Further Analysis of LoTT in Bridge

[0].   Introduction
       Interest in the LoTT (Law of Total Tricks) has led me thus far.
       It is more than 40 years ago, that Jean-René Vernes proposed his revolutionary concept Total Tricks, together with two law formulas, one for suit contracts and the other when one side bids in notrump.   If we remember that, in 1950 and 1960's, he had no personal computers available (so common to us) nor any kind of calculators in his pocket, we understand how great he was with a pencil and paper in his hand and a tremendous endurance in mind.
       In the age of computers, 30 years later, Matthew Ginsberg validated the first law using a computer analysis.   His work is based on a powerful double-dummy solver developed in his study of CI (computational intelligence).   At the same time, he generously released a huge library of  717,102 deals fully analyzed by his solver.  
       Chief results he obtained on  446,841 random deals are summarized elsewhere in my homepage, but one figure will convey its essence, which I have drawn from the data in his paper.  
       As may be seen from his paper (total.ps.gz), as well as from this figure, he chose longest suits as trump suits.   Indeed, longest suits mostly coincide with trump suits, but they are not always the same.   For example, when you have 4ª and 4§, both makeable, with 8 spades and 9 clubs, Ginsberg automatically chooses the longer, clubs as the trump suit.
       Anyway, apart from such minor details, his result supported Vernes's pioneering work strikingly well.   Computer analysis for 446,841 deals did a full credit to the handwork of Vernes for 340 deals.

[1].   Object of This Work
       So far, so good for the first law (for suit contracts).   Now, the second law (for notrumps) remains to be studied in the same way.   For this purpose, I have developed two softwares ViewDDLlib and LoTTanalyzer, which are now open for public use.   But, wait for a moment !   My software should work as nicely as Ginsberg's does, when his setting (or, description) is followed.   It must reproduce previous results for it to be called scientific.   So, I aim:
       (A)  To reproduce Ginsberg's result according to his description,
       (B)  To work out statistics, with notrump and suit deals treated separately.

[2].   Reproducing Ginsberg's Work
       In this section, I will try to reproduce the result of Ginsberg, as far as possible.
       Although the simple description of Ginsberg contains some uncertainties (or artifacts), I chose the options called "Ginsberg" and "Length" in my LoTTanalyzer (read the HTML file inside the ZIP file for detail).   Here, all the 717,102 deals are played in suit contracts.  
       Notrump calls are forbidden at all.   As a result, for example, balanced hands with 3NT are forced to bid some suits, which cannot possibly attain a game.  
       Trump suits are selected solely by Length.   Score or priority of contracts are never considered.   For example, when 4§ and 4ª are both makeable, Clubs are automatically selected if longer than Spades.   Furthermore, Clubs will remain as such even when opponents compete with 4¨ or 4©.
       Artificial declarers, although quite dubious, possessing fewer trumps than partner, are allowed to declare in this virtual Bridge (#1485).  
       The result is shown below in a table and compared with Ginsberg's,

Reproducibility check of LoTTanalyzer for Ginsberg's result
Ginsberg (total.ps.gz),				LoTTanalyzer ( Ginsberg + Length Option)
Total Length	number of samples	average of deviation	average error	Total Length	number of samples	average of deviation	average error
14	46,944	−0.15	0.63	14	55,461	−0.15	0.60
15	47,281	−0.14	0.64	15	55,594	−0.12	0.60
16	120,525	0.10	0.70	16	143,014	0.14	0.68
17	102,184	0.02	0.75	17	122,050	0.07	0.73
18	69,792	−0.01	0.83	18	84,481	0.04	0.81
19	37,561	−0.22	0.87	19	46,129	−0.17	0.86
20	15,845	−0.50	0.99	20	19,876	−0.46	0.97
21	5,041	−0.89	1.20	21	6,434	−0.84	1.16
22	1,286	−1.31	1.48	22	1,627	−1.22	1.40
23	237	−1.78	1.83	23	320	−1.81	1.86
24	45	−2.22	2.27	24	62	−2.21	2.21
Total	446,741	−0.05	0.75	Total	535,049	−0.02	0.73
				Remaining 717,102−535,049=182,053 (24,4%) deals, having HCPs outside the range 15 to 25, have been ignored.

       Obviously, the two results agree very well (as they should) despite the difference in sample size.   Statistics is really reliable, when average is taken over a vast ocean of ensemble.
       This good agreement means the following two:
       (1)   My LoTTanalyzer is running nicely (at least in the Ginsberg setting), and perhaps the data handling is right.  
       (2)   It has now become apparent that Ginsberg did not pay due attention to trump suits.   He simply chose longest suits as trumps.
       Having established this, we go on to the next step.
       I am yet wondering about the difference in the sample size, 446741 and 717102.   Didn't he use this library in his analysis ?

[3].   Separate Treatment
       We now do the statistics, after dividing the deals into two categories, suit and notrump.   Here, notrump hands are required to take 7 tricks or more in notrump and less score in suits.   Otherwise, deals are classified into the former.   As a natural consequence, 4ª and 4© are preferred to 3NT, while 3NT is preferred to 5§ and 5¨.
       As a result, 717102 deals in total are divided into 501591(suits) and 215511(NT).
       Several artifacts mentioned above in [2] are now removed.   They are mostly overcome by considering score rather than length in determining the strain.   Surely, longer suits do not necessarily bring better score, as obvious from the above mentioned example of 4ª and 4§.
      Once competition takes place, however, (i.e., when our high-score contract is overwhelmed by their contract), other strains with more tricks (but lower score) are pursued.   Say, we have 3© and 4¨, both makeable.   We will remain in 3© so long as competition does not take place.   But if opponents are able to bid 4§, we change our denomination to 4¨.   This is an example where competition changes trump suits.
       In addition, artificial declarers are forbidden, by requiring them to have

With these improvements on Ginsberg's work, we obtained the following result, which will be shown separately for suits and notrump.

