[orlam]

Wow interesting thread, but so much nonsensical math has been posted...

[gnasher]

mikeh, on Jan 23 2009, 04:36 PM, said:

a typical match sees 2-4 imps being scored per board.

Do you play a lot of junior bridge?

Edit: Sorry, maybe Mike's right: I've just looked at the scorecard for what I thought was a fairly unexciting match last night, and 74 IMPs seem to have changed hands over the last 16 boards.

[whereagles]

Just for the record, I once lost a 20 board international match by 20-10 solely on overtricks.

[jdonn]

gnasher, on Jan 24 2009, 05:05 AM, said:

mikeh, on Jan 23 2009, 04:36 PM, said:

a typical match sees 2-4 imps being scored per board.

Do you play a lot of junior bridge?

It seems he simply plays good bridge.

In the last Bermuda Bowl, these were the average imps scored per board in the knockout phase:
Quarterfinals: 3.6, 3.3, 3.9, 3.3
Semifinals: 4.1, 5.4
Finals: 4.4
The average of those (rounded) totals: 4.00

Venice Cup:
Quarterfinals: 3.4, 3.6, 4.3, 3.6
Semifinals: 4.3, 5.1
Finals: 4.1
The average of those (rounded) totals: 4.06

It looks like you were correct to suggest Mike was off on his estimate, but because it was far too low, not too high. Not 2-4 imps a board, more like 3.3-5.4 imps a board.

Contrary to popular prejudice, juniors are far more sane, as they range from 3.2-4.7 imps a board.

World Mind Sports Games under 28 event:
Quarterfinals: 4.7, 3.2, 3.6, 4.2
Semifinals: 3.7, 3.7
Finals: 3.6
The average of those (rounded) totals: 3.81

But those were the old juniors, I should be fair and examine those crazy young juniors instead!

Under 20 event:
Quarterfinals: 4.1, 3.5, 3.4, 3.8
Semifinals: 4.5, 3.7
Finals: 4.2
The average of those (rounded) totals: 3.89

Oh, never mind...

[Rossoneri]

mikeh, on Jan 23 2009, 03:36 PM, said:

Assume that in a long match, the situation arises 6 times.. where the odds seem to be 11-1 or 12-1 in favour of the overtrick. I don't remember enough math to work out the probability that you will lose one of these plays if you go for every one of them, but I think that it will be significant.. if it happens, then even if you get every other one right, you can't make up the loss of 10 imps when the play fails... but if the match were infinite, you could. Jlol's analysis would then be valid.

In the meantime, the 4 or 5 or 6 imps you pick up by 4 or 5 or 6 gambles are, usually, on the level of noise in the totality of the imps scored.. a typical match sees 2-4 imps being scored per board.

I remember a particular time more than 1 year ago when I was in camp doing duty one day and I thought of something along these lines.

My assumptions were:
1) You either make the overtrick, or go down
2) Assume this decision happens every board
3) This is an 8-board match

Given the fact that it was nearly 2 years since I had stopped doing maths at that point in time, I attempted to calculate this using a binomial model. My conclusion then was that the payoff of an imp wasn't worth it.

The paper on which I did my workings that day has since been lost, but after slightly more than one term of restarting school again, I am now still pretty sure my simplistic model calculations were correct.

[ceeb]

Simply on the question of IMP odds versus VP odds I've done some analysis. In summary, under typical assumptions there's very little difference but to my surprise VP does not encourage risk.

Why is that? Doesn't the VP scale (like the IMP scale) compress large swings? As Helene said, it's sigmoid isn't it? I always assumed so. But a close look shows this isn't a very good approximation.

Take for example the WBF 14-board scale. There are 5 imp differences (-2 to +2 imps difference) that correspond to a VP tie. And 5 imp differences (3-7 imps) that give one more VP. But the next bracket (8-10 imp) is smaller. The sequence of imp bracket sizes for winning 15-25 VP is (5 5 3 4 4 4 4 4 4 4 infinite). Hence the relation between imp and VP is better described as roughly linear but irregular, than concave upward.

As a concrete example I assume the WBF 14-board VP scale, and that the imp consideration is risking 10 imps to gain 1.

Hence in IMP terms it is break-even to try for the overtrick provided the chance of success is 10/11. A few comparisons of this number with the VP situation:

Typical example: you estimate the match state as variously anywhere from even to plus or minus 40 imps with a standard deviation of imp uncertainty (and this includes both uncertainty in guessing the match state as well as volatility of future boards) of 17 imp. Break-even to try for the overtrick is 10.003/11 chance of success when behind to as much as 10.2/11 when far ahead -- you should be microscopically more conservative at VP.

Extreme example: Suppose in the artificial extreme that your estimate of 10 imp lead has a standard deviation of only one imp. Then 7.3/11 chance is sufficient to justify taking the risk at VP.

[skjaeran]

mikeh, on Jan 23 2009, 02:10 AM, said:

Or our opps reach slam.... and make it... they score 1370 and we lose either 740 or 770.. again, the overtrick is irrelevant. Or they go down.. if we score 600 or 630.. no difference, but if we go down when we had 600 cold... we lose big.

