you can find here an interesting article
on EMG (Equivalent to Money Game) equities.
It maylook complicate, but its introduction
explains clearly what EMG equities are.
Recap of the problem (scores are X-O):
GNU Backgammon Position
Match ID : MAGgADAAEAAA
| O O
O O O | |
| 3 points
O O O | |
| On roll
O O O | |
O O | |
O | |
|^ 5 point match (Cube: 1)
X X | |
X X X | |
X X X | |
| X X
X X X | | X
X | 2 points
For the above position, let's imagine
the 3 following cases (scores are X-O):
1) 3aw-2aw, X is doubled to 2
2) 3aw-3aw, X is doubled to 4
3) 3aw-5aw, X is doubled to 8
In all the cases, the match is at stake
in this game (in case 1, X has an automatic redouble
next turn). So, the choice is to pass
and go 3aw-1aw (in all cases) or to play for the match
from the current position (and a dead
cube). GnuBg eval are (in MWC and EMG, g11 MET):
This is pretty disturbing, since all
the 3 errors are identical.
What's happening is explained in J.Bagai's
paper: the problem is, to me, that computing
EMGs we are extrapolating the linear
approximation given by the two points [MWC for a single
win, +1] and [MWC for a single loss,
-1]. In all the cases above, taking would give a MWC
which is outside the interval (hence
we extrapolate instead of interpolate).
I've made a suggestion which wouldn't
be too complicate to put in place:
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
NE (Normalized Equities)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 NE.
It's like having a different interpolation
depending on the magnitude of the error you're
trying to normalize.
- 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
- 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 solves the issue above: all the 3 errors wil have the same normalized
for "small errors" (leading to MWC that are in the interval [single
loss, single win]),
my suggestion would return the good old EMG.
it's no longer linear: if a -X% MWC error corresponds to -Y normaliwed
a -a*X% MWC error does not necessarily correspond to a -a*Y normalized
will be true for small errors, but not for large ones.
I don't think we really care about
linearity. The goal is to compare errors magnitudes at
different match scores. In fact, my
suggestion introduces some non-linearity inherited by
the intrinsically non-linear behavior
of MWC in match-play.
Apparently D.Zare already discussed
the topic (GammonViallage, July 2006) and made another
suggestion: "Adjust the errors
by the ratio of the size of the error of misplaying an opening
3-1 8/4 for money play and and at the
match score [and cube situation]."
Very interesting too, but it misses
one nice property of EMG (and of my suggestion): a
normalized equity of -1 corresponds
to a borderline take/pass.
* Anybody with comments on all that
* Would it be possible to have the
3 methods in gnubg ?
In the hint panel (for example), there's
a "MWC" button that alternates between EMG and MWC:
we could have 4 "radio buttons"
(only one of them pressed at a time) for MWC, EMG, NE1 and NE2.
I think it would be interesting to
play around with them ...
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[Bug-gnubg] Interesting article on EMG,