I started this thread based on comments from another:

https://groups.google.com/g/rec.games.backgammon/c/n0Xcxt5SliA/m/6EQdoSIVAAAJ

On January 2, 2024 at 6:49:05 PM UTC-7, peps...@gmail.com wrote:

> I have just finished an 11 point match against XG where I had
> a PR of 2.01 I think the fact that this match was longer makes
> my achievement more impressive than yours.
> I lost the match 19-1 (over 5 games) so I don't think it would
> impress Murat.
> I may have impressed Murat by winning the previous 11 point
> match against XG but my PR in the match I won was horrible.
> I can't remember it but somewhere in the 8-9 range, I think.

He won with a PR of 8 but he lost with a PR of 2. In other words,
he lost while playing more "like the bot" but he won while playing
more "unlike the bot". How interesting!

Do you guys remember my having said many times over the years
that you can't beat the bots at their own game, i.e. trying to play
like the bots, but that you can beat them by playing unlike the bots.

Now then, let's see... I'm sure many/most of you guys keep a good
record of all the games you play against the bots. I'd bet Tim does,
if nobody else.

What would you guys guess you would see if you calculated your
average PR's for only the games you won and for only the games
you lost separately, to compare them?

Doing this may require a little work but surely not excessive when
considering the possible value of the discovery that you may make.

Let's say as an example that you win 25% and lose 75% of the time.

You may rarely achieve very low, perhaps 0, PR's while winning or
losing, and eXtremely high PR's while winning or losing as well. It
may help to discard the highest and lowest 5% of PR's both in the
winning and losing categories.

Do you think your winning and losing PR's will be the same or at
least very close to your overall PR?

God forbid, what if your winning PR comes out to be 7 (within 6-8
range) and your losing PR comes out to be 3 (within 2-4 range)...?

I'm just making up numbers for the sake of the argument but I'm
sure at least 50% of you can understand what I'm getting at. And
I bet none of you wouldn't dare find that out. ;)

MK

On 1/3/2024 10:17 PM, MK wrote:
> I bet none of you wouldn't dare find that out. ;)

This sounds like a favorable bet to me. So if I understand
correctly, if at least one person reading this doesn't dare
to find out, then I win the bet?

---
Tim Chow

On Wednesday, January 3, 2024 at 10:17:25 PM UTC-5, MK wrote:
> Do you think your winning and losing PR's will be the same or at
> least very close to your overall PR?

Even if your "winning PR" is higher than your "losing PR," it doesn't necessarily mean that when you play with a higher PR, you're more likely to win.

Here's an oversimplified example to illustrate the point. Suppose I play 25 games, and my PR in each game is either 3, 6, or 9. More precisely:

PR = 3, win 1 game, lose 0 games
PR = 6, win 1 game, lose 6 games
PR = 9, win 8 games, lose 9 games

Restricting attention to the 10 games that I won, my average PR is (3*1 + 6*1 + 9*8)/10 = 8.1.

Restricting attention to the 15 games that I lost, my average PR is (3*0 + 6*6 + 9*9)/15 = 7.8.

So my "winning PR" is higher than my "losing PR." But does that mean I'd rather play with a PR of 9 than a PR of 3? No. If I play with a PR of 9, then my winning percentage is 8/17, which is less than 50%, but if I play with a PR of 3, then I win 100% of the time.

Of course this example is contrived, but it illustrates the difference between your expected PR conditional on winning, and your expected win rate conditional on your PR.

---
Tim Chow

Tim Chow <tchow12000@yahoo.com> writes:

> On Wednesday, January 3, 2024 at 10:17:25 PM UTC-5, MK wrote:
>> Do you think your winning and losing PR's will be the same or at
>> least very close to your overall PR?
>
> Even if your "winning PR" is higher than your "losing PR," it doesn't
> necessarily mean that when you play with a higher PR, you're more
> likely to win.

There is a nice article with some statistics done in R:

http://freerangestats.info/blog/2016/03/19/elo-pr-luck

Bottom line: I you are asked to predict the outcome of a game and may
choose whether you are give the luck rating or the PR to do so, choose
the luck rating.

Best regards

Axel

On January 4, 2024 at 6:27:38 AM UTC-7, Timothy Chow wrote:

> On 1/3/2024 10:17 PM, MK wrote:

>> I bet none of you wouldn't dare find that out. ;)

> So if I understand correctly, if at least one person
> reading this doesn't dare to find out, then I win the bet?

