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BGonline.org Forums
Paper on EMG inconsistencies
Posted By: MaX In Response To: Paper on EMG inconsistencies (Jeremy Bagai)
Date: Tuesday, 27 November 2007, at 9:58 p.m.
Hi Jeremy, good work ! I've always been under the impression that something could be wrong in the EMG computations but, lazy as I am, I never took the time to take a close look. Now, with your example at hand, I think I have something.
WARNING for non-mathemagicians: please read on !! You'll see that what I'm proposing is really easy.
As Jeremy has clearly explained, EMG are nothing more than a linear interpolation between two points, namely the MWC at the score given by a single win (EMG = +1.0) and the MWC at the score given by a single loss (EMG = -1.0), taking into account the current cube value.
In all the cases Jeremy has presented, the problem comes from the fact that if 3-away takes, its MWC are 19.822% and this point is *outside* the interval used to do the normalization since its lower bound is 24.923%MWC (EMG -1, at -3,-1) for all the 3 cases. Mathematically speaking, the computed EMG is, in this case, an extrapolation.
Now that's what was perturbing me (in a subliminal manner though): interpolating inside the interval is OK, but extrapolating outside the interval is *clearly wrong*. Actually, I should say it's bad instead of wrong, since there's no clear definition of what a correct normalization is. However, if the goal of normalization is to help comparing mistakes, then extrapolating can be very bad (as shown by Jeremy).
So, here's my submission to the NNE (New Normalized Equity) contest:
1- let's call W1/2/3 (L1/2/3) the MWC at the scores of a single/gammon/backgammon win (loss) respectively. They are associated to NNE of +1/2/3 (-1/2/3) respectively. The six points [L3,-3], [L2,-2], ... , [W3,+3] form a poly-line with 5 segments (at most, at some scores two point may be identical because gammons/backgammons may not count).
2- draw the poly-line, then use it to convert MWC to NNE.
It's like having a different interpolation depending on the magnitude of the error you're trying to normalize.
Three examples:
- I'm leading 3-0 to 5 cube at 1, what can happen ? With a simple/gammon/backgammon win I go to 4-0/5-0/5-0 while with a simple/gammon/backgammon loss I go to 3-1/3-2/3-3.
- I'm leading 4-1 to 5 post-Crawford (I owe the cube at 2), what can happen ? With a simple/gammon/backgammon win I go to 5-1/5-1/5-1 while with a simple/gammon/backgammon loss I go to 4-3/4-5/4-5.
- I'm leading 3-0 to 5 owing the cube at 2, what can happen ? With a simple/gammon/backgammon win I go to 5-0/5-0/5-0 while with a simple/gammon/backgammon loss I go to 3-2/3-4/3-5.
In any of the above situation, just associate the w/wg/wb scores with NNE +1/+2/+3 and the l/lg/lb scores with NNE -1/-2/-3, reads the MWC of the different scores from your favourite MET, put the points on a graph and draw the poly-line (attention: in some cases you have to use post-Crawford METs).
It seems to me that this way to normalize equities:
- solves the problem in Jeremy's 3 examples: in all the cases, a NNE of -2 means that black has lost the match, hence has MWC equal to 0%. So, in all the 3 cases, you would use a segment with extreme points [0%,-2] and [24.923%,-1]: the NNE of MWC 19.822% would be (19.822%/24.923%) -2 = -1.205, which means that passing would be a -0.205 NNE error, in all the 3 cases.
- it's symmetrical for the two players, since the poly-line for player A is just the one of player B with the values on the Y axis (NNE) of opposite sign and the values on X axis (MWC) symmetrical wrt 0%-100%. I agree that symmetry is not mandatory, but when symmetry exist, there's non need to hide it.
- it's somehow intuitive (at least to me)
One final remark: in each interval, is it right to use a linear approximation ? Now that we have multiple points (up to 6), we may use polynomial interpolation, b-splines and other very cool-looking smooth shapes.
My answer: there's no answer. EMG are linear: if an error of -X% MWC has an EMG of -Y, an error of -2*X% MWC has an EMG of -2*Y. With my NNE, this is still true, but only for errors that are in the same interval (the function converting MWC to NNE is piecewise linear, but not linear overall). Since we have lost linearity, any other shape is as justified as the linear one. Problem is no one is more justified than the linear one (at least to me). Let's keep it simple and have it linear. If nothing else, engineers will like it more :)
MaX.
P.S.
I'm an engineer.
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