More then 50 years have passed, since the Law of Total Tricks (LoTT) was discovered in 1955. So, let me briefly
sketch the history of LoTT.
We owe this history to three big names, Jean-René Vernes, Larry Cohen, and Matthew Ginsberg.
Their contributions may be characterized by Discovery
, and Verification
History of LoTT (1). Jean-René Vernes (1966)
It was the genius of Vernes that invented a revolutionary concept
According to an interview
held in Paris on 10, September, 2000, he answers,
“J'ai decouvert la loi des levees totales vers 1955. J'ai
commence a en parler, a partir de 1958, dans une serie d'articles,
et je l'ai publiee sous sa forme actuelle en 1966 dans « Bridge Moderne de la Defense »
(Editions du Bridgeur, 4e edition, juin 1987).”
Bridge is really a game where players
compete in the difference in tricks they take, so it is difficult to
imagine what significance the sum will have.
As for the Total Tricks, Vernes discovered the following law and named it the
Law of Total Tricks (LoTT, la Loi des Levées Totales) .
 In the Suit vs Suit case, (p.38, French original)
Total Tricks = Total Number of Trumps in both sides.
 In the Notrump vs Suit case, (p. 45-46)
Total Tricks = Number of Trumps (in Opponents) + 8, 7, 6.
Here, the three numbers on the right-hand side correspond to a void,
singleton, or doubleton in a side suit of either hand of suit-contractors.
In his book, the above formula  for Notrump was
given in a table of 3×
4 = 12 cells, which makes the Law formula seem somewhat complex, but
In expectation of how Vernes himself theoretically
accounts for his Laws above,
I purchased his book for € 31.37. The result was totally against my expectation.
His study was not theoretical. He gathered and studied world-championship deals over 10 years,
and discovered that the Law holds statistically.
If we are to do such a study today, we could use personal computers and collect
lots of information through Internet. In 1950's, no personal computers were
available, nor even pocket calculators. But it appears that he was quite determined.
He writes in his preface (my translation).
“Knowledge of a precise law is necessary, which is comparable (in accuracy) to average 26 points
for attaining 3NT and 33 pts for 6NT. Publication of this book
has become possible only because we have found such a law.”
In addition to the above Laws  and , he went as far as to discover
the Rule for the Safe Level (p.44)
[Rule] You can safely bid to the level equal to the number of trumps held by your side.
This Rule is now so popular among Bridge players, but in this article, we will focus on
the LoTT itself and seldom return to the Rule.
History of LoTT (2) Larry Cohen (1992)
Vernes's book written in French remained unnoticed for a long time,
perhaps because it was not in English. Almost 30 years later, LoTT has got
all of a sudden popularized in the hands of Larry Cohen.
His arguments based on the Chart logic was very convincing to explain
the relationship between the Law and the Rule.
As to this relation between them, Vernes gave no good explanations. He simply proposed
a practical Rule and gave no theoretical support to it based on the Law.
Cohen clearly showed that the Chart logic bridges a gap between them.
In good applications, separate understanding of
, Rule (for the Safe Level)
, and Chart logic
is necessary. For example, LoTT is independent of HCP, vulnerabilities,
and scoring system (IMPs or MatchPoints), whereas
the Rule and the Chart depend on them.
As for the Law formulas, Cohen agrees with Vernes on  above in the Suit-Suit case.
However, in Notrump-Suit case, he simplifies (as he writes)  to
Total Tricks = Number of Trumps (in Opponents) + 8, 7½, 7.
History of LoTT (3) Matthew Ginsberg (1996)
In the computer age, rather than studying the live hands actually played (as Vernes did),
Ginsberg tried on statistical
verification of the Law
on a large number of random deals generated on a computer.
In this respect, he was in a singular position since he had developed a
double-dummy solver for the purpose of using it in his software GIB
He has put 446,841 random deals into his solver, and found that the result
closely verifies the Law formula 
for the Suit-Suit case.
gives only numerical result in two tables,
from which I have constructed a figure (click)
Here, the ordinate and abscissa correspond to the left and right-hand sides of eq .
