[han]

An argument for the fact that 4S is more likely the best contract when you have a 4-4 fit is that a ruff in either hand produces an extra trick. With a 5-3 fit, only a ruff in the short hand gives you an extra trick. Unfortunaltely, the hand with 5 trump is more likely to have shortness.

This simple argument is perhaps reflected in the numbers that Zar and Tysen provide.

If I understand it correctly, Tysen has done this calculation for all possible hands with 5-3 and 4-4 spade fits. Zar is only talking about hands that might produce game, therefore the 13% that Tysen gives does not contradict Zar's statement.

I wish these phd-math people would make more precise statements.

[Zar]

>
1. I have wondered about Zar's statement that 5-3 plays better in 3NT, 4-4 in suit. How do you guys feel about this?
<

Feelings don’t cut it here, POJC. It’s all about testing and observation (if you are not blind :-)

I am just communicating my observation and provide the supporting data. Yet, you do NOT have to trust me – just make your own testing and observations and see for yourself.

>
Simply take a look at world class hands from top events, and see how weak people open hands with a serious bid (7 hcp? 9 hcp?), and if they occasionally open weak, when do they pass 12 hcp hands (never? when balanced? when points are in short suits? with buncho queens and jacks?) and see what ZAR evaluation for opening bid would have done.
<

Ben, this reminds me of my definition for expert’s opening (from my first “Zar Points Hand Evaluation” book). It goes like this;

1) if an expert fails to open a 26-count, the expert is asleep.
2) if an expert opens a hand with a below-26-count, the expert is drunk.

Zar Points are being considered as a replacement of the standard for opening hand since they encapsulate the COMBINED RESTRICTIONS of the WBF (the Rule of 18 AND the Rule of the Queen) – not for ACBL of course, where basically you have to have 15 HCP and a 6-carder to open (since this is what constitutes the minimum that makes a beginner comfortable opening).

>
Your writing style is at best idiosyncratic and often verges on obtuse.
<

My style is for intelligent readers – that means people who are:
1) readers;
2) intelligent

I hope you fit the bill.

>
World class players don't need ZAR, this discussion isn't how to improve world class player judgement.
<

This is actually explicitly stated in the book despite the fact that I think it goes without saying.

>
I think ZAR would work much better in a limited bid system a la Precision instead of a wider ranging 2/1 setup.
<

Agree 100%.

Simply a matter of “2-levels-wide” opening range vs. a “4-levels-wide” opening range.

>
I agree with Ben; ZAR in its aggressiveness with shape also attempts to depict in a rather linear way double fits, misfits, and so on. I rather like it, and I use ZAR without my pard knowing it.
<

I know people who did exactly that on the Bermuda Bowl finals 2 years ago when Zar Points were at their infancy, really.

>
This hand is a “miss” according to ZAR’s statistics, that is he reports no grand slam bid. This hand is a miss also, when Tysen studies hands. In fact, this hand is a “hit” or possible “hit” if FIT points are applied.
<

All the files you are referring to are from the first book, Ben. You are correct – no fit / misfit points and no Blackwood.

HOWEVER, the 105,000 board match DOES apply Blackwood for ALL the 9 methods evaluated AND has 2 types of fit calculations (marked as 2 different entries) so people can see the behavior in either case, and chose accordingly.

I have to tell you that I am surprised by the emails I get where people claim they continue using the “3-points-per-trump” DESPITE the fact that they see that this method is #2 and refuse to calculate the Zar Ruffing Power points which you discuss in your post (which is #1 in all categories and sub-categories as you may have already noticed).

>
If I see how you did it, then it's quite possible that I can see some sort of mistake I've made.
<

I have no idea how YOU did it, but I know the mistake you made (boy, do I see through :-)

The problem is that to use Misfit Points you should be able to “calculate” the M2, and from there to approximate M4 (as described in the book).

Now, the computer can ALWAYS do that directly, no questions asked. To do a relevant for “at-the-table” test though (if a theory is not for at-the-table use, I don’t even want to hear about it), you should test ONLY boards that can show the misfit during the bidding process. I know this sounds a bit idiotic so let me elaborate a bit. Please read through the entire thing first, rather than shooting off-the-hip.

