Some New Ideas Concerning
The Law of Total Tricks
by David Promislow
In this article, we investigate the quantity R, defined on any bridge deal as: R = total tricks - total trumps, where total tricks equal the sum of the number of tricks taken by each side if it declares a contract in its longest suit, and total trumps equals the number of cards in these two suits.
If either side has more than one suit of maximum length, we choose the suits and the particular declarers that maximize the total tricks. For total tricks we assume double-dummy play, so this total is what would be deduced from duplicate deal records. It should be close to, although not always equal to, what one would expect from normal play.
The quantity R originates from an article by Jean-René Vernes, which appeared in the June 1969 issue of The Bridge World. Vernes used somewhat different wording but essentially postulated that the expected value of R = 0, a statement he called The Law of Total Tricks, commonly referred to in the literature as just The Law.
Vernes went on to suggest a practical use for this result, that a possible method of predicting the total trick figure, which can be a valuable piece of information in a contested auction, would be to use the total trumps if that could be deduced. This is tantamount to assuming that R actually equals 0 on any particular deal, an assumption that some people have incorrectly taken to be the statement of The Law.
That the question of whether total trumps can be used to estimate total tricks is controversial is indicated by the fact that some of our most eminent bridge theorists have come down on opposite sides of the issue. Larry Cohen is an enthusiastic supporter and is no doubt largely responsible for the current popularity of this principle through his two books To Bid or Not to Bid and Following the Law. Cohen, like Vernes, recognizes the fact that R is obviously not equal to 0 on every deal, but nevertheless argues that the total trump figure can still be used effectively to predict total tricks, provided one invokes a series of adjustments, some of which had been suggested by Vernes. In contrast, Mike Lawrence, together with Anders Wirgren, have many reservations, as outlined in their book I Fought the Law of Total Tricks, in which they argue that counting total trumps is not in general a satisfactory method for arriving at the number of total tricks.
Some information regarding the distribution of R can be obtained from a study done by Matt Ginsberg in a computer analysis of close to half a million deals. The data is subdivided by the number of total trumps, and for those deals where this number is less than 19, about 82 percent of the total, the results provide some confirmation of the Law. The average of R on this set is 0.03. Moreover, R = 0 on about 41 percent of the deals and is off by more than one only 12 percent of the time. Some may take this as a justification for estimating total tricks by total trumps, although Lawrence and Wirgren argue that a method that is wrong close to 60 percent of the time cannot be considered as an effective bidding tool. Things do not work so well for larger numbers of trumps. When the total is 19 or more, R is not close to 0 and is in fact concentrated on negative values.
The main purpose of this article is to give a formula for R that could be of some help in interpreting and obtaining more information about this quantity. The formula involves the difference in the number of trumps in the two hands, indicating that it would be of interest to subdivide the results of a simulation of R not just on the number of total trumps but also on how this total is divided between the two hands. For example, there may be a very different behaviour of R on deals with a 9-9 split rather than a 10-8 split. The formula also provides some explanation for the negative values in the case of a large number of trumps. It provides conditions under which the assumption that R = 0 does or does not hold. Moreover, the methods used to derive the formula suggest an alternative procedure for estimating total tricks that does not rely on knowing total trumps.
To begin, note that there is an exact expression for total tricks that is well-known to many players: The number of total tricks is equal to the excess of the number of tricks your side can take on offense over the number of tricks your side can take on defense, plus 13. For example, suppose that this excess is four. If your side can take ten tricks on offense, then you can take six tricks on defense, so the opponents can take seven tricks on offense, giving a total of 17 = 4 + 13.
The excess is itself the sum of four quantities, each of which can be positive, negative or zero. For each suit, there is a contribution to the excess that is the number of tricks that can be taken in that suit when playing on offense, less the number that can be taken in that suit when playing on defense. This total can be expressed as a sum of two terms: Excess = ET + EN, where ET = that portion of the excess attributable to your trump suit, EN = that portion of the excess attributable to the non-trump suits.
The split is important, since the factors affecting the two quantities are different. An ambiguity arises, because the contribution of the individual suits to the excess depend on how the play unfolds. It is possible to make the definitions precise, but any such attempt is quite complicated. Most deals have normal lines of play that make a suitable division readily apparent. The most common decision occurs on offense when there are cards that could be ruffed, so they would be counted in ET, but don't need to be ruffed since they are winners or can be discarded on high cards in the opposite hand. In the latter two cases, such tricks would be counted in EN. In analyzing deals to determine such a division, a suitable method is to count such a trick in ET if the ruff can be taken safely--that is after the opponent's trumps are extracted. This seems to work best in conjunction with with the estimation procedure shown below.
How might one estimate these two quantities? ET will normally increase with the number of trumps your side holds. An additional trump means that you may well take an extra trick on offense, and, since the opponents are short that trump, you may take one less trick with your suit on defense, thereby possibly adding two to the excess. Deals in which extra trumps do not account for extra tricks are atypical deals. If we accept the two-trick addition, a simple formula would estimate ET as 2n-k for some constant k, where n is the number of trumps your side holds. Here are some examples to show that this formula works well in many instances if we take k = 13.
