Introduction

Gordon Bower's research into when to raise 2NT to 3NT caught my eye the other day, and I thought it would be an interesting area to look into further.

Specifically, since the types of hands that are "marginal" for a 3NT contract after a 2NT opening are remarkably limited, I thought I'd look into those.

Here is some example data. Assume partner opens 2NT with any 5332, 4432, or 4333 pattern and 20-21 points as the Walrus counts them.

As usual all data was generated with Deal, my bridge hand generator.

Also as usual, all data is double-dummy, so take it with a big grain of salt.

Sample Sets

The tabs at the top of this file list different, overlapping data sets.

In all cases, partner is assumed to be allowed to open 2NT on any 5332, 4432, or 4333 shapes, even with bad doubletons.

hcp

Assumes partner opens 2NT holding 20 or 21 points in standard 4321 High Card Points.

hcpNBF

Same as "hcp," but with the added condition that partner doesn't have an 9-card or longer major suit fit. (No Big Fit.)

hcpNF

Same as "hcp," but with the added condition that partner doesn't have an 8-card or longer major suit fit. (No Fit.)

binky

Partner has between 7.94 and 8.94 points using my "Binky Points" notrump evaluator. This is the rough equivalent of 20-21 points. I should have simplified and used 8.00 to 9.00, but it's a lot of data to regenerate.

binkyNBF

Same as "binky," plus no major fit of 9 cards or more.

binkyNF

Same as "binky," plus no major fit.

fifths

Partner has 19.2 to 21 points using the "fifths evaluator," where A=4 points, K=2.8, Q=1.8, J=1, and 10=0.4.

fifthsNBF

Same as "fifths," plus no major fit of 9 cards or more.

fifthsNF

Same as "fifths," plus no major fit.

Key

Each section is a table of hands matching the condition of the title of the section. So "Two Queens, One Jack" means that the table consists of hands where you hold two queens, one jack, and no ace, king, or ten.

The colors of each row indicate whether you want to be in 3NT with this hand in various conditions.

So, red means stop, green means go, violet means vulnerable and blue means non-vulnerable.

The letter codes in the left column are for the color-blind.

Marginal?

Marginal just means that we cannot say with 99% certainty, from the data we have, whether you ought to bid 3NT or pass with the given hand.

At Matchpoints scoring, we want to be in 3NT when the odds are greater than 50%.

At IMPs, we want to be in 3NT when the expected number of IMPs we can make by bidding 3NT is positive, compared with remaining in 2NT. That's a little more complicated.

Here's a little math for you. Let p_7- be the probability that you can't make even eight tricks in notrump. Let p₈ be the probability that you can make 8+ tricks in notrump, and let p₉₊ be the probability that you can make nine or more tricks in notrump.

Then when you bid 3NT at IMPs versus passing at IMPs:

Vulnerable

• You lose 100 points, hence 3 IMPs, with probability p_7-
• You lose 220 points, hence 6 IMPs, with probability p₈
• You gain 450 points, hence 10 IMPs, with probability p₉₊

We can write the expected IMPs score as: 10p₉₊ - 6p₈ - 3p_7-

Or, using that the sum of the three probabilities is equal to one, we can remove p₈ from the formulate, and we get:

16p₉₊ + 3p_7- - 6

Non-Vulnerable

By similar reasoning, we get that the expected IMPs score non-vulnerable is:

11p₉₊ + 3p_7- - 5

These formula explain an oddity you'll see in the tables. The rows are sorted by p₉₊, the probability that you can make 3NT, but there are times when a marginal hand at IMPs, vulnerable, will follow a hand that is marked as as certain for vulnerable IMPs. That's because the second hand has a lower chances of not making even 2NT. Often, this happens when the following hand is 4333 pattern. The 4333 hand might be slightly more likely to make 3NT, but much less likely to go down in 2NT, and so the 4333 hand is a net negative compared to the previous hand.

That oddity sometimes happens for another reason - one row might have more sample data, in which case its error ranges will be smaller. In particular, then, it can make the hand non-marginal, when hands later in the table are marginal. So a hand which makes 3NT 51.4% with a margin of error of 1.2% is marked "M," while a hand which makes 3NT 51.6% with a margin of error of 1.8% would still be considered marginal at matchpoints.

