rhetoric.
--bks
MK
Jul 22, 2023, 5:37:59 AM7/22/23
to
On July 19, 2023 at 9:51:40 AM UTC-6, Axel Reichert wrote:
> | Match length | Average Number of Games |
> | 1 | 1.00 |
> | 3 | 2.35 |
> | 5 | 3.83 |
> | 7 | 5.02 |
> | 9 | 7.24 |
> | 11 | 7.81 |
> | 3 | 2.35 |
> | 5 | 3.83 |
> | 7 | 5.02 |
> | 9 | 7.24 |
> | 11 | 7.81 |
> | 13 | 10.35 |
> | 15 | 10.00 |
> | 17 | 11.60 |
> | 19 | 14.00 |
> | 21 | 13.20 |
> | 23 | 13.85 |
> | 25 | 15.80 |
> | 15 | 10.00 |
> | 17 | 11.60 |
> | 19 | 14.00 |
> | 21 | 13.20 |
> | 23 | 13.85 |
> | 25 | 15.80 |
> 20 matches each is certainly to few, which
> possibly explains the non-monotoneous growth.
What the heck "non-monotoneous" means? Can't you
> possibly explains the non-monotoneous growth.
bring yourself to say say non-linear? ;)
> But I can easily imagine, as Paul put it, some
> nonlinear effects,
words ease your pain in denying your own findings?
You don't need to imagine anything. It only takes two
minutes to put the above columns into a spreadsheet
and click an icon to create a chart. Here it is for you:
https://montanaonline.net/backgammon/ml.pdf
It's an unmistakable curve just as I had predicted and
described. If you run long trials, it will look as smooth
as Tennessee whiskey...
> due to "overshooting" or "undershooting" the match
> length with a particular (higher) cube level.
a foot is all the same.
I can't even begin to wonder what "undershooting the
match length may mean and/or be meaningful in this
context..?!
And what does "particular (higher) cube level" means?
Can you give some "particular" examples...?
Stop piling bullshit upon bullshit please! Is it really that
hard to accept that I am right about cube's magnifying
luck more than skill, even after your own findings show
that...? :(
> Which for me immediately raises the question which
> match lengths are particularly interesting in the sense
> of having many match scores at which the cube action
> deviates strongly from money sessions.
MK
MK
Jul 22, 2023, 7:02:42 AM7/22/23
to
On July 19, 2023 at 9:51:40 AM UTC-6, Axel Reichert wrote:
>. .... But I can easily imagine, as Paul put it,
> some nonlinear effects, due to .....
I remembered that I forgot to comment on this,
after I posted.
Paul hadn't put it as "average number of games
in a match" but as "relationship between expected
match length (in terms of total number of plays)".
After I clarified it, I would have expected a decent
human to rephrase himself but surely not Axel. :(
I was the one who first suggested that the average,
i.e. "effective", gamblegammon match lengths are
non-linear and perhaps Paul got his idea, (which is
not really clear and which he failed to explain thus
far), from my that suggestion.
What made me participate in this thread was that
he spoke as "(in terms of total number of plays)"!
I'm not sure if he misspoke or if he knew what he
was talking about. So, I found it interesting and I
inquired about it.
Frankly, I never put any thought about the total
and/or average number of "plays" in matches of
various lengths. I was just curious to ask if it
followed a curve similar to the one of average
number of games in various match lengths.
But, by now I know that I am again wasting my
time trying to have an intelligent discussion with
half-brained members of a mentally ill dog pack
of addicted gamblers, who not only sniff one
another butt but lick the shit off of one another... :((
MK
Timothy Chow
Jul 22, 2023, 8:16:05 AM7/22/23
to
On 7/22/2023 5:37 AM, MK wrote:
> On July 19, 2023 at 9:51:40 AM UTC-6, Axel Reichert wrote:
>> 20 matches each is certainly to few, which
>> possibly explains the non-monotoneous growth.
>
> What the heck "non-monotoneous" means? Can't you
> bring yourself to say say non-linear? ;)
I think he meant "non-monotonic".
