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interests / rec.games.backgammon / Re: St. Patersburg Paradox in gamblegammon

SubjectAuthor
* St. Patersburg Paradox in gamblegammonMK
`* Re: St. Patersburg Paradox in gamblegammonTimothy Chow
 `* Re: St. Patersburg Paradox in gamblegammonpeps...@gmail.com
  `* Re: St. Patersburg Paradox in gamblegammonpeps...@gmail.com
   +- Re: St. Patersburg Paradox in gamblegammonpeps...@gmail.com
   `* Re: St. Patersburg Paradox in gamblegammonMK
    `* Re: St. Patersburg Paradox in gamblegammonpeps...@gmail.com
     +* Re: St. Patersburg Paradox in gamblegammonTimothy Chow
     |`* Re: St. Patersburg Paradox in gamblegammonMK
     | `* Re: St. Patersburg Paradox in gamblegammonpeps...@gmail.com
     |  `* Re: St. Patersburg Paradox in gamblegammonTimothy Chow
     |   `* Re: St. Patersburg Paradox in gamblegammonMK
     |    +* Re: St. Patersburg Paradox in gamblegammonTimothy Chow
     |    |`* Re: St. Patersburg Paradox in gamblegammonMK
     |    | +* Re: St. Patersburg Paradox in gamblegammonTimothy Chow
     |    | |`* Re: St. Patersburg Paradox in gamblegammonMK
     |    | | +* Re: St. Patersburg Paradox in gamblegammonpeps...@gmail.com
     |    | | |`- Re: St. Patersburg Paradox in gamblegammonMK
     |    | | `* Re: St. Patersburg Paradox in gamblegammonTimothy Chow
     |    | |  `* Re: St. Patersburg Paradox in gamblegammonMK
     |    | |   `* Re: St. Patersburg Paradox in gamblegammonTimothy Chow
     |    | |    +- Re: St. Patersburg Paradox in gamblegammonpeps...@gmail.com
     |    | |    `* Re: St. Patersburg Paradox in gamblegammonMK
     |    | |     `- Re: St. Patersburg Paradox in gamblegammonTimothy Chow
     |    | `* Re: St. Patersburg Paradox in gamblegammonAxel Reichert
     |    |  `- Re: St. Patersburg Paradox in gamblegammonMK
     |    `* Re: St. Patersburg Paradox in gamblegammonAxel Reichert
     |     `* Re: St. Patersburg Paradox in gamblegammonMK
     |      `* Re: St. Patersburg Paradox in gamblegammonAxel Reichert
     |       +- Re: St. Patersburg Paradox in gamblegammonpeps...@gmail.com
     |       `- Re: St. Patersburg Paradox in gamblegammonMK
     `- Re: St. Patersburg Paradox in gamblegammonMK

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St. Patersburg Paradox in gamblegammon

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Subject: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
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 by: MK - Tue, 10 May 2022 09:04 UTC

St. Patersburg Paradox (SPP from here on), was mentioned
in RGB a few times around 20 some years ago but only once
directly related to backgammon. Here's the link for it again:

https://groups.google.com/g/rec.games.backgammon/c/pnQ1s76ih5s/m/VAYl_JE3e74J

Even then, Zare says: "This position is a generalization of the
Petersburg Paradox." and goes on to explain the difference:
"In the classical paradox,....."

What makes that position similar to SPP is not because of the
equities, (which can be whatever), but because of the odds of
entering being the same for both players, which turns it into a
pure luck game like coin tossing. And when the loop ends, the
game doesn't necessarily end immediately but can end in just
another roll or two.

That same example position could come up in a cubeless game
also and the same loop would happen. Thus, it has nothing to
do with the cube either. In a cubeless game, when the loop ends
the game is most likely to be played out to the end.

Then, it was quite puzzling to see Axel refer to SPP during the
"mutant experiment" discussions, in almost every other post,
talking about cube going too high, especially when beavers and
raccoons are allowed.

Cube actions are based on equities. Cube can go very high in
games without a "mutant strategy" also. That's just part of the
gamblegammon.

In the experiment, cube skyrocketed because Gnubg beavered
at MWC's < 50% while at the same time the mutant doubled
and raccooned at MWC's >50%.

So, the problem was not caused by the acts of beavering and
raccooning but caused by the cube skill/strategy/theory being
fundamentally flawed.

I say fundamentally because just changing some jackoff-ski
formulas to not beaver at MWC's < 50% won't solve it.

If a certain "mutant-b" uses a strategy of doubling/racooning
at MWC's >55%, then Gnubg will need to not beaver at MWC's
<55% and so on, in order to prevent what Axel call SPP...

Not allowing raccoons, beavers or even just doubles at some
local games will change the fact that the so-claimed "cube
skill theory" is mostly elaborate bullshit.

The cube can go "too high" in match play also, depending on
what one considers too high relative to match length.

If the cube goes to 16 or 32 in a 25-point match, would Axel
also call that SPP?

I had the idea of defeating the cube skill almost as soon as
I was exposed to it. Here again the link to an article I posted
all the way back in 1999:

https://groups.google.com/g/rec.games.backgammon/c/o4qnefr7XeU/m/oVZ4DeF0rcsJ

I suggest you read it again and carefully and try to undertand
what others couldn't back then.

Because I didn't know how to propose a bet based on actual
vs. expected wins at that time, I came up with betting on my
beating Gnubg 25-0 in 25-point matches and I was soliciting
what odds would people offer me.

Since even losing 1 point would mean my losing the match,
some people proposed that I could start by spotting Gnubg
24 points thinking it would be the same thing. Obviously they
couln't understand my purpose of defeating the cube skill and
that playing with a dead cube wouldn't be the same thing.

BTW: I keep using the word "defeat" not to mean "win more
than 50% against cube skill" (as may be misunderstood by
some) but to mean "nullify", "destroy" the "cube skill theory".

Some people (including Zare in that same thread) actually
argued that unless I spotted Gnubg 24 points, the bot wouldn't
know what I was trying to do and wouldn't do well... Duh! :)

My frustration in RGB from the very beginnings has beeb that
people here don't have enough brains to understand what I'm
talking about. At times I thought it may have been a language
problem but obviously that's not it since they parrot back the
same rotes again and again, based on some trigger words,
regardless of the context. :(

So, what Axel call SPP is not due to beavers and critters but
simply due to the flawed "cube skill theory". Simply banning
critters or putting all sort of caps in money or match plays
is not going to solve the problem.

Just shovel out the bullshit and get done with once for all. :)

BTW: I asked Axel in another thread to reword his statements
about SPP without using the words "Petersburg Paradox" but
by using some alternative words so that we can understand
better but he refused. Well, at least I can understand how it
would be difficult to replace meaningless expressions with
meaningful ones... ;)

MK

Re: St. Patersburg Paradox in gamblegammon

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From: tchow12...@yahoo.com (Timothy Chow)
Newsgroups: rec.games.backgammon
Subject: Re: St. Patersburg Paradox in gamblegammon
Date: Tue, 10 May 2022 06:53:16 -0400
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 by: Timothy Chow - Tue, 10 May 2022 10:53 UTC

On 5/10/2022 5:04 AM, MK wrote:
> If the cube goes to 16 or 32 in a 25-point match, would Axel
> also call that SPP?

No. The essence of SPP is that the expected value does not exist.
When there is a maximum possible value (here, 25 points) then the
expected value always exists.

---
Tim Chow

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: pepste...@gmail.com (peps...@gmail.com)
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 by: peps...@gmail.com - Tue, 10 May 2022 16:30 UTC

On Tuesday, May 10, 2022 at 11:53:18 AM UTC+1, Tim Chow wrote:
> On 5/10/2022 5:04 AM, MK wrote:
> > If the cube goes to 16 or 32 in a 25-point match, would Axel
> > also call that SPP?
> No. The essence of SPP is that the expected value does not exist.
> When there is a maximum possible value (here, 25 points) then the
> expected value always exists.

I disagree with this. In the form of the SPP that I know, the expected value is infinite.
This is not really a paradox at all. The expected value is infinite: so what?

But the reason I disagree with you is that you can variantize the SPP to make the expected
value finite but still retain its essential features.

For example: Let the total dollar value of all the money on Earth = N.
Now, assume that your winnings are capped at N, regardless of how many coin tosses go in your direction.

