Zar Points are based on extensive
research on boards in the Game – Slam area.

This includes:

-Over 100,000 boards with 4
/
contracts;

-Over 70,000 boards with 5
/
contracts;

-Over 30,000 boards with 6
/
/
/
contracts;

-Over 10,000 boards with 7
/
/
/
contracts;

-Over 20,000 boards with NT contracts;

ALL these boards have been played on Double-Dummy to ensure accurate and independent assessment
of the playing level of the board.

You can download all the research boards with the detailed
Zar Points analysis of each
and everyone of them from
the website.

I have run different sets of boards, between 10,000
and 60,000 portions to ensure that the results provided
by the software I’ve written for the analysis
is stable. Below, I discuss the results of the research
and these samples of hundreds of thousands of boards
and the behavior of Zar Points in these experiments.

First let’s have a look
at the facts presented by the computer report.

For the Games in minors I ran
2 sets of boards, all played in double-dummy as usual.

The first set comprised 63,057
boards whose Best Contract indication by the Double-Dummy
play was either 4 or 4. Here is the very bottom of the report, including the last board (so
you get a feel of what’s actually reported on
each board) and the final statistics. NOTE: control
points are the points you get from the card that
brig you controls – 6 for A and 4 for K.

Board
#63057

North:

KT

KJ86

AQ

AQJ92

South:

Q863

T92

T532

T5

BEST
contract, actually played on double-dummy, is 4HE,

Controls = 6, ControlPoints
= 20

SP_len N = 2, HE_len N = 4, DI_len N = 2, CL_len N = 5

SP_len S = 4, HE_len S = 3, DI_len S = 4, CL_len S = 2

The fitin this contract
is 7-card, N has 4, S has 3

a_N = 5, b_N = 4, c_N = 2, d_N = 2

a_S = 4, b_S = 4, c_S = 3, d_S = 2

N - Zar Points (a+b) + (a-d)
= 12

Controls N = 6, HCP N = 20----------------------->
NZar Points = 38

S - Zar Points (a+b) + (a-d)
= 10

Controls S = 0, HCP S = 2------------------------>
S Zar Points = 12

Hands
below 50 Zar + FitPointsfor 63057 boards is 2573

Hands below 48 Zar + FitPointsfor 63057 boards is 662

So the average Zar Points with
the 3 pt. Fit calculations (rather that the Zar Ruffing
Power calculations which are a bit more sophisticated
and depend on both your fit and shortness in the side
suits) is about55 Zar Points.As you remember, the 4-level mark was 52,
which is 3 points below the average. This means that
the bottom level is only 5% below the average! So you have
an idea about the precision Zar Points work with.

Let’s have a look at
the “missed” boards. When you calculate
the rough 3-points-per-additional-trump, the missed
boards fell more than 2 times to only 2,500 out of 63,000+ boards. If you
download the 63,057 analyzed boards from the website,
you’ll see that 90% of the “missed”
boards are boards with double-fit which the program does not calculate. If you add the calculations
for double-fit, you
are in the 99% Game Bidding coverage!

Let’s have a look at
the second sample of 4-in-Major Games. It contains 42,756 totally different
Games:

Board
#42756

North:

KQJ53

A973

9

Q42

South:

976

K84

AJ3

AT85

BEST
contract is 4SP, Controls = 8, ControlPoints
= 26

SP_len N = 5, HE_len N = 4, DI_len N = 1, CL_len N = 3

SP_len S = 3, HE_len S = 3, DI_len S = 3, CL_len S = 4

The fitin this contract
is 8-card, N has 5, S has 3

a_N = 5, b_N = 4, c_N = 3, d_N = 1

a_S = 4, b_S = 3, c_S = 3, d_S = 3

N - Zar Points (a+b) + (a-d)
= 13

Controls N = 3, HCP N = 12------------------------>
NZar Points = 28

S - Zar Points (a+b) + (a-d)
= 8

Controls S = 5, HCP S = 12------------------------>
SZar Points = 25

Hands
below 50 Zar + FitPointsfor 42756 boards is 2947

Hands below 48 Zar + FitPointsfor 42756 boards is 901

Again the drop of “missed”
boards is over 2 times, if counting the fit-points.

Also, the “missed”
boards are about the
same 5%!

When you add the double-fit
adjustments (you can see that the majority of the “missed”
boards are with double-fit when you download the database),
you again end-up in the 99% coverage!

