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# Double Dummy Bridge evaluations

## Introduction

The goal of gathering this data was to get enough information about the trick-taking values of holdings that we could come up with an optimal evaluation technique for gauging the the trick-taking potential of a hand.

I am using double dummy data from Matt Ginsberg's double dummy library. Double-dummy tricks are, of course, not the same as real-world tricks, because, for example, if you hold AKQJ9, you're trick expectations are increased because you'll be able to pick up the Txxxx with RHO even when partner has the stiff. Similarly, unsupported queens are undervalued on defense, because the double-dummy analysis assumes your opponent always guesses your queens right.

Still, double dummy data has some advantages.

• There is a lot of it - 700,000+ deals in the library, and each deal is analyzed with each hand declaring each suit and notrump.
• It is, in some sense, a 'pure' measurement - if, for example, we had a library of 700,000 deals played in the real world, we'd still have vagueries like 'is this the value against expert opponents?' Here, we know exactly the definition of the data. In any event, I'm trying to measure trick-taking potential, and real deals played in the real world often are not played in the contract that takes the most tricks - for example, 3NT is preferred to 4C/D.

In the tables below, there are seven columns of data. The first is a count of the number of samples which match the value. The next six columns represent the average double-dummy tricks you make:

• In offense, in your best contract
• In defense, against opponent's best contract
• In offense, in notrump
• In defense, against opponent's notrump
• In offense, in your best suit contract
• In defense, against opponent's best suit contract
'Best suit contract' means the suit contract in which declaring side makes the most double-dummy tricks. Occasionally this might mean playing in a 6-card fit. I know of one deal where you make 5 tricks in your 8-card fit, 6 tricks in your 7-card fit, and 7 tricks in your 6-card fit. And the opponents make 3NT.

Below, I've kept data which has a low sample count. In theory, we can generate new samples for each low-data row, or those rows for which there is no data (mostly, information about very long suits and really odd shapes.)

The deals were read and evaluated with iDeal, the latest version of my bridge hand generator which has become a real open-ended programming tool. I wrote a pure Tcl reader for the GIB library data, and walked through each deal in the file.

## What is the point?

Originally, I was quite cautious about making any "conclusions" about this data. I intended this article to provoke thoughts about hand evaluation, and specifically to make the point that Notrump evaluation and Suit evaluation are very different beasts.

Trying to come up with strict conclusions from this data is a bit difficult, but a number of people have revised and extended the ideas of this technique.

• Alex Martelli has used this data as further evidence in support of a 6-4-2-1 evaluation scheme (or some variation.) Indeed, this table indicates that the relative strengths of aces, kings, queens and jacks are quite close to these values. Alex also used the sampling idea used here for a pair of articles published in The Bridge World.
• The GIB program uses a heavily revised variation of this idea during bidding. GIB's variation has the ability to adjust the evaluator as you find out more about partner's pattern. The jury is still out on whether this has improved GIB's bidding - experiments show it works well, but I'm not entirely convinced. I'd love to see GIB or another program have a "dynamic" evaluator, where it learned over time and adjusted its evaluations according to past experience.
• This data can also be seen as a first step to producing a better "additive" valuation - one where the sum of your hand's value and the value of partner's hand closely corresponds with the number of tricks that can be taken. I've written an outline of this idea which I call "Binky points" for obscure reasons. The article requires some minimal knowledge of matrices and linear algebra.

## About hand evaluators

The very first hand evaluation techniques we learn at bridge can be termed 'pure holding evaluations.' That is, we learn how to evaluate our holding in each individual suit, and then add up the resulting values. Take the hand:
```S: AKJ2
H: 873
D: QT654
C: 9
```
 Holding HCP HCPwithdistribution Losers AKJ2 8 8 1 873 0 0 3 QT654 2 2 2 9 0 2 1 Total 10 12 7
There are other, slightly more complicated evaluators. Marty Bergen's "Rule of 20" says to add the HCP of your hand to the lengths of your two longest suits, and if they add up to 20 or more, open the hand. Twist and turn as we might, we cannot come up with a way to this do with a pure holding function. We instead add an adjustment at the end, based on the hand pattern:
 Holding Bergen AKJ2 8 873 0 QT654 2 9 0 Pattern 5-4-3-1 9 Total 19
Many other complex evaluations are similar. For example, Kaplan and Ruben's CCCC measurement can mostly be viewed as a holding measurement, but it has a -0.5 adjustment with 4333 patterns.

Note: There are certainly evaluations which don't fit this pattern. For example, a standard adjustment to HCP is to add a point for holding all four aces.

This is of great interest to bridge programmers, because there are only 8192 different suit holdings, and only 39 hand patterns, so we can implement such evaluations quite quickly as lookups into two tables. In reality, we can often simplify the holding table lookup because most evaluators do not distinguish between spots. For example, for 'HCP' we only need a table of 16 entries. For losers, we need a table of size 256=64 * 8. (There are 64=2^6 subsets of "AKQJT9", and there can be between 0-7 'spots.')

We will call this sort of evaluator a 'shape-adjusted holding evaluator.' It consists of a pair of functions (h,p) with h evaluating the holdings and p evaluating the patterns.

