The Law of Total Tricks
by JeanRenĂ© Vernes
As we have seen [in the preceding chapter], the main purpose of hand evaluations is to determine
to what level we can afford to bid.
However, a more exacting analysis indicates that we can find ourselves
in two different bidding situations:
West  North  East  South

1
 1
 Pass
 4

South's 4
bid means simply,
“Partner, my hand is such that, even
if you are minimum for your overcall, we can probably make 4
”
To come to this conclusion, South has simply applied the classical [pointcounting]
method of hand evaluations.
[We say that the auction is not competitive, because
South is speaking as above without knowing adverse bidding.]
But suppose, instead, that the auction went this way:
West  North  East  South

1
 1
 4
 4

Here, the significance of South's bid is quite different. Perhaps
he expects well to make 4
. But it could equally
be that he is expecting to take a one or twotrick set, even doubled,
thinking that EastWest will make 4
. We are now
in the domain of competitive auction.
In an extremely frequent situation like this, the classical rules are helpless
to solve our problem. Certainly, it is easy to figure out that
with equal vulnerabilities, it will pay to go down two, doubled, in order to stop
an opponents' game; and that it is sometimes advantageous to go down one
to stop a partscore. Classical rules will easily let us
work out how many tricks we expect to make, if partner is minimum for
his bid. By contrast,
we have no precise way to determine whether or not
the opponents will make their contract. And nothing is more costly
than to take a sacrifice against a contract that would have gone down.
[Furthermore, it is no less costly to let opponents play a contract which they make,
when we could have covered their contract and could have made it.]
How, in fact, do good players determine, in these positions, whether
to pass, or double, or bid on ? We know, from our long experience,
that the prime factor is distribution: the more unbalanced it is, and the more trump
cards each side has, the higher is competition
justified. Ever since we were beginners in Bridge, we have learned
that the more Aces and Kings we have, the greater is the chance to make a game.
The discovery of a precise [pointcounting] scale,
fixing the relative values of various honors,
has been a great step forward.
But we do not have, yet today, a scale to tell us
how high we can bid by virtue of our distribution.
Could it be that there exists no such a scale, that in this area
we have to pride ourselves on our own intuition ?
No − my aim is to
show that competitive decisions are subject to a precise law −
a particularly simple law, what's more.
And just as the constructive bidding of opener's side depends on hand
evaluations, we believe it impossible to investigate competitive auctions
without reference to this law, at the least indirectly.
The Law of Total Tricks
Examine the following deal, which was actually played in the 1958 World
Championship (No. 93).
 North

 A 6  Dealer: North

 9 7  Both Vulnerable

 K 9 6 4

 A Q 9 3 2

West   East

Q J 10 9 2

 K 8 7

10 8 5 4

 A J 6 2

A Q

 J 10 8 5 2

10 4

 6

 South

 5 4 3

 K Q 3

 7 3

 K J 8 7 5

In the closed room, Italians NorthSouth arrived at a contract 4
;
in the other room, EastWest were allowed to play
2.
Analysis of this deal shows that the result is right. North made
10 tricks in
, losing one spade and two red Aces,
while West made 8 tricks in
at the other table,
losing one spade, one diamond, one club and two hearts.
Now I will ask the reader to consider an unfamiliar concept that I call
“total tricks” − the total of the tricks made by the
two sides, each playing in its proper trump suit. In the deal above,
the number of total tricks is 18
(10 for NorthSouth in
,
plus 8 for EastWest in
).
Now, even though it is impossible, in the course of a competitive auction,
to determine exactly how many tricks the opponents will make, can't it be possible
to predict the average number of total tricks ?
Such a prediction, as we will see shortly, is of lively interest in making
competitive decisions.
In fact, this average exists, and can be expressed in an extremely simple
law
: the number of total tricks of a deal is
approximately equal to the
total number of trumps held by both sides, each in its respective
suit. In the example above, NorthSouth have 10 clubs,
EastWest 8 spades. Thus, the total number of trumps is 18, the same as
the total number of tricks.
You may notice that in this deal the number of trumps held by each side
was equal to the number of tricks it actually made − 10 for NorthSouth,
8 for EastWest. That is a pure coincidence. It is only the
equality between the total number of trumps and the total number of tricks
that obeys a general law.
[Another example, omitted in Bridge World is reproduced here from
the French original, which would help understand the coming paragraph.
What a wonder that Vernes allowed this omission.]
In order to fix these ideas more precisely, we borrow a second example
deal from the 1956 World Championship (No. 111).
 North

