How many dice rolls to establish dodgy dealings?

[Nasti Chestikov]

So I am playing on ZooEscape and I strongly suspect that most of the games I lose are due to dodgy dice.

The site admin obviously refute claims that they're running hooky dice.

How many dice rolls from games I lose would constitute a valid sample size? 1000, 10000?

I intend to push the rolls through a chi-squared test (thank you to Tim on here for alerting me to the fact that it exists).

Anything else I need to be aware of?

Thank you for reading.

[Nasti Chestikov]

Just as a starter, I've analysed 3,670 rolls and the distribution of opponent doubles is

1-1 = 110
2-2 = 113
3-3 = 102
4-4 = 114
5-5 = 129
6-6 = 117

Excel 2010 has a CHISQ.TEST function and using that gives me a chi-squared test value of 0.0267 but I don't know how to interpret that number? Is that telling me there's a 2.67% chance of the dice being fair?

[peps...@gmail.com]

Your opponent is getting more doubles than expected. Indeed all six doubles occurred more often than average (although the issue is ultra-close with 3-3).
So we see an at least slightly unusual pattern.
How unusual is it? It's about as unusual as something that happens 2.67% of the time -- for example an event of someone rolling a 66.
It would be completely wrong to interpret that as there being only a 2.67% chance of the dice being fair.

Suppose you produce another set of 3670 rolls that has the same pattern, and these rolls are different to the ones you've just shown.
And suppose that you're giving all your data and not excluding data that doesn't fit your theory.
If, for this new set of 3670 rolls, we again get the result that all six of your opponent's doubles occur more often than average, then my personal opinion will be that there's a significant problem with the
dice on that site. But I won't necessarily be sure.

Paul

[peps...@gmail.com]

The number of doubles your opponent rolled was approx 3.25 standard deviations higher than the mean.

Paul

[peps...@gmail.com]

On Sunday, March 21, 2021 at 11:52:38 AM UTC, peps...@gmail.com wrote:

The thing that seems unusual to me is that every single one of the doubles occur more often than expected?
How likely is that? A naive estimate would be 1/64 because every double either occurs more often than expected or less often than expected,
and you're approximately equally likely to get fewer than average as you are to get more than average.

However, the real likelihood of this is actually significantly less than 1/64 because the probabilities are not independent.
In other words, if you're getting more 11's and more 22's and more 33's and more 44's than you bargained for, then you are getting
fewer of the outcomes that are outside 11/22/33/44 so from those numbers being excessive, you expect more 55's and 66's to compensate.

Tim knows the exact probability of all six doubles occurring more often than expected, under the null hypothesis of fair dice.
I hope he isn't too shy to reveal this number?

Paul

[peps...@gmail.com]

I mean "you expect fewer 55's and 66's to compensate."

[Nasti Chestikov]

I am taking every game and simply recording the number of doubles rolled by my opponent and how many rolls in total they had; there is no agenda, I strongly suspect the dice are hooky but needed some "science" to, if nothing else, reaffirm to me what my eyes were telling me.

Its a work in progress.

Updated numbers:

4743 total rolls

1-1 = 145
2-2 = 138
3-3 = 128
4-4 = 151
5-5 = 159
6-6 = 155

This now gives me a chi-squared value of 0.01388 (in Excel 2010).

I have been told that a binomial test is a better fit for what I'm doing, luckily Excel has that functionality as well. My binomial value for these dice = 0.00047......apparently you subtract that from 1 and if the result is > 0.5 then you run a mile from the dice.......if I do that, my number is 0.00053!

[Timothy Chow]

On 3/21/2021 4:55 AM, Nasti Chestikov wrote:
> On Saturday, 20 March 2021 at 18:32:35 UTC, Nasti Chestikov wrote:
>> So I am playing on ZooEscape and I strongly suspect that most of the games I lose are due to dodgy dice.
>>
>> The site admin obviously refute claims that they're running hooky dice.
>>
>> How many dice rolls from games I lose would constitute a valid sample size? 1000, 10000?

What matters most isn't the sample size. What matters most is that you
write down, in full detail, *before collecting any data*, exactly what
statistical experiment you intend to carry out, and then carry it out to
the letter, without being influenced in any way by how the data turn
out.

For something like a chi-squared test, 1000 rolls should be enough. I'm
not crazy about your proposal to take rolls only from games you lose.
Such a procedure is vulnerable to systematic bias. For example, whether
consciously or unconsciously, you could "throw" games where you notice
a lot of doubles, and try hard to win only when the number of doubles is
low or medium. The games you lose would then be biased toward having a
lot of doubles, even if the dice themselves were fair. So I would
strongly recommend that you consider *all* rolls, not just rolls from
games that you lose.

