Adjusting the Law of Total Tricks


Since the Law of Total Tricks will come up with the right answer at best 40% of the time, any player who wants to use it simply has to adjust the formula. On this page, we will take a closer look at what Jean-René Vernes and Larry Cohen have written on the subject.


What Vernes said

When Jean-René Vernes published his article in The Bridge World, June 1969, he mentioned a few factors which he deemed important. This is what he wrote:


Corrections

We have established a formula for predicting total tricks that is both very simple, and quite accurate in a majority of instances. Still, just as we have to make corrections, occasionally, in a good point-count method, so too must the law of total tricks be modified. There are three extra factors:


The double fit

Let's see if Vernes was right regarding these corrections. If we start with his first factor, the double fit, we agree with him. But have you ever wondered why there should be extra tricks? In his excellent book Matchpoints (Devyn Press, 1982), Kit Woolsey writes (on page 170) "... the second suit is sort of another trump suit, so both you and your opponents may take more tricks than the law of total tricks would predict even on relatively balanced hands."

Thinking of the second fit as "another trump suit" is a wide spread belief. But it happens to be wrong. Actually, there is nothing good for your side to have two suits of eight or more cards – but it is good for your side when the opponents have it...

Let's use Woolsey's example from page 171 to show what we mean:

S A K 5 4
H 5 3 2
D A Q 8 2
C 10 6
S 10 7 6TableS 9 2
H A 7 4H K Q J 10 6
D 9 4 3D 10 5
C K J 8 4C A Q 5 3
S Q J 8 3
H 9 8
D K J 7 6
C 9 7 2

Both sides can take nine tricks with eight trumps, so this is a +2 deal. Now we give South one of East's clubs and return a diamond to get:

S A K 5 4
H 5 3 2
D A Q 8 2
C 10 6
S 10 7 6TableS 9 2
H A 7 4H K Q J 10 6
D 9 4 3D 10 6 5
C K J 8 4C A Q 5
S Q J 8 3
H 9 8
D K J 7
C 9 7 3 2

The deal is no longer a double fit, but North-South still take nine tricks. East-West, on the other hand, only take eight. Why?

The answer is simple. It has to do with distribution. In both these deals, North-South's SST is 4, and they have the same number of working points. If South has four diamonds and three clubs, or the opposite, doesn't matter to his side. But for East-West, East's getting a diamond instead of a club meant their SST went from 4 to 5, which is the same as adding a loser. Had South swapped cards with West instead, nothing would have happened:

S A K 5 4
H 5 3 2
D A Q 8 2
C 10 6
S 10 7 6TableS 9 2
H A 7 4H K Q J 10 6
D 9 6 4 3D 10 5
C K J 8C A Q 5 3
S Q J 8 3
H 9 8
D K J 7
C 9 7 4 2

No double fit any longer, but the same number of tricks as in the original deal. Since the swap didn't change the SST for either side, we shouldn't be surprised that the number of tricks hasn't changed. It is still a +2 deal.

The reason there often are extra tricks when both sides have two suits of eight or more cards has to do with distribution, nothing else. And you find the answer if you look at your short suits. When the opponents have two suits of eight or more cards, your SST will never be more than 4 – but if they have one suit of eight or more cards and their second longest suit is seven cards or shorter, your SST may be 5.

S K 8 7 6
H K J 8 3
D 8 7 6
C J 2
S 9 2TableS 10 3
H A 7 4H 10 9 2
D K Q 9 3D J 10 5 4
C K 10 9 3C A Q 8 6
S A Q J 5 4
H Q 6 5
D A 2
C 7 5 4

If you ever thought it was good to have two suits of eight or more cards, this example shows that to you have been wrong. Here, the two sides are equally strong, but North-South take nine tricks, while East-West take eight. Why?

The fact that North-South have nine trumps and East-West eight has nothing to do with it. If you think so, move a heart from North to East and move a club back. Then, both sides take eight tricks only, but total trumps are still 17.

Since both sides have their share of the deck (North-South have 20 WP; East-West have 19 WP), how many tricks they take is reflected in their SST. And since North-South have 4, East-West 5, it is easy to understand why North-South take one more trick than their opponents in the diagram above.


When one side has two suits of eight or more cards, the other side is assured of an SST of at most 4.


So when the deal is a double fit, i.e. both sides have two suits of eight or more cards, an extra trick doesn't come from your second eight-card suit but from the fact that your SST is guaranteed to be 4 or less. The gain comes from the opponents' second eight-card suit, not from yours.

