THE ACCURACY AND SCOPE OF THE WINNING TRICK COUNT

To measure is to know

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In the previous chapter I described the two procedures, which I have used to assess the effectiveness of bridge-hand evaluation systems.

As we shall see below, they were applied systematically to an unusually detailed analysis of several hundred documented bridge deals with the aim of assessing the evaluation performance of the Winning Trick Count and comparing it with that of the long-established and broadly accepted Losing Trick Count.

The first stage of this exercise – the identification of the Winning Trick prospects for every one of these deals - involved an initial appraisal of the card distribution, the point count and the honour-card strength for each of their four hands. Following which the information was used to identify the best prospective trump fit and the total point and honour-card counts for both the N/S and the E/W pairs of hands.

In all over 600 contracts were individually assessed in the study. They were taken from two quite distinct sets of randomly dealt hands:

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The first – Over 180 of the grand total of 255 boards that were played in the 1991 World Bridge Championships. These provided 264 of the contracts and they were assessed by directly comparing the Winning Trick forecasts with the number of tricks that were actually made – the first procedure described in the previous chapter.

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The second – 240 deals of eight consecutive BBL (British Bridge League) Open Competitions. These provided 346 of the contracts and in this case the EstimatedTrickValue – the second procedure described in the previous chapter - was used to assess the accuracy of the Winning Trick forecasts.

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The results of these two complementary exercises are described later, but before moving on to these somewhat detailed commentaries the broad conclusions of the overall study and the related comparison of the accuracy of the Winning Trick Count and the Losing Trick Count are outlined below.

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8.1AN OVERVIEW OF THE EVALUATION STUDY.

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The table below provides a summary of the outcome of the assessment of the accuracy of the Winning Trick estimates for all 610 contracts. In order to provide a very simple, but meaningful measure of the success of the evaluation procedure the results are broken down in terms of the number of forecasts, which were correct, or within one trick of the right result.

Although, as we shall see later I judge that that bridge-hand evaluation procedures are broadly irrelevant to the majority of no-trump hands, I have nevertheless included such deals in this consideration in order to assess the validity and accuracy of the Winning Trick concept over a significantly wider range of boards than would otherwise have been possible.

To that end, I have simply assessed the Winning Trick prospects for such no-trump hands as being made up of honour-card or point-count tricks as defined previously, but with suit-length tricks in place of trump tricks [1].

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EVALUATION PROCEDURE	NUMBER OF CONTRACTS	ESTIMATE CORRECT   No. %		ESTIMATE CORRECT WITHIN +/- ONE TRICK No.%
Winning Trick Count (Honours)	610	336	55	553	91
Winning Trick Count (Points)	610	352	58	567	93

8.2THE EVALUATION STUDY AND THE LOSING TRICK COUNT.

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The next stage of the exercise was a comparison of the accuracy of the Winning Trick Count and the Losing Trick Count. In contrast, however, with the overview above theno-trump contracts were excluded from this Winning Trick consideration in order to ensure that the comparative study of the two evaluation procedures was based on similar deals.

A similarly detailed appraisal of the Losing Trick Count prospects for every deal was then undertaken. In this case it was found that the more limited scope of this traditional evaluation procedure with hands characterised by an adverse trump fit, meant that the number of contracts, which were considered, was somewhat smaller. The results of the ensuing comparison are set out in the table below.

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EVALUATION PROCEDURE	NUMBER OF CONTRACTS	ESTIMATE CORRECT   No. %		ESTIMATE CORRECT WITHIN +/- ONE TRICK No. %
Winning Trick Count (Honours)	545	273	50	488	90
Winning Trick Count (Points)	545	288	53	502	92
Losing Trick Count	430	173	40	359	83

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Although the exercise provided confirmation of the expectation that the Losing Trick Count would be successful in predicting the outcome of a sizeable fraction of the contracts it also showed that this widely-used evaluation procedure failed to match the performance of the two versions of the Winning Trick Count. A conclusion, moreover, which was consistently confirmed in all of the subsequent comparative studies that were undertaken.

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8.3THE EVALUATION STUDY AND THE 1991 WORLD CHAMPIONSHIPS

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As noted above one of the primary aims of the evaluation study was to undertake a direct appraisal of the accuracy of the Winning Trick Count by comparing the forecasts for a wide range of World Championship deals with the number of tricks which were actually made when the hands were played.

The exercise, which was based on the published proceedings of the Semi-Finals and Finals of the 1991 World Bridge Championships for the Bermuda Bowl and Venice Cup, was undertaken only for those deals where it was possible to make an unambiguous judgement about the eventual outcome.

This meant that a number of boards were excluded from the consideration because of the variation in the contracts, which were reached, or in the number of tricks that were made at different tables.

