So it is posted on the yahoo.com CCC also. I appologise to the PBers for this, but it was simply too complex in more nice versions. The labels are more inline with ML's originals and ML is still to be congratulated for finding a non-Graham-Stokes proof of the bow effect even as his proof is flawed by the problems I give below:
I tried to develope a nice version of this post and could not. It
simply got too complex in nice versions. Perhaps Ted Forester or T-
Hopper can put it gently and still have it make sense. I either have
to base this on ML's recent post on bj21.com and evidence of clone
prevaricvation or rewrite everything.
First the two significant Theorems of ML's post:
Basic Strategy expectation is invariant for the original size of the
shuffled original pack of cards--and the rules etc.--with
penetration. Proof: any penetration can be transformed into any other
by simply cutting the pack.
The mean for all possible subsets of a given pack is the same as the
mean of that given pack.
ML took these two --and then strangely added a third not in his
original post in a reply to me--and demonstrated the bow effect
without needing to appeal to the Graham-Stokes formula. From that
formula the true count and actual edge cover the same range of
possible pack combinations but the true count has less accuracy. The
integral of their differences must thus involve equal areas of less
and more predicted edge.
ML took a different and unique approach of stepwise summation of the
In either case the bow effect curve of the true count versus actual
edge is just the required curve needed for the true count
underestimating edges with TCs near zero, and overestimating the edge
at more extreme TCs, and getting more bent with deeper penetration.
But the application by ML of Theorem II is incorrect and even
dishonest. Theorem II applies to the actual subsets and not the
perfect mean subsets of each True Count or other pack composition. ML
did not apply this theorem continuously.
The proper equation, for how such edges average out is: E(m,i)=edge
for the perfectly mean pack(s) m; E(m,a)=edge for actual packs:
E(m,a)=average summa subsets E(m,a)
ML's version is to substitute E(m,i)=average summa subsets E(m,a)
ML gives a very good, non-Graham-Stokes, proof for how E(m,i)>average
E(m,a), which points to dishonesty in the following (this is where I
had to get blunt to stay simple and clear) in substituting E(m,i) for
E(m,a), in that this is only true for pack compositions where
composition E(m,a)= composition E(m,i), which is very rare except for
the original pack in that the original full pack is the only
composition E(m,i) certain to occur.
The rarity of composition E(m,i)=composition E(m,a) is a binomial
distribution for predicting the probability that a subset will be
some perfect mean density. For blackjack this involves a reduction in
probability of E(m,i) by approximately 13!/10! for every level of
penetration where a perfect mean is possible. Thus every possibility
of a perfect mean is reduced in probability by 1/1716, at every such
Don Schlesinger used a deep knowledge of the Graham-Stokes equation
for years, on Wall Street, to weed out bad predictive models for
market trading--presumably sucessfully. ML was able to demonstrate
the bow effect without G-S being invoked. They know what they are
doing. It is simply too far-feteched for such skilled mathematicians
to fail to apply Theorem II continuously. Every subset has to be
considered the same as the original pack set, where the claim that
edges average summa E(m,i)=edges average summan E(m,a) simply smells.
The distribution of subserts is a distribution from prior actual
subsets and cannot and does not involve substitution of perfect mean
subsets at the current penetration levels.
Donald Schlesinger knows this. Seek out a publication from Morgan-
Stanely, variously titled over the years, two past titles are, Where
Your Portfolio Is, and Where Do You Go From Now?, which were largerly
written by Don for private clients. That publication advised private
clients in how to decide on changing their strategy goals, and to
only include past results if they involved using the same strategies
in the past. ML and DS are violating these principles when they jump
from current situations to ideal starting points for a distribution.
It simply adds to the evidence that they know better than their
The ML post is thus flawed and dishonest, despite its success in
achieving a seperate derivation of the bow effect, by changing
distributions in this way.