Statistically speaking..
Dodd
I am sure that this question has been asked many times before, but since I
just found this newsgroup I would like to ask it again.
Statistically, how many games of backgammon must two people play before one
can definitively say they are the better player?
Thanks for your anticipated responses.
Best regards,
Dodd
--
Patrick Dodd
Founding Partner, BigSmile, LLC
"Internet Solutions for Dental Professionals."
do...@bigsmile.com
http://www.bigsmile.com/dentist/
602.777.0178
Greycat Sharpclaw
Meow, all...
There is an allegation that do...@bigsmile.com (Dodd) wrote:
>I am sure that this question has been asked many times before, but since I
>just found this newsgroup I would like to ask it again.
>Statistically, how many games of backgammon must two people play before one
>can definitively say they are the better player?
Before I try a numerical answer, let me make 3 points.
1st point:
That would depend on how much "better" the better player is.
If the "better" wins 51% of the points (good for 2% average profit in
point-competition, enough to tell in the long haul) it will take a
long time for their record to be statistically significant.
But if the better player is enough better to win 60% of the time, the
difference will become obvious much quicker.
2nd point:
Are you depending on the score to tell you statistically? Or will you
accept the opinion of the better player when he says "I can tell I'm
better, he's making too many mistakes". I've gotten to this point in
a 9-point match against a few players, and once I was sure of it in
the first game. And I am not very good (currently arround 1430 on
FIBS); an expert might be able to tell faster.
3rd point:
Assuming you are depending on the score and statistics, are you
playing for games, points (i.e. with doubling, etc.), or match? Each
has it's own statistics.
---
OK... now for a crack at the answer. I skip the math for brevity,
I'll send it E-Mail if anyone wants to see it (it's a bit long).
Assumptions used are at end.
I estimate for 3-sigma confidence:
better player's average luck: 3 sigma: result
percentage of matches to get will be available
wins in long run the result within:
51% 22,500 45,000
55% 900 1,800
60% 225 450
.5+d (1.5/d)^2 (2.12/d)^2
This is to say that it takes about twice as long before we are
reasonably sure to have statistically significant result, then it
takes on the average. This is because if the better player has bad
luck, we got to wait for the luck to average out enough for his skill
to become evident.
Of course, this is not _absolute_ certainty. 3-sigma is a common
stabdard for reliable results. 2-sigma would give reasonable results
(right 98% of the time) in 44% of the time. 6-sigma, in 4 times as
long, will give us one-error-per-million-experiements level of
confidence.
I have assumed match play, with only the winner of the match being
recorded... forget the score. If you want to use 1-game matches,
fine, but you are eliminating gammons and the cube, which effects the
skill factor.
And I insist on a statistically significant score result only (i.e. no
opinion on how the games are played, just the numbers please).
I assumed statistical independence of results... forget someone having
a good or bad say mentally, and going on a streak that is other than
random luck.
Oh yeah, and I used a bunch of pretty-good-but-not-exact
approximations, in my math. So the answers are not good to more than
about 2 digits accuracy.
----
Greycat
Gre...@tribeca.ios.com
Does anyone have any spare tunafish??
Stephen Turner
Dodd wrote:
>
> I am sure that this question has been asked many times before, but since I
> just found this newsgroup I would like to ask it again.
>
> Statistically, how many games of backgammon must two people play before one
> can definitively say they are the better player?
>
Unfortunately, it's not quite as simple as that. It depends how far you are
apart after those games. The standard way to set it up a a statistical
problem is to play a pre-determined number of 1-point games -- let's say 100.
If you are the same standard, there's only a 2.5% chance that you will win
at least 60 of them, which is enough to reckon that you are probably the
better player.
Three further comments though:
1) If you are only a tiny bit better, this test is unlikely to pick up the
difference. In that case, we say that the test isn't very powerful. Conversely,
if you are much better, the test will usually pick up the difference, and it's
said to be a powerful test.
2) There's a thing called a "sequential test" whereby you don't have to play
the full 100 games if someone is way ahead after (say) 50 of them. This is
often used in medical experiments, for obvious ethical reasons. But it's
rather complicated to work out what the criterion for stopping and deciding
should be with a sequential test.
3) I assumed 1-point games. The real skill is in the cube, and I've also
neglected gammons. Unfortunately, it's not possible to work out an exact
statistical test for money games without knowing what proportion of each type
of score you would normally get between two equal players. For match play,
however, you're OK: if you play 100 5-point matches, instead of 100 1-point
games, and just worry about the winner not the final score, everything else
works the same way.
That's probably more than you wanted to know. But I wouldn't want to mislead
you with an incomplete answer. :)
--
Stephen Turner sr...@cam.ac.uk http://www.statslab.cam.ac.uk/~sret1/
Statistical Laboratory, 16 Mill Lane, Cambridge, CB2 1SB, England
"This store will remain open during modernisation. We apologise
for any inconvenience this may cause" Topshop, Cambridge
Daniel Murphy
do...@bigsmile.com (Dodd) writes:
>Statistically, how many games of backgammon must two people play before one
>can definitively say they are the better player?
A timely question, because I just learned in another
pseudo-philostatistical thread that the answer is 6, 11, 36, or O or 1
(but not both -- or maybe it *was* both), or "what do you mean by better?"
--
_______________________________________________________
Daniel Murphy | San Francisco | rac...@cityraccoon.com
Monthly tourneys in San Mateo: See www.gammon.com/bgbb/ for details
and some excellently annotated matches. On-line: telnet fibs.com 4321.
