Modelling bridge results (stats)

[Dean]

Hi

 

I'm looking at trying to model our local clubs pairs results. In particular I want to know if after applying a handicap (+2.5%, -5%etc) whether the handicapping makes the results more equivalen/equal.

 

Do pairs results follow any sort of well-known statistical distribution e.g. normal, Posisson etc?

 

Has anyone seen or done any work in this area? Is a simple standard deviation a good measure here or is it not applicable to this sort of data.

 

 

Thanks

Dean

[Guest Jlall]

I'd bet¿fold

[beatrix45]

:blink: To see if data follow a normal distribution, the first thing to do is to calculate the moments of the data - the first moment is the mean, the second is the variance (derived from the standard deviation). These can be anything.

 

The third, fourth and fifth moments will be (or be close to) certain specific numbers if the underlying data are normal. I suspect that scores from a 26 board duplicate session may tend toward the normal since they are based on 26 separate events, but, no doubt, it ultimately depends on the nature of the field.

 

Testing to see if they follow other distributions uses similar techniques.

 

If you have some data files you can send me, let me know. We can exchange e-mail addresses, and you can send them to me. It's not much effort to run the numbers once you have them in the computer.

[helene_t]

If there are no ties, matchpoints from a single board will follow a uniform distribution. The averages over many boards will approximate a normal distribution. I'm not sure what effect ties will have, I suspect they make the convergence towards the normal distribution slightly faster than without ties. In any case, for some 4-6 boards an upwards, the normal distribution should be reasonable.

 

To see if the handicaps make the distribution more equal, I think the most sensible thing to do would be to look at the correlation between results scored at different nights by the same pair and see if that correlation becomes lower when handicaps are applied. Consider the table of matchpoints:

-------Night1 Night2 Night3 ...

Pair1

Pair2

Pair3

.....

You construct this table without handicaps, and summarize the between-rows or between-columns correlations in that table (for example, report the average of the correlation coefficients), and do the same with handicaps. If the handicaps are perfect there would be zero correlation

 

You can also look at the standard deviation of the logarithm of the matchpoints for a single night. This is a direct measure of "equality" but I wonder if that is really what you want to know. For example, shrinking the matchpoints towards 50% by applying the transformation

MP -> (MP+50%)/2

would reduce the standard deviation but would have no effect on the ranking, i.e. it would not improve the weaker pairs' odds of winning.

 

Btw, Gerben has published some work on the distribution of IMPs.

[MikeRJ]

Many years ago when I played at the Cardiff bridge club (Wales) they used a simple handicapping system which seemed to work well.

 

Each player was assigned a handicap ranging from -4 (very good, Camrose standard) to about +6 (corresponded to a player coming out of the "Improver" classes). A pair's combined handicap was simply the sum of the two individual handicaps. After scoring the duplicate in the normal way the scores were adjusted by adding 1% for each handicap point (or deducting for negative handicaps).

 

A small committee reviewed the records and adjusted handicaps periodically.

 

As I recall there were sometimes hurt feelings when a player's handicap was increased, but overall the system worked well and is still in use I believe.

 

Mike

[helene_t]

What are the handicaps used for? I suppose masterpoints and team selection are based on real results.

 

One could give a compensation to pairs that have the misfortune of playing against or in the same direction as many strong pairs. Or one could give wine and chocolate to pairs who perform better than hey used to, i.e. who score positive including handicap.

 

For example, in a Mitchell tournament one could make sure that the EW and NS directions are equally strong, or if they are not, one could adjust the results accordingly.

[hrothgar]

If there are no ties, matchpoints from a single board will follow a uniform distribution.

Its unclear to me that the assumption of a uniform distribution is necessarily warranted. Coincidentally, I was just looking at something called a "Binned Kernel Density Function" yesterday afternoon.

 

http://projecteuclid.org/DPubS/Repository/...d.bj/1078681378

 

Admitted, this assumes that you're starting with a histogram or some such, but...

[Rossoneri]

Interesting. This thread especially caught my eye because I just finished playing a handicapped teams tournament last weekend, and during the round which I sat out (partner had something on, and being the team captain I chose to stay a bit to give teammates some moral support) I was looking at the results from the first 2 rounds and trying to make sense of the handicaps given.

 

Of course I couldn't really make much statistical sense out of it, but one thing that struck me was that individual handicaps might be inaccurate when two unfamiliar people partner each other.

[han]

If there are no ties, matchpoints from a single board will follow a uniform distribution.

Could you post some evidence? I don't trust this.

[MikeRJ]

What are the handicaps used for? I suppose masterpoints and team selection are based on real results.

I think small cash prizes were awarded on the basis of handicapped results. Also I think some improving players liked to have a means of monitoring their progress...

 

Mike

[hotShot]

If there are no ties, matchpoints from a single board will follow a uniform distribution.

Could you post some evidence? I don't trust this.

Like Hannie I have some doubts that the will be a uniform distribution.

At least not at the top or bottom.