740 converts to 12 IMPs, 770 to 13 IMPs....
But that's a minor issue. I agree with Mike.

[dburn]

Jeff Rubens, who is a more than competent enough bridge-playing mathematician to understand these things, has referred more than once to a "parabolic utility function" in respect of overtricks. By this I think he means that you should play for overtricks if the match is either "sufficiently short" or "sufficiently long" to make doing so worthwhile.

Obviously you should take a 51% line for an overtrick as opposed to a 100% line that will always make exactly if the match is one board long - but I wonder: how long does a match have to be so that you should play for "percentage overtricks" (that is, say, taking a 91% line for an overtrick that will gain 1 IMP or lose 10) at every available opportunity?

[pretzalz]

A psychological aspect that I don't think has been mentioned: I never feel more demoralized at the half then for it to seem like we've lost 1 or 2 on almost every board. Most overtricks aren't taking a chance; they are simply better technique or careless defense. I'd rather be down 20 with the dribs and drabs coming our way than down 10 with the dribs and drabs ebbing away from us.

[ceeb]

dburn, on Jan 25 2009, 04:43 PM, said:

Jeff Rubens, who is a more than competent enough bridge-playing mathematician to understand these things, has referred more than once to a "parabolic utility function" in respect of overtricks. By this I think he means that you should play for overtricks if the match is either "sufficiently short" or "sufficiently long" to make doing so worthwhile.

Obviously you should take a 51% line for an overtrick as opposed to a 100% line that will always make exactly if the match is one board long - but I wonder: how long does a match have to be so that you should play for "percentage overtricks" (that is, say, taking a 91% line for an overtrick that will gain 1 IMP or lose 10) at every available opportunity?

I love a paradoxical result as much as anyone, but I don't see where such a result comes from. The behavior of the model that makes sense to me is not so intriguing. Rather --

In a 1-board match, I agree that a 51% chance merits playing for the 1 imp gain at the risk of a 10 imp loss.

However, as the number of boards increases, the necessary odds simply increase monotonically asymptotically to the imp odds. That is, the longer the match, the closer to 90.9090...% must be your chance of success before it's worth risking the contract for an overtrick. I don't see any increase then decline, or the opposite.

My model is as follows. The object is to end the match plus imps, no matter the margin. If the IMPs finish even, assume a coin-flip playoff. Other than the present board, assume the teams are even with a normal probability distribution assigned to the various IMP margins -- i.e. the largest probability for a tie, slightly less for + or - one imp, etc. The spread of this distribution -- i.e. the standard deviation -- is identically 0 for the zero other boards of a one-board match, and the longer the match the broader the spread.

It doesn't require many boards before the threshold percentage to justify risking the contract is within a trivial margin of the imp odds. I.e. at a standard deviation of 10 imps, you need an 89.5% chance to justify gambling for an overtrick.

Charles Brenner

[orlam]

I bet on Charles Brenner against Jeff Rubens.

[Fluffy]

On Knock out matches, since we would assume our team is better than the opponents, we should worry more about 10 IMP swings than about 1 IMP swings, at least at the early boards.

[FrancesHinden]

ceeb, on Jan 26 2009, 03:10 AM, said:

Other than the present board, assume the teams are even with a normal probability distribution assigned to the various IMP margins -- i.e. the largest probability for a tie, slightly less for + or - one imp, etc. The spread of this distribution -- i.e. the standard deviation -- is identically 0 for the zero other boards of a one-board match, and the longer the match the broader the spread.

When you say a 'normal probability distribution', do you mean a Normal probability distribution, or the distribution that you commonly see?

Because empirically the distribution of IMP margins per board is not normal - I did some research on this a while ago, it's at home, but I could look it out again. The chance of a flat board is relatively too high compared to a normal distribution, and it has more than one peak - 7/8 imp swings are rare than some higher swings.

This may not change the conclusion. I just thought I'd mention it.

[ceeb]

FrancesHinden, on Jan 26 2009, 04:45 AM, said:

ceeb, on Jan 26 2009, 03:10 AM, said:

Other than the present board, assume the teams are even with a normal probability distribution assigned to the various IMP margins -- i.e. the largest probability for a tie, slightly less for + or - one imp, etc. The spread of this distribution -- i.e. the standard deviation -- is identically 0 for the zero other boards of a one-board match, and the longer the match the broader the spread.

When you say a 'normal probability distribution', do you mean a Normal probability distribution, or the distribution that you commonly see?

I meant Normal, i.e. Gaussian. However, so long as the distribution is unimodal and symmetric I can't imagine the conclusion would be different.

Quote

Because empirically the distribution of IMP margins per board is not normal - I did some research on this a while ago, it's at home, but I could look it out again.  The chance of a flat board is relatively too high compared to a normal distribution, and it has more than one peak - 7/8 imp swings are rare than some higher swings.

That is interesting. But regardless of the distribution for one board, the Central Limit Theorem seems to say it will be normal in the limit, i.e. arbitrarily close to normal for a sufficiently long match. Is your point that real matches aren't "sufficiently" long? Or that the CLT doesn't apply for some other reason?

