If the cube goes to 16 or 32 in a 25-point match, would Axel
also call that SPP?
On Tuesday, May 10, 2022 at 11:53:18 AM UTC+1, Tim Chow wrote:
On 5/10/2022 5:04 AM, MK wrote:I disagree with this. In the form of the SPP that I know, the expected value is infinite.
If the cube goes to 16 or 32 in a 25-point match, would AxelNo. The essence of SPP is that the expected value does not exist.
also call that SPP?
When there is a maximum possible value (here, 25 points) then the
expected value always exists.
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
On Tuesday, May 10, 2022 at 5:30:21 PM UTC+1, peps...@gmail.com wrote:"leads" -> "needs"
On Tuesday, May 10, 2022 at 11:53:18 AM UTC+1, Tim Chow wrote:
On 5/10/2022 5:04 AM, MK wrote:I disagree with this. In the form of the SPP that I know, the expected value is infinite.
If the cube goes to 16 or 32 in a 25-point match, would AxelNo. The essence of SPP is that the expected value does not exist.
also call that SPP?
When there is a maximum possible value (here, 25 points) then the expected value always exists.
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.
PaulIn 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
On 5/10/2022 5:04 AM, MK wrote:
If the cube goes to 16 or 32 in a 25-point match, would AxelNo. The essence of SPP is that the expected value does not exist.
also call that SPP?
When there is a maximum possible value (here, 25 points) then the
expected value always exists.
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.
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.
If a total solution to the SPP was "But the amount of money
is finite", the "paradox" would have less content than it does.
For that, the expected value only leads to be sufficiently large,
not infinite. It's infinitudinizationness is hardly "the essence".
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?
I wasn't asking if it would be SPP but if "Axel would callNo. 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.
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.
Are you saying that this is what Axel is doing? I wish heBut 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.
himself would also try to explain what he is doing.
Just when I think I undestand, I don't. :(If a total solution to the SPP was "But the amount of money
is finite", the "paradox" would have less content than it does.
For that, the expected value only leads to be sufficiently large,"Infinitudinizationness"... Thanks for making me smile. :)
not infinite. It's infinitudinizationness is hardly "the essence".
MK
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.
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:
.....
The essence of what Axel means by SPP *in the current
context* is that the expected value does not exist.
On May 11, 2022 at 7:17:45 AM UTC-6, Tim Chow wrote:
The essence of what Axel means by SPP *in the currenthttps://usbgf.org/backgammon-glossary/
context* is that the expected value does not exist.
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."
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 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.
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.
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.
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.
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.
(For sticklers like Paul, I'm implicitly using the law
of large numbers to define expected value, rather
than using the standard definition.)
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.
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 .....
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?
(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(?).
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?
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(??).
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?
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??
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"?
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:
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.
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.
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.
I don't think we're talking about totally different things.
You asked about undefined equities and that's what I'm
talking about.
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.
On May 16, 2022 at 8:05:55 AM UTC-6, Tim Chow wrote:...
On 5/16/2022 3:53 AM, MK wrote:
...Theoretically, this is not correct. Even after the cube getsIt means that cube skill theory is completely fine inmatch play, because there is a maximum value that
the cube can take.
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
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.
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?
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?
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 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.
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.
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"...
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"...
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:))
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..?
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
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.
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.
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.
I don't draw the same conclusions that you do from
this obvious fact, but the fact itself is obvious.
..... 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).
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).
..... 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Who cares? Why does it need to exist? Whatever points
each player wins or loses is the only thing that matters.
You're now digressing into the question of whether SPP is...
relevant to backgammon.
On May 19, 2022 at 2:04:21 PM UTC-6, Axel Reichert wrote:
MK <mu...@compuplus.net> writes:
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?
MK <mu...@compuplus.net> writes:
On May 19, 2022 at 2:04:21 PM UTC-6, Axel Reichert wrote:
MK <mu...@compuplus.net> writes:
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?It is called "math", you know, and it can be used to prove things.
MK <mu...@compuplus.net> writes:
On May 19, 2022 at 2:04:21 PM UTC-6, Axel Reichert wrote:
MK <mu...@compuplus.net> writes:
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?
It is called "math", you know, and it can be used to prove
things.
On 5/19/2022 6:28 PM, MK wrote:
Who cares? Why does it need to exist? Whatever points
each player wins or loses is the only thing that matters.
Your initial question was whether Axel would call it SPP
if the cube goes to 16 or 32 in a 25-point match. You said,
"I'm trying to understand what he means by SPP in the
gamblegammon context."
These are the questions I was addressing.
"Whatever points each player wins or loses" is *not*
the only thing that matters *if what you are trying to
understand is what Axel means*.
You're now digressing into the question of whether SPP
is relevant to backgammon.
I don't expect that discussion to go anywhere.
If you still don't understand *what Axel means* then I
can continue to explain. If you now understand what
he means and just disagree, then my job is done here.
To that, I had replied, as you partially quoted: "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". Why did you skip the part that
shows you had "essentially" given an irrelevant answer?
| Sysop: | Keyop |
|---|---|
| Location: | Huddersfield, West Yorkshire, UK |
| Users: | 293 |
| Nodes: | 16 (2 / 14) |
| Uptime: | 220:02:02 |
| Calls: | 6,622 |
| Calls today: | 4 |
| Files: | 12,171 |
| Messages: | 5,317,887 |