Comments on Zar Points

The 4-3-2-1 HCP scale has become a standard descriptor in the definition of opening bids. They represent hard information on the basis of which the opponents can make deductions on which to base their actions. The HCP content by itself is not a good method of hand evaluation in the case of a hand with shortage, so points have been added to the HCP scale that reflect that fact. These points do not represent hard information, as the opponents cannot know their source until after the hand is played. Thus, even a 2/1 player may open ‘light’ with 10 HCP upon occasion when the shape is particularly attractive. Some literally-minded opponents may be deceived as to the defensive potential of the opening bid and allow the opponents to steal the hand. Generally their protests fall on deaf ears, but they may still feel injured by a bid that did not fall within the limits prescribed on a convention card. It then becomes a question of how far a player may carry this action of adding points for distribution.

In the case of so-called Zar points, the answer is: a long way. Even a hand with 8 HCP can be given consideration for constructive action. For details of the development of the Zar point count process, the reader is referred to the reference by the originator, Zar Petkov of Ottawa (2003) available from the Bridge Guys site under ‘zar points’.

The Statistical Basis
Before we discuss the theoretical basis for the Zar formula, we will critique the arguments  that Petkov gives to justify his claims of superiority to the traditional Goren methods. First we should say that criticizing Goren is akin to beating a dead horse, as for years experienced have rejected it except as a very rough initial guide to hand evaluation.

Suppose TV reporters interview Occupy Wall Street protesters 100 of whom have beards. They ask the question, ‘do you support a special tax on CEO’s earning over $1 million?’ After the first 50 are questioned, the interrogators believe they are on to something as 49 have confessed that they do support such a tax. It was not surprising that 98 out of the 100 bearded protesters felt the same.

Armed with this information, those who do opinion polls decide to stop bearded men in the vicinity of Wall Street and ask the same question. Can they expect 98% accuracy in predicting that bearded men support such a tax? No. After a few no answers, the analysts add a modification: they exclude bearded men in suits who carry brief cases. Accuracy improves after the police release some protesters and a number of them return to the area. So there is some validity to the conclusion that bearded New Yorkers tend to support a super-tax, but it would be wrong to think it applies in most cases – that would constitute prejudicial judgement based on tainted evidence.

This is a simple example of how deductions from a test group cannot always be taken as predictors over a wider sample. The Petkov statistical results come from a narrowly chosen set of hands that satisfy a certain criterion. Let’s take as an example the set of 70,000 hands in which the correct contract is 3♥ or 3♠. By correct we mean that on a double dummy basis 9 tricks and only 9 tricks are made. The Goren points method overbids on 21931 boards (30%) whereas the Zar points method overbids on 2439 boards. In that sense the Zar method provides a much more accurate evaluation. However, there is an advantage to overbidding at IMP scoring where a vulnerable game should be bid with only a 3 out of 8 chance of success. At matchpoints, one gains by bidding impossible games that come home on a defence that falls short of double dummy status. The more uncertainty in the bidding, the better the chance of a faulty defence, so the Goren methods may work advantageously in practice. So if we choose a sample of 70,000 results from hands played by those with a wide variety of skills, we can expect quite different results with a greater degree of fluctuation.

The above arguments against the validity of statistical justification for Zar points as predictors does not mean that they do not constitute a good method of evaluation. There are theoretical reasons why they should work better than the Goren points, and we shall go into those next. The first advantage and perhaps the greatest, is that the method allows for light opening bids – a clear practical advantage. Petkov points out that there are more hands that fall in the narrow range of 8-11 HCP than as fall in the wider range 12 to 37 HCP. Traditionally the former fall in the category of an initial pass, while the latter are divided into 5 main categories of opening bids. On an information-theoretic basis, this is a bad arrangement. It is a better situation if those 8-11 HCP hands were also divided into 5 categories, which increase the average information of an opening call. This is not feasible, but the more passed hands that can be moved to opening bid status, the more informative the system becomes. This is a justification for aggressive systems in general – being aggressive also means being more informative, and more accurate in prediction as well. So, some special arrangements for those ‘good-bad’ hands have been made at the 2-level, while the HCP limits to 1-level bids have been lowered.