 J 8 7  Dealer: East

 Q J 7 5  Both Vul.

 A 10 6 5

 9 2

West   East

Q 2

 A K 6 5 4

10 9

 6

K Q 8 7 3 2

 J 4

Q 10 5

 A 8 7 6 4

 South

 10 9 3

 A K 8 4 3 2

 9

 K J 3

In the closed room, West plays
4.
With this position of
K, he cannot hold opponents to
take one heart, one club and two diamonds, going one down.
In the other table, South defends themselves with
4,
losing three spades, one club, and hence will take only 9 tricks.
We observe that the number of total tricks is 9 + 9 = 18, which is
equal to the number of total trumps (10 hearts
in NS plus 8 diamonds in EW = 18).
This “law of total tricks” surely seems very surprising at first sight.
An analysis of the deals I have presented will show why it works.
In the
first example, EastWest didn't know which opponent held
K.
Had it been with South, West would have been able to make one trick more
playing in
. But then, clearly, North would have
made one trick less playing in his
contract.
Thus, the actual number of tricks taken by each side depends on
the location of a key card, but the number of total tricks remains the same.
In the
second example, one may regard it unhappy for West
to play 4
and find that the trumps are split 41 in the opponents.
Nevertheless, suppose, for example, North has one less card in diamonds and
one more in clubs, South one more diamond and one less club.
In that case, West will gain one more trick playing in
,
while North will be unable to avoid one diamond loser playing in
, thus
taking one trick less.
Here again, we cannot predict the exact numbers of tricks taken by each side, but,
the number of total tricks remains unchanged in any case.
In this way,
the two major uncertainty elements,
which no classical evaluation methods can resolve (first for success in finesse,
second for split in opponent' hands) will
disappear, when we calculate the total tricks.
These are the considerations that led to the discovery of the law of total
tricks. Even though the above considerations will clarify the mechanism of the
law,
only a thorough statistical study can bring us sufficient proof of the law.
[The details of early analyses by Vernes and, independently,
by The Bridge World staff, now superseded by more complete surveys,
have been omitted.Ed.]
Detailed analysis of the deals on which the statistics were based
verifies this conclusion. It shows that, had the cards been played
perfectly, that is, doubledummy, the totaltrick formula would have given
an exactly accurate prediction in well over half the cases.
What is more, it showed that the number of total tricks would often have
been lower than that actually won  the knowledge that declarer had
of his side's full resources gave him an appreciable edge.
At doubledummy, the number of total tricks closely approximates the
theoretical number indicated by the formula. The supplementary
quarter of a trick per deal at the table may well be, in large part,
"declarer's advantage."
[In a paragraph omitted in this extract, he has remarked that the actual
average number of total tricks was a little larger (by 0.275 tricks) than
the law prediction, as a result of his statistical analysis.]
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 to calculation of
High Card Points and Distribution Points,
a more precise study shows that the number of total tricks is affected by three
extra factors:
(1) Existence of a « double fit », each side having 8 cards or more
in two suits. When this happens, the number of total tricks is
frequently one trick greater than the general formula would indicate.
This is the most important of the extra factors.
(2) Possession of trump honors. The number of total tricks is
often greater than predicted, when each side has all the honors in its
own trump suit. Likewise, the number is often smaller than predicted,
when these honors are owned by the opponents. (It is the middle
honors − King, Queen, Jack − that are of more importance than Ace and 10.) Still,
the effect of this factor is considerably smaller than one might suppose.
So it appears impossible to find a practical formula to improve on the
calculation,
and one may take account of this factor only in limited cases.
(3) Distribution of the side suits. Up to now,
we have considered only how the cards are divided between the two sides,
not how the cards of one suit are divided between two partners.
This distribution has a very small, but not completely negligible, effect.
Safety Levels
The law of total tricks has many practical uses. The principal one
is that it allows us to distinguish between two forms of safety.
We may call them “security of honors” and “security of distribution.”
Suppose the bidding goes like this (A):
Case (A)
West  North  East  South

