A Mathematical Model on LOTT

[keylime]

Greetings friends...here's somehting I think you'd be interesting in relating to LOTT.

 

The Theory of the Law of Total Tricks: A Simplistic Mathematical Model

 

Introduction

 

The Law of Total Tricks (LOTT) was first introduced into bridge literature in an issue of the Bridge World in 1969 by Jean Rene Vernes and brought forward to the mainstream by Larry Cohen?s book To Bid or Not to Bid. In essence, what both authors propose is that the total number of tricks that is possible in any bridge hand equals the number of trumps that the partnership possesses. It is also a facet of LOTT that double fits, extra trumps, slam bidding, wastage of honors, etc. et al. all factor positively or negatively in the overall approximation of LOTT. In my article I discuss to you a generic formula and some of the other mathematical elements that I didn?t include in my posts discussing LOTT.

 

A Generic Formula

 

When competing for a partscore or a game contract, there exists I believe a linear formula that can predict with a fair level of accuracy what the total number of tricks one side can take. After extensive analysis and painstaking work involving data modeling and double-dummy studies, the curve closely fits this equation:

 

N = t + .5(p-20)

 

N = the total number of tricks available

T = total number of trumps held

P = high card point (HCP) totals of the two hands

 

Two features that crop up with the model that I found interesting. Firstly, when the pair holds the balance of the high card points AND when honor cards are working (i.e. no singleton Kings or Q-x opposite a weak suit), there exists an extra trick for each two points over average (20). Secondly, once fit is show, any extra trumps are worth a trick separately.

 

Normally for each HCP held over 20, there exists a solid correlation of ? trick gained. This equivalence is mutually exclusive of the number of potential trumps held. However, when the pair holds 31 or more HCP, the model loses its integrity. There are some reasons for this: defenders are on lead and normally are going to lead a suit OTHER than trump or more often the distributions of the hands prevents being able to implement trumps at the first trick, thus temporarily preventing the full effectiveness of trumps from being implemented. Also, as the hands become more shapely, there exists a greater chance for wastage and for higher degrees of fit.

 

Adjustments in the Generic Formula

 

The data model is quite consistent for hands that are not of a high level of freakness up to game. Beyond game, two concepts appear that require the adjustment in the formula:

 

1.      When the HCP total exceeds 30, it normally requires 3 points for each extra trick. This warrants caution when exploring slam.

2.      Wastage is critical when the hands are shapely. Stiff honors opposite weak holdings do not carry their full weight.

 

Does this formula work?

 

The formula?s strength is its simplicity. I?ve used a population of nothing less than 10,000,000 deals, comparing the results with other works relating to the subject and with double-dummy solvers. My results were that the error was 0.0145 and the standard deviation of 1.1862, which compares favorably to the accuracy of LOTT (published results indicate 0.126 as the mean and 1.039 for standard deviation). If we were dealing with a normal distribution, the deviation infers that roughly 60% of all trials fall between ?1 and +1.

 

Using the Formula

 

Does the formula substitute for other factors? No, it simply serves as an approximation to aid in bridge judgment. At MPs I feel this formula excels because of the very competitive nature of the event, while at IMPs if suffers a little because of the scoring that?s involved. This formula is suited for suit contracts. At no-trumps, the formula doesn?t make any distinction between long running minors and majors, and long running suits in general. When there exists a double fit, normally ? to one additional trick exists for the partnership.

 

Closing

 

I?ve attempted to articulate some form of measure that could be used with a fair degree of success. I?m not one to take something at face value due to my background in mathematics. These trials have served to validate LOTT as a decent theory that can gauge with relative predictability the total trick taking power at low-level contracts. However, with distribution and slam explorations, LOTT does not have the mathematical restraints needed to predict with accuracy the degree of fit coupled with the elevated nature of the contract. Thusly, use this as you will for the run of the mill contracts, but work on those slam bidding skills, for even mathematical modeling can?t predict if a slam will succeed or not.