Hmm, well, I'm not sure teacher. If two negatives
make a positive, wouldn't it mean "all of you would
dare"..? ;)

MK

On January 4, 2024 at 9:50:42 AM UTC-7, Tim Chow wrote:

> On January 3, 2024 at 10:17:25 PM UTC-5, MK wrote:

>> Do you think your winning and losing PR's will be
>> the same or at least very close to your overall PR?

> Even if your "winning PR" is higher than your "losing
> PR," it doesn't necessarily mean that when you play
> with a higher PR, you're more likely to win.

Of course not but how does that relate to my point?

> Here's an oversimplified example to illustrate the point.
> Suppose I play 25 games, and my PR in each game is
> either 3, 6, or 9. More precisely:
> PR = 3, win 1 game, lose 0 games
> PR = 6, win 1 game, lose 6 games
> PR = 9, win 8 games, lose 9 games
> So my "winning PR" is higher than my "losing PR." But
> does that mean I'd rather play with a PR of 9 than a
> PR of 3? No.

If higher PR meant more wins in the past, why wouldn't
you keep doing what you have been doing?

> If I play with a PR of 9, then my winning percentage
> is 8/17, which is less than 50%, but if I play with a PR
> of 3, then I win 100% of the time.

The point here is to compare two average PR's, one for
games won, one for games lost; not to tally how many
games you won or lost at a specific PR.

In your above 25 games, you can end up with 25 unique
PR's with two decimal accuracy. With a 4.61 PR, you may
have played only one game and if you lost it, you would
have lost 100% of the time. With a 7.19 PR, you may have
played only one game and if you won it, you would have
won 100% of the time. And so on...

> Of course this example is contrived, but it illustrates the
> difference between your expected PR conditional on
> winning, and your expected win rate conditional on your PR.

It surely does not. I was about to say "you're no good logic
and math; go back to correcting people's grammar errors",
but you're no good at that either... :(

MK

On January 4, 2024 at 3:19:34 PM UTC-7, Axel Reichert wrote:

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

>> On January 3, 2024 at 10:17:25 PM UTC-5, MK wrote:

>>> Do you think your winning and losing PR's will be
>>> the same or at least very close to your overall PR?

>> Even if your "winning PR" is higher than your "losing
>> PR," it doesn't necessarily mean that when you play
>> with a higher PR, you're more likely to win.

> Bottom line: I you are asked to predict the outcome of
> a game and may choose whether you are give the luck
> rating or the PR to do so, choose the luck rating.

While trying to approach from the opposite direction in
believing that it will help you cling on to your dogmatic
fallacies, both of you are actually supporting my point.

If I tell you that a player has won the game and asked
you whether you think he played with a low PR or he
got lucky, per your above comments, you would choose
that he got lucky.

And that fits perfectly with the bots' calculations that
the winner is almost always the luckier player, as it
must be so in order to validate luck+skill=1 mathshit.

Thus, the luckier winner will be the lesser skilled one
with the higher PR.

Since bots are perfect players with 0 PR, their opponents
will almost always incur a higher PR, whether they win or
lose. Then, the question is reduced to whether their PR's
are the same, higher or lower when winning and losing.

As they say: you can't have your PR and eat it too...

MK

On 1/4/2024 7:59 PM, MK wrote:
> On January 4, 2024 at 6:27:38 AM UTC-7, Timothy Chow wrote:
>
>> On 1/3/2024 10:17 PM, MK wrote:
>
>>> I bet none of you wouldn't dare find that out. ;)
>
>> So if I understand correctly, if at least one person
>> reading this doesn't dare to find out, then I win the bet?
>
> Hmm, well, I'm not sure teacher. If two negatives
> make a positive, wouldn't it mean "all of you would
> dare"..? ;)

Right! So you're betting that all of us would dare. If at least
one of us doesn't dare, then I win the bet.

---
Tim Chow

On 1/4/2024 8:19 PM, MK wrote:
> The point here is to compare two average PR's, one for
> games won, one for games lost

That's what I did. In my example, PR for games won was 8.1;
PR for games lost was 7.8.

---
Tim Chow

On January 4, 2024 at 8:51:38 PM UTC-7, Timothy Chow wrote:

> On 1/4/2024 7:59 PM, MK wrote:

>> Hmm, well, I'm not sure teacher. If two negatives
>> make a positive, wouldn't it mean "all of you would
>> dare"..? ;)

> Right! So you're betting that all of us would dare. If
> at least one of us doesn't dare, then I win the bet.

