Law of Similarity?

[pljr]

I recall an empirical "law" that said that if you had a long suit then maybe that was the same pattern in other hands.

 

I have looked for more info without success.

 

Any ideas please?

 

I believe it wasn,t true.

[hrothgar]

This is a weird claim from 60+ years ago, back before computer dealt hands.

 

It has absolutely no validity (assuming fair shuffling)

[pilowsky]

It's a great urban myth though.

I used to believe that if the cards were new then I was more likely to get a distributional hand. I got flamed for that thought.

 

Perhaps we should have a Bridge Urban Myths Section? (BUMS).

 

 

[pescetom]

I recall an empirical "law" that said that if you had a long suit then maybe that was the same pattern in other hands.

 

I have looked for more info without success.

 

Any ideas please?

 

I believe it wasn,t true.

 

Most postulates proposed as "laws" are unreliable at best and this one (the "law of similarity") was clearly nonsense even in its time.

The originator was of course Ely Culbertson, somewhere between bridge genius and con-man.

[nige1]

It was Culbertson's "Law" of Symmetry.

e.g. Singleton kings :)

[Cyberyeti]

This is a weird claim from 60+ years ago, back before computer dealt hands.

 

It has absolutely no validity (assuming fair shuffling)

 

The fair shuffling comment matters, I recall a deal with new cards where all 4 hands were (6421) with identical pips within each suit.

[thepossum]

Suppose you have a 7-2-2-2 hand vs a 4-3-3-3 hand

 

Surely the distribution of the remaining cards is such to shift the chance of others having a long suit - who knows how much by - I doubt you could assume much about others having similar distributions, but come on it seems obviousl, albeit hard to calculate.

 

It was always a bit depressing when a night seemed to become defined by a boring pack of cards - time to try another one sometimes

 

But then again I imagine boring distributions are the bulk of the bridge deal space with occasional bright interesting patches - much like life, the universe and everything really :) In fact come to think of it, when I watched a video of John Conway discussing surreal numbers he seemed to have a similar philosophy towards infinity, that only a relatively small part of it was interesting - sorry for the paraphrase - apologies, it was more along the lines of levels or types of interestingness of different parts so he concentrated on the most interesting bit

 

There may be something useful in this page by Pavileck on Freakness

 

I heard once that if you had a 13-0-0-0 hand it was likely someone else had a void somewhere

 

 

Has anyone written anything interesting on shuffling theory :) Maybe an idea (not serious) for bridge software would be to have different shuffling algorithms to make it more interesting (has it been done)

[mikeh]

I recall an empirical "law" that said that if you had a long suit then maybe that was the same pattern in other hands.

 

I have looked for more info without success.

 

Any ideas please?

 

I believe it wasn,t true.

You are correct

 

 

Culbertson, one of the pioneers of the game, believed in what he called the Law of Symmetry

 

I dont believe any good player believes in it these days.

[blackshoe]

According to P.E.I. Bonewits, the Law of Similarity is one of the laws that govern the creation of magical spells. Bonewits was the only person who had a degree in Magic from a major Western university (B.A., U.C. Berkeley, ca. 1970). B-)

[pescetom]

I heard once that if you had a 13-0-0-0 hand it was likely someone else had a void somewhere

As this hand has a generic frequency of 6 x 10 to the power of -10, it seems somewhat academic :)

[mycroft]

Ah, it's more common than that - it's shown up in at least 4 books that I own.

 

Almost always when it's shown up, of course, it's because a new deck was put in a board by suits (or, back in the day, when we suited boards on last play to be able to create the hands from handouts) and not shuffled/made/dealt.

[pescetom]

Almost always when it's shown up, of course, it's because a new deck was put in a board by suits (or, back in the day, when we suited boards on last play to be able to create the hands from handouts) and not shuffled/made/dealt.

Back in the day? We still do that.

Indeed my alternative reply was that it was even likely that each other hand had 3 voids.

