I've performed a Chi-squared test on the hypothesis that the probabilities of responder's first roll having 0, 1 and 2 dice the same as opener's roll are as they should be, that is 16/36, 18/36 and 2/36 respectively.

This is a test with 2 degrees of freedom, so the chi-squared statistic has:
 a 5% chance of exceeding 6.0 if the probabilities are correct,
 a 1% chance of exceeding 9.2
 a 0.5% chance of exceeding 10.6, and
 a 0.1% chance of exceeding 13.8

alanback's result (from 55 games) is 9.2.  A result this high or greater would happen only 1% of the time, so this is enough to cause suspicion that the dice aren't following the desired probabilities.  But it's not proof.  Also, this test is reckoned to give very accurate results only if the expected outcomes are all greater than 5.  So, with our smallest probability being 2/36, this means we need at least 90 games in our sample for me to be happy beyond reasonable doubt about the conclusion.

So, moving on to my results (100 games), I get a statistic of 102.

And lastly, wetware's results (137 games) give a statistic of 139.

Remember, if the dice rolls are working properly, there's only a 1 in a 1000 chance of this chi-squared statistic being 13.8 or higher in any individual test, so the conclusion can't really be in doubt - something is wrong somewhere.

playBunny: I think we all know the answer to that.

Just out of curiosity I looked at the 55 games in matches I have completed in 2010. There are 8 in which the first two rolls were the same (same two dice, order not considered), versus the predicted 3 and change. Both dice different, predicted 24, actual 19.

http://brainking.com/en/Board?bc=26&plla=1050529 - 1st May 2010 with reference to November 2009 and as early as 2008.

The crucial question, I believe, is which standard deviation will be the greater - how far from chance the return opening rolls are or how far Fencer is from caring!

Resher:  Thanks so much for the SD calculations!  I knew that was critical for us to express just how extreme these results are, but I've forgotten too much of my statistics coursework and tools.

FYI: my next data set (matches from 2008) will be ~10 times the size of the results I reported from 2009.

wetware: Just finished looking through my first 100 BG games from 2009 and:

Number of games where responder's first roll matches on both dice = 27 (expected number = 6, so more than 9 sd's away)

Number of games where responder's first roll differs on both dice = 16 (expected number = 44, so more than 5 sd's away)

Resher and Alan:

Thanks, guys!  I am now planning to conduct a similar review of my 124 matches from 2008.  Expected completion: some time this weekend.

Alan: I didn't care about the order of the dice for these purposes.  (I.e., I considered 52 and 25 to be a match.)

Another oddity noticed during review, but not yet reported:  the "misses" show a strong tendency to be "near-misses".  For example, if an opening roll of 42 is not exactly matched by responder's roll, the responder's roll will show excessively high occurrences of 52, 43, 41, or 32.  That is to say: even when you do manage to miss, you're too often "off by 1".

The most extreme outlier was a match with Hrqls (Backgammon (Hrqls vs. wetware) ), in which 9 games out of 14 were exact matches, and the remaining 5 non-matches were all of the "off by 1" variety.  Note: I'm inclined to consider 1 to be "next" to 6.  I think it's reasonable, especially if some "remainder" function is at play in the dice generation routine(s), as is often the case.

Orlandu: There is even a link to the perl script that is used, on the site's Help page:

http://www.dailygammon.com/help/#dice


Now THAT is transparency!

Orlandu: Yes. Dailygammon.com uses random numbers from random.com

alanback: random.org has a great dice roll system...

Resher: Too bad, as it would probably be more meaningful to measure only the same order case. Still, based on your analysis, there does seem to be something odd going on here!

alanback: I've just tested Wetware's data against a Binomial distribution with n=137, p=1/18.

I think getting 38 double matches is more than 11 standard deviations away from the expected number!

#sd's = [38-np]/sqrt[np(1-p)] = 11.33

alanback: I don't think we can distinguish between the two situations (ie same order match v different order match) as the game records the order the dice were played rather than the order they were rolled.

I've also downloaded my 2009 BG games and am working on the same stats as Wetware has produced, to increase the sample size.