> On July 19, 2023 at 9:51:40 AM UTC-6, Axel Reichert wrote:
>> 20 matches each is certainly to few, which
>> possibly explains the non-monotoneous growth.
>
> What the heck "non-monotoneous" means? Can't you
> bring yourself to say say non-linear? ;)
https://en.wikipedia.org/wiki/Monotonic_function
---
Tim Chow
MK
Jul 22, 2023, 6:15:00 PM7/22/23
to
keep moving towards better understanding and
improving at least my own arguments.
I couldn't wrap my head around "non-monotoneous"
within the context, (not to claim that I could in any
other context), but "non-monotonic" makes sense
and is easier to understand within our subject.
So, "non-monotonic" necessarily means "non-linear"
but I understand that curves can be increasing both
monotonically and non-monotonically..? Like these:
https://i.stack.imgur.com/OriGC.png
The only decreases in Axel's number are 10.35 to 10
and 14 to 13.2 which may not jump at one's face but
they become easier to see looking at my chart.
I don't really understand Axel's usage of "nonlinear
effects" in this context either. Specifically, I can't tell
if they can apply only to some sections of a line or a
curve or if they must apply to the entire function.
Assuming that by "nonlinear effects" Axels refers to
those two decreases, simply dismissing them trying
to still call it "linear" won't work for him but I certainly
can live with a "non-monotonic curve", increasing at
a decreasing rate.
MK
Axel Reichert
Jul 23, 2023, 4:35:51 AM7/23/23
to
Timothy Chow <tchow...@yahoo.com> writes:
> As for odd match lengths, a surprising number of people will suggest
> that it's to avoid ties. This of course makes no logical sense, but
> there are of course many contests in sports that have a "best-of-n"
> form, where n needs to be odd to avoid ties. So perhaps people just
> got used to match lengths being an odd number in other sports, and
> carried over this tradition to backgammon even though there's no
> mathematical reason for it.
Might be. In Germany, the "best-of-n" wording came up only some decades
> As for odd match lengths, a surprising number of people will suggest
> that it's to avoid ties. This of course makes no logical sense, but
> there are of course many contests in sports that have a "best-of-n"
> form, where n needs to be odd to avoid ties. So perhaps people just
> got used to match lengths being an odd number in other sports, and
> carried over this tradition to backgammon even though there's no
> mathematical reason for it.
ago, probably as an "Americanism". The standard wording for, say, a
best-of-5 tennis match was "3 Gewinnsaetze", roughly translating as "the
winner will need 3 won sets". I still remember being puzzled by the
"best-of-n" way, because I needed to do calculations to come up with the
needed number of won sets and also because in many cases not all n sets
are played. So this terminology felt weird (and still does), the
backgammon way feels more natural.
Best regards
Axel
Timothy Chow
Jul 23, 2023, 5:10:55 AM7/23/23
to
On 7/23/2023 4:35 AM, Axel Reichert wrote:
> So this terminology felt weird (and still does), the
> backgammon way feels more natural.
Not only that, there's no reasonable way to describe a
> So this terminology felt weird (and still does), the
> backgammon way feels more natural.
backgammon match in "best-of-n" language, because one can
win or lose more than one point per game.
By the way, here's a puzzle that I think I have posted on
rec.games.backgammon before, but which you may not have seen.
Two teams are playing in the World Series, which is a best-of-7
match ("4 Gewinnsaetze"). One team, the Slow Starters, always
loses the first game. The other team, the Late Chokers, always
loses *if* the series reaches a score of 3-3. Otherwise, the
two teams are evenly matched, and are equally likely to win any
particular game. Which team is more likely to win the series?
---
Tim Chow
peps...@gmail.com
Jul 23, 2023, 10:12:30 AM7/23/23
to
Both candies are exactly the same size!"
The late chokers start with a 1 0 lead.
They win the match if they win 3 of the next 5.
Because of the Tim-Termination condition, they lose if they win only
two of the next 5.
In a 50/50 context, winning at least 3 out of 5 is a 50/50 parlay.
So you can take either candy you want.
