The Law of Total Tricks is currently the most
widely used guideline in competitive bidding situations. Although only
fragmentary statistical evidence in support of the law was available
when the original Bridge World article, excerpted below, was
published, recent research, spurred by the availability of cheap
computing power, has shown that the law is remarkably accurate. Details
of its precision (how much the estimate it provides varies from deal to
deal) are also available, and most current experimentation in this
field deals with guidelines for recognizing situations in which
deviations from the global total-trick average are likely to occur.
by Jean-Rene Vernes
As we all realize, the aim of point-count valuation
is to determine the precise level to which we can afford to bid.
However, a more exacting analysis indicates that we can find ourselves
in two entirely different bidding situations:
WEST | NORTH | EAST | SOUTH |
1 | 1 | Pass | 4 |
South's bid means simply, "Partner, my hand
is such that, even if you are minimum for your overcall, we can
probably make four spades." To come to this conclusion, South has only
to apply the classical methods of hand evaluation. But suppose that the
bidding went this way:
WEST | NORTH | EAST | SOUTH |
1 | 1 | 4 | 4 |
Here, the significance of South's bid may be
quite different. Perhaps he expects to make four spades. But it could
equally be that he is expecting to take a one- or two-trick set, even
doubled, thinking that East-West will make four hearts. We are in the
domain of competitive bidding.
Now, in this extremely common position the
classical rules are helpless to solve our problems. Certainly it is
easy to figure out that with good vulnerability it will pay to go down
two, doubled, to stop an enemy game; and that it is sometimes
advantageous to go down one to stop a part-score. Point-count valuation
will easily let us work out how many tricks we expect to make if
partner is minimum for his bid. But 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.
How, in fact, do good players determine, in
these positions, whether to pass, or double, or bid on? We know, from
long experience, that the prime factor is distribution: the more
unbalanced it is, the more cards each side has in its trump suit, the
higher is competition justified. Beginners learn that the more high
cards they have, the greater is their chance to make game. The
discovery of an exact scale, fixing the relative value of the various
honors, was a great step forward. But we do not have, today, a scale to
tell us how high we can bid by virtue of our distribution.
Could it be that there is no such scale,
that in this area we must pride ourselves on our intuition? No--my aim
is to show that competitive decisions are subject to a precise law, and
a particularly simple one what's more. And just as it is impossible to
talk of constructive bidding without reference to accurate hand
valuation, it is impossible to investigate competitive bidding without
at least indirect consideration of this law.
The Law of Total Tricks
Examine the following deal, Number 93 of the 1958 World Championship.
North dealer
Both sides vulnerable
| NORTH
A 6
9 7
K 9 6 4
A Q 9 3 2 |
WEST
Q J 10 9 2
10 8 5 4
A Q
10 4 | |
EAST
K 8 7
A J 6 2
J 10 8 5 2
6 |
| SOUTH
5 4 3
K Q 3
7 3
K J 8 7 5 |
In one room, the Italians arrived at a
contract of four clubs, North-South; in the other room, they were
allowed to play two spades, East-West. Analysis shows that the result
was never in doubt. North made ten tricks in clubs, losing only one
spade and two red aces, while West made eight tricks in spades 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 best trump suit. In the deal
above, the number of total tricks is 18 (10 for North-South in clubs,
plus 8 for East-West in spades).
Now, even though it is not possible, in the
course of a competitive auction, to determine how many tricks the
opponents will make, can it be possible to predict, on average, the
number of total tricks? If so, this average figure cannot help but be
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 in a hand is approximately equal to the total
number of trumps held by both sides, each in its respective suit.
In the example above, North-South have ten clubs, East-West eight
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--ten for North-South, eight for East-West. That is pure
coincidence. It is only the equality between the total number of trumps and the total number of tricks that obeys a general law.