PS maybe I don't understand what you are tracking here. Can you describe exactly what you are looking for? As I understand it, you are comparing the two dice of the first roll of the game, i.e. of the first player to move, with the two dice on the next roll, i.e. the first roll of the second player to move.

Are you counting situations in which the same dice occur, but in a different order, as a match or non-match?

The probability of the same two dice occurring in the same order on the second roll is 1/36. However, the probability of the same two dice occurring in any order is 2/6 x 1/6 = 1/18, as you observe.

wetware: Good work. There certainly is a suspicious pattern. Your sample size is still quite small, however. What is needed now is a statistical analysis of the probability that this distribution could occur randomly. I.e., is it within a couple of standard deviations of the norm, or is it a one in a billion chance?

[cross-posting from the "Feature requests" board, where a discussion developed recently]

I wish the system made it easier to gather the data that would demonstrate the severity of this problem.  As I've said elsewhere (I think), it could also help pinpoint when the problem began (or was it always this way?), which should help identify what changed to cause it.  I also wish the problem occurred 100% of the time, which would make it impossible to dismiss.  As it is, we have only the beginnings of statistical support for our claim, and gathering the aggregate data that would provide more solid support is a daunting task...for us the members.

Fencer is quite correct about the groundless wailing over poker or dice cheats.  There's plenty of that online; such complainers are easy to find, and the vast majority deserve our scorn and satire.  But the genuine exceptions (in online poker especially) that have come to light should give one pause.

Hats off to those who stood their ground in such cases, labored to gather the data, and revealed the truth at last.

(added comment Wednesday evening / Thursday morning):  I plan to make one last good-faith effort to present some useful data.  I plan to examine all of my backgammon (but no variants) games stored here from 2009--that's 13 matches, and a total of 137 games in which at least 2 rolls took place.  I have no reason to believe that that year is appreciably better or worse than any other year of mine.  I'm sorely tempted to separately track the opening rolls that I "won" and "lost", but I don't plan to do so for this exercise.  If I'm understanding the situation correctly (and there are good "numbers" people here who will be able to correct me if I'm wrong), the responder's roll should theoretically match the opener's roll 1 in 18 times (or 5.555_ %), on average.  I'm also going to report the percentage of responder's rolls where both dice differ from the opener's roll.  Theoretically, I think that should happen 4 times in 9 (or 44.444_%), on average.  But I think the actual observed value is going to shock and convince even the most diehard skeptics here.


Average expectation of opener's and responder's both dice exactly matching out of 137 played = 7.6111_ games.  Observed number=38 games (27.737%)

That's about 5 times the expected frequency!

Average expectation of responder's dice both differing from opener's dice out of 137 played = 60.888_ games.  Observed number=28 games (20.438%)

That's less than half of what's expected.

If I feel energetic, I'll analyze my 124 matches from 2008! 

A monthly series of fast-paced Hypergammon tourneys.


I like it fast, 5-point Hyper, 1st August 2010

I like it fast, 5-point Hyper, 1st Sep 2010

A monthly series of fast-paced Hypergammon tourneys.


I like it fast, 5-point Hyper, 1st July 2010

I like it fast, 5-point Hyper, 1st August 2010

Quick movers Random tourney.

Random game tourney with 2 backgammon variants

nabla:  I almost feel dirty playing an opening 21$ here.  It's like stealing equity.  (I normally play it anyway, but here it's a gift.)

Nothingness: I sure hope you're using autopass! ;-)

AlliumCepa: I don't think it's a "bad" for anyone, lol. In any case, I've changed my home page link to log me in without using the web subdoimain.

AlliumCepa: I am not sure if the "web" part should be there at all.

That's what I have in my address bar and it works for me, so it's legitimate. I just took the web. bit out and that worked too, except that I wasn't logged in. I'm not sure when or why Fencer created the web subdomain but, whenever it was, I updated my home page, which has links to login me in directly. It's been like that ever since.

i have passed 32 turns in a row

AlliumCepa: Thanks for that. Interesting, it's the right page but you're not logged in!