[thepossum]

As this hand has a generic frequency of 6 x 10 to the power of -10, it seems somewhat academic :)

 

I know but some people may want a special convention to cover it

 

Would a transfer work

 

It would be sad to get excited and bid no trumps

 

I do imagine a lot of people would pass in case they were accused of psyching

 

But seriously getting back to the OP have there been attempts at analysing or proving such things. Could you not prove it on a simplified game

[HardVector]

The "law" I had heard of postulated that hand patterns repeated in the suits. So if you have a 5332 hand, there will be a suit that is 5332. I heard of it about 30 years ago as hearsay and have never heard it again. It was in conjunction with the idea that if you have a void, there will be another hand with a void out there.

[thepossum]

I reckon you could try something by induction

 

Consider a simple game. 2 players. 2 cards each, 2 suits etc

[hrothgar]

 

But seriously getting back to the OP have there been attempts at analysing or proving such things.

 

Yes

 

It is trivial to demonstrate that this is complete nonsense for either

 

A. Computer dealt hands or

B. Hands that a hand shuffled in "good" manner

 

It is possible that there is some truth for this for hands that are manually shuffled using some "poor" technique. The problem here is that almost any claim is consistent with bad shuffling.

 

The only way to really analyze this is to collect all sorts of data about hands that are manually shuffled in bridge clubs and use this to test the hypothesis. There was a data set that got posted a few months back from one bridge club in the Netherlands. As I recall, it didn't show anything particularly interesting.

 

Could you not prove it on a simplified game

 

The issue here is not the complexity of the game, but rather the availability of the data

[pescetom]

The "law" I had heard of postulated that hand patterns repeated in the suits. So if you have a 5332 hand, there will be a suit that is 5332. I heard of it about 30 years ago as hearsay and have never heard it again. It was in conjunction with the idea that if you have a void, there will be another hand with a void out there.

 

You are right. To quote from a 1933 newspaper article by Ely Culbertson:

"It was original with me, and I have long recommended it to all bridge players because I know how valuable it has been to me, and thus can be to them. The player must simply remember that whatever the pattern of any hand of 13 cards dealt, there will probably be a suit pattern which corresponds to it. Thus if a player's hand is divided five-four-three-two-one(sic), the distribution of some suit in the four hands will probably be five-four-three-one also. This varies, of course, but only to the extent of one card. That is, the equivalent suit may be divided five-four-two-two."

 

Almost one hand in four is either 5431 or 5422, so allow space for "exceptions" and this "law" might be seen to work.

[thepossum]

Yes

 

It is trivial to demonstrate that this is complete nonsense for either

 

A. Computer dealt hands or

B. Hands that a hand shuffled in "good" manner

 

It is possible that there is some truth for this for hands that are manually shuffled using some "poor" technique. The problem here is that almost any claim is consistent with bad shuffling.

 

The only way to really analyze this is to collect all sorts of data about hands that are manually shuffled in bridge clubs and use this to test the hypothesis. There was a data set that got posted a few months back from one bridge club in the Netherlands. As I recall, it didn't show anything particularly interesting.

 

 

 

The issue here is not the complexity of the game, but rather the availability of the data

 

Come on Richard, until now I gave you credit for a fair bit of understanding of maths and the concept of proof..it's nothing to do with shuffling at all. It's a mathematical proofs, whether possible analytically or not I don't know

 

Unless of course by shuffling we are discussing complete randomness with any hand equally likely etc

 

And in terms of data you would need however many times 10^28 hands and a large database. Sorry the number of hands has been reduced considerably by the OPs hand

 

But it's obviously and selfevidently true. But I don't know if it's provable

 

 

Just by way of an aside I read something about a law of similarity in Gestalt psychology. It seemed to me a statement of the obvious too

[smerriman]

Come on Richard, until now I gave you credit for a fair bit of understanding of maths and the concept of proof..it's nothing to do with shuffling at all. It's a mathematical proofs, whether possible analytically or not I don't know

Nothing to do with it? The shuffling is the entire point and premise of the 'law of symmetry' - the first paragraph of Culbertson's article:

 

To say that mathematics is only a small part of bridge is not to impugn the validity of mathematical law, but the laws of simple probabilities must be modified and corrected when applied to cards. In calculating the probabilities of various hand and suit distributions the mathematicians presuppose an abstract perfect shuffle which is nonexistent in practice. This fact renders many of the current mathematical tables, which are sacred to so many experts, at best of problematical value.