S 10 9 8
H K Q 10 4
D A 3
C Q 9 8 7
S Q J 7 6TableS A K 4 3
H 9 3 2H A 5
D Q 10 9 2D J 7 6 5
C 3 2C A 6 5
S 5 2
H J 8 7 6
D K 8 4
C K J 10 4

If East-West play a spade partial, is their second eight-card fit of any good to them? No. The only effect is that the defenders are assured of a ruff. If we move one of East's diamonds to another side-suit, they may still run into a ruff, of course, but at least they have the chance that there is none: North-South's diamonds may be 3-3. And they aren't better off in diamonds, since then North-South can arrange a ruff in spades. Having two eight-card fits was no good for East-West.

The same can be said for North-South. They too will run into a ruff no matter which suit they choose as trumps. But had they had seven cards in their second longest suit, they would have gained one trick if the opponents' cards in that suit were 3-3.

Now you probably wonder "If both sides have half the deck in working points (North-South have 19 WP, East-West have 21 WP) and each side has an SST of 4, why do they only take eight tricks? What's wrong with your formula?"

There's nothing wrong with it. The formula predicts how many trick you can take if nothing bad happens. When there is a ruff, something bad does happen, and no matter which method of valuation you use, you can't predict it. Another way of looking it is this: When there is a ruff, and that ruff is taken with an otherwise useless trump, what happens is that your side loses some of its WP. If East-West play a spade contract and North-South take their diamond ruff, the effect of that is that East-West didn't have any WP in diamonds. The first, second and third diamond rounds were all won by North-South. So the effect of the ruff was that East-West had 18 WP and an SST of 4, equalling eight tricks. And the same goes for North-South: if East-West arranges their ruff, North-South haven't 6 WP in their second longest side-suit, only 3 WP. The fact that both sides win a trick with a fourth-round winner isn't reflected in their WP; it's a part of the distribution.


Trump honors

What about Vernes' second factor, possession of trump honors? Would you belive us if we told you that it isn't needed and that looking at WP and SST is much better? OK, here we go.

West East
S K QS J 4

North-South's trump suit is spades. If East-West's spades are 2-2, as here, both sides will lose one spade trick if they declare the hand.

West East
S K Q 4S J

If spades are 3-1, East-West will still lose one spade trick but North-South will lose two. Since East-West have the potential for two (or one) discards they may gain from being 3-1 instead of 2-2, but quite often those discards can't be used (the defenders cash out early). Then East-West don't gain anything. Assuming their second shortest suit is 3, their SST in the first case is 5, in the second case 4. But they don't take more tricks because what they gain in distribution in the second case (going from 5 to 4 in SST) is balanced by a loss in WP (going from 3 WP in the first case to 0 WP in the second). Since 3 WP represent one trick, the loss in WP is compensated by the gain in distribution.

For North-South the difference between 3-1 and 2-2 is one trick. And that is reflected in their WP. When trumps break evenly they have the equivalent of 6 WP in trumps (losing one trump trick only), but if trumps are 3-1, they only have 4 WP in trumps (losing two trump tricks).

If North-South instead have all trump honors, the difference between 2-2 and 3-1 is worth a trick, but not to them. The gain comes to the other side, when East-West declare the hand and North-South's 10 HCP in spades only take one trick on defense instead of two. Here, the gain is purely distributional. The extra trick doesn't come from one side's having all trump honors, it comes from a reduction of losers in the opponents' strong suit. And it goes to the other side.

There's nothing bad in having all trump honors, of course, but no extra tricks pop up from that. If there are more tricks than trumps on a deal where one side or both have all trump honors, then it's a function of the distribution.


The distribution of a suit between the partners

What shall we say about Vernes' third and final correction, that the distribution in a suit between two partners "has a small, but not completely neglible, effect"? To put it mildly, we think this is a serious understatement.

Sometimes, a gain in distribution for one side is a gain both on defense and on offense. But often the gain is either only on defense (like when you had KQx opposite J singleton instead of KQ opposite Jx) or only on offense (like when you had xxx opposite x instead of xx opposite xx). Are these situations really rare? Of course not.

If a gain in distribution is a loss in WP, nothing happens. Suppose you have a side-suit like AKx opposite Qxx and an SST of 5. If we change that suit to AKxx opposite Qx (and keep everything else as it is), your SST goes down to 4, but if you can't use the suit for discards, the gain in distribution is balanced by a loss in WP (going from 9 WP in this suit to 7 WP). Then, nothing happens. This is not unusual, but it is definitely not the norm, as Vernes seems to think.