First let’s have a look
at the facts presented by the computer report.

For the Games in minors I ran
2 sets of boards, all played in double-dummy as usual.

The first set comprised 27,776
boards whose Best Contract indication by the Double-Dummy
play was either 5
or 5.
Here again is the very bottom of the report, including
the last board and the final statistics.

Board
#27776

North:

Q654

A7

J5

KT632

South:

---

KJT5

K8642

AJ94

BEST
contract, actually played on double-dummy, is 5CL,

Controls = 7, ControlPoints
= 24

SP_len N = 4, HE_len N = 2, DI_len N = 2, CL_len N = 5

SP_len S = 0, HE_len S = 4, DI_len S = 5, CL_len S = 4

The fitin this contract
is 9-card, N has 5, S has 4

a_N = 5, b_N = 4, c_N = 2, d_N = 2

a_S = 5, b_S = 4, c_S = 4, d_S = 0

N - Zar Points (a+b) + (a-d)
= 12

Controls N = 3, HCP N = 10------------------------>
NZar Points
= 25

S - Zar Points (a+b) + (a-d)
= 14

Controls S = 4, HCP S = 12------------------------>
SZarPoints = 30

Hands
below 54 Zar + FitPointsfor 27776 boards is 1618

Hands below 52 Zar + FitPointsfor 27776 boards is 493

Hands below 50 Zar + FitPointsfor 27776 boards is 100

Hands below 48 Zar + FitPointsfor 27776 boards is 16

So the average Zar Points with
the 3 pt. Fit calculations (rather that the Zar Ruffing
Power calculations which a bit more sophisticated and
depend on both your fit and shortnesses
in the side suits) is 59 Zar Points.As you remember, the 5-level mark was 56,
which is 3 points below the average. So the bottom level
is only 5% below the average! This gives
you an idea about the precision interval Zar Points
work with.

Let’s have a look at
the “missed” boards. When you calculate
the rough 3-points-per-additional-trump, the missed
boards fell 3 times to only 1,600 out of 27,776. If you download the 27,776
boards from the website, you’ll see that 95% of
the “missed” boards are boards with
double-fit which the program does not calculate.
If you add the calculations for double-fit, you
are in the 99% Game Bidding coverage.

Let’s have a look at
the second sample of 5-in-Minor Games. It contains 43,014 totally different
Games:

Board
#43014

North:

JT8642

Q

KT843

9

South:

AQ3

832

A9765

J6

BEST
contract is 5DI, Controls = 5, ControlPoints
= 16

SP_len N = 6, HE_len N = 1, DI_len N = 5, CL_len N = 1

SP_len S = 3, HE_len S = 3, DI_len S = 5, CL_len S = 2

The fitin this contract
is 10-card, N has 5, S has 5

a_N = 6, b_N = 5, c_N = 1, d_N = 1

a_S = 5, b_S = 3, c_S = 3, d_S = 2

N - Zar Points (a+b) + (a-d)
= 16

Controls N = 1, HCP N = 6------------------------>
NZar Points = 23

S - Zar Points (a+b) + (a-d)
= 11

Controls S = 4, HCP S = 11------------------------>
SZarPoints = 26

Hands
below 54 Zar + FitPointsfor 43014 boards is 2642

Hands below 52 Zar + FitPointsfor 43014 boards is 823

Hands below 50 Zar + FitPointsfor 43014 boards is 192

Hands below 48 Zar + FitPointsfor 43014 boards is 26

You see how consistent the
behavior is – again 59 Zar Points average, again
the drop of “missed” boards is over 2 times,
if counting the fit-points.

Also, the “missed”
boards are about 2,000 out of 40,000, or the same 5%!

When you add the double-fit
adjustments (you can see that the majority of the “missed”
boards are with double-fit when you download the database),
you again end-up in the 99% coverage!

The behavior of Zar Points
is surprisingly (or "not-surprisingly"
:-) consistent!!!

This concludes the presented
research on Game Bidding – the next 2 sections
deal with Small Slams and Grand Slams bidding.

