Slam Bidding in Zar Bridge
So I’ve been doing some statistical analysis of bridge hands, and I realise this may come a bit out of left-field on a blog otherwise devoted to pictures of ducks.
I’ll try to get around to explaining where this came from in another post later. For now, I want to write about numbers.
Numbers
Quick note on the methodology:
I’ve built up a set of randomly generated partnerships and assigned each of them an ‘ideal contract’ based on a set of baseline criteria, e.g.
7NT: Total points = 67+, no singleton or void in either hand, 4 aces, 8+-card fit
Contracts can have multiple criteria, e.g there is also:
7NT: Total points = 72+, no singleton or void in either hand, 4 aces, no fit
Points are counted using Zar, because it doesn’t suck. It does not include any adjustments for superfit, honours in Partner’s suit etc. It does include an adjustment down for unsupported honours.
Each partnership is then tested against its ideal contract across 50 samples, with the remaining cards dealt randomly to the opposing partnership in each sample. Cardplay is simulated with a double-dummy engine, and the number of tricks taken in each sample recorded to provide a distribution for each hand.
The whole dataset includes 27,819 partnerships, with a 98.9% sample resolution rate.
I’ve then sliced down the dataset to ask questions like ‘of all the partnerships assigned 6NT, how did the ones with 3 kings compare to the ones with 4 kings?’ by looking at the total % of samples which made the contract.
I started out writing this as a set of notes to the slam bidding system I’d come up with, only to realise as I was writing the notes that the system didn’t really match the conclusions I was drawing, so the conventions themselves will get written up in a new post when I have, yet again, completely revised them.
So for now, numbers.
Small Slam
My baseline contract criteria follow the standard Bridge precept that if a partnership has no singleton or void, it should be in no-trumps; if it has one, it should in a trump contract, so that’s how trump/no-trump has been divided. It would be interesting to test some of these partnerships against the other contract at some point, but for now I’m assuming the received wisdom is true.
Percentages below show the % of samples which make the contract under each set of conditions. ‘N’ is the total number of partnerships in that slice.
62-66 points, 8-card fit, 6-trump slam (n=318):
4 aces, K of trumps: 79.9% (missing Q, 70.2%; with Q, 87.3%)
4 aces, missing K of trumps: 54.0% (missing Q, 37.1%; with Q, 58.4%)
3 aces, K of trumps: 53.4% (missing Q, 34.5%; with Q, 64.7%)
3 aces, < 4 kings, K of trumps: 49.2% (missing Q, 31.2%; with Q, 60.0%)
62-66 points, 9+-card fit, 6-trump slam (n=382):
4 aces, K of trumps: 95.8% (missing Q, 93.4%; with Q, 96.8%)
4 aces, missing K of trumps: 82.1% (missing Q, 66.6%; with Q, 88.2%)
3 aces, K of trumps: 81.2% (missing Q, 66.7%; with Q, 84.0%)
3 aces, < 4 kings, K of trumps: 80.7% (missing Q, 69.9%; with Q, 82.5%)
3 aces, missing K of trumps: 55.2% (missing Q, 40.0%; with Q, 56.1%)
62-66 points, 8-card fit, 6NT (n=65):
4 aces: 79.9%
4 aces, < 4 kings: 79.3% ( < 3 kings, 64.2%; no data for < 2)
3 aces: 67.8%
3 aces, 4 kings: 73.4% (3, 54.7%, n=6)
62-66 points, 9+-card fit, 6NT (n=45):
4 aces: 82.3%
4 aces, < 4 kings: 79.9% ( < 3 kings, 75.6%; no data for < 2)
3 aces: 77.7%
3 aces, 4 kings: 83.1% (not enough data for < 4 kings)
Standard bridge wisdom, based on the usual scoring systems, is that if you’ve got a 50% chance of making a small slam you should go for it. Conversely, you need a 75-80% chance of making a grand slam to bid it over a small slam, as the rewards are relatively less and the small slam, with this kind of point count, basically certain.
So this leads to a few rules:
62-66 points, 8 card fit, unbalanced:
If you’ve got 4 aces and either the K or Q of trumps, go for it.
If you’ve got 3 aces and both the K and Q of trumps, go for it.
If you’ve got 4 aces and don’t have space to ask about the K or Q of trumps, you should probably bug out. If you need to ask about them, it means you can’t see either of them, and that 54% includes all the partnerships where they’re in your hand too.
62-66 points, 9+-card fit, unbalanced:
If you’ve got 4 aces, you’re good, even missing both K and Q you’ve got a 2/3 chance of success.
If you’ve got 3 aces, you’re probably still good, holding either the K or Q will put you over 50%.
62-66 points, balanced:
Bid the slam with 3+ aces. You’ll have a better shot at making if you’ve got the superfit or have most of the kings – but even without the superfit you make it more often than not, and with a point count this high hands without at least 2 kings are ridiculously rare.
Grand Slam
With Grand Slams things are a bit different. We’re going to assume you always have 4 aces here, if you’re missing one you should be bugging out.
I’ll spare you the full breakdown for this, if only because it just means less – the sample sizes are getting pretty damn thin for hands of this size. But the tl;dr:
8-card fit:
For a 75% hit rate, you need both the K and Q of trumps. You can get away with 3 kings total as long as one of them is the trump, but you still need the Q to put you over the 75% chance of making. 4 kings without the Q doesn’t get you there.
9-card fit:
3+ kings, one of which is the king of trumps, is the ballgame. Having the Q of trumps always makes things better, but she never puts you over 75% if you weren’t there already.
3 kings but missing the king of trumps is interesting – in the data these partnerships get wrecked (24% make) but there’s only 2 partnerships out of the whole dataset with this combination, so not really good for a conclusion. I would avoid, regardless.
7NT:
Not enough data. Only 9 partnerships meet the baseline of 67+ points and balanced, so there’s nowhere near enough to slice up to see what’s really needed.
No Fit:
If you’ve got 67+ points and don’t have a fit, you’re basically always in 6NT. 4 aces is best, 3 still has a 55.4% make-rate.
***
Right now what’s standing out most to me is that the biggest deciding factor is the fit. If you’ve got a 9+-card fit, you can get away with having far less, so you want to know if that’s there before you get to the slam-bidding part of the auction.
Next up, I’ll need to adjust the bidding system so that it actually makes sense with these numbers.
And after that, I’ll go back to my ducks.