S A J 8 7
H A K 4 3
D 5 4 3
C J 8
S 9 6TableS 5 2
H J 9 2H 10 7 6 5
D A K 9 6D J 10 2
C 9 7 6 2C A 10 5 3
S K Q 10 4 3
H Q 8
D Q 8 7
C K Q 4

27 HCP, nine strong spades, and two doubletons, but since the defenders can win the first four tricks, there is no game. When that happens, North-South may realize that they only have 20 WP (the HQ, the DQ and the CQ-J don't take tricks; therefore, they are not working). Add that to their SST of 4, and you understand why these cards aren't good enough for game against best defense.

Another case when a change in the side-suit distribution changes the total tricks is when ruffs are out. Then, total tricks go down.

S A J 8 7
H K J 10 2
D 5 4 3
C 8 7
S 9TableS 6 5 2
H 6 5H A 9 4 3
D Q 10 8 2D J 9 6
C Q 10 9 6 5 2C A J 3
S K Q 10 4 3
H Q 8 7
D A K 7
C K 4

North-South take 11 tricks in spades. They have 1 HCP less than in the previous example, and still an SST of 5, but since declarer has time to discard his diamond loser on dummy's fourth heart, that fourth heart should be counted as 3 WP. Therefore, North-South have 29 WP. Note that if South's diamond king had been the club ace instead (and the distribution the same), North-South gains 1 HCP, but loses 2 WP (and a trick): Then, after a diamond lead, dummy's fourth heart isn't worth anything.

East-West take 8 tricks. They have 14 WP and an SST of 3, so the result is expected. If we swap one spade and one heart between East's and West's hand, we keep their SST at 3 (and their tricks at 8), but that swap removes two tricks from North-South.

S A J 8 7
H K J 10 2
D 5 4 3
C 8 7
S 9 6TableS 5 2
H 6H A 9 5 4 3
D Q 10 8 2D J 9 6
C Q 10 9 6 5 2C A J 3
S K Q 10 4 3
H Q 8 7
D A K 7
C K 4

East-West take the same 8 tricks if they declare, but North-South suffer two heart ruffs and only take 9 tricks.

What has happened is that North-South no longer have 9 WP in hearts as before. Two winners were ruffed away, so North-South lose 6 WP. Since 6 WP is the same as two tricks, we know why North-South take two tricks less in this scenario. The fourth heart is still useful for a discard, which means they have 3 WP in the suit.


What Cohen said

Larry Cohen also talks about corrections in his books, but he uses the term adjustments instead, and since these books have made a big impact on the bridgeplaying community of today we will stick with his term.

He divides the adjustment factors into two groups: Negative (suggest total tricks will be less than total trumps) and Positive (suggest total tricks will be greater than total tricks). The factors are:

  1. Purity
  2. Fit
  3. Shape

Purity

Negative Purity is defined as "minor honors in opponents' suits and/or poor intermediates in your own suits", while Positive Purity is "No minor honors in opponents' suits and/or good intermediates in your own suits". What he means is that if one side, say North-South, has a slow loser in one of their long suits if they play the hand, that trick often isn't worth anything to East-West when they declare.

S K 10 8 3
H K 6 5 4
D A 7 6
C 9 8
S Q 5 4TableS J 2
H A 9 2H 8 7 3
D K Q J 9D 10 8 5
C Q 3 2C A J 10 7 6
S A 9 7 6
H Q J 10
D 4 3 2
C K 5 4

In this example, East-West's spade combination is worth one trick on defense, but is worthless if they declare (if a spade is led, the defenders can shift to hearts to stop declarer from establishing the queen of spades for a useful discard).

Which side gains?

One problem with looking at all tricks together, as the Law of Total Tricks does (instead of taking one side at a time), is that it doesn't say which side is affected. In this example, East-West has a slow trump trick on defense, so it is natural to think that the loss if for North-South. But it isn't. It is easy to understand if you do as we teach and view the deal in terms of WP and SST.

North-South have 20 HCP, and all of them are working. Their SST is 5. If we put them together the formula says "eight tricks". And the formula is right, again.

East-West also have 20 HCP, but since their spade honors are useless if they declare, they only have 17 WP. Since they too have an SST of 5, we can figure out that they should take seven tricks. And they do.