Clearly, the expected value is now finite.
But we still have an SPP.
If a total solution to the SPP was "But the amount of money is finite", the "paradox" would have less content than it does.

Paul

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: pepste...@gmail.com (peps...@gmail.com)
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 by: peps...@gmail.com - Tue, 10 May 2022 16:46 UTC

On Tuesday, May 10, 2022 at 5:30:21 PM UTC+1, peps...@gmail.com wrote:
> On Tuesday, May 10, 2022 at 11:53:18 AM UTC+1, Tim Chow wrote:
> > On 5/10/2022 5:04 AM, MK wrote:
> > > If the cube goes to 16 or 32 in a 25-point match, would Axel
> > > also call that SPP?
> > No. The essence of SPP is that the expected value does not exist.
> > When there is a maximum possible value (here, 25 points) then the
> > expected value always exists.
> I disagree with this. In the form of the SPP that I know, the expected value is infinite.
> This is not really a paradox at all. The expected value is infinite: so what?
>
> But the reason I disagree with you is that you can variantize the SPP to make the expected
> value finite but still retain its essential features.
>
> For example: Let the total dollar value of all the money on Earth = N.
> Now, assume that your winnings are capped at N, regardless of how many coin tosses go in your direction.
>
> Clearly, the expected value is now finite.
> But we still have an SPP.
> If a total solution to the SPP was "But the amount of money is finite", the "paradox" would have less content than it does.
>
> Paul

In fact, I just googled it and got:
"The St. Petersburg paradox is a situation where a naive decision criterion which takes only the expected value into account predicts a course of action that
presumably no actual person would be willing to take. It is related to probability and decision theory in economics."

For that, the expected value only leads to be sufficiently large, not infinite. It's infinitudinizationness is hardly "the essence".

Paul

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: pepste...@gmail.com (peps...@gmail.com)
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 by: peps...@gmail.com - Tue, 10 May 2022 16:48 UTC

On Tuesday, May 10, 2022 at 5:46:19 PM UTC+1, peps...@gmail.com wrote:
> On Tuesday, May 10, 2022 at 5:30:21 PM UTC+1, peps...@gmail.com wrote:
> > On Tuesday, May 10, 2022 at 11:53:18 AM UTC+1, Tim Chow wrote:
> > > On 5/10/2022 5:04 AM, MK wrote:
> > > > If the cube goes to 16 or 32 in a 25-point match, would Axel
> > > > also call that SPP?
> > > No. The essence of SPP is that the expected value does not exist.
> > > When there is a maximum possible value (here, 25 points) then the
> > > expected value always exists.
> > I disagree with this. In the form of the SPP that I know, the expected value is infinite.
> > This is not really a paradox at all. The expected value is infinite: so what?
> >
> > But the reason I disagree with you is that you can variantize the SPP to make the expected
> > value finite but still retain its essential features.
> >
> > For example: Let the total dollar value of all the money on Earth = N.
> > Now, assume that your winnings are capped at N, regardless of how many coin tosses go in your direction.
> >
> > Clearly, the expected value is now finite.
> > But we still have an SPP.
> > If a total solution to the SPP was "But the amount of money is finite", the "paradox" would have less content than it does.
> >
> > Paul
> In fact, I just googled it and got:
> "The St. Petersburg paradox is a situation where a naive decision criterion which takes only the expected value into account predicts a course of action that
> presumably no actual person would be willing to take. It is related to probability and decision theory in economics."
>
> For that, the expected value only leads to be sufficiently large, not infinite. It's infinitudinizationness is hardly "the essence".
>
> Paul
"leads" -> "needs"
"It's" -> "its"

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
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 by: MK - Tue, 10 May 2022 21:15 UTC

On May 10, 2022 at 10:46:19 AM UTC-6, peps...@gmail.com wrote:
> On May 10, 2022 at 5:30:21 PM UTC+1, peps...@gmail.com wrote:
>> On May 10, 2022 at 11:53:18 AM UTC+1, Tim Chow wrote:
>>> On 5/10/2022 5:04 AM, MK wrote:

>>>> If the cube goes to 16 or 32 in a 25-point match,
>>>> would Axel also call that SPP?

>>> No. The essence of SPP is that the expected value does
>>> not exist. When there is a maximum possible value
>>> (here, 25 points) then the expected value always exists.

I wasn't asking if it would be SPP but if "Axel would call
it SPP". I'm trying to understand what he means by SPP
in the gamblegammon context.

Also, would it make a difference if it were a 10,000-point
or 4,000,000,000-point match? Again, I'm just trying to
understand how cube and SPP can be linked in any way.

>> But the reason I disagree with you is that you can variantize
>> the SPP to make the expected value finite but still retain its
>> essential features.

Are you saying that this is what Axel is doing? I wish he
himself would also try to explain what he is doing.

>> If a total solution to the SPP was "But the amount of money
>> is finite", the "paradox" would have less content than it does.

Just when I think I undestand, I don't. :(

> For that, the expected value only leads to be sufficiently large,
> not infinite. It's infinitudinizationness is hardly "the essence".

"Infinitudinizationness"... Thanks for making me smile. :)

MK

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: pepste...@gmail.com (peps...@gmail.com)
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 by: peps...@gmail.com - Wed, 11 May 2022 07:30 UTC

On Tuesday, May 10, 2022 at 10:15:32 PM UTC+1, MK wrote:
> On May 10, 2022 at 10:46:19 AM UTC-6, peps...@gmail.com wrote:
> > On May 10, 2022 at 5:30:21 PM UTC+1, peps...@gmail.com wrote:
> >> On May 10, 2022 at 11:53:18 AM UTC+1, Tim Chow wrote:
> >>> On 5/10/2022 5:04 AM, MK wrote:
>
> >>>> If the cube goes to 16 or 32 in a 25-point match,
> >>>> would Axel also call that SPP?
>
> >>> No. The essence of SPP is that the expected value does
> >>> not exist. When there is a maximum possible value
> >>> (here, 25 points) then the expected value always exists.
> I wasn't asking if it would be SPP but if "Axel would call
> it SPP". I'm trying to understand what he means by SPP
> in the gamblegammon context.
>
> Also, would it make a difference if it were a 10,000-point
> or 4,000,000,000-point match? Again, I'm just trying to
> understand how cube and SPP can be linked in any way.
> >> But the reason I disagree with you is that you can variantize
> >> the SPP to make the expected value finite but still retain its
> >> essential features.
> Are you saying that this is what Axel is doing? I wish he
> himself would also try to explain what he is doing.
> >> If a total solution to the SPP was "But the amount of money
> >> is finite", the "paradox" would have less content than it does.
> Just when I think I undestand, I don't. :(
> > For that, the expected value only leads to be sufficiently large,
> > not infinite. It's infinitudinizationness is hardly "the essence".
> "Infinitudinizationness"... Thanks for making me smile. :)
>
> MK

Yes, I enjoy lengthening already long words like "variantization" or ""Infinitudinizationness" -- glad you appreciate this.
I don't know what Axel is doing because I haven't got time / don't what to spend the time to read what he wrote.

The standard presentation of the SPP has two features:
1) An illustration of the fact that you sometimes don't want to max your expected value, but take the utility of your wealth into account.
2) Infinite expectation.

Of these two features, 1) is far more important than 2). Furthermore, 1) and 2) are independent of each other. Either can be illustrated
without illustrating the other.

Paul

Re: St. Patersburg Paradox in gamblegammon

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From: tchow12...@yahoo.com (Timothy Chow)
Newsgroups: rec.games.backgammon
Subject: Re: St. Patersburg Paradox in gamblegammon
Date: Wed, 11 May 2022 09:17:43 -0400
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 by: Timothy Chow - Wed, 11 May 2022 13:17 UTC

On 5/11/2022 3:30 AM, peps...@gmail.com wrote:
> I don't know what Axel is doing because I haven't got time / don't what to spend the time to read what he wrote.

The essence of what Axel means by SPP *in the current context* is
that the expected value does not exist.

---
Tim Chow

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
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 by: MK - Fri, 13 May 2022 00:40 UTC

On May 11, 2022 at 1:30:23 AM UTC-6, peps...@gmail.com wrote:

> I don't know what Axel is doing because I haven't got
> time / don't what to spend the time to read what he wrote.

This is so unfortunate. Axel made considerable efforts
for moths and posted some calculations that probably
only mathematicians could understand but was ignored
by all, including the chest-beating mathshitters except
Chow who only superficially participated in the threads.