A chi-squared test is most naturally used to test whether there is *any*
deviation from uniform random, not just whether the doubles are wonky,
but it can be used to test just doubles. This again is something you
need to decide ahead of time, before collecting any data. That is, are
you going to create a list of 21 numbers (the number of times each roll
occurs) or 2 numbers (doubles versus non-doubles) or maybe 7 numbers
(one for each of the 6 doubles, plus one big category for all non-
doubles)? Also, are you going to separate the rolls you get from the
rolls the bot gets? All these things should be decided ahead of time.

I'd also recommend picking just one test to do, rather than multiple
tests. If you run multiple tests then it can get quite complicated to
compute significance values---if you run 3 tests then the chances that
at least one of them shows something unusual is going to be higher than
the chances that a single test shows something unusual.

You should also decide ahead of time exactly how much data you're going
to collect. 100 games? 1000 rolls? It doesn't matter too much as
long as you write down the number *in advance* and stick to it
religiously. You can't decide, mid-stream, that some particular game
"doesn't count" or that "that's enough data; I don't need to collect
more" or "I'm on a roll; let's collect an extra 20 games." All such
"cheats" can open the door to statistical bias. Once again, I cannot
stress enough that by far the most important thing is to specify
absolutely every detail of what you plan to do *in advance*, writing
it down, and following the plan to the letter as if your life depended
on it. Only then will the conclusions be statistically valid.

> Just as a starter, I've analysed 3,670 rolls and the distribution of opponent doubles is
>
> 1-1 = 110
> 2-2 = 113
> 3-3 = 102
> 4-4 = 114
> 5-5 = 129
> 6-6 = 117
>
> Excel 2010 has a CHISQ.TEST function and using that gives me a chi-squared test value of 0.0267 but I don't know how to interpret that number? Is that telling me there's a 2.67% chance of the dice being fair?

I am not able to reproduce a number of 0.0267 from the six numbers
above. You'll need to read the documentation for an explanation
of what CHISQ.TEST does. Generally, a chi-squared test requires
as input two lists; one is the list of actual observations, and
the other is the list of the expected observations if the dice
were random. The two most natural things to output are the statistic
itself and the p-value; you're going to be interested in the p-value.

---
Tim Chow

[Timothy Chow]

On 3/21/2021 8:18 AM, Nasti Chestikov wrote:
> I am taking every game and simply recording the number of doubles rolled by my opponent and how many rolls in total they had; there is no agenda, I strongly suspect the dice are hooky but needed some "science" to, if nothing else, reaffirm to me what my eyes were telling me.

If you're taking only the games you lose and only the
doubles rolled by your opponent then I would expect to
see more than 1/6 doubles even if the dice were fair.

For example, suppose you were to have a bot play itself
100 games, and in each game, you were to take the winning
side and count how many doubles the winning side rolled.
I would expect that the answer would be significantly
higher than 1/6. The bias would be coming not from the
dice, but from the fact that doubles tend to be good rolls
that help you win the game.

---
Tim Chow

[Nasti Chestikov]

On Sunday, 21 March 2021 at 14:03:11 UTC, Tim Chow wrote:
> On 3/21/2021 4:55 AM, Nasti Chestikov wrote:
> > On Saturday, 20 March 2021 at 18:32:35 UTC, Nasti Chestikov wrote:
> >> So I am playing on ZooEscape and I strongly suspect that most of the games I lose are due to dodgy dice.
> >>
> >> The site admin obviously refute claims that they're running hooky dice.
> >>
> >> How many dice rolls from games I lose would constitute a valid sample size? 1000, 10000?
> What matters most isn't the sample size. What matters most is that you
> write down, in full detail, *before collecting any data*, exactly what
> statistical experiment you intend to carry out, and then carry it out to
> the letter, without being influenced in any way by how the data turn
> out.

Thank you for your input.

This is exactly what I have done; my hypothesis is that I believe I lose a lot of the games that I do due to disproportionate double rolls for my opponent.

Hence, I am testing for doubles rolled in games I lose.

>
> For something like a chi-squared test, 1000 rolls should be enough. I'm
> not crazy about your proposal to take rolls only from games you lose.
> Such a procedure is vulnerable to systematic bias. For example, whether
> consciously or unconsciously, you could "throw" games where you notice
> a lot of doubles, and try hard to win only when the number of doubles is
> low or medium. The games you lose would then be biased toward having a
> lot of doubles, even if the dice themselves were fair. So I would
> strongly recommend that you consider *all* rolls, not just rolls from
> games that you lose.

Point taken; however, this is a retrospective exercise (and, in any event, I'm not into throwing games when it looks like my opponent has gotten lucky).

So I am going back over time and analysing the dice rolls.

> You should also decide ahead of time exactly how much data you're going
> to collect. 100 games? 1000 rolls? It doesn't matter too much as
> long as you write down the number *in advance* and stick to it
> religiously. You can't decide, mid-stream, that some particular game
> "doesn't count" or that "that's enough data; I don't need to collect
> more" or "I'm on a roll; let's collect an extra 20 games." All such
> "cheats" can open the door to statistical bias. Once again, I cannot
> stress enough that by far the most important thing is to specify
> absolutely every detail of what you plan to do *in advance*, writing
> it down, and following the plan to the letter as if your life depended
> on it. Only then will the conclusions be statistically valid.