The blue straight line with inclination 45°represents eq , while the
stand for the average Total Tricks available for give Total Length.
Concurrently with his work, he generously released a library of 717,102 random deals (together
with results of double-dummy analysis) and left it in his
, which is now unfortunately closed
because of spin off.
. Object of This Work
So far, so good for the first law (for suit contracts).
Now, the second law (for Notrumps) remains to be studied in the same way.
For this purpose, I have developed two softwares ViewDDLlib and LottAnalyzer,
which are now open for public use.
But, wait for a moment ! My software should work as nicely as Ginsberg's does, when his setting (or, description)
is followed. It must reproduce previous results for it to be called scientific.
So, I aim:
(A) To reproduce Ginsberg's result according to his description,
(B) To work out statistics, with Notrump and Suit deals treated separately.
. Reproducing Ginsberg's Work
In this section, I will try to reproduce the result of Ginsberg, as far as possible.
Although the simple description of Ginsberg contains some uncertainties (or artifacts),
I chose the options called "Ginsberg" and "Length" in my
Here, all the 717,102 deals are played in suit contracts.
Notrump calls are forbidden at all. As a result, for example, balanced hands with 3NT are forced to bid
some suits, which cannot possibly attain a game.
Trump suits are selected solely by Length.
Score nor priority of contracts are never considered.
For example, when 4
and 4 are both makeable, Clubs are automatically selected if longer than Spades. Furthermore, Clubs will remain as such
and will have to concede to opponents, if they compete with 4 or 4,
even though we can make 4 with shorter Spades as the trump suit.
Artificial declarers, although quite dubious, possessing fewer trumps than partner, are allowed to declare in this
virtual Bridge (#1485).
The result is shown below in a table and compared with Ginsberg's,
|Reproducibility check of LottAnalyzer for Ginsberg's result
||LottAnalyzer ( Ginsberg + Length Option)
|Total Length||number of samples||average of deviation||average error
||Total Length||number of samples||average of deviation||average error
|14 ||46,944||−0.15||0.63 ||14 || 75,608||−0.15||0.63
|15 ||47,281||−0.14||0.64 ||15 || 75,592||−0.14||0.64
|16 ||120,525||0.10 ||0.70||16 ||193,690||0.10 ||0.70
|17 ||102,184||0.02 ||0.75||17 ||163,632||0.02 ||0.75
|18 || 69,792||−0.01 ||0.83||18 ||111,997||−0.01||0.83
|19 || 37,561||−0.22 ||0.87||19 || 60,416||−0.21||0.86
|20 || 15,845||−0.50 ||0.99||20 || 25,545||−0.50||1.00
|21 || 5,041||−0.89 ||1.20||21 || 8,123||−0.89||1.20
|22 || 1,286||−1.31 ||1.48||22 || 2,035||−1.28||1.46
|23 || 237||−1.78 ||1.83||23 || 396||−1.83||1.87
|24 || 45||−2.22 ||2.27||24 || 68||−2.22||2.25
|Total||446,741||−0.05||0.75||Total||717,102 ||−0.05 ||0.75
Obviously, the two results agree very well (as they should)
difference in sample size.
Statistics is really reliable, when average is taken over a vast ocean of ensemble.
This good agreement means the following two:
(1) My LottAnalyzer is running nicely (at least in the Ginsberg setting)
and perhaps the data handling is right (in particular, correctly reading his double-dummy library file).
(2) It has now become apparent that Ginsberg did not pay due attention
to trump suits, because neither did I in the above analysis (right).
He simply chose longest suits as trumps and paid no attention to score nor to Notrump play.
Having established this, we go on to the next step.
I am yet wondering about the difference in the sample size, 446741
Didn't he use this library in his analysis ?
. Separate Treatment
We now take the statistics, after dividing the deals into two categories, Suit and Notrump.
Here, Notrump hands are required to take 7 tricks or more in Notrump and less score in
suits. Otherwise, deals are classified into the former.
As a natural consequence, 4 and 4
are preferred to 3NT, while
3NT is preferred to 5 and 5.
