--- A bit about the LoTT.

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

--- 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.

--- 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.

--- 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.

--- 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).

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

--- 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 ….

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

--- 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.

--- 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.

--- 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 ?

--- 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.

--- 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.

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

--- 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).

--- Exactly.

--- 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.

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

--- No.

--- 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.

--- 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?

--- 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.

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

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.

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.

--- 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.

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

--- 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.)
      Does this make an answer to your question ?

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

--- 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.

--- 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.

--- 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:

--- Yes. He writes:

--- 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:

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.

--- 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.

--- 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,

--- 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:

--- 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, …

--- 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.

--- 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.

--- 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.

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

--- 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.

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

--- 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.

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

--- 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.

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

--- 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.

--- 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

--- 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

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

--- You've already got it through your work. Now, suppose you go down in doubled contract 2♥X, 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

--- 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.

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

--- 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).

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

--- 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,

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.

--- About what?

--- 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.

--- 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.

--- 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.

--- 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.

--- 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

--- and how many 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♥?

--- and, now compare the HCP.

--- 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♠:

--- Yes, it's very important.

--- 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.

--- 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?

--- 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.

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

--- Yes.

--- 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.

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.

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

--- 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.

--- Standard requirements for weak two are:

--- 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

--- 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.

--- 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.

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

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

--- 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%).

--- 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.

--- 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 )

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

3

--- 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)

--- Do it.

--- 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 )

Partner	RHO	You	LHO
1	Pass	4

--- Did you hear some good reason ?

--- Then, consider in context of LoTT.

--- 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, … ?

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)

--- 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.

--- 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.

--- 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.

--- 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.

--- Thank you. So long for today.
And many thanks to readers who have read up to here.