Hands
below 54 Zar + FitPointsfor 34700 boards is 303

Hands below 52 Zar + FitPointsfor 34700 boards is 61

So the average Zar Points with
the 3 pt. Fit calculations (rather that the Zar Ruffing
Power calculations which a bit more sophisticated and
depend on both your fit and shortness in the side suits)
is 63 Zar Points.As you remember, the 6-level mark was 62,
which is 1 points below the
average. This means that the bottom level that Zar Points
have is only 2% below the average! Even closer than the results in the Game levels.

Let’s have a look at
the “missed” boards. When you calculate
the rough 3-points-per-additional-trump, the missed
boards fell more than 2 times to only 300 out of 34,700. And again it’s
the double-fit boards that the program doesn’t
calculate the fit for. If you add the calculations for
double-fit, you are in the 99% Game Bidding coverage again.

As I have mentioned several
times during the Bermuda Bowl discussions, the small
bidding requires careful examination of the location
of the controls, if you do not want to get into the
“nasty” calculations of footprints. Slam-bidding
conventions are there to be used.

Hands below 54 Zar + FitPointsfor 10344 boards is 12

Hands below 52 Zar + FitPointsfor 10344 boards is 2

So the average Zar Points with
the 3 pt. Fit calculations (rather that the Zar Ruffing
Power calculations which a bit more sophisticated and
depend on both your fit and shortnesses
in the side suits) is 67 Zar Points.As you remember, the 7-level mark was 66,
which is 1 points below the
average. This means that the bottom level that Zar Points
have is only
2% below the average – a good indication about
the precision Zar Points work with.

When you have a look at the
“missed” boards again - the “missed”
boards are boards with
double-fit which the program does not calculate.
If you add the calculations for double-fit, you are in the 99% Game Bidding coverage.

For Grand Slams the precision
of getting the controls right is vital – and it
is worth considering the rule I introduced during the
Bermuda Bowl discussions – you have to be able
to mentally play the entire hand before the dummy opens
up. That’s the accuracy you target in the Slam
bidding zone.

So much for the trump contracts.

As you know, Zar Points were developed targeting trump
Games and Slams. So it was even more interesting to
see how they behave in the NT are.

The next section discusses the results for over 2,000
NT contracts exactly at level 3 – 3NT.

So the average points for level
3 is 47 - as you remember, the 3-level mark was 46,
which is 1 point below the average. This means that
the bottom level that Zar Points have is only 2% below the average! And we are
talking 3NT game here.

In case you are wondering why even for 3NT Zar Points
give accurate results, remember that:

- there is virtually no Fit-premium
here;

- the distributional points
are relatively small in these balanced hands;

- the HCP-portion of the Zar
Points calculations is heavy.

I have run 60,000 Game Boards in Major and 40,000 Slam
Boards in 3 different “modes”:

-Goren Points – HCP plus the 3-2-1 points for void, singleton, doubleton (HCP plus 321)

-Zar Basic Points – HCP plus Controls plus (a+b)
+ a-d)

-Zar Points with general Fit correction - (3 points for extra trump regardless
ofside-suit holdings).

For Goren points, the 3-2-1 points were added to both the declarer and for the dummy, of course,
and I have used 25 points for Game boundary and 32 points
for Slam Boundary.

For both types of Zar Points I certainly used the corresponding
Game and Slam boundaries for
Zar Points that you are familiar with – 52 Zar
Points got the Game calculation and 62 Zar Points for
Slam calculation.

When it says, for example, “reach 38975 Games,
39K/58K”, this means that from the total of 58,000
boards all of which are makeable
on double-dummy, the method actually reaches only 38975
of them (certainly, all of them makeable
on double-dummy).

That’s the settings.

First, let’s have a look at the Game-level performance.
ALL results presented in K (thousands) are rounded UP
for every one of the 3 methods.

The restrictions were measured via HCP only, meaning
that neither Basic Zar Points, nor Goren
nor Zar Points with general Fit were involved in the
selection process. So here you go:

1) Running the boards with less than 27 HCP count (total of 58K boards under this restriction):

These
slam-bidding results also do not involve calculations
for fit-honors, for double-fit etc. Nevertheless, Zar
Points manifest results which are times better than
the Goren Points results you are accustomed to.

If we do the average behavior, we have the following
numbers for the 3 approaches:

Average achievement for Goren
in the Slam Bidding zone 25-33 HCP Games: 17%

Average achievement for basic Zar Points in the Slam
Bidding zone 25-33 HCP Games: 41%

Average achievement for Zar Fit Points in the Slam
Bidding zone 25-33 HCP Games: 67%

The
poor behavior of Goren is a result of virtual inability to bid Slams with the
lower end HCP
like 25, 26, 27 HCP count.

