LoTT is a parabola

[jogs]

LoTT is a parabola

 

Larry Cohen first said total tricks equal total trumps.

Mike Lawrence objected.

Then Cohen said total tricks approximately equal total trumps.

That statement is still too powerful.

E(tricks) = trumps

That statement would imply the relationship between tricks and trumps is a straight line.

When total trumps is greater than 18, total tricks is less than total trumps.

Each trump pass 18 produces less than one additional trick.

26 trumps obviously doesn't always produce 26 tricks. If one example where 26 trumps produces fewer than 26 tricks is found then the statement is proved.

 

[hv=pc=n&s=shakqjt98d432cjt9&w=sakqjt98hdjt9c432&n=sh765432d8765cakq&e=s765432hdakqc8765]532|400[/hv]

 

The parabola is a better fit for tricks to trumps.

 

http://i57.tinypic.com/2ew1aoi.jpg

 

This is only an illustration of what the relationship looks like. I do not know the exact parabola.

[mikeh]

I have no idea what your point is.

 

Besides, nobody ever said, as far as I can recall, that the LOTT was anything other than an approximation. I have the original book by Cohen, and while I can't be bothered to check it I am fairly confident that his point was that it was a pretty good tool for estimation, not calculation, and that even in the first book he acknowledged that there were factors that would tend to make the estimation less reliable in some situations than in others.

 

I seem to recall that you like to spend a lot of energy on metrics. My advice: quit worrying so much about metrics and learn to 'play' the game itself.

 

You are like a golfer who, rather than learning how to swing the club properly, focuses on things like the stiffness of various shafts, and the drag coefficient of various clubhead designs.

[Trinidad]

There are laws in all sizes. Many laws in physics have allmost absolute accuracy: E=mc² and V=IR.

 

In other fields laws are less accurate. Bridge is such a field. The Law of Total Tricks should not be interpreted as absolutely accurate. Nevertheless is the LOTT a good indicator for the total amount of tricks, particularly at the partscore level. At higher levels, the expected number of total tricks is obviously lower than the number of total trumps (as you point out: 26 trumps do not mean 26 tricks).

 

But probably more important than that the law overestimates the number of tricks at high level is the fact that the variation (or error) in the prediction gets much larger at higher levels. This means that in practice one simply shouldn't use the LOTT (or a parabolic correction to it) at high levels. On these distributional hands other factors are more important, particularly secondary fits and the correct placement of honors and controls. This is why fit bids are popular in this type of auctions.

 

Rik

[johnu]

You are like a golfer who, rather than learning how to swing the club properly, focuses on things like the stiffness of various shafts, and the drag coefficient of various clubhead designs.

 

You say that like it's a bad thing :o But certainly there are more important things like coefficient of restitution, type of finish on your wedges, and having the hottest putter on tour in your bag.

[Bbradley62]

Borrowing a line from Pirates of the Caribbean, maybe we should call it the Guideline of Total Tricks B-)

[whereagles]

LoTT is a parabola

E(tricks) = trumps

That statement would imply the relationship between tricks and trumps is a straight line.

 

It's better said "relationship between average tricks and trumps". But anyway, while the parabola-like relationship actually seems plausible, making a case for it it needs more evidence than what you present. Some data and statistics is needed to prove it at, say, 95% confidence level.

 

1. There are laws in all sizes. Many laws in physics have allmost absolute accuracy: E=mc² and V=IR.

2. The Law of Total Tricks should not be interpreted as absolutely accurate. Nevertheless is the LOTT a good indicator for the total amount of tricks,

 

1. V=IR is actually a bad example lol. It only works for ohmic materials.

 

2. Lawrence/Wirgren did some statistics and found that the LOTT works, as in "tricks = trumps", in about 40% of the cases. In 60% of the cases it is off by at least 1 trick. Not sure you can call that a good indicator. Still, the LOTT seems to hold if you include the word "average" (which is actually how it was originally postulated).

[jogs]

It's better said "relationship between average tricks and trumps". But anyway, while the parabola-like relationship actually seems plausible, making a case for it it needs more evidence than what you present. Some data and statistics is needed to prove it at, say, 95% confidence level.

That 5% significance level is reserved for testing new prescription drugs. In bridge we settle for being right clearly more often than the other guys.

 

 

1. V=IR is actually a bad example lol. It only works for ohmic materials.

 

2. Lawrence/Wirgren did some statistics and found that the LOTT works, as in "tricks = trumps", in about 40% of the cases. In 60% of the cases it is off by at least 1 trick. Not sure you can call that a good indicator. Still, the LOTT seems to hold if you include the word "average" (which is actually how it was originally postulated).

Lawrence/Wirgren concluded that was not sufficient for a system based solely on LoTT.

