### Are You LoTTer ?

(C) boco_san (2003/08/15) 2012/07/26.
[ Translated into English and appended, 2009/May-August.]
Stop abstract asking whether or not LoTT is valid in general.
once you've learned the mechanism of LoTT
[quoted from below].

### « The Law of Total Tricks (LoTT) »

−−−  Introduction  −−−
1.  Just to start
2.  How to get on with the LoTT;  the Rule for the Safe Level
3.  References
−−−−  Basics  −−−
4.  What is LoTT ?  (Chart #1)
5.  Accuracy of LoTT
6.  Accuracy of LoTT, a little more
7.  Why does LoTT hold ?  −− the Mechanism of LoTT
8.  Fundamental Postulates of LoTT
10.  LoTT in Notrumps  (Chart #2)
11.  The Rule for the Safe Level, or Vernes Rule
12.  LoTT Holds Independent of HCP.
13.  Watch Chameleons
−−−  Very Simple Applications  −−−
14.  Watch the Number of Trumps
15.  LoTT in Weak Two  (Chart #3)
16.  LoTT in Higher Preempts  ( Chart #4,   #5,   #6 )
So, you, who have understood the basics of LoTT,
you'll learn bidding better than ever.

Just to start

1. Hi, dear boco_san, what are you going to talk about, this evening ?

--- A bit about the LoTT.

2. Nice to hear from you. In the Social Lounge, we are playing in, we sometimes hear about LoTT, after bitter competitive auctions.

--- For sure, the law (the Law of Total Tricks, LoTT) is useful in judging to what level one can safely bid.

3. But there are too many things to learn in bridge bidding, and quite a bit on card-play. Does a beginner like me have to learn LoTT now ?

--- Well, it's a difficult question to answer. Larry Cohen, who distributed the LoTT world-wide, says in his book (1992) that LoTT is the fundamental concept, even when teaching a novice.
In the current bidding system, point-counting method (you count A=4, K=3, …) is essential, as you know. You can't bid without point-counting. However, it works as far as the auction remains uncontested (i.e., opponents don't overcall). In contested auctions, the point-counting becomes less useful, and LoTT takes its place. This is just what Jean-René Vernes declares in his book (1966), who discovered and established the LoTT.
So, these days, I've come to speak often about the LoTT in the Social Lounge. Perhaps I've made a LoTTer myself.

4. Is it really necessary for starters?

--- I think no. What's really fundamental for starters is to learn the bidding system according to the point-counting. Only after you've learned the system through once, switch yourself to learning the LoTT.

How to get on with the LoTT

1. And, … what on earth is the Law ? What does it benefit me, if I come aware of it ?

--- I thought I could write well what you want, when I started writing. But, … it turns out to me a hard work, and I find myself in difficulties.

--- Sorry. References below may be of some help to you. I'm yet unsure of myself writing in full on LoTT. I promise I'll write in full for you after having considered much more on LoTT and having finished probability calculations.

3. Probability? Does it have something to do with LoTT ?

--- Yes, something for sure, you'll know finally. At the moment, let me talk a bit something which I've ever learned from my experience.
[1] Concerning the LoTT, I did read explanations on various web sites. I did also read the "25" book below. However, such a short (a few page) explanations were utterly insufficient to convince me. That was my first experience.
I think I used to understand rather quickly what I read on bridge bidding, say, written by Mike Lawrence. However, LoTT was exceptional in this regard. So, I went on to the two volumes of Larry Cohen (1992, 1994), and even to the French original of Jean-René Vernes (1966).

4. What do you mean, boco_san ? What's important for begin-ners like me ?

--- [2] What's important is the TWO points you just asked me above:

(a) What is LoTT,

(b) How to use LoTT.
Theoretically, you should start with (a) understanding LoTT, and then (b) apply LoTT. This is certainly the right way, but it doesn't work.

5. Why ?

--- Because you'll take much time in (a), which will be boring for you. So, you learn (a) and (b) in parallel, and go back and forth in between ….

6. I don't quite understand what you mean. But, how about (b) using it, without knowing details ?

--- That's quite easy. You've only to follow this rule:

(c) [Rule for the Safe Level]
You can safely bid to the level equal to
the number of trumps held by your side.
This rule was first enunciated by Jean-René Vernes in 1966, but it remained almost unnoticed for a long time, perhaps because the book was written in French. It became all of a sudden popular in the hands of Larry Cohen, in 1992. This rule, (the Rule for the Safe Level, to be detailed below), gives us a good guideline in defensive or competitive auctions, as the title of Vernes's book indicates.

7. Then, are there TWO independent concepts I'm going to learn ? The LoTT (on which I know yet nothing) and the above rule (c) ?

--- Right TWO, but not independent of eath other. You'll see later how LoTT guarantees the above Rule (c).
For now, suppose you and your partner have 9 cards in spades, then you can safely bid up to 9 tricks, or 3, in contested auctions.
This is the easiest way to start learning the LoTT.   Without details, just try and use the above "Rule for the Safe Level" in contested auctions. You'll sometimes succeed. Then, consider why you've succeeded. You'll sometimes lose following the above rule. Then, consider why you've lost.

8. You speak a little too easy, don't you ?

--- No. However my voice may sound irresponsible, I believe this is the best way to start learning the LoTT.
Well … , during you're practicing the Rule (c) above, I'll teach myself more on LoTT and will talk later again. So long for today.

9. Oh dear! Don't flee! Just a moment, please. What do you mean by your phrase "can safely bid"? How can I be SAFE ? Even if we have 9 spades in our hands, the 3 contract will go down, if we don't have enough HCP ….

--- Let us see a sample auction from Vernes's book:
 West North East South 1 1 3 3?
This is a good example of contested auction. West opened 1 and North overcalled 1. In this situation, how many hearts do you think West has, and how many Spades North does have ?

10. That's so easy. In the 5-card major system, West should have 5 or more in hearts. North's overcall guarantees at least 5 cards in spades.

--- In that situation, East has now bid 3, which may be understood either as a limit-raise or as a pre-emptive raise. He has perhaps 4 (or more, or good 3) cards in hearts. It doesn't matter here.
Now, suppose you sit on South and consider what to bid.

11. How do you want me to consider ?

--- If the auction went uncontested (i.e. if East had passed), you'd judge according to the point-counting system.
However, in contested auctions like this, the point-counting doesn't matter. Number of trumps becomes all important. You just add the partner's 5 and your number of cards in spades, and will bid 3 if the sum becomes equal to 9, and will pass if it's smaller.