Paul
peps...@gmail.com
Jul 23, 2023, 10:18:30 AM7/23/23
to
and in many contexts, that's exactly right.
For example, you might be a potential spectator who would be unwilling to leave
in the middle of the match. So you focus on the max length to judge whether you
can spend the time watching.
You might want to promote the event to fans and entice them with the prospect of
a long match. So the max length figures prominently. In tennis, audiences generally prefer longer matches.
This contrasts to soccer, I think, where the aficionados generally like the match to
be concluded in the regulation 90 minutes rather than hoping for extra time and penalties.
Paul
Axel Reichert
Jul 23, 2023, 12:56:53 PM7/23/23
to
difference between non-monotonic and non-linear, which is why I tried to
use the former on purpose.
Meanwhile, I have automated things with GNU Backgammon (IMHO /the/
killer feature compared to XG) and run 100 matches for all match lengths
from 1 to 64 (the maximum allowed in GNU Backgammon). Here are the
results (a plot shows games and moves to fit very nicely to straight
lines), which do not bear out the hypothesis that there might be a
different relation between the match length and the number of games
played than O(n):
|--------+-------+--------------+--------+------------|
| Length | Games | Games/length | Moves | Moves/game |
|--------+-------+--------------+--------+------------|
| 1 | 1.00 | 1.000 | 55.0 | 54.95 |
| 2 | 1.00 | 0.500 | 54.0 | 53.96 |
| 3 | 2.24 | 0.747 | 95.8 | 42.78 |
| 4 | 2.59 | 0.647 | 105.9 | 40.90 |
| 5 | 3.38 | 0.676 | 145.2 | 42.96 |
| 6 | 3.70 | 0.617 | 162.7 | 43.98 |
| 7 | 4.69 | 0.670 | 199.3 | 42.49 |
| 8 | 5.39 | 0.674 | 223.6 | 41.48 |
| 9 | 5.98 | 0.664 | 257.3 | 43.03 |
| 10 | 6.24 | 0.624 | 256.8 | 41.15 |
| 11 | 7.05 | 0.641 | 307.1 | 43.56 |
| 12 | 8.36 | 0.697 | 338.6 | 40.50 |
| 13 | 8.29 | 0.638 | 350.6 | 42.30 |
| 14 | 8.92 | 0.637 | 380.1 | 42.61 |
| 15 | 9.40 | 0.627 | 401.3 | 42.69 |
| 16 | 10.58 | 0.661 | 445.2 | 42.08 |
| 17 | 11.46 | 0.674 | 475.2 | 41.46 |
| 18 | 11.80 | 0.656 | 492.4 | 41.73 |
| 19 | 12.49 | 0.657 | 529.5 | 42.40 |
| 20 | 13.14 | 0.657 | 564.6 | 42.97 |
| 21 | 14.25 | 0.679 | 608.2 | 42.68 |
| 22 | 14.88 | 0.676 | 612.7 | 41.18 |
| 23 | 14.35 | 0.624 | 600.9 | 41.87 |
| 24 | 16.15 | 0.673 | 665.9 | 41.23 |
| 25 | 16.61 | 0.664 | 680.5 | 40.97 |
| 26 | 17.04 | 0.655 | 703.1 | 41.26 |
| 27 | 17.63 | 0.653 | 746.1 | 42.32 |
| 28 | 17.59 | 0.628 | 754.0 | 42.87 |
| 29 | 19.29 | 0.665 | 796.6 | 41.30 |
| 30 | 20.72 | 0.691 | 869.5 | 41.97 |
| 31 | 20.89 | 0.674 | 868.4 | 41.57 |
| 32 | 20.44 | 0.639 | 862.9 | 42.22 |
| 33 | 22.07 | 0.669 | 910.2 | 41.24 |
| 34 | 23.21 | 0.683 | 958.5 | 41.30 |
| 35 | 23.76 | 0.679 | 974.3 | 41.01 |
| 36 | 24.18 | 0.672 | 1017.4 | 42.08 |
| 37 | 25.57 | 0.691 | 1046.1 | 40.91 |
| 38 | 25.65 | 0.675 | 1075.5 | 41.93 |
| 39 | 25.99 | 0.666 | 1081.4 | 41.61 |
| 40 | 27.40 | 0.685 | 1143.7 | 41.74 |
| 41 | 27.15 | 0.662 | 1144.9 | 42.17 |
| 42 | 28.91 | 0.688 | 1204.2 | 41.65 |
