T-Hopper on the CCC questioned part of my claims for the bow-effect versus the usual (or Strong) Floating Advantage. I submit that the ffollowing is the only way that the basic strategy edge can be invariant with penetration--as proved by simply cutting the pack--and how the bow-effect results from an invariant basic strategy edge and an increasingly higher amplitude and distorted prediction in your edge coming from your true count, as penetration increases.
T-Hopper took interest in how seperating a bow-effect from a strong FA could be controversial. (BTW he is always a far better IMHO "devil's advocate" for discussion and far more polite than ML in such roles). The arrows mark T-Hoppers comments; the rest is my reply amplified for here with some new material:
> It looks to me like you just proved the opposite of your
> conclusion.
The perfect mean subdecks follow a Strong Floating Advantge effect (that your edge is the same as if you started with the same number of decks as you have left as you count along...).
The spread of probabilities for all subdecks however, whatever the
count, contains variations that can only come from their origin from
a larger parent deck or pack. Approached from the derivation (1) I
gave for the bow-effect, a contraint to the mean still matching the
basic strategy edge combined with the true count getting more extreme
etc., you have the same bow-effect that you get from examining each
subdeck/subpack, which I gave in (2), which is that the general
effect of variations from a perfect mean subdeck, for any count, is
to raise the edge in middle count ranges and cut the edge in more
extreme ranges. This is clearly seen with hi-lo and varrying the
density of aces and tens versus mean subpacks. If you increase the
aces, you get more splits overall, more profit in them at middle
count ranges, less profit at extreme true counts, and less
blackjacks. Increase the tens instead and you get more 20s and less
blackjacks and the same behavior at middle and extreme TCs. Show these effects with the high cards and that whatever effects there are with low cards still comes to something like the basic strategy edge, with balanced counts, and you demonstrate the same must occur with the low cards.
That has both the bow-effect and the spreading away from perfect
means you get as a deck depletes. Adjust the probabilities of such
deviations from the perfect mean, with the penetration and the rise
in the expectation of the perfect mean is exactly canceled by the
decline in the probability of the mean.
My original post Proof of the Strong and Weak means was to show how the Strongest mean is the mean that occurs with the largest possible pack that can come from a given starting pack, for a given true count, and the original pack, for basic strategy edge, and that the means of all subdecks are going to match that original pack edge. The perfect means for ALLsubpacks are a massively small portion of all possible subpacks. They do not provide any "over-ride" of the overall pack mean.
For ML:
All variations from a perfectly mean pack do not result in a lower edge if you comeup with a pack with the same densities but a larger size. They very very often result in a higher edge (Griffin's TOB example of the 8s and 7s packs is actually fairly typical) and the larger pack that has the same skew as a smaller pack is vastly more improbable if you consider how often they occur when drawn from their common larger orginal pack. You tried to pull one over on us and this is not a courtroom, but a discussion board.
Only with these above conditions met is it ever possible to have, an invariant basic strategy edge, and an imperfect true count as a predictor of basic strategy edge, coexist!