12. What a simple decision! Bidding relying on the number of trumps alone.

--- Yes, it's all simple, and it's what I'm asking you to prac-tice.

13. … when I bid 3, we will score 9 tricks with good hands. But, with bad hands … .

--- will go down. You'll go down in some cases. But when your hands are so bad, opponents could have easily achieved 3, or even could have bid 4 to gain more than you lost in the 3 contract.
So, you'd better bid 3 rather than pass, in order to make your loss smaller (when you've 4 spades in your hand).

14. Is that what the Rule (c) above means by "can safely bid" ?

--- Exactly.

References

1. Barbara Seagram & Marc Smith: "Bridge 25 Ways to Compete in the Bidding" (Master Point Press, 2000) pp. 220.
In Chap 24 (pp. 205-212), the authors give a brief account of the LoTT and the Rule for Safe Level, with a few exercises.
2. Karen's Bridge Library: A simplified guideline is provided for beginners to compete based on Rule for the Safe Level.
3. Paul Lanier's web site:(expired)
The author used to give a series of lectures on application of LoTT in match-point competitions.
4. Jean-René Vernes: "Bridge Moderne de la Défense" (Éditions Le Bridgeur, 1966, 4-th Ed., 1987) pp.195, €11.37.
The original book written in French by discoverer and investi-gator of the LoTT.
5. Jean-René Vernes: Bridge World, retrieved in my web page.
A résumé of Chap 2 of the above original book is given in English. You'll see here everything about LoTT, both funda-mentals and applications.
6. Bridge Guy's Glossary, Jean-René Vernes:
Same as above, but in larger font.
7. BridgeHands: Encyclopedia of Bridge Terms, Law of Total Tricks
Another résumé of Chap 2, followed by counter-examples raised by Anders Wirgren.
8. Larry Cohen: "To Bid or Not to Bid: The LAW of Total Tricks" (Natco Press, 1992) pp. 286.
This book introduced and popularized the LoTT to all over the world.
9. Larry Cohen: "Following the Law: The Total Tricks Sequel" (Natco Press, 1994) pp. 173.
Another one by the same author, which stresses on deviations (adjustments) from LoTT.
10. Larry Cohen: "Larry and the Law", three articles in Better Bridge, Vol. 9 (2004/2005), for intermediate players, #1, #2, #3 (PDF).
11. Colin Ward: Kaleidoscope Series, Lesson 4,
« LOTT versus "Got More, Bid More" »
Two ways of bidding are explained in contrast. One based on LoTT, the other conventional.
12. Matt Ginsberg: "On the Law of Total Tricks" Bridge World (1996 Nov). Now that his site has expired, read this paper retrieved in my homepage as an HTML file.
13. Anders Wirgren: Discussing the Law of Total Tricks.
The author discusses how the LoTT doesn't hold in many cases. Much to learn about LoTT from anti-LoTTers, Lawrence and Wirgren, than from LoTTers.
14. Mike Lawrence & Anders Wirgren: "I Fought The Law of Total Tricks" (2004) \$17.95, available from the above site.

What is LoTT

1. Oh, good! You've come back to continue … .

--- Yes, I've written what I had planned at the outset.
I tried to write this so as to be understandable to beginners. But anyhow the LoTT is not so easy to grasp, and I ask you to read carefully.
It's pretty long, moreover.

2. Then, what in the world is LoTT ?

--- The answer is simple. The LoTT claims this equation,

(d) Number of Total Tricks  =  Number of Total Trumps.
Let us go back to the last sample auction. Here, suppose West-East have 8 cards in hearts, and North-South 9 spades. Then, the right-hand side of this equation (d) is
Number of Total Trumps  =  8 + 9  =  17.
This is easy to see.
On the other hand, the Number of ToTal Tricks on the left-hand side is a novel concept put forth by Vernes, nearly 50 years ago. It means
the sum of the Number of Tricks which West-East can take on their hearts contract, and the Number of Tricks which North-South can take on their spades contract.
It was a revolutionary concept in bridge in the sense that no one had ever thought of it, or its usefulness. LoTT declares that both sides should be equal. Since this law concerns the number of Total Tricks, it was named as such by Vernes (the Law of Total Tricks, la Loi des Levées Totales).
It's important to notice here that LoTT says something only on the ToTal Tricks. It says nothing about how many tricks each side will take. That's why the concept was so revolutionary.

3. Oh, LoTT doesn't tell us how many tricks we can take, really ?

--- No.

4. If no, how can I profit from the LoTT? How will LoTT affect my bidding?

--- That's one of the central questions in learning the LoTT. Although LoTT itself is concerned with Total Tricks, you can judge, using LoTT, to what level you can safely bid in con-tested auctions.

5. Still very unclear. Go back to the sample auction and explain.

--- With pleasure. In this sequence, suppose that the Number of ToTal Trumps is known to be 17, as we counted above.
 West North East South 1 1 3 3?
LoTT tells you that the Number of Total Tricks should then be equal to 17. What do you know from this?

6. Is there some way to make use of that value 17 in my bidding?

--- Yes, certainly.
Since the deal has 17 tricks in all,
 (A) If W-E have 2 (8 tricks), N-S will have 3 (9 tricks). (B) If W-E have 3 (9 tricks), N-S will have at most 2 (8 tricks). (C) If W-E have 4 (10 tricks), N-S will have at most 1 (7 tricks).
This is all LoTT tells you. Think about the case (C) now, assuming both are vulnerable.

7. Well … if they bid and make 4, they will gain (30 x 4 + 500) = 620 pts.

--- In the event that they feel unsure of their game, and double your 3 ?

8. In that case, we will take 7 tricks, going down 2, and lose 500 pts, … oh, yes, our loss 500 is less than their gain 620, so I should definitely bid 3 here.
Now, I've understood what "can safely bid" means. But I need to consider the other cases as well, … isn't there some easier way to compare ?

 Chart for 17 Total Tricks Both Vulnerable N-S play 3 W-E play 3 N-S Tricks N-S Score W-E Tricks N-S Score (A) 9 +140 8 +100 (B) 8 −100 9 −140 (C) 7 −200 (3) 10 −620 (4) (C') 7 −500 (3X) 10 −620 (4)
--- Larry Cohen wisely devised what he calls a chart.   At the bottom of this chart, you'll see the score you've just counted.