| 43 | 29.73 | 0.691 | 1241.3 | 41.75 |
| 44 | 29.76 | 0.676 | 1268.0 | 42.61 |
| 45 | 31.24 | 0.694 | 1299.0 | 41.58 |
| 46 | 30.49 | 0.663 | 1264.4 | 41.47 |
| 47 | 31.58 | 0.672 | 1306.6 | 41.38 |
| 48 | 34.06 | 0.710 | 1429.5 | 41.97 |
| 49 | 33.37 | 0.681 | 1383.9 | 41.47 |
| 50 | 33.59 | 0.672 | 1404.7 | 41.82 |
| 51 | 35.79 | 0.702 | 1474.2 | 41.19 |
| 52 | 35.77 | 0.688 | 1479.1 | 41.35 |
| 53 | 35.89 | 0.677 | 1477.9 | 41.18 |
| 54 | 36.32 | 0.673 | 1491.2 | 41.06 |
| 55 | 37.31 | 0.678 | 1547.9 | 41.49 |
| 56 | 37.79 | 0.675 | 1557.7 | 41.22 |
| 57 | 40.29 | 0.707 | 1645.4 | 40.84 |
| 58 | 40.60 | 0.700 | 1684.1 | 41.48 |
| 59 | 41.50 | 0.703 | 1723.9 | 41.54 |
| 60 | 42.20 | 0.703 | 1737.5 | 41.17 |
| 61 | 43.32 | 0.710 | 1790.3 | 41.33 |
| 62 | 43.99 | 0.710 | 1828.3 | 41.56 |
| 63 | 43.96 | 0.698 | 1822.4 | 41.46 |
| 64 | 44.05 | 0.688 | 1822.3 | 41.37 |
So the number of games is pretty constant at about 2/3 of the match
length. The fit given in
https://www.bkgm.com/rgb/rgb.cgi?view+1241
seems to consider only the data for shorter matches from
https://www.bkgm.com/rgb/rgb.cgi?view+712
Even though longer matches would offer the the opportunity for
higher cubes, thus drastically reducing the number of games played, it
seems that the likelyhood of higher cubes (with proper cube skill)
diminishes faster than the match length increases. See
https://www.bkgm.com/rgb/rgb.cgi?view+662
for how rare already a cube of 8 is.
Also, the number of moves is about 42, apart from the obvious special
cases of 1- and 2-pointers (about 54). This confirms the first table in
https://www.bkgm.com/rgb/rgb.cgi?view+712
I have done a similar thing for cubeless backgammon (but only for
matches up to 25 point), the corresponding data look like this:
|--------+-------+--------------+--------+------------|
| Length | Games | Games/length | Moves | Moves/game |
|--------+-------+--------------+--------+------------|
| 1 | 1.00 | 1.000 | 54.6 | 54.65 |
| 2 | 1.95 | 0.975 | 107.5 | 55.10 |
| 3 | 3.00 | 1.000 | 163.8 | 54.61 |
| 4 | 4.69 | 1.173 | 247.5 | 52.76 |
| 5 | 5.97 | 1.194 | 323.8 | 54.24 |
| 6 | 6.99 | 1.165 | 374.3 | 53.55 |
| 7 | 8.70 | 1.243 | 465.4 | 53.49 |
| 8 | 9.89 | 1.236 | 521.6 | 52.74 |
| 9 | 11.51 | 1.279 | 609.8 | 52.98 |
| 10 | 12.40 | 1.240 | 668.3 | 53.90 |
| 11 | 14.00 | 1.273 | 759.0 | 54.21 |
| 12 | 15.88 | 1.323 | 848.9 | 53.46 |
| 13 | 16.61 | 1.278 | 886.8 | 53.39 |
| 14 | 18.56 | 1.326 | 990.5 | 53.36 |
| 15 | 20.16 | 1.344 | 1082.6 | 53.70 |
| 16 | 20.89 | 1.306 | 1124.0 | 53.81 |
| 17 | 22.71 | 1.336 | 1218.6 | 53.66 |
| 18 | 24.14 | 1.341 | 1295.7 | 53.67 |
| 19 | 25.40 | 1.337 | 1358.3 | 53.47 |
| 20 | 26.46 | 1.323 | 1418.1 | 53.60 |
| 21 | 28.57 | 1.360 | 1527.2 | 53.46 |
| 22 | 29.75 | 1.352 | 1603.1 | 53.89 |
| 23 | 31.37 | 1.364 | 1678.8 | 53.52 |
| 24 | 32.91 | 1.371 | 1770.3 | 53.79 |
| 25 | 33.93 | 1.357 | 1795.3 | 52.91 |
The number of moves per game is again about 54, no surprise here. The
number of games per match is pretty constant again, this time at about
4/3 of the match length.
