GNU Backgammon against its Murat Mutant: The first 1000 games

Axel Reichert

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Oct 27, 2021, 2:24:33 PM10/27/21

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Hello,

I gave it a try. Since I am programming a bot anyway (based on rules of thumb,
for personal experimentation and gathering of statistical data, not as a
serious contender to world class players), it was a rather simple
exercise to implement a strategy as follows:

1. Checker play: Do what GNU Backgammon does with checker play set to
"Expert" level.

2. Cube decisions:

a) Double with more than 50 percent cubeless winning chances

b) Take or pass according to GNU Backgammon's assessment ("World Class")

c) Raccoon if beavered ("higher order" rodents forbidden)

I hope this is more or less what Murat politely suggests us to do to
become better backgammon players, supposedly on "alpha" level.

I pitted this strategy against GNU Backgammon, of course likewise with
"Expert" checker play and "World Class" cube decisions. So both
opponents essentially did the same checker plays, only the cube
strategies were different. Think of my bot as a mutant with some genetic
changes in its doubling brain.

The score after 1000 games (money session, Jacoby) is

GNU Backgammon: 14182 (564 wins)
Murat Mutant: 7582 (436 wins)

Now the score is very high for "only" 1000 games (of course, since the
mutant drives up the cube value), so some statistical checks were in
order.

The average game value was 21.764 and the variance was 19263.4, hence
its standard deviation was a whopping 139. There were 34 games with at
least 128 points, the most points won in a single game were 4096.

Now with

https://bkgm.com/rgb/rgb.cgi?view+709

we can assess whether GNU Backgammon's win was already statistically
significant with such a volatile/erratic opponent. We assume that the
mutant's strategy is as good as GNU Backgammon's, so expect with 95 %
probability that the absolute difference between the two players is
smaller than

2 * sigma * sqrt(1000) =

2 * 139 * 32 =

8778

This is larger than 6600 (GNU Backgammon's net result), so the jury is
still out. I will continue the run and keep you posted. In the mean
time, statistical advice is very welcome.

Best regards

Axel

MK

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Oct 27, 2021, 5:34:00 PM10/27/21

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On October 27, 2021 at 12:24:33 PM UTC-6, Axel Reichert wrote:

> I gave it a try.

At last someone tried! Great news. Hopefully you will correct,
refine and improve on the experiment to go beyond what I had
suggested. Surely we all will learn new things either way.

> 1. Checker play: Do what GNU Backgammon does with checker
> play set to "Expert" level.

How long did the 1000 games take? Can you set it to the highest
level from now on? If it would take too long to run each iteration,
I would be willing to contribute CPU time.

> 2. Cube decisions:

> a) Double with more than 50 percent cubeless winning chances

Can we run this using both cybeless and cubeful winning chances?

And also cubeless winning chances calculated by other bots for at
least the opening and reply rolls?

In my own experiments, for example, I double agressively after early
62, 63, 64 and 21 depending on how they are played by either side.
Did your mutant double immediately after gnubg opened with 63 and
took the beaver?

> b) Take or pass according to GNU Backgammon's assessment
> ("World Class")

I don't remember exactly what I had proposed years ago but can we
make this based on more than 50 percent also (in light of what Paul
is using in his other thread in a related discussion)?

> c) Raccoon if beavered ("higher order" rodents forbidden)

Again can you run this either way? In my later experiments I tried to
raccon also but I never did enough to publish the results since I lost
overall interest.

Can you explain why the limitation? How would it help the experiment?

> I hope this is more or less what Murat politely suggests us to do to
> become better backgammon players, supposedly on "alpha" level.

Let's be clear that I didn't equate the two. I argued that an alpha-bg
bot will handily beat the current bots and debunk all current dogmas.
In my own experiments, I tried to play like how I best predicted that
an alpha-bg bot may play but never claimed that I knew exactly how
an alpha-bg bot will play.

With that said, I would actually like to see your experiments be based
on improved predictions than mine (i.e. closer to an alpha-bg bot) but
I don't know how we could know that without having already access
to an alpha-bg bot.

> The score after 1000 games (money session, Jacoby) is

I never use Jacoby myself. Can you turn it off even if it may not make a
difference in this kind of experiments?

> This is larger than 6600 (GNU Backgammon's net result), so the jury
> is still out. I will continue the run and keep you posted.

Honestly, I couldn't make sense of your statistical calculations and/or
conclusions but I am glad that the jury is still out and you will continue
the "experiments" (not just "run" the same one longer) with my above
corrections/suggestions.

> In the mean time, statistical advice is very welcome.

As well as any other advice, I would add. Hopefully with efforts toward
not just proving me wrong (which is okay also) but to learn new things.

MK

MK

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Oct 27, 2021, 7:18:27 PM10/27/21

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On October 27, 2021 at 12:24:33 PM UTC-6, Axel Reichert wrote:

> In the mean time, statistical advice is very welcome.

Here are a few from me. When I analysed my experiments,
I tabulated how many times which side started first, cubed
first, cubed last, lost while holding the cube, last cube value,
won by race, etc.

I had found all these more or less relevant/meaningful to
try learning from. You may want to keep similar statistics.

MK

Timothy Chow

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Oct 28, 2021, 11:42:22 PM10/28/21

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On 10/27/2021 2:24 PM, Axel Reichert wrote:
> The score after 1000 games (money session, Jacoby) is
>
> GNU Backgammon: 14182 (564 wins)
> Murat Mutant: 7582 (436 wins)

[calculation omitted]

> This is larger than 6600 (GNU Backgammon's net result), so the jury is
> still out. I will continue the run and keep you posted. In the mean
> time, statistical advice is very welcome.

This is an interesting experiment. I think it's going to be a little
tricky to attach a number like "95% confidence" because we're dealing
with two very different players. I mean, you could say that the null
hypothesis is that GNU is the same as the mutant, and you could reject
that hypothesis very quickly with high confidence, but that's not very
interesting---we *know* that GNU is not the same as the mutant without
collecting any statistics at all.

Or your null hypothesis could be that the mutant's expected net score
against GNU is zero. But rejecting this hypothesis with high confidence
isn't too interesting either, since that wouldn't say anything about the
*magnitude* of the difference between the two players, which is what you
are really interested in.

What I would suggest is to forget about 95% confidence and just plot a
histogram of the results---for each possible game outcome (by which I
just mean the score---e.g., a gammon with the cube on 2 is treated the
same as a single win with the cube on 4), plot the number of games that
have that outcome. Then just play enough games so that even the rarer
outcomes are achieved more than a handful of times.

---
Tim Chow

MK

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Nov 4, 2021, 1:27:42 PM11/4/21

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On October 28, 2021 at 9:42:22 PM UTC-6, Tim Chow wrote:

> .... Then just play enough games so that even the rarer

> outcomes are achieved more than a handful of times.

How many would be "enough" and should that number
be decided beforehand? Otherwise, if one is not happy
with the results after 5000 games (undeclared but
initially considered enough), what prevent going on to
6000 games and stoppig at that if the results prove a
certain point (which may not be so again after 7000
games)?

Another suggestion I have is to make the mutant make
cube decisions based on not just expected 50%+ winning
chances but on other criteria such as "whether enough
checker play is left in the game".

I think those are the words I had used in one occasion
when I was explaining (to Michael?) how I made cube
decisions.

Can a bot determine not just the complexity of a position
but also expected number of turns left to play in the game,
allowing tome enough for luck to swing (perhaps more
than once) and more importantly for the checker skill to
make a decisive difference??

If any bot mutant is to be called a Murat mutant, it needs
to be able to do this! (and my other similar strategies).

Otherwise, just call it Jacoffsky's mutant...

MK

Axel Reichert

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Nov 4, 2021, 2:29:29 PM11/4/21

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MK <mu...@compuplus.net> writes:

> How long did the 1000 games take?

Over night.

> Can you set it to the highest level from now on?

No. "Expert" is certainly strong enough checker play for this
experiment, especially since both sides will have the same
setting. "World Class" likewise is strong enough for cube decisions
("Expert" will not "see" market losers)

> Can we run this using both cybeless and cubeful winning chances?

No. The percentages that I use for the doubling criterion of the
mutant are given by GNU Backgammon as cubeless. Cubeful values are
given with equities.

> And also cubeless winning chances calculated by other bots for at
> least the opening and reply rolls?

No. I do not have other bots. All are strong enough for this kind of
experiment. I neither care for the opening and reply rolls, just
whether GWC is over the 0.5 threshold.

> Did your mutant double immediately after gnubg opened with 63 and
> took the beaver?

No. According to "World Class", after GNU Backgammon splits with 62
and 63 or runs with 64 (which is was "Expert" checker play does) the
mutant is a slight underdog. Hence no double. But there are other
"Expert" plays where "World Class" thinks the replier is favourite:

21S, 41S, 51S, 43Z => Double by the mutant

And of course, Beaver by GNU Backgammon, and depending on my settings,
Raccoon by the mutant.

> can we make this based on more than 50 percent

Perhaps I am willing to do a test with 0.618 (golden ratio, to throw in
esoterics for good measure), which is between 0.6 and 0.625: Leaving
gammons/backgammons aside, one could make a simplistic case for these
numbers based on live/dead cube assumptions and the football analogy:

https://bkgm.com/articles/Kleinman/FootballFields/index.html

>> c) Raccoon if beavered ("higher order" rodents forbidden)
>
> Again can you run this either way?

No. I never play with Raccoons, and in our club we had discussions of
banning Beavers as well (they tend to attract the "wrong" players).

> Can you explain why the limitation? How would it help the experiment?

Axel Reichert

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Nov 4, 2021, 3:09:02 PM11/4/21

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Timothy Chow <tchow...@yahoo.com> writes:

> plot a histogram of the results

Done, see my other post in this thread.

Axel

MK

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Nov 6, 2021, 3:00:26 AM11/6/21

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On November 4, 2021 at 12:29:29 PM UTC-6, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:

>> How long did the 1000 games take?

> Over night.

So, it sounds like we can run numerous quick experiments
and even longer ones within a reasonable amount of time.

For the questions that answered "no", (asked for my reasons
and answered for your reasons), I guess there is no point in
wasting time by discussing further but thanks for answering.

>> Did your mutant double immediately after gnubg opened
>> with 63 and took the beaver?

> No. According to "World Class", after GNU Backgammon splits with
> 62 and 63 or runs with 64 (which is was "Expert" checker play does)
> the mutant is a slight underdog. Hence no double. But there are other
> "Expert" plays where "World Class" thinks the replier is favourite:

This one is important and that's why I asked if you could use
opening equity calculations by other bots (like TD-Gammon).
I thought the opening book was user editable, no? Also, bots
like XG++ split the 64 and slot the 21. If you want to imitate
Murat's experiments, you need to do these. Otherwise, it's an
experiment of your own and has nothing to do with what I had
suggested (as the thread title implies).