[4].   Suit versus Suit
       Although I tried best improvements (so I think) on Ginsberg's work, the result turned out to be similar to his.   Comparison will be made now in the format below: Here, deviation will mean

as usual.

Comparison between Ginsberg and LoTTanalyzer for Suit Contracts
	Ginsberg (total.ps.gz)	LoTTanalyzer (Standard)
+1 trick deviation	22.4%	31.6%
0 trick deviation	40.0%	38.8%
−1 trick deviation	24.5%	14.9%
Average Deviation(tricks)	−0.05	+0.37
Average Error(tricks)	0.75	0.79
Sample Size	446,741	389,908

       For example,  "+1 trick"  means that LoTT underestimates Total Tricks by one trick.  
       "Average deviation" means its average over the entire deals.  
       "Average error" stands for average of its absolute magnitude (|deviation|).   It is different from the standard deviation commonly used in statistics, but we follow here the convention started by Jean-René Vernes.  
       From this table, we observe that both yield an almost equal value for the average error, 0.75 and 0.79 tricks, respectively.   So, an error of  0.8  tricks is the best value known from the double-dummy analysis.   I started this work in the hope of reducing the error (or rather, variation), by proper selection of trump suits.   Notrump hands have been excluded in LoTTanalyzer, while Ginsberg includes them as suit contracts.   Nevertheless, my efforts did not reduce the average error.  
       So far for error.   As for the magnitude of deviations, LoTTanalyzer tells us that the LoTT underestimates Total Tricks by 0.37 tricks on the average.   This is most clearly seen in the figure output from LoTTanalyzer (on the left).   In most frequent cases of 15-18 total trumps, the LoTT almost constantly underestimates by 0.4 tricks, and will tend to overestimate with increase in total trumps (for more details, ask LoTTanalyzer).  
       Overall behavior is quite similar to the one you have already seen above (to appear now in a separate window), except a vertical constant.   Just view them in parallel and compare.  

[5].   Notrump versus Suit
       Now, we proceed to deals where one side bids notrump.
       Here arises a problem as to which formula to take as the LoTT for notrump.   Most popular version is the one due to Larry Cohen, which says,

where the Total Trumps on the right-hand side means the trumps held by opponent suit contractors.  
Earlier, however, Jean-René Vernes had proposed according to the side suits in suit contractors.
       Here, I would confuse to add one more (and call it Modified Vernes) and compare all the three on LoTTanalyzer.   The result is interesting:

Results of LoTTanalyzer for Notrump Contracts
Law Formula	Cohen	Vernes	Modified Vernes
+2 trick deviation	13.5%	21.5%	5.0%
+1 trick deviation	32.5%	40.8%	21.5%
0 trick deviation	37.3%	25.4%	40.8%
−1 trick deviation	11.4%	5.9%	25.4%
Average Deviation	0.62 tricks	0.95 tricks	0.05 tricks
Average Error	0.87 tricks	1.09 tricks	0.73 tricks
Sample Size	145,140	145,140	145,140

      I added a row for the +2 deviation, since it occurs so often in Cohen and Vernes.   If you are to choose one among the three according to a good sense in statistics, Modified Vernes is best, not because it best accords with the law (with the 0.05 average), but because it has least average error (or least variations).   The figure on the right plots the result for this case.   You see the crosses in wine color nicely sit on the blue line predicted by the law formula.
      However, to a great regret of Vernes, his correction turned out to be worst.   It was perhaps unavoidable since only 73 deals were available to him (which were played on two tables, one in NT and the other in a suit, and the suit had to be common).  
[6].   Independence of LoTT from HCP
      This note is for someone who believes that the LoTT holds for only a limited range of HCP, or believes that Vernes so said.
      In truth, Jean-René Vernes admits three kinds of corrections in the LoTT.   However, he speaks nothing on HCP concerning LoTT.   It is only when he comments on his Rule (la Règle de Sept à Douze, Rule for the Safe Level): So, the requirement for HCP is not for his Law, but for his Rule.  
Remember that Vernes is very careful in his writing.
LottAnalyzer easily confirms this:

Result of LoTTanalyzer (for suit contracts) under the Four Requirements for HCP
	0 - 40 HCP	15 - 25 HCP (à la rigueur)	17 -23 HCP (de préférence)	0-14, 26-40 HCP
+2 trick deviation	9.7%	9.5%	9.3%	10.2%
+1 trick deviation	31.5%	31.5%	31.5%	31.0%
0 trick deviation	38.4%	38.8%	39.2%	37.0%
−1 trick deviation	15.2%	14.9%	14.9%	16.0%
Average Deviation	0.36 tricks	0.37 tricks	0.36 tricks	0.34 tricks
Average Error	0.79 tricks	0.79 tricks	0.78 tricks	0.82 tricks
Sample Size	506,581	389,908	280,383	116,673

      The software LoTTanalyzer, on which this article is based, is open.  
            Download and use it.   You will find more tables and graphs.

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