speachless: Please always give the move number in such cases, so that the situation in question can be easily found later, when the game has been played on.

Hrqls: You are of course right that statistics is required, but it is a) considerable work, and I don't want to bother if fencer seemingly is not interested anyway, and b) it is not always possible. One can't tell afterwards in which order the dice in different games have been rolled. So there is no way but recording during play. I once tried to create a tally sheet counting the equal and the different rolls when going from one BG game to the next, but I quickly lost track ... and see point a) above.

Ananju mentioned there is a way to reproduce the opponent's roll with higher probability than normal. This might be proven by analysing many old games, but if only few players know about this method and use it only in few situations, the statistical effect is very small and difficult to detect.

It seems we just have to accept the questionable dice rolling here. We may play here for fun, but for serious play one should go elsewhere.

moistfinger: I have played on Dailygammon for some time, and didn't notice any strange or unnatural behaviour of the dice. Also, they have documented their dice rolling, and this method, if implemented correctly (bugs are always possible of course) should give fair and statistically correct rolls.

moistfinger: The first reason is: there seemingly is still the bug causing to repeat the same dice roll with too high probability, when you go from one Backgammon game directly to another (with "submit and go to next game of this type").

Second reason: many players have the impression that the dice "just feel wrong" on this site. Me too. (That can be mistaken, of course; it is difficult to tell without statistical analysis.)

Third reason: there have been other bugs in the past, concerning dice rolling. At least one was proven by statistics: the first and second roll in a game have too often been equal. This bug is removed, however, but it shows that the implementation of the dice was erroneous from the beginning, and it still is not ok yet (see first point above). That thing seems overly complicated or inherently bad and therefore difficult to fix.

Fourth and most important reason: the random number generation is not documented, so that the players don't know what they are using. It is as if someone brought a bag full of self-made dice to the Backgammon world championship and claims these are better than any other existing dice (or at least not worse), and the tournament director decides to use them without testing. I think you agree that this would be a bad idea.
It is possible (and not very difficult) to create a cryptographically secure random number generator to roll the dice, that uses for every game a secret key which can be published after the game is over, and the generator code can also be public, so the players know the mechanism and can test it for it's general quality and as well for it's correct implementation, because once the key is known, the dice rolls can be re-calculated with own implementation of the generator. The rolls are however not predictable as long as the key is secret.
Such a method would IMO be required for serious online play of games with random.

Aganju: You are right about the match length. There is a reason why serious BG tournaments go for, typically, 21 points.

But the dice rolling of BK is questionable, to say the least. Serious play of Backgammon - or any other game with dice - is not possible here.

Pedro Martínez: This bug is repaired, but not the following: if one submits a move in backgammon and goes directly to the next backgammon game, the probability to get the same roll again as in the previous game seems to be higher than it should be.

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 %)

pedestrian: This is well possible, I must only download more files. I'm working on it...

I'm also going to do a chi-square test, which calculates a probability for an event to occur by coincidence, under the assumption that the dice are ok (equally distributed and independent).

TC: I also had the impression that equality of first and second roll occurs much too often.

(For this topic, see the posts starting 3. Nov. 2009)

To check this, I downloaded the MAT files of all Backgammon games I have played so far, and wrote a programm which counts the games where the first rolls of both players are equal. Result:

206 games total
79 of them have equal first and second rolls (38.3 %)

This is 6.9 times the expected value (5.556 %).

At the beginning of every game, the initial position is determined randomly. The pieces are set up symmetric like in usual backgammon, i.e. if white has n pieces on point x, then black has n pieces on point 25-x.

Besides from this, there can be various detailed rules to set up the pieces. The following is an example:

There is no blot (single stone on a point). every player has 15 pieces like in normal Backgammon. Every player has at least 2 stones in every quarter of the board. The rearmost stones of a player are not passed by any opponent's stones (i.e. if the white rearmost stones are on point 4, then there are no black on points 1 to 3).

All other rules are the same as in standard Backgammon.