Monday, December 29, 2008

Pressure (Warning: contains math)

I'm often asked the main differences between club-level bridge and high-level bridge. My answer's always the same: "Pressure." Expert opponents will attempt to give you a problem on every hand.

Here's an example:

You're West, holding

AQT
Q65
QJT
T954

In an all-expert IMP game, you deal (nobody's vulnerable). North opens 1H. Partner (former junior standout and current all-around great guy Charlie Garrod) leaps to 3S. South thinks for a bit and bids 4H. What's your call?

First of all, how many defensive tricks do you have? I'd say right around one and a half. Most of the time the opponents' spades will split 2-1, and the diamond holding will usually be worth a trick. Once in a very great while, you'll be able to take two spade tricks-- remember, partner's under 30 and won't always have a seven-bagger for a nonvul three bid.

Next, is it safe to bid? Well, I'd argue more that it's not safe to pass! I think we have seven spade tricks and a diamond for at worst -300 in four spades doubled. So it's certainly right to bid.

But here's the pressure part. When he held this hand against me, Stan Schenker made the gutsy call of 5S, forcing the last guess on his opponents!

We guessed correctly -- 5S went for 500 when we could only make 5H -- but let's look at the mathematics of the situation:

Let's say that the opponents will double 5S 80% of the time, and bid a failing 6H 20% of the time. So, 20% of the time we win 11 imps for +50 opposite +450. The other 80% of the time would be split up like this:

3/4 of the time, 5S will go for 500. We'll lose 2 imps for this.
1/8 of the time, partner has a better hand than we expected and will only go for 300. We now win 4 imps.
1/16 of the time, partner has a worse hand than we expected-- maybe a six-card suit. We go down 800, and lose 8 imps. Ouch!
1/16 of the time, partner has a real dog. They defend well and we go for 1100! Lose 12! Oof!

So here's the whole table:

20% win 11
60% lose 2
10% win 4
5% lose 8
5% lose 12

Put all these wins and loses together and the expected imp gain is plus 0.4 imps! This means that this action could work out well, it could work out poorly, but in the long run, it's a winning action.

The percentages I used to weight the actions are totally subjective, and only a product of my at-the-table experience. If you think that the numbers are wildly different, plug them in to a spreadsheet and play around with them. You can download the Excel spreadsheet I used for this exercise by clicking this link.

Two-fifths of an imp may not sound like that much, but if you make sure all of your bids and plays have a positive imp expectancy, you'll find yourself winning most of your matches.

Nice bid, Stan!

2 comments:

woodsythefriendlyowl said...

So you make a good point that in the long run it's worth your while to sustain a very likely small minus (down an extra X) for the chance of a large plus (6H going down).

But shouldn't you compare that (admittedly positive) result to result from your other natural action, 4S? Given they are experts, they're likely to go on to 5H (for a push the way you're scoring), but they might double 4S.
Repeating your weighted average of down 1,2,3, or 4 I got 4S* being worth slightly more than 3 IMPs to you. Meaning that if there's a better than 2/15 chance that the opponents will sit for 4S*, then you will gain *more* bidding 4S than 5S.

Stan said...

There are actually a couple of other cases which need to be added to the equation.

In some cases, the opponents actually make 6 hearts. You then need to factor in the times you push them to 6 hearts making, the times you prevent them from bidding 6 hearts, and the various scores relating to 480 rather than 450.