### Normalizing the evaluator

Notice that in my original table above, the dstributional value of the stiff nine was determined as part of the holding computation, and therefore we didn't need a pattern adjustment value. Still, we could have evaluated the stiff '9' as zero, and then had a 'pattern adjustment' which picked up the value of the stiff.

In general, we can have multiple pairs (h,p) which evaluate to the same value for all hands.

One way we can normalize a generalized evaluator of this sort is by setting it up so that the holding valuations average zero for each length. We do this as follows.

Start with an evaluator (h,p). For each length l, we define

```	avg(l) = average of h(holding) for all holdings of length l.
```
Now define:
```	h'(holding)=h(holding)-avg(length(holding))

p'(u-v-w-x)=p(u-v-w-x)+avg(u)+avg(v)+avg(w)+avg(x)
```
This new pair, (h',p'), evaluate to the exact same value as (h,p) for all hands, and has the property that h'(holding) averages zero for holdings of fixed length.

You can see p'() as measuring the 'expected value' for the valuation, for each hand pattern. Then h' determines the degree to which the holding increases or decreases that value.

There's a specific reason I note this.

## The DDtricks evaluators

The DDtricks evaluators are an attempt to build evaluators that best approximates the trick-taking potential of a hand.

We are seeking functions p and h.

We can use the patterns table to determine a p' which gives us the expected number of tricks for each hand pattern. We are seeking an h' which is normalized, but which matches roughly the holding values in the holdings table. But, given that we know the average values of each length, from the lengths table, we can determine h' as the difference of the value from the holding table and the value from the lengths table.

This actually gives six evaluators, one per column in the table:

• ddtricks_offense_best
• ddtricks_defense_best
• ddtricks_offense_nt
• ddtricks_defense_nt
• ddtricks_offense_suit
• ddtricks_defense_suit

You can test these evaluators in my Evaluators demo.

For example, to determine the value of the hand:

```S: AKJ2
H: 873
D: QT654
C: 9
```
Compute h' for each holding:
```                      Best          NT          Suit
Off   Def    Off   Def    Off   Def

+h(AKJx)         9.52  5.86   7.59  8.43   9.49  5.87
-avg(4):         8.34  4.65   6.08  6.86   8.32  4.67
-----------------------------------------------------
h'(AKJx)        +1.18 +1.21  +1.51 +1.57  +1.17 +1.20

+h(QTxxx)        8.08  4.08   5.40  6.22   8.07  4.10
-avg(5):         8.58  4.58   6.03  6.92   8.57  4.60
-----------------------------------------------------
h'(QTxxx)       -0.50 -0.50  -0.63 -0.70  -0.50 -0.50

+h(xxx):         7.64  3.97   5.23  6.13   7.62  3.98
-avg(3):         8.21  4.56   6.09  6.91   8.19  4.58
-----------------------------------------------------
h'(xxx)         -0.57 -0.59  -0.86 -0.78  -0.57 -0.60

+h(9)            8.82  4.42   5.85  6.89   8.81  4.46
-avg(1):         8.90  4.53   6.00  7.04   8.89  4.56
-----------------------------------------------------
h'(9)           -0.08 -0.11  -0.15 -0.15  -0.08 -0.10
```
Then add p' to h':
```  p'(5-4-3-1)      8.69  4.68   6.03  6.88   8.68  4.70
h'(AKJx)        +1.18 +1.21  +1.51 +1.57  +1.17 +1.20
h'(QTxxx)       -0.50 -0.50  -0.63 -0.70  -0.50 -0.50
h'(xxx)         -0.57 -0.59  -0.86 -0.78  -0.57 -0.60
h'(9)           -0.08 -0.11  -0.15 -0.15  -0.08 -0.10
-----------------------------------------------------
ddtricks         8.72  4.69   5.90  6.82   8.70  4.70
```
So this hand's holder expects to have 8.72 tricks on offense, and 4.69 tricks on defense. In notrump, this hand expects to take 5.9 tricks on offense, and 6.82 tricks on defense. In the best suit contract, this hand expects to take 8.70 tricks on offense and 4.70 tricks on defense.

Incidentally, the average values are:

```average          8.44  4.56   6.06  6.94   8.42  4.58
```
So the hand above is 0.28 tricks better than average offensively, most of that being picked up by being a good hand to play a suit contract - it's a worse-than-average hand for notrump, both on offense and defense.

## Comments on data

By its nature, notrump is less influenced by overpowering values than suit contracts. For example, if you hold AKQJxxxx in a suit, you could still take zero tricks declaring in NT. It's highly unlikely that you'll take fewer than 8 tricks in your best suit contract.

Because of this, the NT data tends to be more 'variable.' I wish I had computed standard deviations for this data, but I neglected to - I think it would have shown a much higher deviation for the NT columns than for the suit columns.

As a rule, since the average number of suit tricks in your best suit is 8.42, and the average number of tricks in notrump is 6.06, the 'Best' column is dominated by the suit contract values. So, for example, the 5-4-3-1 column:

```p'(5-4-3-1)      8.69  4.68   6.03  6.88   8.68  4.70
```
the 'best' contract is only 0.01 tricks better than the best suit contract. That indicates that it's fairly rare that NT is the denomination where you can take the most most tricks.

In fact, almost the only hands where my DDtricks evaluation favors notrump is very strong (25+-point) balanced hands.