If we put this together we end up with
54 * 4/3 * m = 72 * m
checker plays for cubeless backgammon (match length m) and
42 * 2/3 * m = 28 * m
for cubeful backgammon. For the latter (let us assume that the cube gets
turned in the middle of the game, at move 21) there are 21 cube decision
with the centered cube and about 10 (the half of the remaining 21 moves)
with the cube owned by either of the players. All on average, of course,
and a very rough estimate. If we factor in any take/pass decision there
are about 60 decisions (checker or cube) in cubeful backgammon.
This of course does not amount to cubeless backgammon being the more
skillful game:
1. Cashing a boring (in the sense of low equity difference between
candidate plays) race or other low-skill games cuts away the luck.
2. Imperfect human players might squander more equity getting cube
decisions wrong than checker plays.
Best regards
Axel
MK
Jul 24, 2023, 3:36:12 AM7/24/23
to
On July 23, 2023 at 10:56:53 AM UTC-6, Axel Reichert wrote:
> Timothy Chow <tchow...@yahoo.com> writes:
>> On 7/22/2023 5:37 AM, MK wrote:
>>> On July 19, 2023 at 9:51:40 AM UTC-6, Axel Reichert wrote:
>>>> 20 matches each is certainly to few, which
>>>> possibly explains the non-monotoneous growth.
>>> What the heck "non-monotoneous" means? Can't
>>> you bring yourself to say say non-linear? ;)
Let me start by being nice to say that I appreciate
> Timothy Chow <tchow...@yahoo.com> writes:
>> On 7/22/2023 5:37 AM, MK wrote:
>>> On July 19, 2023 at 9:51:40 AM UTC-6, Axel Reichert wrote:
>>>> 20 matches each is certainly to few, which
>>>> possibly explains the non-monotoneous growth.
>>> What the heck "non-monotoneous" means? Can't
>>> you bring yourself to say say non-linear? ;)
your below work and I did my contributing part by
copy-pasting it into spreadsheets and generating
pretty looking charts to help you folks see things
more clearly. As opposed to Axel's misinterpreting
his own findings, once again, they indeed bolster
my arguments more clearly, decisively. Let's begin.
>> I think he meant "non-monotonic".
>> https://en.wikipedia.org/wiki/Monotonic_function
> Yes, indeed. And of course, but at least you know
> this, there is a difference between non-monotonic
> and non-linear,
"non-monotonic" necessarily meant "non-linear",
which doesn't mean that there is no difference
between them but that once you say something is
"non-monotonic", you can no longer use the word
"linear"! But of course you will keep arguing that it
is "non-monotonic" and "linear" at the same time.
> which is why I tried to use the former on purpose.