Absolutely; which is why I am analysing *every* game that I have lost. I'm not going to cherry pick (after all, why should I, I know what my eyes tell me, I'm just wanting the satisfaction of having some scientific approach validate that).

Thank you again for your thoughts, I do appreciate it.

[Timothy Chow]

On 3/21/2021 10:14 AM, Nasti Chestikov wrote:
> Absolutely; which is why I am analysing *every* game that I have lost.

See my other comment. If you select only games you have lost
and only your opponent's doubles, then I would expect significantly
more than 1/6 doubles, even with perfectly fair dice.

How much more than 1/6 is unfortunately not something that can be
mathematically calculated. It requires answering questions
such as, "How much does rolling doubles help you win a game
of backgammon?" not to mention questions about how your style
of play and your opponent's style of play affect the outcome.

---
Tim Chow

[Peter]

The answer to the question in your Subject header is that it all depends
on the roll. A roll totalling 13 just once establishes dodgy dealings.


--
When, once, reference was made to a statesman almost universally
recognized as one of the villains of this century, in order to
induce him to a negative judgment, he replied: "My situation is
so different from his, that it is not for me to pass judgment".
Ernst Specker on Paul Bernays

[Nasti Chestikov]

On Sunday, 21 March 2021 at 14:29:42 UTC, Tim Chow wrote:

> See my other comment. If you select only games you have lost
> and only your opponent's doubles, then I would expect significantly
> more than 1/6 doubles, even with perfectly fair dice.
>
> How much more than 1/6 is unfortunately not something that can be
> mathematically calculated. It requires answering questions
> such as, "How much does rolling doubles help you win a game
> of backgammon?" not to mention questions about how your style
> of play and your opponent's style of play affect the outcome.
>
> ---
> Tim Chow

This I don't understand?

The dice don't know the board position, doubles should still come down approximately 1-in-6 rolls regardless of any other factors?

[badgolferman]

Nasti Chestikov wrote:

>So I am playing on ZooEscape and I strongly suspect that most of the
>games I lose are due to dodgy dice.

I find it easier to just expect funky dice rolls. That way I won't get
mad when the opponent receives them and I can laugh when I get them.

[Timothy Chow]

Let's take a simpler example. Instead of "how many doubles?" let's
consider the question "who wins the opening roll?"

I'm assuming we agree that if I win the opening roll, it gives me a
slight but definite advantage in the game. Do you agree?

Now suppose I play a whole bunch of games---let's say for simplicity
that my opponent plays exactly the same way I do---and I focus my
attention only on the games that my opponent wins. Of the games that
my opponent wins, do you expect that half the time I'll win the opening
roll and half the time my opponent will win the opening roll? I claim
that the answer is no.

For example, let's assume for the sake of argument that winning the
opening roll gives me a 52-to-48 edge. I play 200 games with my
opponent. Say I win the opening roll in 100 games and my opponent
wins the opening roll in 100 games. We expect something like this:

52 games: I win opening roll, I win game
48 games: I win opening roll, opponent wins game
52 games: opponent wins opening roll, opponent wins game
48 games: opponent wins opening roll, I win game

Now limit your attention to the 100 games where the opponent wins
the game. You'll see that in 52 of those games, the opponent wins
the opening roll, and in 48 of those games, I win the opening roll.
Yet the dice are completely fair. The bias arises not from the dice
but from the fact that winning the opening roll gives you an advantage.

Now just repeat this argument with "rolling more doublets" in place
of "winning the opening roll."

---
Tim Chow

[Nasti Chestikov]

On Sunday, 21 March 2021 at 17:14:08 UTC, Tim Chow wrote:
>
> I'm assuming we agree that if I win the opening roll, it gives me a
> slight but definite advantage in the game. Do you agree?
>

>

> ---
> Tim Chow

That's an interesting supposition probably worthy of a debate in itself.

I'll consider your other points and come back with a response.

But to address the first roll advantage.......your first roll cannot be a double whereas mine can.

So you roll 6-2 first up and play 24-18, 13-11. I roll 6-6 first up and play 24-18(2), 13-7(2). You're already on the bar and I have an 8-7-6 prime....who is ahead after one roll?

[Timothy Chow]

On 3/21/2021 1:20 PM, Nasti Chestikov wrote:
> But to address the first roll advantage.......your first roll cannot be a double whereas mine can.
>
> So you roll 6-2 first up and play 24-18, 13-11. I roll 6-6 first up and play 24-18(2), 13-7(2). You're already on the bar and I have an 8-7-6 prime....who is ahead after one roll?

Obviously the the 66 puts that player ahead.