You're unconfusable. :( Fair is fair, you win the bet
and so does everyone for that matter.

Even though, because of the way I play against the
bots, my PR's wouldn't be meaningful in the same
way as you people's, out of curiousity I calculated
my PR's in my last 100 money games against XGR++
(see https://montanaonline.net/backgammon/xg.php)

Here are some stats:

I won 33 games with an average PR of 16.83 and lost
67 games with an average PR of 24.97.

So my losing PR is the higher one but that's because
I was trying to win more points than win more games
by "losing low and winning high cubes". 27 of the 67
games I lost were 1-point drops, mostly after only a
few rolls into the game. with an average PR of 40.24.

Amazingly, one of those incurred 0 PR! and another
incurred 100 PR! :) The rest were between 9.94 and
92.40 PR. In the end, I had managed to win 126 vs
XGR++'s 119 points. So, if you want to win more than
50% against the strongest bots, you may want to try
a similar "buy low, sell high" cube strategy...

What I always argue to be more interesting is how
much I won compared to how much I should have
won according to the bot, based on my error rate. In
this case I won 7 point when I should have lost 107
for a total surplus of 114 points in 100 money-games!
which proves that, not only the bots aren't unbeatable,
but also that your luck/skill/error/etc. calculations are
totally unfounded/inaccurate pile of horse muffins.

Now that I pawed the way, let's see if some of you
will dare share your winning/losing PR numbers...?

MK

On January 4, 2024 at 8:55:39 PM UTC-7, Timothy Chow wrote:

> On 1/4/2024 8:19 PM, MK wrote:

>> The point here is to compare two average PR's,
>> one for games won, one for games lost

> That's what I did. In my example, PR for games
> won was 8.1; PR for games lost was 7.8.

Okay, but you also added a bunch of other stuff to
it and claimed that it proved something that it didn't.

MK

On 1/5/2024 4:02 PM, MK wrote:
> On January 4, 2024 at 8:55:39 PM UTC-7, Timothy Chow wrote:
>
>> On 1/4/2024 8:19 PM, MK wrote:
>
>>> The point here is to compare two average PR's,
>>> one for games won, one for games lost
>
>> That's what I did. In my example, PR for games
>> won was 8.1; PR for games lost was 7.8.
>
> Okay, but you also added a bunch of other stuff to
> it and claimed that it proved something that it didn't.

Specifically, which sentence that I wrote do you say is wrong?

---
Tim Chow

On January 6, 2024 at 2:23:25 PM UTC-7, Timothy Chow wrote:

> On 1/5/2024 4:02 PM, MK wrote:

>> On January 4, 2024 at 8:55:39 PM UTC-7, Timothy Chow wrote:

>>> On 1/4/2024 8:19 PM, MK wrote:

>>>> The point here is to compare two average PR's,
>>>> one for games won, one for games lost

>>> That's what I did. In my example, PR for games
>>> won was 8.1; PR for games lost was 7.8.

>> Okay, but you also added a bunch of other stuff to
>> it and claimed that it proved something that it didn't.

> Specifically, which sentence that I wrote do you
> say is wrong?

Okay, I will quote again and answer. You had said:

> PR = 3, win 1 game, lose 0 games
> PR = 6, win 1 game, lose 6 games
> PR = 9, win 8 games, lose 9 games
> Restricting attention to the 10 games that I won,
> my average PR is (3*1 + 6*1 + 9*8)/10 = 8.1.
> Restricting attention to the 15 games that I lost,
> my average PR is (3*0 + 6*6 + 9*9)/15 = 7.8.
> So my "winning PR" is higher than my "losing PR."

Thus far, it's all good. You fall victim to you own
fallacy when you ask and answer:

> But does that mean I'd rather play with a PR of 9
> than a PR of 3? No.

Here you switched from average PR's to a specific
pair of invidual PR's and tried to generalize but you
can't generalize based on two arbitrarily picked PR's
the same way you can do with two averages which
are themselves generalized PR's for wins and losses.

The logical question would be: "Does that mean I'd
rather play with an average PR of 8.1 than an average
PR of 7.8?" Then the answer would would be: "Yes".

> If I play with a PR of 9, then my winning percentage
> is 8/17, which is less than 50%, but if I play with a
> PR of 3, then I win 100% of the time.

If you had lost the game you played with a PR of 3,
then your average winning PR would be 8.666 and
your average winning PR would be 7.5 and so your
"winning PR would be higher than your losing PR"
by even a wider margin, but then you would say that
when you play with a PR of 3, you lose 100% of the
time. There is no end to such illogical deduction...