1) First, what is the goal of introducing the Misfit points? The answer is – to let you better evaluate your trick-making potential in a “crazy” distribution. Keep this in mind when we are going through the eval of both NT and Trump trick-making potential (based on misfit) below.
2) Let’s focus on the M2 (since M4 is a derivative anyway). Can you tell me the value of M2 after a sequence like 1NT-2NT-3NT? Or on the trump-contract side 1S-2S-4S? The answer is no. Do you care about misfit in either case? The answer is no again. The Misfit points come into play where:
a. for NT contracts you end-up in NT just because “nothing better is in sight”;
b. for Trump contracts you are in “invitational mode” (Game or Slam) and at least one of the partners (NOT the guy that makes the decision for the Slam) has shown his side suit during the bidding. Quick example: 1S – 2S – 3C – 4S. Note that guy that MAKES the decision to jump to a Game has NOT shown his side long suit – and there is no need to. BUT he is in a position to calculate the M2 since he can project the differences in BOTH side suits (say, + or – 1).
c. finally, please note that for NT contracts you have to calculate the entire M4, while for Trump contracts you only need to calculate the M2. Why? Because in NT ALL suits are equal and participate in the trick-making (or trick-losing for that matter) process, while in trump contract you do not care for the misfit value coming from the TRUMP suit itself and sit #4 that you don’t even know the difference of. In other words, in a trump contract, the trick-making “duo” are the 2 side suits of the 2 partners so you only calculate the M2 for these side suits.
d. If you calculate M4 for trump contract by a computer, you’ll jump way off on the trump contract and will be disappointed. Admittedly, the book doesn’t pay enough attention to these considerations and focuses on the idea of the Misfit points rather than the details mentioned here.

Hope this helps:

ZAR

[POJC]

Zar, on Sep 2 2005, 04:41 AM, said:

>
1. I have wondered about Zar's statement that 5-3 plays better in 3NT, 4-4 in suit. How do you guys feel about this?
<

Feelings don’t cut it here, POJC. It’s all about testing and observation (if you are not blind :-)

I am just communicating my observation and provide the supporting data. Yet, you do NOT have to trust me – just make your own testing and observations and see for yourself.

Ok Mr jokster answer this:
How much better a priori is it to play 3NT than 4M with a 5-3 fit
What are the percentages? No feelings involved ;-)

[Zar]

>
Ok Mr jokster answer this:

How much better a priori is it to play 3NT than 4M with a 5-3 fit
What are the percentages? No feelings involved ;-)
<

I see about the feelings :-)

The point is not in the joke, POJC. The point is in the POINT of the joke :-) No joke shows-up on an empty space.

As I mentioned before, run your own tests (seriously) and eyeball the results – point being that it is a bit counter-intuitive (and most of the time our intuition is wrong) since you “feel” (your favorite word) safer holding a 5-carder than a 4-carder and a 5-carder looks “more like a trump-suit”. Admit it – if you have a 5-card Spade suit and your PD supports you, it probably doesn’t even cross your mind to play 3NT. And since in football, automobiles, and bridge everybody’s an expert, it is hard to convince anyone that in football you basically kick the ball with your foot throughout the field while in handball you throw it with your hand, if your intuition tells you otherwise (for whatever reason).

My observations about the 5:3 vs. 4:4 as well as 5:2 vs. 4:3 (with the percentages you are interested in) are presented in the tables which you can find on pages 74 - 77. Not ducking the question – just don’t want to pollute the space with tables which are readily available and wouldn’t even print properly here (tried that before). You’ll even find how the percentages change in the cases of double-fit, the dependencies on side-distributions etc. It’s hard to give “one number” for a multi-dependency thing because you can always say “You are wrong, dude (no feelings involved) – look at this hand. I found a counter-example” :-)


Cheers:

ZAR

[POJC]

Zar, on Sep 2 2005, 08:35 PM, said:

My observations about the 5:3 vs. 4:4 as well as 5:2 vs. 4:3 (with the percentages you are interested in) are presented in the tables which you can find on pages 74 - 77.