In practice, however, this restriction did not prove to be a significant constraint and in fact, 186 of the total of 255 boards that were played in the two events were fully analysed.

Moreover, since a number of these boards were played by both N/S and E/W they provided a total of 264 contracts for consideration.

The availability of such a large database of real hands had three important advantages:

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-The first, that it combined a wide selection of randomly-dealt deals with the reasonable presumption that the results reflected a consistently high standard of play and hence that they provided an accurate basis on which to judge the evaluation forecasts. Moreover, the use of the same hands for both competitions provided an additional check on the outcome of many of the contracts. In fact, the 95 Semi-Final boards were all played eight times, while most of the 160 Final boards were played four times.

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-The second that the highly competitive nature of the competition meant that a number of boards were played in contracts by both sides. The availability of these results meant that the assessment of the evaluation procedures was not only able to cover a wide range of weak, as well as strong pairs of hands, but that they also enabled the study to be extended to a consideration of the link between the Winning Trick Count and the Law of Total tricks.

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-The third, as we shall see below, that the broad range of real deals and results also provided the basis for a detailed appraisal of the validity and accuracy of the Estimated Trick Value concept for the evaluation of bridge hands.

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The results of this detailed study of these World Bridge deals are summarised below

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EVALUATION PROCEDURE	NUMBER OF CONTRACTS	ESTIMATE CORRECT   No. %		ESTIMATE CORRECT WITHIN +/- ONE TRICK No.%
Basic Winning Trick Count (Honours)	264	138	52	232	88
Winning Trick Count (Points)	264	159	60	238	90

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It can be seen that the exercise provided a convincing demonstration of the strength and relevance of the Winning Trick Count for the evaluation of real bridge contracts.

In particular, the basic honour-card version of the WTC was again within one trick of the right result around 90% of the time and it predicted the correct outcome for over 50% of the contracts. And, as before, the point-count version scored equally well.

Moreover, the outcome was broadly the same when the Semi-Final and Final deals which made up this total were assessed separately and also when the appraisals were based on the shape and competitiveness of the hands. In view of this we will consider just one more feature of this detailed comparative analysis.

This was aimed at comparing the success of the Winning Trick Count in evaluating the trick-taking prospects of both the weak and the strong pairs of hands in competitive auctions. To that end, the 264 World Championship contracts were divided into those with a point count of at least twenty, and those (over 27%) where the side, which won the auction and actually played the hand, had a total of nineteen points or less.

The results of this particular study of the World Bridge hands, which are summarised in the two Tables below, show clearly that the estimates for such low-point contracts are quite comparable in accuracy with those for the stronger pairs of hands

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EVALUATION PROCEDURE	NUMBER OF CONTRACTS	ESTIMATE CORRECT   No. %		ESTIMATE CORRECT WITHIN +/- ONE TRICK No.%
Basic Winning Trick Count (Honours)	192	97	51	169	88
Winning Trick Count (Points)	192	114	59	175	91

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EVALUATION PROCEDURE	NUMBER OF CONTRACTS	ESTIMATE CORRECT   No. %		ESTIMATE CORRECT WITHIN +/- ONE TRICK No.%
Basic Winning Trick Count (Honours)	72	41	57	63	88
Winning Trick Count (Points)	72	45	62	63	88

8.4THE EVALUATION STUDY AND THE LAW OF TOTAL TRICKS

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The World Championship results described above also provided a convenient database for a consideration of the link between Jean-René Vernes’ Law of Total Tricks and the Winning Trick Count, which I described in Chapter 4 above. This correlation arises from the fact that:

On the one hand, we have the LAW which maintains that ‘The number of total tricks in a hand is approximately equal to the number of trumps held by both sides’

On the other, if we add together the two Winning Trick estimates for any deal we find similarly that - the total number of tricks, which would be available if each side were to play in their best contract, is simply the sum of the combined trump holdings.

When I tested this link with the broad selection of 1991 World Championship boards above I found that 69 (almost 30% of the total) had been played in contracts by both North-South and East-West.

As expected the sum of the two Winning Trick estimates corresponded in every case with the total number of trumps. In addition, however, the exercise also showed that for around half of these deals this sum of the two Winning Trick estimates was exactly the same as the number of tricks which actually made and that it was within one trick of the correct result on over 85%.

This further measure of the accuracy of the Winning Trick evaluation procedure compared well with Vernes’ extensive study of World Championship deals, which provided the basis for his LAW, where he found that the total trick count was exactly the same as the total number of trumps on one third of the deals and that it was within one trick on 80% of them.

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8.5THE EVALUATION STUDY AND THE ESTIMATED TRICK COUNT

Although the comparison above of the Winning Trick Count forecasts with the actual World Championship results provide a valuable pointer to the scope and accuracy of the evaluation procedure they only meet one of the aims of this study.