## Evaluating evaluators

The following tables determine the correlation between various evaluators and expected tricks in various contracts.

In the analysis below, most evaluators correlate with the expected number of tricks with a factor of roughly 0.5. That's not surprising - we'd expect on average that the number of tricks we can take is only correlated about 0.5 with our hand, since partner's hand is another, often roughly equal, factor. Even then, with complete information about partner's hand, we can't determine total tricks exactly because where the opponent's cards lie is relevant, as well. So when we get correlations higher than 0.5, it's because we have better information that we'd expect. Usually, that's when we have a strong hand either in raw power, or in shape.

### Explanation of evaluators.

Some of the most basic evaluators simply assign values to cards. For example, `A=4, K=3, Q=2, J=1` is the standard high card points evaluator. `A=2, K=1` is the standard control count. I'll call these 'vector' evaluators. Many of the evaluators below are vector evaluators:

 controls 2,1 hcp 4,3,2,1 p5321 5,3,2,1 p6421 6,4,2,1 tricksvec.suit 82,51,27,14,6,3,1 tricksvec.nt 115,74,43,23,10,4,2

The last two are explicitly based on the double-dummy data in the cards table.

The other evaluators take distribution into account to some extent:

losers
A crude losing trick count
hcpplus
High card points plus distribution values (doubleton=1,singleton=2, void=3.)
CCCC
The Kaplan-Rubens evaluator
offense.suit
My evaluator for offensive value in a suit.
offense.nt
My evaluator of offensive value in NT.
defense.suit,defense.nt
My double dummy evalutors estimating defensive values.

### Notrump offense

```           Valuation Correlation
losers     -0.3767
offense.suit      0.4573
controls      0.4791
CCCC      0.4812
hcpplus      0.4904
hcp      0.5071
defense.nt      0.5083
defense.suit      0.5094
p5321      0.5107
p6421      0.5110
tricksvec.suit      0.5123
tricksvec.nt      0.5128
offense.nt      0.5174

```
Note that CCCC is actually worse than HCP for estimating notrump offensive worth. In fact, our 'vector' evaluators do quite well in this table. That's because shape, as noted in the patterns table, is only a small contributing factor to notrump offensive value.

### Offense suit

```           Valuation Correlation
controls      0.4487
defense.suit      0.4602
hcp      0.4669
offense.nt      0.4695
p5321      0.4717
p6421      0.4729
tricksvec.nt      0.4739
tricksvec.suit      0.4743
losers     -0.4888
defense.nt      0.4917
hcpplus      0.5051
CCCC      0.5182
offense.suit      0.5311

```
Here is where CCCC shows its worth. It is by far one of the best evaluators. Vector evaluators perform poorly here, for obvious reasons, but hcpplus is a surprising second best. CCCC is better than hcpplus by 0.013, while offense.suit is better than CCCC by the same 0.013.

In fact, offense.suit is the highest correlation in the entire dataset.

### Defense

```
Defense
Value      NT    Suit
losers -0.4140 -0.3386
controls  0.4791  0.4487
offense.suit  0.4838  0.4181
defense.suit  0.4955  0.4835
hcpplus  0.5046  0.4489
CCCC  0.5069  0.4372
hcp  0.5071  0.4669
p5321  0.5107  0.4717
p6421  0.5110  0.4729
tricksvec.suit  0.5123  0.4743
tricksvec.nt  0.5128  0.4739
offense.nt  0.5132  0.4761
defense.nt  0.5226  0.4578

```

First note that none of the measures correlates bettern than 0.484 with suit defense. Estimating suit defense appears inherently difficult.

In notrump, the vector evaluators are great estimators, only beaten by the double-dummy evaluators of notrump offense and defense.

The success of defense.nt is partly because defense.nt sees 'AKQJxxx' and expects to take 7 top tricks. This is clearly a case where the holder of the hand might find a double of a notrump contract, even if that was all there was in his hand :-) In other words, the casual user of 'hcp' knows when to take this sort of thing into account, and abandon the cruder evaluator.

### Other areas to investigate

While this is a good evaluator for determining the number of tricks we expect to take, looking at our own hand, it's not very useful when added to partner's valuation - the two numbers added together are not much better at predicting playing strength than just adding the "hcpplus" values of the two hands.

I've found a nifty mathematical solution to this, which caused me to develop something called "Binky Points."

## Lengths table

```

Best          NT          Suit
Length   Count    Off   Def    Off   Def    Off   Def
0  146816   9.62  4.54   6.14  7.25   9.62  4.58
1  919147   8.90  4.53   6.00  7.04   8.89  4.56
2 2360473   8.40  4.51   6.04  6.96   8.39  4.52
3 3284394   8.21  4.56   6.09  6.91   8.19  4.58
4 2740694   8.34  4.65   6.08  6.86   8.32  4.67
5 1430496   8.58  4.58   6.03  6.92   8.57  4.60
6  476133   8.91  4.38   6.01  7.17   8.90  4.41
7  100949   9.33  4.12   5.99  7.57   9.32  4.16
8   13488   9.84  3.82   5.96  8.04   9.84  3.86
9     993  10.40  3.39   5.72  8.43  10.40  3.44
10      48  11.00  3.29   5.90  9.29  11.00  3.35
11       1  12.00  4.00   6.00  6.00  12.00  4.00

