Estimating Our Tricks

[jogs]

POWER

 

Power is one of two multi-dimensional independent random variables

used for estimating tricks. High card point count is the main component

of this estimator. Most systems treat high card points as if it were the

estimator, power. The location of the HCP and how honors interact are

also components of the power estimator. Honors in long suits are worth

more than honors in short suits. Honors working together are worth

more than honors standing alone.

 

Point count: ace=4, king=3, queen=2, and jack=1

♠ AKQJ ♥ 432 ♦ 432 ♣ 432

♠ A432 ♥ K32 ♦ Q32 ♣ J32

♠ AK32 ♥ QJ2 ♦ 432 ♣ 432

Each is an example of a 10 point hand. They are obviously not

of exactly equal value. It is difficult to measure the exact effects

of the honors in each hand. 10 is the approximately correct value

for each hand. In statistics there is the error component in every

model to account for the deviations.

Power is the independent random variable. High card points is

the dependent variable of power which is proportional to tricks.

Therefore it is easier to use HCP to estimate tricks than the

actual independent random variable, power.

[Scarabin]

:)

POWER

 

Power is one of two multi-dimensional independent random variables

used for estimating tricks. High card point count is the main component

of this estimator. Most systems treat high card points as if it were the

estimator, power. The location of the HCP and how honors interact are

also components of the power estimator. Honors in long suits are worth

more than honors in short suits. Honors working together are worth

more than honors standing alone.

 

Point count: ace=4, king=3, queen=2, and jack=1

♠ AKQJ ♥ 432 ♦ 432 ♣ 432

♠ A432 ♥ K32 ♦ Q32 ♣ J32

♠ AK32 ♥ QJ2 ♦ 432 ♣ 432

Each is an example of a 10 point hand. They are obviously not

of exactly equal value. It is difficult to measure the exact effects

of the honors in each hand. 10 is the approximately correct value

for each hand. In statistics there is the error component in every

model to account for the deviations.

Power is the independent random variable. High card points is

the dependent variable of power which is proportional to tricks.

Therefore it is easier to use HCP to estimate tricks than the

actual independent random variable, power.

Of the 3 example hands 1&3 clearly have less losing tricks than 2, because the high cards are concentrated. Culbertson honour tricks, Bissell point count, and surely all losing trick counts reflect this.

 

Milton Work HCP are only good for:

ease of calculation

balanced hands

locating missing honours.

 

I feel I must be missing some deep point, otherwise what is the problem? :)

[FM75]

Estimating Our Tricks

<snip>

The estimate of our tricks for the general case.

 

E(tricks) = trumps + (HCP-20)/3 + e

 

Trumps is the total combined trumps of the partnership.

(HCP-20)/3 means for every HCP over 20 assign another 1/3 trick to the estimate.

The curve generated by this equation is approximated by the normal curve(bell curve).

'e' is the error of the observations. E(e)=0.

For large samples the average error will approach zero.

<snip>

 

Strain = NT, HCP = 29,

E(tricks) = 3

 

Huh? Does that mean we get 9 tricks, 3 tricks, or the formula is garbage for NT? (The assumption is that e = 0, because this is a large sample case). Or does it mean we expect 6.5 +3 = 9.5?

 

Strain NT, HCP = 40, 6.23 trick, 12.73 tricks?

Strain some suit, HCP=40, trumps = 13, 19.23 tricks or 25.73 tricks?

 

Strain some suit, hcp =11, trumps = 13, 16 tricks, 22.5 tricks? I suspect that the "average" is closer to making 7 tricks (1 suit) than 16 tricks.

 

If the formula does not predict the simple cases, toss it in the garbage bin.

 

(Ok, I studied physics, and expected the friction free limit of a formula including friction to reduce to the friction-free formula.)

[jogs]

Huh? Does that mean we get 9 tricks, 3 tricks, or the formula is garbage for NT? (The assumption is that e = 0, because this is a large sample case). Or does it mean we expect 6.5 +3 = 9.5?

 

This model is for suit strains and the general case.

E(e) = 0, The expected error is zero.

The standard deviation of the error is approximately

1.25 tricks/boards.

 

Turns out E(e) <0 for combined trumps >= 10 or expected tricks >= 10.

This model works best when the expected tricks is between 3 to 10.

Our expected tricks fluctuated wildly depending on whether we

declare in our suit or defend in their suit.

 

When trumps >= 10 or expected tricks >= 10 it requires a much more

complex polynomial model. For a specific board in high level auctions

we should attempt to count the tricks.

[jogs]

test: trying to post west/east hands

 

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[jogs]

retest: trying to post west/east hands

 

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[jogs]

:) Of the 3 example hands 1&3 clearly have less losing tricks than 2, because the high cards are concentrated. Culbertson honour tricks, Bissell point count, and surely all losing trick counts reflect this.

 

Milton Work HCP are only good for:

ease of calculation

balanced hands

locating missing honours.