If one combines LoTT with SST, one can get better estimates than either method on a stand alone basis. Our partnership make more tricks when partners holds singletons and voids than when both partners hold flat patterns. Here a flat pattern is defined as any pattern without a singleton or void.

[mikestar13]

This is nothing new -- Cohen himself in his books says that LOTT tends to break down for extreme numbers of trumps. "If there were 26 total trumps, do you think that both sides could make a grand slam?" -- LC, IIRC. Your parabola look like it says LOTT is very accurate up to 19 total trumps and not bad with 20, the drop off gets noticeable around 21.

 

By the way, I have read both Cohen and Lawrence and think they both have portions of the truth. Trumps are not "everything", but neither are they "irrelevant".

[Siegmund]

I would be very surprised if the relationship between total trumps and expected tricks were fit well by a parabola.

 

The relationship between (for instance) your side's HCP and your side's expected number of tricks in notrump is extremely close to linear, all the way up to 11.5 tricks at 31 HCP, and then rather abruptly flattens out. A parabola captures neither the long straight portion nor the sharp elbow well. It would not surprise me at all if the LoTT relationship were linear up to about 21 or 22 total trumps and then rather abruptly flattened out.

[nige1]

In 1969, Jean-René Vernes wrote a Bridge World article about his discovery, made in the 50s.

In 1981, Dick Payne and Joe Amsbury wrote TNT and Competitive Bidding.

In 1992, Larry Cohen popularised it among US players with To Bid or Not to Bid: The LAW of Total Tricks.

In 2004, Mike Lawrence and Anders Wirgren wrote I Fought The Law Of Total Tricks but it's unclear whether or not the law won.

[mike777]

What was most surprising was how little traction fought the law gained in the minds of the bridge public or bridge writers, not sure why.

 

 

I have often posted about the fought the law theory.

-----------------

When we talk about nonlinearity or asymmetry we are talking about the properties of an option.

 

Think of an option as= asymmetry + rationality

 

(asymmetry is a form of nonlinearity.)

 

One property of the option: When we are discussing about nonlinear payoffs we don't care about average outcomes, only the favorable ones.(the downside doesn't count beyond a certain point).

[SteveMoe]

The issue I've had with both the LoTT and Lawrence/Wirgren is both are descriptive/correlational, not causal, in how they relate tricks to fit/shape/power. Steve Bloom did a better job in his 7 part Theory of Total Tricks series March 2013 - http://bridgewinners.com/article/series/theory-of-total-tricks/

 

What I believe Steve was able to do is define tricks in terms of our side's 2 suit fit and the purity of our hands. This comes closest to cause of any approach I've seen. Pure hands TT=SF+3 while impure hands TT=SF+2. Steve concludes "counting short suit losers and using second fit gives a better estimate of total tricks than counting the number of trumps".

[mike777]

perhaps this is better than Lawrence but we see no evidence presented

 

it would be nice to see what top class says on live deals.

 

It would be nice to ignore average outcomes

[yunling]

LoTT is not very accurate, and so is HCP. People use them because their accuracy is ok most of the time and they are very easy to remember and use.

Sophisticated methods are, of course, theoretically more sound, but you have to do some extra work to put it into use. For most players it is just not worth the effort.

[GreenMan]

26 trumps obviously doesn't always produce 26 tricks. If one example where 26 trumps produces fewer than 26 tricks is found then the statement is proved.

 

I played a deal at the club some time back that had 26 trumps but only 25 tricks. QED. Although the temptation to lead the wrong suit meant that there were +1770s in both directions.

 

That 5% significance level is reserved for testing new prescription drugs. In bridge we settle for being right clearly more often than the other guys.

 

Eh? New prescription drugs get 0.1% or less, don't they?

[yunling]

Simulation by Matt Ginsberg(published in Bridge World, Nov 1996)

Length Samples Total tricks

14 46944 13.85±0.63

15 47281 14.86±0.64

16 120525 16.10±0.70

17 102184 17.02±0.75

18 69792 17.99±0.83

19 37561 18.78±0.87

20 15845 19.50±0.99

21 5041 20.11±1.20

22 1286 20.69±1.48

23 237 21.22±1.83

24 45 21.78±2.27

 

LoTT works fairly well when total trumps is 18 or shorter, but with longer trumps, LoTT overestimates the total tricks and has a high variance. At this level I think SST is more important.

[Trinidad]

It's better said "relationship between average tricks and trumps". But anyway, while the parabola-like relationship actually seems plausible, making a case for it it needs more evidence than what you present. Some data and statistics is needed to prove it at, say, 95% confidence level.