>> can we make this based on more than 50 percent

> Perhaps I am willing to do a test with 0.618 (golden ratio, to throw
> in esoterics for good measure), which is between 0.6 and 0.625

Okay, this is good news. Just try whatever numbers you fancy
but try some different things. In fact, I had asked/suggested
many times in the past that any arbitrary and/or calculated
constants used by the bots should be made user selectable
variables in the settings or in a editable config file. I will be
curiously waiting to see what you come up with 0.618, etc...

I wish I could volunteer to help with CPU time and such but
you won't share your utility.

>> Again can you run this either way?

> No. I never play with Raccoons, and in our club we had discussions
> of banning Beavers as well (they tend to attract the "wrong" players).

But aren't Raccoons, Rats, Bats, Cats, etc. all part of what you all
call "cube skill"...?

This is not human play at your club. This an experiment about the
"cube skill" thing. Beavers may attract the wrong players at your
club but bots are asexual... ;)

I'm grinning from ear to ear though, at your idea of banning Beavers.
Go ahead. In fact, I have been advocating banning the doubling cube
altogether and I now have raised hopes that with your help it may
come to that sooner than later...

>> Can you explain why the limitation? How would it help the experiment?
> More on this in a different thread.

I'll look for it.

>> I never use Jacoby myself. Can you turn it off

> I always use Jacoby. This is crucial in chouettes, because without
> Jacoby the team might be bored to death while the captain, having
> slept during the cube decisions, tries to squeeze out a Gammon.

This has nothing to do with boring chouettes but fine with me. When
the common complaint that the cube gets too high, I don't think we
need to worry too much about games ending without a cube action.

> My gut feeling says that this is almost certainly not because your
> doubling strategy is competitive, but rather due to the St Peterburg
> Paradoxon. More of my thoughts on this in a different thread.

I'll look for it.

> Out of curiosity I tested some other doubling "strategies" (all with
> Jacoby and without Beavers, Null hypothesis as before):

> - Double with 50 % winning chances, always take

Should be at least 51% and no point in always taking without any
chace of winning left. Meaningless test.

> - Always double if legal, take according to GNU Backgammon "World
> Class"

Always doubling without any chace of winning is also pointless.
So, another meaningless test.

> - Always double if legal, always take

> This could be rejected after 200 games.

Why are you even wasting time with these??

> Note that even this last, maniac strategy needed 200 games to get
> rejected with 95 % certainty! This means that you should be extremely
> suspicious regarding results from a mere 100 games, especially if the
> strategy tends to drive the cube up, up and away.

If you are referring to my 100 games, I can't very well play 1000
games in one night and that's exactly why we are making bots
to play "long enough" sessions.

As for "jacking up the cube" (the old popular expression), I always
used the saying "It takes two to tango". If the bot's "extraterrestrial
cube skill" dances along, how can you blame for using my own
"human cube skill" to my advantage??

> That's it, I will not spend more time on things like this

Sounds like the results went against your expectations? Oh well,
at least you tried by doing more than anyone else has done yet.

MK

Axel Reichert

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Nov 6, 2021, 5:23:08 AM11/6/21

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MK <mu...@compuplus.net> writes:

> On November 4, 2021 at 12:29:29 PM UTC-6, Axel Reichert wrote:
>
>> No. According to "World Class", after GNU Backgammon splits with 62
>> and 63 or runs with 64 (which is was "Expert" checker play does) the
>> mutant is a slight underdog. Hence no double. But there are other
>> "Expert" plays where "World Class" thinks the replier is favourite:
>
> This one is important and that's why I asked if you could use opening
> equity calculations by other bots (like TD-Gammon). I thought the
> opening book was user editable, no? Also, bots like XG++ split the 64
> and slot the 21. If you want to imitate Murat's experiments, you need
> to do these. Otherwise, it's an experiment of your own and has nothing
> to do with what I had suggested (as the thread title implies).

To my knowledge, GNU Backgammon has no opening book, it just plays
according to the current settings, be that Expert (for speed reasons) or
World Class (too slow for these mass runs). According to XG's opening
book, the replier has an advantage only after 41S (with or without
Jacoby). My understanding is that you double after all 6x splits and
maybe for other opening rolls, I don't know. My mutant doubles after
1x splits and 43Z. In all these cases the winning chances of the replier
should be between and 49.48 % and 50.16 %. So in my opinion it does not
matter in which of these cases you raise the stakes, it is too early
anyway, because you are foregoing the possibility to double your
opponent out. This is the value of cube ownership, see the football
field analogy by Danny Kleinman.

> any arbitrary and/or calculated constants used by the bots should be
> made user selectable variables in the settings or in a editable config
> file. I will be curiously waiting to see what you come up with 0.618,
> etc...

Which is also "arbitrary". If you start to come up with ideas about
gammonish positions and "play left in the game" I should start to get
worried, because then you would reinvent the (mathematical) concepts of
equity and volatility. (-;

>> No. I never play with Raccoons, and in our club we had discussions of
>> banning Beavers as well (they tend to attract the "wrong" players).
>
> But aren't Raccoons, Rats, Bats, Cats, etc. all part of what you all
> call "cube skill"...?

The interesting question I try to research here is not any "maniac" cube
strategy, but whether a Petersburg Paradoxon occurs in backgammon with
unlimited cube (depending on number of beavers allowed). If yes, I think
we will have an interesting (but mostly philosophical) discussion about
skill.

But as soon as the cube is capped (e.g., in match play), no Peterburg
Paradoxon can occur and a "maniac" cube strategy will hurt the maniac
and thus prove his inferior cube handling. It follows immediately that
skill is involved.

>> - Double with 50 % winning chances, always take
>
> Should be at least 51%

Why not 50.000001 %? I should have written (and have implemented) "more
than 50 % winning chances".

> Why are you even wasting time with these??

See above. My interest is the Peterburg Paradoxon, not the cube
strategy.

>> Note that even this last, maniac strategy needed 200 games to get
>> rejected with 95 % certainty! This means that you should be extremely
>> suspicious regarding results from a mere 100 games, especially if the
>> strategy tends to drive the cube up, up and away.
>
> If you are referring to my 100 games, I can't very well play 1000
> games in one night and that's exactly why we are making bots
> to play "long enough" sessions.

Sure. But for precisely this reason you should be cautious with claims
that your "unorthodox" doubling strategies are superior.

> Sounds like the results went against your expectations?

No, but they indicate that another approach might make more sense: If a
Petersburg Paradoxon occurs, it no surprise that strategies jacking up
the cube can not be proven any more to be worse (or better). So in that
case you might be claiming superior cube skill but ignoring the pink
elephant in the room (by the way, I liked your Commodore story).

Axel

MK

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Nov 11, 2021, 5:28:21 PM11/11/21

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On November 6, 2021 at 3:23:08 AM UTC-6, Axel Reichert wrote:

> To my knowledge, GNU Backgammon has no opening book

> .....

> According to XG's opening book

Ah, okay, I got them confused.

> the replier has an advantage only after 41S

How about we go by the Gnubg rollouts from this link:

https://bkgm.com/openings/rollouts.html

> (with or without Jacoby).

Jacoby never interested me beyond wondering what do bots
play differently other than using different doubling windows?
I really don't even care to know anything about it.

> My understanding is that you double after all 6x splits and
> maybe for other opening rolls, I don't know.

Yes 62, 63, 63 because they give me 2 direct 1 indirect shots,
(ignoring my 1 point), and I like how the games develop then.

Lately I added 41 if slotted but without raccoon for now since
it gives me only 1 direct 1 indirect shots.

And of course, I play certains ways on the 2nd and 3rd rolls
also, as well as the rest of the games.

I just looked at some TD-Gammon articles and noticed that
63 was one of the rolls v1 had "misplayed" (24/21 13/7) and
found it quite interesting. Perhaps it did that not because the
24/21 13/7 was a good play but 24/18 13/10 ended up being
worse based on how it played against itself.

In other word, the "styles" or "strategies" of players also matter
in determining whether certain rolls and moves are conducive
to them... Maybe that's why I still love this game.

> My mutant doubles after 1x splits and 43Z. In all these cases
> the winning chances of the replier should be between and
> 49.48 % and 50.16 %.

I don't understand the S's, Z's, etc. after the rolls but I can agree
with going by the winning percentages because it's easier to
program those into the bot. I don't understand why the 49.48%
to 50.16% range but according to the summary table at the link
above 41, 43, 64, 32 result in < 50% winning chances.

> So in my opinion it does not matter in which of these cases
> you raise the stakes, it is too early anyway, because you are
> foregoing the possibility to double your opponent out.

Well the "too early" part is exactly my point for a different reason,
which is that you can't calculate cubeful equities until towards the
end of the game. I thought the common teaching of "cube skill"
was that it was better used to maximize your winning and not
necessarity to double your opponent out.

> Which is also "arbitrary". If you start to come up with ideas about
> gammonish positions and "play left in the game" I should start to
> get worried, because then you would reinvent the (mathematical)
> concepts of equity and volatility. (-;

I don't know what to say other than you may as well start getting
worried, not because I want to reinvent any such "mathematical
concepts", but to debunk them alltogether.

>> But aren't Raccoons, Rats, Bats, Cats, etc. all part of what you all
>> call "cube skill"...?

> The interesting question I try to research here is not any "maniac"
> cube strategy, but whether a Petersburg Paradoxon occurs in
> backgammon with unlimited cube

I didn't invent doubling, nor beavering, raccooning, etc. not do I mind
your calling my or any other strategy "maniac" as long at it results in
winning more.

And I have no idea what Petersburg Paradox has anything to do with
the subject where more than just the probabilites of luck, i.e. "skill" is
involved, especially with some people arguing more for skill than luck
in backgammon.

>... If yes, I think we will have an interesting (but mostly philosophical)
> discussion about skill.

The only practical thing you can do about the "cube skill fantasy" is
just counting the potatoes and living with the result. I think it's you
guys who are making more out of it than what it is.

>> Should be at least 51%

> Why not 50.000001 %?

Sure, as long as you have enough potatoes to count... My argument
had initially started based on the fact that winning the opening roll
gives an advantage (according to the link above +.0393 on average).

Checker+cube skills being equal, the player who will win more opening
rolls will win more. You can't argue against this simple statistics fact.

What I'm interested in is to find out if luck+checker skills are equal,
how much does the so called "cube skill" matter after 4 billion games?

>> Why are you even wasting time with these??

> See above. My interest is the Peterburg Paradoxon, not the cube
> strategy.

In that case we're just wasting time. That is, I am anyway... :(

MK

Axel Reichert

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Nov 12, 2021, 2:22:44 PM11/12/21

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MK <mu...@compuplus.net> writes:

> On November 6, 2021 at 3:23:08 AM UTC-6, Axel Reichert wrote:
>
> How about we go by the Gnubg rollouts

As I wrote, this will not matter. Your thinking seems to be that as soon
as you are even a tiny favourite, you should raise the stakes. This is
correct if you cannot get redoubled. But if the opponent has the
opportunity to redouble, he could perhaps double you out. So by giving
the cube (giving it away!) you forgo your chances of doubling out the
opponent and thus make it harder for you. You have move the goal posts,
and this is a huge cost for you, hence doubling early with only a tiny
advantage is very wrong. So it does not matter whether you have 49.89 %
according to bot A or 50.03 % according to bot B, since this difference
will be dwarfed by the difference between having access to the cube or
not.