9. Yes, the row (C') clearly shows that we lose less by bidding 3 than they gain with 4.
When I go up and look at the row (B), …. In this case, 3 goes down one, and we lose 100 pts, but if I pass, they'll make the 3 contract and will gain (3 x 30 + 50) = 140 pts. Again their gain surpasses our loss. In every case, the chart invites me calling 3.

--- That's the way to look at the chart. I'll recommend you to make this chart for some other situations, for your sake. In his book, Larry Cohen is so kind to assign readers an exer-cise to work with a chart. Such a handwork will make your understanding deeper, Cohen surely assures.

10. Yep, but it's impossible to prepare a chart every time I bid.

--- Sure, but what's really required in bidding is the reason-ing you acquire from preparing and looking at charts. Cohen gave us a nice tool to enter "LoTTology".
And, for the above example, be sure to confirm that you can safely bid to the 3 level (9 tricks), holding 9 trumps, which means validity of the Rule for the Safe Level.

Accuracy of LoTT

1. When the above law is stated (repeated here for conven-ience) so definitely
Number of Total Tricks  =  Number of Total Trumps,
and is called Law, it sounds like a theorem in mathematics. Does it work so well ?

--- When Jean-René Vernes discovered and put forth the LoTT, he examined a number of hands and confirmed its validity.

2. A number of …, how many ?

--- He says, after having invented the law in 1955, he searched among 2444 deals played in 1953-1963 ten [sic] world-championships. Just think about, there were no personal computers at all, nor even any pocket calculators in the world, in 1966. The handwork would have needed consid-erable patience, I can't even imagine.
Out of those 2444 deals, he found 340 in all, which were declared by both sides with different suit contracts. His final examination was done for those 340 deals, and he found that 272 deals (80%) obey the LoTT within tolerance of one trick. In other words, LoTT holds with accuracy 80%, within an error of one trick. (The remaining 20% is for deviations of 2 tricks or more. He also obtained an average error (écart moyen) of 0.93 tricks.)

3. Yes, thank you.
Now I understand the situation well, and I do understand why so often you like to refer Vernes, as well.

--- Your thank had better be turned to the labor of Jean-René Vernes, I think.

4. I'd also thank the labor of Vernes. But the error of one trick, you say, doesn't seem a bit too large ?

--- It will depend on players whether one trick tolerance is too large or not. Anyway, one has to compare it with something else before saying it's large or small.
A good statistician Vernes again examined all the deals in 1953-1959 world-championships, which were, this time, played with the same contract (same strain on the same level) and the same declarer, and then he compared differ-ences in the tricks taken. In this way, he intended to remove from his data (0.93 tricks) ambiguities arising from different card-plays of declarer and defenders. He obtained an aver-age of 0.52 tricks for this difference.

5. What all does this mean? I'm not a statistician.

--- It means that his data (0.93 tricks) contains an unavoid-able variation of 0.52 tricks, since tricks will depend on players or card-plays.
Vernes admits the difference 0.93 − 0.52 = 0.40 [sic] tricks as an intrinsic error in his law. This interpretation of Vernes may be right as a statistician. However, for many bridge players, it's quite natural to understand the 0.93 trick average-error as significant (because it exceeds 0.52) and that the LoTT holds with an average error of 1 trick, as underlined above.

More precisely, he found,
(1)  LoTT is  +1 trick off in 28% deals (underestimates),
(2)  LoTT is just on in 33% deals,
(3)  LoTT is  −1 trick off in 19% deals (overestimates),
(4)  Average deviation is +0.275 tricks,
(5)  Average error is 0.93 tricks.
Compare with an accurate one, which appeared after 30 years.

On Accuracy of LoTT, a little more

1. Have you something to append, to start a new section ?

--- Yes. As you see from the above-mentioned study of Vernes, he frankly admits an average error of 1 trick in his law, which is, in part, unavoidable. So, instead of the above equation, which may sound rigid, he modestly says:

The Number of Total Tricks in a deal is approximately equal (approximativement égal) to the Total Number of Trumps.
This is his original statement (French original, p. 38).

2. His work is coherent, it seems to me. But, Larry Cohen appears to have a more popular version.

--- Yes. He writes:

The Total Number of Tricks available on any deal is equal to the Total Number of Trumps.
Obviously, he removed the approximate character from the LoTT and seems to believe it to hold rather rigid. This version is adopted in some bridge books. I'll not digress here, but if you're interested, visit the web site of Anders Wirgren, who gives a heavy criticism on this difference.

3. What do you think about the difference ?

--- Oh, unexpected question.
The difference is superficial, I think.
You see, Cohen believes it to hold on any deal after two volumes of adjustments.
In fact, he adds «approximately» in his second volume, and later in his "Better Bridge" article (PDF) cited above, he goes as far as to define the Law as follows:

The total number of tricks available on any deal is approximately equal to the total number of trumps.
[This version may sound as if he believes the Law is always approximate.]

4. One more question on this occasion. What do you think about opinions of anti-LoTTers ?

I quite disagree with anti-Lotters.
A number of bridge players know from their experiences that the LoTT is working, either to some extent or to a great extent, that may depend on players. To raise a simple example, LoTT gives a good theoretical support to pre-emptive bids, which no one will deny. Weak 2 bids are good examples. We have really much to learn from as well as with LoTT.

Still a Little More

In 1996, Matthew Ginsberg, well known for his Bridge software GIB, published a paper in Bridge World on the Law of Total Tricks, armed with his powerful double-dummy solver. This paper is open in his site (see References above).
Chief results he obtained from 446,741 deals are as follows:
(1)  LoTT is  +1 trick off (underestimates) in 22.4% deals,
(2)  LoTT is just on in 40.0% deals,
(3)  LoTT is  −1 trick off (overestimates) in 24.5% deals,
(4)  Average deviation is −0.05 tricks,
(5)  Average error is 0.75 tricks.
His result (1)-(5) supports the work of Vernes astonish-ingly well in view of the huge difference in the sample size, 340 versus 446,741, which will mean a 36.2 times difference in statistical accuracy.
Actually, he gave his result only in his numerical tables. To see the graph I prepared for you from his paper, click here, and come back. You'll observe that the crosses sit nicely on the straight line predicted by the Law formula.
Together with his paper, Ginsberg left a big library of 717,102 deals analyzed by his double-dummy solver. Now, a viewing software is available for this precious library elsewhere in my homepage.

