Timothy Chow <tchow12000@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

MK <murat@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

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

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

On November 18, 2021 at 7:46:07 PM UTC-7, Tim Chow wrote:

> 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

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

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

MK <murat@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

MK <murat@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 <mail@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

On November 22, 2021 at 11:09:43 AM UTC-7, Axel Reichert wrote:

> 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?

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

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

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

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

MK <murat@compuplus.net> writes:

> 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.

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

Axel

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

MK <murat@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

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 <mail@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

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

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.