 

I feel I must be missing some deep point, otherwise what is the problem? :)

 

The point is the estimator isn't linear. It isn't a straight line. It is more like

multi-dimensional blob. It isn't only our honor count that matters. The

location of those honors and whether they are working together also

matter. During the initial evaluation of the hand there is no need to be

precise.

Bissell points and workcount are just loose guidelines for valuation.

I use Milton Work HCP because of ease of calculation.

Other methods are much harder to calculate. Have never seen evidence

any of those methods are more than a small marginal incremental

improvement. The variance of the calculations from those methods are

just as large as the variance of workcount.

Assuming we are dealer, what do we need to know? First, whether or

not to open. If we decide to open, what to open. Why do we need

a more precise valuation of our hand?

During the auction when we learn more about how partner's hand fits

with our hand the valuation can change dramatically. This negates the

efforts of a complex initial valuation. This model is in terms of our tricks

rather than my points.

 

Point count: ace=4, king=3, queen=2, and jack=1

♠ AKQJ ♥ 432 ♦ 432 ♣ 432

♠ A432 ♥ K32 ♦ Q32 ♣ J32

♠ AK32 ♥ QJ2 ♦ 432 ♣ 432

 

These hands are 10 +/- e points and this error is large. As we learn

more about partner's hand the error will be reduced. During the

initial evaluation, it is sufficient to know to pass these hands as dealer.

 

E(tricks) = trumps + (HCP-20)/3 + e

Std dev is about 1.25 tricks/board.

E(tricks) = trumps + (HCP-20)/3 + SF + e

For flat hands the std dev can be as low as 1 trick/board.

This SF term is skewness/flatness as suggested by Lawrence/Wirgren.

Flat boards reduce the trick estimates. Skewed boards increase the

trick estimates.

 

These two models are for fitting partner's hand with our hand.

With luck we will know the value of the terms by the second

or third call of the auction.

 

These models are best for assisting the contested auctions 3 over 2

and 3 over 3. In general the flat hand should not bid 3 over 3.

5332 should rarely compete 3 over 3.

.................

In another post I will introduce the other independent random

variable, pattern.

[jogs]

PATTERN

 

Pattern is the ordered configuration of the four suits in one hand.

Joint pattern is the joint pattern of the two partnership hands.

Trumps is usually the suit with the longest combined length. Larry

Cohen has chosen trumps as his parameter from estimating tricks.

SST(short suit totals) is the sum of the two shortest suit holdings of

the partnership. Lawrence/Wirgren uses the shorter suit holding of

each partner. Cohen and Lawrence/Wirgren are just using a

different end of the same variable(pattern) to estimate tricks.

Trumps is the coarse estimate. SST is the fine tuning. Using

both gives us better estimates than either on a stand alone basis.

 

Now for a naive test of joint pattern. Each side will be given

20 points with all points in two suits.

 

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This example has 18 trumps and only 16 tricks.

 

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Do not assume that when we hold flat patterns, opponents will also

hold flat patterns. In this example EW is 2434//2533, while NS is

4342//5134. EW makes 8 tricks in hearts. NS makes 10 tricks

in spades. 18 trumps producing to 18 tricks. This is compensating

errors. LoTT is credited for being 'right' when it is 'right' for the

'wrong' reasons.

 

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In this example the points between the diamonds and clubs have

been exchanged. EW is 2434//2533, while NS remains

4342//5134. EW makes 8 tricks in hearts. NS makes 9 tricks

in spades. 18 trumps producing to 17 tricks.

 

[hv=pc=n&w=s42hakt76d642caqt&e=s53hqj98d753ckj98]266|200[/hv]

 

On all three examples EW held 9 hearts and made only 8 tricks.

So maybe the expected tricks are usually only dependent on our power

and pattern. Their pattern and trump length has little effect on our tricks.

There are exceptions where their pattern reduces out tricks. Usually a

singleton to an ace, followed by a ruff. It is rare. We should just

ignore their hands and use only ours to estimate tricks.

 

E(tricks) = trumps + (HCP-20)/3 + SF + e

 

-----------------

Aside: this model applies mostly to when we expect to win 7 to 10

tricks. Very useful for making contested part score auctions. At higher

levels controls play a larger role.

 

Examined hand records from BBO minis, most part score boards had

a std dev between 0.5 to 1 when played in the same strain. Admittedly

the range of playing abilities on these minis was huge. Even in flight A

events it was rare to see the std dev fall under 0.5.

Could not find any evidence knowing their trump length assisted us in

better competitive decisions. Knowing their joint distribution would

have been useful.

[jogs]

The objective of hand valuation should be to find the best contract

for the partnership. It is not to find the most precise valuation of

my hand in a vacuum. The initial point count is just a transitory

value which will be readjusted with every successive bid in the

auction.

Power is the independent random variable used to estimate tricks.

Point count is a dependent component of power.