Why does LoTT hold? Mechanism of LoTT

1. After having learned what LoTT is, I'd like to ask you why LoTT does hold.
Why on earth? This is a simple question, isn't it ?

--- That's the core question.
In search for its answer, I bought two books of Cohen, and even a French one by Vernes, and furthermore, searched among web pages. However, I've found none.

2. None? Didn't Vernes himself give some accounts for his law ?

--- None. Rely me on my French. Theoretical accounts are nowhere available in the 195 page volume.
He invented the LoTT and verified it on a number of deals; he relied on a solid basis of statistics, indeed, but he deduced nothing. This absence of accounts may be under-stood from the web page of Anders Wirgren, where he says,

"Lots have been written about the Law, but practically no explanation has been given."

3. A bit astonished …, because I hear LoTT so often these days.

--- So, I was forced to consider by myself. I started my reasoning from Sec.2.7 (p.46) of Vernes's book and Chap 7 (p.173-174) of Cohen's first book (where they commonly explain the formula for Notrumps). Read below, please. The reason why LoTT works may be deduced this way:

[a]  In order to argue on a solid ground, consider first a Notrump contract, and assume that cards are dealt evenly. Either side has no trumps. In this case, one side takes 7 tricks, and the other 6. The number of Total Tricks is, of course, equal to 13. So far, the law doesn't appear.
[b]  Consider next a suit contract, and again assume that cards are dealt evenly. Every player has a 4-3-3-3 balanced hand with an even HCP. Both sides will then have 7 trumps in their own contracts. So, we have
The Number of Total Trumps = 7 + 7 = 14.
Count then the number of Tricks each side will take on their own contracts. Under the above assumption of even distribu-tion, either declarer will draw trumps (possibly some losing in trumps), and will take 6 tricks and finally use his last trump to ruff, gaining 7 tricks. Therefore,
The Number of Total Tricks = 7 + 7 = 14.
Here, we find that the LoTT equation holds precisely. Note that the seventh trump used for ruff is the key in LoTT.
[c]  From the above card distribution that you imagine in your mind, switch (swap) any cards any way you like, but keep the number of trumps unchanged each player has. Winners will then go out from this side to the other, and losers from that to this. The number of tricks which this and that sides take will change by your switching. However, the total remains the same, 14. So, the LoTT holds again.
Here, you're observing an all important basis (or an assumption, as one may call) of LoTT. The total number of tricks is invariant against card switching.
[d]  Continue your card switching, but now allow change in the number of trumps. Both sides will then have more trumps, and can use them for ruff, which will increase the number of tricks.
Here, you're observing another basis of LoTT. It assumes that the seventh and further (or, extra) cards in trump suit are useful for ruff and will increase the number of tricks. Every time you've more trumps, you've so more tricks.
[e]  We arrive at the law in this way. --- Some comments are in order, see below.

4. Thank you for a good explanation. I think I've understood at least something. Is this all you thought by yourself?

--- Yes.
From the above reasoning, you'll know that the LoTT rests on some assumptions. And most disputable is the invariance (so I call) of Total Tricks under card-switching. Bridge books give beautiful examples in which the invariance is maintained, but, …

5. What do you want to say to me, boco_san ?

--- Oh, sorry. I have to explain why I told you the mechan-ism of LoTT. If you understand that the LoTT rests on some assumptions which you already know, you'll take LoTT in your own way, and you'll be able to better judge in your bidding. This is all I mean.
Stop abstract asking whether or not LoTT is valid in general. Rely on your bidding judgments, once you've known the mechanism of LoTT.

From the above mechanism of LoTT, we understand:

[f] (neglect of length). An important feature of LoTT, apparent from above, is neglect of length (except in trumps). Every card, except trumps, is expected to win by its own strength, never by length of its suit. Hence, LoTT underestimates total tricks for hands with a long side-suit, or a running suit in NT.
[g] (ruff in dummy). It counts ruffs only in the long, declarer's hand, and never in the short, dummy's hand, again underestimating when the latter (cross-ruff) takes place. This is particularly important for the player who is supposed to be dummy.
[h] (mirrored hands). When declarer and dummy have the same length in the same suits, extra trumps will not get ruff, giving an overestimate, sometimes considerable.

Fundamental Postulates of LoTT

1. I'm so sorry to say, … , but, your lengthy explanation is a little boring and is rather confusing. Couldn't you give me a more straight explanation ?

--- To put it straight, it's best to raise the posutulates on which the LoTT rests:

1. The total trick count is invariant under interchange of any cards between any players.
2. Honors win through their own strength (never by length), except that 7-th and further trumps win trick(s) by ruffing.

1. So far, so good. I've learned what LoTT is. Although it's called Law, it doesn't always work precisely. The mechanism and postulates of LoTT you talked to me may be of some help in my bidding. However …, if it sometimes works wrong, aren't there some way to predict in advance ?

--- That was the matter of good concern for both authors. What you mean by prediction is called correction by Vernes and adjustment by Cohen, and the latter author considers the problem in quite a depth. However, as we read the adjust-ments in detail, the simple LoTT becomes dressed up in full, and we are led into confusion, remembering the original simplicity of LoTT.

2. And, what does Vernes say, at least ?

--- I'll quote below all the three factors raised by Vernes (which are perhaps less accessible to you), and add a few.

[double fit] Double fit gives a positive correction, often one trick. Vernes believes this is the most important factor. {This is quite understandable from neglect of length in LoTT. When this happens, winners in long-side suit increase the gain in tricks. As regards my impression and study on double-fit, see the column below.}
[honors in trumps] When both sides have all of the honors in respective trumps, total tricks become more than expec-ted from total trumps. When some of the trump honors are held by opponents, total tricks become less (see the example hand below). But this factor is not so large as one might think, and it's at most one trick, says Vernes.
[holding in side-suits] When both you and your partner have shapely hands, e.g., you bid 4 with singleton , and partner with singleton , total tricks will be more than total trumps. {This is also understandable from the mechanism of LoTT. Before drawing out trumps, declarer can get some tricks by ruff in dummy.}
[adverse ruff]  That's all three Vernes counts. Another factor I'd like to add here is the adverse ruff. Adverse ruff takes place when a defender has a void (or singleton) in a side-suit. It decreases total tricks. As you see from the mechanism of LoTT, it assumes that extra trumps are used in declarer's hand, but it never counts ruff by defenders, and worse, the adverse ruff is unpredictable.
[control in side-suit] Another interesting factor is the control in side-suits as pointed out by Anders Wirgren.
Say you declare 4 with a long solid and weak , and wish to discard under . To achieve this, you need to have a control in . If you, declarer, have Kx, then you lose only 1 trick in . However, when your partner, dummy, has Kx, you can possibly lose 2 tricks in . In this case, number of tricks depends on which (you or partner) declares the contract. Invariance of total tricks doesn't hold in this case. This is another clear counterexample to the LoTT.