As a result, 717102 deals in total are divided into 501591(suits) and 215511(NT).
Several artifacts mentioned above in  are now removed.
They are mostly overcome by considering Score rather than Length in
determining the strain. Surely, longer suits do not necessarily bring better score,
as obvious from the above mentioned example of 4 and 4.
Once competition takes place, however, (i.e., when our high-score contract is
overwhelmed by their contract), other strains with more tricks (but lower score) are pursued.
Say, we have 3 and 4, both makeable.
We will remain in 3 so long as competition does not take place.
But if opponents are able to bid 4, we change our denomination to 4.
This is an example where competition changes trump suits.
In addition, artificial declarers are forbidden, by requiring them to have
longer trump suit than partner,
(more HCP, when equal in length),
more HCP than partner in Notrump.
With these improvements on Ginsberg's work, we obtained the following result, which will be shown
separately for Suits and Notrumps.
. Suit versus Suit
Although I tried best improvements (so I think) on Ginsberg's work,
the result turned out to be similar to his.
Comparison will be made now in the format below:
Here, deviation will mean
Number of Total Tricks − Number of Total Trumps,
|Comparison between Ginsberg
and LottAnalyzer for Suit Contracts
|Ginsberg (total.ps.gz)||LottAnalyzer (Standard+Score Option)
|+1 trick deviation ||22.4% ||31.6%
|0 trick deviation ||40.0% ||38.8%
|−1 trick deviation|| 24.5% ||14.9%
|Average Error(tricks)||0.75 ||0.79
For example, "+1 trick" means that LoTT underestimates Total Tricks by
"Average deviation" means its average over the entire deals.
"Average error" stands for average of its absolute magnitude (|deviation|).
It is different from the standard deviation commonly used in statistics, but we follow here the
convention started by Jean-René Vernes.
From this table, we observe that both yield an almost equal value for the average error,
0.75 and 0.79 tricks, respectively.
So, an error of 0.8 tricks is the best value known from the double-dummy analysis.
As a matter of fact, I have started this work in the hope of reducing the average error (or rather, variation), by proper selection of trump suits.
Notrump hands have been excluded in LottAnalyzer, while Ginsberg includes them as suit
contracts. Nevertheless, my efforts did not reduce the average error.
So far for error. As for the magnitude of deviations, LottAnalyzer tells us that
the LoTT underestimates Total Tricks by 0.37 tricks on the average.
This is most clearly seen in the figure output from LottAnalyzer (on the left)
In most frequent cases of 15
total trumps, the LoTT almost constantly underestimates
by 0.4 tricks, and will tend to overestimate with increase in total trumps
(for more details, ask LottAnalyzer).
Overall behavior is quite similar to the one
you have already seen above
(to appear now in a separate window),
except a vertical constant 0.4 tricks. Just view them in parallel and compare.
Conclusion to , Suit Contracts
So, what to conclude ?
Ginsberg's work revealed that if you choose longest
suits as trumps, you
can most profitably expect the LoTT to hold on the average.
In bridge table, however, trump suits are determined through more deliberate considerations.
Length is certainly important, but it cannot be all.
My best treatment of trump selection together with exclusion of Notrump hands in
LottAnalyzer tells us that LoTT will underestimate total tricks by
You might say, "Oh, that is too simple. I am already doing positive adjustments
with a long side suit".
You are indeed right in doing so,
You have to remember here that Ginsberg's analysis and mine are statistical by nature.
The conclusions drawn from them are almost exact because of the huge sample size, but
they can tell us nothing for each deal. You have to do right judgments for each deal at
Ginsberg's result shows that if you take the longest suit (instead of
the trump length) in the LoTT formula,
you will need no more corrections on 40% deals. Further corrections are necessary on
60% deals. Since the frequencies of positive and negative corrections are nearly equal,
they cancel out on the average
and makes it more appealing.
As for the conventional LoTT formula, the LottAnalyzer tells us that
we need no corrections on 39% deals. The positive
average deviation 0.37
tricks will mean that one has to do positive corrections more often than negative ones.
You will pick up hands which need +1 correction twice more often than those which need −1 correction.