Greycat Sharpclaw
There is an allegation that Stephen Turner <sr...@cam.ac.uk> wrote:
>Dodd wrote:
Some basically good points, including one I wish to comment on...
>3) I assumed 1-point games. The real skill is in the cube, and I've also
>neglected gammons. Unfortunately, it's not possible to work out an exact
>statistical test for money games without knowing what proportion of each type
>of score you would normally get between two equal players. For match play,
>however, you're OK: if you play 100 5-point matches, instead of 100 1-point
>games, and just worry about the winner not the final score, everything else
>works the same way.
True, but probably match play will favor the "better" player more than
game play. Time _within_ the match for luck to average out while
skill doesn't, plus the complexities of cube, gammon chances, and
match poiny/equity issues add more skill elements.
This of course doesn't take into account that player A may be better
at some aspect of the game, and player B at others. So "A is better
than B" may in truth vary depending on the rules of the contest... A
can be a better game player, while B is better at 7-point matches.
True, skill factors tends to correlate (the better player _tends_ to
be better at each and every aspect), but it is hardly absolute.
So while match play and game (not point) play are mathematically
similar, in that they use the same statistical formulae, the "inputs"
(the probability that either player will mark up a win in an attempt).
Stephen Turner
Greycat Sharpclaw wrote:
>
> There is an allegation that Stephen Turner <sr...@cam.ac.uk> wrote:
>
> >3) I assumed 1-point games. The real skill is in the cube, and I've also
> >neglected gammons. Unfortunately, it's not possible to work out an exact
> >statistical test for money games without knowing what proportion of each type
> >of score you would normally get between two equal players. For match play,
> >however, you're OK: if you play 100 5-point matches, instead of 100 1-point
> >games, and just worry about the winner not the final score, everything else
> >works the same way.
>
> True, but probably match play will favor the "better" player more than
>
> So while match play and game (not point) play are mathematically
> similar, in that they use the same statistical formulae, the "inputs"
> (the probability that either player will mark up a win in an attempt).
>
You're right of course, and I blurred this distinction. This will make the
test use the same numbers, but it will be a more powerful test, i.e., it's
more likely to spot the difference between the players, because the
difference is magnified. (I also didn't make it explicit that if the test
doesn't show a difference, you can't conclude anything).
Of course, as you also point out, it's possible that one player is better at
one-pointers and one player is better at 5-pointers, so really you're testing
two different things.
Alexander Nitschke
Stephen Turner wrote:
>
> Dodd wrote:
> >
> > Statistically, how many games of backgammon must two people play before one
> > can definitively say they are the better player?
> >
>
> apart after those games. The standard way to set it up a a statistical
> problem is to play a pre-determined number of 1-point games -- let's say 100.
> If you are the same standard, there's only a 2.5% chance that you will win
> at least 60 of them, which is enough to reckon that you are probably the
> better player.
>
>
> 3) I assumed 1-point games. The real skill is in the cube, and I've also
> neglected gammons. Unfortunately, it's not possible to work out an exact
> statistical test for money games without knowing what proportion of each type
> of score you would normally get between two equal players. For match play,
> however, you're OK: if you play 100 5-point matches, instead of 100 1-point
> games, and just worry about the winner not the final score, everything else
> works the same way.
>
I worked out a test for money game: You collect all results in a large
number of games between two players (in this case a friend of mine and
JellyFish Level5).
Then you assume a normal distribution for the sum of n games with n a
large number (here 300) with a mean of zero and an estimated standard
deviation. The estimation for the standard deviation was deducted from
the collected data of course and resulted in about 2.67. Generally I
would say that this is a good value for the sd, in real life between two
humans the sd might be higher if the guys double early and take a lot,
resulting in a sd of 3.
Then you can say with a standard test how many points one must be ahead
after n games to be said that he is the better player at a given
confidence level.
In my example my friend was 9 points ahead after 300 games. 73 points
were needed for a confidence level of 95% percent. You see that a lot of
games are needed or one must be really better to get a significant
result :-)
--
Alexander Nitschke
(acey_deucey in FIBS)
Donald Kahn
do...@bigsmile.com (Dodd) wrote:
>I am sure that this question has been asked many times before, but since I
>just found this newsgroup I would like to ask it again.
>
>Statistically, how many games of backgammon must two people play before one
>can definitively say they are the better player?
>
>Thanks for your anticipated responses.
>
>Best regards,
>
>Dodd
>
>--
>Patrick Dodd
>Founding Partner, BigSmile, LLC
>"Internet Solutions for Dental Professionals."
>do...@bigsmile.com
>http://www.bigsmile.com/dentist/
>602.777.0178
There are any number of really difficult middle game positions -
usually with four or more pieces back - any expert can supply dozens
of them off the top of his head.
Let the two players play one of these twenty times - ten times from
each side. At the end, there will be no doubt in anyone's mind who is
the better player.
Donald Kahn
Lawrence J. Gier
The answer is ONE!
On Tue, 25 Mar 1997 14:15:12 GMT, don...@easynet.co.uk (Donald Kahn)
wrote:
Lawrence J. Gier
Chris W.
What color is the sky in YOUR world Lawrence?
Chris
Lawrence J. Gier <ljg...@kdsi.net> wrote in article
<33468ef0...@news.kdsi.net>...
Lawrence J. Gier
Blue and in your world? Using the words statistically and DEFINITELY
brought about my reply. After one game, the winner is DEFINITELY
better until the next game. YOU SEE, and I hope you do see, terms such
as "definitely" and "better" are relative terms so DEFINE the question
in more EXACT terms.
Lawrence J. Gier