3. You write below that you've got perplexed with double fits, but I've got now perplexed much more.
How can I deal with these corrections or adjustments?

--- Don't worry so much about corrections.
You have to first learn the mechanism with which the LoTT works. Then, you'll see how and when it goes right, and how and when it goes wrong, at the bridge table. Only when you've got interest in deviations of LoTT, attack the corrections. For the moment, it's not clever to delve into details for starters.

[double double-fit]   Both authors (Vernes and Cohen) agree on importance of the double fit, but it's not a simple double fit (8 or more cards in two suits in the opposite two hands), but it's a double double-fit, as Cohen calls, which means the double fit on BOTH sides.
When I first learned both authors speaking of double double-fit, I was got in a perplex, wondering how I may detect double double-fit from the bidding sequence. Perhaps both authors found many exam-ples of double double-fit a posteriori. That's OK. But, even if I know OUR double fit from my cards and partner's bidding, how can I know THEIR double fit from their bidding? If I can't detect it a priori through bidding, I 'll learn nothing from them.
Interest in double double-fit led me to an exten-sive study of fitting probabilities (see elsewhere in this web site, for details), and I discovered at last (stupidly enough, after long probability calculations) the Double-Fit Rule (so I call), which says
When you have an 8-card double fit [8d] in your two hands, opponents will neces-sarily have an [8d], or a 9-7 double fit, or more. Their hands can never be a simple 8-card fit.
This rule means that if I confirm a double fit with partner, I don't have to mind their hands. They surely have shapely hands with an 8-card double fit, or a 9-7 fit, … wow.

LoTT in Notrump

1. So far, I've learned a lot about LoTT, which is written as a formula (repeated again for convenience),
Number of Total Tricks  =  Number of Total Trumps.
However, the right-hand side would mean that the contracts are in different two suits. What about when one side calls Notrump? Is there some similar formula available for Notrumps?

--- Yes. Cohen gave us a simple formula,

Number of Total Tricks  =  Number of Trumps + 7.

2. How do you explain it?

--- Cohen himself explains it in his first book. It's very simple, especially since you've understood the mechanism of LoTT.
Start again with a Notrump contract. The number of ToTal Tricks is equal to 13, independent of how many tricks each side will take.
When the same deal is played in a suit contract, declarer can use their 7-th and further trumps for ruff and can gain more tricks. As a result, the number of ToTal Tricks increases by an amount (number of Trumps − 6). This explains the above formula.

3. OK. Nice. And the invariance you said is assumed here?

--- Yes. The assumptions are the same as before. Number of ToTal Tricks is assumed to be invariant against switching cards.

4. Notrump contracts are sometimes declared on a solid long suit with stoppers in other suits. Does LoTT consider such long suits ?

--- No. As for long suits, LoTT takes account of only the length of trumps. It takes no care of other long suits irrespective of strain of contracts. This is evident from the mechanism of LoTT. It totally neglects length except in trumps. So, in such an event, you have to judge and consider positive corrections.

5. By the way, doesn't Vernes have his own formula for Notrumps?

--- Yes, he has. But Cohen's simple formula is practical.
I'll put the detail in a box, to avoid interference.

Vernes's Formula for Notrumps

In his book (1966), Vernes had put forth:
Number of Total Tricks = Number of Trumps + 8, 7, 6,
where, the three numbers on the right-hand side cor-respond to a void, singleton, and doubleton in any side suit of opponents.
As regards accuracy of his formula, Vernes sur-veyed 73 deals where Notrump is declared in one room and a suit contract in the other. In this way, he obtain-ed an average error of 1.03 tricks in his formula, (which is a little larger than 0.93 in suit competitions.)
But he knew from his experience that Notrump plays are more liable to happenings, so he proceeded to compare Notrump deals in general, which were played with the same level on two different tables.
As a result, he obtained an average 0.90 for the difference in tricks taken by Notrump declarers from the world championship hands (greater than 0.52 of suit declarers.)
As a whole, the Notrump formula of Vernes was found to hold within an average error of one trick.

6. That's OK for the formula itself, but how do I use it, when I call Notrump myself, or overcall against Notrumps?

--- The situations in which you use the above two formulas are very different.
When both sides try contracts in different suits, the number of Total Tricks may be sometimes as small as 16 (say, 2 versus 2) ; it goes sometimes up to 20 (4 vs 4), or even reaches as large as 23 (5 vs 6).
On the other hand, when one side calls Notrump, bidding remains mostly on the 2 and 3 levels. So, the number of Total Tricks is not so large.

7. How about overcalling against a 1NT opening with 2 level in a suit?

--- That's a good example to try and use the LoTT.
Let us assume that the 1NT bid is a standard, sound 1NT opening (balanced hand with 15 -17 HCP). Against this 1NT opening, if you have 8 trumps in your two hands, LoTT tells you

Number of Total Tricks  =  8  +  7  =  15,
and that you can safely bid to the level 2.

8. Oh, once more again "can safely bid!" How can you conclude it from the number 15 ?

--- Remember! Cohen gave us a nice tool called chart.
It's a good exercise for you to prepare the chart. Try, assuming both are non-vulnerable.

 Chart for 15 Total Tricks Both non-vulnerable We play 2 They play 1NT Our Tricks Our Score Their Tricks Our Score 9 +140 6 +50 8 +110 7 −90 7 −50 8 −120 6 −100 9 −150

9. So, I assume we've 8 hearts in our two hands, and I'm over-calling 2 against 1NT.
I find the chart nowhere in Cohen's book, … and I'm work-ing it out by myself, … I'm going to calculate both our and their possible gains and losses, … and arrange them on left and right so that the total tricks be 15 (This last is perhaps important, … it seems, … for sure.) Is this right, sir ?

--- Good, and how do you read it ?

10. That's too easy. All the 4 lines above suggest me strongly to bid 2, because the score on the left is always greater than the score on the right.