Take the common case of a four-four trump division. If you have the top three honors, you will normally be able to make five trump tricks on offense, assuming one ruff, and two tricks on defense, given the usual three-two split, for an ET value of three, which agrees with the formula. If you have king-queen-jack in the combined hands and are missing the ace, you should similarly acquire four tricks on offense and one on defense, again for an excess of three. Suppose you have ace-queen-jack in one hand and low cards in the other. If the king is onside, the five-two figures should still hold most of the time, while if the king is offside you will have one less trick on offense, but also one less on defense. (This latter point is an example of a familiar principle, first illustrated by Vernes, and elaborated by Cohen, that switching two non-trump cards, which keeps total trumps unchanged, often changes the tricks for each side but leaves that total unchanged as well. It provides a possible explanation for the somewhat surprising conclusion that one may be able to estimate total tricks without first estimating the particular tricks taken by each side. However, this principle does not always hold, as noted by Lawrence and Wirgren, who devote an entire chapter to examples where it fails.)
The case of a five-three split shows similar results. You can expect one more offensive trick in the long trump hand, but this is offset by the fact that you might not be able to obtain a ruff in the hand with only three trumps.
Now suppose you have nine cards in your proposed suit, say a five-four split, holding the top three honors. You should normally get six tricks on offence, allowing for one ruff, and one trick on defence, giving the formula answer of five. Sometimes there will be a two-two split in the opposing hands, so you can get two tricks on defence, but then you might freely obtain two ruffs, again achieving an excess of five. With a six-three split and the top honours, you can expect six tricks on offense (a ruff is not always possible) and one on defense (in the most-probable case of a three-one split).
In the less likely situation of ten trumps, say a six-four division, holding the top two honors, you expect one trick on defence, and if you can obtain two ruffs, you have the formula result of seven.
For seven trumps, the formula gives an excess of 1, which seems appropriate,
Now consider EN. It is not unreasonable to start with a preliminary estimate of zero. As pointed out by other writers, if you have an ace in a suit, it will normally take a trick on either offense or defense. There are several cases where EN is not zero, but if some of the three terms making up EN are positive or negative, these might cancel out to leave EN = 0.
Let's consider the effect of using ET = 2n -13, EN = 0, which I will call the standard estimates. Let e be the error made in using the standard estimates. That is, e = (true value of excess - standard estimate of excess). Let d = (the number of your trumps - the number of opponents' trumps). We know that total tricks is equal to the true value of the excess + 13, so we can write: total tricks = standard estimate + e + 13 = 2n - 13 + e + 13 = 2n + e, while total trumps = n + n - d. Subtracting, this leads to the promised formula: R = e + d.To clarify: We defined R on a particular deal, but for each deal these there are two values of e, one for the North-South hands, and the other for the East-West hands. Thus, e is defined on a pair of items--a deal and a specific direction. R could be defined similarly, but as it involves totals, its value will be the same for both directions. The the value of d for one direction is the negative of the value of d for the other, so the above formula shows that the two values of e on a deal will differ by the absolute value of the difference in trump holdings.
This formula expresses the error in estimating total tricks by total trumps as the sum of three quantities, which allows one greater scope in determining whether this procedure might or might not be appropriate. The three quantities are the error in using the standard estimate for ET, the error in using in the standard estimate for EN, and the difference in the number of trumps. (The values of the first two of these will depend on how one makes the division into ET and EN.)
The formula shows that the assumption that R = 0 on a deal with equal trump holdings is equivalent to assuming that the combined standard estimate of the excess gives the correct total value of the excess for each side. In fact, correctness for one side implies that of the other.
For unequal trump holdings, the assumption that R = 0 is equivalent to assuming that the combined standard estimate for the side with the larger number of trumps overstates the true excess by the absolute difference in the number of trumps. Equivalently, the combined standard estimate for the side with the smaller number of trumps understates the true excess by the absolute difference in the number of trumps. Each of these statements implies the other.
Another perspective: Estimating total tricks by total trumps is first using the standard estimates, then modifying that by making use of a second opinion. For example, suppose you have nine trumps to the opponents' eight. You will first estimate 18 total tricks. You then realize that if the opponents use the same method they will estimate 16 total tricks. It is reasonable for you to conclude that you have overestimated, and they may similarly conclude that they have underestimated, so if you both split the difference, you will arrive at the so called Law estimate of 17 tricks.
The distribution of e = R - d could be obtained from a study that subdivided results by the trumps for each side. The expected value of d is clearly equal to 0, so if the expected value of R really is equal to 0, then the same must hold for e. However, for deals with unequal trump holdings, e can equal 0 on at most one of the two directions. Observation suggests that e is negatively correlated with d, which provides one reason for the fact that R is generally close to 0 on most deals.
