Back to Hand Evaluation Articles
Double Dummy Bridge evaluations
- Introduction
- What is the point?
- About hand evaluators
- The DDtricks evaluators
- Evaluating evaluators
- Comments on data
- Lengths table
- Patterns table
- Holdings table
- Cards table
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
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 | HCP with distribution |
Losers |
| AKJ2 | 8 | 8 | 1 |
| 873 | 0 | 0 | 3 |
| QT654 | 2 | 2 | 2 |
| 9 | 0 | 2 | 1 |
| Total | 10 | 12 | 7 |
| Holding | Bergen |
| AKJ2 | 8 |
| 873 | 0 |
| QT654 | 2 |
| 9 | 0 |
| Pattern 5-4-3-1 | 9 |
| Total | 19 |
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: 9Compute 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 4.70So 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 4.58So 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 4.70the '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