[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]

JLOL, on Jan 28 2009, 06:07 PM, said:

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.