Ok i'm looking the wrong place since it's not in the ZarPointsBOOK. Pls give me a link.
P

[Zar]

I am currently going through a different exercise which would be of interest to you. To understand what’s going on, let’s have a real-quick crash-course on statistics for “normal people”. The subject is how you evaluate a set of observations and draw behavioral conclusions.
The Max and Min are the corresponding maximum and minimum values within the observation set and determine the Range of the observed values. The Median is the middle between the Max and Min. The Mean (Average) is the sum of the observed values divided by their number and is used to calculate the central tendency of an observation, while the Variance is a measure of how spread out the observed data is – it is the average of the squared deviations from the Mean, engendering that the unit of measure is also squared. Taking the square root of the variance gets us back the units used in the original scale (tricks in our case) and results in called standard deviation (STD). Standard deviation tells you how tightly a set of values is clustered around the average of those same values. It's a measure of dispersal, or variation, in a group of numbers. Since STD measures spread around the mean, for data with the same mean, the greater the spread, the greater the standard deviation - if on the other hand all the values are the same, then the mean equals this “same” value and the STD is 0 (the absolute min). Err of Mean or Coefficient of Variation gives us some sense of how much the Average represents the set of numbers it comes from – it is calculated as Standard Deviation / Mean. Finally, the Mode is the most frequent value in the observation set.
For discrete values (like tricks in our case), measuring the above summary statistics is typically done via Frequency Tables. Here is the frequency table for Goren Points (rows for only 26 and 27 shown) where the columns are the possible tricks taken with the corresponding points for that row:

Goren 9 tricks 10 tricks 11 tricks
26 50% 50% 0%
27 22% 73% 5%

Now the squared deviations are multiplied by each frequency's value, and then the total of these results is calculated before dividing to the sum of the frequencies. Let us see how all this works based on the example Goren table of 2 rows above. Pay special attention to the Variance calculation.

Max = 10 tricks
Min = 9 tricks
Median = 1/2 * [ 9 + 10] = 9.5 tricks
Mean = (50*9 + 50*10) / 100= 9.50 tricks
Var = [ (9.5 – 9)**2*50 + (10 – 9.5)**2*50]/100 = [ 12.5 + 12.5]/100 = 0.25 tricks**2
STD= Sqrt( 0.25) = 0.5 tricks (rather than tricks**2 – you don’t measure distance in square ft, right?)
Mode = 9
Err = 0.5 / 9.5 = 0.05

Max = 11 tricks
Min = 9 tricks
Median = 1/2 * [ 9 + 11] = 10 tricks
Mean = 1/100 * (22*9 + 73*10 + 5*11) = 9.83 tricks
Var = 1/100* [(9-9.83)**2*22+(10-9.83)**2*73+(11-9.83)**2*5]=1/100*(15.16+2.11+6.84)=0.24 tricks**2
STD= Sqrt( 0.24) = 0.49 tricks
Mode = 10
Err = 0.49 / 9.83 = 0.05

So we obtain the following extended table:

Goren 9 tricks 10 tricks 11 tricks Min Max Med Mean Var STD Mode Err
26 50% 50% 0% 9 10 9.5 9.50 0.25 0.50 9 0.05
27 22% 73% 5% 9 11 10 9.83 0.24 0.49 10 0.05

We will do these stat calculations for every point-amount of every method and compare the results. After that we will see how we can use these findings to tune-up the methods depending on IMP vs. Match points, Vulnerable vs. Non-vulnerable etc.

We will study the Value of the Aces and Kings (rounded to 6 and 4 respectively now in Zar Points) and find the ones that minimize the STD thus maximizing performance.

We will also see how we can “squeeze” the STD further by pushing the Variance towards the Mean, thus making the method optimized for precision. The way to do that is to study the influence of upgrading and downgrading considerations on the STD, like Concentration of HCP, Duplication of Distributional and Honor Values, short honors, honors in opponents’ suits, etc.

To that end, I am preparing a list of Upgrade / Downgrade features to evaluate the influence of. You can post here YOUR view on what MAY be important to consider. You suggest a consideration, I post back the Value of the this consideration (for example, “Singleton Ace” is worth a Downgrade of 2 Zar Points or 1 Goren Point and this adjustment reduces the initial STD by 6%).

Feel free to post all the features you want to see evaluated. We will prepare a Common List and after that I’ll post the results here.

Cheers:

ZAR

[david_c]

Zar, on Sep 6 2005, 02:50 PM, said:

Median  = 1/2 * [ 9 + 11] = 10 tricks

This seems to be using a non-standard definition of "median". Not that I can see much point in looking at the median in this context anyway. Or the midrange, for that matter.

