Likelihood?

[kenrexford]

Years ago, I had a theory that I have not been able to test out in any sort of reliable manner. Maybe someone else can help with this?

 

The idea was that if, say, West holds 14 HCP, North 14 HCP, East 4 HCP, and South 8 HCP, then North-South (given a few parameters) was odds-on to make game. I am curious as to what the odds actually are for North-South in that scenario, perhaps limited to when they have an 8-card major fit or perhaps when known to have sufficient stoppers (or no threat suit) for 3NT.

 

The 14-14-4-8 layout might also not be the maximized position. For instance, there might be a higher game yield at 16-14-2-8.

 

If I had the program to handle it, I could imagine creating an odds chart, always with North-South having 22 HCP (of course, other numbers might be done too, but 22 is a nice start).

 

For example, if LHO is known to have 14 (a 14-4 split), then vertical could be odds of game making and horizontal the HCP for South. This would show the ideal split, which might be 14-8 but might be something else. 14-8 seems intuitively right, because South has enough strength to have entries, but North has more strength behind the stronger West.

 

A series of charts might then be built, with each split of the opposition strength. Chart 1 might be West 11, East 7. Chart 2 West 12,East 6. And so on. This might tell where the magic spot lies, which might be 18-0, but there might be a threshold, where for example the odds shift at maybe 12-6 or at 14-4.

 

I could also imagine, then, expanding this to see the effect of East growing larger than West (which seems functionally identical to South growing larger than North), and it seems possible that a double curve might be generated, perhaps with a slighter bump in the inverse.

 

Is anyone following what I am mentioning? If so, any ideas on generating this sort of chart?

[barmar]

In general, when most of the defense's strength is concentrated in the hand in front of your strong hand AND you know this, it's worth an extra trick or two. You can play the hand practically double dummy, knowing which finesses are likely to make.

 

This is why I balance aggressively in auctions 1NT-P-P.

[chasetb]

Ken, I understand the theory and tend to agree with it - I go hyper aggressive when I know my points are partners are sitting correctly. The big thing isn't so much points, as fit and/or entries to the weaker hand to take said finesses. Unfortunately, I am not a programmer, and I have drank enough tonight to where I can't think too much about bridge.

 

On a lesser note, how close are you to Columbus OH? I might head over there for the upcoming sectional, and you sound like a good/fun guy to play with (coming from Conner, of Wheeling WV).

[JLOGIC]

Playing with Ken is probably somewhere between Epic and Completely Epic, but you might never be the same again! :)

[Phil]

Playing with Ken is probably somewhere between Epic and Completely Epic, but you might never be the same again! :)

 

Epic Bridge Time!

[rbforster]

I agree with Ken's premise and further add that if you manage to put West on lead away from his strength, esp against 3N, this will be an even bigger benefit. This come up a lot in Power Double auctions that right side NT - (1suit)-X-(P)-123NT. here double usually shows a strong NT overcall, and it's been my experience to bid your 3NT games on 23ish combined due to favorable honor placement and lead considerations.

[Jboling]

I tested a double dummy study, but realized that it probably was pointless. DD is DD, so you already know the location of every single card from the start, so nothing is gained from that you know that LHO has 14 points from the bidding. The average number of tricks, under the conditions Ken suggested, with opponents points 4-14 and 9-9 were 9.0 and 8.8 respectively. And the likelihood for making at least ten tricks (with a 4-4 spade fit, with spades as trumps) were 31% and 28% respectively. Or maybe this was what Ken actually was asking for? Most likely you gain much more in a single dummy setting from knowing more about the point distribution based on bidding.

[gwnn]

Jboling, it is not entirely pointless to look at DD data. Even double dummy it is better to have our king after their ace rather than before it.

[Cascade]

Double Dummy

 

14 (north) opposite 8 (south)

 

At least 8 spades, not a better (combined length) fit in hearts.

 

These were the frequencies of 10+ tricks in spades from either north or south

 

1000 hands for each number of HCP with West

 

West  Frequency
HCP

0	408
1	406
2	457
3	421
4	423
5	412
6	415
7	411
8	435
9	395
10	424
11	457
12	442
13	443
14	455
15	468
16	497
17	499
18	446

[gwnn]

Looks like it would be perhaps better to try 10 000 hands each (or maybe even more). It's hard to believe, for example that 2 is equally good as 14. But maybe that would take ages to calculate?

[han]

Note that all numbers are below 500. I think 19 times 1000 hands is enough to dismiss the premise.

[gwnn]

Oh sure, the premise is disproven, but it would still be cool to see the actual dependence (with the flaws of double dummy analysis, but without the fluctuations from a low sample size).

[Cascade]

I will try a larger data set tonight.

 

For 0 18 for example the simulation with hand selection and double dummy analysis took around 20 minutes.

 

So about three hours for 10000 hands for that one. The simulations with more even distributions of hcp were around 5 minutes for 1000 hands. I think it could be around 24 hours to get the simulation complete for 10000 hands.

[kenrexford]

Thanks, Cascade!