In the end, I'm feeling sad/bad for him. I also feel sorry
that I melself haven't been nice enough to him. :( I tried
to be though and would have if he were to break away
from the "pack" as I repeatedly encouraged him to do.
He chose not and I kept treating him as one of the pack.
Instead, his ilk deserted him. Too bad. :( Maybe another
time. It's never too late.
> The standard presentation of the SPP has two features:
> .....

What I would like to know specifically is if and how SPP
applies to gamblegammon? With real examples, please.

MK

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
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 by: MK - Fri, 13 May 2022 00:58 UTC

On May 11, 2022 at 7:17:45 AM UTC-6, Tim Chow wrote:

> The essence of what Axel means by SPP *in the current
> context* is that the expected value does not exist.

https://usbgf.org/backgammon-glossary/

defines "equity" as "One’s value in the current game,
mathematically equivalent to the expected value".

So, then, it means (its "mathematically equivalent")
"equity does not exists".

Can someone explain what "equity does not exists"
means?

If it comes to a point that equity can't be calculated,
wouldn't any cube action dependent of equity stop
there also?

Any cube actions prior to that, correctly done based
on equities can't be SPP by any strectch of definition.

Unless cube actions continue with unknown equities,
there can't be SPP after that either. So, do you all say
that beavers, etc. continue even after the equities (or
"expected values") cease to exist? And if so, how?

MK

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: pepste...@gmail.com (peps...@gmail.com)
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 by: peps...@gmail.com - Fri, 13 May 2022 10:16 UTC

On Friday, May 13, 2022 at 1:58:29 AM UTC+1, MK wrote:
> On May 11, 2022 at 7:17:45 AM UTC-6, Tim Chow wrote:
>
> > The essence of what Axel means by SPP *in the current
> > context* is that the expected value does not exist.
> https://usbgf.org/backgammon-glossary/
>
> defines "equity" as "One’s value in the current game,
> mathematically equivalent to the expected value".
>
> So, then, it means (its "mathematically equivalent")
> "equity does not exists".
>
> Can someone explain what "equity does not exists"
> means?
>
> If it comes to a point that equity can't be calculated,
> wouldn't any cube action dependent of equity stop
> there also?
>
> Any cube actions prior to that, correctly done based
> on equities can't be SPP by any strectch of definition.
>
> Unless cube actions continue with unknown equities,
> there can't be SPP after that either. So, do you all say
> that beavers, etc. continue even after the equities (or
> "expected values") cease to exist? And if so, how?

In SPP, the expected value does exist and is infinite.
Tim probably meant "does not exist as a finite number."

There's no paradox about this. Some numbers just are infinite.
For example, the number of prime numbers is infinite.

Paul

Re: St. Patersburg Paradox in gamblegammon

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From: tchow12...@yahoo.com (Timothy Chow)
Newsgroups: rec.games.backgammon
Subject: Re: St. Patersburg Paradox in gamblegammon
Date: Fri, 13 May 2022 09:04:06 -0400
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 by: Timothy Chow - Fri, 13 May 2022 13:04 UTC

On 5/13/2022 6:16 AM, peps...@gmail.com wrote:
> In SPP, the expected value does exist and is infinite.
> Tim probably meant "does not exist as a finite number."

You're right that sometimes it can be convenient to say
that the expected value of a random variable is +infinity.
However, in the context of a game, I prefer to say that the
expected value, or equity, does not exist. We're in the realm
of applied math rather than pure math here. To say that the
expected value exists and is +infinity makes it sound like
assigning that value to the game allows you to draw meaningful
conclusions, but it doesn't, really. It's clearer just to say
that there is no meaningful or useful number that one can attach
to the game.

To answer Murat's question, "equity" is a number that we assign
to a position (where by a "position" I include the information
about the cube value and location, and whose turn it is) which
is supposed to predict how much money we would earn (or lose,
if the equity is negative) per session if we were to play out
that exact same position over and over again. There is a tacit
assumption that if we play out the position often enough, then
our wins and losses will "average out" and settle down to some
long-term average payoff per session.

To say that the equity does not exist means that there is something
about the position that undermines the tacit assumption---no matter
how often you play out the position, the payoff per session does not
"average out in the long run and settle down." In the case at hand,
what is happening is that if you play a large number of games, then
there's a good chance that in at least one game, there will be a huge
cube value that throws off the overall average. To smooth out the
effect of that huge cube value, you have to play even more games, but
when you do that, there's a chance of an even huger cube value showing
up at least once, which throws things off again. No matter how many
times you repeat this process, things never settle down.

This can happen only if the cube value is allowed to be arbitrarily
large. If it's capped at (say) 1024, then the average payoff per
session will settle down once the number of sessions you play is large
compared to 1024.

---
Tim Chow

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
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 by: MK - Sat, 14 May 2022 01:51 UTC

On May 13, 2022 at 7:04:11 AM UTC-6, Tim Chow wrote:

> However, in the context of a game, I prefer to say
> that the expected value, or equity, does not exist.

At the risk of shocking you all, I say thank you for
making the effort to post this. It shows that when
we really want, we can communicate. Why was it
so hard to talk about SPP "in the context at hand"?

> We're in the realm of applied math rather than pure
> math here.

This is the key. I too admire the eloquence in math
formulas but they may not always apply to real life
situations.

> To answer Murat's question, "equity" is a number
> that we assign to a position (where by a "position"
> I include the information about the cube value and
> location, and whose turn it is) which is supposed to
> predict how much money we would earn (or lose, if
> the equity is negative) per session if we were to play
> out that exact same position over and over again.

I'll agree with the clarification that equity can be and
(as far I understand), is currently determined cubeless
first, then cubeful equity is calculated using a formula.

> There is a tacit assumption that if we play out the
> position often enough, then our wins and losses will
> "average out" and settle down to some long-term
> average payoff per session.

Again, I'll agree if I understand you correctly that when
luck evens out after enough trials, wins and losses will
settle down to some values "assuming" that the players
are also of the same skill level and play consistently.

Then, only if we also "assume" that these "benchmark
values" result from best/perfect/optimum strategy, we
can assess other players' skills by comparing to these.

In order to not distract from the main subject here, I'll
just say that I won't dispute the accuracy of cubeless
equities beyond my argument that more than only one
best/perfect/optimum strategies are possible, which
may or may not lead to the same equity values.

What I'm disputing here is that cubeful equities except
for positions towards the end of games are bogus and
can be demostrated by experiment like Axel has done
(just to make me happy:))

> To say that the equity does not exist means that there
> is something about the position that undermines the
> tacit assumption---no matter how often you play out the
> position, the payoff per session does not "average out
> in the long run and settle down."

I want to mention here that this won't happen in cubeless
positions just to preserve the contrast.

> In the case at hand, what is happening is that if you play
> a large number of games, then there's a good chance
> that in at least one game, there will be a huge cube value
> that throws off the overall average. To smooth out the
> effect of that huge cube value, you have to play even
> more games, but when you do that, there's a chance of
> an even huger cube value showing up at least once,
> which throws things off again. No matter how many
> times you repeat this process, things never settle down.

This, I don't understand and pursue further. Many people
offered statistics in the past about the average number
of moves in a money game, from 21 to 27. Let's say 24
is good enough for the "real life gamblegammon context".

Since all other numbers mentioned are also averages, in
games of average 24 moves, the cube can't keep going
higher past a natural limit. This, if you play a large enough
number of games, all possible cube values should settle,
no??

If Axel had played 10,000,000 actual games instead of
only 10,000 and using Markov Chains to extrapolate to
5,000,000,000 games, I wonder if "things would settle"
better in an experiment closer to real life..?

> This can happen only if the cube value is allowed to be
> arbitrarily large. If it's capped at (say) 1024, then the
> average payoff per session will settle down once the
> number of sessions you play is large compared to 1024.

Why should large cube values matter? Math is math and
numbers are numbers. The problem is in the so-called
"best/perfect/optimum cube strategy theory" which can
be defeated even by such a crude "mutant" as Axel used
in the experiment that I had proposed. You'll just have to
accept this sooner or later.

The 1024 limit in XG and 4096 limit in Gnubg were not
in order to prevent the "equity does not exist" problem
but for a more realistic cube strategy closer to idea of
"table stakes" limit. I doubt it (and apparently no sign of
it either) that they had any idea that cube skill debates
would ever come to this. They were not just thinking of
numbers but of "numbers of units" (like dollars, degrees,
grams, meters, etc.) We can treat numbers as numbers
only and stil stay within the realm of applied math...