```

## Patterns

```
Best          NT          Suit
Pattern   Count    Off   Def    Off   Def    Off   Def
4-3-3-3  302661   7.80  4.59   6.15  6.83   7.76  4.59
4-4-3-2  619409   8.09  4.67   6.10  6.80   8.07  4.68
4-4-4-1   85946   8.62  4.83   6.09  6.79   8.61  4.85
5-3-3-2  443513   8.14  4.52   6.07  6.90   8.13  4.53
5-4-2-2  303105   8.41  4.56   6.01  6.87   8.40  4.57
5-4-3-1  371483   8.69  4.68   6.03  6.88   8.68  4.70
5-4-4-0   35658   9.38  4.80   6.19  6.99   9.37  4.84
5-5-2-1   91467   9.03  4.53   5.94  6.95   9.03  4.56
5-5-3-0   25826   9.51  4.66   6.12  7.07   9.51  4.71
6-3-2-2  161401   8.51  4.30   6.00  7.16   8.50  4.32
6-3-3-1   98854   8.78  4.43   6.03  7.17   8.77  4.46
6-4-2-1  134700   9.02  4.41   5.97  7.13   9.02  4.45
6-4-3-0   37917   9.51  4.59   6.18  7.25   9.51  4.63
6-5-1-1   20206   9.61  4.28   5.83  7.15   9.61  4.32
6-5-2-0   18685   9.88  4.38   6.03  7.27   9.88  4.43
6-6-1-0    2098  10.51  4.08   5.83  7.40  10.51  4.14
7-2-2-2   14761   8.91  4.03   5.95  7.54   8.91  4.05
7-3-2-1   53849   9.14  4.09   5.94  7.53   9.13  4.13
7-3-3-0    7522   9.65  4.29   6.16  7.64   9.64  4.35
7-4-1-1   11107   9.67  4.16   5.98  7.61   9.66  4.20
7-4-2-0   10387   9.89  4.25   6.14  7.69   9.88  4.30
7-5-1-0    3149  10.50  4.11   6.00  7.75  10.50  4.15
7-6-0-0     174  11.18  3.80   6.14  7.59  11.18  3.86
8-2-2-1    5565   9.57  3.76   5.87  7.96   9.57  3.80
8-3-1-1    3403   9.83  3.84   5.96  8.11   9.82  3.88
8-3-2-0    3076  10.00  3.87   6.08  8.07   9.99  3.92
8-4-1-0    1333  10.49  3.88   6.06  8.16  10.49  3.92
8-5-0-0     111  11.26  3.70   6.02  7.98  11.26  3.71
9-2-1-1     468  10.19  3.31   5.61  8.34  10.19  3.36
9-2-2-0     228  10.56  3.54   5.99  8.73  10.55  3.60
9-3-1-0     268  10.56  3.39   5.63  8.31  10.55  3.44
9-4-0-0      29  11.24  3.41   6.17  8.62  11.24  3.41
10-1-1-1       9  10.89  3.33   4.89  9.11  10.89  3.56
10-2-1-0      38  10.97  3.21   5.95  9.24  10.97  3.24
10-3-0-0       1  13.00  6.00  13.00 13.00  13.00  6.00
11-1-1-0       1  12.00  4.00   6.00  6.00  12.00  4.00
```