 

I am not so sure that the premise is defeated by the numbers, gwnn. Getting the numbers near 50% is what I was hoping for.

 

The full parameters of the analysis to me seems to be one of what I have thought of as "finessable positions." In other words, consider where the split if 14-8. Now, you look at the 14-count hand. Holdings like AQ10 and KJx obviously grow up. But, 10's and 9's generally take on added value in that scenario.

 

Equally, not all 8's are equal. Primed 8's and 8's with positional strengths (like internals or J10's or such) take on added strengths.

 

If the general rules get us at near 50% and just below, then judgment adjustments fill in the blanks. However, the "without adjustments" analysis starts to tell when to start thinking along the lines of aggressive game analysis. The general trend seems to be to have a steady increase as West gets stronger, which seems intuitive. The slight bump at the 2-16 split was predicted to some degree, suggesting that 14 in front of a strong 1NT opening might produce games more often than expected, if partner has 8 or so. (This plays out in practice, as bidding 3NT in some auctions like this seems to yield 9 tricks more often than it "should.")

 

Also, the odds without adjustments seem to favor game bidding at IMP scoring if vulnerable, where a slightly lower percentage than 50% is productive.

 

Add to this that IMO the double dummy analysis actually decreases the tricks taken in real life, as West, with strength to lead from, has a real hard time making a non-costly lead and probably defends on average worse than double dummy more than Declarer is hurt by not declaring double-dumnmy.

[gwnn]

We probably have different concepts of "odds-on" then. Anyway, it is an interesting concept and I am very curious as to what the bigger sample size will bring.

[Jboling]

I did the full table as well, but only with 1000 deals like Cascade:

hcp west :  10+ tricks : average tricks
        0  279.0000    8.9210
   1.0000  302.0000    8.9730
   2.0000  321.0000    9.0280
   3.0000  310.0000    8.9660
   4.0000  296.0000    8.9840
   5.0000  269.0000    8.8920
   6.0000  274.0000    8.9280
   7.0000  312.0000    8.9450
   8.0000  286.0000    8.9590
   9.0000  294.0000    8.9190
  10.0000  307.0000    8.9670
  11.0000  294.0000    8.9210
  12.0000  313.0000    9.0030
  13.0000  332.0000    9.0330
  14.0000  319.0000    9.0060
  15.0000  334.0000    9.0410
  16.0000  343.0000    9.0740
  17.0000  334.0000    9.0350
  18.0000  326.0000    9.0600

Looks quite similar to Cascades table, except that I limited it to a 4-4 fit in spades, so my numbers are lower. Having more than 8 cards in trumps increases the number of tricks, when I redid the 9hcp with west situation with a 4-4+ fit the number of 10+ deals increased from 294 to 391, and the average number of tricks increased from 8.9 to 9.2. So the effect of additional trumps is clearly bigger than the distribution of points with the opponents (in a DD setting, it is probably the other way around single dummy). Keeping everything else but hcp constant would be the ideal when you study the effect of hcp.

 

By the way I got the same strange maximum at 2 points with west. The reason for that you get more tricks with the majority of points with east is probably that he is often endplayed when leading. Why this trend does not last to the end seems strange.

 

But still I think the connection between these investigations and real play is rather weak, as in a single dummy setting you really gain a lot single dummy, if you know based on the bidding where the majority of the points are.

[kenrexford]

The "bump" at the 2-16 split is really interesting to me, as well. Notice how with the 4-4 split the 16-2 has a slight bump also, as 343 is the largest number.

[hrothgar]

The "bump" at the 2-16 split is really interesting to me, as well. Notice how with the 4-4 split the 16-2 has a slight bump also, as 343 is the largest number.

 

The bump simply means that queens are worth a lot more than Jack

 

The 2-16 split requires either a pair of Jacks or a Queen in one hand.

 

It's not at all surprising that this section of space looks a lot bumpier than areas with many more legal card combinations.

[Cascade]

10000 hands in each simulation

 

Using dealer

 

With the following constraints:

 

 

hcp(west)==0 and
hcp(north)==14 and
hcp(east)==18 and
hcp(south)==8 and

spades(north)+spades(south)>=8 and
spades(north)+spades(south)>=hearts(north)+hearts(south)

 

and checking for either hand to be able to make 10 tricks:

 

 

game = tricks(south,spades)>=10 or tricks(north,spades)>=10?1:0

 

The frequencies for 10 tricks were:

 

 

0	4144
1	4313
2	4500
3	4429
4	4167
5	4197
6	4312
7	4238
8	4165
9	4388
10	4423
11	4319
12	4491
13	4626
14	4495
15	4809
16	4984
17	4916
18	4756

[kenrexford]

One thing seems clear, then. IMP scoring CLEARLY favors bidding the game if vulnerable. Fine-tuning seems to help for MP's.

 

Thanks much for the data!

 

The bump at 2-3 HCP seems to be confirmed, as well. Using 45% as a fair value, perhaps, with 14-8 HCP, then, the game is very approachable if the 14 is behind a sound opener or in front of a strong 1NT opener.