--- You've already got it through your work. Now, suppose you go down in doubled contract 2X, and rewrite the bottom two lines.

 Chart for 15 Total Tricks Both non-vulnerable We play 2X They play 1NT Our Tricks Our Score Their Tricks Our Score 9 +140 6 +50 8 +110 7 −90 7 −100 8 −120 6 −300 9 −150

11. Still, our loss turns out to be smaller than their gain, as far as we go down one. But, if we go down two, …

--- Don't worry. In that case, opponents could have bid 3NT to get 400 pts.
You've learned here again what "can safely bid" does mean.

12. Yes. The chart invites me to overcall, when our expected loss is smaller than their expected gain. That's what "can safely bid" means.

--- and also that you can safely bid to the 2 level (8 tricks), having 8 trumps.

13. Oh, yes. I can safely bid 2 with 8 hearts. The LoTT and the chart guarantee it to me.
But, here remains a last, important question: I've so far assumed 8 trumps in our hands. However, it's almost impossible to have 8 cards in my single hand. How many can I expect in partner's hand ?

--- It's no cause for your worry. Practically, you need to have 6 cards and a good hand (with which you could have opened if RHO had passed).

14. Why do 6 cards suffice ?

--- You need here knowledge of fitting probability (see elsewhere for details). Good bridge statistics tells us that

If you have 6 cards in a suit, then your partner will have 2 or more cards in that suit on about 3 deals out of 4 (or, more precisely, with probability 76.3%).
So, you can reasonably rely on your partner for 2 cards.

The Rule for the Safe Level, or Vernes Rule

1. I've seen two examples for the rule you mentioned earlier (repeated here for convenience),
[Rule for the Safe Level]: You can safely bid to the level equal to the number of trumps held by your side.
I wonder how this Rule is related to LoTT. I've seen only two examples where the Rule is confirmed by using the LoTT and the chart logic. Can you demonstrate the Rule from the LoTT?

--- No, impossible to demonstrate or deduce.
Vernes discovered this rule empirically after examining various competitive auctions (on various levels) on the basis of LoTT, which means naturally the chart logic. On this Rule, he comments,

1.  It holds on all levels up to small slams.
2.  It holds when the HCP of the two sides are preferably (de préférence) in the range 17 to 23,
Or, in the range 15 to 25, in [at] a pinch (à la rigueur).
3.  Vulnerability must be equal or favorable.
In case of unfavorable vulnerability, strength of cards must be equal or almost equal (original text in French, p.44-45).

2. So, it doesn't always hold, even though you suggested me to use it so eagerly.

 Chart for 17 Total Tricks N-S Vul,   E-W Non-Vul N-S play 3 W-E play 3 N-S Tricks N-S Score W-E Tricks N-S Score (A) 9 +140 8 +50 (B) 8 −100 9 −140 (C) 7 −200 (3) 10 −420 (4) (C') 7 −500 (3X) 10 −420 (4)
--- Sorry.
You have to be particu-larly careful in unfavor-able vulnerabilities. Here on the right, the Rule doesn't protect you bidding 3, because the opponents, having really good hands, will double your 3, and will gain more than they get in 4. (Compare with the first chart.)
With these reservations in mind, make full use of the Rule. Surely, it's quite easy to use, and no bridge experts have ever provided us with such a handy tool to rely on in competitive auctions.

Notes on the Rule

[Note 1]: The Rule above is referred to as "Rule of 7 to 12" (la Règle de Sept à Douze) by Vernes, implying its usefulness on a wide range of levels up to the small slam. This name sounds awkward, however.
In English, "distributional security", "rule of securi-ty", "law protection", or even (after Cohen) "LAW's competitive guideline" are commonly used, all again awkward. I love the more direct naming, "Rule for the Safe Level", but "Vernes Rule" would be most appro-priate in honor of his great achievements, just as we qualify various conventional bids with inventor's proper names.
[Note 2]: Many bridge players inadvertently credit the above Rule to Larry Cohen, who popularized it, although we owe actually this simple Rule (which is a corollary from the LoTT) also to the genius of Jean-René Vernes.
[Note 3]: Sometimes, the Rule is mistakenly identified with the LoTT. However, they should be understood with distinction. As a matter of fact, the LoTT holds independent of vulnerabilities as well as of HCP, whereas the Rule advises you to care about vulnerability and HCP.

LoTT Holds Independent of HCP

2. In the last section, you explained me some conditions (A through C) for use of the Rule for the Safe Level. I accept some of them quite reasonable. But the condition B, which warrants HCP of the hands, is confusing me. I wonder how the condition is related to the LoTT.
Obviously, there can be two possibilities:
[1]  Validity of the LoTT depends on this condition B,
or
[2]  The LoTT holds for any HCP, but the Rule requires this condition B.
Which is true, boco_san?

--- Well, this is an educated question. As a starter, I used to be confused with this distinction. But, now I can definitely answer: [2] is right.
I'll affirm. LoTT holds independent of HCP.

3. How can you be so definite?

--- Well, for one thing, you can be sure about it from the context of Vernes's writing. He says nothing on HCP when he presents his Law. HCP appears only when he comments on his Rule.

4. …, and for the other?

--- For the other, it is evident from the two postulates of LoTT, which say nothing on HCP.
Furthermore, I'll bet my boots I can show you a clear evidence based on double-dummy analysis. For that purpose, I've developed a free software LottAnalyzer.
Just look at the table below obtained over a half million deals (which were fully double-dummy analyzed by Matthew Ginsberg. For detail, see another article).
 Result of LoTTanalyzer (for suit contracts) under the Three Requirements for HCP 0 - 40 HCP 15 - 25 HCP(à la rigueur) 17 -23 HCP(de préférence) +2 trick deviation 9.7% 9.5% 9.3% +1 trick deviation 31.5% 31.5% 31.5% 0 trick deviation 38.4% 38.8% 39.2% −1 trick deviation 15.2% 14.9% 14.9% Average Deviation 0.36 tricks 0.37 tricks 0.36 tricks Average Error 0.79 tricks 0.79 tricks 0.78 tricks Sample Size 506,581 389,908 280,383
You see above that the LoTT holds the same way in the three cases. It doesn't depend on whether or not you limit HCP of the hands.

5. This is surely convincing.
But, then, what does the comment B mean on the Rule?

--- It means a warning not to overuse the Rule.
If you bid without sufficient strength up to a supposed safe level, relying only on the number of trumps, you'll go down a lot.
The LoTT holds still, but the Rule no more protects you.