MK

Re: St. Patersburg Paradox in gamblegammon

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From: tchow12...@yahoo.com (Timothy Chow)
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Subject: Re: St. Patersburg Paradox in gamblegammon
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 by: Timothy Chow - Sat, 14 May 2022 13:33 UTC

On 5/13/2022 9:51 PM, MK wrote:
> On May 13, 2022 at 7:04:11 AM UTC-6, Tim Chow wrote:
>> To answer Murat's question, "equity" is a number
>> that we assign to a position (where by a "position"
>> I include the information about the cube value and
>> location, and whose turn it is) which is supposed to
>> predict how much money we would earn (or lose, if
>> the equity is negative) per session if we were to play
>> out that exact same position over and over again.
>
> I'll agree with the clarification that equity can be and
> (as far I understand), is currently determined cubeless
> first, then cubeful equity is calculated using a formula.

That is how bots *estimate* equity, but it's not how equity
is *mathematically defined*. It's defined the way I stated
above, as the long-term average. (For sticklers like Paul,
I'm implicitly using the law of large numbers to define
expected value, rather than using the standard definition.)

>> There is a tacit assumption that if we play out the
>> position often enough, then our wins and losses will
>> "average out" and settle down to some long-term
>> average payoff per session.
>
> Again, I'll agree if I understand you correctly that when
> luck evens out after enough trials, wins and losses will
> settle down to some values "assuming" that the players
> are also of the same skill level and play consistently.

Yes, it is important to assume that the players play consistently.
It's actually not necessary to assume that they have the same skill
level as long as they play consistently, but let's go ahead and
assume that. In fact, the usual assumption is that the players
have "solved" backgammon (think AlphaZero backgammon, if you like)
and play accordingly.

> This, I don't understand and pursue further. Many people
> offered statistics in the past about the average number
> of moves in a money game, from 21 to 27. Let's say 24
> is good enough for the "real life gamblegammon context".
>
> Since all other numbers mentioned are also averages, in
> games of average 24 moves, the cube can't keep going
> higher past a natural limit. This, if you play a large enough
> number of games, all possible cube values should settle,
> no??

If there were an upper limit on the length of a game then what
you say here is correct. But 24 is an *average* number. An
individual game could potentially last an arbitrarily long time.

> If Axel had played 10,000,000 actual games instead of
> only 10,000 and using Markov Chains to extrapolate to
> 5,000,000,000 games, I wonder if "things would settle"
> better in an experiment closer to real life..?

This is the key question. If the equity does not exist,
then it doesn't matter how many games you play; the average
payoff per game won't settle down.

>> This can happen only if the cube value is allowed to be
>> arbitrarily large. If it's capped at (say) 1024, then the
>> average payoff per session will settle down once the
>> number of sessions you play is large compared to 1024.
>
> Why should large cube values matter? Math is math and
> numbers are numbers. The problem is in the so-called
> "best/perfect/optimum cube strategy theory" which can
> be defeated even by such a crude "mutant" as Axel used
> in the experiment that I had proposed. You'll just have to
> accept this sooner or later.

Suppose we have a sequence of numbers, each of which is between
-3072 and +3072 (representing a backgammon with the maximum cube
value of 1024). Suppose they represent the outcomes of a sequence
of games played by consistent, equally skillful players as we
discussed above. As we play more and more games, we can compute
a running average---the net payoff after n games, divided by n.
This running average will fluctuate, perhaps rather dramatically
at first, but as n gets larger, I claim that it will eventually
settle down to some number (in fact, if we are using the standard
starting position of backgammon, which is symmetrical, then the
running average will settle down to zero, but if we start with an
asymmetrical position that favors one player, then it will not
settle down to zero, but it will still settle down to *some*
number). In particular, there's no way that the running average
can get bigger and bigger as n gets larger, for the simple reason
that the most I can win in a single game is 3072, so no matter
how many games I play, my running average can never get higher
than 3072 points per game.

On the other hand, suppose there is no limit on the cube value.
Then there is no reason in principle why the running average
couldn't get larger and larger as n gets larger, without ever
settling down. Here's an artificial example that wouldn't
actually happen in practice but which illustrates the idea and
which you can check yourself with a simple computer program.
If n is a natural number, let v(n) be the highest power of 2
that divides n. So for example if n is a power of 2 then
v(n) = n; if n is even but not a multiple of 4 then v(n) = 2;
if n is odd then v(n) = 1; etc. The sequence v(1), v(2), ...
starts off like this:

1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4,
1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, ...

Now imagine that these numbers represent the payoffs of a sequence
of backgammon games. We can try to compute the running average.
For example, after 4 games, our net payoff is 1 + 2 + 1 + 4 = 8,
so our average payoff per game is 8/4 = 2. After 16 games, our
net payoff is

1 + 2 + 1 + 4 + 1 + 2 + 1 + 8 + 1 + 2 + 1 + 4 + 1 + 2 + 1 + 16

which works out to 48, so our average payoff per game is 48/16 = 3.
You can check with a computer that after 4^k games, the average
payoff per game is k+1, which gets bigger and bigger as k gets
bigger. It never settles down. It is important that the value of
a single game can be arbitrarily large, or else this can't happen.

---
Tim Chow

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
Injection-Date: Mon, 16 May 2022 07:53:55 +0000
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 by: MK - Mon, 16 May 2022 07:53 UTC

On May 14, 2022 at 7:34:04 AM UTC-6, Tim Chow wrote:

> On 5/13/2022 9:51 PM, MK wrote:

>> I'll agree with the clarification that equity can be and
>> (as far I understand), is currently determined cubeless
>> first, then cubeful equity is calculated using a formula.

> That is how bots *estimate* equity, but it's not how
> equity is *mathematically defined*. It's defined the
> way I stated above, as the long-term average.

Okay, what will it take for you to accept that the bots
do not *estimate* equity as *mathematically defined*
and that their estimates are inaccuarte by unknowable
amounts?

Especially because traing your steps backwards from
end positions "combinatorially explode" and because
in backgammon the same positions can be arrived at
through more than one path.

> (For sticklers like Paul, I'm implicitly using the law
> of large numbers to define expected value, rather
> than using the standard definition.)

I brought in the USBGF's definition that "equity" is
"One’s value in the current game, mathematically
equivalent to the expected value". I hope USBGF
isn't trying to accommodate sticklers also(?).

>> Since all other numbers mentioned are also averages,
>> in games of average 24 moves, the cube can't keep
>> going higher past a natural limit. This, if you play a
>> large enough number of games, all possible cube
>> values should settle, no??

> If there were an upper limit on the length of a game
> then what you say here is correct. But 24 is an
> *average* number. An individual game could
> potentially last an arbitrarily long time.

I don't understand why you have a problem with this
average number when you don't have any problems
with all kinds of other average numbers?

People came up with average number of roll numbers
looking at tens of millions of games. In real life, it seems
that it does "settle" to some average numbers. In this:

https://groups.google.com/g/rec.games.backgammon/c/n5rR1MmDdSY/m/rdKdAYxV-7MJ

Stick, for eample, says "I did this study a long time
ago and believe my results were 23 rolls was the
length of the average money game." I don't want to
needlessly crow but I can give many links if you want.

So, let me ask you this: if you were to agree to 24 or 23,
etc. would you change your above argument? Surely the
"cube skill theory" formulas must assume that games
don't go arbitrarily long(??).

>> If Axel had played 10,000,000 actual games instead
>> of only 10,000 and using Markov Chains to extrapolate
>> to 5,000,000,000 games, I wonder if "things would
>> settle" better in an experiment closer to real life..?

> This is the key question. If the equity does not exist,
> then it doesn't matter how many games you play; the
> average payoff per game won't settle down.

After all this discussion, I still don't understnad when
equity does not exist? If you don't put a cap on the cube
does the equity stop existing at some point or does it
not exist from the very beginning? How about giving
some real life examples of equity not existing?

>>> This can happen only if the cube value is allowed to
>>> be arbitrarily large. If it's capped at (say) 1024, then
>>> the average payoff per session will settle down once
>>> the number of sessions you play is large compared
>>> to 1024.

