Hi all,
you can find here an interesting article
on EMG (Equivalent to Money Game) equities.
http://www.fortuitouspress.com/emg.html
It maylook complicate, but its introduction
explains clearly what EMG equities are.
Recap of the problem (scores are X-O):
GNU Backgammon Position
ID: 6t4LAQDWvQ8AAA
Match ID : MAGgADAAEAAA
+-1--2--3--4--5--6-------7--8--9-10-11-12-+
O: white
| O O
O O O | |
| 3 points
| O
O O O | |
| On roll
|
O O O | |
|
|
O O | |
|
|
O | |
|
|
|BAR|
|^ 5 point match (Cube: 1)
|
| |
|
|
X X | |
|
|
X X X | |
|
|
X X X | |
|
| X X
X X X | | X
X | 2 points
+24-23-22-21-20-19------18-17-16-15-14-13-+
X: black
For the above position, let's imagine
the 3 following cases (scores are X-O):
Case
1) 3aw-2aw, X is doubled to 2
Case
2) 3aw-3aw, X is doubled to 4
Case
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):
As you can see, the error of a take
in terms of MWC is the same in all cases, which is
natural since the 3 positions are essentially
identical. But the EMG is not:
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.
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).
Upside:
-
it solves the issue above: all the 3 errors wil have the same normalized
equity
-
for "small errors" (leading to MWC that are in the interval [single
loss, single win]),
my suggestion would return the good old EMG.
Downside:
-
it's no longer linear: if a -X% MWC error corresponds to -Y normaliwed
equity error,
a -a*X% MWC error does not necessarily correspond to a -a*Y normalized
error. This
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 ...
MaX.
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[Bug-gnubg] Interesting article on EMG,
Massimiliano Maini<=