6. Give me an example deal.

--- Consider this deal. Count first the number of trumps.
 North Both A4 Non-Vul K86 J853 West K983 East KQJ8753 T962 J9 Q74 Q96 T4 6 JT42 --- AT532 AK72 AQ75 South

7. Well, W-E have 11 spades, and N-S 8 hearts, so the total is
Total Trumps = 11 + 8 = 19.

--- and how many tricks?

8. W-E will take 7 tricks (perhaps fewer than they expect), while N-S 11 tricks.

--- This deal has then
Total Tricks = 7 + 11 = 18.
The LoTT over-estimates the Total Tricks in this case, but the error remains only one.
Now, what will happen, if W-E try 5 over 5?

9. If W-E pass 5, then N-S will get 5 × 30 + 300 = 450 points. In the event W-E bid 5 and get doubled, they go down four and lose 800.

--- and, now compare the HCP.

10. The HCPs are 12 and 28, far outside the range recommended above. This is certainly a good warning not to overuse the Rule.

--- Yes. In this deal, the Law itself is still valid (with 1 trick error), but the Rule no longer protects W-E to bid on to 11 tricks, owing to their poor HCP.
You'll learn more from this deal why West and East should never bid 5:

Watch Chameleons

1. What do you mean by Chameleons, boco_san?
Do you have something to say on the above deal?

--- Yes. In that deal, W-E expects to take 11 tricks because of their 11 spade cards. In fact, they can take only 7.
Now consider this deal the other way: How many tricks do N-S win in their 5 contract?

2. An easy question! N-S obviously have losers in and . The Club loser may be discarded on A. So, they will take eleven (back to the last hand, if necessary).

--- Then, which cards of (defending) W-E will win ?

3. Q of East (or J of West) and Q of East will win.

--- So, on that basis, W-E should have judged whether to go up to the 5 level. Will those two Queens gain tricks in their own contract 5?

4. Oh, definitely no. The Queens gain no tricks in their own contract. The two queens may be replaced by spot cards (which may be ruffed on partner's spade).

--- I propose to call such cards Chameleons. They gain no tricks in their own contract, but will gain in defense.

5. Then, what do Chameleons have to do with the LoTT?

--- That's the point. Chameleons will reduce the total tricks. Therefore, as is evident from the chart logic, they will tend to reduce the safe level. In the above deal, W-E should deduct the two chameleons from the number 11.

Watch the Number of Trumps

1. I've so far learned that LoTT is clever enough to tell me the number of Total Tricks available in a deal, from the number of Total Trumps. For it to work, however, I'll have to know the number of trumps exactly from the bidding sequence …

--- Yes, it's very important.

2. How can I do that ?

--- You'll learn by yourself, if you try and use LoTT in your bidding. Of utmost importance is that you and your partner bid in such a way that you two have a good communication on the number of trumps.

3. ?!

--- Suppose that the auction has proceeded this way:
 West North East South 1 1 2 2 3 ?
Now it's your turn to bid, you sitting North. Will you bid 3 or Pass?

4. How do you want me to consider?

--- In conventional bidding system, you'll judge according to the point counting. You'll bid 3 on a good hand, and will pass on a bad one. However, if you agree with your partner on bidding based on the LoTT, what's all important is the number of trumps.
So, consider again here how many spades you and your partner each will have.

5. That's easy. My overcall is based on 5 (or more) spades, and my partner's support 2 promises at least 3 spades.

--- In that way, you're communicating! At this point, you apply the Rule for the Safe Level.

6. h'm, … then, I'll bid 3 with 6 (or more) spades, and pass otherwise, … is this right ?

--- Yes.

7. Have I to pass with 5 spades, even when having a really good hand ?

--- Yes, you have to.
If you go out of the Rule and bid 3 (perhaps because you've a good hand), then your partner will misunderstand you've 6 spades (and hence, 9 spades in total). Partner will then be unable to do a right decision whether he should bid more or pass on the basis of the Rule, when East further competes with 4.

8. Couldn't you show me another sample?

 Q 7 3 9 8 Q 10 2 A 9 6 4 3 ( 8 HCP )
 West North East South 1 Pass 1 X Pass ?
--- OK.
You picked up this hand, sitting North.
West opened 1, and you passed.
East then responded 1, and partner doubled to take you out.

9. In that case, I'll bid 2.

--- Are you then prepared to bid 3 on your next turn of bidding ?

10. Oh, yes. Partner's double shows 4 cards in both of unbid and . The Rule will then protect me bidding 3 on the three level with nine cards. Thank you for a nice example hand.

LoTT in Weak Two

1. Do you mean the usual Weak-Two opening by the above title ?

--- Yes. I'm going to show you how LoTT already occupies some grounds of the conventional bidding system.
Say, holding a hand like this,
 K Q 9 7 6 2 9 Q 10 8 4 7 3 ( 7 HCP )
you'll open 2, or jump overcall against one-level openings.

2. I know well.
But, could you please remind me of requirements for weak-two bids ?

--- Standard requirements for weak two are:

(1)  5-10 HCP,
(2)  good 6 cards (preferably 3 of top 5),
(3)  no 4 cards in either major,
(4)  no void.

3. I'm not sure about some of them.

--- If you have as much as 11 HCP, then Marty Bergen's rule of 20 will allow opening on one level. His count will give

HCP + 6 (longest) + 3 (next longest) = 20.

4. Oh, yes. And, what about the third?

--- That is in case partner also has 4 cards in that major suit.
If you have an 8-card fit with partner (which occurs with probability 33.7%), your long (side) suit will help you taking tricks a lot. As to this probability, see elsewhere.

5. And the fourth? Why avoid hands with a void?

--- In bridge, appearance often deceives our judgement: Hands with a void are stronger than they appear. Suppose you have this hand:
 K Q 9 7 6 2 9 7 3 Q 10 8 4 − ( 7 HCP )
It depends on partner's hand, but sometimes, you may have 4 or 5. Void in clubs will then work quite a bit. If you open with 2, partner will regard your hand as utterly useless and will not seek game. So, it's best to pass with this hand.

6. OK. That's all.

--- Then we'll return.
You've 6 spades. How often can you expect partner to hold 2 spades ?

7. That has appeared somewhere above, … here, 76.3%. So, I can expect fitting with partner on three deals out of four.

--- Assume now that you've an 8-card fit.