> I don't understand the S's, Z's, etc. after the rolls

https://bkgm.com/articles/Keith/nactation.html

> I don't understand why the 49.48% to 50.16% range

These are the winning chances after said rolls according to

http://www.extremegammon.com/OB/Opening_in_unlimited_game.html

> I thought the common teaching of "cube skill" was that it was better
> used to maximize your winning and not necessarity to double your
> opponent out.

Precisely. And because you give away the powerful weapon of the cube you
should not double to early, even if you are a favourite. Please read

https://bkgm.com/articles/Kleinman/FootballFields/index.html

> not because I want to reinvent any such "mathematical concepts", but
> to debunk them alltogether.

I know. But it won't be easy. (-:

> not do I mind your calling my or any other strategy "maniac" as long
> at it results in winning more.
>
> And I have no idea what Petersburg Paradox has anything to do with the
> subject

See below.

> winning the opening roll gives an advantage (according to the link
> above +.0393 on average).

Sure, but this is not enough to give the weapon away.

> Checker+cube skills being equal, the player who will win more opening
> rolls will win more.

Sure. But in the long run no player will win the opening roll more often
than the other.

> What I'm interested in is to find out if luck+checker skills are equal,
> how much does the so called "cube skill" matter after 4 billion games?

This can be answered as long as you can put numbers on the value of a
position. If you run into a Petersburg Paradox you cannot do this any
more, so at that point discussions about the pros and cons of particular
cube strategies become meaningless, because there are no numbers to
compare. Now if your cube strategy turns backgammon into a Petersburg
Paradox than you can neither claim that your cubing is better than the
bot's nor could someone else claim that it is worse than the bot's. It
cannot be proven any more.

If, on the other hand your cube strategy does not turn backgammon into a
Petersburg Paradox (e.g., because it is too timid for this, or because
it is prevented by rules, be it match play, a cap on the cube in money
sessions, forbidding beavers, ...), then there exists a number for the
value of a position, and so immediately one can debunk one strategy or
the other, even if it takes lots of games.

I think without beavers (as shown in one of my previous posts in this
thread) the case against the "mutant" strategy was already settled after
5000 games. With one beaver or, even more so, with raccoons allowed, it
was not yet settled, because due to the high cubes the scores were
wildly fluctuating and I was expecting an awfully high number of games
needed to settle the question. Hence I decided to first run a "stability
check" using Markov chain reasoning fed with the data accumulated from
10000 games. By now I am quite sure that with raccoons forbidden
backgammon does not turn into a Peterburg Paradox, whereas with raccoons
allowed it probably does.

This would mean that the mutant strategy can be debunked in the
beavers-only case by spending more CPU time. And it would also mean that
with raccoons it does not make sense to spend more CPU time, because it
will just produce random noise. If so, there is an easy explanation for
your success against the bot's in a particular session of only 100 games
(not your mistake, of course, I do understand the reasons).

Best regards

Axel

Frank Berger

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Nov 12, 2021, 6:14:57 PM11/12/21

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I don't expect to convince anyone that the cube has a value, but I found this article https://plus.maths.org/content/os/issue15/features/doubling/index very convincing. As Janowsji writes in a comment the model is to simplistic as it treats BG as a continous model, but I think it illustrates the facts excellently. As soon as you include non steady stuff it get's complicated.

Axel Reichert

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Nov 13, 2021, 5:02:18 AM11/13/21

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Frank Berger <bgbl...@googlemail.com> writes:

> I don't expect to convince anyone that the cube has a value

(-:

> https://plus.maths.org/content/os/issue15/features/doubling/index
>
> very convincing

Yes, linked on bkgm.com, where I have read it long ago. Probably to
probabilistic for some target audience ...

This is why I like the football/tug of war analogy: Somewhere above the
60 mark you feel strong enough to shift your own target from 80 to 100,
because then you still have 40 to go (from 60 to 100), whereas the taker
has to tug you from 60 to 20 (live cube, no (back)gammons). With a dead
cube you shift your own target from 75 to 100, so you should ensure that
you have less to go than the taker (from x to 0, not from x to 25,
because the cube is dead), which is the case at x=50, obviously.

All this is certainly more or less trivial for you, but as you can see
from this thread I still have not given up explaining. Even elementary
cube theory is not intuitive for beginners (as can be seen when trying
to teach them the 25 % take point for the dead cube).

So one could characterize my mutant's doubling "strategy" as treating
himself the cube as dead but hoping that the opponent treats it as very
much alive. (-;

Greeting from the Isar to the Rhine!

Axel

MK

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Nov 13, 2021, 8:14:55 PM11/13/21

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On November 12, 2021 at 12:22:44 PM UTC-7, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:

>> How about we go by the Gnubg rollouts

> As I wrote, this will not matter. Your thinking seems to be
> that as soon as you are even a tiny favourite, you should
> raise the stakes. This is correct if you cannot get redoubled.

> .....

In your experiment, the mutant plays normally after the first
cube. In my proposal, the mutant never drops except when
it has no chance of winning. Thus, it tries to turn the games
into "cubeless" games as much as it can, by forcing then to
be played out to the last roll. I propose that this will counter
the value of cube ownership. This has been one of my many
arguments for 20 years.

> So it does not matter whether you have 49.89 % according to
> bot A or 50.03 % according to bot B

I didn't put much importance on this. I only proposed using
the numbers from that link since Gnubg has no opening book.

> since this difference will be dwarfed by the difference between
> having access to the cube or not.

That's where we differ. I'm not so sure about the value of cube
ownership and I believe it can be overcome or even surpassed
by the strategy I proposed above.

>> I don't understand the S's, Z's, etc. after the rolls

> https://bkgm.com/articles/Keith/nactation.html

I guessed so but wasn't sure because I had the impression that
it was more complicated. I never even took a casual look at it,
as part of my trying to not keep my bg brain uninfected and to
not mutate into one of you flock. This seems to be a harmless
clever shorthand but I probably will keep avoid using it.

>> I don't understand why the 49.48% to 50.16% range

> These are the winning chances after said rolls according to ...

Okay. One thing I find very interesting, intrigueing is tha XGR++
slots with opening 41, as was TD-G v1 doing. At the time it was
considered "misplayed" by the human experts, along with the
other "bad" opening roll 63. I find it curious that the mother of
all later bg bots, and the only one without human bias! would
misplay the two worst opening rolls. I wonder if time may prove
otherwise...?

>> I thought the common teaching of "cube skill" was that it was
>> better used to maximize your winning and not necessarity to
>> double your opponent out.

> Precisely. And because you give away the powerful weapon of
> the cube you should not double to early, even if you are a favourite.
> Please read https://bkgm.com/articles/Kleinman/FootballFields

This and its variations, as well as other analogies have been used
many times here in the past. To me, resorting to such analogies
only shows a person's inability make his mathematical argument
strictly related to bg alone.

I myself sometimes use analogies but not to sustitute facts, such
as likening your guys' elaborate yet inapplicable equity, skill, etc.
calculations to pre-Copernican astronomy when they had refined
their formulas to calculate and predict some planets' retrograde
movements exactly, even though planets never moved backwards.

So now, let's assume Jeffrey Epstein is playing against Mocky,
for stakes high enough for Mocky but peanuts for Jeffy. I and
John Wayne are advising him, looking over his shoulder. When
Mocky rolls and opening 63 and splits, I urge him to immediately
double. He does. Mocky beavers. Jeffy raccoons. Mocky now has
"the powerful weapon of the cube" ownership.

Later, Mocky get a chance to redouble. Oops. But Duke says:
"Damn to torpedos! Full speed ahead!". After all, losing a few
million bucks would only be a mosquito byte for Jeffy... So, in
short, it's all relative and unverified, unproven.

>> not because I want to reinvent any such "mathematical concepts",
>> but to debunk them alltogether.

> I know. But it won't be easy. (-:

Yes, but I stuck to my guns (and my puns) as a Lone Ranger for 20
years. Lately I feel like I'm finally getting some traction. If you can
hang in there, in the end you may get some credit as Tonto. :)

>> And I have no idea what Petersburg Paradox has anything to do with
>> the subject

> See below.

>> winning the opening roll gives an advantage (according to the link
>> above +.0393 on average).

> Sure, but this is not enough to give the weapon away.

How do you know? Have you tested and verified how much is the
weapon worth?

>> What I'm interested in is to find out if luck+checker skills are equal,
>> how much does the so called "cube skill" matter after 4 billion games?

> This can be answered as long as you can put numbers on the value
> of a position. If you run into a Petersburg Paradox you cannot do this
> any more, so at that point discussions about the pros and cons of
> particular cube strategies become meaningless, because there are no
> numbers to compare. Now if your cube strategy turns backgammon
> into a Petersburg Paradox than you can neither claim that your cubing
> is better than the bot's nor could someone else claim that it is worse
> than the bot's. It cannot be proven any more.

My argument goes back to the stage before the "numbers" are
calculated. I propose that even the cubeless equity calculations
after TD-G v.1 are human biased and inaccurate by an unknown
magnitute. Thus, cube double/take points, etc. calculated based
on those equities are also inaccurate, in addition to being plain
wrong for other reasons and to being partially inapplicable to bg.

Have you read my discussions with Chow about HypestGammon?

It's a variant that I created to isolate the cube skill, in oder to test,
quantify and define it. Since it's played with only one die, there is
zero checker skill involved. It's pure cube skill game.

Chow claimed he could calculate the equities for all possible
positions and shared his findings. Since there are only a small
number possible positions, even with desktop CPU power, we
could create an alpha-bot that would be trained through "cubeful
self-play" and then we could compare the calculated equities to
the statistical equities, in order to see if the formulas used in the
calculations were accurate. For reasons/excuses, questionable
to me, he never finished the experiment.

If the numbers matches, it wouldn't necessarily prove anything
about "real bg" but if they didn't match, it would mean that further,
more complex expriments would be needed and be worth doing
to test the accuracy of Jackoffsky, etc. formulas used in "real bg".

My opening 63 proposal was an alternative, a substite test that
we could do with limited CPU power, just to get a glipse of what
may be a much bigger problem to investigate. And this is where
you came into the picture, offering to do an experiments but I
don't think it was what I had proposed to begin with and then
further deviated into something else completely.

> If, on the other hand your cube strategy does not turn backgammon
> into a Petersburg Paradox (e.g., because it is too timid for this, or
> because it is prevented by rules, be it match play, a cap on the cube
> in money sessions, forbidding beavers, ...), then there exists a number
> for the value of a position, and so immediately one can debunk one
> strategy or the other, even if it takes lots of games.

As I had said, I don't think the Petersburg Paradox applies here
and don't understand your arguments related to it. But I think I
understand part of what you are saying above.