> Of course this example is contrived, but it illustrates
> the difference between your expected PR conditional
> on winning, and your expected win rate conditional
> on your PR.

Again, this is not true. PR is a result calculated based
on the luck+skill=1 fallacy.

So, if you tell me that you played 25 games and that
in N1 of them your average PR was 8.1 and in N2 of
them your average PR was 7.8, then ask me in which
set I would "expect" you to have won more, I would
say the set with the higher average PR.

Similarly, if you tell me that you played 25 games and
that you won 10 of them and that you lost 15 of them,
then ask me in which set I would "expect" your average
PR to be higher, I would say the set of 10 that you won.

If you don't confuse apples and oranges, there is no
difference between "PR conditional" and "win rate
conditional", if you can ask the questions correctly.

MK

On 1/7/2024 5:08 AM, MK wrote:
> On January 6, 2024 at 2:23:25 PM UTC-7, Timothy Chow wrote:
>> But does that mean I'd rather play with a PR of 9
>> than a PR of 3? No.
>
> Here you switched from average PR's to a specific
> pair of invidual PR's and tried to generalize but you
> can't generalize based on two arbitrarily picked PR's
> the same way you can do with two averages which
> are themselves generalized PR's for wins and losses.

I agree that one can't generalize. But in my original post,
I did not attempt to generalize. I simply asked the question,
does that mean I'd rather play with a PR of 9 than a PR of 3?
And the correct answer to that question is no, as I said.

>> If I play with a PR of 9, then my winning percentage
>> is 8/17, which is less than 50%, but if I play with a
>> PR of 3, then I win 100% of the time.
>
> If you had lost the game you played with a PR of 3,
> then your average winning PR would be 8.666 and
> your average winning PR would be 7.5 and so your
> "winning PR would be higher than your losing PR"
> by even a wider margin, but then you would say that
> when you play with a PR of 3, you lose 100% of the
> time. There is no end to such illogical deduction...

But in my original post, I did not make any illogical deductions.
The illogical deductions were introduced by you.

>> Of course this example is contrived, but it illustrates
>> the difference between your expected PR conditional
>> on winning, and your expected win rate conditional
>> on your PR.
>
> Again, this is not true. PR is a result calculated based
> on the luck+skill=1 fallacy.

What I said was true. The example was indeed contrived. It
also illustrated the difference that I stated.

> So, if you tell me that you played 25 games and that
> in N1 of them your average PR was 8.1 and in N2 of
> them your average PR was 7.8, then ask me in which
> set I would "expect" you to have won more, I would
> say the set with the higher average PR.
>
> Similarly, if you tell me that you played 25 games and
> that you won 10 of them and that you lost 15 of them,
> then ask me in which set I would "expect" your average
> PR to be higher, I would say the set of 10 that you won.
>
> If you don't confuse apples and oranges, there is no
> difference between "PR conditional" and "win rate
> conditional", if you can ask the questions correctly.

I won't quibble with what you say here, but none of it
contradicts what I said in my original post. It only
contradicts what you imagined I said, not what I actually
said.

---
Tim Chow

On January 8, 2024 at 7:02:15 AM UTC-7, Timothy Chow wrote:

> On 1/7/2024 5:08 AM, MK wrote:

>> On January 6, 2024 at 2:23:25 PM UTC-7, Timothy Chow wrote:

>>> But does that mean I'd rather play with a PR of 9
>>> than a PR of 3? No.

>> Here you switched from average PR's to a specific
>> pair of invidual PR's and tried to generalize but you
>> can't generalize based on two arbitrarily picked PR's
>> the same way you can do with two averages which
>> are themselves generalized PR's for wins and losses.

> I agree that one can't generalize. But in my original
> post, I did not attempt to generalize.

Actually, by misgeneralizing using arbitrary PR's, you
were trying to show generalizing using average PR's
was also wrong but you failed at it.

> I simply asked the question, does that mean I'd
> rather play with a PR of 9 than a PR of 3? And the
> correct answer to that question is no, as I said.

The word "THAT" in your question referred to your
observation immediately above your question. It
was a nonsensical, stupid question with no logical
transition from what preceeded it.

Instead of using PR 9 and PR 3 numbers, if you had
used PR 9 and PR 6 numbers, your winning percents
would be 8/17 and 1/7, thus the answer to whether
you should better play with a PR of 9 than a PR of 6
would be "Yes".

Once again you demonstrate that you are incapable
of logic and coherent thoughts.