Sample Rows

Here are some sample rows from the section "One Queen, Two Jacks, and Three Tens" under the "hcpNF" tab:

Key

The letter code used above, blank being "don't bid."

Hand

The hand pattern, where the x vacancies can be any spot of denomination nine or lower, and the suits can come in any order.

Makes in NT 9+ tricks

The probability that your partner can make nine tricks in notrump opposite this hand type, based on our sample, and the 98% margin of error.

Makes in NT <8 tricks

The probability that your partner cannot make even 8 tricks opposite this hand time. This affects IMPs scoring in interesting ways, see below.

Expected IMPs Score, Vulnerable

The expected IMPs value for bidding 3NT with this sort of hand when vulnerable.

Expected IMPs Score, Non-vulernable

The expected IMPs value for bidding 3NT with this sort of hand when non-vulnerable.

Sample Size

The number of deals dealt with this hand type where partner has a 2NT opener and you don't have an 8-card major fit.

Average Tricks

The average number of tricks partner can take

Analysis

First of all, try not to take the prescriptions of whether to bid on or not literally. I believe mining this data for patterns is more likely to yield more interesting ideas than just treating these numbers as gospel.

When holding an Ace or King

In marginal hands holding an ace or king, you generally seem to have the best chance when you have a five-card suit. That's not surprising, because the five card suit is a source of tricks. In some cases, it seems like you want your honors in the five card suit, and in other cases, it seems like you want honors in separate suits.

For example, the five hand types most likely to make 3NT when you hold a king and a queen, and no ace, jack or ten, are all of the 5332 patterns with no honors in the doubleton.

4333 Pattern

Holding a completely flat had is a disadvantage in two ways:

1. You do not have a source of tricks.
2. You are more likely to make exactly 2NT.

The latter seems counter-intuitive, but the problem with making exactly 2NT is that, at IMPs, it is the worst time to be in 3NT. If you are only going to make seven tricks in notrump, the cost of bidding 3NT is not nearly as high after partner opens 2NT.

One thing you'll notice is that the 4333 patterns have a consistently lower standard deviation in the "Average Tricks" column. That means that the number of tricks we can make is more likely to be "closer to" the average. In particular, that means that, when you are in the 8.0 to 8.5 average tricks range, you are more likely to make exactly eight tricks in notrump, which is precisely the worst situation for bidding 3NT over a 2NT opener.

Let's look at two rows from the "One Queen, Two Jacks, Three Tens" table:

The second row makes 3NT about 1.7% of the time more than the first row but it is 8% less likely to go down in 2NT. That means that the expected IMPs score for bidding 3NT, non-vulnerable, is higher for the first row than it is for the second.

In a few cases, a 4333 pattern is the "best" hand to hold in these tables. When holding four tens and three or four jacks, you prefer to hold 4333 pattern. Same with holding a queen and four jacks. These cases have a few things in common: Few, if any, entries, so holding a long suit is not useful as a source of tricks. Without a source of tricks, what is really useful is (1) Helping partner stop suits, and (2) helping partner set up tricks in his hand. Both of these are aided best with the 4333 pattern.

Honors in doubletons

Obviously, honors in doubletons are bad, but sometimes it is better to have the queen in the doubleton than the jack. For instance, when dealt a queen, two jacks, and three tens, the worst hands you can have are QJTxx-Txx-xxx-JT and QJTxx-Txx-Txx-Jx. That means that having the queen in the doubletion - even the QJ in the doubleton - is better than these two hands. Why is that?

My guess is that the queen doubleton is more likely to aid partner in running her suit. When your entries are few-to-none, the advantage of your hand is to help partner set up tricks in his own suits and stop suits. In the case of helping partner with stoppers, holding the queen in a long suit is a duplication of values - the long suit is already a stopper.

Three Jacks, Three Tens

Three Jacks, Four Tens

Four Jacks

Four Jacks, One Ten

Four Jacks, Two Tens

Four Jacks, Three Tens

Four Jacks, Four Tens

One Queen, One Jack, Three Tens

One Queen, One Jack, Four Tens

One Queen, Two Jacks

One Queen, Two Jacks, One Ten

One Queen, Two Jacks, Two Tens

One Queen, Two Jacks, Three Tens

One Queen, Two Jacks, Four Tens

One Queen, Three Jacks

One Queen, Three Jacks, One Ten

One Queen, Three Jacks, Two Tens