Yes, after the opening 63 and split, for example, we need to
play lots of games, i.e. the proverbial 4 billion games, and just
count potatoes... But the mutant needs to play as I descibed
above, trying to minimize and ideally surpass any value of cube
ownership.

Deciding most games by checker play is very inportant to my
proposed experiment. In fact, I had dome my own personal
experiment by playing against the bot with me making the worst
first turn move in each game but to play normally after that, and
especially trying to recover from the huge blunder.

See the first experiment (with real-time Youtube videos and all) at:

https://www.montanaonline.net/backgammon/xg.php

I was surprised that I wasn't totally decimated by the bot as I had
expected to happen.

What that experiment proved to me that checker errors early in the
games are not as important as errors in late stages of games.

I propose that the same applies to cube errors also and that they
should be rated on a sliding scale of some sort. (This doesn't mean
that I acknowledge their accuracy but just saying that even as they
are wrong, they should be still wrongly calculated by differently.)

> This would mean that the mutant strategy can be debunked in the

> beavers-only case by spending more CPU time....

But your mutant is not mutated enough to match the experiment
I had proposed. Cubing early aggressively at 50%+, only to drop it
later according to Jackoffsky's take/drop points is meaningless...

> If so, there is an easy explanation for your success against the
> bot's in a particular session of only 100 games
> (not your mistake, of course, I do understand the reasons).

I don't know what you are insinating but this is experiment is apart
from my a few dozen short sessions of 100 games.

If there is any link at all, it may be that I raised the question and
proposed various experiments indeed because I myself had no
clear explanation (which I admitted openly in the past) for my
"success". And also, even if I had, the overall total number of
games I had played wasn't statistically sicnificant enough for
most of you. And since I couldn't play 4 billion games myself,
I proposed that we make the bots do it instead.

From my stand, I too do understand the reasons for your resistance
or reluctance, and I do give you credit for having tried this much,
(which is more than anyone else has done), and I hope that you will
keep doing more logical experiments. Because, you know, it's just a
matter of time...

MK

MK

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Nov 13, 2021, 9:11:40 PM11/13/21

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On November 13, 2021 at 3:02:18 AM UTC-7, Axel Reichert wrote:

> Frank Berger <bgbl...@googlemail.com> writes:

>> I don't expect to convince anyone that the cube has a value

> All this is certainly more or less trivial for you, but as you
> can see from this thread I still have not given up explaining.

The feeling of frustration is mutual. To me, you guys sound
like pre-Copernican astronomers showing me all kinds of
elaborate formulas and trying to impress me with how so
exactly you can predict retrograde movements of planets.

And I still have not given up explaining to you guys that your
calculations have no value applicable to reality because
planets don't travel backwards.

Interesting how you validate each other so eagerly. I guess
misery likes company...

> Even elementary cube theory is not intuitive for beginners
> (as can be seen when trying to teach them the 25 % take
> point for the dead cube).

Maybe it's just difficult to "teach" (convince of) something
that doesn't add up..?

Also, your "cube hypothesis" must have self-verified itself
into "cube theory" without the use of any empirical data,
test or experiment. Convincing a small number of mentally
ill gamblers that they will win more by doubling/taking at
certain calculated equities, and then "observing" that they
all try to play like that but only the ones most capable of
it (i.e. achieving low PR's) win more, is not enough to make
your cube hypothesis a cube theory.

> So one could characterize my mutant's doubling "strategy"
> as treating himself the cube as dead but hoping that the
> opponent treats it as very much alive. (-;

Well enough with the clarification that it is "your" mutant
alone and not mine nor anyone else's. So, yes, you should
be given full credit for "your mutant's" silly doubling strategy.

MK

Message has been deleted

Axel Reichert

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Nov 14, 2021, 6:32:49 AM11/14/21

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MK <mu...@compuplus.net> writes:

> In your experiment, the mutant plays normally after the first cube. In
> my proposal, the mutant never drops except when it has no chance of
> winning.

So you always double with winning chances > 50 % and always take with
winning chances > 0 %?

My mutant did essentially this (beavers forbidden) and got
thrashed. Your comment one week ago was "Meaningless test". Note that it
does not matter much whether I use "> 0 %" (your suggestion) or ">= 0 %"
(my test, = "always take"). I admit that beavers will make a difference
(if only ending up as Petersburg paradox, probably like your Epstein
thought experiment, in which the deeper pockets will trivially win).

> How do you know? Have you tested and verified how much is the weapon
> worth?

Would about this proposition?

https://www.bkgm.com/rgb/rgb.cgi?view+838

Axel

Nasti Chestikov

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Nov 14, 2021, 12:07:19 PM11/14/21

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What these inbred cocksuckers can't grasp is the chances of GnuDung rolling a 4-5 are 2-in-36......but the chances of rolling a 4-5 when it's *exactly* the roll it needs is.....?

MK

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Nov 14, 2021, 4:54:06 PM11/14/21

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On November 14, 2021 at 4:32:49 AM UTC-7, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:

>> In your experiment, the mutant plays normally after the first cube. In
>> my proposal, the mutant never drops except when it has no chance of
>> winning.

> So you always double with winning chances > 50 % and always take with
> winning chances > 0 %?

This is where things get complicated. The answer may depend on
what we are trying to achieve with our experiment.

The goal of my 63 experiment was not to provide a comprehensive
answer by proving an alternative cube theory but just to poke a hole
in what you call a "cube theory". So, we can try different ways to see
if one achieves this.

My proposals weren't revelations to me. I'm not attached to any of
them and I'm willing to emend. I'm trying to come up with something
from my own way of cube strategy which is progressive, "depending
on the amount of checker play left in the game".

The paper linked by Frank acknowledges and explains that a little
but preceeds as if not anyway. I realize "chance of winning" doesn't
seem to work too well for explaining myself. Maybe another word
like "hope" would work better? You double and take as long as you
still have "hope of winning". At the start of the game your hopes
are high (d/t point is low), towards the end of the game your hopes
are low (d/t point is high).

If I understand it correctly, this is similar to the difference between
live and dead cube points(??) but more precise. If your math phd's
can come up with a formula for this, more power to you. In the past,
I had suggested many variants of cubefull bg, anywhere from raising
the cube by an arbitrary number as in poker (i.e. not just doubling it
but raising it by 100 or 1,500 etc.) to rasing the cube by fractional
values like 3.5 or 10.62 or 0.29 etc.

In this latter one, if played with ultimately precise cube skill, the cube
("stakes") would be raised at each and every turn, by the exact equity,
by both sides, until the last roll. This would take all the gambling fun
out of doubling and cubefull backgammon.

Since we can't change the numbers 2, 4, 8... on the cube, we need to
adjust the d/t points instead according to "how much hope we still
have left" or "how much checker play is still left in the game". Since
I'm not a math phd tempted to nail everything with a math hammer,
I would gladly settle for being a simple potato counter and run an
experiment with 4 billion trials to see what works best.

>> How do you know? Have you tested and verified how much is the weapon
>> worth?

> Would about this proposition?
> https://www.bkgm.com/rgb/rgb.cgi?view+838

11 can't be an opening roll unless playing by the middle-eastern rules
but even ignoring that, the argument relies on calculations using the
results of previous calculations in a circular manner. If the previous
calculations are biased and inaccurate, then the ensuing ones will
necessarily be so as well.

The question is: how much are you willing to buy back the "weapon"
you previously sold? If I may use a non-numerical analogy here, this
is like not being able to buy your own house back for the same price
in a raising real estate market but onlyif/because you have no means
to control/effect the market. In the "cube market" of backgammon,
however, you can manipulate the market by checker play, which is
"the other weapon"...

MK

MK

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Nov 14, 2021, 5:01:27 PM11/14/21

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On November 14, 2021 at 10:07:19 AM UTC-7, Nasti Chestikov wrote:

> ... the chances of GnuDung rolling a 4-5 are 2-in-36......but the

> chances of rolling a 4-5 when it's *exactly* the roll it needs is.....?

Unrelated to your point, this reminds me of my own ancients
arguments that the luck calculations need to be progressive
through the stages of games and proportionate to positions
complexities.

MK

Axel Reichert

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Nov 15, 2021, 2:29:18 PM11/15/21

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MK <mu...@compuplus.net> writes:

> On November 14, 2021 at 4:32:49 AM UTC-7, Axel Reichert wrote:
>
> You double and take as long as you still have "hope of winning". At
> the start of the game your hopes are high (d/t point is low), towards
> the end of the game your hopes are low (d/t point is high).

Standard cube theory has the concepts of dead/life cube and volatility
for this.

> If I understand it correctly, this is similar to the difference
> between live and dead cube points(??) but more precise.

Less. Quantify hope. (-:

But your thinking reminds me on a player in our club who was eager to
take the most desperate positions if only the volatility was sky-high.

> Since we can't change the numbers 2, 4, 8... on the cube

This is what

https://bkgm.com/rgb/rgb.cgi?view+429

is about. Imagine a tripling cube (3, 9, 27, ...) and you end up with a
Petersburg paradox. Which is why I am so eager to find out whether
aggressive cube strategy, beavers, or raccoons have the same effect as
the tripling cube. And because of the high volatility this cannot be
done by just running long sessions with the bot (they would be too
long), but we need to have a surrogate model (Markov chains), which is
fed with the data from shorter sessions with the bot. The surrogate
model can then easily be run a billion times. This is what I am doing.

> "how much checker play is still left in the game". Since I'm not a
> math phd tempted to nail everything with a math hammer, I would gladly
> settle for being a simple potato counter

How would this potato counter look like? We need to quantify things, not
because we like our math hammer, but because otherwise we cannot test
hypotheses.

> In the "cube market" of backgammon, however, you can manipulate the
> market by checker play, which is "the other weapon"...

With both sides playing their checkers like the bot, this weapon is
cancelled with the argument from symmetry. So my experiment leaves just
the cube skill in the game, as desired.

Best regards

Axel

MK

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Nov 16, 2021, 1:58:24 PM11/16/21

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On November 15, 2021 at 12:29:18 PM UTC-7, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:

>> You double and take as long as you still have "hope of winning". At
>> the start of the game your hopes are high (d/t point is low), towards
>> the end of the game your hopes are low (d/t point is high).

> Standard cube theory has the concepts of dead/life cube and
> volatility for this.

Yes, but the range is not wide enough. Also, I maintain that there
is no such thing as a "standard cube theory".

>> If I understand it correctly, this is similar to the difference
>> between live and dead cube points(??) but more precise.

> Less. Quantify hope. (-:

I meant more precise than live/dead cube points. Did you mean
the same by saying less in the opposite direction?

I could try to use other words like "expectation", etc. but I don't
think I can quantify. Isn't that what we are trying to do, with me
arguing that you can't quantify cube skill..?

> But your thinking reminds me on a player in our club who was
> eager to take the most desperate positions if only the volatility
> was sky-high.

Why not? Especially against completely predictable opponents
like bots? Their predictability allows you to "steer" them through
tactical checker moves. People like Chow bring up repeatedly
that bots can be beaten by forcing them into backgames, etc.
It may be harder to do with unpredictabe human opponents but
arguably it can be done.