 

Let us take an example with ties:

If you think of a 7 table 13 round Howell movement, you are a close to a perfect movement as you can get.

Now take 14 pairs:

camp1 and camp2

average1 ... average12

Now enter the results.

campX vs. averageY => campX gets the top (use +-100 as result)

camp1 vs. camp2 or averageX vs. averageY=> average (use +-50 as result)

 

Now calculate the MP's and you will find that the averageX plaver don't get the same percentage. You will find that the results differ about +-3 percentage points.

This is caused by the fact that if both camps play a board on the same side the best score for the average player (5 equal results) is different from that they get if the camps are on different sides (6 equal results).

 

My experience at club level is that the seating is not random. For example some players insist to sit e.g. at the smokers corner ....

 

The size of the distortion caused by the movement varies from movement to movement and it depends on the number of strong/weak pairs.

[hrothgar]

Like Hannie I have some doubts that the will be a uniform distribution.

At least not at the top or bottom.

I think that folks are getting overly excited about symantics:

 

Lets assume for the moment that we have a set of results from a board in which there are no ties. In this case, we have a set of unique scores, each of which occured with an equal frequency.

 

One might argue that this suggests a uniform distribution across the set of observed scores.

 

In any case, back around a year ago I computed the standard deviation of all the boards from the finals of the Open Pairs at the Verona chamionships in 2006. (Paul Marston was wondering whether it was appropriate to apply the 68-95-99 rule to board results). Here's a copy of an email that I sent to him.

 

I just ran some actual numbers.  I used the results from the Final of the Open Pairs at Verona in 2006.  (Please note:  In theory, the skill level of the remaining pairs should be fairly close since a lot of the really weak pairs should have been eliminated before the finals)

 

The maximum Standard Deviation for the 30 boards was 30.06 (Board 13)

 

The minimum Standard Deviation for the 30 boards was 22.85 (Board 27) -

This was a freakishly flat board compared to most of the others.

 

The average of the Standard Deviations was 28.22

 

I also lumped all of the board results together and calculated the stdev of the entire set.  This came out to be 27.9

 

There is a lot of clumping in the data set, so its probably inappropriate to apply the 68-95-99 rule.  However, if you wanted to apply this, then you'd be saying that

 

68% of all observations fall between the range 22.1 and 77.9

 

(Note that the width of the interval is 27.9 * 2 = 55.8.  Its not all

that much smaller than 68)

 

If anyone wants, all of the results are available at

 

http://www.swangames.com/main/Bridgecast/R...erona_2006.html

 

The specific boards that I used to calculate these statistics are available at

 

http://www.swangames.com/rama/eventinfo.php?eventid=282807

[hrothgar]

For anyone who cares, I just ran the board results through dear old MATLAB.

 

I have some pretty pictures that the forums system won't let me post. I tried fitting a number of different distributions to the Verona scores. Nothing looked that good.

 

Its entirely possible that a single 30 board tournament isn't sufficient to generate enough samples.

 

If anyone would like to collect all the necessary data, I'd be happy to analyze the results.

 

Ideally, I'd like to get my hands on a decade or so worth of data from the Blue Ribbon Finals or some such.

[Mbodell]

What are the handicaps used for? I suppose masterpoints and team selection are based on real results.

We have a once a week game that gives master points primarily to overall results but also gives results based on handicaps. It is IMP scoring and based solely on masterpoints and the handicaps range from around +1 IMP/board for teams of people with 0 master points to 0 IMP/board for teams of life masters to around -1 IMP/board for teams of the 5-digit master point crowd. The game tends to have people with <1000 points so the handicaps stay relatively close, and the handicap is just the sum of the two players. So a rookie and a life master might be +14 (rookie) and +0 for a handicap of +14 IMPs across 27 boards.

 

But this is just for local club games, not for selection or anything.

[helene_t]

If there are no ties, matchpoints from a single board will follow a uniform distribution.

Could you post some evidence? I don't trust this.

This is obvious so if you dispute this we must be talking about different things.

 

The matchpoints from a, say, top-22 board (12 tables, to MPs per win) without ties are

22

20

18

..

4

2

0

 

I would call this uniform.

 

Coincidentally, I was just looking at something called a "Binned Kernel Density Function" yesterday afternoon.

What this procedure does is it constructs a candidate distribution of the real data, given some binned data. I don't think one can talk about the "real" data as opposed to the discretized data when analyzing matchpoint results. The problem I see is that the discreteness of matchpoints is related to sampling, not to binning. But I can imagine the Kernel Density Function could still be useful for some purposes where one for some reason wants to model the matchpoints directly without seperating the sampling from the distribution of per-board score probabilities.

 

Sr if this doesnt make sense, it's early morning here.

[gerry]

Hi,

 

last time I did this exercise I used a triangle distribution for each board. The resulting distributions passed the most important (IMHO) statistical test: they looked right :blink: .

 

Triangle distribution has the practical advantage too that it is easy to modify the mode (point of triangle) to try and simulate pairs of various strengths