## Holdings

```                                  Best          NT          Suit
Holding   Count    Off   Def    Off   Def    Off   Def

-  146816   9.62  4.54   6.14  7.25   9.62  4.58

x  495231   8.81  4.41   5.84  6.89   8.80  4.44
9   70427   8.82  4.42   5.85  6.89   8.81  4.46
T   70675   8.82  4.43   5.86  6.91   8.81  4.46
J   71048   8.85  4.49   5.92  6.99   8.85  4.52
Q   70924   8.89  4.58   6.02  7.08   8.88  4.61
K   70241   8.97  4.75   6.26  7.28   8.96  4.77
A   70601   9.62  5.31   7.23  8.16   9.61  5.33

xx  635907   8.10  4.17   5.55  6.51   8.09  4.19
9x  212118   8.11  4.19   5.56  6.52   8.10  4.21
Tx  212445   8.13  4.22   5.60  6.56   8.11  4.24
T9   30256   8.13  4.23   5.62  6.57   8.12  4.25
Jx  211292   8.18  4.30   5.72  6.67   8.17  4.32
J9   29947   8.19  4.32   5.75  6.69   8.18  4.33
JT   30091   8.23  4.35   5.76  6.72   8.22  4.37
Qx  211693   8.31  4.45   5.95  6.87   8.29  4.47
Q9   30585   8.31  4.46   5.93  6.86   8.29  4.47
QT   29998   8.35  4.51   6.00  6.95   8.33  4.52
QJ   30423   8.39  4.56   6.05  7.00   8.37  4.57
Kx  211718   8.71  4.84   6.56  7.40   8.69  4.86
K9   30151   8.73  4.86   6.58  7.43   8.71  4.88
KT   30110   8.75  4.90   6.63  7.47   8.73  4.91
KJ   30345   8.83  5.01   6.77  7.62   8.80  5.02
KQ   30192   8.94  5.14   6.91  7.79   8.92  5.15
Ax  211106   9.09  5.20   7.10  7.95   9.08  5.22
A9   30596   9.10  5.22   7.10  7.95   9.08  5.24
AT   30558   9.12  5.26   7.14  7.99   9.10  5.27
AJ   30125   9.20  5.34   7.26  8.11   9.18  5.35
AQ   30399   9.43  5.58   7.65  8.42   9.40  5.59
AK   30418   9.65  5.85   7.85  8.62   9.63  5.86

xxx  401886   7.64  3.97   5.23  6.13   7.62  3.98
9xx  241434   7.65  3.99   5.24  6.14   7.63  4.00
Txx  241139   7.70  4.04   5.32  6.21   7.68  4.05
T9x   80254   7.72  4.06   5.34  6.23   7.70  4.07
Jxx  241111   7.80  4.16   5.51  6.37   7.78  4.17
J9x   80790   7.83  4.20   5.54  6.40   7.81  4.21
JTx   80365   7.88  4.23   5.60  6.46   7.86  4.24
JT9   11637   7.92  4.26   5.61  6.48   7.90  4.27
Qxx  240850   8.02  4.38   5.89  6.69   8.00  4.39
Q9x   80391   8.06  4.42   5.94  6.74   8.03  4.43
QTx   80633   8.12  4.52   6.07  6.86   8.10  4.52
QJ9   11575   8.20  4.62   6.23  7.01   8.17  4.63
QJx   80296   8.21  4.62   6.24  7.01   8.18  4.62
QJT   11524   8.26  4.67   6.29  7.06   8.22  4.67
Kxx  241066   8.36  4.71   6.31  7.10   8.33  4.72
K9x   80122   8.40  4.76   6.37  7.16   8.38  4.77
KTx   80536   8.47  4.84   6.48  7.26   8.44  4.84
KT9   11304   8.52  4.90   6.57  7.34   8.49  4.91
KJx   80827   8.62  4.99   6.70  7.46   8.58  5.00
KJ9   11500   8.62  5.04   6.73  7.48   8.59  5.05
KJT   11448   8.66  5.08   6.79  7.54   8.63  5.09
Axx  240375   8.71  5.03   6.79  7.59   8.69  5.04
A9x   80825   8.76  5.09   6.85  7.65   8.74  5.10
KQx   80907   8.76  5.17   6.90  7.66   8.73  5.17
KQ9   11528   8.76  5.19   6.91  7.66   8.73  5.20
ATx   79893   8.82  5.18   6.95  7.73   8.79  5.19
AT9   11583   8.85  5.21   6.97  7.75   8.82  5.22
KQT   11434   8.85  5.25   7.01  7.76   8.82  5.25
KQJ   11431   8.85  5.32   7.03  7.78   8.82  5.32
AJx   80441   8.97  5.32   7.19  7.93   8.94  5.33
AJ9   11585   9.03  5.36   7.23  7.99   9.00  5.37
AJT   11452   9.05  5.44   7.32  8.06   9.03  5.44
AQx   79757   9.18  5.52   7.44  8.18   9.15  5.53
AQ9   11600   9.20  5.57   7.49  8.24   9.18  5.58
AQT   11377   9.27  5.64   7.57  8.30   9.23  5.65
AQJ   11643   9.34  5.74   7.65  8.38   9.30  5.75
AKx   80396   9.38  5.76   7.67  8.44   9.35  5.77
AK9   11370   9.43  5.84   7.73  8.51   9.40  5.84
AKT   11542   9.49  5.91   7.83  8.58   9.46  5.92
AKJ   11595   9.57  6.02   7.93  8.70   9.54  6.02
AKQ   11659   9.68  6.15   7.99  8.81   9.65  6.16