But I think it was widely agreed, long before any bots arrived on the
scene, that winning the opening roll conferred an advantage. The bots
put some numerical value on this advantage, of course, and one can
debate whether to believe these numerical values, but the overall
advantage seems to be big enough that it didn't take too long for human
experts to notice it.

The topic has of course been discussed frequently before, e.g.:

https://groups.google.com/g/rec.games.backgammon/c/Gode2Kzinfg/m/yrYUVSI32icJ

---
Tim Chow

[Axel Reichert]

"badgolferman" <REMOVETHISb...@gmail.com> writes:

> I find it easier to just expect funky dice rolls. That way I won't get
> mad when the opponent receives them and I can laugh when I get them.

That's the right attitude, if "funky" refers to your *perception* of
randomness, not any statistical *quantities*. As I wrote previously,
humans are very poor with randomness, since evolution has formed us into
pattern recognition machines.

Axel

[Peter]

If you look at \S 3.3 in Knuth's AoCP you'll find an interesting discussion.

[Bradley K. Sherman]

Peter <peterxp...@hotmail.com> wrote:
> ...

>If you look at \S 3.3 in Knuth's AoCP you'll find an interesting discussion.

Which is in Volume 2, strangely enough.

--bks

[Nasti Chestikov]

On Sunday, 21 March 2021 at 21:00:59 UTC, Axel Reichert wrote:

> That's the right attitude, if "funky" refers to your *perception* of
> randomness, not any statistical *quantities*. As I wrote previously,
> humans are very poor with randomness, since evolution has formed us into
> pattern recognition machines.
>
> Axel

Hence the reason I'm simply plugging the numbers into an Excel data model and letting the built-in functions CHISQ.TEST and BINOM.DIST tell me the *facts*.

Latest = 5,705 rolls
1-1 = 165
2-2 = 160
3-3 = 159
4-4 = 185
5-5 = 187
6-6 = 182

CHISQ.TEST value = 0.02027
BINOM.DIST value = 0.99897 !! (Apparently you subtract this from 1 and if the result is > 0.05 you run a mile......)

[MK]

Thread title sounded like it would be waste of time to read but
20+ posts in a thread also indicate that it may be interesting.

And, indeed, how interesting to hear Chow say such things... :)

I would like to see him spending more of his time with trying to
answer questions like "How much does rolling doubles help win
a game of backgammon?" instead of wasting it with discussing
positions.

I can't believe he is willing to even ask questions like "How one's
and his opponent's styles of play affect the outcome?" He sure
has come a long way... ;)

Once you acknowledge that there may be other "styles" than the
"bot style" to win at backgammon, it's time for you folks to quit
that endless anal position discussions and get deeper into more
intelligence stimulating discussions like these.

I will try to participate and contribute my humble share...

MK

[Nasti Chestikov]

On Monday, 22 March 2021 at 07:22:46 UTC, MK wrote:
>
> I would like to see him spending more of his time with trying to
> answer questions like "How much does rolling doubles help win
> a game of backgammon?" instead of wasting it with discussing
> positions.
>

> MK

I actually wrote a dice rolling dll for XG that specifically wouldn't roll doubles just to see how I fared against the bot.

I was surprised that there was very little difference in the match outcomes whether I used my own "normal" dice rolling dll or the non-double dll which led me to believe that doubles don't play that big a part in backgammon.

Your mileage may differ, of course.

[MK]

On March 21, 2021 at 9:49:19 AM UTC-6, Nasti Chestikov wrote:

> On 21 March 2021 at 14:29:42 UTC, Tim Chow wrote:

>> See my other comment. If you select only games you have lost
>> and only your opponent's doubles, then I would expect significantly
>> more than 1/6 doubles, even with perfectly fair dice.

> This I don't understand?
> The dice don't know the board position, doubles should still come
> down approximately 1-in-6 rolls regardless of any other factors?

Looking onwards and looking at all games, yes; but looking
backwards and at only the games you have lost, no. Try to
understand what he is explaining.

But why are you wasting your time with this to begin with?

Starting from the FIBS and Jellyfish days of 25 years ago, I
have tried to argue how dice "frequency" and "distribution"
are useless in proving/disproving that a bg server or bot,
cheats or soesn't cheat.

Let's go from primitive to sophisticated. Even though players
can complicate the race by recycling, etc. BG is still a game
of race and doubles, being played 4 times, help win even if
they are not always as useful and can indeed be bad rolls in
some positions. Similarly, there is no difference between big
or small doubles.

With that, if a server or bot is good enough to know that one
side needs a nudge but is not good enough to know how or
by how much, (i.e. temperature map of rolls), it can just give
more doubles as a fail-safe primitive cheating. But even so,
such a server or bot would at least equally intelligent/dumb
to give unusable, worthless doubles to the same player in
order to hide its tracks.

A little more advanced cheater server or bot would give out
hitting numbers but would also balance those similarly.