>> Since we can't change the numbers 2, 4, 8... on the cube

> This is what
> https://bkgm.com/rgb/rgb.cgi?view+429
> is about.

Were you curious enough to look up the RGB thread that it was
extracted from? I did. And wow! Hundreds of long and detailed
articles written by perhaps 40-50 different participants, many of
the apparently mathematicians. They are broken into 5 pages. I
spent a couple of hours last night and I could only read hald of
one page. I'll keep reading and will post on this subject again.

If I were you, I wouldn't just rely on posts hand-picked by bkgm.

> Imagine a tripling cube (3, 9, 27, ...) and you end up with a
> Petersburg paradox. Which is why I am so eager to find out
> whether aggressive cube strategy, beavers, or raccoons have
> the same effect as the tripling cube.

Regardless of my opinion on Petersburg paradox in backgammon,
what will the result of your experiment mean regarding what you
call "standard cube theory"? You need to state this ahead of time,
not make it fit retroactively.

> And because of the high volatility this cannot be done by just
> running long sessions with the bot (they would be too long),

I don't understand what high volatility has to do with it but still, I
think the hard way is the only way.

> but we need to have a surrogate model (Markov chains), which is
> fed with the data from shorter sessions with the bot. The surrogate
> model can then easily be run a billion times. This is what I am doing.

This to me is like saying that a high resolution poster would require
to much work and resources, thus you will take a snapshot and blow
it up to poster size. I say it will end up very grainy, blurry.

>> "how much checker play is still left in the game". Since I'm not a
>> math phd tempted to nail everything with a math hammer, I would
>> gladly settle for being a simple potato counter

> How would this potato counter look like? We need to quantify things,
> not because we like our math hammer, but because otherwise we
> cannot test hypotheses.

You just run long enough sessions and tally the potatoes. I'm not
trying to make less of math but just saying that complex math is
not always necessary and can even be counter productive. If you
can run 10,000 games overnight, we should be able to tackle this
without questionable substitutions.

BTW: did you mean that you can test hypotheses by quantifying
things with math?

>> In the "cube market" of backgammon, however, you can manipulate
>> the market by checker play, which is "the other weapon"...

> With both sides playing their checkers like the bot, this weapon is
> cancelled with the argument from symmetry. So my experiment
> leaves just the cube skill in the game, as desired.

Well enough as far as your specific experiment. But you can do
other experiments with bots playing the cube moves the same
and the mutant bot making maniac checker moves... :) We are
anly using bots in the experiment because we can't use humans.
We just need to wait until we can have an alpha-bot to see it
decimate all current bots, as well as humans (including even me:),
by making maniac cube and checkers plays.

MK

Message has been deleted

Axel Reichert

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Nov 17, 2021, 2:10:38 PM11/17/21

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MK <mu...@compuplus.net> writes:

> On November 15, 2021 at 12:29:18 PM UTC-7, Axel Reichert wrote:
>
> Were you curious enough to look up the RGB thread that it was
> extracted from? I did. And wow! Hundreds of long and detailed articles
> written by perhaps 40-50 different participants, many of the
> apparently mathematicians.

Thanks for the hint, I might take a look.

> what will the result of your experiment mean

If raccoons turn out to end up as Petersburg paradox, it would just be an
incentive for some clubs (mine, for example) to have them forbidden in
order to keep backgammon a mind sport, not a gambling amusement. Same
for beavers (if Petersburg kicks in there as well). If not, then maniac
cube strategies (contradicting standard cube theory) can be dismissed by
investing CPU time, be it "real" sessions or Markov chain runs.

By the way, in 10000 games with 1 beaver allowed, double > 0.5 and
take > 0.0 the mutant lost 62117 against gnubg's 84870. In one game the
cube reached 4096 (gnubg's limit), so I checked this game manually from
the session file. In fact gnubg held the cube after beavering the
mutant's redouble to 2048. The game turned around, so gnubg would have
had a redouble, take to 8192. The game turned such that the mutant was
over 50 percent again, so another redouble beavered by gnubg. Cube now
at 32768 and a win for gnubg. So the final result would have been rather
113542 versus 62117 for gnubg, which is quite a margin.

But before declaring victory over the mutant's strategy, I need to
ensure that the expectation settles, of which I was not yet sure after 5
billion games (Markov chain runs).

>> And because of the high volatility this cannot be done by just
>> running long sessions with the bot (they would be too long),
>
> I don't understand what high volatility has to do with it

The higher the volatility (the one of the whole process, not the
volatility of an individual position), the longer it takes until the law
of large numbers kicks in.

> just run long enough sessions

From my Markov chain runs it is quite certain that even several million
games are not enough. I will not spent half a year of CPU time if smart
surrogate methods yield a robust result much quicker. "There nothing
more practical than a good theory."

> did you mean that you can test hypotheses by quantifying things with
> math?

Sure. This is called simulation, my field of expertise for 25 years.

Axel

Timothy Chow

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Nov 17, 2021, 10:48:40 PM11/17/21

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On 11/17/2021 2:10 PM, Axel Reichert wrote:
> If raccoons turn out to end up as Petersburg paradox, it would just be an
> incentive for some clubs (mine, for example) to have them forbidden in
> order to keep backgammon a mind sport, not a gambling amusement. Same
> for beavers (if Petersburg kicks in there as well). If not, then maniac
> cube strategies (contradicting standard cube theory) can be dismissed by
> investing CPU time, be it "real" sessions or Markov chain runs.

I still find this line of reasoning peculiar. The St. Petersburg
is an arcane mathematical oddity; why would anyone care about it in
the context of a practical decision (like the rules for a club)?
The positions with undefined equity already demonstrate that the
paradox arises with ordinary backgammon, but apparently that does
not dissuade your club from allowing money games. So why would
showing that the paradox arises with raccoons dissuade your club from
allowing raccoons? I don't follow the logic.

Again, if the problem is that money games smell of gambling, then don't
play money games. What's wrong with that simple logic?

I can see that you might not want raccoons or even beavers if it
causes the cube to get so high *in practice* that it starts to
have negative effects on how people behave (maybe they lose more
money than they can afford). But that again has nothing to do with
the St. Petersburg paradox.

---
Tim Chow

MK

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Nov 18, 2021, 5:20:51 AM11/18/21

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On November 17, 2021 at 12:10:38 PM UTC-7, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:

>> Were you curious enough to look up the RGB thread that it was
>> extracted from? I did. And wow! Hundreds of long and detailed
>> articles written by perhaps 40-50 different participants, many
>> of the apparently mathematicians.

> Thanks for the hint, I might take a look.

It's not a hint. It's what anyone capable of independent thinking
should be doing. I believe that bkgm.com is maintained by Tom
Keith who is apparently another one of the dime a dozen math
authorities in gamble-gammon. I have nothing against him and
I benefoted from some of his articles but I have to assume that
he is biased as one of the butt sniffing mutts pack of mentally
ill gamblers. His web site only includes selected articles without
opposing arguments, even is they were taken from RGB which is
an open, unmoderated forum. "The real world".

I estimated the number of articles in that thread which is more
likely to be around 80-90 but from a few dozen participants for
sure. You should read them. Most, if not all, of the articles are
from mathematicians with some of them dissenting at times.
Like David Ullrich. Personally, I kind of liked quite a few of the
things that he wrote in RGB. You can't find a single reference
to him in bkgm.com or bgonline.org. You should ask why??

>> what will the result of your experiment mean

> If raccoons turn out to end up as Petersburg paradox, it would
> just be an incentive for some clubs (mine, for example) to have
> them forbidden in order to keep backgammon a mind sport, not
> a gambling amusement.

Can you reword that in terms of what it means about "cube skill"?

Would it support "cube skill" or not, regardless of whether you
prefer to call it a "theorie" or a "hypothesis"?

> Same for beavers (if Petersburg kicks in there as well). If not, then
> maniac cube strategies (contradicting standard cube theory) can
> be dismissed by investing CPU time, be it "real" sessions or Markov
> chain runs.

I had never seen the term "maniac" used in this context before and
thought it was your own coining until I just ran into it in an article
by Gary Wong, (whom I detest as one of the biggest assholes in
gamble-gammon world), who had also estimated that by 2048 (or
so??) we should have enough CPU power to solve calculate all
cubeless equities in 3 minutes...

> By the way, in 10000 games with 1 beaver allowed, double > 0.5
> and take > 0.0 the mutant lost 62117 against gnubg's 84870.

What's important here is how that compares to what would be
expected. Did you use your high math to calculate a prediction
before you started? Would you have expected that the mutant
would lose by ten fold, twenty fold, etc...? You haven't. And the
above numbers are incredibly good towards proving that the
so-called "cube skill" is bullshit. When you refine your doubling
and especially taking points from >0 to more logical/practical
one (such as deriving from my tests), you will see the the mutant
will decimate gnubg...

The test we are doing now is a preliminary one to see if there is
any sense of going further and your numbers above are 10 times
more than what anyine would have expected to say "yes".

> In one game the cube reached 4096 (gnubg's limit), so I checked
> this game manually from the session file. In fact gnubg held the

> cube after beavering the mutant's redouble to 2048.... blah blah

Was gnubg wrong to hold the cube? Then argue for it why. And
you may convince yourself against yourself...

> But before declaring victory over the mutant's strategy, I need to
> ensure that the expectation settles, of which I was not yet sure
> after 5 billion games (Markov chain runs).

Take your time, I've got the beer chilling... ;)

>> I don't understand what high volatility has to do with it

> The higher the volatility (the one of the whole process, not the
> volatility of an individual position), the longer it takes until the
> law of large numbers kicks in.

Okay, thanks for clarifying.

>> just run long enough sessions

> From my Markov chain runs it is quite certain that even several
> million games are not enough.

That's funny. You substitute Markov Chain for real trials and then
turn around to extrapolate how many real trials you woul have
needed based on your Markov Chain runs..? Too much jacking
off going on in the gamble-gammon world for my taste.. :(

> I will not spent half a year of CPU time if smart surrogate
> methods yield a robust result much quicker.

I suggested that we could distrubute the needed CPU time and
offered to contribute to it myself. What's you problem with that?!

> "There nothing more practical than a good theory."

Is that a quote from the bible?? Hallelujah! Amen!

>> did you mean that you can test hypotheses by quantifying
>> things with math?

> Sure. This is called simulation, my field of expertise for 25 years.

Okay, we'll just make a note of this and remeber it when time comes.

MK

Axel Reichert

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Nov 18, 2021, 1:49:12 PM11/18/21

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Timothy Chow <tchow...@yahoo.com> writes:

> The positions with undefined equity already demonstrate that the
> paradox arises with ordinary backgammon, but apparently that does not
> dissuade your club from allowing money games. So why would showing
> that the paradox arises with raccoons dissuade your club from allowing
> raccoons?

