# BBO Discussion Forums: LoTT is a parabola - BBO Discussion Forums

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## LoTT is a parabola not a straight line

### #1jogs

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Posted 2014-November-04, 15:47

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.

The parabola is a better fit for tricks to trumps.

This is only an illustration of what the relationship looks like. I do not know the exact parabola.
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### #2mikeh

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Posted 2014-November-04, 16:02

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.
'one of the great markers of the advance of human kindness is the howls you will hear from the Men of God' Johann Hari
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Posted 2014-November-04, 16:28

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
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### #4johnu

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Posted 2014-November-04, 16:56

mikeh, on 2014-November-04, 16:02, said:

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 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.
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Posted 2014-November-04, 16:59

Borrowing a line from Pirates of the Caribbean, maybe we should call it the Guideline of Total Tricks
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### #6whereagles

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Posted 2014-November-04, 17:38

jogs, on 2014-November-04, 15:47, said:

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).
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### #7jogs

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Posted 2014-November-04, 19:44

whereagles, on 2014-November-04, 17:38, said:

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.

Quote

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.
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### #8mikestar13

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Posted 2014-November-04, 20:07

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".
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### #9Siegmund

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Posted 2014-November-04, 20:19

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.
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### #10nige1

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Posted 2014-November-04, 21:47

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.
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### #11mike777

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Posted 2014-November-04, 23:15

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).
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### #12SteveMoe

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Posted 2014-November-05, 00:13

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".
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### #13mike777

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Posted 2014-November-05, 00:35

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
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### #14yunling

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Posted 2014-November-05, 01:19

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.
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### #15GreenMan

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Posted 2014-November-05, 01:25

jogs, on 2014-November-04, 15:47, said:

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.

jogs, on 2014-November-04, 19:44, said:

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?
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### #16yunling

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Posted 2014-November-05, 01:42

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.
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Posted 2014-November-05, 02:10

whereagles, on 2014-November-04, 17:38, said:

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
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The most exciting phrase to hear in science, the one that heralds the new discoveries, is not “Eureka!” (I found it!), but “That’s funny…” – Isaac Asimov
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### #18helene_t

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Posted 2014-November-05, 04:04

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?
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### #19whereagles

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Posted 2014-November-05, 04:21

SteveMoe, on 2014-November-05, 00:13, said:

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.

yunling, on 2014-November-05, 01:42, said:

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.]

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.
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### #20campboy

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Posted 2014-November-05, 04:28

jogs, on 2014-November-04, 19:44, said:

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.
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