I think the most helpful posts about the non-random rolls (including stats) were posted on this DB in July 2010. For example see:
http://brainking.com/en/Board?bc=26&bscx=1066337#1066337


The particular issue discussed there may have been fixed by now - does anyone know?

Re: taking advantage - I think I remember Nabla posting a set of opening moves that took advantage of the unusual opening roll distribution, but again - this might no longer apply.

pgt: I think your initial post mentioned the wrong player, so might confuse people if it were still visible.  I think you posted the name of the opponent in those 15 games you mentioned.

Aganju: Can you choose to offer a draw?  If not, then you could PM Fencer, who should be able to end the game as a draw.

skipinnz: Thanks, good to know

skipinnz: Oh, have the rules changed? I don't remember that caveat being there when the game was first introduced here. Thanks for pointing it out.

wetware: Your expected numbers of the different types of responder roll look right to me.  Using them, I get a chi-squared statistic of 274, when I'd  expect a figure of 13.8 or above for only 1 in a 1000 samples (years in this case) if the dice rolls were totally independent and generated fairly.  So we're talking odds of many, many millions to one against this being the case.

I think by now that most of us are agreed on this being caused by non-independence of the opening rolls rather than non-fair "dice" being used.  But data and stats are fascinating, so feel free to produce more!

Hopefully some of your analysis will give someone some insight as to when at least part of the opener's roll is used as part of the responder's roll too.  My guess would be that the actual rolling is simulated and so re-rolls will be generated if both players are assigned the same first roll, and it's this re-rolling that isn't working properly.  I think pB's already suggested this.  Hard to test though ....

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.

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)

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.

playBunny: The reported order of the numbers within a roll isn't relevant is it?  Doesn't this just depend on whether a player swapped the numbers before using them?

I also checked my most recent two 21-pointers and found 6 out of 17 same initial rolls for both matches, so 35% compared with an expected percentage of less than 6% if they were independent.  I think this is statistically significant and wonder whether there is some circumstance where the first two dice rolls are determined at the same time (and so would always be the same), rather than when each player clicks on the game at different times?

TC: I, too, have noticed that the first two rolls are the same much more often than I would expect.  And I don't think it's just me remembering these occasions that is distorting my perception.  Any other players noticed this?

spirit_66: Some games don't include any difficult decisions to make and so could easily show up no fault.  But I agree if it happens on more complex games, then it's either suspicious or you're dealing with a very good player.  I would expect most games against nabla to show no fault on his part, but only occasionally for lesser players like myself.

Can you post a link to a game or two you're suspicious of so we can take a closer look. 

I don't think you should be suspicious of aaru - I really don't believe he's the cheating sort (based on my experience, I enjoy playing him) and anyway, he has far too many games on the go, to have time for it!