The formula above for total tricks shows that we can effectively estimate this quantity without knowing total trumps as long as we know the number of trumps for our side and we have a good estimate of the error e. To do this, we must recognize situtations where the standard estimates do not hold. There are four different cases. We have e < 0 in cases 1 and 3, and e > 0 in cases 2 and 4. These incorporate many of the ideas of Cohen and Vernes and show how the adjustments proposed to the total-trumps figure when estimating total tricks can be framed as correcting errors in the standard estimates for ET and EN.
1. ET is less than 2n - 13:
This is quite common and usually occurs when the distribution is such that the required number of ruffs cannot be obtained. Lawrence and Wirgren give several examples where total tricks is less than expected, and this is the primary reason. Indeed, their first example is a deal in which total tricks is three less than total trumps. Two of these short tricks follow from the five-five trump holding, where both hands have 5-3-3-2, so ruffs will provide at most one extra trick.
Cohen is well aware of this situation and lists balanced distributions as one of the features requiring a negative adjustment when estimating total tricks. Lawrence and Wirgen also stress the importance of distribution in determining total tricks, so everybody is in agreement.
Since the standard estimate of ET for a five-five division assumes three ruffs, a shortage will almost always occur with this holding. This is true as well for six-four splits missing the top two honors and nearly all holdings of more than 10 trumps. This explainx the results in the Ginsburg study, where R tends to take negative values for large number of total trumps.
Another major reason for a deficit in ET is a paucity of high cards in the trump suit. For example, holding five to the queen opposite three to the jack with poor intermediates might well mean two tricks on offense versus none on defense. Four to the king opposite four to the queen might result in three tricks on offense versus one on defense. This adjustment is mentioned by both Cohen and Vernes.
2. ET is more than 2n -13:
As a companion statement to that of case 1, the most common reason here is when shortness provides the opportunity for many ruffs. A typical example is a four-four split with the top three honors, where the distribution and communication are such that you can obtain two ruffs.
An excess also can arise where one hand has a large number of trumps. For example, a six-two split with the top three honors will usually mean a value of four for ET.
3. EN is negative:
The primary reason here is a lack of purity in the non-trump suits. Purity means that your side's honors combine well with each other and work to produce tricks, as when you hold queen-third and partner has king-jack-low. In contrast, if you hold queen-third opposite two low cards, that is a lack of purity. The queen may well take a trick on defense but not on offense, causing that suit's contribution to EN to be negative. Cohen has an extensive discussion of this phenomena in both of his books, under the heading of
Lawrence and Wirgren are somewhat dismissive of this concept and present a deal that is highly pure but for which total tricks is less than expected. However, this example is one of those mentioned in case 1, where the deficiency arises from the lack of ability to get sufficient ruffs. Purity or lack of it in side suits will affect EN rather than ET.
Another source of a negative value in the contribution of the opponents' trump suit to EN is tricks obtained by ruffs when you are defending.
4. EN is positive:
The most common reason here is a second long suit, generally of length four or more, that produces tricks on offense but not on defense. Vernes and Cohen both recognize this by mentioning a double fit as a positive adjustment factor. An article by Steven Bloom and Mel Colchamiro, The Second Fit (Bridge World, December 2011), is largely concerned with estimating the amount of adjustment that arises from these extra tricks.
Positive contributions to EN can also occur in a short suit, when a high card will be ruffed on defense, either because of original shortness or when losers are discarded on an opponent's long suit.
The other situations causing errors in the standard estimates are minor in comparison.
Here is the alternative procedure for estimating total tricks: Forget about the number of your opponents' trumps. It can be often difficult to determine this anyway, since you can't see their hands. You do need to guess at your own. Then try to deduce e using the adjustments described in cases 1 to 4. For the error in ET, take into account such items as short suits (the key factor, according to Lawrence and Wirgren) or a five-five trump split with its likely negative contribution. Consider each of the three non-trump suits separately and use the ideas in cases 3 and 4 to estimate the error in EN. Admittedly, this will not be easy. A major problem is that the values of ET and EN can depend on the unknown distributions of the opponents. Moreover, one must avoid double-counting, as when cards in a long suit may be also ruffed. Clearly, there will be much guesswork involved but not more so than needed in counting total trumps.
The difference between this procedure and Cohen's use of adjustments is that the focus is on excess rather than on total trumps, so one doesn't need the opponents' trump count. More importantly, ET and EN are considered separately, which will provide greater accuracy.
The last paragraph of case 2 suggests an interesting question for further investigation: It could well be that one can get a much better estimate of ET from a formula dependent on the number of trumps in each hand, rather than on just the total number for the side. Consider your side's holding 13 trumps, with no tricks in the suit on defense. With a 13-0 split, the value of ET is 13. With a seven-six split, the value will most likely be seven, plus the number of ruffs available in the short hand, probably around eight or nine.
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