It should be okay from now on. I've been doing each month by editing the previous month's post and copying it into a new one. For some reason that gives me a fully-specified url (http://web.brainking.com/en/Tourneys/...) instead of the relative one (Tourneys/...). I'll make sure that I delete the unwanted portion in future.

what is the record for the most amount of consecutive passes?

A monthly series of fast-paced Hypergammon tourneys.


I like it fast, 5-point Hyper, 1st June 2010

I like it fast, 5-point Hyper, 1st July 2010

AlliumCepa: Thanks Ivaylo.

Where does the link with "web" take you?

still going after nearly six years wow

Neznalek's private tournament in Gammon Race #12

AbigailII:His opponents do not have infinite Vacation Days,so his Days on the bar can be much shorter then anticipated.

AbigailII: Life's too short to drink bad wine or play cloning backgammon!

AbigailII: I think you should get achievement points for that!    :-)

wetware: Pip count 521 - 340.

Nothingness:I think that 503 was my highest pip count ever:  Cloning Backgammon (Peter 63 vs. wetware)

Nothingness: they way it seems to be worded i could feesably end up with an infinite amount of pieces to bear of my bar.

No, that would require an infinite amount of moves. But you're always complaining your opponents move only a few times a month, so while the number of pieces on your bar may grow, you don't live old enough to get an infinite amount. Now in theory, the maximum number of pieces on the bar is unbounded.

Nothingness: I think you'll find the stratergy is that if you hit your oppnent then you will gain an extra man, therefore you have to way up the concequences of taking men off.

could anyone explain cloning backgammon for me? the rules are a bit confusing. they way it seems to be worded i could feesably end up with an infinite amount of pieces to bear of my bar. if i keep catpturing my enemys pieces i have to bear off more and more pieces.

Pedro Martínez:
LOL

alanback:
Point taken.

moistfinger: Your credibility is already suspect when you say you never whined before -- all real backgammon players whine about the dice ;-)

A chi-square test is almost pointless; obviously this is almost impossible to happen just by coincidence, if the probability of equal start rolls is 1/18 = 5.556 %. But I have done it anyway.

Every one of the n = 630 games falls in one of two classes:
class 1: first and second roll equal (expected probability p1 = 1/18)
class 2: first and second roll not equal (p2 = 17/18)

Y1 = 218 is the number of games in class 1.
Y2 = 412 the games in class 2.

now is calculated:
V = (Y1 - n * p1)^2 / (n * p1) + (Y2 - n * p2)^2 / (n * p2)
= (218 - 35)^2 / 35 + (412 - 595)^2 / 595
= 1013.1

If the dice were not biased (probabilities p1 and p2 as expected), then the value V would be lower than 6.635 with probability 99 %, and lower than 10.84 with probability 99.9 %.

(edit:) see this table for the values to compare V with. With 2 classes one must look in the line with DF=1 (degrees of freedom).

Meanwhile I have downloaded more games:

630 games total
218 games with equal starting rolls (34.6 %)


All games in the sample are played between Summer 2009 and today.

Games in 2009:

306 total
99 with equal start rolls (32.4 %)

Games in 2010:

324 total
119 with equal start rolls (36.7 %)

Pedro Martínez: Great machine, and here was i thinking they paid 1000 people to sit at their cubes and roll dice!

If there is a defect it would seem not to be the random number generator but that the code doesn't use the generator in the particular case mentioned, but instead just gets the last number... sounds like a caching problem?
I wonder if those games were played with 2 people on the same computer? moves happening very quickly? We need more of this type of data most likely, in order to replicate the problem.

moistfinger: However, it is quite surprising because BK uses a very sophisticated system to generate the dice. :)
http://www.youtube.com/watch?v=7n8LNxGbZbs

Pedro Martínez: Well, that explains my searches not finding it. :)
It was closed in 2008... i'd say it should be re-opened and investigated. I'm sure anyone with access to the historical database could publish the statistics for a start.

wetware: The dice is a total joke on this website. It just doesn't feel right. I have never been one to whinge about backgammon dice before playing on Brainking. I know it is supposed to be random but it just isn't, it is skewed.