[han]

Zar, on Sep 6 2005, 08:50 AM, said:

Goren 9 tricks 10 tricks 11 tricks
26 50% 50% 0%
27 22% 73% 5%

Sounds interesting Zar, a low standard deviation indeed seems to indicate a good evaluation method. I'm looking forward to see how your point count compares to methods like HCP, BUMRAP+321+fit, and Binky+fit.

I wonder if the table I'm quoting here is yet another joke, or really something you found when you were adding and dividing (just kidding). The actual numbers seem rather unlikely.

[Zar]

>
The actual numbers seem rather unlikely.
<

Sorry - forgot to mention that the Goren numbers in the example 2 lines are indeed just for explanation purposes.

Pleas do not use them at the table :-)

On the median and the mean - it does make sense to compare "how far" the two numbers are and see how "shifted" the sample is. But I do agree that it is not THAT important here. In fact, I even gave kind-of "simistic" view of some of the terms (the median itself would be an example - I used just the symetrical case) in order to make it easier for the "normal people" (as I tagged them :-) to understand the idea of the exercise.

Cheers:

ZAR

[inquiry]

Hannie, on Sep 6 2005, 11:34 AM, said:

Zar, on Sep 6 2005, 08:50 AM, said:

Goren            9 tricks              10 tricks            11 tricks           
26                    50%                  50%                  0%                 
27                    22%                  73%                  5%

Sounds interesting Zar, a low standard deviation indeed seems to indicate a good evaluation method. I'm looking forward to see how your point count compares to methods like HCP, BUMRAP+321+fit, and Binky+fit.

I wonder if the table I'm quoting here is yet another joke, or really something you found when you were adding and dividing (just kidding). The actual numbers seem rather unlikely.

The numbers were to illustrate how the statistics would be calculated. The "chances" these are the real numbers are of course ZERO....Since some 26 point hand make 13 tricks, some make 6 or 7 tricks. This is jsut to highlight he approach, in simple terms.

As for average (mean) versus median question.

The average is easy, total the number of tricks, divide by the number of hands. Zar showed this as "Mean = (50*9 + 50*10) / 100= 9.50 tricks" What this means, hand played 100 times, won 9 tricks 50 times (add 50 nines) and 10 tricks 50 times (add 10 50 times), add those together, and divide the sum by 100 (the number of hands). IF it was played 101 times and the 101st time won 5 tricks, you would add 5 to the total and divide by 101.

For the Median, ZAR presented the equation as.... Median = 1/2 * [ 9 + 10] = 9.5 tricks, which was questioned. is well, 1/2 right and 1/2 wrong. For some applications, my understanding is the Medium (in unranked data), is the middle item when data is ranked from lowest to highest. Here, 9 is the lowest, 10 is the highest, so by this definition the Median is 9.5. Another definition is the value where exaclty half the data is higher than the current number, and half is lower the number. Here, just taking half the range (as ZAR did), could lead to a false number.

Lets imagine, for sake of arguement this data set...

9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 11

The Medium ZAR way would be 10 (1/2 * (11+9) = 10, but the value with exaclty half the number above, and have below would not be 10. (10 had one above, and 8 below). According to my guestimate (and recollection) the calculated Median for this set of numbers would be 9.00000, not some 9 point something like the mean (which is... 9.36)

It looks liek ZAR intends to use the Median is the less descriptive way which may distort the finding. I believe I would want to know the average number of tricks with the SD rather than the median. But he is giving us both, and he is showing the way he chooses to calculate it. Which is one acceptable way, but it give a different number than if you use the more traditional value.

[Zar]

>
It looks liek ZAR intends to use the Median is the less descriptive way which may distort the finding. I believe I would want to know the average number of tricks with the SD rather than the median. But he is giving us both, and he is showing the way he chooses to calculate it.
<

I mentioned in my previous post that it is a simplistic (the "simistic" is a typo there) way to look at the Median for symetrical distributions. And I agree with David that calculating the Median is not "in the heart" of this research but rather just to allow a reader to have another view (when comparing it to the Mean for example).

The correct deficintion is exactly the way you present it, Ben. I may actually even discard the Median at the end, since there is too much info there anyway, and the Median is not dirctly related to the "core" stuff - the Mean, Variance, and STD.