Axel concluded that beavers and raccoons end in SPP.
According to you, that's a false conclusion since without
a limit on the cube, cubeful games without beavers and
raccoons allowed can also go arbitrarily long. So then,
wouldn't you say that even only one double per move is
allowed, gamblegammon can end in SPP? And why do
you think Axel didn't find that??

> Suppose we have a sequence of numbers, each of
> which is between -3072 and +3072 (representing a
> backgammon with the maximum cube value of 1024).
> Suppose they represent the outcomes of a sequence
> of games played by consistent, equally skillful players
> as we discussed above. As we play more and more
> games, we can compute a running average---the net
> payoff after n games, divided by n. This running average
> will fluctuate, perhaps rather dramatically at first, but as
> n gets larger, I claim that it will eventually settle down
> to some number (in fact, if we are using the standard
> starting position of backgammon, which is symmetrical,
> then the running average will settle down to zero, but if
> we start with an asymmetrical position that favors one
> player, then it will not settle down to zero, but it will still
> settle down to *some* number). In particular, there's no
> way that the running average can get bigger and bigger
> as n gets larger, for the simple reason that the most I
> can win in a single game is 3072, so no matter how many
> games I play, my running average can never get higher
> than 3072 points per game.

Okay, very good. I understand all that. What I'm asking is
"what is the meaning/significance/implication of all this
in relation to "cube skill theory"?

> On the other hand, suppose there is no limit on the cube
> value. Then there is no reason in principle why the running
> average couldn't get larger and larger as n gets larger,
> without ever settling down.

This may be the reason for my getting confused. I looks
like when you say "equity" or "expected value", you mean
"running average of all equities in all games played".

If so, this is a huge disconnect. :( If you remember, while
talking with Axel, I kept talking about "counting potatoes",
meaning just tallying the games and the points won by
each player. The reason was that I (the mutant) was trying
to "defy the torpedos" and turn the games into cubeless
and force longer games by causing them to be played out
to the end. It's rather discouraging that we were talking
about totally different things. Actually, I think Axel started
out right but midway through the experiment he strayed
into the pastures of SPP. :( Now I see...

> Here's an artificial example that wouldn't actually happen
> in practice but which illustrates the idea and .....

Frankly, I would prefer real life examples to artificial ones.

BTW: There is a paragraph that summarizes Tim's long
explanation above in a very simple language and a very
simpled example:

==============================================https://en.wikipedia.org/wiki/Expected_value#Definition

Let X represent the outcome of a roll of a fair six-sided die.

More specifically, X will be the number of pips showing on
the top face of the die after the toss. The possible values
for X are 1, 2, 3, 4, 5, and 6, all of which are equally likely
with a probability of 1/6. The expectation of X is:

E[X]=1.1/6+2.1/6+3.1/6+4.1/6+5.1/6+6.1/6=3.5.

If one rolls the die n times and computes the average
(arithmetic mean) of the results, then as n grows, the
average will almost surely converge to the expected
value, a fact known as the strong law of large numbers.
===============================================
I just quoted it hoping it may help somple people better.

MK

Re: St. Patersburg Paradox in gamblegammon

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From: tchow12...@yahoo.com (Timothy Chow)
Newsgroups: rec.games.backgammon
Subject: Re: St. Patersburg Paradox in gamblegammon
Date: Mon, 16 May 2022 10:05:51 -0400
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 by: Timothy Chow - Mon, 16 May 2022 14:05 UTC

On 5/16/2022 3:53 AM, MK wrote:
> On May 14, 2022 at 7:34:04 AM UTC-6, Tim Chow wrote:
>> That is how bots *estimate* equity, but it's not how
>> equity is *mathematically defined*. It's defined the
>> way I stated above, as the long-term average.
>
> Okay, what will it take for you to accept that the bots
> do not *estimate* equity as *mathematically defined*
> and that their estimates are inaccuarte by unknowable
> amounts?

That their estimates are inaccurate by unknowable amounts
is an obvious fact.

>> (For sticklers like Paul, I'm implicitly using the law
>> of large numbers to define expected value, rather
>> than using the standard definition.)
>
> I brought in the USBGF's definition that "equity" is
> "One’s value in the current game, mathematically
> equivalent to the expected value". I hope USBGF
> isn't trying to accommodate sticklers also(?).

The USBGF definition is fine. My comment was intended only
for sticklers, not for you. I don't expect you to know the
official mathematical definition of expected value or to know
the official statement of the law of large numbers. That is
not important for the current discussion.

> I don't understand why you have a problem with this
> average number when you don't have any problems
> with all kinds of other average numbers?

I have no problem with this average number. I am only saying
that using the average number does not let you draw the conclusion
that you want to draw.

> So, let me ask you this: if you were to agree to 24 or 23,
> etc. would you change your above argument? Surely the
> "cube skill theory" formulas must assume that games
> don't go arbitrarily long(??).

No, the cube skill formulas do not assume this.

> After all this discussion, I still don't understnad when
> equity does not exist? If you don't put a cap on the cube
> does the equity stop existing at some point or does it
> not exist from the very beginning? How about giving
> some real life examples of equity not existing?

Figuring out whether the equity of a given position exists is
in general not easy. Here's an example of a position whose equity
does not exist:

http://timothychow.net/cg/undefined-1.html

But if you're not willing to take the time to understand the
artificial example I gave, then you're unlikely to understand
this example.

> Axel concluded that beavers and raccoons end in SPP.
> According to you, that's a false conclusion since without
> a limit on the cube, cubeful games without beavers and
> raccoons allowed can also go arbitrarily long. So then,
> wouldn't you say that even only one double per move is
> allowed, gamblegammon can end in SPP? And why do
> you think Axel didn't find that??

I don't disagree with anything Axel said. Just because you
can get undefined equities without beavers and raccoons doesn't
mean you can't get undefined equities with them as well.

> Okay, very good. I understand all that. What I'm asking is
> "what is the meaning/significance/implication of all this
> in relation to "cube skill theory"?

It means that cube skill theory is completely fine in match play,
because there is a maximum value that the cube can take.

> If so, this is a huge disconnect. :( If you remember, while
> talking with Axel, I kept talking about "counting potatoes",
> meaning just tallying the games and the points won by
> each player. The reason was that I (the mutant) was trying
> to "defy the torpedos" and turn the games into cubeless
> and force longer games by causing them to be played out
> to the end. It's rather discouraging that we were talking
> about totally different things. Actually, I think Axel started
> out right but midway through the experiment he strayed
> into the pastures of SPP. :( Now I see...

I don't think we're talking about totally different things.
You asked about undefined equities and that's what I'm talking
about. The long-term average is the same as the equity. If
the long-term average does not settle down then the equity does
not exist. The only way this can happen is if the cube value
has no upper bound.

>> Here's an artificial example that wouldn't actually happen
>> in practice but which illustrates the idea and .....
>
> Frankly, I would prefer real life examples to artificial ones.

Sure. However, do you understand the artificial example? If
you don't even understand the simple, artificial example, then
you're not going to understand the real life examples either.

> BTW: There is a paragraph that summarizes Tim's long
> explanation above in a very simple language and a very
> simpled example:

There is nothing wrong with this simple explanation, but you
asked why the cube value has to be unlimited for the expected
value not to exist. To answer that question requires going into
more detail than the Wikipedia summary provides.

---
Tim Chow

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
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 by: MK - Thu, 19 May 2022 09:13 UTC

On May 16, 2022 at 8:05:55 AM UTC-6, Tim Chow wrote:

> On 5/16/2022 3:53 AM, MK wrote:

>> Okay, what will it take for you to accept that the
>> bots do not *estimate* equity as *mathematically
>> defined* and that their estimates are inaccuarte
>> by unknowable amounts?

> That their estimates are inaccurate by unknowable
> amounts is an obvious fact.

It wasn't obvious neither always nor to all and it still
isn't. Even to you, only in temporary/easy words but
not in essence or with any consequences. You just
make detached statements in order to save the day
and then continue from where you left off before that.

> The USBGF definition is fine.

Of course, it's fine but in a different context than what
Axel, you, etc. have been using it in this thread.

> My comment was intended only for sticklers, not
> for you. I don't expect you to know the official
> mathematical definition of expected value or to
> know the official statement of the law of large
> numbers. That is not important for the current
> discussion.

Stop trying to put me down by mathshitting and try to
be more useful by making a case for why a converged
"running total expected value" is needed in addition to
just adding up the wins and losses for each player and
leave it at that?