xxxx  134325   7.53  3.84   4.92  5.76   7.52  3.86
9xxx  133916   7.57  3.88   4.99  5.82   7.56  3.90
Txxx  134132   7.63  3.94   5.11  5.91   7.62  3.96
T9xx   80515   7.66  3.98   5.17  5.96   7.65  4.00
Jxxx  134518   7.75  4.07   5.33  6.10   7.74  4.08
J9xx   80574   7.80  4.11   5.41  6.16   7.78  4.13
JTxx   80373   7.83  4.16   5.49  6.23   7.81  4.17
JT9x   26964   7.85  4.18   5.52  6.27   7.83  4.19
Qxxx  134360   7.94  4.24   5.59  6.35   7.92  4.26
Q9xx   80247   8.00  4.31   5.70  6.45   7.99  4.33
QTxx   80282   8.05  4.38   5.78  6.52   8.03  4.39
QT9x   26750   8.11  4.45   5.88  6.61   8.09  4.45
QJxx   80871   8.16  4.49   5.94  6.68   8.13  4.50
QJ9x   26680   8.18  4.53   5.99  6.73   8.15  4.54
QJTx   26374   8.23  4.56   6.02  6.78   8.20  4.57
Kxxx  133668   8.26  4.56   5.98  6.73   8.24  4.58
QJT9    3851   8.27  4.63   6.09  6.88   8.24  4.64
K9xx   80561   8.32  4.63   6.07  6.81   8.30  4.64
KTxx   80481   8.41  4.72   6.21  6.95   8.39  4.73
KT9x   26815   8.45  4.78   6.28  7.01   8.42  4.79
KJxx   80533   8.54  4.84   6.37  7.12   8.52  4.85
KJ9x   26653   8.59  4.90   6.44  7.20   8.57  4.91
Axxx  133756   8.61  4.92   6.43  7.20   8.60  4.93
KJT9    3793   8.61  4.88   6.43  7.18   8.58  4.89
KJTx   26936   8.63  4.93   6.47  7.25   8.61  4.94
KQxx   80503   8.67  4.99   6.53  7.30   8.65  5.00
A9xx   80552   8.68  4.99   6.53  7.29   8.67  5.00
KQ9x   26862   8.72  5.03   6.58  7.38   8.69  5.04
KQT9    3864   8.75  5.12   6.66  7.43   8.72  5.12
ATxx   80823   8.76  5.07   6.65  7.40   8.74  5.08
KQTx   26799   8.79  5.11   6.67  7.47   8.76  5.11
AT9x   26668   8.81  5.13   6.74  7.47   8.79  5.14
KQJx   26727   8.83  5.17   6.73  7.54   8.80  5.17
KQJT    3726   8.86  5.25   6.81  7.66   8.82  5.25
AJxx   80271   8.90  5.20   6.83  7.59   8.88  5.21
KQJ9    3816   8.91  5.25   6.79  7.63   8.89  5.25
AJ9x   26980   8.98  5.29   6.96  7.71   8.96  5.30
AJTx   27168   9.03  5.33   7.01  7.79   9.00  5.34
AJT9    3879   9.05  5.39   7.06  7.80   9.02  5.40
AQxx   80642   9.10  5.40   7.07  7.84   9.09  5.41
AQ9x   26929   9.15  5.43   7.13  7.90   9.13  5.44
AQTx   26813   9.21  5.52   7.23  8.01   9.19  5.53
AQT9    3866   9.28  5.59   7.28  8.09   9.26  5.59
AQJ9    3843   9.29  5.62   7.32  8.12   9.27  5.63
AKxx   80322   9.30  5.65   7.29  8.09   9.28  5.66
AQJx   26978   9.30  5.61   7.31  8.12   9.28  5.62
AK9x   27156   9.36  5.71   7.37  8.17   9.34  5.72
AQJT    3810   9.38  5.69   7.38  8.19   9.35  5.70
AKTx   26814   9.41  5.77   7.44  8.26   9.39  5.79
AKT9    3914   9.45  5.74   7.47  8.29   9.42  5.75
AKJx   26984   9.52  5.86   7.59  8.43   9.49  5.87
AKJT    3777   9.55  5.93   7.60  8.45   9.52  5.93
AKJ9    3819   9.59  5.94   7.67  8.52   9.56  5.95
AKQx   27008   9.63  5.98   7.73  8.62   9.60  5.98
AKQT    3889   9.69  6.05   7.81  8.71   9.66  6.05
AKQJ    3734   9.74  6.02   7.80  8.77   9.70  6.02
AKQ9    3830   9.74  6.04   7.82  8.77   9.72  6.04

xxxxx   23435   7.61  3.73   4.67  5.50   7.60  3.77
9xxxx   38712   7.62  3.73   4.71  5.53   7.62  3.77
Txxxx   38924   7.71  3.78   4.83  5.65   7.71  3.82
T9xxx   39308   7.72  3.80   4.84  5.65   7.71  3.83
Jxxxx   39174   7.79  3.88   5.01  5.80   7.78  3.90
J9xxx   38607   7.87  3.91   5.09  5.88   7.87  3.94
JTxxx   38886   7.89  3.91   5.09  5.90   7.88  3.94
JT9xx   23432   7.93  3.93   5.14  5.94   7.92  3.96
Qxxxx   39018   7.99  4.01   5.26  6.07   7.98  4.03
Q9xxx   38439   8.06  4.06   5.35  6.17   8.05  4.09
QTxxx   39015   8.08  4.08   5.40  6.22   8.07  4.10
QT9xx   23360   8.14  4.13   5.46  6.30   8.13  4.14
QJxxx   38915   8.21  4.20   5.57  6.41   8.20  4.21
QJ9xx   23083   8.23  4.18   5.57  6.42   8.21  4.19
QJTxx   23391   8.27  4.23   5.64  6.51   8.26  4.24