 

 

 

1. V=IR is actually a bad example lol. It only works for ohmic materials.

 

2. Lawrence/Wirgren did some statistics and found that the LOTT works, as in "tricks = trumps", in about 40% of the cases. In 60% of the cases it is off by at least 1 trick. Not sure you can call that a good indicator. Still, the LOTT seems to hold if you include the word "average" (which is actually how it was originally postulated).

Whether being exactly correct in 40% of the cases is a good indicator or not depends on the field and the alternatives. In bridge there is a "law" that is used much wider than the LoTT. It uses HCP to predict the number of tricks.

 

In the late '90s I simulated a large amount of deals, let the bridge playing program that I had play them in the longest trump fit for both sides, and wrote down how many tricks were taken by each side, how many trumps each side had and how many HCPs each side had. Much to my surprise, it turned out that the LoTT was a much better predictor for the amount of total tricks than the amount of HCPs was to predict the number of tricks for one side. Nevertheless, bridge teachers all over the world are teaching their students the Milton Work count. Why don't they stop teaching them this rubbish?!?

 

The reason is that in an uncertain game like bridge, being absolutely correct in 40% of the cases is pretty darn good. Even the less accurate Milton Work count method is good, because there are few alternatives. Lawrence and Wirgren can whine that the LoTT is off by at least one trick in 60% of the cases, but they forget to mention that being off by one trick is in most cases perfectly acceptable. It will rarely lead to disasters. Now, if Wirgren and Lawrence had come up with a better alternative for the LoTT, as the buzz was when the book came out, then they would be entirely correct to attempt to throw out the LoTT. But they didn't. Instead the book was basically on hand evaluation when you know something about the opponents hands. It tells you how many tricks you can take, but not how many tricks the opponents can take. It doesnot say anything about the question: Bid one more or defend? As a result, I found it to be the worst book -by far- with Mike Lawrence's name on it.

 

Rik

[helene_t]

Cohen said that the law overestimates the value of the tenth trump and the data Yunling presented confirm this.

 

The +/- numbers are standard deviations, I suppose?

[whereagles]

The issue I've had with both the LoTT and Lawrence/Wirgren is both are descriptive/correlational, not causal, in how they relate tricks to fit/shape/power. Steve Bloom did a better job in his 7 part Theory of Total Tricks series March 2013 - http://bridgewinners...f-total-tricks/

 

What I believe Steve was able to do is define tricks in terms of our side's 2 suit fit and the purity of our hands. This comes closest to cause of any approach I've seen. Pure hands TT=SF+3 while impure hands TT=SF+2. Steve concludes "counting short suit losers and using second fit gives a better estimate of total tricks than counting the number of trumps".

 

Thx. I'll have a look at it.

 

Simulation by Matt Ginsberg(published in Bridge World, Nov 1996)

Length Samples Total tricks

14 46944 13.85±0.63

15 47281 14.86±0.64

16 120525 16.10±0.70

17 102184 17.02±0.75

18 69792 17.99±0.83

19 37561 18.78±0.87

20 15845 19.50±0.99

21 5041 20.11±1.20

22 1286 20.69±1.48

23 237 21.22±1.83

24 45 21.78±2.27

LoTT works fairly well when total trumps is 18 or shorter, but with longer trumps, LoTT overestimates the total tricks and has a high variance. At this level I think SST is more important.

 

I didn't know of this study. Nice stuff there. I did a graph of it and we do get to see the claimed incline. The incline is probably due to increase in the likelihood of HCP wastage (K/Q opposite singleton, or A opposite void). [The graph didn't come out as I wanted (got no excel mad skillz), but it should show the point.]

 

 

http://i61.tinypic.com/5p0x9t.png

 

1. it turned out that the LoTT was a much better predictor for the amount of total tricks than the amount of HCPs was to predict the number of tricks for one side. Nevertheless, bridge teachers all over the world are teaching their students the Milton Work count. Why don't they stop teaching them this rubbish?!?

 

2. if Wirgren and Lawrence had come up with a better alternative for the LoTT

 

1. Well, HCP count works fine until a fit is found. That's why people teach it :) After fit is found, HCP needs corrections (points for singletons, voids, etc). In fact, it is much like the LOTT + corrections.

 

2. They did present an alternative: the SST/WP stuff. Just that it's a bit too complicated to use at the table. But yeah, I tend to agree that LOTT + corrections, while not ideal, should be good enough for most practical cases.

[campboy]

That 5% significance level is reserved for testing new prescription drugs. In bridge we settle for being right clearly more often than the other guys.

On the contrary, I think we should expect more convincing proof in this case than for a medical study. It should be trivial to get huge amounts of data on the LoTT, just running random hands through a double-dummy analyser. Even with the resources of a massive multinational company, you just can't do medical trials on enough people to compete.

[helene_t]

It's better said "relationship between average tricks and trumps". But anyway, while the parabola-like relationship actually seems plausible, making a case for it it needs more evidence than what you present. Some data and statistics is needed to prove it at, say, 95% confidence level.