These are the conclusions we learn from the statistical LoTT analysis.
. Notrump versus Suit
Now, we proceed to deals where one side is going to play Notrump.
Here arises a problem as to which formula to take as the LoTT for Notrump.
Most popular version is the one due to Larry Cohen (in To Bid or
Not To Bid), which reads,
Total Tricks = Number of Trumps (in Opponents) + 7.
to which he recommends +1, or +½ corrections according as opponents
have a void or a singleton.
Earlier, however, Jean-René Vernes
had proposed his own formula:
Total Tricks = Number of Trumps (in Opponents) + 8, 7, 6.
which depends on the distribution of side suits (of either hand) of opponents.
Here, I would confuse to add one more (and call it Modified Vernes)
Total Tricks = Number of Trumps (in Opponents) + 9, 8, 7.
by adding One to the Vernes's formula,
and compare all the three on
LottAnalyzer. The result is interesting:
|Results of LottAnalyzer
for “Notrump vs Suit” Contracts under the Three Different Law Formulas
|Law Formula||Cohen||Vernes||Modified Vernes
| +2 trick deviation ||13.5% ||21.5% ||5.0%
| +1 trick deviation ||32.5% ||40.8% ||21.5%
|0 trick deviation || 37.3% ||25.4% ||40.8%
| −1 trick deviation || 11.4% ||5.9% ||25.4%
|Average Deviation|| 0.62 tricks || 0.95 tricks || 0.05 tricks
|Average Error || 0.87 tricks || 1.09 tricks || 0.73 tricks
I added a row for the +2 deviation, since it occurs so often in Cohen and Vernes.
If you are to choose one among the three according to a good sense in statistics,
is best, not
it best accords with the law (with the 0.05 average)
because it has least average error (or least variations).
The graph on the right plots the result for this case. You see the crosses in wine color
nicely sit on the blue line predicted by the proposed law formula.
So, it is the best among the three.
However, to a great regret of Vernes, his correction has turned out to be worst.
It was perhaps unavoidable since only 73 deals were available to him (which were played
on two tables, one in NT and the other in a suit, and the suit had to be common).
. Independence of LoTT from HCP
Conclusion to , Notrump versus Suit
We have found that the Modified Vernes
most reliable in Notrumps, with an average error (0.73 tricks) comparable to that in suits (0.79 tricks).
It works better than the Cohen formula
Nevertheless, I will recommend here Cohen's law formula, which is simple and easy to keep in memory.
You have only to learn the magic number 7
by heart, and consider
corrections according to your hand and the bidding sequence.
You will sometimes try a correction of TWO
tricks with a void, remembering the Modified
Vernes law formula.
This note is for someone who believes that the LoTT holds only in a limited range of HCP,
or believes that Vernes so said.
In truth, Jean-René Vernes admits three kinds of corrections in the LoTT.
However, he speaks nothing on HCP concerning LoTT.
It is only when he comments on his Rule (la Règle de Sept à Douze, the Rule for the Safe Level)
, English version
that he mentions something on HCP.
So, the requirement for HCP is NOT
for his LAW, BUT
for his RULE.
Remember that Vernes was careful in his writing.
LottAnalyzer easily confirms this:
Here, following his advice, we pick up the ranges 15-
(for suit contracts only)
under the Four Different Requirements for HCP
|| 0-14, 26-40
| +2 trick deviation ||9.7% ||9.5% ||9.3% ||10.2%
| +1 trick deviation ||31.5% ||31.5% ||31.5%||31.0%
|0 trick deviation || 38.4% ||38.8% ||39.2%||37.0%
| −1 trick deviation || 15.2% ||14.9% ||14.9%||16.0%
|Average Deviation|| 0.36 tricks || 0.37 tricks || 0.36 tricks||0.34 tricks
|Average Error || 0.79 tricks || 0.79 tricks || 0.78 tricks||0.82 tricks
|Sample Size||506,581 ||389,908 ||280,383 ||116,673
LottAnalyzer, on which this article is based, is open.
Download from my site below and use. You will find more tables and graphs.