[obscurans]

I have a heretic assertion to make (and I disbelieve it as strongly as everyone else): 23HCP makes for a 3NT bid.

For simplicity, let's assume matchpoints and ignore extra undertricks for extreme overbidding, so what you want is: IF we have X total HCP between us, THEN we have a 50% chance of making 3NT. Find X.

I'm using the double dummy library provided by Prof. Matt Ginsberg (of GIB fame) found at http://www.cirl.uore...ibresearch.html. I took every deal, computed N/S and E/W total HCP, then recorded their double-dummy tricks possible for all 4 declarers. So an entry is in the form (total HCP of pair, tricks possible at NT). Whether the deal should be played in NT, whether they have slam, or if they should be even declaring (read: we have 3HCP together) is ignored. If the tricks differ depending on right-siding, the result for the stronger hand declaring is weighted 2/3, the weaker one 1/3 (if equal, 1/2 each). All entries are collected and summed up for statistics.

Fundamentally, this is a binary classification problem. You have a hidden variable (whether we have 9+ tricks at NT) and an observed variable (total HCP), and you want to predict the hidden using the observed. Unless you have a better idea, we predict all hands with X+ total HCP as making 3NT and any less as not making 3NT. This gives rise to a confusion matrix:

True negative (number of hands where we have LESS than X HCP and LESS than 9 tricks at NT)
False negative (number of hands where we have LESS than X HCP and AT LEAST 9 tricks at NT)
False positive (number of hands where we have AT LEAST X HCP and LESS than 9 tricks at NT)
True positive (number of hands where we have AT LEAST X HCP and AT LEAST 9 tricks at NT)

Then we can calculate a boatload of different statistics from the matrix. The most important one, corresponding to the question asked above, is the positive predictive value, TP/(FP+TP) - verify it matches the description. We want the minimum X such that PPV is at least 50%. And here's where the wheels come off - the magic number is not 25, it's 23. If you restrict it to hands where double-dummy shows 7-11 NT tricks... it drops to 22.

After a couple hours of not believing and hence fruitlessly debugging the program, I gave up. Simply put, when people state that 25HCP is required for NT game, they seem to be answering the question: IF we have 3NT, THEN we have X+ HCP 50% of the time, where X is indeed 25. But that is backwards, and in fact is calibrating the sensitivity, TP/(FN+TP), which measures how many games have 25+HCP.

Now, if you haven't written me off as a complete lunatic yet, is that why bridge is getting more and more aggressive? The people who bid closer to 23HCP games do better in the long run? Is the benefit of double dummy (remember, defenders also get to play perfectly) really a full trick on average?

PS: Data.

HCP #no 3NT  #hv 3NT
 0        0        0
 1       42        0
 2      204        0
 3      498        0
 4     1416        0
 5     3726        0
 6     8292        0
 7    16254        0
 8    30240        0
 9    51648        0
10    84996        0
11   131166        0
12   185604        0
13   258696        0
14   341816       10
15   430580       46
16   517026      126
17   599910      576
18   661065     2007
19   700184     6088
20   709570    16598
21   668130    38142
22   584836    78236
23   464634   135852
24   322037   195115
25   196346   234280
26   105431   236395
27    50292   208404
28    22352   163252
29     9610   121556
30     3762    81234
31     1310    50338
32      409    29831
33       74    16180
34        2     8290
35        0     3726
36        0     1416
37        0      498
38        0      204
39        0       42
40        0        0

[matmat]

HCP #no 3NT  #hv 3NT
23   464634   135852

Doesn't this line imply that going off in 3NT is 3 times more likely than making when holding 23 high? what am I missing?

Quote

I took every deal, computed N/S and E/W total HCP

*every* deal?

[obscurans]

OK, I've regained partial sanity. MP measures only differences, so in fact you are asking: IF we have EXACTLY X HCP, THEN we have a 50+% chance of making 3NT: find the minimum X that still fits (back to 25, yay). Since we suppose the 'normal' people will bid 25HCP games anyway, your anti-field behavior is limited to 24HCP and 23HCP bids... which are less than 50% shots. Solved.

Now, at IMPs vul, doing the same thing and reading off the columns, 24HCP gives a 37.7% chance of making 3NT, so it's possible that you can bid 24pt 3NT red. This number is so close to the mark (you stand to gain 10 for 3N= vs 2N+1, lose 6 for 3N-1 vs 2N=, so you need a chance of 6/(6+10) or 37.5%) that it's well within experimental error. I'm running a couple million more hands first.