Kxxxx   38921   8.30  4.33   5.66  6.49   8.29  4.35
QJT9x    7792   8.32  4.26   5.68  6.56   8.31  4.28
K9xxx   38740   8.37  4.36   5.74  6.57   8.37  4.38
KTxxx   39207   8.45  4.42   5.86  6.69   8.43  4.44
KT9xx   23135   8.50  4.46   5.92  6.76   8.49  4.48
KJxxx   39145   8.56  4.52   6.01  6.88   8.55  4.53
KJ9xx   23249   8.62  4.55   6.06  6.96   8.60  4.56
KJTxx   23266   8.65  4.58   6.12  7.03   8.64  4.59
Axxxx   39445   8.67  4.74   6.11  6.95   8.67  4.78
KJT9x    7718   8.69  4.60   6.17  7.09   8.68  4.60
A9xxx   38952   8.71  4.77   6.18  7.01   8.70  4.80
KQxxx   38979   8.71  4.65   6.21  7.12   8.70  4.66
KQ9xx   23514   8.76  4.68   6.25  7.18   8.74  4.69
ATxxx   38910   8.80  4.84   6.31  7.16   8.79  4.86
KQTxx   23299   8.82  4.73   6.36  7.31   8.80  4.74
KQT9x    7934   8.83  4.74   6.34  7.33   8.81  4.75
AT9xx   23167   8.85  4.91   6.41  7.25   8.84  4.94
KQJxx   23100   8.87  4.77   6.41  7.42   8.85  4.78
KQJ9x    7677   8.91  4.78   6.44  7.50   8.89  4.78
KQJTx    7650   8.93  4.81   6.49  7.56   8.91  4.82
KQJT9    1137   8.94  4.78   6.46  7.60   8.91  4.78
AJxxx   38828   8.96  4.96   6.53  7.40   8.95  4.98
AJ9xx   23184   9.01  4.99   6.62  7.51   8.99  5.01
AJTxx   23230   9.07  5.03   6.71  7.63   9.06  5.05
AJT9x    7762   9.11  5.05   6.72  7.66   9.10  5.06
AQxxx   39193   9.15  5.13   6.77  7.67   9.13  5.15
AQ9xx   23417   9.22  5.19   6.89  7.80   9.20  5.20
AQTxx   23262   9.29  5.22   6.98  7.92   9.27  5.24
AQT9x    7794   9.29  5.24   6.97  7.92   9.28  5.25
AKxxx   38890   9.35  5.39   7.02  7.96   9.34  5.41
AQJxx   23241   9.37  5.27   7.05  8.05   9.35  5.28
AQJ9x    7730   9.40  5.31   7.12  8.13   9.38  5.32
AK9xx   23241   9.41  5.41   7.10  8.06   9.40  5.43
AQJTx    7670   9.44  5.33   7.18  8.20   9.42  5.33
AQJT9    1099   9.44  5.34   7.16  8.19   9.42  5.35
AKTxx   23269   9.47  5.46   7.21  8.19   9.46  5.48
AKT9x    7676   9.51  5.46   7.26  8.25   9.49  5.48
AKJT9    1100   9.58  5.47   7.37  8.39   9.56  5.48
AKJ9x    7934   9.58  5.50   7.36  8.42   9.56  5.51
AKJxx   23406   9.60  5.56   7.41  8.43   9.58  5.57
AKJTx    7876   9.65  5.57   7.44  8.54   9.63  5.58
AKQxx   23248   9.69  5.60   7.55  8.65   9.67  5.61
AKQ9x    7710   9.73  5.63   7.59  8.74   9.72  5.63
AKQT9    1156   9.79  5.66   7.75  8.86   9.76  5.66
AKQTx    7814   9.79  5.65   7.70  8.86   9.77  5.66
AKQJx    7910   9.84  5.68   7.82  9.02   9.81  5.69
AKQJT    1142   9.85  5.67   7.86  9.04   9.82  5.67
AKQJ9    1078   9.90  5.79   7.99  9.12   9.87  5.79

xxxxxx    7826   7.82  3.54   4.41  5.31   7.82  3.60
Txxxxx   15523   7.92  3.58   4.54  5.44   7.91  3.64
Jxxxxx   15505   8.00  3.63   4.68  5.60   8.00  3.68
JTxxxx   19384   8.10  3.67   4.77  5.70   8.10  3.72
Qxxxxx   15529   8.16  3.77   4.97  5.92   8.16  3.81
QTxxxx   19569   8.28  3.81   5.06  6.05   8.27  3.85
QJxxxx   19400   8.40  3.87   5.21  6.23   8.40  3.90
QJTxxx   15478   8.45  3.86   5.19  6.25   8.45  3.89
Kxxxxx   15645   8.49  4.02   5.38  6.38   8.48  4.05
KTxxxx   19433   8.59  4.07   5.54  6.57   8.58  4.10
KJxxxx   19269   8.69  4.13   5.71  6.80   8.68  4.15
KJTxxx   15640   8.80  4.17   5.79  6.95   8.79  4.19
Axxxxx   15508   8.83  4.49   5.85  6.90   8.83  4.56
KQxxxx   19400   8.85  4.26   5.91  7.08   8.84  4.28
KQTxxx   15623   8.94  4.32   6.01  7.26   8.93  4.33
ATxxxx   19345   8.96  4.57   6.07  7.13   8.96  4.62
KQJxxx   15574   9.06  4.33   6.14  7.47   9.04  4.34
AJxxxx   19209   9.10  4.64   6.28  7.42   9.10  4.69
KQJTxx    7917   9.11  4.37   6.16  7.51   9.10  4.38
AJTxxx   15500   9.19  4.69   6.41  7.59   9.18  4.73
AQxxxx   19364   9.24  4.77   6.52  7.71   9.23  4.80
AQTxxx   15428   9.38  4.82   6.71  7.95   9.37  4.85