When you have slow winners in the opponents' suits, which are useless if you declare the hand, the effect is that your WP goes down – and that you will take less tricks on offense than if those honors had been working.

Now suppose you have four cards in their trump suit, the queen, the jack, the ten and the deuce, and that these cards are useless to you if you declare the hand. The first effect is that 3 of your HCP are not working. The second effect is that how your cards in their trump suit are distributed says how many WP the opponents have in that suit. If your cards split 2-2, their nine trumps headed by ace-king is worth 10 WP (no losers), which means that they will take one more trick than if they are 3-1 (and you take a trump trick on defense).

What about your side? Gain one, you say? Maybe. The difference between 2-2 and 3-1 is worth one trick, if and only if, your SST is reduced by one. Suppose your suit is spades and their suit is clubs. If you have 5-4 in the majors and your partner a flat hand, it doesn't matter if you are 2-2 or 3-1 in the minors; in both cases your SST will be 4. So, if you go from 2-2 to 3-1 in the opponents' suit (by trading a club for a diamond with your partner), that change isn't worth a trick if you at the same time go up from 2 to 3 in another suit. By looking at the two shortest suits, as we do, we avoid mistakes like thinking "one less loser in their suit = one more trick to us". Since each hand consists of 13 cards, it all depends on which card you got instead. If a gain in one suit is compensated by a loss in another, nothing happens.

To summarize: If your cards in their suit are 2-2, the opponents gain a trick on offense, since they gain 3 WP. If your cards in their suit are 3-1, you gain a trick if your SST goes down by one. In the first case the gain came from extra WP, in the second it came from improved distribution.

Why 'purity' doesn't explain anything

In To Bid or Not to Bid, Larry Cohen wants us to look at a "totally pure deal" (page 225):

S A K 10 9 3
H 8 5
D A 2
C 8 7 6 5
S 8 7TableS 2
H 7 4H K 10 9 6 3
D J 10 8 5 4D K Q 9 7
C K Q J 4C A 10 3
S Q J 6 5 4
H A Q J 2
D 6 3
C 9 2

Here, North-South can take 11 tricks, while East-West take 9 tricks. 19 trumps, but 20 tricks. According to Cohen this is attributed to the 'purity' factor. It isn't. There is no magic in bridge, only logic. Tricks come from a combination of distribution and honors, nothing else. In this example too.

What about North-South? They have 21 HCP, but since the king of hearts is onside, the heart jack is just as good as the king, and that "king" will provide a useful discard, so they have 23 WP. Their SST is 4, which suggests ten tricks, but there is eleven. That extra trick comes from the fact that North-South have three side-suit doubletons, and all of them are useful. Therefore, we adjust the SST to 3. We discussed this topic in more detail in I Fought The Law of Total Tricks.

East-West also have an SST of 3, but only 16 of their HCP are working. That is the same as nine tricks.

If we move the heart king from East to West, does that make the deal "less pure"? Cohen probably thinks so, because now there is no extra trick. For us it is easy to explain why: When West has the king, it becomes useful for East-West, so they can add 3 WP. 19 WP and an SST of 3 is ten tricks. At the same time, North-South lose 5 WP. In the first case, South's queen and jack of hearts were just as good as king-queen, but now they are no better than deuces. With 18 WP and an adjusted SST of 3, it's easy to understand why they now take two tricks fewer than when they had another 5 WP.

S J 9 4 3
H A Q 6
D 10 8
C 10 8 3 2
S ATableS 10 2
H 7 2H 10 9 8 4
D A K Q 7 4 3D J 6 2
C K 9 5 4C A Q J 6
S K Q 8 7 6 5
H K J 5 3
D 9 5
C 7

This deal is published in Following The Law (page 72). Cohen writes "Notice that there were 19 trumps and 20 tricks. The extra trick comes from the extreme purity of the deal – no wastage in any suit." But that explanation is false.

This is the truth about the deal: North-South have 16 WP and an SST of 3, which is the same as nine tricks. East-West have 24 WP and an SST of 3, which is the same as 11 tricks. It's all in the cards: distribution and useful honors.

Now let's swap a few cards between the partners (a heart and a club between East and West, a heart and a club between North and South). Then we get:

S J 9 4 3
H A Q 6 3
D 10 8
C 10 8 3
S ATableS 10 2
H 8 7 2H 10 9 4
D A K Q 7 4 3D J 6 2
C K 9 5C A Q J 6 4
S K Q 8 7 6 5
H K J 5
D 9 5
C 7 2

Now, the same 19 total trumps only produce 18 tricks, two less than before. Does moving a few spot cards around mean that the "extreme purity" disappeared? Then, why? Or is the truth that talking about 'purity' is not talking about the real reason for the loss of these two tricks...