> But in my original post, I did not make any illogical
> deductions. The illogical deductions were
> introduced by you.

You asked a question and answered it. What would
you call that if not a "deduction", (i.e. drawing of a
conclusion by reasoning)?

> What I said was true. The example was indeed
> contrived. It also illustrated the difference that
> I stated.

There is no problem with it being contrived. It failed
despite being intentionally contrived to illustrate a
difference because your trying to contrast a pair of
arbitrary PR's against a pair of average PR's failed.
It's just as simple as that but you either don't make
an effort to understand, incapable of understanding
or too conceited to accept even if you understand.

> I won't quibble with what you say here, but none of
> it contradicts what I said in my original post. It only
> contradicts what you imagined I said, not what I
> actually said.

You said: "Of course this example is contrived, but it
illustrates the difference between your expected PR
conditional on winning, and your expected win rate
conditional on your PR."

Maybe what you don't understand is that I wasn't
arguing against what you were stating/claiming,
which may wery well be true but your example failed
to illustrate it.

I could have just left it at saying this much but I went
one step further to use your own example to ask the
same question, in both diresctions (i.e. PR conditional
and win rate conditional), to show you that your example
illustrated the opposite of what you had intended to.

You sure can try to use a different example, but this
one simply didn't work. :(

MK

On 1/8/2024 2:12 PM, MK wrote:
> On January 8, 2024 at 7:02:15 AM UTC-7, Timothy Chow wrote:
>
>> On 1/7/2024 5:08 AM, MK wrote:
>
>>> On January 6, 2024 at 2:23:25 PM UTC-7, Timothy Chow wrote:
>
>>>> But does that mean I'd rather play with a PR of 9
>>>> than a PR of 3? No.
>
>>> Here you switched from average PR's to a specific
>>> pair of invidual PR's and tried to generalize but you
>>> can't generalize based on two arbitrarily picked PR's
>>> the same way you can do with two averages which
>>> are themselves generalized PR's for wins and losses.
>
>> I agree that one can't generalize. But in my original
>> post, I did not attempt to generalize.
>
> Actually, by misgeneralizing using arbitrary PR's, you
> were trying to show generalizing using average PR's
> was also wrong but you failed at it.

That is not what I was trying to show.

>> I simply asked the question, does that mean I'd
>> rather play with a PR of 9 than a PR of 3? And the
>> correct answer to that question is no, as I said.
>
> The word "THAT" in your question referred to your
> observation immediately above your question. It
> was a nonsensical, stupid question with no logical
> transition from what preceeded it.

The question makes sense and its answer is no.

> Instead of using PR 9 and PR 3 numbers, if you had
> used PR 9 and PR 6 numbers, your winning percents
> would be 8/17 and 1/7, thus the answer to whether
> you should better play with a PR of 9 than a PR of 6
> would be "Yes".

But that is not the question I posed.

> You asked a question and answered it. What would
> you call that if not a "deduction", (i.e. drawing of a
> conclusion by reasoning)?

I call it asking a question and answering it.

>> What I said was true. The example was indeed
>> contrived. It also illustrated the difference that
>> I stated.
>
> There is no problem with it being contrived. It failed
> despite being intentionally contrived to illustrate a
> difference because your trying to contrast a pair of
> arbitrary PR's against a pair of average PR's failed.
> It's just as simple as that but you either don't make
> an effort to understand, incapable of understanding
> or too conceited to accept even if you understand.

It did not fail to do what I intended to do.

> You said: "Of course this example is contrived, but it
> illustrates the difference between your expected PR
> conditional on winning, and your expected win rate
> conditional on your PR."
>
> Maybe what you don't understand is that I wasn't
> arguing against what you were stating/claiming,
> which may wery well be true but your example failed
> to illustrate it.

I know you weren't arguing against what I was stating/claiming.
Nor was I arguing against what you were stating/claiming.
It's only in your own mind that there is such an argument.

> I could have just left it at saying this much but I went
> one step further to use your own example to ask the
> same question, in both diresctions (i.e. PR conditional
> and win rate conditional), to show you that your example
> illustrated the opposite of what you had intended to.

It illustrated what I intended it to illustrate. If you misunderstood
what I intended to illustrate, and found that my example failed to
illustrate what *you* thought it intended to illustrate, then that's
your problem and not mine.

