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:       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 [point-counting] 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:       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 two-trick set, even doubled, thinking that East-West 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 part-score. 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 [point-counting] 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.

      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 North-South arrived at a contract 4; in the other room, East-West 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 North-South in , plus 8 for East-West 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, North-South have 10 clubs, East-West 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 North-South, 8 for East-West.  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.

      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, East-West 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 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, 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."
[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.]

      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.

      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):

      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 East-West from making 3. 

1. 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.
2. 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.
3. 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 total-trick minimum with which we can overcall up to 3.   Thus, we say that such a competitive bidding is protected by “security of distribution.”

      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 major-suit 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, East-West 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 East-West 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 17-23 HCP, in/at a pinch, no greater than 15-25 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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