Pass
 1
 Pass
 3



Case (B)
West  North  East  South

1
 1
 3
 3


North and South could have a mediocre distribution
: for example,
4 trumps in each hand. But then, they must
have enough points (surely, minimum 23) to expect to make the
contract. The 3
bid is then protected by
“security of honors.”
In contrast, if the bidding goes like this (B),
South could be bidding 3
with a good fit even with a
weak hand in honors, to stop EastWest from making 3
.
 This 3 overcall will mostly be a little profitable,
if 3 makes while 3
goes down only one. In this case, the deal has 17 total tricks.
 Of course, that will result in a net loss, in case both 3 and
3 go down. The deal will then have 16 total tricks.
 Third, if the four hands are such that both 3 and 3 make,
the overcall is extremely profitable. The deal has now 18 total tricks.
The figure 17 is the totaltrick minimum with which we can overcall up to 3
.
Thus, we say that such a competitive bidding is protected by “security of distribution.”
Practical Rule
The most delicate problem in practice is to determine the number
of total trumps. (Paradoxically, this calculation is often easier for
the defending side than for opener's side. For example, you can usually
work out the total trumps with great precision, when a reliable partner
makes a takeout double of a majorsuit opening.) Most often, however,
players can tell exactly how many trumps their own side has, but not how many
the opponents have. Nevertheless, this knowledge itself suffices to allow
the law of total tricks to be applied with almost complete certainty
:
Consider, for example, the
second bidding sequence (B), and suppose that
South has 4 cards in
. After partner's 1
overcall, he
can count on partner for at least 5 spades, or 9 spades in total for his side.
Thus, EastWest have at most 4 cards in
among their 26 cards.
In other words, they must have a minimum of 8 trumps in one of the three
remaining suits [(26−4)/3]. Thus, South can count for the deal a minimum of 9+8=17
total tricks. So a bid of 3
is likely to show
a profit, and at worst will break approximately even.
A similar analysis shows that the situation is entirely different, when
South has only 3 cards in
, so that his side has a considerable
chance of holding only 8 trumps. To reach the figure of 18 total
tricks, it is now necessary for EastWest to hold 10 cards in their
suit − which is not impossible, but hardly likely. It is much more reasonable
to presume that the deal will yield only 16 or 17 total tricks. Thus,
it is wrong to go beyond the two level.
As we examine one after another of the competitive problems at various levels,
we find that the practical rule appropriate to each particular case can be
expressed as quite a simple general rule: You are protected by
“security of distribution” in bidding for as many tricks as your side holds
trumps. Thus, with 8 trumps, you can bid practically without danger
to the two level, with 9 trumps to the three level, with 10 to the four level,
etc., because you will have either a good chance to make your contract
or a good save against the opponents' contract.
This rule holds good at almost any level, up to a small slam (with only one
exception: it will often pay to compete to the three level in a lower
ranking suit, when holding 8 trumps). Of course, the use of this
rule presupposes two conditions:
(1) difference in HCP must not be too great between the two sides,
preferably no greater than 1723 HCP, in/at a pinch, no greater than 1525 HCP;
(2) the vulnerability must be equal or favorable. For this rule to
operate on unfavorable vulnerability (Vul. vs NonVul.), your side must have as many high cards
as the opponents (or more).
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