Current servers and bots are sophisticated enough to know,
(according to their own wisdoms), not only the equities of
all possible rolls at the current position but also for n-ply
positions ahead. Thus, they can give any side any numbers
that may help them to any degree without even worrying
about being detected. There is really no point in clarifying
this any further since it's simply impossible to prove it,
especially not in the case of servers with no static compiled
executabe files distributed.

So, anyway, I just don't understand the point in discussing
this except if there is a money bet (or an aternative method
of inflicting pain) in a practically possible way to determine
who wins and who loses...

MK

[MK]

On March 21, 2021 at 11:37:28 AM UTC-6, Tim Chow wrote:

> On 3/21/2021 1:20 PM, Nasti Chestikov wrote:

>> But to address the first roll advantage.......your first roll
>> cannot be a double whereas mine can.
>> So you roll 6-2 first up and play 24-18, 13-11. I roll 6-6
>> first up and play 24-18(2), 13-7(2). You're already on the
>> bar and I have an 8-7-6 prime....who is ahead after one roll?

You have to put that in statistically significant context.

> Obviously the the 66 puts that player ahead. But I think it was
> widely agreed, long before any bots arrived on the scene, that
> winning the opening roll conferred an advantage.

Incidentally, this is so even is games of no-race, like getting to
play whites. Can we generalize this even wider to include other
games like checkers, tic-tac-toe, hopscotch, etc...?

> The bots put some numerical value on this advantage,

Yes, and only because, (as indicated in the discussions
you linked), bots came to be deemed perfect players
and the difference could only be calculated if both sides
were played by the same identical player.

I am making less of this but notice that it still serves my
argument.

> of course, and one can debate whether to believe these
> numerical values, but the overall advantage seems to be
> big enough that it didn't take too long for human experts
> to notice it.

Well, actually, humans didn't need infinitesimal calculations
to know this. Simple logic suffices.

What bots did was to prove that even the smallest advantages
can add up after 4 billion trials...

I have argued for years that this whatever small advange
gained by winning the opening roll will never be lost for
the rest of the game, no matter what checker play and/or
cube play decision are made, as long as you sample 4
billion or by now better yet 4 tetragazillion tries.

In other words, a bot jacking off playing against itself,
should double after gaining the slightest advantage
and the other bot itself should drop, thus, I had argued
that if there was such a thing as "cube skill" no BG game
would last more than a few rolls!!!

I'm still making the same argument. I see that you folks,
at least Chow, are progressing closer to understanding
it even while discussing seemingly unrelated subjects.

Some time ago I had done and published my experiments
with making the very worst opening or second move
against XG++++++ (depending on who wins the opening)
move. If you didn't make an effort to understand it, go
throw yourself off a bridge. My findings were that even
the worst initial equity lost didn't determine the outcome,
since there was a lot of game left to turn things around.

The cube shortens BG games but my argument holds,
since 4 trillion games will be statistically significant
whether cubeful (premeturely ejaculated) or cubeless
(boringly played out to the end by the sick gamblers).

My purpose in doin that was of course to show that the
cubeful rollouts for the very first moves, let alone the
opening moves, were plain bullshit!

It's just a question of time, and hopefully soon enough
to be in our lifetimes, that Chow will come around full
circle to join me in my arguments and will greatly help
the herds accepts them due to his "unearned credit" in
BG (just for being a math phd).

Come on Chow! Enough with the baby steps for years
past. You are a big boy now. Take bigger steps. Come
to papa...! ;)

MK

[Timothy Chow]

On 3/22/2021 4:45 AM, MK wrote:
> In other words, a bot jacking off playing against itself,
> should double after gaining the slightest advantage
> and the other bot itself should drop, thus, I had argued
> that if there was such a thing as "cube skill" no BG game
> would last more than a few rolls!!!
>
> I'm still making the same argument. I see that you folks,
> at least Chow, are progressing closer to understanding
> it even while discussing seemingly unrelated subjects.

Don't worry, I'm not getting any closer to agreeing with you,
R.B.

---
Tim Chow

[Timothy Chow]

By the way, you never commented on the Hypestgammon results.
There are plenty of examples there where the bot doubles
and takes. What kind of convoluted excuse can you dream up
to explain that away?

---
Tim Chow

[Timothy Chow]

On 3/22/2021 3:22 AM, MK wrote:

> I can't believe he is willing to even ask questions like "How one's
> and his opponent's styles of play affect the outcome?" He sure
> has come a long way... ;)

Your memory is evidently getting worse with age. I haven't said
anything different from what I've always said.

---
Tim Chow

[Timothy Chow]

I'm still not sure exactly what calculation you're doing in Excel,
but it doesn't matter. As I said before, you haven't excluded the
possibility that the only thing these tests are picking up is the fact
that rolling more doubles than your opponent tends to give you an
advantage.