That was the whole purpose of the experiment (i.e. to
collect empirical data) before it turned into SPP bullshit
(which still wasn't based on correctly applied Markov).

>> I don't understand why you have a problem with this
>> average number when you don't have any problems
>> with all kinds of other average numbers?

> I have no problem with this average number. I am only
> saying that using the average number does not let you
> draw the conclusion that you want to draw.

Of course, it does. It's puts an even more realistic limit
on the cube value than your artificial 1024, 4096, etc.

>> So, let me ask you this: if you were to agree to 24 or
>> 23, etc. would you change your above argument?
>> Surely the "cube skill theory" formulas must assume
>> that games don't go arbitrarily long(??).

> No, the cube skill formulas do not assume this.

So then, the bots using the cube skill formulas should
be able to handle large cube values. Why don't they?

I have used the saying "it takes two to tango" all along
and saw that you started using it also recently. So why
Gnubg does tango with the mutant and end up in SPP?

That it the problem isn't as bad in match play doesn't
save the "cube skill theory" from being bullshit. If you
dare call something skill and theory, it needs to work
in all types of cubeful games. And it needs to work
especially well in cubeful gamblemmon money games
because it's whole purpose was expediate gambling!

>> After all this discussion, I still don't understnad when
>> equity does not exist? If you don't put a cap on the
>> cube does the equity stop existing at some point or
>> does it not exist from the very beginning? How about
>> giving some real life examples of equity not existing?

> Figuring out whether the equity of a given position
> exists is in general not easy. Here's an example of
> a position whose equity does not exist:
> http://timothychow.net/cg/undefined-1.html
> But if you're not willing to take the time to understand
> the artificial example I gave, then you're unlikely to
> understand this example.

This example is also "artificial". It's not a legal position.

Just like other artificial examples you concoct such as
imaginary footbal games played by tossing coins, etc.

You never were able to make a case for cube skill except
by giving real life examples from positions at the last two
or three rolls in a game.

Even in the other "infamous example" you made yours
look like, you debate the correct cube actions based on
equities estimated by the bots that you acknowledged
just above as being inaccuarte by unknowable amounts.

What comes first? The chicken or the egg? Well, who
gives a shit? All your arguments are teflon coated. You
are an artificial example of a mathematician yourself.

> I don't disagree with anything Axel said.

I didn't ask you to disagree with him. Strawman... :(

> Just because you can get undefined equities without
> beavers and raccoons doesn't mean you can't get
> undefined equities with them as well.

This is a nonsensical sentence. My argument was that
if the reason for getting undefined equities was games
lasting arbitrarily long and that games without beavers
and raccoons can also last arbitrarily long, then in all
those "one cube only" games in Axel's experiment, he
should have also ran into undefined equities but he
hasn't.

Estoy preguntando: porque no, hombre? Comprendes??

>> Okay, very good. I understand all that. What I'm asking
>> is "what is the meaning/significance/implication of all
>> this in relation to "cube skill theory"?

> It means that cube skill theory is completely fine in
> match play, because there is a maximum value that
> the cube can take.

Theoretically, this is not correct. Even after the cube gets
past the match value, there is always a chance that your
opponent may drop, so both players should keep doubling
and the cube can get just as high as in money games but
I'll spare you the agravation and won't dwell on this, "Mr.
gambletician mathematician"...

> I don't think we're talking about totally different things.
> You asked about undefined equities and that's what I'm
> talking about.

I meant that the experiment's purpose wasn't to find out
if undefined equities existed or not. It was to see what
would happen in real life, if a very very crude "bot mutant"
was to play a very very long series of games aginst the
bot that played according to the "cube skill theory".

Then the discussion strayed into undefined equities, etc.

> There is nothing wrong with this simple explanation, but
> you asked why the cube value has to be unlimited for the
> expected value not to exist.

No, I didn't ask that at all. In response to your argument
that the cube needs to be limited by an arbitrary constant
like 1024, 4096, etc. I offered that "in real life" and average
money games last about 23-24 rolls, which puts a natural
limit on the cube.

> To answer that question requires going into more detail
> than the Wikipedia summary provides.

No, it doesn't. "You need" to show off your mathshitting
by convoluting the discussion. It's clear in the example
that there is no problem with expected value because
the maximum number in a dice roll is limited to 6! Much
simpler and enough to understand without all you guys'
"maths and mirrors"...

MK

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: pepste...@gmail.com (peps...@gmail.com)
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 by: peps...@gmail.com - Thu, 19 May 2022 10:55 UTC

On Thursday, May 19, 2022 at 10:14:00 AM UTC+1, MK wrote:
> On May 16, 2022 at 8:05:55 AM UTC-6, Tim Chow wrote:
>
> > On 5/16/2022 3:53 AM, MK wrote:
....
> > > It means that cube skill theory is completely fine in
> > match play, because there is a maximum value that
> > the cube can take.
> Theoretically, this is not correct. Even after the cube gets
> past the match value, there is always a chance that your
> opponent may drop, so both players should keep doubling
> and the cube can get just as high as in money games
....
I'm not a rules expert, but I think that, if a player owns the cube,
and the cube has a high enough value so that winning the current game
at single value wins the match for the cube owner, then the cube owner
is not allowed to double.

This prevents the trick of offering the cube and then claiming the match
if your opponent rejects it.

Paul

Re: St. Patersburg Paradox in gamblegammon

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From: tchow12...@yahoo.com (Timothy Chow)
Newsgroups: rec.games.backgammon
Subject: Re: St. Patersburg Paradox in gamblegammon
Date: Thu, 19 May 2022 09:40:26 -0400
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 by: Timothy Chow - Thu, 19 May 2022 13:40 UTC

On 5/19/2022 5:13 AM, MK wrote:
> On May 16, 2022 at 8:05:55 AM UTC-6, Tim Chow wrote:
>> That their estimates are inaccurate by unknowable
>> amounts is an obvious fact.
>
> It wasn't obvious neither always nor to all and it still
> isn't. Even to you, only in temporary/easy words but
> not in essence or with any consequences. You just
> make detached statements in order to save the day
> and then continue from where you left off before that.

No. I have always said this. I don't draw the same
conclusions that you do from this obvious fact, but the
fact itself is obvious.

> Stop trying to put me down by mathshitting and try to
> be more useful by making a case for why a converged
> "running total expected value" is needed in addition to
> just adding up the wins and losses for each player and
> leave it at that?

Suppose I have a fair coin, and I gain $2 whenever it comes
up heads and $0 whenever it comes up tails. Then the expected
value per flip is $1. You can't get reliably get this answer
of $1 by flipping the coin just a few times and adding up the
wins and losses. Say I flip it 10 times. It might come up
heads 6 times and tails 4 times, so I'd win $12, and dividing
by 10 would yield $1.20, not $1. I could flip it 100 times,
or 1000 times, but it still wouldn't guarantee that I would
get an answer of $1. To get $1, I have to keep flipping it
and see what value the answer settles down to (or "converges"
to, to use the standard term).

>> I have no problem with this average number. I am only
>> saying that using the average number does not let you
>> draw the conclusion that you want to draw.
>
> Of course, it does. It's puts an even more realistic limit
> on the cube value than your artificial 1024, 4096, etc.

Again, if the game *never* exceeded 20 rolls then you would
be correct. But the *average* length can be 20 and the
expected value could still be infinite (i.e., not exist).

Suppose that 2/3 of the time, a game lasts 10 rolls, and
2/9 of the time, a game lasts 20 rolls, and 2/27 of the time,
a game lasts 40 rolls, and 2/81 of the time, a game lasts
80 rolls, and so on, with a game lasting 5*2^n rolls 2/3^n
of the time. Then the average length of a game will be
20 rolls.

Now suppose that you win 4 points when the game lasts 10 rolls,
16 points when the game lasts 20 rolls, 64 points when the game
lasts 40 rolls, 256 points when the game lasts 80 rolls, and so
on, where you win 4^n points when the game lasts 5*2^n rolls.
Then the expected value will not exist.

>>> So, let me ask you this: if you were to agree to 24 or
>>> 23, etc. would you change your above argument?
>>> Surely the "cube skill theory" formulas must assume
>>> that games don't go arbitrarily long(??).
>
>> No, the cube skill formulas do not assume this.
>
> So then, the bots using the cube skill formulas should
> be able to handle large cube values. Why don't they?

