IMP tactics

[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.

[Finch]

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]

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.

 

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?

[Finch]

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.

 

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]

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]

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]

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

[ceeb]

I see. The chance of winning goes up for a while then comes down. However it plunges down then immediately reverses back up (especially if you count a tie as a 50% chance for a win, which I think makes sense), so it's more a sawtooth than a parabola.

 

Too, the sawteeth continue. Plunging occurs at roughly 21 boards, 32 boards ... and in the long run of course the graph smooths and rises approaching 100% chance of winning. So I wouldn't bet that's what Jeff Rubens had in mind.

 

Besides as you say it's unrealistic. The right point of view is what Stephen Tu said -- we only need to consider this choice arising once in a match, and we can ask the expected result over a long series of matches.

 

Charles

[helene_t]

I couldn't find a 7 board table so I use the 8 board table, 30 VPs to divide, maximum 25. Hopefully it doesn't matter too much. Further assume the assumption of 93% chance of a 1 IMP gain and 7% chance of a 11 IMP loss is correct.

 

If the remainder of the boards are a tie, a one IMP win will give us 15 VP while an 11 IMP loss will give us 12, while we get 15 if we play safe. That is a loss of 0.07*3 = 0.21 VP on average. In the same matter, the average gain from going for the overtrick can be computed for each putative IMP score for the remaining boards:

-20.00 -0.28

-19.00 -0.28

-18.00 0.72

-17.00 -0.28

-16.00 -0.28

-15.00 0.72

-14.00 -0.28

-13.00 -0.28

-12.00 0.72

-11.00 -0.28

-10.00 -0.28

-9.00 0.72

-8.00 -0.28

-7.00 -0.28

-6.00 0.72

-5.00 -0.28

-4.00 -0.28

-3.00 -0.21

-2.00 0.72

-1.00 -0.28

0.00 -0.21

1.00 0.72

2.00 -0.28

3.00 -0.21

4.00 -0.21

5.00 0.72

6.00 -0.21

7.00 -0.21

8.00 0.72

9.00 -0.28

10.00 -0.21

11.00 0.72

12.00 -0.28

13.00 -0.21

14.00 0.72

15.00 -0.28

16.00 -0.28

17.00 0.72

18.00 -0.28

19.00 -0.28

20.00 0.72

 

Assuming all the 41 putative scores are equally likely (while we cannot be leading or trailing by more than 20 IMPs), the average gain from going for the overtrick is 0.05.

 

Break even happens at 91.6666...% or 11/12. This happens to be exactly the same as the break even for the expected IMP gain. You can get slightly different results by making different assumptions wrt to the score for the remaining boards. For example, if we take -30 to +30 instead, the expected gain from going for the overtrick is only 0.026, while break even is 91.304%.

 

Is the conclusion different if we assume we are already leading? If I take a range from 0 to 20 IMPs, the average gain from going for the overtrick is 0.076666... If I take 0 to 50, it becomes 0.005294118. Maybe some trends could be found if I seek long enough, but to be practical: forget about this VP thing, maximizing expected IMPs is fine.

[helene_t]

Oops -700 is a 12 IMPs loss, not 11.

 

Anyway, a (maybe) more interesting question is whether optimizing the raw score is a good surrogate for optimizing IMPs.

 

If we assume the other table has -600, we have 93% chance of +1 IMP and 7% chance of -12 IMPs which is 0.09 on average. Break-even is 12/13=92.3% while raw score optimizing would give a break even of 700/730 = 95.9%. For alternative scores at the other table, the average IMP gain is

-1100 0.65

-800 -0.63

-630 0.16

-600 0.09

-500 -0.12

-200 -0.19

-140 -1.12

-110 -0.12

-100 -1.12

100 -0.84

110 -0.84

140 0.16

200.0 -0.7

300 -0.63

500 -0.42

 

So although the IMP gain was positive under the assumption that the other table will score -600, for most alternative scores at the other table, it is better to play safe.

 

So it seems that it does matter what the score is at the other table. I must admit I never consider this when playing teams. I thought the advantage (or disadvantage, depending on your taste) of IMPs versus matchpoints is that you don't have to worry about the field ....

[JLOL]

Helene, I bet you knew at least intuitively that if you are in a game the other table won't be in at least some reasonable %age of the time you should be less apt to risk that game for an overtrick.

[helene_t]

Helene, I bet you knew at least intuitively that if you are in a game the other table won't be in at least some reasonable %age of the time you should be less apt to risk that game for an overtrick.

Well although the results are not surprising I must say I just play safe at IMPs (or at least try to :) )

 

Actually my calculations are not quite realistic since the score at the other table may be correlated to the one at the other table. It could happen that both declarers go for the overtrick and one makes it while the other goes down, but more often than not they will both make it (when splits are friendly) or both go down. I thought of doing similar calculations for the problem of aggressive vulnerable game bidding but that gets even more complex as the alternative score (140 or 170) will be correlated with the game score (620 and -100 as well) so I will leave it for now.