---
Tim Chow

[Nasti Chestikov]

On Monday, 22 March 2021 at 13:28:55 UTC, Tim Chow wrote:

> I'm still not sure exactly what calculation you're doing in Excel,
> but it doesn't matter. As I said before, you haven't excluded the
> possibility that the only thing these tests are picking up is the fact
> that rolling more doubles than your opponent tends to give you an
> advantage.
>
> ---
> Tim Chow

But to what order?

The latest numbers (+6000 rolls) I have show 5-5 running at +16% occurrences than expected, 6-6 running at +15%.

Those numbers are suspect to say the least.

Even if you assume and accept a 10% increase across the board, the numbers are nowhere near what I'm seeing.

So, 6000 rolls you expect 1000 doubles; if you accept 10% then it's 1100 doubles.

My findings on real data are considerably in excess of that (as observed by the CHISQ.TEST and BINOM.DIST values).

[Axel Reichert]

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

> I would like to see him spending more of his time with trying to
> answer questions like "How much does rolling doubles help win
> a game of backgammon?" instead of wasting it with discussing
> positions.

While

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

is an interesting read, it would be much more rewarding to any dice
"sceptic" to not spend his time on obsessing with things not under his
control (dice rolls). The play of these rolls is under his control, so
at least theoretically there is something to be gained: Understanding of
the backgammon, which is a game of luck management.

The errors in play and the whining about luck, "rigged dice"
etc. correlate positively and very strongly. Likewise, the more people
lament on the internet that a particular program cheats, the more likely
it is that it plays really well. As a heuristic, of course, not as a
causal relation.

The time needed to learn enough about the sometimes quite ambitious
statistical concepts could be (probably better) invested, say, into
learning the reply rolls and get race doubles correct.

Axel

[Timothy Chow]

On 3/22/2021 11:34 AM, Nasti Chestikov wrote:

> But to what order?

There's a simple test you can do to partially answer this
question. Repeat your calculations, but now restrict to
games that *you* won, and tabulate how many doubles *you*
rolled in those games.

---
Tim Chow

[peps...@gmail.com]

Yes, that's his point. The dice are biased so that the winner gets more doubles.

Another interesting thought experiment is what would happen if the rules of backgammon were changed
to make it a game of pure luck -- the first player to roll a double wins. Then the proportion of doubles in
won games would be hugely more than 1/6. Indeed any game where the winner had 1/6 doubles or fewer would
correspond to a sequence of 11 non-doubles in a row which is quite a small parlay.
The problem is that not everyone listens to or thinks about the thought experiments.

Paul

[peps...@gmail.com]

On Tuesday, March 23, 2021 at 3:42:17 AM UTC, Tim Chow wrote:

Sorry, I misread the thread somewhat. More progress was made in communication
than I thought. Everyone agrees that sampling only the games of the winner gives
a positive bias to the doubles proportion, assuming fair dice.
The hypothesis is then made that this positive bias is too small to account for the
actual doubles proportion observed.

However, this too-small-bias hypothesis just appears out of thin air, with no evidence
or even intuition to support it.

Paul

[Nasti Chestikov]

On Tuesday, 23 March 2021 at 07:56:50 UTC, peps...@gmail.com wrote:
>
> Sorry, I misread the thread somewhat. More progress was made in communication
> than I thought. Everyone agrees that sampling only the games of the winner gives
> a positive bias to the doubles proportion, assuming fair dice.
> The hypothesis is then made that this positive bias is too small to account for the
> actual doubles proportion observed.
>
> However, this too-small-bias hypothesis just appears out of thin air, with no evidence
> or even intuition to support it.
>
> Paul

I'd be interested in your thoughts (or anyone's for that matter) as to what you'd perceive as a fair positive bias?

My latest numbers:

289 games analysed
7,056 total rolls
1-1 198 actual, 196 expected (+1.02%)
2-2 200 actual, 196 expected (+2.04%)
3-3 192 actual, 196 expected (-2.04%)

now this is where it gets silly

4-4 221 actual, 196 expected (+12.76%)
5-5 226 actual, 196 expected (+15.31%)
6-6 225 actual, 196 expected (+14.80%)

Would you expect the winner of 289 backgammon games to be enjoying those "lucky" rolls? (Specifically 5-5 and 6-6).

It's interesting that 1-1, 2-2 and 3-3 are in line with expectations, you'd expect, surely, a uniform distribution of excess doubles across all six doubles?

[Timothy Chow]

On 3/23/2021 4:19 AM, Nasti Chestikov wrote:
> It's interesting that 1-1, 2-2 and 3-3 are in line with expectations, you'd expect, surely, a uniform distribution of excess doubles across all six doubles?

Not really. Backgammon is fundamentally a race, so we'd
expect big doubles to be more helpful than small doubles
when it comes to winning a game.

Again, I recommend that you do the experiment I suggested.
Investigate the hypothesis that there is cheating in your
favor because you're getting inordinately many doubles
when *you* win.