[inquiry]

Your model suggest that a 4-3 fit would have game if the combined holding is 26 hcp

 

N=T+0.5(p-20) = 7+0.5(26-20)=7+0.5*(6) = 7+3 = 10

 

Since you always have at least one 7 card fit, your equation says bid game on 26 point hands. That seems just about right.

 

It also says you should bid game with a 9 card fit and 22 hcp. This works assuming with 22 hcp, you have some useful distribution. So a quick look suggest it may be a fair approximation. Now, if I can only figure out how many trumps and how many hcp my side have doing the bidding I might be able to apply it at the table.   :-*

[Lestat]

Dwayne Hoffman wrote:  When competing for a partscore or a game contract, there exists I believe a linear formula that can predict with a fair level of accuracy what the total number of tricks one side can take. After extensive analysis and painstaking work involving data modeling and double-dummy studies, the curve closely fits this equation:

 

N = t + .5(p-20)

 

N = the total number of tricks available

T = total number of trumps held

P = high card point (HCP) totals of the two hands

 

*****

 

Very interesting and good work.  It appears that the formula works for part-scores and games, perhaps for normal distributions and fits.  Valuable information!  However, it's easy to create extremish examples where the formula doesn't work.  e.g.

SAKQJTxx  HAKQ Dxx Cx opposite S_ Hxxx Dxxxx Cxxxxxx

 

7 trumps, 19 HCP.  Formula predicts just under 7 tricks instead of the likely 10.  But of course for that hand, I wouldn't need a formula to bid game!  lol

 

It would be nice to expand the formula to handle other distributions including non-fits and slamish hands.  I suspect that as the point count increases, a non-linear model would be more accurate because with increasing point count, more options exist to make slams.  And in any event, one cannot make more than 13 tricks, even though one might have 14+ tricks in hand.

[keylime]

I am developing a general mathematical model for hands that are freakish in nature because with higher levels of distribution it longer is a linear formula that will fit the data sets. I have reason to believe it's along the lines of a hyperbola, but more tests need to be run of course.

 

I can imagine players popping out these two formulae in order to bid or pass lolol  ;D

[mikestar]

I look forward to seeing a more detailed model--this is excellent for something so simple. Areas where corrections may be needed (besides the ones Dwayne mentioned):

 

1. over 10 trumps (LOTT starts to break down according to Larry Cohen and others).

2. HCP very uneven between the two hands. 24 HCP and 8 trumps are a good game if they are somehwere near 12-12 and a really bad game if 20-4.

3. Is the distribution as unbalanced as is normal for the number of trumps (freakiness only cover this in part).

 

An example of the last point:

With 10 trumps the formula suggests that 20 HCP are enough for game. With 5-3-3-2 opposite 5-2-3-3 I'd wager it will go down more often than not, with 5-4-3-1 opposite 5-2-3-3 it will be close, and 5-4-3-1 opposite 5-1-3-4 will be easy. Yet none of these shapes are freaky.

[keylime]

Everyone will be happy to know that in the very near future I will be publishing my continuing work for a set of equations to predict nearly all cases of LOTT. I am still in the works about certain distributions (i.e. very wild ones, got about 80-85 percent of the entire tree determined already).

 

I fully intend to publish in a scientific journal my findings, since it deals directly with Boye's theroem (we know it as the theory of restricted choice) and conditional probability. ;)

[flytoox]

Everyone will be happy to know that in the very near future I will be publishing my continuing work for a set of equations to predict nearly all cases of LOTT. I am still in the works about certain distributions (i.e. very wild ones, got about 80-85 percent of the entire tree determined already).

 

I fully intend to publish in a scientific journal my findings, since it deals directly with Boye's theroem (we know it as the theory of restricted choice) and conditional probability. ;)

 

 

That would be great and I certainly want to be the first reader:) Actually, I also thought about modelling partnership bidding using game theory.

[mishovnbg]

Hi Dwayne!

 

LOTT=35%. Another opinion, also tested, even over real boards from world championship!!! You can download it from here:

 

http://mnet.bg/~mfn/TotalLaw.zip

 

Misho