8. Then, with certainty, I can bid 2 on the basis of the Rule for the Safe Level.

--- You can see the deal in more detail.
How the opponent's hands will look like?
 Chart for 16 Total Tricks Both Vulnerable N-S play 2 W-E play 2 N-S Tricks N-S Score W-E Tricks N-S Score 9 +140 7 +100 8 +110 8 −110 7 −100 9 −140 6 −200 10 −620 (4)
Here will work another piece of knowledge on fitting probability: When you have an 8-card fit, they are likely to have also an eight-card fit, or more. So, most certainly, the deal has 16 Total Trumps (with probability100−14.8= 85.2%).

9. If it's so certain, then LoTT will predict 16 Total Tricks in the deal, and I can use the chart logic, … Oh, the chart again supports the Rule for the Safe Level. It nicely supports Weak-Two opening based on 6 cards. What have we learned about Weak Two's here ?

--- Everybody has learned Weak-Two bids, without knowing the LoTT. We were taught just to do so without theoretical accounts. However, we have found that Weak-Two bids on 6 cards are clearly understood in terms of the LoTT (or rather, the Rule for the Safe Level) together with some knowledge of fitting probabilities.

LoTT in Higher pre-empts

1. Does that story work in higher pre-empts, too?

--- Yes. Let's pick up two deals.
Suppose now you've got this hand:
 K Q 10 7 6 5 2 8 4 10 9 4 3 ( 5 HCP )

2. I'll open 3, with 7 cards in spades.

--- Yes, for sure. And let's consider the fitting probabilities, as we did above for Weak Two's.
Partner will have

1+ spades, with probability 92.9% (Table (1))
2+ spades, with probability 66.7% (again from Table (1), 100 −(7.1+26.2)=66.7).
The upper probability is almost 100%, as everyone expects. So, we can reasonably assume the lower case: Partner will support you with 2 (or more) spades, and hence you've 9 trumps, twice in three deals.

3. In that case, the Rule supports me strongly to pre-empt 3. Oh, yes. Now, I've learned why I can safely pre-empt on the 3 level with seven cards. The Rule and the fitting probability support me.

--- Right, I'm so happy to hear you speaking "safely" so naturally, but the story goes a bit farther.
We ask, what sort of hands opponents are likely to have? Again look up my table (Table (5)). When you two have 9 trumps, opponents will have at least an 8-card fit in their own trumps. They may have even an 8-card double fit, or a 9-card fit, or more …

 Chart for 17 Total Tricks Both Vulnerable N-S play 3 W-E play 3 N-S Tricks N-S Score W-E Tricks N-S Score 10 620 (4) 7 +200 9 +140 8 +100 8 −100 9 −140 7 −200 10 −620 (4)

4. In that case, the deal has at least 9+8=17 tricks accord-ing to LoTT, and I can pre-pare the chart.

--- Do it.

5. I see here again that the chart supports the Rule for the Safe Level.

--- So we observe here another example where the chart logic is playing a good background for the current bidding system, even if many beginners do not take notice of it.
Having said this conclusion, let us pick up another prob-lem, perhaps familiar to you. Auction has proceeded this way:
 Partner RHO You LHO 1 Pass ?
Partner opened 1, and RHO passed. What will you bid with this hand?
 J 10 5 J 10 7 3 2 Q 5 4 3 4 ( 4 HCP )

6. Well, it's a very weak hand (4 HCP) with 5 hearts and a sin-gleton in clubs.
With this kind of hands, I was taught to bid 4, even with a Yarborough (no honors).
 Partner RHO You LHO 1 Pass 4

--- Did you hear some good reason ?

7. Just to pre-empt in case opponents may have 4

--- Then, consider in context of LoTT.

8. That's too easy. The Rule for the Safe Level will protect me on the 4 level, because we have 10 trumps.

--- Quite so, but one step more, you can go.
When you've a 10-card fit, what the opponent's hands are likely to be at least?
Do they have a 7-card fit, or 8, or 9, … ?

How many Total Trumps do you expect in this deal ?
This is the question you always ask when you're bidding on the basis of LoTT.
For now, look up the probability table. You'll see there that opponents will have at least a double 8-card fit (22.68%). This is not a simple 8-card fit; it means 8-card fits in two suits. If you remember that both authors of LoTT recommend positive correction / adjustment, …

 Chart for 19 Total Tricks Both Vulnerable N-S play 4 W-E play 3 N-S Tricks N-S Score W-E Tricks N-S Score 11 +650 8 +100 10 +620 9 −140 9 −200(x) 10 −620 (4) 8 −500(x) 11 −650 (4)

9. Oh, well, I now understand what you mean. We have 10 trumps, and they have 8, so there are 10+8=18 trumps, in total. LoTT then tells me 18 total tricks. However, their fit is at least double 8, which strongly suggests a positive correction. So, I'll assume 19. This is minimum, it can be more, but I assume 19, for the moment, and make up the chart …

--- The chart is so simple, and perhaps you didn't have to make it up. But, here again, we have learned a good reason to bid 4 with 5 hearts. In these two examples as well as in Weak Two, competition is not yet happening, but you can predict it most likely to happen from the fitting probabilities. Surely, bridge is a game of probability.

10. Are these two hands all you're to pick up?

--- As I remember, I wrote more in Japanese original, but translation into English cleaned off some digressions.
Before closing, let me pick up one more. Now, you're going to respond partner's 2 opening,
 Partner RHO You LHO 2 Pass ?
with this hand.
 10 9 4 2 J 8 7 3 K 9 3 2 4 ( 4 HCP )
What do you call? Competition is not yet happening.

11. Well, … anyway, it's a very weak hand with 4 HCP. So, I'll bid 3, … or, bid 4 because we have 10 spades.

--- You're now in a good position to decide. As for myself, while I was totally ignorant of the LoTT, I learned quite a lot on bidding among books and softwares of Mike Lawrence (to whom I owe much). In the above situation (4 trumps and a singleton), he recommends as a TIP to bid 4, irrespective of strength of the hand, particularly, however weak it may be.

12. What do you want to mean finally ?

--- Oh, sorry. Once we have learned the LoTT and something related to it, some TIPs we learn from experts become trivial in the light of LoTT. So, you, who have understood the basics of LoTT, you'll learn bidding better than ever.

13. Most probably. Thank you, boco_san.

--- Thank you. So long for today.