This is because the primary role of such hand evaluation is to help us to make a judgement at the bridge table about the number of tricks we are likely to make if we contend the auction.

As we saw in the earlier chapters we can clearly make such Winning Trick forecasts – and often at an early stage of the contest. What we usually lack, however, is a measure of the accuracy of these estimates for that substantial fraction of deals where we may make such an assessment of our prospects but where the opponents subsequently win the auction.

It is here that the Estimated Trick Value, which I described in Chapter 7, proved invaluable.

This alternative procedure is simply to examine our partnership hands for such a deal and to then estimate the number of tricks, which we could have expected to make if we had in fact played in our best suit fit. The resulting Estimated Trick Value can then be compared with the evaluation forecasts for the same pair of hands

This Estimated Trick procedure is well established as a means of bridge-hand assessment and it has the advantage that it can be used to assess the trick-taking prospects for almost any bidding situation.

Thus, as I noted in the previous chapter, Ron Klinger describes it as the ‘Partnership Playing Strength’ and he defines it aptly as ‘the number of tricks you can expect to win if suits break normally and half of your finesses work’

On the other hand, given that the reality of bridge is that seemingly reasonable contracts do sometimes fail because of bad suit breaks, or finesses that don’t work, or opponents who conspire to make more tricks than you expected, we clearly need a measure of the accuracy of the procedure.

What was fortunate, therefore, from the viewpoint of this exercise, was that I was able to use the large number of World Bridge results described above to make such an appraisal by simply comparing the Estimated Trick Count for each contract with the number of tricks that were actually made.

Although the estimation of the Partnership Playing Strength sometimes involved an element of judgement it was relatively easy to apply and I found that I was able to use the procedure for 240 of the 266 World Bridge contracts considered above.

What the consideration highlighted, however, was the fact that a large number of the contracts (around 40%) were characterised by an odd number of finesses, and hence by two equally plausible estimates differing by one trick of the possible outcome. And as we shall below this quite neglected, phenomenon constitutes a significant and quite fundamental limitation to the accuracy of bridge-hand evaluation procedures

However, in the case of this particular appraisal, I was able to avoid this quite significant odd-trick uncertainty by simply restricting the analysis to that substantial fraction of contracts where the estimate of the Partnership Playing Strength was in fact based on an even number of finesses. The results of the subsequent comparison of the Estimated Trick Count with the number of tricks, which were actually made for these 151 World Bridge contracts, are set out in the table below.

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EVALUATION PROCEDURE	NUMBER OF CONTRACTS WITH AN EVEN NUMBER OF FINESSES	ESTIMATE CORRECT   No. %		ESTIMATE CORRECT WITHIN +/- ONE TRICK No.%
Estimated Trick Count	151	129	85	150	99

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As can be seen the Estimated Trick Countcorrectly predicted the number of tricks, which were actually made for 85% of the contracts, and it was within one trick of the result, on all but one of the remainder. This reassuring measure of the accuracy of the procedure meantthat I could move on with confidence to use the concept for the much broader range of contracts described below.

8.6THE APPLICATION OF THE ESTIMATED TRICK COUNT TO A MUCH BROADER RANGE OF CONTRACTS

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As I noted earlier the primary role of an evaluation procedure is to provide us with an estimate of our partnership trick-taking prospects – and if possible those of our opponents – while we are bidding a particular hand.

As we saw above the singular advantage of the Estimated Trick Count is that it provides us with an accurate measure of the merits of such evaluations for both pairs of hands on a very wide range of bridge deals. 

In turn this enabled me to extend my consideration of the scope and accuracy of the Winning Trick Count to an additional 240 boards which had been randomly dealt with the aim of comparing the Estimated Trick Count estimates with the evaluation forecasts for as many of the pairs of hands as possible.

In fact, it transpired that the estimation procedure was relatively straightforward and I was able to make such a comparison for 346 out of the 480 pairs of hands.

As before I found that a significant fraction of the Estimated Trick Count results involved an odd number of finesses and hence two equally plausible estimates - again differing by one trick - of the possible outcome. In this case, however, I judged that this uncertainty simply reflects the real situation when you are playing such a hand and I included these contracts in the analysis by assuming that the odd finesse would only succeed half the time and adjusting the score accordingly.

The results are summarised below:

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EVALUATION PROCEDURE	NUMBER OF CONTRACTS	ESTIMATE CORRECT   No. %		ESTIMATE CORRECT WITHIN +/- ONE TRICK No.%
Basic Winning Trick Count (Honours)	346	198	57	321	93
Winning Trick Count (Points)	346	193