The "eternal redoubling" position is, as you state, an oddity, so that,
if it ever comes up during practical play, it would be more of a reason
to celebrate this rare event than frown upon the Petersburg paradox.
(-:

But there is in my opinion a more fundamental thing: It should be hardly
possible to deliberately strive for this kind of positions by employing
particular checker play "techniques". Whereas my "maniac" cube strategy
is comparatively simple to implement and would be annoying throughout
the evening.

So the one has the taste of lucky occurrence, while the other has the
smell of abuse (from the subjective view of someone who wants to ensure
the "mind sport" aspect).

Best regards

Axel

Axel Reichert

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Nov 18, 2021, 3:51:16 PM11/18/21

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MK <mu...@compuplus.net> writes:

> On November 17, 2021 at 12:10:38 PM UTC-7, Axel Reichert wrote:
>

>> If raccoons turn out to end up as Petersburg paradox [...]

>
> Can you reword that in terms of what it means about "cube skill"?

I have tried to explain this before. Without existing expected value I
do not think there is a way to quantify the difference between cube
strategies (Big O notation, anyone?). As long as you have an expected
value, it is possible to dismiss particular cube strategies as
inferior.

>> By the way, in 10000 games with 1 beaver allowed, double > 0.5 and
>> take > 0.0 the mutant lost 62117 against gnubg's 84870.
>
> What's important here is how that compares to what would be
> expected. Did you use your high math to calculate a prediction before
> you started? Would you have expected that the mutant would lose by ten
> fold, twenty fold, etc...? You haven't.

As in my other trials the Null hypothesis was that the mutant cubes as
good as gnubg. Assuming a normal distribution of the cube value (which
is not quite correct, since it should be closer to a geometrical
distribution) the lead should be between -13296 and +13296 with 95 %
probability. However, gnubg's lead was nearly twice as high. But see
below.

> And the above numbers are incredibly good towards proving that the
> so-called "cube skill" is bullshit.

Why so? I am eagerly waiting for your irrefutable reasons.

> When you refine your doubling and especially taking points from >0 to
> more logical/practical one (such as deriving from my tests), you will
> see the the mutant will decimate gnubg...

Show the results once you have them. I am eagerly waiting for your
statistically significant experiments.

>> gnubg held the cube after beavering the mutant's redouble to 2048
>

> Was gnubg wrong to hold the cube?

With "held" I meant "was possessing". It could not redouble any
more. Like I wrote, without the hard-wired limit of 4096, it would have
redoubled a couple of rolls later according to standard cube theory.

>> But before declaring victory over the mutant's strategy, I need to
>> ensure that the expectation settles, of which I was not yet sure
>> after 5 billion games (Markov chain runs).
>
> Take your time, I've got the beer chilling... ;)

Good, me too. (-:

So in the meantime I have arrived at 100e9 games, see the results from
beyond 10000 games here:

| Games | Avg. game value | Avg. advantage for gnubg (PPG) |
|-----------------+-----------------+--------------------------------|
| 20.000 | 28 | 4.22185 |
| 50.000 | 28 | 5.79926 |
| 100.000 | 119 | 93.25437 |
| 200.000 | 34 | 1.63743 |
| 500.000 | 45 | -0.795082 |
| 1.000.000 | 44 | 2.359327 |
| 2.000.000 | 83 | -37.00945 |
| 5.000.000 | 41 | 5.281775 |
| 10.000.000 | 44 | 8.318696 |
| 20.000.000 | 73 | 35.509407 |
| 50.000.000 | 49 | 10.718045 |
| 100.000.000 | 79 | 36.462276 |
| 200.000.000 | 103 | -26.76603 |
| 500.000.000 | 57 | 6.1772475 |
| 1.000.000.000 | 71 | 21.412815 |
| 2.000.000.000 | 70 | 8.383391 |
| 5.000.000.000 | 63 | 14.327376 |
| 10.000.000.000 | 125 | 74.427704 |
| 20.000.000.000 | 73 | 5.6380033 |
| 50.000.000.000 | 71 | 3.6864755 |
| 100.000.000.000 | 79 | 23.954538 |

As you can see, GNU Backgammon was leading most of the time, on
average with an advantage amounting to roughly 1/5th of the average
game value (this is a lot!).

However, if you plot the second column over the (logarithmic) number
of games, then to me it seems that the average value DOES NOT SETTLE,
BUT KEEPS ON INCREASING. My interpretation of all this is that you
have successfully entered Petersburg country, so your cube strategy
cannot be dismissed easily (big O notation, anyone? Perhaps this could
be applied in this context), because no expected value
exists. However, with all due respect, you will almost certainly not
be able to back your strategy with cash in a real money session, since
the highest cube I encountered was roughly 500e9. Even at a modest
stake of 1 Euro/point this is a fairly ambitious amount of money.

It goes without saying that (if I am right about the beavers) allowing
raccoons will end up in Petersburg country as well.

Without beavers, it is a different story: From my experiments with the
more conservatively taking mutant ("take like gnubg") I am very sure
that for the current mutant's cube strategy there exists an expected
value, and the strategy will fail miserably. But I will test this and
report on it. I do not believe that even further restrictive measures
(such as capping the cube at 64, "beginner-friendly") are called for
to get out of Petersburg country.

>> "There nothing more practical than a good theory."
>
> Is that a quote from the bible?

No. https://en.wikiquote.org/wiki/Kurt_Lewin

Axel

Timothy Chow

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Nov 18, 2021, 9:46:07 PM11/18/21

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On 11/18/2021 5:20 AM, MK wrote:
> Like David Ullrich. Personally, I kind of liked quite a few of the
> things that he wrote in RGB. You can't find a single reference
> to him in bkgm.com or bgonline.org. You should ask why??

He is mentioned on bkgm.com a few times.

https://www.bkgm.com/rgb/rgb.cgi?view+983
https://www.bkgm.com/rgb/rgb.cgi?view+1254
https://bkgm.com/rgb/rgb.cgi?view+1310
https://bkgm.com/articles/Zare/UndefinedEquity/index.html

---
Tim Chow

MK

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Nov 19, 2021, 4:05:37 AM11/19/21

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On November 18, 2021 at 1:51:16 PM UTC-7, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:

>> Can you reword that in terms of what it means about "cube skill"?

> I have tried to explain this before. Without existing expected value I
> do not think there is a way to quantify the difference between cube
> strategies (Big O notation, anyone?). As long as you have an expected
> value, it is possible to dismiss particular cube strategies as inferior.

I can't speak your language and was asking to reworded in simpler
English but maybe it's not easy for you to do either.

Let me try another way. If you ran sessions with different "cube
strategies", (BTW: I like that you are using these words), against
the "standard cube theory" that gnubg uses, can you compare if
some do better than others? If they all give similar results, maybe
this will be the best argument that show that the "standard cube
theory" us flawed (or in my exaggerated words, that the "cube
skill" is mostly bullshit).

>>> By the way, in 10000 games with 1 beaver allowed, double > 0.5
>>> and take > 0.0 the mutant lost 62117 against gnubg's 84870.

>> What's important here is how that compares to what would be
>> expected. Did you use your high math to calculate a prediction
>> before you started? Would you have expected that the mutant
>> would lose by ten fold, twenty fold, etc...? You haven't.

> As in my other trials the Null hypothesis was that the mutant cubes
> as good as gnubg. Assuming a normal distribution of the cube value
> (which is not quite correct, since it should be closer to a geometrical
> distribution) the lead should be between -13296 and +13296 with 95 %
> probability. However, gnubg's lead was nearly twice as high. But see
> below.

All this sounds like unnecessarily overcomplicating things. If more
cube skill wins more, then a defying, almost nonexistent cube skill
should lose incomparably more. Didn't you have a "gut feeling", an
rough guess about how the results would come out? I did. :)

>> And the above numbers are incredibly good towards proving that
>> the so-called "cube skill" is bullshit.

> Why so? I am eagerly waiting for your irrefutable reasons.

Before running the experiment, if you took bets from gamblers here
on what would be gnubg's lead, I would've bet that they would've bet
that it would be much more than merely twice.

I don't think either that you are comfortable enough with obtaining
the same results if you ran another 10,000 games. Gnubg may win
by more but may also lose this time.

>> When you refine your doubling and especially taking points from
>> >0 to more logical/practical one (such as deriving from my tests),
>> you will see the the mutant will decimate gnubg...

Okay, two things here. Let me first correct myself that of all people,
I shouldn't have said things like "more logical/practical take points
than > 0" because I am the one arguing against a "certain cube skill"
to begin with. :( I don't know what was I thinking. Unless proven, I
don't think there is any way to claim that a take point of > 10% will
win more than a take point of > 0% or 20% or 50%. It's much more
complicated than that for me anyway.

> Show the results once you have them. I am eagerly waiting for your
> statistically significant experiments.

I already said no human can live long enough to satisfy you guy's
requirement for statistically significant. All I can offer is that I'm
fairly confident that, if given the opportunity to play an observed
100 games long demonstration session, I can substancially
duplicate the results of my previous several dozens.

>>> gnubg held the cube after beavering the mutant's redouble to 2048

>> Was gnubg wrong to hold the cube?

> With "held" I meant "was possessing". It could not redouble any
> more. Like I wrote, without the hard-wired limit of 4096,

Okay, I understand now.

> it would have redoubled a couple of rolls later according to standard
> cube theory.

But even then, there is no way to know what else would ensue, no?
How can you tell that mutant wouldn't take and win or redouble?
Especially in a long enough run, if the situation arose multiple times
with different results for either side, maybe mutant would win more?

> ... My interpretation of all this is that you have successfully entered

> Petersburg country, so your cube strategy cannot be dismissed
> easily (big O notation, anyone? Perhaps this could be applied in this
> context), because no expected value exists.

I suppose this is good news for my argument even if the experiment
wasn't exactly what I had proposed?

> However, with all due respect, you will almost certainly not be
> able to back your strategy with cash in a real money session,

Why not? Wouldn't anyone who believed in math do so??

> since the highest cube I encountered was roughly 500e9. Even
> at a modest stake of 1 Euro/point this is a fairly ambitious
> amount of money.

What if I have the money? Wouldn't you if you had the money?
I bet Zare would.

>>> "There nothing more practical than a good theory."

> > Is that a quote from the bible?

> No. https://en.wikiquote.org/wiki/Kurt_Lewin

There it says: "A business man once stated that there is nothing
so practical as a good theory". Regardless, it sound like a cheap
slogan, if not an oxymoron, to me.

MK

MK

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Nov 19, 2021, 4:12:34 AM11/19/21

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Thanks for looking up. I'm not very familiar with how to search
that site. The first three are unrelated to math, just about rules.

Zare gives him credit by saying "See the rec.games.backgammon
archive. I’d like to thank Chris Yep and David Ullrich for their work
and helpful discussions" but there is nothing from him at the link.

Anyway, I value and encourage dissent in any area. I don't think
there much in rgb anymore. When everyone agrees, you get dogma.

MK

MK

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Nov 19, 2021, 4:46:59 AM11/19/21

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On November 18, 2021 at 1:51:16 PM UTC-7, Axel Reichert wrote:

> Show the results once you have them. I am eagerly
> waiting for your statistically significant experiments.

