Errors in the estimation

Even though the WP+SST method will predict accurately most of the time, we would be lying if we said the method does not has its downsides. There are a few cases where the estimation will be wrong, but if you are aware of them you will know when to upgrade or downgrade your hand.

(1) Aces versus queens

Take a look at this deal. Do you find something strange with it?

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

If we start with North-South, we see that all their honors are working, but that they don't have much distribution. So following the Law and contracting for 9 tricks wouldn't be wise. With 22 WP and an SST of 5, we would expect 9 tricks, but there are only 8. How come?

In our dicussion on the page "Working points", we said that when you take a trick with an ace, you in fact use a little too much power to win a trick. An average trick is 3 points, but an ace is valued to 4 points. In this example North-South take three side-suit tricks with aces, which means their 12 WP in the side-suits don't work optimally. Since 12 / 3 = 4, one would expect 12 WP to produce four tricks, but they only take three.

At the same time, East-West will do very well with their 18 HCP. They have an SST of 4, which suggests 8 tricks – but they take 9. And they do it because their side-suit honors work optimally. They have no aces, so they don't waste 4 WP to win a single trick; they win tricks with kings and queens, using up 3 WP or 2 WP, respectively. If we say that the lower red-suit honors, the heart jack and the diamond jack, are used to promote the higher honors in those suits (using the same reasoning as we did on the page "Working points"), East-West have only 10 WP in the side-suits, but those 10 WP take one more trick than North-South's 12 WP in their side-suits.

A digression: The 18 total trumps on the deal produce 17 total tricks, even though the deal is perfectly pure (which is supposed to lead to extra tricks).

You may have read that aces and kings are better for suit play than queens and jacks. To some extent that is wrong. What the writers mean is that queens and jacks in side-suits are often useless, therefore valued too high. But – and this is an important but – when a queen or a jack is working, they are always undervalued. And an ace is always overvalued.

For a given sum of working points, say 12, you will take more tricks the more honors those points consists of. Therefore, if you have lots of aces and few queens, you may have less WP than you think; while if you have many queens and few aces, you may have more WP than you think.

This is nothing strange. If you have three aces, those three honors will take three tricks. If you have two kings and three queens, those five honors may take five tricks. If one or more of them don't, it means they weren't working and therefore should be valued to 0 WP. So, if you only take tricks with the two kings and one of the queens, it means you only have 8 WP in those suits – but these 8 WP take just as many tricks as the (overvalued) three aces.

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

This is the kind of deal where most pairs would bid too low. After all, they only have 30 HCP and two balanced hands. But 12 tricks will be there almost always.

The WP+SST formula gets it right, though. SST is 5, but since the club jack provides a useful discard we value it to 3 WP. Therefore, East-West have 32 WP, which coupled to an SST of 5 is the same as 12 tricks.

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

Conversely, this is the kind of deal where many pairs will reach a terrible 6S contract (only an unlikely squeeze will save them). Now East-West have 31 HCP, and the strong hand with only aces, kings, trump honors and four-card trump support will gladly cooperate in any slam hunt, thinking he has a golden hand. Still, trading the queen-jack of clubs for the trump ace was losing a full trick. The previous West hand was much better.

But if you realize that you lost two undervalued trick-taking cards and gained one overvalued trick-taking card, and that the fourth trump didn't do East-West any good, the outcome of the swap is predictable.

WP+SST says "31 HCP and an SST of 5 = 11 tricks", so we would get this one right too.

Our next two examples show the real reason why aces are useful. It is easy to understand once you have grasped the concept of Working Points.

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

Here, West's heart singleton means East only has 7 WP (3 in spades, 4 in hearts). Add that to West's 7 WP and an SST of 4, and the result – 7 tricks – is predictable.

Let's add the ace of diamonds to East's hand, to see what that means.

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

Adding the ace of diamonds didn't add one trick; it added three tricks. Does that mean the ace of diamonds was worth more than 4 WP?

No. But it means that the added ace gave East-West 9 extra WP, since now both the heart king and the heart queen are working. The big thing about aces is that they bring life to other honors, not that they take more tricks than their point-count suggest.

(2) Not enough potential

When your side has a very low SST, say 0 or 1, you may overestimate your tricks if you neither have lots of trumps nor a long side-suit. Even if you have few losers it doesn't automatically follow that you have lots of winners. Here is a simple case:

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

North-South has 20 WP and an SST of 0 – but they don't take 13 tricks. On a minor suit lead they can take 12 tricks (by setting up a long card in the suit led), but on a trump lead 11 tricks is all they get. But add an extra trump or two to North-South, and the estimation will be OK. As it was, North-South didn't have enough potential.

