> Hi all;
> Hi Max,
>
> Thanks for providing the link to my paper:
> >> Hi all,
>> you can find here an interesting article on EMG (Equivalent to
Money
>> Game) equities.
>>
>>
http://www.fortuitouspress.com/emg.html >
> I had posted to GammonU and was about to post here but you beat me
> to it. Thanks also for your introduction and summary of the
issue.
> I hope people on this list get interested in the problem.
>
> I'm curious about your poly-line suggestion. You wrote:
>
>> 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. >>
>>
> Can you please provide a full numerical example of your proposal?
> You say that "it solves the issue above: all the 3 errors
wil have
> the same normalized equity," but you don't say what that equity
will
> be. Can you show the calculation?
Hi Jeremy, I will take your example from the beginning,
to make this post self-contained. I'll use G11 MET, like in your article, to get the
very same figures.
You presented 3 situations where black faces a take/pass
decision:
Situation A: black
is led 3away-2away and is doubled to 2 Situation B: black
is 3away-3away and is redoubled to 4 Situation C: black
leads 3away-5away and is redoubled to 8
In all your 3 cases, black has cubeless GWC of 19.822%
(no gammons, no backgammons), and the cube is dead after this decision (in situation
A black has a mandatory double the next turn). White having doubled, the alternatives
for black are:
- pass and go
3away-1away (crawford), with 24.923% MWC - take and play
for the match with 19.822% GWC, i.e. 19.822% MWC
Hence, the error in all three situations is (19.822%-24.923%)
= -5.101% MWC.
Now in order to normalize this, the usual procedure
(EMG) is the following:
- draw the line passing via the points (ML,-1) and
(MW,+1) where ML and MW are, respectively, the MWC of a simple loss and
win - use the drawn line to obtain the EMG of the decision
If you put on X axis the MWC and on Y axis the EMG,
the equation of the line passing via the points (ML,-1) and (MW,+1) is:
Situation A (black is led 3away-2away and is doubled
to 2): - a simple loss
will give black a score of 3away-1away, or 24.923% MWC = ML - a simlpe win
will give black a score of 2away-2away, or 50.000% MWC = MW Situation B (black is 3away-3away and is redoubled
to 4): - a simple loss
will give black a score of 3away-1away, or 24.923% MWC = ML - a simlpe win
will give black a score of 1away-3away, or 75.077% MWC = MW Situation C (black leads 3away-5away and is redoubled
to 8): - a simple loss
will give black a score of 3away-1away, or 24.923% MWC = ML - a simlpe win
will give black a win in the match, or 100.000% MWC = MW
Substituting the ML and MW values in the equation
and computing for x = 19.822% MWC you will find the following EMG:
Sit. A:
EMG(Take) = -1.4068 ==> EMG error = -0.4068 Sit. B:
EMG(Take) = -1.2034 ==> EMG error = -0.2034 Sit. C:
EMG(Take) = -1.1359 ==> EMG error = -0.1359
The above is exactly what you exposed in your article.
My point is that, in all three cases, the take decision is so bad that the corresponding
MWC are outside the interval used for the normalization. In fact the 3 cases have
the same left end-point for the normalization interval (MWC of a simple loss = 24.923%
in all cases) while the MWC of a take are 19.822% < 24.923%.
The idea is then to use another interval, the one
with end-points MWC of a double loss (or gammon loss) and MWC of a simple loss, associated
respectively to -2 and -1 "new normalized equities" (NNE). Notice
that the rightmost point of this new segment is the leftmost point of our original EMG segment.
In all 3 cases, if black loses a gammon *at the current
cube level* (i.e. 1 in situation A, 2 in B and 4 in C), black loses the match. This
means that, in all three cases, the interval used for the normalization is given by the
two points: - 00.000% MWC,
-2 NNE - 24.923% MWC,
-1 NNE
Since the segment is the same in all three cases,
the NNE of the take will be the same. To compute its value you need to write the equation
of of the line pasing via the points (MLg,-2) and (ML,-1), where MLg are the MWC after
a gammon loss:
For ML = 24.923%, MLg = 00.000% and x = 19.822%, the
NNE is (in all 3 cases) -1.2047, which says that the normalized error wrt a pass is
-0.2047.
I've attached a plot of the poly-line in the 3 cases
(X and Y axes are swapped, I have MWC on Y and normalized equities on X, because Excel sucks):
this graphically represents the computations.
NNE.png