Zar Points and the Law
The Law of Total Tricks is a principle that is used by many to guide their bidding. A hand does not exist in isolation, so the playing potential depends on the degree of fit that one expects to encounter with partner’s hand. The most common division of sides is 8-7-6-5. The number of total tricks is the sum of 13 and the difference between the length of the longest combined suit (8) and the shortest combined suit (5), so the number of total tricks (TT) equals 16. Less than 16 and the hands do not fit well, greater than 16, and we are taught to bid ‘em up. Here are 3 examples with their a priori probabilities to consider.

8-7-6-5   TT=16   23.6%
8-8-5-5   TT=16    3.3%
8-8-6-4   TT=17    4.9%

The occurrence of the 8-7-6-5 division of sides greatly outweighs the other 2, and it is reasonable to base action on the assumption of this division, provided that it remains the most probable condition once one sees one’s own hand. To stick with this a priori assumption means that one will sometimes miss the opportunity to act on a more favorable division with a greater number of total tricks.

Judging a hand in isolation, one may consider the difference between the longest suit held and the shortest as an indication of playability in that it represents the maximum available contribution to TT. This sets a limit to what is possible. Thus, a 4-3-3-3 shape can contribute at most 1 to TT, whereas 5-4-3-1 can contribute up to 4, so has more potential.

Players have also learned from statistical studies that hands with a double fit play better than the TT predict. So when one considers opening a hand, one should take into account the probability that a double fit exists. Let’s consider the division of sides when one is dealt a hand with 5-5-2-1 shape.

♠ 5 – 3  (8)          ♠ 5 – 2    (7)
♥ 5 – 3  (8)         ♠ 5 – 3    (8)
♦ 2 – 3   (5)         ♦ 2 – 4     (6)
♣ 1 – 4   (5)         ♣ 1 – 4      (5)
Weight : 16           9

The probability that the division of sides is 8=8=5=5 relative to a division of 7=8=6=5 is in the ratio of 16 to 9 (64%). Based on which is more likely, it makes sense to act as if there is a double fit and the TT equal 17, not 16. That results in a greater than normal motivation to generate action.

Double fits enhance the playing strength of the combinations. In the case above one sees that the 5-5-2-1 shape readily produces a double 8-card fit. Overall the a priori chance of a double fit is 44%. For a 5-4-3-1 shape the chance is 34% and for 4-4-3-2, it is 22%. Petkov has taken this into account by adding as points the sum of the 2 longest suits, 10, 9, and 8, respectively. The greater the sum, the more likely that a double fit exits.

It is possible to calculate the probabilities of Total Tricks and double fits for any given shape of hand, but the problem remains as to how to rank the distributions and provide them with a number of points that will reflect their relative degrees of playability. Petkov has assigned points in a simple manner. 5-4-4-0 is ranked 1 point below 5-5-3-0. Is that a valid assessment when the former has a great probability of encountering a double fit? Is 1 point the correct differential?

Zar Points Formulation
In its simplest version we have this definition:
Zar Points =  Honour Strength + Distribution
     =    HCP + Controls + (Longest – Shortest) + (Two Longest)

Zar points are divided into 2 main categories: honour strength and distribution,  subdivided into the following four factors: the HCP on the scale of 4-3-2-1, the number of controls (Ace=2, King=1), the difference between the lengths of the longest suit and the shortest suit, and the sum of the 2 longest suits. These four are not independent. The sum of the first 2 results in a points scale of 6-4-2-1, which favors the aces and kings over the queens and the jacks. This is appropriate for hands that are distributional in nature and are suitable for play in suit contracts. The third term relates to the potential contribution to the TT, and the fourth term relates to the probability of a double fit. Thus, the basic elements of hand evaluation as described above are included in the Zar evaluation.

There is another factor that so far has not been considered: the losing trick count. This takes into account the placement of the honors. A combination of KQxx in one suit and xx in another counts as 3 losers, whereas a combination of Kxxx in one suit and Qx in another counts as 4 losers. Clearly the coincidence of the KQ in one suit is the more favorable situation. It is more likely that a suit with 4 card has been dealt 2 top honours than it is that a suit with 2 cards has been dealt one top honour, so on that basis alone if one looks at successful combinations more of them will be of the former type than of the latter. Generally hands for which game is likely have a suitable losing trick count, hence a well placed honour structure, so that factor is filtered out in the Petkov selection process.

Integration into a System
To bid is to release information. A major question is how partner can react systemically to the revelations. Opening light in third seat, even on a 4-card major, is a feature of 2/1 systems. The use of such bids has been justified on the grounds that partner has passed and will not over-react to a noise, or that the opponents may be about to enter the auction profitably with the balance of power. What Zar evaluation implies is that one shouldn’t wait for partner to pass – the idea is that one should pre-balance on speculation, as it were.