Hey, let's start suggesting possible candidates for DOWNGRADES / UPGRADES so I can include them in the calculations and give you back the numbers - rather than telling me later "yeah, but you didn't consider the most important one" :-)

Cheers:

ZAR

[inquiry]

Sorry ZAR, I was writing my answer to hannie, and inclucling the comments on Median as you were posting, so I didn't see your reply before mine got posted. I realize that you know the different ways to do median... simplistic (well both are really).

Up valuations....

Superfit - duh
Double fits
PURITY of suit
Well place K or AQ from the bidding.
honors in partners long suit


Down evaluation

Queen and jacks in their suits
Poorly placed K or AQ in their suit or side suit when one of them has shown value
4333
CLEAR MISFITS -- BIG DOWNGRADE.

[tysen2k]

Zar, on Sep 1 2005, 09:41 PM, said:

The Misfit points come into play where:
a. for NT contracts you end-up in NT just because “nothing better is in sight”;

[snip]

c. finally, please note that for NT contracts you have to calculate the entire M4

This is exactly the place where I was looking at the Misfit points. If you look at my original post, I stated that I took a set of hands that have no 8+ fit. And I did use the entire M4 for these hands.

And if you look at the graph I posted, it shows pretty much no correlation between M4 and how many points you should adjust by.

Tysen

[hrothgar]

Zar, on Sep 6 2005, 07:57 PM, said:

Hey, let's start suggesting possible candidates for DOWNGRADES / UPGRADES so I can include them in the calculations and give you back the numbers - rather than telling me later "yeah, but you didn't consider the most important one" :-)

Cheers:

ZAR

I've always preferred to start with the simple and then move to the complex. Its much more efficient to establish a sound foundation and then build upon it rather than investing lots of time and effort thrashing back and forth.

For this example, I'd suggest focusing on one very simple issue:

Tysen's analysis suggested that Zar points are less accurate than BUMRAP + 321, BUMRAP + 531, and Binky points. I'd be most interested in seeing you run this same analysis and comparing results.

Once this is done, we can start worrying about adding more complexity - fit points, misfit points, whatever.

[han]

hrothgar, on Sep 6 2005, 12:33 PM, said:

Tysen's analysis suggested that Zar points are less accurate than BUMRAP + 321, BUMRAP + 531, and Binky points. I'd be most interested in seeing you run this same analysis and comparing results.

Exactly, same for me.

[inquiry]

hrothgar, on Sep 6 2005, 01:33 PM, said:

I've always preferred to start with the simple and then move to the complex. Its much more efficient to establish a sound foundation and then build upon it rather than investing lots of time and effort thrashing back and forth.

For this example, I'd suggest focusing on one very simple issue:

Tysen's analysis suggested that Zar points are less accurate than BUMRAP + 321, BUMRAP + 531, and Binky points. I'd be most interested in seeing you run this same analysis and comparing results.

Once this is done, we can start worrying about adding more complexity - fit points, misfit points, whatever.

Well,

Tysen posted his data here. I asked to see the hands and how he calculated ZAR points. Easy enough, he could have post 50 hands if necessary right here, or he could email me or post the file. He would not (no time I think, read thread for yourselfs).. http://forums.bridge...topic=3385&hl=#


For his "FIT POINTS" he added THREE points of each extra trump in his ZAR+FIT and subtracted a random 3 points if there was no 8 card fit. This is a bastardization of ZAR's methods, which I pointed out at the time (see link above). Zar counts +3 points ONLY if hand with EXTRA trump support also has VOID. So is it surprising that the majority of hands where there is an extra trump YOU DON:T ADD 3 points?

The point was, TYSEN claims (and perhaps rightly, who knows), that he has compared ZAR and ZAR fit points with other methods. ALL I ask is to see the hands to see if he is calcuating these values correctly. ZAR post all his hand RIGHT on the web, so you can see how his calcualtions for each hand type is done. You can look and say, ok... I see. When I looked at ZAR's hands (at least the first large group), I found he wasn't adding points for fit. Antother group added fit, but didn't seem to subtract for misfit (I like the new way better than the old way).

I think before one gets TOO excited about +/- SD for different point counts, there has to be agreeement on how the point counts are calculated. For ZAR, this is now clearly etched in stone (well he might find new fingle factors with the proposed study). Someone saying they USED ZAR METHODS should show us the hands used with their description of the calcuated points. That seems fair and open, yes?