Just be thankful to him and leave it well alone. You
don't need to be pathetic trying to return the favor by
wiping it off of his lips... :(
> I have automated things with GNU Backgammon
> (IMHO /the/ killer feature compared to XG)
to specify a session length to run. I was going to
suggest that they also add the ability to specify the
number of matches to run, i.e. a "session of matches"
like a "session of (money) games", but I decided to
not waste my time with them weird bunch. I'm sure
they will implement it if someone other than Murat
suggests it... Then, others won't need to "automate
things" as you have done with Noo-BG.
> and run 100 matches for all match lengths from 1
> to 64 (the maximum allowed in GNU Backgammon).
your data as for you previous experiments??
> Here are the results (a plot shows games and
> moves to fit very nicely to straight lines),
> which do not bear out the hypothesis that there
> might be a different relation between the match
> length and the number of games played than O(n):
O(n) was spiteful Walt's lack of understanding what
Paul was talking about, i.e. the relation between the
match length and the total number of plays, which
is non-monotonic decreasing but perhaps not by
not enough to warrant different clock rules. See:
https://montanaonline.net/backgammon/mlg.pdf
Paul was clearly talking about clock time which is
depleated per move/play not per game. One must
be a moron to miss that. Interestingly Paul would
have a very strong case in backgammon even if
not in gamblegammon.
In backgammon, number of plays per game is also
non-monotonic but increasing! and significatly. See:
https://montanaonline.net/backgammon/mlb.pdf
I also generated a chat for only 25 gamblegammon
matches for easier visual comparing. See:
https://montanaonline.net/backgammon/mlg25.pdf
For one thing, you are calculating your "moves per
game" column wrongly using the effective match
lengths. One must be twice a moron to miss that,
in his original post, Paul wrote "... give each player
2 * n minutes for the match". So, I added a column
showing plays per stated match length for which
the curves look similar im gamblegammon but not
in backgammon.
> So the number of games is pretty constant at
> about 2/3 of the match length.
> Even though longer matches would offer the the
> opportunity for higher cubes, thus drastically
> reducing the number of games played,
a little disappointed that the curve doesn't bend as
I had envisioned. Maybe because of that there are
increasingly more matches of different lengths
between cube value increments, i.e. between 4 and
8 cube there are 4, 5, 6, 7-points; between 8 and 16
cube there are 8, 9, 10, 11, 12, 13, 14, 15-points,
and the matches of lengths closer after the cube
value will be effected worse, i.e shortened by more
games. Those are the zigzags, non-monotonic drops
along the curve of effective match lengths.
> seems that the likelyhood of higher cubes (with
> proper cube skill) diminishes faster than the match
> length increases. See
How rare in matches of what lengths? Nubers there
seems for money games. Meaningless for matches.
> I have done a similar thing for cubeless backgammon
> (but only for matches up to 25 point),
What you are referring to is a "cubeless variant of
gamblegammon". None of the past or current bots
offer "backgammon", i.e. without 3-point wins, etc.
> The number of moves per game is again about 54,
> no surprise here.
mean but you are wrongly dividing the total number
of moves/plays by the effective match length, not by
the stated match length. When you do it right, it's a
range from 53.75 to 73.76 moves/plays per game.
On a side note, I used the magic 54 in a mutant cube
experiment draft that I never got around to posting.
There are some stats out there which point at about
that average but it's not always clear whether they're
based on match games and/or money games. See:
https://zooescape.com/backgammon-stats.pl
> The number of games per match is pretty constant
> again, this time at about 4/3 of the match length.
linear again! It is a *monotonic* increasing curve!
> If we put this together we end up with
> 54 * 4/3 * m = 72 * m
> 42 * 2/3 * m = 28 * m
False! You are presenting your fantasies as facts.
> This of course does not amount to cubeless
> backgammon being the more skillful game:
"cubeless gamblegammon" is a more skillful game,
as your own experiment has demonstrated, because
without the cube luck isn't magnified.
Your experiment is between equally checker skilled
players. Thus skill is already level and only luck can
fluctuate.
Look at the charts carefully.
In "cubeless gamblegammon" luck fluctuates but
never enough to overtake checker skill. That's why
the effective match length is a *monotonic* curve.
In "cubeful gamblegammon" luck fluctuations are
magnified by the cube and overtake checker skill,
(and cube skill for that matter), causing the dips
in the non-monotic curve.