Trouble does *not* automatically arise when the games go on arbitrarily
long, but only when the game go on arbitrarily long *and* the cube value
gets too high.

>> But if you're not willing to take the time to understand
>> the artificial example I gave, then you're unlikely to
>> understand this example.
>
> This example is also "artificial". It's not a legal position.

You're not going to understand anything if you are focused on
lobbing irrelevant objections instead of actually trying to
learn something. The purpose of the example is to illustrate
a point in a simple way. If you understand the simple example
then you can see how to construct more complicated and realistic
examples.

In this case, the conclusion is exactly the conclusion that you
want to draw...this is a position that proves your point that
it makes no sense to talk about "cube skill" here if the cube
is allowed to get arbitrarily high. Why are you complaining
about this example when it's exactly the type of example that
can be used to prove your point to other people?

> You never were able to make a case for cube skill except
> by giving real life examples from positions at the last two
> or three rolls in a game.

You seem to think that my artificial example above is intended
to prove the existence of cube skill. It's not. It's intended
to prove *your* point, that cube skill makes no sense. No wonder
you never get anywhere in these discussions if you can't even tell
that the other person is agreeing with you, and insist on
contradicting them just because that's your nature.

Any by the way, I did make a case for cube skill by solving
Hypestgammon. Again, you never took the time to study the
results I got. Probably because it proved you wrong and you
refuse to understand anything that reveals your errors.

>> Just because you can get undefined equities without
>> beavers and raccoons doesn't mean you can't get
>> undefined equities with them as well.
>
> This is a nonsensical sentence. My argument was that
> if the reason for getting undefined equities was games
> lasting arbitrarily long and that games without beavers
> and raccoons can also last arbitrarily long, then in all
> those "one cube only" games in Axel's experiment, he
> should have also ran into undefined equities but he
> hasn't.

Again, the fundamental reason isn't that games last arbitrarily
long, but that they last arbitrarily long *and* that the cube
value gets arbitrarily high.

>> It means that cube skill theory is completely fine in
>> match play, because there is a maximum value that
>> the cube can take.
>
> Theoretically, this is not correct. Even after the cube gets
> past the match value, there is always a chance that your
> opponent may drop, so both players should keep doubling
> and the cube can get just as high as in money games but
> I'll spare you the agravation and won't dwell on this, "Mr.
> gambletician mathematician"...

You know that your argument is silly here. Cube skill theory
is completely fine in match play, because if the cube value
goes higher than the match value, it does not increase the
payoff, so there is effectively a maximum value that the
cube can take.

> Then the discussion strayed into undefined equities, etc.

It "strayed" there because you asked for an explanation of SPP.

>> There is nothing wrong with this simple explanation, but
>> you asked why the cube value has to be unlimited for the
>> expected value not to exist.
>
> No, I didn't ask that at all. In response to your argument
> that the cube needs to be limited by an arbitrary constant
> like 1024, 4096, etc. I offered that "in real life" and average
> money games last about 23-24 rolls, which puts a natural
> limit on the cube.

See the example above. The *average* length of a game is 20
but there is no limit on the cube value because *sometimes*
the game lasts arbitrarily long.

>> To answer that question requires going into more detail
>> than the Wikipedia summary provides.
>
> No, it doesn't. "You need" to show off your mathshitting
> by convoluting the discussion. It's clear in the example
> that there is no problem with expected value because
> the maximum number in a dice roll is limited to 6! Much
> simpler and enough to understand without all you guys'
> "maths and mirrors"...

That the maximum value is 6 is exactly why the example does
not illustrate what can happen when the cube value gets arbitrarily
large. A more complicated example is needed to illustrate that
point.

---
Tim Chow

Re: St. Patersburg Paradox in gamblegammon

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From: mai...@axel-reichert.de (Axel Reichert)
Newsgroups: rec.games.backgammon
Subject: Re: St. Patersburg Paradox in gamblegammon
Date: Thu, 19 May 2022 22:04:19 +0200
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 by: Axel Reichert - Thu, 19 May 2022 20:04 UTC

MK <murat@compuplus.net> writes:

> What I'm disputing here is that cubeful equities except
> for positions towards the end of games are bogus and
> can be demostrated by experiment like Axel has done
> (just to make me happy:))

You misinterpret my experiment.

> Since all other numbers mentioned are also averages, in games of
> average 24 moves, the cube can't keep going higher past a natural
> limit. This, if you play a large enough number of games, all possible
> cube values should settle, no??

No, as Tim has explained with several nice examples by now.

> If Axel had played 10,000,000 actual games instead of
> only 10,000 and using Markov Chains to extrapolate to
> 5,000,000,000 games, I wonder if "things would settle"
> better in an experiment closer to real life..?

No. But the Markov chains helped to clarify my thought and led to
showing the "beaver instability" against mutant cubing. No more, no
less.

> The problem is in the so-called "best/perfect/optimum cube strategy
> theory" which can be defeated even by such a crude "mutant" as Axel
> used

You misinterpret my experiment.

Axel

Re: St. Patersburg Paradox in gamblegammon

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From: mai...@axel-reichert.de (Axel Reichert)
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Subject: Re: St. Patersburg Paradox in gamblegammon
Date: Thu, 19 May 2022 22:15:03 +0200
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 by: Axel Reichert - Thu, 19 May 2022 20:15 UTC

MK <murat@compuplus.net> writes:

> Axel concluded that beavers and raccoons end in SPP.
> According to you, that's a false conclusion since without
> a limit on the cube, cubeful games without beavers and
> raccoons allowed can also go arbitrarily long.

Yes, but everything hinges on the frequency of the swings and the
"power" of the cube. With beavers and mutant cubing the cube on average
does not increase the stakes by a factor of 2, but more. This is
crucial.

Axel

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
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 by: MK - Thu, 19 May 2022 20:20 UTC

On May 19, 2022 at 4:55:20 AM UTC-6, peps...@gmail.com wrote:

> On May 19, 2022 at 10:14:00 AM UTC+1, MK wrote:

>> Theoretically, this is not correct. Even after.....

> ....
> This prevents the trick of offering the cube and
> then claiming the match if your opponent rejects it.

You can make any rules you want for any reason.
As I indicated, I made a theoretical comment and
mainly for the purpose of showing that cube value
is not intrinsicly related to value of the game.

MK

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
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 by: MK - Thu, 19 May 2022 22:28 UTC

On May 19, 2022 at 7:40:31 AM UTC-6, Tim Chow wrote:

> On 5/19/2022 5:13 AM, MK wrote:

>> It wasn't obvious neither always nor to all and it still
>> isn't. Even to you, only in temporary/easy words but
>> not in essence or with any consequences. You just
>> make detached statements in order to save the day
>> and then continue from where you left off before that.

> No. I have always said this.

Okay, fine. No big deal. I'll just try to hold you to it.

> I don't draw the same conclusions that you do from
> this obvious fact, but the fact itself is obvious.

I don't expect you to call the cube skill bullshit. I am
doing it liberally, enough for all of us. ;) But you can't
avoid the implications of what you acknowledge. You
can't keep building more elaborate arguments on top
of inaccurate equities.

>> ..... try to be more useful by making a case for why a
>> converged "running total expected value" is needed
>> in addition to just adding up the wins and losses for
>> each player and leave it at that?

> Suppose I have a fair coin, and I gain $2 whenever .....
> I have to keep flipping it and see what value the answer
> settles down to (or "converges" to, to use the standard
> term).

You keep explaining what I already said I understand
but failed to answer my question. Read it again. I was
asking "what is the need, in real life gamblegammon,
for a running total expected value to converge"? I don't
know how else to word my question. Try to understand.

>> Of course, it does. It's puts an even more realistic limit
>> on the cube value than your artificial 1024, 4096, etc.

> Again, if the game *never* exceeded 20 rolls then you
> would be correct. But the *average* length can be 20 and
> the expected value could still be infinite (i.e., not exist).

So, I would be correct but not correct. You're making
conflicting statements but not of matter to dwell on.

> ..... Then the average length of a game will be 20 rolls.
>
> Now suppose that you win 4 points when the game .....
> Then the expected value will not exist.

Who cares? Why does it need to exist? Whatever points
each player wins or loses is the only thing that matters.

>> So then, the bots using the cube skill formulas should
>> be able to handle large cube values. Why don't they?