 

Example 1. I have an ace, two kings and a jack. My pattern is

1=5=4=3. In work count I have 11 points regardless of the

location of the honors.

 

a) ♠2 ♥AK752 ♦KJ82 ♣642

b) ♠K ♥J8752 ♦K852 ♣A42

 

Many would open hand 'a' 1♥. Few would open hand 'b'. Yet

both are 11 point hands. The location of the honors and whether

they are working together does affect the ability of a hand to generate

tricks. Hand valuation is about how honors interact with one another.

Hand valuation is more complex than assigning a fixed value to each

honor.

 

Example 2.

[hv=pc=n&w=s2hak752dkj82c642&e=s653hqj98daq53c98]266|200[/hv]

 

West has 11 points. East has 9 points. 4♥ should make nearly

every time. The joint pattern 1543//3442 fits well. No wasted

honors in the short suits.

 

Example 3.

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West has the same 11. East still has 9 points. 1543//4234 This pair of

hands do not fit nicely. If everything goes wrong west may make only 3

or 4 tricks in hearts. It is about power, points, location of those points,

and how those points fit together. It is also about joint pattern.

Too many systems go to great pains to evaluate individual hands to too

many decimal places. It isn't about points. Taking tricks depends on

how well those points are working together and how well the patterns

of the partnership hands fit.

.............

The main theme is this thread is

The objective of hand evaluation should be to find the best contract

for the partnership.

The best method to achieve this objective is to estimate partnership

tricks. Finding a more precise initial point count will do little to help

us achieve this objective. We need better methods to determine

the fit of the partnership, and thus would be better able to estimate

the partnership tricks. We must realize that expected tricks may vary

radically with every successive bid.

 

E(tricks) = trumps + (HCP-20)/3 + e

Std dev is about 1.25 tricks/board.

E(tricks) = trumps + (HCP-20)/3 + SF + e

For flat hands the std dev can be as low as 1 trick/board.

 

There are four models for expected tricks, one for each suit. Expected

tricks fluctuate wildly depending on strain of the final contract. We want

to be in the strain which maximizes our expected tricks.

[Scarabin]

OK so far. We can easily agree on the general principles but I expect problems only arise when we get to the details of your proposed complete evaluation system. ;)

[jogs]

OK so far. We can easily agree on the general principles but I expect problems only arise when we get to the details of your proposed complete evaluation system. ;)

 

There will be no proposed detailed complete evaluation system presented

here. The theme of this thread is think in terms of tricks for our partnership

rather than points in our hand.. These statistical models are for the general

case. Often completed by the third bid of the auction. Our expected tricks

is X +/- e. All estimates come with errors and this e >=1.

 

There are three phases to the evaluation process. The first phase is the

initial point count. The two our tricks models are the second phase.

By using the model which consist of HCP, trumps, and skewness/flatness

one can improve his starting point for the third phase. The first two

phases are for the general case. This third phase is counting the tricks

for the actual board. This will require uncovering the effects of addtional

parameters. This thread will make no suggestions on how to accomplish

this task of counting. Future posts will suggest how to utilize the models

for better bidding decisions, often at the part score level.

[jogs]

Power and Pattern

 

Review of the material.

Power and pattern are two multi-dimensional independent random

variables used to estimate tricks.

Power is high card points, the location of those HCP, and how they

interact with each other. This would require a complex polynomial

too difficult to solve at the table. HCP is the component of power

which is roughly proportional to tricks. HCP will be used as a

proxy for power in the our tricks model.

Pattern is the ordered configuration of the four suits in one hand.

Joint pattern is the joint pattern of the two partnership hands.

All four suits have effects on tricks. Combined trumps is the

component of pattern which is roughly proportional to tricks.

Trumps is the proxy for pattern.

 

E(tricks) = trumps + (HCP-20)/3 + e

 

Skewness/flatness. Singletons and voids increases the expected

tricks. The absence of singletons and voids(or flatness) reduces

the expected tricks. Singletons and voids are components of

pattern, not power.

 

E(tricks) = trumps + (HCP-20)/3 + SF + e

 

Interaction. There are effects due directly to each individual

parameter. There are also effects due to the interaction of those

parameters.

Most systems assign a fixed value to a singleton. Singletons are

components of pattern, not power. The true value of a singleton

is dependent on how it reacts with other parameters. Therefore a

singleton should have a variable value expressed in expected tricks.

 

5431 // 3343

The singleton is valuable when we play in spades. Singletons in the

hand with the long trumps limits opponents to one trick in that suit.

The ruffing in spades is probably tricks we already had.

 

5431 // 3523

Singletons in the hand with the short trumps are additional trump

tricks, provided there are sufficient trumps. Played in the 5-4

heart fit we are able to ruff clubs with the short hearts. That is

potentially seven trump tricks with hearts as trumps.

 

These examples show that singletons do have a variable value

depending on how it interacts with other parameters and in the

general case that value should be measured in expected tricks.

On any particular board attempt to count the actual tricks

during the auction.