This "law of total tricks" surely seems very
surprising at first sight. An analysis of the deal I have presented
will show why it works. East-West could not know which opponent held
the king of diamonds. Had it been with South, West would have been able
to make one trick more playing spades. But then, clearly, North would
have made one trick fewer playing in his club contract. Thus, the
actual number of tricks made by one declarer varies according to the
location of a key card, but the number of total tricks remains the
same.
Two major elements of uncertainty (will a
finesse work? will a suit split well or badly?), uncertainty that no
classical method of valuation can possibly resolve, [often] disappear
when we calculate the total tricks. These were the considerations that
led to the discovery of the law of total tricks. Still, although they
seem to clarify how the law operates, only a thorough statistical study
could bring us sufficient proof of its accuracy. [[The details of early
analyses by Vernes and, independently, by The Bridge World staff, now superseded by more complete surveys, have been omitted.--Ed.]]
A more detailed analysis of the deals on
which the statistics were based verified this conclusion. It showed
that had the cards been played perfectly, that is, double-dummy, the
total-trick 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 double-dummy, 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."
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.
(1) The existence of a double fit, each side
having eight 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) The 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
lower than predicted when these honors are owned by the opponents. (It
is the middle honors--king, queen, jack--that are of greatest
importance.) Still, the effect of this factor is considerably less than
one might suppose. So it does not seem necessary to have a formal
"correction," but merely to bear it in mind in close cases.
(3) The distribution of the remaining
(non-trump) 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:
WEST | NORTH | EAST | SOUTH |
Pass | 1 | Pass | 3 |
North and South could each have mediocre
distribution, but then they must have a high enough point count (say,
24 points) to expect to make the contract. The bid of three spades is
protected by "security of honors." In contrast, if the bidding goes:
WEST | NORTH | EAST | SOUTH |
1 | 1 | 3 | 3 |
South could be bidding
three spades with a good fit even with a low point-count, to stop
East-West from making three hearts. This three-spade bid will usually
be slightly profitable if either side can make its contract, even if
the other's contract would be down one. Of course, it will show a loss
if both three spades and three hearts go down. And it will be most
successful when both contracts make. In the first case, the deal has 17
total tricks; in the second case, 16; and in the third case 18. The
figure 17 is the total-trick minimum at which we can outbid the
opponents to the three-level. Thus, we may say that such a competitive
bid is protected by "security of distribution."
A Practical Rule
Unfortunately, it is very difficult in
practice to determine the total number of trumps. (Oddly, this
calculation is often somewhat easier for the defending side than for
opener's. For example, you can usually work out the total trumps with
great precision when a reliable partner makes a takeout double of a
major-suit opening.) Most often, though, players can tell exactly how
many trumps their side has, but not how many the opponents have.
However, this itself is sufficient to allow the law of total tricks to
be applied with almost complete safety.
Consider, for example, the second bidding
sequence above, and suppose that South has four spades. After partner's
one-spade overcall, he can count on him for at least five spades, or
nine spades for his side. Thus, East-West have at most four spades
among their 25 cards. In other words, they must have a minimum of eight
trumps in one of the three remaining suits. Thus, South can count for
the deal a minimum of 9+8=17 total tricks. So a bid of three spades 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 three spades, so that his
side has a considerable chance of holding only eight of its trumps. To
reach the figure of 18 total tricks, it is now necessary for East-West
to hold ten cards in their suit--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; three
spades must lose or break even.
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 a quite simple
general rule: You are protected by "security of distribution" in bidding for as many tricks as your side holds trumps.
Thus, with eight trumps, you can bid practically without danger to the
two level, with nine trumps to the three level, with ten to the four
level, etc., because you will have either a good chance to make your
contract or a good save against the enemy 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 eight trumps).
Of course, the use of this rule presupposes two conditions: (1) the
point-count difference must not be too great between the two sides,
preferably no greater than 17-23, certainly no greater than 15-25; (2)
the vulnerability must be equal or favorable. For this rule to operate
on unfavorable vulnerability, your side must have as many high cards as
the opponents (or more).
|