---
Tim Chow

[Nasti Chestikov]

On Tuesday, 23 March 2021 at 13:24:50 UTC, Tim Chow wrote:

> Not really. Backgammon is fundamentally a race, so we'd
> expect big doubles to be more helpful than small doubles
> when it comes to winning a game.
>
> Again, I recommend that you do the experiment I suggested.
> Investigate the hypothesis that there is cheating in your
> favor because you're getting inordinately many doubles
> when *you* win.
>
> ---
> Tim Chow

Yep, I'm onto that although it's quite a painstaking exercise, Zooescape has no facility (that I'm aware of) to export games.

[peps...@gmail.com]

On Tuesday, March 23, 2021 at 1:24:50 PM UTC, Tim Chow wrote:
> On 3/23/2021 4:19 AM, Nasti Chestikov wrote:
> > It's interesting that 1-1, 2-2 and 3-3 are in line with expectations, you'd expect, surely, a uniform distribution of excess doubles across all six doubles?
> Not really. Backgammon is fundamentally a race, so we'd
> expect big doubles to be more helpful than small doubles
> when it comes to winning a game.

That's not what I'd expect. I'd expect 11 to be the best roll on the whole.
Whenever there's significant contact, it works beautifully to fill in points or to creep to the edge of a prime.
If you're struggling to enter from the bar, it usually works well, because the ace point is less likely to be made than other points.
Even in a race, it often works well to remove four checkers on the ace point.
If contact has only just broken and you're bearing in, it ain't the best chicken I've eaten.
But that's not a huge percentage of positions.
However, if it's really not such a good roll after all, then I accept that data finding, but it is a bit surprising (to me).

Paul

[Nasti Chestikov]

On Tuesday, 23 March 2021 at 13:48:35 UTC, peps...@gmail.com wrote:
>
> That's not what I'd expect. I'd expect 11 to be the best roll on the whole.
> Whenever there's significant contact, it works beautifully to fill in points or to creep to the edge of a prime.
> If you're struggling to enter from the bar, it usually works well, because the ace point is less likely to be made than other points.
> Even in a race, it often works well to remove four checkers on the ace point.
> If contact has only just broken and you're bearing in, it ain't the best chicken I've eaten.
> But that's not a huge percentage of positions.
> However, if it's really not such a good roll after all, then I accept that data finding, but it is a bit surprising (to me).
>
> Paul

So I intend analysing a lot more rolls than this but, for starters, (and based on Tim's earlier post that he believes 1,000 rolls is a large enough sample size for a chi-square test) here are my figures so far:

Opponent rolls (only in matches where the opponent has won) : 7,648 (312 games)
1-1 : expected 212.44, actual 219 (+3.09%)
2-2: expected 212.44, actual 216 (+1.67%)
3-3: expected 212.44, actual 201 (-5.39%)
4-4: expected 212.44, actual 232 (+9.21%)
5-5: expected 212.44, actual 244 (+14.85%)
6-6: expected 212.44, actual 242 (+13.91%)

Chi-Squared value via Excel 2010: 0.0427

My rolls (only in matches where I have won) : 1,001 (40 games)
1-1 : expected 27.81, actual 29 (+4.30%)
2-2: expected 27.81, actual 31 (+11.49%)
3-3: expected 27.81, actual 27 (-2.90%)
4-4: expected 27.81, actual 26 (-6.49%)
5-5: expected 27.81, actual 26 (-6.49%)
6-6: expected 27.81, actual 30 (+7.89%)

Chi-Squared value via Excel 2010: 0.97378

So simply validating what my own eyes have told me.

Any thoughts anyone?

[Axel Reichert]

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

> Backgammon is fundamentally a race, so we'd expect big doubles to be
> more helpful than small doubles when it comes to winning a game.

Indeed, see here for a quantification of the average luck per roll:

https://www.bkgm.com/rgb/rgb.cgi?view+1122

66, 55, 44 are the luckiest rolls.

Best regards

Axel

[Timothy Chow]

Another article along the same lines:

https://bkgm.com/rgb/rgb.cgi?view+1576

---
Tim Chow

[Timothy Chow]

On 3/23/2021 10:08 AM, Nasti Chestikov wrote:

> Opponent rolls (only in matches where the opponent has won) : 7,648 (312 games)
> 1-1 : expected 212.44, actual 219 (+3.09%)
> 2-2: expected 212.44, actual 216 (+1.67%)
> 3-3: expected 212.44, actual 201 (-5.39%)
> 4-4: expected 212.44, actual 232 (+9.21%)
> 5-5: expected 212.44, actual 244 (+14.85%)
> 6-6: expected 212.44, actual 242 (+13.91%)
>
> Chi-Squared value via Excel 2010: 0.0427
>
> My rolls (only in matches where I have won) : 1,001 (40 games)
> 1-1 : expected 27.81, actual 29 (+4.30%)
> 2-2: expected 27.81, actual 31 (+11.49%)
> 3-3: expected 27.81, actual 27 (-2.90%)
> 4-4: expected 27.81, actual 26 (-6.49%)
> 5-5: expected 27.81, actual 26 (-6.49%)
> 6-6: expected 27.81, actual 30 (+7.89%)
>
> Chi-Squared value via Excel 2010: 0.97378
>
> So simply validating what my own eyes have told me.
>
> Any thoughts anyone?

