Discussing the Law of Total Tricks

Ever heard of the Law of Total Tricks? We guess you have. Formulated by French bridge theoretician Jean-René Vernes in an article in The Bridge World in June, 1969, it spread like wildfire when American expert Larry Cohen popularized it with his two books To Bid or Not to Bid and Following the Law, published in 1992 and 1994, respectively. Today, few people, whether experts or just average players, seem to question its accuracy.

The Law of Total Tricks, or the Law, as its adherents usually call it, is a theory which says:

"The number of tricks your side can take, if you play in your best trump suit, added to the number of tricks the opponents can take, if they play in their best trump suit, is approximately equal to the number of trumps held by both sides."

So, if North-South's best trump suit is spades, where they hold eight cards, and East-West's best trump suit is clubs, where they hold nine cards, the total number of trumps is 17. If the sum of the tricks North-South can take in spades and the tricks East-West can take in clubs – the total tricks – also is 17, the deal is according to the Law.

What is important to remember is that Vernes was talking about average values. And if you take a large number of deals and compare the average total tricks with the average total trumps, they will indeed be approximately equal. So far, so good.

What Vernes didn't say – and definitively didn't mean – but which many people think the Law says, is that "Total tricks and total trumps are equal on any deal". One of the reasons why this false claim is so popular is that Larry Cohen formulated the Law that way in his first book To Bid or Not to Bid. Even The Official Encyclopedia of Bridge has got it wrong, by including the expression "on any given bridge deal", words which weren't present in the original and correct definition of the Law. It is important to understand this, because the truth is that it is more likely that trumps and tricks are not equal (roughly 60% versus 40%).

If you are a questioning person you may wonder why trumps and tricks should go hand in hand. Lots have been written about the Law, but practically no explanation has been given. Larry Cohen skips the issue, just like the other main proponent of the theory, Cohen's former partner Marty Bergen. And there is a very simple reason for their silence: There is no connection. If we have a deal where the total tricks are 16 and the total trumps also are 16, the tricks are not a function of the trumps. They are related to each other only tenuously and indirectly.

We who say so are three-time world champion Mike Lawrence of the USA and bridge theoretician Anders Wirgren of Sweden, and we can prove that the number of trumps is a poor guide to the number of tricks. Devotees of The Law have no proof to back up their claim. We will show you what is important in estimating your potential tricks and we will give you a brand new tool that will make you cry: "The Law is dead. Long live (the new) Law!".

All this is presented in our book, I Fought the Law of Total Tricks, which tells you all you need to know about competitive bidding. Still, you may have questions and doubts, which we understand very well. After all, this thinking is new. We will try to help you along the way, so don't hesitate to contact us with questions and deals you want to have analyzed. If we find your question to be interesting, we will publish it here on this site together with an answer. And don't worry. You won't see any answers like "You can't expect the Law to be right on every deal," which used to be the stock retort of the other side. We will tell you exactly why. To submit a question and/or an interesting deal, use the links "Questions" or "Deals" in the left frame.

Let us look at two examples, where both sides are vulnerable:

1)

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

North-South have eight spades and can take nine tricks. East-West have eight hearts and can also take nine tricks. The total tricks are 18. Both sides have eight trumps, so the total number of trumps is 16. Therefore, this deal is not according to the Law. Since there are two more tricks than trumps, we call it +2.

And before the Law guys start screaming about "the need for adjustments" and mutter something about a factor called 'purity' (which, by the way, is just nonsense), we keep the same honors, give each side an extra trump and change the distribution slightly. Then we get:

2)

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

Now the situation is the reversed. Here, the total trumps are 18, but the total tricks are 16. Since we now have two tricks fewer than the total trumps, we call the deal -2. And since the 'purity' of the deal is exactly the same as in the first diagram, we have just proved that purity does not explain the difference. Else how could two tricks disappear into thin air?

A corollary to the Law of Total Tricks, often called The Law of Total Trumps and sometimes confused with it, is that you should "always bid to the same level as your number of trumps". And a few years ago Larry Cohen won a prize for a bridge tip titled "Eight never, nine ever." With that he meant that in deciding whether to bid 3-over-3, say 3 spades over the opponents' 3 diamonds, you shouldn't do it with eight trumps but you should with nine.

Applying this principle to the deals above, we realize that North-South will do the wrong thing on both occasions if East-West compete to 3 hearts. In diagram 1, they will let East-West play (eight never), and in diagram 2 they will bid 3 spades (nine ever). And in doing so, they have gone -140 instead of +140, and -100 instead of +100. Uh-oh!

