Being Sloppy

Let's say you had a convention where a 1 opening showed either 10 points or 30. You want to run a simulation based upon this opening and so you need to deal out a bunch of hands which fit. So you tell the computer to give you some hands with 10 points and some hands with 30 points and go on your merry way.

Ah, but the fact is that 30 point hands are much rarer than 10 point hands. Much rarer. So if you want to do a proper investigation, you need to deal out the hands with the same proportion as they would occur in real life, otherwise you'll be skewing the data massively. Anyone care to work out how many 10 point hands we should deal for every 30 point hand? No, I can't be bothered either.

The simple way to do this is to just deal out completely random hands with no restrictions (random deals actually, but we'll let the distinction slide) and then pick out all the ones which don't fit. We'll count the points on every one of possibly millions of hands that come our way and only include it in the investigation if it has 10 or 30. This will ensure that the proportion of 10pt hands to 30pt hands in our sample set is more or less as it would be in real life. Computers are super-fast at dealing random bridge hands so the fact that we're dealing a heck of a lot more than we should isn't an issue.

It took my computer about 30 seconds to deal out 10 million completely random deals. Of these, South held 10 points 940,000 times and 30 points a mere 22 times. We've wasted over 9 million deals! We can now perform our analysis on the ones left, knowing that it will broadly mirror reality.

This is the same thing that we did with our example hand. We dealt out millions and millions of random deals and only kept the ones which fit the knowledge we had been given. If we did it right then the deals which we kept will be consistent with the action so far (LHO passed, partner opened 1NT, RHO passed). The full set of rules that we used were as follows:

• North has 12-14 points
• North is balanced
• West has 0-11 points
• West has an 8 loser hand or worse
• East has 0-14 points
• East has a 7 loser hand or worse
• South (you) has the problem hand, 4 A Q 2 A J 7 6 2 J 9 4 2

"But", you say, "I would compete over 1NT with lots of 7-loser hands! Your analysis is flawed and I'm never going to look at your stupid blog again!". And that's a fair point. We originally stipulated that oppo aren't hyper-aggressive but they surely would act with, say:

Q J 10 2 A K 8 7 2 4 J 6 5

That's seven losers so our analysis will include it. But so is this:

4 2 K Q 4 3 K Q 6 2 J 4 2

I'm not completely sure what you're supposed to call with that if not Pass!

The conclusion is, unsurprisingly, that the losing trick count isn't a great way to assess hands for overcalls. Perhaps we should do something more sophisticated like use the Rule of 20/21/22 or count shortage points or implement Binky points or Zar points or a number of other hand evaluation methods. Perhaps we should, and our analysis would undoubtedly be stronger if we did. The question is, though, how much? We don't only want to model the kinds of hands which East can hold, we want to model them with the correct frequency, so how much does it really matter if the odd attractive 7-loser 11-count slips in?

Over 100,000 deals I counted the HCP and the loser count of all the East hands. They were distributed as follows:

HCP

3    :   991
4    :  3123
5    :  7750
6    : 11703
7    : 14802
8    : 15785
9    : 15105
10   : 12677
11   :  8917
12   :  5325
13   :  2827
14   :   995

Losing Trick Count (in half-tricks)

14   : 22143
15   :  4988
16   : 29206
17   :  7738
18   : 21251
19   :  5338
20   :  7104
21   :  1433
22   :   755
23   :    44

You will see that a healthy majority of hands are clear passes. In terms of HCP, you might consider overcalling on some 11-14pt hands but these only account for 18.1% of the sample. If we're looking at losers, we might act on some 7-loser hands (like the example above) and also some 7.5-loser hands which makes up 27.1% in total.

What does it look like if we use the Rule of X (HCP + length of longest suit + length of second longest suit) for overcalling? Here are the stats:

Rule of X

10   :   121
11   :   800
12   :  2591
13   :  5822
14   :  9629
15   : 12991
16   : 15580
17   : 15605
18   : 14158
19   : 10900
20   :  6956
21   :  3536
22   :  1172
23   :   136
24   :     3

We can probably say that we'd be acting on a Rule of 24 hand (say, 14 HCP with a 5-5 shape) but note that, because of the losing trick count restriction, it won't be anything like as good as:

K Q J 10 2 A K 8 7 2 4 J 5

More likely it'll be:

J 6 5 4 2 A J 8 7 2 K K Q

Ok, we'd be bidding but it's not exactly beautiful. We might also get involved with the Rule of 23 hands and some of the Rule of 22 hands. But not all of them. A balanced 14-count fulfills the Rule of 22 and our oppo won't be bidding with that (I just asked them). And even if we add in the Rule of 21 hands (some of which we'd act on; most of which we wouldn't), it only amounts to 4.8% of the sample.

We're not getting anywhere very fast like this, and we won't either unless we sit here and painstakingly map out a detailed hand evaluation metric for overcalling a weak NT when vulnerable. But that's hard to do and I'm too lazy anyway. And I don't believe that we have to. If more than 5% of our sample size is flawed because we're including hands which would have overcalled and therefore do not accurately mirror the problem scenario then I'd be very surprised (at a guess, I'd put it at more like 1-2%).

And what does it hurt us if we include these hands, anyway? Instead, for example, partner will open 1NT, RHO will overcall 2 naturally and we'll still be in a similar position, wondering whether to bid 3NT or try for 4 or perhaps look for a minor game. The decision doesn't change — we're just handicapping ourselves by pretending we didn't see the overcall. The same could be said if RHO overcalls something else. It becomes harder the higher he overcalls, but even if he bids 2 we can still wonder to ourselves whether to bid 3NT (if partner has a stop) or look for a heart game (perhaps by cue-bidding 3 in a way that denies a stop).

What I'm trying to say, in a very long-winded way, is that it would be great if we could model the problem scenario in perfect detail such that every hand is consistent with the knowledge we would have at the table, but that's hard. Much easier is to be sloppy and use crude evaluation systems such as high-card points or the losing trick count or both. If we do this, we'll find that the majority of hands work out just fine and, of the hands which we analyse when we shouldn't, most of them won't make any difference anyway. We don't have to be perfect — getting it right 95-99% of the time is more than enough.

The moral: try not to be sloppy but if you are then don't worry too much!