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## [Bug-gnubg] Interesting article on EMG

 From: Massimiliano Maini Subject: [Bug-gnubg] Interesting article on EMG Date: Mon, 3 Dec 2007 16:52:50 +0100

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
+-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):

Case 1)
Cube analysis
2-ply cubeless MWC  70.11% (Money:  +0.604)
0.802 0.000 0.000 - 0.198 0.000 0.000
Cubeful equities:
1. Double, pass         75.08%
2. Double, take         80.18%  (  5.10%)
3. No double            74.17%  ( -0.91%)
Proper cube action: Double, pass
Cube analysis
2-ply cubeless equity  +0.604 (Money:  +0.604)
0.802 0.000 0.000 - 0.198 0.000 0.000
Cubeful equities:
1. Double, pass         +1.000
2. Double, take         +1.407  ( +0.407)
3. No double            +0.928  ( -0.072)
Proper cube action: Double, pass

Case 2)
Cube analysis
2-ply cubeless MWC  65.14% (Money:  +0.604)
0.802 0.000 0.000 - 0.198 0.000 0.000
Cubeful equities:
1. Double, pass         75.08%
2. Double, take         80.18%  (  5.10%)
3. No double            73.39%  ( -1.69%)
Proper cube action: Redouble, pass
Cube analysis
2-ply cubeless equity  +0.604 (Money:  +0.604)
0.802 0.000 0.000 - 0.198 0.000 0.000
Cubeful equities:
1. Double, pass         +1.000
2. Double, take         +1.203  ( +0.203)
3. No double            +0.933  ( -0.067)
Proper cube action: Redouble, pass

Case 3)
Cube analysis
2-ply cubeless MWC  60.20% (Money:  +0.604)
0.802 0.000 0.000 - 0.198 0.000 0.000
Cubeful equities:
1. Double, pass         75.08%
2. Double, take         80.18%  (  5.10%)
3. No double            73.10%  ( -1.98%)
Proper cube action: Redouble, pass
Cube analysis
2-ply cubeless equity  +0.604 (Money:  +0.604)
0.802 0.000 0.000 - 0.198 0.000 0.000
Cubeful equities:
1. Double, pass         +1.000
2. Double, take         +1.136  ( +0.136)
3. No double            +0.947  ( -0.053)
Proper cube action: Redouble, pass

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:

Case                MWC                EMG
1                -5.10%        -0.407
2                -5.10%        -0.203
3                -5.10%        -0.136

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.