AQJxxx   15605   9.47  4.90   6.86  8.16   9.46  4.92
AQJTxx    7712   9.48  4.86   6.85  8.19   9.47  4.87
AKxxxx   19384   9.54  5.05   6.93  8.20   9.53  5.07
AKTxxx   15645   9.61  5.05   7.06  8.38   9.60  5.07
AKJxxx   15651   9.73  5.14   7.28  8.67   9.72  5.15
AKJTxx    7758   9.77  5.10   7.34  8.79   9.76  5.11
AKQxxx   15573   9.85  5.16   7.50  9.04   9.83  5.17
AKQTxx    7698   9.92  5.21   7.65  9.23   9.90  5.21
AKQJxx    7773   9.95  5.17   7.70  9.36   9.93  5.18
AKQJTx    2265   9.96  5.18   7.79  9.42   9.94  5.19

xxxxxxx    2129   8.19  3.35   4.22  5.22   8.19  3.45
Jxxxxxx    4858   8.38  3.38   4.39  5.48   8.37  3.46
Qxxxxxx    4977   8.55  3.51   4.63  5.81   8.55  3.57
QJxxxxx    7351   8.70  3.57   4.80  6.03   8.70  3.63
Kxxxxxx    4986   8.75  3.65   5.11  6.38   8.75  3.69
KJxxxxx    7326   8.98  3.75   5.36  6.77   8.98  3.78
Axxxxxx    5010   9.16  4.21   5.69  7.10   9.16  4.29
KQxxxxx    7467   9.17  3.87   5.57  7.12   9.17  3.89
KQJxxxx    7402   9.31  3.89   5.71  7.38   9.30  3.91
AJxxxxx    7563   9.38  4.31   6.11  7.61   9.37  4.36
AQxxxxx    7440   9.54  4.42   6.38  8.00   9.53  4.46
AQJxxxx    7422   9.74  4.50   6.71  8.43   9.73  4.52
AKxxxxx    7277   9.79  4.61   6.81  8.62   9.78  4.64
AKJxxxx    7284   9.94  4.63   7.12  9.09   9.93  4.64
AKQxxxx    7443  10.10  4.69   7.45  9.59  10.09  4.69
AKQJxxx    5014  10.17  4.70   7.60  9.78  10.15  4.70

xxxxxxxx     460   8.71  3.09   3.90  5.11   8.71  3.20
Qxxxxxxx    1275   9.05  3.18   4.33  5.72   9.05  3.28
Kxxxxxxx    1257   9.39  3.42   5.07  6.69   9.39  3.47
KQxxxxxx    2165   9.61  3.42   5.12  6.98   9.61  3.45
Axxxxxxx    1258   9.72  3.99   5.69  7.57   9.72  4.08
AQxxxxxx    2209  10.05  4.10   6.40  8.47  10.04  4.15
AKxxxxxx    2207  10.18  4.14   6.73  9.22  10.18  4.16
AKQxxxxx    2657  10.41  4.17   7.32 10.07  10.40  4.18

xxxxxxxxx       6   8.50  1.83   2.50  2.83   8.50  2.50
Qxxxxxxxx      55   9.31  2.71   3.58  4.91   9.31  2.85
Kxxxxxxxx      58   9.84  3.07   4.79  6.26   9.84  3.16
Axxxxxxxx      70  10.00  3.17   4.74  7.30  10.00  3.31
KQxxxxxxx     159  10.07  2.97   4.35  6.26  10.07  3.04
AQxxxxxxx     171  10.39  3.64   6.07  8.58  10.38  3.67
AKxxxxxxx     173  10.66  3.56   6.68  9.75  10.66  3.57
AKQxxxxxx     301  10.88  3.62   6.54 10.16  10.88  3.65

Kxxxxxxxxx       2   9.50  2.50   3.50  6.00   9.50  2.50
Axxxxxxxxx       2  10.00  2.50   3.00  9.00  10.00  2.50
KQxxxxxxxx      10  10.60  2.90   3.80  5.50  10.60  3.20
Qxxxxxxxxx       1  11.00  4.00   5.00  5.00  11.00  4.00
AKxxxxxxxx       6  11.00  3.67   7.83 10.33  11.00  3.67
AQxxxxxxxx       7  11.29  4.14   7.00 11.71  11.29  4.14
AKQxxxxxxx      20  11.35  3.20   6.55 10.60  11.35  3.20

AKQxxxxxxxx       1  12.00  4.00   6.00  6.00  12.00  4.00

```

## Cards

[ Analysis terminated after running through only part of the data ]
```
Best          NT          Suit
Card     Count    Off   Def    Off   Def    Off   Def
2   2039996   8.30  4.42   5.85  6.74   8.28  4.43
3   2039996   8.29  4.42   5.85  6.73   8.28  4.43
4   2039996   8.29  4.42   5.85  6.73   8.28  4.43
5   2039996   8.29  4.42   5.85  6.74   8.28  4.43
6   2039996   8.30  4.42   5.85  6.73   8.28  4.44
7   2039996   8.30  4.42   5.85  6.74   8.28  4.44
8   2039996   8.31  4.43   5.87  6.75   8.29  4.45
9   2039996   8.32  4.45   5.89  6.77   8.31  4.46
T   2039996   8.36  4.48   5.95  6.83   8.34  4.50
J   2039996   8.44  4.56   6.08  6.96   8.42  4.58
Q   2039996   8.57  4.69   6.28  7.16   8.55  4.71
K   2039996   8.82  4.94   6.59  7.48   8.79  4.95
A   2039996   9.11  5.24   7.00  7.88   9.10  5.25

```
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Copyright 1997-2005.
Thomas Andrews (bridge@thomasoandrews.com.)