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THEORY

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.

The Law of Total Tricks

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).