This is how we explain it: The swaps didn't affect the WP for any side, but the SST went up from 3 to 4 for both sides. Therefore, each side lost a trick.

Good fit versus misfit

Cohen's second asjustment factor, "fit", was discussed when we looked at Vernes' corrections. We then showed why a double fit often produces extra tricks. It was not because the second fit was some sort of a "second trump suit"; it had to do with distribution. Go back and take a look if you forgot what we wrote.

When Cohen says a misfit often produces fewer tricks, we agree with him. If you have a singleton opposite a suit where partner has KQJxx, it is possible that those honors are useless in your contract, while they take two tricks on defense. Then, the effect is that your side will take fewer tricks in your contract, because 6 of your HCP are not working.

Shortness in partner's suit

What happens when there is a shortness opposite nothing wasted? If you have a singleton in partner's suit, where he has five small or ace fifth, for example, your side has no wasted honors in that suit – and still there may be fewer tricks than the Law of Total Tricks suggests.

This happens when there are ruffs out. For some reason, Cohen skips this issue. But we won't. When you lead a singleton and get one, two or even three ruffs with otherwise useless trumps, what happens is that some of the opponents' WP in that suit lose their value. If they have KQx opposite J10x in hearts, and we start the defense with heart ace, heart ruff and later score a second heart ruff, what has happened is that the opponents' hearts weren't better than xxx opposite xxx, and what looked like 6 WP in reality was 0 WP. This is another reason why HCP can't automatically be translated to WP. If an honor doesn't take a trick, it isn't worth anything.

Shape is important

Cohen's third factor, shape, is perfect. Actually, it's the most important of them all. It's so important that it has made the Law of Total Tricks obsolete.

S K Q J 2
H K Q 3
D 8 5 4
C 8 6 2
S 8 5TableS 6 4 3
H 8 6 5H 9 7 4 2
D K Q 2D J 10 6
C A J 10 4 3C K Q 7
S A 10 9 7
H A J 10
D A 9 7 3
C 9 5

This is the very first example in To Bid or Not to Bid (page 10). The deal looks "extremely pure" to us, and still there are no extra tricks (an indication that 'purity' is something that is best forgotten). Now we will do two changes. First, we swap two cards between East and West (heart five and diamond jack).

S K Q J 2
H K Q 3
D 8 5 4
C 8 6 2
S 8 5TableS 6 4 3
H 8 6H 9 7 5 4 2
D K Q J 2D 10 6
C A J 10 4 3C K Q 7
S A 10 9 7
H A J 10
D A 9 7 3
C 9 5

If you had shown this deal to a Law guru and he had responded that the extra trick comes from something called 'purity' you might even have believed him. Now you know better. The extra trick comes from East-West's improved distribution, nothing else. In the first example, East-West had 16 WP and an SST of 5. Here they have 16 WP and an SST of 4. Can anybody be surprised that the difference is one trick? Since the swap didn't affect North-South, they still take nine tricks with their 23 WP and an SST of 5.

From the original example, we now swap two cards between North and South (spade deuce and club five).

S K Q J
H K Q 3
D 8 5 4
C 8 6 5 2
S 8 5TableS 6 4 3
H 8 6 5H 9 7 4 2
D K Q 2D J 10 6
C A J 10 4 3C K Q 7
S A 10 9 7 2
H A J 10
D A 9 7 3
C 9

Now, North-South can take ten tricks, while East-West take the same seven tricks as before. The difference is distributional again. The SST for North-South has been improved from 5 to 4, and the result is one trick for them. Note that the swap didn't do East-West any harm; they still take seven tricks.

Both these swaps show that one of the bases for the Law of Total Tricks, namely the claim that "Bad breaks for one side translate into good breaks for the other. The Total Trick count is constant. (To Bid or Not to Bid, page 19)" is false. It may be that changes work both ways, but often they only affect the tricks for one side.

The small changes are the important ones

An interesting fact is that small changes are usually the ones affecting one side only: going from 3-3 to 4-2, or from 2-2 to 3-1 will reduce our SST, but it will often have no bad effect on the opponents. But bigger changes, like going from 3-3 to 5-1 will quite often have an effect on defense as well as on offense.


Copyright © 2016, Mike Lawrence & Anders Wirgren