When a partner discovers a fit, he may jump preemptively (Bergen style) or he may ask for further definition through a check-back bid, such as Drury. So one merely moves Drury to the third seat and the best hand at the table may end up doing the asking. A problem may arise when there is no apparent fit. The probability of a fit with one of the longer suits has not been realized, which in the case of a shapely hand goes against the a priori odds. In this exceptional case more must be known of the distribution and relays may be an effective solution, but there is still a danger of getting too high. Once the Distribution Points are known, a lower limit is set on the total of high card points.

There are some, myself included, who will go against the strictures of the 2/1 system by occasionally opening light in first seat. One danger is that the opponents may overcall and the auction becomes competitive, in which case partner may feel obliged to double the opponents in a contract that may prove unbeatable. To guard against this happening I prefer to open light on suits that I want led, if it comes to that. The same applies to my overcalls. Another danger is that partner may take us to 3NT. Again, if I can provide a good suit that represents a potential source of tricks, I am more inclined to open light.

Zar points do not provide a means of distinguishing good suits and bad suits, so in that respect they share a fault with Goren points. My qualification for a light opening bid is to possess at most 7 losers and at least 3 controls in the long suits. The more points I have outside my best suit, the less inclined I am to take action with less than the normal compliment of HCP. Here is a hand given by Petkov that qualifies by my standards:  ♠ KQxxx ♥ KJxxx ♦ xxx ♣ —, 9 HCP, 2 controls, but only 6 losers.  I would be inclined to wait-and-see with this 7-loser hand: ♠ Qxxxx ♥ KJxxx ♦ Kxx ♣ —. One consideration: if we defend at a high level I am less sure that a spade lead will get us off to the right start.

What is a Void Worth?
To examine the difference in evaluation between a void and a singleton, let’s compare the 5-4-3-1 shape to the 5-4-4-0 shape by looking at light opening bids with 10 HCP.

♠ KQxxx	♠ AQxxx	♠ KQxxx
♥ KQxx	♥ KJxx	♥ KQxx
♦ xxx	♦ xxx	♦xxxx
♣ x	♣ x	♣ —
25 Zar points	26 Zar points	26 Zar points
7 losers	7 losers	6 losers

Because of the void, the hand on the far right has one less loser. That should be worth about 5 Zar points because a game bid in hearts or spades (10 tricks) requires 52 Zar points. Well, the void represents a contribution of 5 points in that it applies to Zar points through the term (Longest – Shortest). That is almost as good as an ace on the 6-4-2-1 point scale. The singleton contributes 4 points, only 1 point less, as good as a king. What is the significance? The hand on the left is not an opening bid by Zar standards, but the middle hand is. The difference in these borderline hands lies in the number of controls held, 2 on the left and 3 in the middle. The void delivers the equivalent of a difference between an Ace and a King. Next we examine some frequent division of sides.

Hand	Opposite   (Division)
♠ 5	3    (8)	2    (7)	2    (7)		3    (8)
♥ 4	3    (7)	3    (7)	4    (8)		4    (8)
♦ 3	3    (6)	4    (7)	3    (6)		5    (8)
♣ 1	4    (5)	4    (5)	4    (5)		3    (4)
TT	16	15	16		17

Hand	Opposite   (Division)
♠ 5	3    (8)	2    (7)	2    (7)		3    (8)
♥ 4	3    (7)	3    (7)	4    (8)		4    (8)
♦ 4	3    (7)	4    (8)	3    (7)		5    (9)
♣ 0	4    (4)	4    (4)	4    (4)		3    (4)
TT	17	17	17		18

The a priori odds of at least an 8-card fit with a 5-4-3-1 shape is 74%, which is why it is generally considered a shape with which one strives to bid.  One sees the mundane 8-7-6-5 division of sides is common, and there is a danger of a misfit division, 7=7=7=5.

A common division with a 5-4-4-0 shape is 8-7-7-4. Overall the a priori odds of at least an 8-card fit is 84%, so the prospects are clearly better than for 5-4-3-1 by an average of 1 card. That one card extra in a fit is equivalent to one less loser. It is not clear that Zar points give sufficient weight to the difference at the game level. However, one must keep in mind that 3 losers (xxxx) is not typical of a 4-card suit, and that QTxx is much better.