> Trouble does *not* automatically arise when the games
> go on arbitrarily long, but only when the game go on
> arbitrarily long *and* the cube value gets too high.

I'm okay with your clarification.

>> This example is also "artificial". It's not a legal position.

> You're not going to understand anything if you are focused
> on lobbing irrelevant objections instead of actually trying to
> learn something.

Same goes for you...

> The purpose of the example is to illustrate a point in a
> simple way. If you understand the simple example then
> you can see how to construct more complicated and
> realistic examples.

As you yourself said recently, what we are talking about
here is applied math to gamblegammon, not theoretical
math. Your using an example that can't arise in real life
is useless. Furthermore, it's not simple. If so, and since
you must necessarily understand your own simple artificial
example, I dare you to "construct more complicated and
realistic examples". Let's see how you will do...

But, my point wasn't even this at all. First you concocted
an illegal position. Then you did an XG rollout of it, which
produces some "inaccurate equity estimates". But you
didn't mind this and went on the disagree with the bot's
cube decision based on its "inaccurate equity estimates".

Then with your own "maths and mirrors" calculations, you
came up with different equities and different correct cube
decisions. Finally you conluded "Therefore "perfect play"
doesn't make sense in this position", meaning "this illegal,
artificial position. What have you accomplished? What are
the implications of your argument and conclusion? Sorry,
but I just don't get it. All I see is circular mathshit. :(

> In this case, the conclusion is exactly the conclusion that
> you want to draw...

Similar to mine but not exactly or even remotely the same.

> this is a position that proves your point that it makes no
> sense to talk about "cube skill" here if the cube is allowed
> to get arbitrarily high. Why are you complaining about this
> example when it's exactly the type of example that can be
> used to prove your point to other people?

Except that I use it to generalize and say that the entire cube
skill theory is bullshit. For me, it's either all valid or not at all.
You, Axel, etc. limit it to only certain positions and otherwise
call it good. That's where we diverge.

>> You never were able to make a case for cube skill except
>> by giving real life examples from positions at the last two
>> or three rolls in a game.

> You seem to think that my artificial example above is
> intended to prove the existence of cube skill. It's not.
> It's intended to prove *your* point, that cube skill makes
> no sense.

I should be elated to hear this but I'm not because I know
that you don't mean it. It's just another temporary, detached
statement from you that you won't hold yourself to. You'll
just go back making your old arguments as though you've
never said this.

> Any by the way, I did make a case for cube skill by solving
> Hypestgammon. Again, you never took the time to study
> the results I got. Probably because it proved you wrong and
> you refuse to understand anything that reveals your errors.

I had looked at the results and had responded to it, although
with a long delay, because of our misunderstanding. What
you did was through the same calculations used for equities
that you acknowledged as "inaccurate by unknown amounts".

You need to do a real life experiment by a "cubeful training"
of a bot and see if you and that bot would come up with the
same equities. Then you will have proven that you solved it.

> You know that your argument is silly here. Cube skill theory
> is completely fine in match play, because if the cube value
> goes higher than the match value, it does not increase the
> payoff, so there is effectively a maximum value that the
> cube can take.

Yes but you need that "cube skill" to know this, don't you? Or
else, a player can drop a theoretically allowed and correct
double past the game value because of lack of cube skill...

Anyway, you are right though that this isn't worth dwelling on.

>> Then the discussion strayed into undefined equities, etc.

> It "strayed" there because you asked for an explanation of
> SPP.

It strayed before I asked anything about it. SPP should have
never been mentioned durins the experiment. I continuously
objected that it didn't apply to gamblegammon.

And also, I didn't ask for "an explanation of SPP" which I had
sufficiently understood. What I asked was how Axel and you
were using it in relation to gamblegammon becaus it doen't
exist in gamblegammon. Only some rare similar sequences
occur within games that don't amount to SPP by any definition.

> See the example above. The *average* length of a game is 20
> but there is no limit on the cube value because *sometimes*
> the game lasts arbitrarily long.

For the purpose that I had suggested the experiment, it doesn't
matter. Especially in "real life" which I keep stressing. In fact,
during Axel's first 5,000+3,000+3,000 games everything was
fine in the cube value stats that he had posted.

Only when he started extrapolating with Markov chains that
things went berzerk, most likely because he made incorrect
assumptions. Then, since he couldn't post a tally of actual
cube values encountered, he switched to posting "average
game values" and started talking about SPP...

> That the maximum value is 6 is exactly why the example
> does not illustrate what can happen when the cube value
> gets arbitrarily large. A more complicated example is
> needed to illustrate that point.

It clearly says: ".....as n grows, the average will almost surely
converge to the expected value, a fact known as the strong
law of large numbers".

Okay, so the reader needs to at least know what "converge"
means, if not "strong law of large numbers" (which I can't
say that I know much about myself). In your example, you
were explaining that the value will conver until a very large
cube occurs, that if you keep playing it will converge again
until an even larger cube occurs, and so on... Isn't that the
weak law of large numbers? That it will repeatedly and
periodically converge to "some value"? If so, what's wrong
with accepting those "some values"?


Click here to read the complete article
Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
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 by: MK - Thu, 19 May 2022 23:07 UTC

On May 19, 2022 at 2:04:21 PM UTC-6, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:
>> What I'm disputing here is that cubeful equities except
>> for positions towards the end of games are bogus and
>> can be demostrated by experiment like Axel has done
>> (just to make me happy:))

> You misinterpret my experiment.

You called it "Murat mutant experiment". Clearly you were
trying to do what I had suggested (even if you ended up
not doing take >0% part correctly).

>> If Axel had played 10,000,000 actual games instead of
>> only 10,000 and using Markov Chains to extrapolate to
>> 5,000,000,000 games, I wonder if "things would settle"
>> better in an experiment closer to real life..?

> No.

How do you know?

> But the Markov chains helped to clarify my thought and
> led to showing the "beaver instability" against mutant
> cubing. No more, no less.

Your first statement is your business but you can't say
"no more, no less"! You don't know that. You didn't keep
your game files for anyone to look at to see if there was
anything more or less in them. You weren't even capable
of doing a clean, proper experiment.

But, as I said before, it's never too late. You already have
the tools. Do the experiment again with real games only.
Without any Markov, etc. "maths and mirrors". Make sure
you save the games in a practical format to share with us
and let's see what comes out of it.

If you are not willing to do the work, share your scripts so
that we can do it. Unless you may be afraid to expose that
your scripts were flawed to begin with.

>> The problem is in the so-called "best/perfect/optimum
>> cube strategy theory" which can be defeated even by
>> such a crude "mutant" as Axel used

> You misinterpret my experiment.

Perhaps you mean to say that this wasn't your intention
but you did the experiment that I had suggested for my
own purposes. After any experiment is done, anyone
can interpret the results any way they want... I was fair.
I never intended to nor did "use you" but to "utilise you".
And I do sincerely appreciate that you have done it. Now
if I can only get you to do it again and better this time... ;)

MK

Re: St. Patersburg Paradox in gamblegammon

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Subject: Re: St. Patersburg Paradox in gamblegammon
From: mur...@compuplus.net (MK)
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 by: MK - Thu, 19 May 2022 23:29 UTC

On May 19, 2022 at 2:15:05 PM UTC-6, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:
>> Axel concluded that beavers and raccoons end in SPP.
>> According to you, that's a false conclusion since without
>> a limit on the cube, cubeful games without beavers and
>> raccoons allowed can also go arbitrarily long.

> Yes, but everything hinges on the frequency of the swings
> and the "power" of the cube. With beavers and mutant
> cubing the cube on average does not increase the stakes
> by a factor of 2, but more. This is crucial.

Okay, so with no beavers SPP is much less likely. Now,
try to look at it from my point of purpose, which is not
about SPP but about debunking "cube skill theory".

Beavers and raccoons supposedly require even more
more cube skill than simple doubles. After debunkin
raccon skill and beaver skill, is it still all that neccesry
to debunk simples doubles skill?

This remind me of a folk story about "Nasreddin Hodja
and the stork". He thinks that he stork's exaggeratedly
long beak and legs must be defects of nature. He takes
a pair of scissors and clips off its long beak and legs.
He then looks at it and says: “Now you look like a bird.”

As you keep clipping the defective parts of the "cube
skill theory" maybe it will look like a bird but of a much
reduced size... :)

MK

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