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Notes on hand evaluation by Richard Cowan

My recent article on hand evaluation has started to generate quite a correspondence and just a little controversy. In response to the many questions and comments, I have prepared a few notes. These are set out in a question and answer style. Hopefully, those who struggled a little with the mathematics in my article will find these notes more to their taste.
Some of my answers are incomplete, but time limitations have prevented me from further explorations at this stage. The published study itself took about 12 years of part-time effort. Alas, bridge research has always been over-shadowed by my normal scientific commitments.

Q. What made you embark on the study?
A. I was actually seeking a justification for the 4-3-2-1 scheme. I was completely unconvinced by existing arguments that purported to justify 4-3-2-1. The scheme is simple, it seems to work and it has had 60 years of empirical testing, so surely it could be supported by a convincing theory.
In many respects my study is now the best justification of 4-3-2-1. It shows that, amongst evaluation schemes that simply score points for individual honour cards (additive schemes), 4-3-2-1 is quite reasonable on jointly flat hands. Initially I looked for the optimal weightings for the top 5 honours as an academic exercise, but when the weightings (which were not strictly integers) turned out to be so close to the proportions 5:4:3:2:1, I was quite excited. I found it interesting that 4-3-2-1 was OK, but not the best, and that the best was equally simple.

Q: So how would you summarise the conclusions for a non-mathematical audience? The paper was, after all, published in a specialist mathematical journal.
A. There are some types of hands where one places great dependence on high-card point count. These are, of course, the flat hands. A partnership which devotes most of its bidding space below game level to shape description (a policy which I support strongly), is totally dependent on point count when both partners are flat. On the more distributional hands one
depends less on point count. In this situation one assesses controls, losercounts, lengths, shortnesses, texture of the long suits, the role of the short suits and «fit». The mildly distributional hands without a trump fit but with reasonable «communication», say 5-3-3-2 opposite 2-4-2-5, use point count as a guide tempered by long-suit texture and the knowledge that fewer points are needed to bid game in NT.
So, I believe that the point-count system should be tailored to perform best where it is most needed, on the commonly occurring jointly-flat hands. A good bidding system and the other evaluation tools take over when distribution is evident. So the study focusses on jointly flat hands. To evaluate the worth of honour cards in this restricted context, one needs to evaluate the number of tricks that each jointly flat combination will produce, averaged over all possible ways that the remaining 26 cards are distributed. But how do we evaluate the average number of tricks for even one example, say
(a) North: AJxx Kxx xxx Axx
South: Qx ATxx KJTx Qxx.

One needs to state a playing strategy for each potential lead and each contingency that evolves as the tricks are played, and use this to calculate the average number of tricks made. One must also average over the two cases where North or South are dummy. If we had the analytic powers to perform this task, we might come up with a statement like this: the pair of hands will produce 6 tricks 1% of the time, 7 tricks 5%, 8 tricks 40%, 9 tricks 42% and 10 tricks 12%. (I have made up these numbers for illustrative purposes.).
Thus the average number of tricks would be 8.59. Let us call this the worth of the two hands. Another (easier) example is
(b) North: KQxx QTx Axx Kxx
South: JTx KJxx Kxx Axx.

In this case, we can calculate the worth as 9.75, assuming that the leader leads his longest suit or chooses from equally long suits. Under this assumption there is a 36.27% chance of a Spade or Heart lead and hence 10 tricks. On the remaining 63.73%, a Diamond or Club is led. Suppose West is on lead and leads Clubs. Declarer’s best line is to hold-up on the first trick, winning the second round of Clubs. He/she forces out the Spade Ace (say). At this stage we can list the outcomes of the hand for each possible split of the Clubs and each location of the two missing Aces. If West has both Aces, declarer makes 9, 8, 7 or 6 tricks when the Clubs split 4-3, 5-2, 6-1 or 7-0 respectively. If West has AH and Clubs are 4-3, 9 tricks result. Otherwise, 10 tricks are assured. Formal probability calculations show that declarer makes 6 tricks with chance 0.05%, 7 with 0.83%, 8 with 4.69%, 9 with 13.17% and 10 with chance 81.27%, thus yielding the average of 9.75.
Whilst we can calculate the worth of some hands, we cannot do so for each of the 5.8 x 1020 jointly flat holdings. So I have taken the approach of analysing the average number of tricks for each possible suit holding (see Table 3 in the paper) and adding these to give a rating to full hands. I call this rating «trick taking potential», or TTP. Consulting Table 3, you will see that (a) and (b) have TTPs of 8.41 and 10.00 respectively compared with their worths of 8.59 (based on my guessed outcome) and 9.75. The TTP rating is not the same number as the worth, but I feel that it is a useful number on
which to base a study of point count. It is sometimes up, sometimes down on the worth. I believe that TTP and worth are reasonably close provided declarer retains reasonable control of the play.

The paper shows that, within the context of additive schemes, the TTP of two flat hands is best estimated (prior to dummy’s disclosure) by a 5-4-3-2-1 scheme for the top 5 honours. The paper arrives at this conclusion by
analysing all jointly flat holdings that a partnership might have and using some substantial mathematics. The study makes no assumptions about the locations of key cards with the opponents, nor about suit splits. The TTP of a pair of hands is a measure of trick generating potential averaged over all possible splits and key-card locations.
Perhaps more importantly, the paper establishes a method for exploring the issues of hand evaluation. When time permits, I propose to use the basic data of the paper (Table 3) and the mathematical method (Appendix) to evaluate honour combinations in the style of Culbertson or, more recently, Steen. I anticipate big improvements in predictive
accuracy over simple additive schemes. In due course I will be able to quantify well known maxims: honours in combination are better than honours separated; honours in long suits are more valuable than those in short suits. (Here we have only 4, 3 or 2 card suits, but there is a clear gradation of worth.)

I published in the Journal of The Royal Statistical Society partly because the paper was of a suitable standard for a refereed scientific journal, partly because I wished to place the mathematical details on record, but mainly because I wanted a journal held in libraries throughout the world. Every university library will subscribe to this journal, but (sadly) it is hard to find even municipal libraries which keep the well-known bridge magazines.

Q. It is hard to break a lifetime habit. Counting with 5-4-3-2-1 will be tricky.
A. Yes, but one correspondent, Danny Kleinman, has suggested a simple way. Score initially using the familiar 4-3-2-1, then count the number of honours A through T. Add the two entities together. He remarks that one can think of adding quality of honours to quantity of honours to yield my pointcount score.

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