Whether statistically significant or not, I just played a
100-games session today, trying to refine my strategy,
to play more carefully and consistently, accepting XG's
raccons this time around, and the results were:

Errors: checker = 9.19 / cube = 32.95 / overall = 13.26
Wins: expected = -302 / effective = +182 / difference = +484

The results for the previous three similar sessions were:

Errors: checker = 18.56 / cube = 25.21 / overall = 19.55
Wins: expected = -114 / effective = +7 / difference = +121

Errors: checker = 13.89 / cube = 31.09 / overall = 16.91
Wins: expected = -79 / effective = -56 / difference = +23

Errors: checker = 8.97 / cube = 29.39 / overall = 12.80
Wins: expected = -140 / effective = +56 / difference = +196

Now that I have become interested again, I'll try to play
more sessions just trying my "new and improved strategy".

MK

Axel Reichert

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Nov 22, 2021, 1:09:43 PM11/22/21

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MK <mu...@compuplus.net> writes:

> accepting XG's raccons

So this means you beavered XG's doubles. According (roughly) to what
criterion? Being above (estimated) 40 % winning chances? Or always?

Axel

Axel Reichert

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Nov 22, 2021, 1:38:57 PM11/22/21

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MK <mu...@compuplus.net> writes:

> On November 18, 2021 at 1:51:16 PM UTC-7, Axel Reichert wrote:
>
> I can't speak your language and was asking to reworded in simpler
> English but maybe it's not easy for you to do either.

Right. Some complex concepts require a certain language, which is why
mathematician have invented formula notation.

> If you ran sessions with different "cube strategies" [...], can you

> compare if some do better than others?

This is what I did. And by now I also have ideas what to do in the cases
where we end up with a Petersburg paradox (that is what the "big O"
notation referred to).

> I shouldn't have said things like "more logical/practical take points
> than > 0" because I am the one arguing against a "certain cube skill"

Well, a very good test to check whether strategic elements are involved
(and, by the way, also used in German court rulings about games of luck
versus games of skill) is to compare against a random player. So you
could roll a dice for every cube decision (1, 2: pass, 3, 4: take, 5, 6:
beaver and 1, 2, 3: double, 4, 5, 6: roll). For checker play, copy what
the bot would do. If you think this foolish, then apparently you believe
in cube skill. (-;

> I already said no human can live long enough to satisfy you guy's
> requirement for statistically significant.

Of course you can. But the length of the trial depends on the
volatility. Once we are "in Petersburg country", a more or less trivial
strategy is to get the cube to a high level, win one of these games and
protect the lead from then on by passing.

>> My interpretation of all this is that you have successfully entered
>> Petersburg country, so your cube strategy cannot be dismissed easily
>> (big O notation, anyone? Perhaps this could be applied in this
>> context), because no expected value exists.
>
> I suppose this is good news for my argument even if the experiment
> wasn't exactly what I had proposed?

Not too fast. Let us put it this way: In Petersburg country we need to
employ other techniques (big O notation, analytical approaches) to
dismiss foolish strategies. I might have a 66 in my dice cup ...

More to come!

Axel

Axel Reichert

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Nov 22, 2021, 5:21:45 PM11/22/21

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Axel Reichert <ma...@axel-reichert.de> writes:

> by now I also have ideas what to do in the cases where we end up with
> a Petersburg paradox

[...]

> In Petersburg country we need to employ other techniques (big O
> notation, analytical approaches) to dismiss foolish strategies.

[...]

> More to come!

So here we go. All of this is on a maths level for (almost) adults, so
the later high school years.

Probabilities for cube decisions with the mutant doubling above 50 and
taking above 0 percent winning chances, gathered from 10000 games played
against gnubg:

| Cube decision | Probability |
|---------------------------------+-------------|
| gnubg doubles, mutant takes | 0.1530 |
| gnubg doubles, mutant beavers | 0.0000 |
|---------------------------------+-------------|
| gnubg redoubles, mutant takes | 0.5447 |
| gnubg redoubles, mutant beavers | 0.0000 |
|---------------------------------+-------------|
| mutant doubles, gnubg takes | 0.1955 |
| mutant doubles, gnubg beavers | 0.6454 |
|---------------------------------+-------------|
| mutant redoubles, gnubg takes | 0.1754 |
| mutant redoubles, gnubg beavers | 0.1439 |

Over-doubling and over-redoubling cube

So with respect to "centered" cube action, gnubg wields a "doubling"
cube with an average value of 2, whereas the mutant wields an
"over-doubling" cube with an average value of

M = (0.1955*2 + 0.6454*2*2^b) / (0.1955 + 0.6454)
= 3.5350 (for b = 1, which stands for 1 beaver allowed)

Likewise, with respect to "owned" cube action, gnubg wields a
"redoubling" cube with an average value of 2, whereas the mutant wields
an "over-redoubling" cube with an average value of

M = (0.1754*2 + 0.1439*2*2^b) / (0.1754 + 0.1439)
= 2.9013 (for b = 1)

Case 1: Mutant starts cubing

Let's assume that the mutant cubes first (this is far more likely than
gnubg doubling first, around 84 % versus 15 %, see the above table).

Then let us assess the probability of an odd number of cubings:

p(n = 1) = (0.1955 + 0.6454) * (1 - 0.5447)

p(n = 3) = (0.1955 + 0.6454) * 0.5447 * (0.1754 + 0.1439) * (1 - 0.5447)

...

p(n = 2k-1) = (0.1955 + 0.6454) * (0.5447 * (0.1754 + 0.1439))^(k-1) * (1 - 0.5447)

= 0.3745 * 0.1739^(k-1)

(k = 1, 2, 3, ...)

How high (on average) is the cube for an odd number of cubings?

c(n = 1) = 3.5350

c(n = 3) = 3.5350 * 2 * 2.9013

c(n = 5) = 3.5350 * 2 * 2.9013 * 2 * 2.9013

...

c(n = 2k-1) = 3.5350 * (2 * 2.9013)^(k-1)

= 3.5350 * 5.8026^(k-1)

(k = 1, 2, 3, ...)

Now the same for an even number, with the mutant starting the cubing:

p(n = 2) = (0.1955 + 0.6454) * 0.5447 * (1 - 0.1754 - 0.1439)

p(n = 4) = (0.1955 + 0.6454) * 0.5447 * (0.1754 + 0.1439)
* 0.5447 * (1 - 0.1754 - 0.1439)

p(n = 6) = (0.1955 + 0.6454) * 0.5447 * (0.1754 + 0.1439)
* 0.5447 * (0.1754 + 0.1439)
* 0.5447 * (1 - 0.1754 - 0.1439)

...

p(n = 2k) = (0.1955 + 0.6454) * 0.5447 * ((0.1754 + 0.1439)*0.5447)^(k-1)
* (1 - 0.1754 - 0.1439)

= 0.3118 * 0.1739^(k-1)

(k = 1, 2, 3, ...)

How high (on average) is the cube for an even number of cubings?

c(n = 2) = 3.5350 * 2

c(n = 4) = 3.5350 * 2 * 2.9013 * 2

c(n = 6) = 3.5350 * 2 * 2.9013 * 2 * 2.9013 * 2

....

c(n = 2k) = 3.5350 * 2 * (2.9013 * 2)^(k-1)

= 7.0700 * 5.8026^(k-1)

(k = 1, 2, 3, ...)

With respect to the existence of the expected value, we can neglect
whether a single game, a gammon, or a backgammon was won. This will be
relevant only for assessing the gains of one cube strategy over the
other (postponed to a later article).

E(n = 2k-1) = p(n = 2k-1) * c(n = 2k-1)

= 0.3745 * 0.1739^(k-1) * 3.5350 * 5.8026^(k-1)

= 1.3239 * 1.0092^(k-1)

E(n = 2k) = p(n = 2k) * c(n = 2k)

= 0.3118 * 0.1739^(k-1) * 7.0700 * 5.8026^(k-1)

= 2.2044 * 1.0092^(k-1)

In sum we have:

E = E(n = 2k-1) + E(n = 2k)

= 1.3239 * 1.0092^(k-1) + 2.2044 * 1.0092^(k-1)

= 3.5283 * 1.0092^(k-1)

This is a geometrical series, and a divergent one, since

1.0092 >= 1

Hence the mutant cube strategy ends up as a Petersburg paradox, if
beavers are allowed. But it is close.

With 0 beavers allowed, we get 0.6957. No Petersburg paradox.
With 2 beavers allowed, we get 1.6363. Clearly Petersburg.

Case 2: GNU Backgammon starts cubing

For the much rarer case that gnubg cubes first (around 15 % versus 84 %
for the mutant cubing first) the mathematics is exactly the same. If you
do this boring exercise (left for the eager reader), it turns out that
the crucial terms (the ones with the exponent k-1) have exactly the same
factor in front of them. So the verdict is the same:

With 0 beavers allowed, we get 0.6957. No Petersburg paradox.
With 1 beaver allowed, we get 1.0092. Close, but still Petersburg.
With 2 beavers allowed, we get 1.6363. Clearly Petersburg.

So my next task will be to use these numbers to calculate the expected
value including gammons and backgammon. This will need some distinctions
between the various cases (cube ownership, even/odd number of cubings,
...), but most likely I will have time for these in the next couple of
days.

Stay tuned!

Axel

MK

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Nov 24, 2021, 1:50:32 AM11/24/21

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Oops, I misspoke. Sorry. I meant I raccooned XG's
beavers. I hardly ever beaver and of I do, it's often
irrational almost self-destructive gambling (which
also applies to some of my plain takes but usually
only if I'm way ahead). :(

MK

MK

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Nov 24, 2021, 3:01:58 AM11/24/21

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On November 22, 2021 at 11:38:57 AM UTC-7, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:

> ... If you think this foolish, then apparently you believe in cube skill. (-;

When I say "cube skill is bullshit", it's an exaggeration but only to
mean that "cube skill" is exaggerated. Whenever needed, I have
always clarified that I acknowledge a cube skill ranging from
barely defensible early in the game to undeniable late in the game.

I wonder if it's even misleading to call it "cube skill" when it's really
based on the skill of estimating one's winning chances at any given
point during the game and that what I said above applies to equity
calculations in early/late positions in the game.

Talking about foolish, while reading old articles in RGB, I saw that
I had even done an experiment over 10 years ago against Gnubg,
doubling at my first opportunity, without any other conditions, to
see if I could overcome the cost of giving away the cube. It looks
like I had some "colorful" discussions with old RGB regulars who
were much more numerous back then and who all liked me. :))

I had even suggested experiments to quantify the skill level of
random checker play while arguing that FIBS formula was too
arbitrary and inaccurate. It was never done and many thought
it was a ridiculous idead but when I insisted, Ullrich at least had
obliged to estimate that random play could achieve 1700 rating... :)

>> I already said no human can live long enough to satisfy you guy's
>> requirement for statistically significant.