Does that mean we have to despair, and accept "the occasional bad result" (in truth, more than occasional), as advocates of the Law say? No, not when there is a better way, that tells you what to look for and helps you do the right thing more often than not. Read our book and find out for yourself. To order it, use the link "Order our book" in the left frame.


Ever heard anyone say, "Not enough trumps to bid, say, 5 hearts over 4 spades"? You surely have. The Law guys say things like that all the time. The examples just discussed showed that explanation to be false – and here is another example, from the Bermuda Bowl in Port Chester, 1981:

North dealer, North-South vulnerable

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

In the match between Pakistan and Argentina, the bidding went as below:

WestNorthEastSouth
pass1HDBL
3H4S5HDBL
passpasspass

South led a heart, won the spade king at trick two and returned another heart, sacrificing his trump trick. When he gained the lead next, he played his last trump to restrict East to five trump tricks in hand and one ruff in dummy. That excellent defense meant declarer was five down; -900 (according to the old scale for doubled undertricks – today it would be -1100), 6 IMPs worse than conceding -680 to North-South.

In To Bid or Not to Bid, Cohen says that East should have figured nine trumps for his side and eight for the opponents. If so, there were 17 total tricks, and if North-South can take 10, East-West only take seven. To bid 5 hearts was a violation of the Law and was severely punished.

According to that reasoning, there weren't enough trumps on this deal. But what about the next one, where we have moved the heart king from East to West and kept everything else as it was.

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

On the same defense, East now takes no fewer than nine tricks if he gets everything right (and the Law assumes that). He wins the lead in hand and gives up a diamond. On the trump return, he sticks in the ten, ruffs a diamond, draws the last trump with the king and ruffs another diamond. All he loses are three black tricks and one diamond, -300. Now, East wins 9 IMPs for bidding 5 hearts over 4 spades.

If North plays in spades, he will take twelve tricks no matter where the king of hearts is. For North-South, it doesn't matter where the heart honors are.

So, if there were "not enough trumps to bid 5 hearts over 4 spades" on the previous layout, but here there are indeed "enough trumps to bid 5 hearts over 4 spades," we have a contradiction, as the number of trumps in each example is the same. Therefore, the argument "Not enough trumps to bid 5-over-4 (or 3-over-2 or whatever you may think of)", is not valid. If you ever hear it again (we hope you don't), dismiss it as irrelevant.

You may have read that any changes which gain tricks for one side if they play the hand are compensated by the same loss for the other side if they declare. That claim is false. Many changes affect one side only. Here, moving the heart king from East to West meant a gain of three tricks for them (actually, only two, because if the defenders lead and continue spades – their best defense in that scenario – East will be held to eight tricks), but it didn't affect how many tricks North-South could take in spades (or clubs). As this example shows, where the honors are located may matter for one side but not for the other.



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

What we said about honor location can also be applied to distribution, so that a change which is favorable to one side may not influence the other side's tricks. The diagram above is one such example. It is clear that East-West can take five tricks in a spade contract, no more no less.

North-South's best trump suit is diamonds, where they have eight cards. Since 8+9=17, the Law says there should be 17 total tricks. Is that so?

Well... The answer is that it depends on North-South's distribution in spades, their weak suit. If they are 2-2, North-South will take 11 tricks, and the total tricks will be 16. If their spades are 3-1, they will take 12 tricks, and if their spades are 4-0 they will take all 13 tricks. Now, the total tricks are 17 or 18, respectively. Assuming that 16, 17 and 18 are equally likely (which they aren't; the correct percentages are: 40.7, 49.7 and 9.6), the average will indeed be 17 as Vernes said, but for any individual deal the Law will be right only one third of the time. And this shows how dangerous it may be to use an average value for prediction.

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

Conversely, East-West's tricks will vary if we change their side-suit distribution. From the last diagram, we have moved a heart to West and a club to East, a swap that means they will take one more trick if North-South's spades are 3-1 or 4-0 (by leading trumps the defenders stop the second club ruff in West's hand), or two more tricks if North-South's spades are 2-2. But the change from 3-3 in hearts and clubs to 4-2 in those suits does not affect how many tricks North-South take in a diamond contract. Once again, the change was a gain for one side only – and the total tricks went up with one or two tricks.


Copyright © 2016, Mike Lawrence & Anders Wirgren