Back to the sensitivity problem, which measures how many games you can safely bid (knowing only combined HCP), at 25 it's 70.9%, so you will miss 30% of games. This is the true yardstick of how good your hand evaluation is - I'll be optimizing that and be back with results (much) later.

[obscurans]

matmat, on 2011-April-12, 00:58, said:

Doesn't this line imply that going off in 3NT is 3 times more likely than making when holding 23 high? what am I missing?

Yes, ninjaed, I missed that. I plead 3am your honor.

matmat, on 2011-April-12, 00:58, said:

*every* deal?

Every one in the given library, all 717102 of them.

[matmat]

obscurans, on 2011-April-12, 01:03, said:

Yes, ninjaed, I missed that. I plead 3am your honor.

Statistics get dangerous after midnight.

Quote

Every one in the given library, all 717102 of them.

Seems like a fairly small sample, tbh.

[obscurans]

Actually 700k is a pretty good sample for global statistics on hands. I can even pin down the occurrence of insanely rare hands (6610 or worse, 16+HCP) with 100M or so hands generated. It's the same thing with pollsters grabbing about 1k people and generalizing to however many million there are in your country. The assumption that your random deals are similarly dealt to those at the table is sufficient to get 1/n variance behavior.

[bluecalm]

Few points:

a)As you count the deals when declaring side has any distribution, then of course hcp threshold will be lower. It's nothing new that if we have 6+card suit then 23hcp is enough for 3nt
b)Double dummy stats are useless imo when thinking about how often 3nt make. The reason is that opening lead gives away a lot (and subsequent play is about even between declarer and defenders usually, at least in real play at high level)
c)I did some stats from real play (from vugraph) counting only hands without 6 card suit and then without 5 card suit. 3nt made more often than not on 24hcp, but of course things like T's and 9's weren't included. If I have time for that later I will count those too.

[paulg]

It is also important to factor in the number of undertricks when you do not make 3NT. The oft-quoted percentage of 37% for a vulnerable game presumes that you only ever go one down.

[1eyedjack]

paulg, on 2011-April-12, 02:40, said:

It is also important to factor in the number of undertricks when you do not make 3NT. The oft-quoted percentage of 37% for a vulnerable game presumes that you only ever go one down.

And of course if you are destined to go 2+ down there is a fair prospect of being doubled, with a rather lower likelihood of being doubled if there are overtricks in the air.

[nige1]

For all its faults, IMO double-dummy analysis is a good way of analysing these problems.

IMO, somebody else studied this recently, using double-dummy analysis of a large deal-database. I thought it was Richard Pavlicek. Whoever it was, I'm afraid that I can't find the link.

AFAIR the result depended heavily on the point-split, between declarer and dummy. 23-24 HCP often sufficed if the points were divided evenly (11-12 or 12-12). More HCP were required, the greater the point-disparity.

I can't remember if the strong hand was made declarer in these simulations but I guess that lead-values would have some effect , too.

[Cyberyeti]

Something which would interest me would be to analyse the 24/23 point hands that make and divide them up by point split and shape. Does 13/11 make a lot more often than 16/8 for example and do you need a 5 card suit to make 3N a goodly proportion of the time with 24 ?

[johnu]

Cyberyeti, on 2011-April-12, 09:42, said:

Something which would interest me would be to analyse the 24/23 point hands that make and divide them up by point split and shape. Does 13/11 make a lot more often than 16/8 for example and do you need a 5 card suit to make 3N a goodly proportion of the time with 24 ?

Good question Cyberyeti. I'm a little hazy on this and most of my old bridge books are still in boxes, but IIRC in the original Kaplan-Sheinwold system, I think they proposed that 12 opposite 12 was enough for game to be a good proposition on average because of the flexibility of transportation between the 2 hands, while 16 opposite 8 wasn't as good. This was way before the age of personal computers and massive databases so I'm assuming this wasn't based on any real statistics, just a general memory of previous individual results.

The 5 (or 6 card) suit question seems particularly relevent, since if the suit can be established and run, it provides one (or 2 in the case of a 6 card suit) extra trick without an increase in required HCP's. So, how do boards where neither hand has a 5 card suit compare to hands where there is at least one 5 card suit compare with the same total point counts?

[Quantumcat]

It would be interesting to re-do the simulations with all hands that have one 8+ card fit that aren't 4-4, and all hands that have a 5-2 fit, and all hands that are 4333 opposite 4333, and see what points you need for making in each of those situations. The ones with really big fits probably won't make because of lack of stoppers in short suits, but the fairly balanced ones with a 5-3 or 6-3 fit will probably make 50% of the time with 23 HCP.