These results surprise me slightly (although I still don't know
what exact calculation you're doing to get your "chi-squared
value") but there are still two confounding issues.

1. The test was done retrospectively, which violates the
cardinal rule of statistics. That is, you noticed your opponent
rolling a lot of doubles, and then retrospectively calculated
the probability. That by itself is going to bias your results.

2. You and your opponent don't play the same way. So even if
you were given the same dice rolls, your winning probabilities
would differ.

To round out the picture, there are two more natural statistics
you could compute---how many doubles do you (respectively, your
opponent) roll when you (respectively, your opponent) lose? This
would help mitigate the effect of #2 above.

---
Tim Chow

[Nasti Chestikov]

On Wednesday, 24 March 2021 at 02:41:54 UTC, Tim Chow wrote:

> These results surprise me slightly (although I still don't know
> what exact calculation you're doing to get your "chi-squared
> value") but there are still two confounding issues.

It's a function in Excel; from Microsoft's site:

"Returns the test for independence. CHISQ.TEST returns the value from the chi-squared (χ2) distribution for the statistic and the appropriate degrees of freedom. You can use χ2 tests to determine whether hypothesized results are verified by an experiment.

Syntax

CHISQ.TEST(actual_range,expected_range)

The CHISQ.TEST function syntax has the following arguments:
Actual_range Required. The range of data that contains observations to test against expected values.
Expected_range Required. The range of data that contains the ratio of the product of row totals and column totals to the grand total."

>
> 1. The test was done retrospectively, which violates the
> cardinal rule of statistics. That is, you noticed your opponent
> rolling a lot of doubles, and then retrospectively calculated
> the probability. That by itself is going to bias your results.
>

> ---
> Tim Chow

I don't understand this bit, it's always as if you're in denial that any website / program could possibly be found out to be cheating! :-)

Take global warming (ha ha, I know it's a joke but humour me) - 5,000 years ago, or whatever, they didn't start taking measurements which have carried on until today to test whether it was happening (clue - it isn't).

They've taken data from now and worked backwards.

And that seems to be accepted as an ok practice. Why is me taking data from today and working backwards by plugging numbers into an Excel model any different? If the dice are biased, they're biased surely?

[Timothy Chow]

On 3/24/2021 4:29 AM, Nasti Chestikov wrote:
> On Wednesday, 24 March 2021 at 02:41:54 UTC, Tim Chow wrote:
>
>> These results surprise me slightly (although I still don't know
>> what exact calculation you're doing to get your "chi-squared
>> value") but there are still two confounding issues.
>
> It's a function in Excel; from Microsoft's site:

Right, but you haven't reported your full "actual_range" and
"expected_range" numbers. You've reported only the counts
for the doubles. So I'm unable to reproduce your calculation.

>> 1. The test was done retrospectively, which violates the
>> cardinal rule of statistics. That is, you noticed your opponent
>> rolling a lot of doubles, and then retrospectively calculated
>> the probability. That by itself is going to bias your results.
>

> I don't understand this bit, it's always as if you're in denial that any website / program could possibly be found out to be cheating! :-)

Funny, you understood it a few days ago, when you said,
"Point taken; however, this is a retrospective exercise."

---
Tim Chow

[Nasti Chestikov]

On Wednesday, 24 March 2021 at 13:20:09 UTC, Tim Chow wrote:
>
> Funny, you understood it a few days ago, when you said,
> "Point taken; however, this is a retrospective exercise."
>
> ---
> Tim Chow

Behave Lamborghini boy, you know exactly what I'm getting at here.

[Axel Reichert]

Nasti Chestikov <nasti.c...@gmail.com> writes:

> You can use χ2 tests to determine whether hypothesized results are
> verified by an experiment.

See? You need a hypothesis before the fact, period. This has nothing to
do with prejudices/convictions, this is simply statistics 101. There are
tons of textbooks out there. Do your homework instead of hinting
vaguely.

See

https://bkgm.com/rgb/rgb.cgi?view+1586

for the general overview and

https://bkgm.com/rgb/rgb.cgi?view+259

for a particularly nice example of some "Cheating!" suspicion, which
could be debunked because someone was using his brain. And after that,
spend time on learning to play better instead of wasting time with
analyzing dice. bkgm.com offers plenty of material worth reading.

Axel

[Timothy Chow]

I do?

---
Tim Chow