[FrancesHinden]

ceeb, on Jan 26 2009, 10:59 AM, said:

FrancesHinden, on Jan 26 2009, 04:45 AM, said:

ceeb, on Jan 26 2009, 03:10 AM, said:

Other than the present board, assume the teams are even with a normal probability distribution assigned to the various IMP margins -- i.e. the largest probability for a tie, slightly less for + or - one imp, etc. The spread of this distribution -- i.e. the standard deviation -- is identically 0 for the zero other boards of a one-board match, and the longer the match the broader the spread.

When you say a 'normal probability distribution', do you mean a Normal probability distribution, or the distribution that you commonly see?

I meant Normal, i.e. Gaussian. However, so long as the distribution is unimodal and symmetric I can't imagine the conclusion would be different.

Quote

Because empirically the distribution of IMP margins per board is not normal - I did some research on this a while ago, it's at home, but I could look it out again.  The chance of a flat board is relatively too high compared to a normal distribution, and it has more than one peak - 7/8 imp swings are rare than some higher swings.

That is interesting. But regardless of the distribution for one board, the Central Limit Theorem seems to say it will be normal in the limit, i.e. arbitrarily close to normal for a sufficiently long match. Is your point that real matches aren't "sufficiently" long? Or that the CLT doesn't apply for some other reason?

It's not unimodal.
It's not necessarily symmetric either, although in the abstract case it is (for some combinations of teams, even of equal absolute standard, it isn't symmetric).

I did say that none of this would necessarily change the conclusion, particularly for long matches (I started to get something resembling a normal distribution after about 10 boards).

However, if you are (e.g.) playing a two-board tiebreak, it changes the odds on the first of the two boards.

[ceeb]

orlam, on Jan 25 2009, 10:42 PM, said:

I bet on Charles Brenner against Jeff Rubens.

Thank you. That's nice to hear. David Burns is certainly right though that Jeff has a very sharp mind about these things, and he is a professional mathematician. There's a good chance that the resolution is misunderstanding of some kind.

[hotShot]

If you leave out stupid misplay, your chances to score in an IM match are:

- collecting overtricks
- playing a higher ranked suit or NT
- bid and make close games and slams
- hope that opps close game/slam fail

The first 2 may not win you many direct IMPs, but they apply pressure to your opps to make a "big" score. The pressure increases in short matches and towards the end of a set/match. Your opps will have to take greater risks to (over)compensate your lead.

So the question is:
Do I bet on a close game with 42% chance and hope that opps fails, or do I bet on several 91+% chances.

Remark:
The ♣ suit can not split 2-2 according to opps bidding. So the possible splits reduce to 3-1 and 4-0. This reduces the chance to make the overtrick to 83% and the chance to make 3NT at all to 91% (4-0 wrong sided).

[ceeb]

FrancesHinden, on Jan 26 2009, 06:13 AM, said:

ceeb, on Jan 26 2009, 10:59 AM, said:

But regardless of the distribution for one board, the Central Limit Theorem seems to say it will be normal in the limit, i.e. arbitrarily close to normal for a sufficiently long match. Is your point that real matches aren't "sufficiently" long? Or that the CLT doesn't apply for some other reason?

It's not unimodal.
It's not necessarily symmetric either, although in the abstract case it is (for some combinations of teams, even of equal absolute standard, it isn't symmetric).

I did say that none of this would necessarily change the conclusion, particularly for long matches (I started to get something resembling a normal distribution after about 10 boards).

However, if you are (e.g.) playing a two-board tiebreak, it changes the odds on the first of the two boards.

Ok, so the point is that we may have to consider the short run.

[ceeb]

Fluffy, on Jan 26 2009, 03:49 AM, said:

On Knock out matches, since we would assume our team is better than the opponents, we should worry more about 10 IMP swings than about 1 IMP swings, at least at the early boards.

For sure. Suppose team A is better by 1/4 IMP per board. Guessing that the variance is about the same, for a 16 board match the better team wants 22:1 odds to gamble 10 imps for 1, but the weaker team will take that gamble at 4:1.

Over 64 boards the better team, though a mere 16 IMP favorite, needs over 1000:1 assurance to gamble for an overtrick. That conforms to the convention wisdom of don't waste your time. For the weaker team, 6:1 is enough -- interesting in that this line of thought begins to speak to the question of strategy for beating a better team.

(I'm still using my simple-minded model of imp uncertainty because it's easier.)

Charles

[pretzalz]

What about this hypothetical. Say every board you played you had a 93% line for a 1 IMP gain/10 IMP loss and a 100% line for a push.
length of match ------------------------- odds of winning
1 board 93%
2 boards 86%
3 boards 80%
4 boards 75%
5 boards 70%
6 boards 65%
7 boards 60%
8 boards 56%
9 boards 52%
10 boards 48%
11 boards 45% ***TIE == 37%
12 boards 80%
13 boards 77%
...
21 boards 56%
23 boards 78%
I know this is wildly unrealistic, but does it have no relevance whatsoever?

Travis