Prove what? Significance levels are used when you are interested in rejecting a null hypothesis. Here the null hypothesis might be that the LOTT is accurate but we know that that isn't true. If you want to prove that the parabola is "correct" then you could postulate an noninferiority hypothesis, i.e. the hypothesis that the optimal parabola is not so much worse than the optimal generic model that it matters for practical purposes.

 

Alternatively, you could test the hypothesis that the lott is not so much worse than the optimal parabola that it matters for practical purposes.

 

Either way, we need to agree on what accuracy level is relevant for "practical purposes".

 

But I agree with Campboy that for simulation studies, 95% confidence is not enough. Just keep sampling until the 99.9% confidence interval is small enough to be insignificant for practical purposes.

[navahak]

2. They did present an alternative: the SST/WP stuff. Just that it's a bit too complicated to use at the table. But yeah, I tend to agree that LOTT + corrections, while not ideal, should be good enough for most practical cases.

 

I think SST+WP is replacement for HCP+SSP instead of LOTT. It is surprising accurate at predicting own side trick taking potential if bidding manages to transfer SST information. Too bad in many sequences SST is unknown value making predictions poor. Of course one could try to predict opponents SST+WP based on trump suit fit and HCP for on side but that has a lot variables to guess and needs a lot work.

 

In table one needs to adapt evaluate method depending on own hand and information available from partner and opponents.

[campboy]

Simulation by Matt Ginsberg(published in Bridge World, Nov 1996)

Length Samples Total tricks

14 46944 13.85±0.63

15 47281 14.86±0.64

16 120525 16.10±0.70

17 102184 17.02±0.75

18 69792 17.99±0.83

19 37561 18.78±0.87

20 15845 19.50±0.99

21 5041 20.11±1.20

22 1286 20.69±1.48

23 237 21.22±1.83

24 45 21.78±2.27

 

LoTT works fairly well when total trumps is 18 or shorter, but with longer trumps, LoTT overestimates the total tricks and has a high variance. At this level I think SST is more important.

It's not clear to me how he's calculated these numbers. Assuming the ± bit gives a confidence interval for the expected number of tricks, the greater uncertainty for the 21+ range would just be a consequence of the low sample size; it doesn't mean there is a higher variance in the number of tricks made. However, the actual numbers given make this interpretation a bit implausible.

[helene_t]

The +/- numbers are much too large to indicate confidence intervals unless the sample size is ridiculously small. Ginbergs public database contains about 800,000 hands.

 

it would be interesting to see how much variance can be explained by specific adjustment factors such as purity, shortness duplication and double fit.

[Trinidad]

It's not clear to me how he's calculated these numbers. Assuming the ± bit gives a confidence interval for the expected number of tricks, the greater uncertainty for the 21+ range would just be a consequence of the low sample size; it doesn't mean there is a higher variance in the number of tricks made. However, the actual numbers given make this interpretation a bit implausible.

The larger range is not due to the sample size.

 

What Ginsberg seems to have done is simply create a lot of deals and let a computer play them. Since it was Ginsberg, he probably used GIB. Then he kept track of the number of tricks and the number of trumps. So, in a way, we can say that he determined the number of tricks experimentally.

 

In my garden, I have an apple tree with about 30 apples. I could weight each apple and calculate the average weight and the standard deviation. Let's say that I find that the average is 110 g with a standard deviation of 15 g. What does that mean?

 

Does it mean that all apples are 110 g, but that my scale is so poor that for one apple it gives 110, whereas for an other it gives 125 g? Or does it mean that some apples weight 110 g and others 125 g? Unless my scale is really poor, and I have a magic apple tree, I would think the latter. That means that the standard deviation represents the standard deviation of the apple weight distribution function. It does not represent the error in my scale.

 

If instead of one tree, I would have an orchard full of similar trees, I could weight all these apples. Would this standard deviation decrease? NO! Because the standard deviation is a property of the apple weight distribution. I will be able to determine the average weight more accurately, and I will be able to determine this standard deviaton more accurately, but there is no reason to assume that their values will increase or decrease.

 

The same holds for Ginsbergs experiment. There are deals with a larger amount of tricks than trumps and there are deals with less tricks than trumps, just like there are heavy apples and light apples. The deviation in the number of total tricks is a property of the total trick distribution of bridge deals. If he would have used more deals, he would have been able to determine the average number of tricks, as well as the standard deviation, more accurately, but there is no reason why the value for the standard deviation would decrease or increase. From the start, the reported standard deviation has been the best estimate for the standard deviation of the total trick distribution. More measurements will lead to a better estimate for this standard deviation, but we cannot say whether it will be higher or lower.

 

Rik