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Have you noticed this before?

Posted By: Chris Yep
Date: Tuesday, 8 March 2016, at 1:21 p.m.

In Response To: Have you noticed this before? (Timothy Chow)

Ok, I think I see what you're saying now. You're comparing P and O in relative terms rather than absolute terms. On the other hand, you're judging the accuracy of O/(P+O) in absolute terms rather than relative terms.

In my second example (9000-away/1000-away), P and O differ by 8000 in absolute terms, but the ratio P/O is "only" 9, so they're not extremely far away from each other in relative terms.

I agree that if P is much larger than O in relative terms (i.e. the ratio P/O is very large) then the O/(P+O) match winning chance (MWC) approximation is close to the actual MWC in absolute terms; both O/(P+O) and the actual MWC will be very close to zero in this case. However, I don't find this fact very useful. (Of course I also note that you didn't claim it was "useful.") For example, if the O/(P+O) approximation predicts a 0.01% MWC, but the actual MWC is 0.0000000001%, I consider this a huge error. In other words, I believe that considering relative errors (instead of absolute errors) is a better measure of a formula's accuracy.

Note that when P/O is very large, Alireza's dead-cube takepoint formula (for initial doubles) predicts a dead-cube takepoint of approximately 50%, i.e. P/(2(O+P-1)) will be approximately equal to P/2P = 50%. But this is way off, even for lopsided match scores. Similarly, when P/O is very small, Alireza's dead-cube takepoint formula (for initial doubles) predicts a dead-cube takepoint of approximately 0%. But this is also way off, even for lopsided match scores.

Another way to think of it is that when P/O is very large, the actual formula for calculating a dead-cube takepoint, Risk/(Risk+Gain) involves very small numbers (both "Gain" and "Loss" are very small numbers). But if "Risk" and "Gain" are very inaccurate compared to each other in relative terms then the Risk/(Risk+Gain) calculation will also be very inaccurate.

So, I guess my final conclusion is that I don't disagree with you (if we judge the O/(P+O) approximation by its MWC error in absolute terms), but I just want to point out that judging the accuracy of O/(P+O) in relative terms is usually more useful in my opinion.

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