I would like all sides if we are compaing methods, to clearly state how they count their points, and then show their data so all can look for themselves if applied fairly.

Ben

[tysen2k]

I also would like to see a little due dilligence applied to the "accuracy vs. aggression" issue. Zar has done much better in his analysis by looking at IMPs performance for both overbidding and underbidding but I think there is still a major issue. Look at this exerpt from Zar's book on game vs. partscore decisions.


Let's focus on the Goren Points (GP). On the hands game was possible, GP underbid 13,088 times and bid game 2,564 times. On the hands where there was no game, GP bid the partscore 23,000 times and bid game 1,107 times. GP loses 130,000 IMPs on the game hands and only 6,600 IMPs on the partscore hands.

This is a huge imbalance (20x more IMPs in underbidding), whereas the Zar is much more evenly distributed. This is because Zar requires the GP bidders to have 26 combined points to bid game. If this were dropped to 25, 24, 23, etc., I bet Goren's performance would improve dramatically.

The same applies for BP and WTC which have very different IMP performances in the two categories.

The point is, Zar needs to apply due dilligence and give all of the other evaluators the "best" performance possible by selecting the appropriate "point levels" for bidding game and slam for each method. Otherwise you are totally ignoring accuracy and just penalizing based on the levels set by the user.

Tysen

[tysen2k]

inquiry, on Sep 6 2005, 12:45 PM, said:

Tysen posted his data here. I asked to see the hands and how he calculated ZAR points. Easy enough, he could have post 50 hands if necessary right here, or he could email me or post the file. He would not (no time I think, read thread for yourselfs).. http://forums.bridge...topic=3385&hl=#

The link you provided even gives reference to the 13,000 hands of data that I posted at a yahoo group. That group got canceled since there was no activity for 90 days, but I could create another one.

Ben, you looked over those 13,000 hands and said that you felt I had calculated the points correctly (using the +3 points per extra trump method).

The reason I used the strict +3 points per trump method was because that was exactly the method Zar was using in his data/calculations at the time. I wanted to use the same method Zar was using in his calculations. He has now repeated his analysis using the more advanced fit calculations.

Tysen

[Zar]

>
BUMRAP + 321, BUMRAP + 531, and Binky points
<

You keep chasing these Binky 521 and 741 and 321 etc. like a dog chasing its tail :-) When was the last time you used Binky or RUP or Gib or any of these AT THE TABLE?

All the 741, 531, and 321 are covered in the Zar Count Machine – go to the website and play with them. The problem with all of these (BESIDES using fractions etc. ) is that they MERGE important distributions like 4441 being equal to 6331, 4432 being equal to 6332, 5440 being equal to 7330 etc. Doesn’t a difference of 2 TRUMPS make any difference to you? Now, pushing it to an STD evaluation only worsens the result since while taking an average may provide some “neutralization”, when taking the STD there is ONLY ACCUMULATION and nothing ever gets “neutralized”, I am sure you understand that. All the errors coming from “merging” distributions with 2-cards-difference in the longest suits will ACCUMULATE – it doesn’t take more than a couple of brain-cells to figure that out and I am sure you have more than that.

Make you own research and see for yourself – I have described the way to do it in simple and straightforward terms, I tend to believe. Haven’t I?

>
Once this is done, we can start worrying about adding more complexity - fit points, misfit points, whatever.
<

Man, your “priorities” and “worries” are ... well … your priorities and worries – feel free to check them. My computer chokes when it starts calculating fractions :-)

ZAR

[Zar]

>
Let's focus on the Goren Points (GP). On the hands game was possible, GP underbid 13,088 times and bid game 2,564 times. On the hands where there was no game, GP bid the partscore 23,000 times and bid game 1,107 times. GP loses 130,000 IMPs on the game hands and only 6,600 IMPs on the partscore hands.

This is a huge imbalance (20x more IMPs in partscores), whereas the Zar is much more evenly distributed. This is because Zar requires the GP bidders to have 26 combined points to bid game. If this were dropped to 25, 24, 23, etc., I bet Goren's performance would improve dramatically.
<

The problem with both you and hrotgar is that you don't read.

"because Zar requires the GP bidders to have 26..." How can you say that?

In the Aggressive Bidding section they are limited to 21HCP!!! Does 21 equal 26 in your arithmetics?

ZAR