So
here is the table conveniently presenting both results:

Slam Zone

Game Zone

Part Score

Goren

17

40

Zar Points Basic

41

66

Zar Points Fit

67

88

As
you can see, there is an empty column to fill –
the part-score zone. This is actually the subject of
the next section.

The
previous section compares the behavior of the 3 approaches
in the so called “Type
1 Error Zone”, i.e. the errors of the type
“underbid” or “misses” of Games
and Slams.

This
section examines the “Type
2 Error Zone”, i.e. the errors of the type
“overbid” in the part-score zone, where
instead of staying at level 3 of a Major the corresponding
approach goes beyond the makeable
contract at Level 3 and consequently goes down.

I
have run 70,000 Boards where the best contract is the
3
/3
zone according to the indicated Double-dummy results.

Here
is the bottom of the report:

Board
#70188

The Zar Points for this Board are 43

The GOREN Points for this Board are 18

North:

AQ82

J8754

8

KT6

South:

JT7543

32

J3

Q95

BEST contract, actually played on double-dummy, is
3SP,

Controls = 3, ControlPoints
= 10

SP_len N = 4, HE_len N = 5, DI_len N = 1, CL_len N = 3

SP_len S = 6, HE_len S = 2, DI_len S = 2, CL_len S = 3

The fitin this contract
is 10-card, N has 4, S has 6

a_N = 5, b_N = 4, c_N = 3, d_N = 1

a_S = 6, b_S = 3, c_S = 2, d_S = 2

N - Zar Points (a+b) + (a-d)
= 13

Controls N = 3, HCP N = 10------------------------>
NZar Points
= 26

S - Zar Points (a+b) + (a-d)
= 13

Controls S = 0, HCP S = 4------------------------>
SZar Points = 17

Basic Zar Points ( no fit,> 52 ) overbid2439 contracts

Fit Zar Points (+3
extr.trmp > 52 ) overbid10543 contracts

So,
out of the 70K boards, Goren
went overboard (bidding Game) in 22K
cases,the more
aggressive Zar Fit Points were overboard in 11K
cases, and the basic Zar Points went overboard in
3K cases.

What
we are interesting in though, is the complimentary result,
which is how many correct contracts they reach, at level
3.

These
numbers are 48K for Goren, 67K for basic Zar Points, and 59K for Zar Points with
Fit.Not surprisingly,
Zar Points with Fit calculations go overboard much more
often than the conservative version of Basic Zar Points.
Nevertheless, they behave more that 2 times better than
Goren Points.

So
here are the results:

Average achievement for Goren
in the Part Score Zone 21-27 HCP Games: 69%

Average achievement for basic Zar Points in the Slam
Bidding zone 21-27 HCP Games: 96%

Average achievement for Zar Fit Points in the Slam
Bidding zone 21-27 HCP Games: 85%

Here
is our table, now presenting the results of all 3 major
Bidding Zones – Part score, Game, and Slam:

The
bottom line is that if you stick with the simple
basic Zar Points:

Zar Points = HCP + Controls + (a+b) + (a-d)

this
will still give you a substantial competitive advantage
over the “widely-accepted” for the last
50 years bidding methods – roughly 166 % better bidding performance.

Going
to Zar Points with “corrections” basically
DOUBLES your
bidding performance.

The research presented here is just a small portion
of the research done while coming-up with the Zar Points
idea. The database I support is huge and getting bigger
on a regular basis – all the World, North-American,
European, Olympic etc. events are entered into the database
on a regular basis.

Here is a screenshot of the database, just to give you
an idea of the flexibility it provides for research
both through flexible SQL queries and little ad-hoc
Java/JDBC programs that I write on as-needed basis.

This is to say that Zar Points is a project that has
not stopped but rather continues with constant research
over new data and new ideas coming from the various
discussions, experiments, software development, tournaments,
commentaries, etc.

If you want to get involved, I can only say –
be my guest :-) Any new idea is worth at least considering
for awhile – if it is not fruitful by itself,
it may trigger side thoughts and push us towards something
fruitful that we wouldn’t have thought about otherwise,
right?

I hope this research would inspire you to keep looking
ahead yourself!