If we swap two cards between North and South, we get:

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

In the first example both North-South's long suits were distributed 4-4. As we saw, that didn't give them lots of tricks. Here, both suits are 5-3 – a big difference. Now they have five tricks in each major, and it is easy to arrange two ruffs in dummy, even if the defenders find the best lead of a trump. Now they take 12 tricks. But even here, the estimation is too optimistic. An extra trump or an extra heart winner would have been useful.

If we make a final change, the formula will be perfect:

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

With both majors 6-2, North-South have no problem taking all tricks. Now, the WP+SST formula predicts accurately.

This example is another illustration to the fact that distribution is the most important factor for estimating your tricks. The SST+WP method looks at your important honors and your short suits, but as this example shows you may also have to look at the distribution of your long suits.

If we should conclude the previous examples, we can say that

It is good for you to have your long suits unevenly distributed.

This may surprise you. After all, didn't Ely Culbertson teach the world of the advantages of playing in the 4-4 fit? And haven't we all heeded his advice ever since?

Yes we have, but somewhere along the line the reasons for playing in the 4-4 fit got lost. What Culbertson said was that if you play on a 4-4 fit instead of in notrump, you often gain one trick. Suppose the suit is solid. Then, it is worth four tricks in notrump. In a suit contract you can usually draw trumps in three rounds later using your remaining trumps separately, to get five trump tricks; and should you be able to ruff more, you may get six or even seven trump tricks. The issue was 4-4 versus notrump, not 4-4 versus some other possible trump suits.

On page 71 in I Fought The Law of Total Tricks we give an example of a 4-4 fit being better than a 5-3 fit. Such deals are not uncommon. But it is important to realize that the extra trick(s) don't come from the 4-4 fit; they come from the discards on the long side-suit. If we can't use those discards, we can just as well play on the 5-3 fit. And when the discards are useful, that is reflected in a higher WP. In this example, North-South have 24 WP if hearts are trumps. Add that to the SST of 4, and we get 10 tricks. But if they play in spades, there are two heart discards coming, so the WP go up to 30. That means there is potential for 12 tricks, which there is, but on a club lead, declarer can only take 11 tricks. The reasons for the missing trick comes in the following section.

(3) Not enough trumps

This one is going to please the Law friends, because the situation we are going to discuss is one where the Law has a good point.

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

North-South have 14 WP and an SST of 4. We would expect 7 tricks, but on the best defense of repeated trump leads, there are only 6 tricks.

That the formula fails here is because one of North-South's short suits (North's doubleton heart) couldn't be used. The result would have been the same had North been 4-3-3-3. The effect of only three trumps in dummy is that SST for North-South was equal to South's short suits, i.e. 5.

This is a fairly common situation, and when one hand is weak with not too many trumps, and no source of tricks in any of the side-suits, the estimation will be too high.

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

It is worth noting that "not enough trumps" may be any number. Here, ten trumps are too few. North-South have 10 WP and an SST of 2 (adjusted to 1, because of a third short suit), but they don't take 9 tricks, they only take 8 if the defenders lead trumps in time. Add a trump in dummy, and South will be able to ruff all his losers there.

Now, let's swap one of North's spades for one of South's trumps, and the formula predicts accurately. In that scenario, dummy has still not enough trumps to ruff South's all losers but he has an extry entry, which permits him to set up and use dummy's long diamonds. And that brings us to the next section...

(4) Entries

Finally, we mustn't forget an important thing like entries. If a contract depends on dummy having three entries and he only has two, for instance, any estimation is likely to fail. Here is a typical example:

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

14 WP and an SST of 3 (adjusted to 2 because of the third short suit) suggest 9 tricks, but if trumps are 3-1, and the defenders do what usually is best when they hold the balance of power, i.e. lead trumps to cut down on ruffs, declarer will be held to 8 tricks.

But if we move one of the spade honors to the East hand (say the East has Q-8-7-6 instead), things change. Now, declarer wins the first two trump leads in hand and plays on hearts. Then, he will indeed take 9 tricks as long as hearts are 3-2.

A related issue is if the contract is played from the right side. On Page 91 in I Fought The Law of Total Tricks, we gave an example of a two tricks difference depending on if South or North was declarer. Such deals are not rare. The Law makes no mentioning of this important fact, but we do. So even if your estimation is correct, you are not guaranteed of a good result – if one of the sides (or both) is attempting to play the contract from the wrong side.

Copyright © 2016, Mike Lawrence & Anders Wirgren