With perfect shuffling it's trivial to prove mathematically. If you fix North and South's hands, for example, you can swap cards between them without affecting any frequencies of East + West's - so the shape of one cannot influence the shape of another.

[hrothgar]

With perfect shuffling it's trivial to prove mathematically. If you fix North and South's hands, for example, you can swap cards between them without affecting any frequencies of East + West's - so the shape of one cannot influence the shape of another.

 

I'd frame it slightly differently

 

With perfect shuffling the odds of being dealt any given deal are equal.

 

Lets filter down the set of hands by fixing the North hand and the South hand.

The odds of being dealt any given pair of East West hands are still equal.

[thepossum]

Its Sunday morning and I'm either not fit for an discussion of this kind or motivated to engage

 

However if you have to divide a suit up between 3 hands, the number of available cards 0 to 13 determines the distribution and possible lengths of those hands

 

Trivial example 0-4, say, cards left the chance of anyone having length in that suit is 0

Any more cards and the chance of anyone having length in that suit increases :)

 

As I said, I caan't be bothered to even think what kind of distribution it is. How you can arrange n cards in the longest remaining suit (say) between 3 etc and how the distribution changes as n increases etc

 

.... and I gave a trivial example of how it works in the simplest case I could think of with 2 hands, 2 suits and 2 cards each :)

.... actually I know you could have a 2 hand, 2 suit, 1 card each game that maybe also demonstrates it but I wanted a game with more than card each

 

But maybe to make it more convincing we need to start with a 4 hand game and the most basic case etc

[hrothgar]

I will note the following:

 

I think that it is a mistake to start by examining the precise algorithms that are used to assign cards to hands.

 

Shuffling is more complicated than it seems (just look at Knuth).

 

I think that a better way to proceed is to look at shuffling in the abstract. What are the characteristics of a "good" shuffling algorithm?

 

To me, a good shuffling algorithm is one which

 

1. Is capable of generating any / all bridge hands

2. The likelihood that any possible bridge hand is generated is uniform

 

As some of us have noted, with this as a starting point, it is trivial to show that the law of similarity is nonsense.

 

Equally significant, if you are starting someplace different (say some specific BAD shuffling algorithm) you can probably justify (pretty much) anything that you want.

 

"I assert that the Law of Similarity holds true because my mother does a dreadful job shuffling hands and I normally play bridge against my mother and her friends and it feels like this happens" really isn't a testable theory.

[pescetom]

I wrote a dealer script to validate this "Law" and posted it in dealer script forum so as not to hijack this discussion with scripting issues.

It allows you to input a shape (e.g. 5-4-3-1) and see the probability that any given suit (or number of suits) has the same shape.

 

The following table shows for each common shape first the generic frequency, then the probability that 0,1,2,3,4 suits have the same shape.

All numbers are rounded percentages, the last five come from running the script.

If the Law was true then the last five numbers should be close to 0 100 0 0 0.

[Note that Culbertson fails even to consider that more than 1 suit may have the hand shape and the example he gives would fail if it happened].

 

Shape %Freq %0 %1 %2 %3 %4

4333 10 34 42 20 2 1

4432 22 20 40 29 10 1

5332 15 31 41 23 4 1

6322 6 48 39 11 8 0

6331 3 62 32 6 0 0

5431 13 34 48 15 3 0

5422 11 37 45 10 2 0

5521 3 67 27 6 0 0

4441 3 70 26 4 0 0

5440 1 69 29 2 0 0

 

If the numbers are correct then it's easy to see why Culbertson may have 'felt like it happens' with 5431 and 5422, although even there it happens in precisely one suit less than half the time.

[smerriman]

If the Law was true then the last five numbers should be close to 0 100 0 0 0.

Just to reiterate, this is not what the law says. Culbertson was well aware that it is not true for mathematically random hands. The law applies to real-life shuffled hands only:

 

The factor X - the artificially formed patterns and the imperfect shuffle - must be seriously reckoned with in calculations, and forms the basis of the Law of Symmetry.

[pescetom]

Just to reiterate, this is not what the law says. Culbertson was well aware that it is not true for mathematically random hands. The law applies to real-life shuffled hands only:

I too imagine he was well aware, and probably had noticed less than 50% too. That did not convince him to mention either fact, nor to mention the issue of shuffling.