With 4 cube, a 4-point match is more likely to end in
a single game than a 5-point match. A 6-point match
is less likely and a 7-point point match is even lesser
like yet. The curve will dip again at 8-point match and
again at 16-point match and again at 32-point match.
If you don't believe me, try to believe yourself. What
more proof do you want that cube add more luck to
gamblegammon than it adds skill.
And look at how smooth the effective match length,
or number of games in a match as you say, curve is
for "cubeless gamblegammon" compared to the one
for "cubeful gamblegammon". You guys need to read
what I had written in the thread "Circular Circus". See:
https://groups.google.com/g/rec.games.backgammon/c/CiG54VoJvq8
Here are a couple of snippets:
"... real question is how quickly the match-winning
"chances increase with match length". I couldn't
"guess how quicky but in my opinion it will increase
"more smoothly and steadily in backgammon, more
"erratically in gamblegammon ...
"As I had pointed out many times in the past,
"doubling cube shortens matches. A 13 or a 17
"point gamblegammon match may be equivalent
"of a 5 point backgammon match, similarly a 19
"or 25 point gamblegammon match may be
"equivalent of a 7 point backgammon match.
Lo and behold! Your total moves for a 5-point cubeless
gamblegammon match is 324, for a 13-point cubeful
gamblegammon match is 351. Similarly, for a 7-point
cubeless gamblegammon match is 465, for a 19-point
gamblegammon match is 530. Backgammon numbers
would be higher as matches would last longer without
3-point wins and thus would prove my guestimates to
be pretty darn accurate, if not almost exact.
> 1. Cashing a boring (in the sense of low equity
> difference between candidate plays) race or other
> low-skill games cuts away the luck.
"cuts away the luck", why don't you make championship
matches 15-points instead of 25-points? Or even why
not 7-points? 3-points? 1-points? If that's still too boring
for you, you can skip playing altogether and just roll dice
or better yet toss a coin to determine the winner. ;)
> 2. Imperfect human players might squander more
> equity getting cube decisions wrong than checker plays.
never made any efforts to check if your scales are not
imperfect, inacurate themselves.
As a last comment, let me say that despite all the harsh
words I use in citicizing you, I still hold you special from
others because you are the only one open minded and
willing enough to conduct experiments. It's obvious that
you are doing them not to discover, learn something new
but to prove that your gambling addiction is actually an
exercise in skill competition, in order to comfort your
sick mind. Still, it's better than nothing. So, I do openly
thank you for that.
MK
Timothy Chow
Jul 24, 2023, 7:58:55 AM7/24/23
to
On 7/23/2023 10:12 AM, peps...@gmail.com wrote:
> As usual, in these things. "It doesn't matter which one you choose.
> Both candies are exactly the same size!"
>
> The late chokers start with a 1 0 lead.
> They win the match if they win 3 of the next 5.
> Because of the Tim-Termination condition, they lose if they win only
> two of the next 5.
> In a 50/50 context, winning at least 3 out of 5 is a 50/50 parlay.
> So you can take either candy you want.
Correct!
> As usual, in these things. "It doesn't matter which one you choose.
> Both candies are exactly the same size!"
>
> The late chokers start with a 1 0 lead.
> They win the match if they win 3 of the next 5.
> Because of the Tim-Termination condition, they lose if they win only
> two of the next 5.
> In a 50/50 context, winning at least 3 out of 5 is a 50/50 parlay.
> So you can take either candy you want.
Here's my solution. Best-of-n matches usually terminate as soon
as one side clinches the win. But let's go ahead and force the
players to play all n matches anyway. Furthermore, I'm going to
stipulate that the Late Chokers always lose the 7th game, whether
or not the score is 3-3 at that point. This won't change any
outcomes, and it will allow us to apply a symmetry argument: the
Late Chokers always lose the 7th game, and the Slow Starters always
lose the 1st game. Otherwise, every game is equally likely to go
either way. So the teams are equally likely to win the series.
---
Tim Chow
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