> Of course you can. But the length of the trial depends on the volatility.
> Once we are "in Petersburg country", a more or less trivial strategy is
> to get the cube to a high level, win one of these games and protect the
> lead from then on by passing.

Are you suggesting that this is what I have been doing? In my last
session the highest I won was a gammon with cube at 32 on the
59th game. On the 70th game I lost with cube at 16. Without them
I would still be +134. Even though I was never under, during the first
10 games I had gained +21 vs the last 10 games only +7. So, you
may have something there... ;)

Bu this is not about me at all. I just shared my experience similar
to your experiment that's all. Just ignore all of my stuff if you wish.
Let's see what will statistically sugnificant bot vs bot experiments
reveal.

>> I suppose this is good news for my argument even if the experiment
>> wasn't exactly what I had proposed?

> Not too fast. Let us put it this way: In Petersburg country we need to
> employ other techniques (big O notation, analytical approaches) to
> dismiss foolish strategies. I might have a 66 in my dice cup ...

I'm still trying to understand how this "Petersburg country" thing is
relevant here? Are we eliminating all (cube) skill so that what is left
is just probabilities? And then going into a gambling frenzy from
there to "Petersburg country"??

MK

MK

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Nov 24, 2021, 3:50:53 AM11/24/21

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On November 22, 2021 at 3:21:45 PM UTC-7, Axel Reichert wrote:

> E(n = 2k-1) = p(n = 2k-1) * c(n = 2k-1)
> = 0.3745 * 0.1739^(k-1) * 3.5350 * 5.8026^(k-1)
> = 1.3239 * 1.0092^(k-1)
> E(n = 2k) = p(n = 2k) * c(n = 2k)
> = 0.3118 * 0.1739^(k-1) * 7.0700 * 5.8026^(k-1)
> = 2.2044 * 1.0092^(k-1)

> This is a geometrical series, and a divergent one, since
> 1.0092 >= 1

> With 0 beavers allowed, we get 0.6957. No Petersburg paradox.
> With 1 beaver allowed, we get 1.0092. Close, but still Petersburg.
> With 2 beavers allowed, we get 1.6363. Clearly Petersburg.

Looks like a lot of "smoke and maths" to me... :)

> So my next task will be to use these numbers to calculate the
> expected value including gammons and backgammon. This
> will need some distinctions between the various cases (cube
> ownership, even/odd number of cubings, ...), but most likely I
> will have time for these in the next couple of days.
> Stay tuned!

Do you or anyone else have any predictions as to what the results
will be?

Or are you trying to make happen certain results to prove one side
of the argument?

MK

MK

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Dec 2, 2021, 8:32:54 PM12/2/21

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On November 22, 2021 at 3:21:45 PM UTC-7, Axel Reichert wrote:

> So my next task will be to use these numbers to calculate the
> expected value including gammons and backgammon. This
> will need some distinctions between the various cases (cube
> ownership, even/odd number of cubings, ...), but most likely I
> will have time for these in the next couple of days.
> Stay tuned!

It has been 10 or "a couple of 5" days since... I hope you are
still intending to do or working on this. I don't know about the
others but I'm all ears, impatiently waiting to hear about it.

MK

Axel Reichert

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Dec 4, 2021, 4:27:49 PM12/4/21

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Yes, I am making progress and think my approach works.

Axel

MK

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Dec 4, 2021, 8:11:00 PM12/4/21

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On December 4, 2021 at 2:27:49 PM UTC-7, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:

>> It has been 10 or "a couple of 5" days since... I hope you are
>> still intending to do or working on this. I don't know about the
>> others but I'm all ears, impatiently waiting to hear about it.

> Yes, I am making progress and think my approach works.

Even as I asked for it twice, you never offered any predictions
nor committed to what will your results mean either way. Can
we at least observe your progress for the sake of openness?

To me, it looks like your are working on a secret receipe in a
dark room, tweaking and calculating, tweaking and calculating
and that you will let us know if/when you finally fabricate your
desired results...

MK

Axel Reichert

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Dec 6, 2021, 1:47:12 PM12/6/21

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MK <mu...@compuplus.net> writes:

> Even as I asked for it twice, you never offered any predictions nor
> committed to what will your results mean either way.

Mathematical proofs are different in nature from statistical hypothesis
testing. Pythagoras did not need to state a hypothesis about the sides
of a triangle before testing it, he proved it. For eternity.

Since my approach is analytical, there is no need for hypothesizing
either. If it works, fine, if not, the jury is still out.

Axel

MK

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Dec 10, 2021, 12:52:06 AM12/10/21

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On December 6, 2021 at 11:47:12 AM UTC-7, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:

>> Even as I asked for it twice, you never offered any predictions
>> nor committed to what will your results mean either way.

> Mathematical proofs are different in nature from statistical
> hypothesis testing. Pythagoras did not need to state a
> hypothesis about the sides of a triangle before testing it,

Your equating the two is laughable but let's see what you come
up with. Any attempt is better than dogmatic complacency.

> Since my approach is analytical, there is no need for
> hypothesizing either. If it works, fine, if not, the jury is still out.

Shouldn't I have a right to ask you to define the "works" before
you declare if it works or not? Also, who is the jury? You?

And, regardless of all that, what harm would it do to allow us
to observe your progress as you are fabricating your "proof"?

MK

Axel Reichert

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Apr 16, 2022, 5:36:20 AM4/16/22

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Axel Reichert <ma...@axel-reichert.de> writes:

> So my next task will be to use these numbers to calculate the expected
> value including gammons and backgammon. This will need some
> distinctions between the various cases (cube ownership, even/odd
> number of cubings, ...)

We need some further results from my older posting and some results
omitted there back then for brevity:

Case 1: Mutant starts an odd number n of cubings

Probability of n cubings
p(n = 2k-1) = 0.3745 * 0.1739^(k-1)

Cube value after n cubings
c(n = 2k-1) = 3.5350 * 5.8026^(k-1)

Case 2: Mutant starts an even number n of cubings

Probability of n cubings
p(n = 2k) = 0.3118 * 0.1739^(k-1)

Cube value after n cubings
c(n = 2k) = 7.0700 * 5.8026^(k-1)

Case 3: GNU Backgammon starts an odd number n of cubings

p(n = 2k-1) = 0.1041 * 0.1739^(k-1)
c(n = 2k-1) = 2 * 5.8026^(k-1)

Case 4: GNU Backgammon starts an even number n of cubings

p(n = 2k) = 0.0222 * 0.1739^(k-1)
c(n = 2k) = 5.8026 * 5.8026^(k-1)

We further need the game values (without cube value factored in) from
the session of 10000 games. The games without any cubing have been
omitted to simplify things: With one of the player so cube-happy, this
is an occurence in the per mille range. The following table lists the
fraction of games (single, gammon, backgammon) won/lost by GNU
Backgammon depending on final cube ownership:

| Cube owner | Win 1 | Win 2 | Win 3 | Lose 1 | Lose 2 | Lose 3 |
|----------------+--------+--------+--------+--------+--------+--------|
| GNU Backgammon | 0.0230 | 0.0210 | 0.0006 | 0.2981 | 0.1076 | 0.0050 |
| Mutant | 0.4427 | 0.1422 | 0.0057 | 0.0901 | 0.0000 | 0.0000 |

So the game values from GNU Backgammon's point of view are

0.0230*1 + 0.0210*2 + 0.0006*3 - 0.2981*1 - 0.1076*2 - 0.0050*3 =

-0.4615

if GNU Backgammon holds the cube at the end and

0.4427*1 + 0.1422*2 + 0.0057*3 - 0.0901*1 - 0.0000*2 - 0.0000*3 =

0.6541

if the mutant holds the cube at the end.

Note that if the mutant starts cubing and we have an odd number of
cubings, then GNU Backgammon owns the cube. For an even number of
cubings, the mutant owns the cube. Vice versa if GNU Backgamman starts
cubing.

With this in place, we can calculate the expected "points per game" for
GNU Backgammon:

Case 1 and 2: Mutant starts cubing

l_m = p(n = 2k-1) * c(n = 2k-1) * e(n = 2k-1) +
p(n = 2k) * c(n = 2k) * e(n = 2k)

= 0.3745 * 0.1739^(k-1) * 3.5350 * 5.8026^(k-1) * (-0.4615) +
0.3118 * 0.1739^(k-1) * 7.0700 * 5.8026^(k-1) * 0.6541

= 1.3239 * 1.0092^(k-1) * (-0.4615) + 2.2044 * 1.0092^(k-1) * 0.6541

= 0.8309 * 1.0092^(k-1)

Case 3 and 4: GNU Backgammon starts cubing

l_g = p(n = 2k-1) * c(n = 2k-1) * e(n = 2k-1) +
p(n = 2k) * c(n = 2k) * e(n = 2k)

= 0.1041 * 0.1739^(k-1) * 2 * 5.8026^(k-1) * 0.6541 +
0.0222 * 0.1739^(k-1) * 5.8026 * 5.8026^(k-1) * (-0.4615)

= -0.0594 * 1.0092^(k-1)

Since the mutant starts cubing in 0.8409 of the games and GNU Backgammon
starts cubing in 0.1530 of the games, we have

l = 0.8409 * l_m + 0.1530 * l_g

= 0.8409 * 0.8309 * 1.0092^(k-1) + 0.1530 * (-0.0594) * 1.0092^(k-1)

= 0.6896 * 1.0092^(k-1)

To summarize: Like for the expected value of a single game (as shown
previously), we have a Petersburg Paradox occuring for the lead of GNU
Backgammon in a session of such games, so the expected value of this
lead does not exist (base of the exponential term > 1 for the math
people, "oscillations too wild" for the non-math people).

But let's assume for the sake of argument that my session of 10000 games
with this cube strategy was rather an outlier, and that the "real" base
of the exponential term is slightly smaller than 1, even if one beaver
is allowed. Then the mutant strategy of doubling above 50 % GWC and
taking above 0 % GWC (checker play like GNU Backgammon) will result in
losing almost 0.7 (cube-normalized) points per game against GNU
Backgammon.

Case closed.

Axel

MK

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Apr 18, 2022, 5:38:34 AM4/18/22

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On April 16, 2022 at 3:36:20 AM UTC-6, Axel Reichert wrote:

> Axel Reichert <ma...@axel-reichert.de> writes:

> Case closed.

Just for the record that I'm not ignoring this post in this thread.

I have some things to say about it all but I'm standing aside for
the moment in order to not be rude by getting ahead of the math
PHD's here. My prediction is that we won't hear a peep from any
of them, for the reasons even you should be able to guess, but I
will patiently wait a to see... (Before slapping you some more ;)

MK

Nasti Chestikov

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Aug 24, 2022, 11:43:42 AM8/24/22

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On Monday, 18 April 2022 at 10:38:34 UTC+1, MK wrote:
> My prediction is that we won't hear a peep from any
> of them, for the reasons even you should be able to guess, but I
> will patiently wait a to see... (Before slapping you some more ;)
>
> MK

Yeah, cocksucking GnuDung fanboys soon went quiet.