---
Tim Chow

On January 8, 2024 at 7:31:10 PM UTC-7, Timothy Chow wrote:

> On 1/8/2024 2:12 PM, MK wrote:

>> On January 8, 2024 at 7:02:15 AM UTC-7, Timothy Chow wrote:

>>> On 1/7/2024 5:08 AM, MK wrote:
>>>> On January 6, 2024 at 2:23:25 PM UTC-7, Timothy Chow wrote:

>> Actually, by misgeneralizing using arbitrary PR's, you
>> were trying to show generalizing using average PR's
>> was also wrong but you failed at it.

> That is not what I was trying to show.

Would you care to clarify what exactly your were trying
to show? Otherwise, this is just an empty assertion.

>> The word "THAT" in your question referred to your
>> observation immediately above your question. It
>> was a nonsensical, stupid question with no logical
>> transition from what preceeded it.

> The question makes sense and its answer is no.

Again, proof by mere repeated assertion without any
new supporting argument.

>> Instead of using PR 9 and PR 3 numbers, if you had
>> used PR 9 and PR 6 numbers, your winning percents
>> would be 8/17 and 1/7, thus the answer to whether
>> you should better play with a PR of 9 than a PR of 6
>> would be "Yes".

> But that is not the question I posed.

I knew that. I asked the same question you asked using
a different pair of PR's in your example but leading to a
different answer, to show you that your stupid question
and its answer did not illustrate anything whatsoever.

>> You asked a question and answered it. What would
>> you call that if not a "deduction", (i.e. drawing of a
>> conclusion by reasoning)?

> I call it asking a question and answering it.

You mean, you asked a question and answered it for
no reason..? Poor you... :( Pitiful you... :((

>> There is no problem with it being contrived. It failed
>> despite being intentionally contrived to illustrate a
>> difference because your trying to contrast a pair of
>> arbitrary PR's against a pair of average PR's failed.

> It did not fail to do what I intended to do.

Yet another proof by mere repeated assertion without
any new supporting argument. I guess you will win the
debate when I get tired of your school yard behavior...

> It illustrated what I intended it to illustrate.

One more proof by mere repeated assertion without any
new supporting argument. But by now, I'm beginning to
guess what may be ailing you.

Maybe you looked at your historical average PR's when
winning and when losing, then saw that your winning
PR was higher than your losing PR, and said: "Oh shit,
Murat may be on to something!"

Then in order to deceive and confort yourself about
you flock's rolledoats horse muffins dogma, you tried
to illustrate that it didn't always need to be true using
a contrived example that blew up in your face... :(

I wonder how many more of you suffer from "Tim's
syndrome", i.e. looked at your winning and losing PR's
(if you dared to, of course) and said: "Oh shit, Murat
may be on to something!"..? ;)

You know I have your cure. Just come to papa... :)

MK

On 1/9/2024 2:03 AM, MK wrote:
> Would you care to clarify what exactly your were trying
> to show? Otherwise, this is just an empty assertion.

I see that you've finally realized that you can't find any errors
in my original post, so my work is done.

---
Tim Chow

On January 9, 2024 at 6:49:36 AM UTC-7, Timothy Chow wrote:

> On 1/9/2024 2:03 AM, MK wrote:

>> Would you care to clarify what exactly your were trying
>> to show? Otherwise, this is just an empty assertion.

> I see that you've finally realized that you can't find
> any errors in my original post, so my work is done.

Whoever told you that must have fooled with you. I have
found errors in your conrived example based illustration,
which didn't illustrate anything, and I have explained why
so in three rounds of detailed arguments.

But if you want to exit, I have nothing more to gain from
tormenting and torturing you further either as I also have
achieved my part. So, bye-bye...

But before you go, would you be nice enough to share
your average winning and losing PR's against the bot?
Would anyone else..??

MK

On 1/11/2024 3:46 AM, MK wrote:
> On January 9, 2024 at 6:49:36 AM UTC-7, Timothy Chow wrote:
>
>> On 1/9/2024 2:03 AM, MK wrote:
>
>>> Would you care to clarify what exactly your were trying
>>> to show? Otherwise, this is just an empty assertion.
>
>> I see that you've finally realized that you can't find
>> any errors in my original post, so my work is done.
>
> Whoever told you that must have fooled with you.

Wouldn't be the first time that you've fooled with me!

---
Tim Chow

On 1/11/2024 3:45 PM, Timothy Chow wrote:

> On 1/11/2024 3:46 AM, MK wrote:

>> Whoever told you that must have fooled with you.

> Wouldn't be the first time that you've fooled with me!

This time I swear that it wasn't me but maybe
you are beginning to think that you are me..?

MK