Re: Clarke Cant on Floating Advantage cont.
Postulate IIb: A cards left penetration count ends in zero and is
subject to the TCT as well. Anytime you specify the start of a round
you specify a penetration point for a shuffled pack of cards.
Penetration cannot have any effect on a strategy that is optimal for
a full pack of cards in that penetration not only is subject to the
TCT. and on average does not change, but the true count of the
penetration, cards left, number absolutely never changes. Axiom V
applies meaning that the effect of setting a boundary for the start
of a round of cards does not change anywhere with penetration.
Penetration by itself cannot change basic strategy edge or basic
strategy recomendations.
Postulate IV: A balanced rank count of all tens, versus all other
ranks is well accepted to result in an underestimate of edge at its
extremes, in that such a count has all pushes at its all tens
extreme, which is all 20 to 20 pushes with the dealer.
A balanced rank count of all aces versus all other ranks will result,
at its all aces extreme, in a player loss in that most rules do not
allow resplitting or rehitting split aces. Thus the ace count
underperforms at its extremes.
A balanced count of a single low card and all others will
underperform predicted edge when it is at its all low card extreme in
that the player will spit and stand on lower totals than the dealer,
with the exception of all 7s, where the player and dealer will push
with 21 totals, the player having split to 4 hands.
A neutral count, especially with more practical counts, implies a
pack that resembles a pack that started with less decks, as
penetration increases.
In all of the above, such bow effects increase with penetration.
As per Axiom II, every practical count that is intended for actual
use has to be a combination of the above, and exhibit the sum of such
characteristics.
Thus the general characteristics of the bow effect have been proven
without any Postulate (I,III,IV) being shown to necessarily modify
Postulate II.
The following Cases refute challenges to the above statement:
CaseI: The most expected composition at any penetration level (though
any penetration level also means predictions involve fractions of
cards compositions and not just integer numbers of cards) matches the
most expected composition, at that level, for a true count =0.
Doesn't this imply that since the edge for a True Count of
0, "floats" upward with penetration, that the basic strategy edge
does as well?
Answer: A TC=0 is a statement that excludes other TCs, where some
have postive bow effects and the extreme TCs have negative bow
effects. Case I does not modify Theorem II in that the sum of all of
the excluded bow effects at those other TCs is allowed to balance the
postive bow effects at TC=0. Imbalance must be demonstrated to
require this modification.
Case II: if basic strategy edge does not change over all TCs, doesn't
the behavior of all likely TCs, as penetration increases, still
require some rise in basic strategy edge?
Answer: Bow effects, as proven in Postulate III, included any
balanced count, and can be postive or negative as per Axiom IV. The
flaw in the Case II objection is that the effects of extreme counts
is being excluded on their probability alone and not the product of
their probability and effect when observed, and it is ignored how
such effects can involve a main count near middle ranges, and
necisarily included side counts, as per Axiom IV, that may be well
into such negative bow effects ranges. Case II is a pure attempt to
evade Axiom IV, by attempting to exclude more extreme count ranges.
Case III: Simulations still appear to show increases in basic
strategy edge with increased penetration. What gives?
Answer: Simulation results are typically reported with several layers
of rounding in how observed true counts are grouped with similar
groups of observed true counts at different penetrations. Such
reports, or reporting modules of code frozen within the popular
simulators, ignore how the true count is a discrete number, and how
the average true counts that are possible, within any rounding range,
creap upward as penetration increases.
Cut card effects are present also that are near universally
overcorrected, and because cut card effects increase with
penetration, such adjustments result in the appearence of more edge
with more penetration for basic strategy playing decisions also.
Case III: is a simple example of equating apples to oranges in two
areas.
Case IV: Sometimes a composition will be formed at random that is the
same as starting out with less decks of cards initially. Gotcha?
Answer: This is a highly improbable event, but it is a limited
exception. There is otherwise a collection of true counts, required
by Axiom II, that have conserved true counts, where there are
pertibations in composition and edge that are only possible from
those subsets originating with a larger initial pack. But this
exception is also overvalued in a reverse of Case II fallacies.
In all of the above the idea of a strong floating advantage is a
stubborn one in that the fallacies decribed in Cases I thru IV often
are combined. They arrise simply because most people are too trusting
or too compartmentalized in their thinking to reason through them.
Notes:
Griffin omitted any sort of Axiom IV in his discussion, in Theory of
Blackjack, of a regression function operating with penetration for
the changes in compostion as a pack depletes.
ML, in his paper posted on bj21.com and bjmath.com, made intersection
of means errors, as shown in Case I and Case II, by omitting any sort
of Axiom IV, in step 4 of his "proof" claiming that the bow effect
required modification of what is given here as Postulate II.
Don Schlesinger, in Blackjack Attack, both editions, used a squeezed
balloon analogy, for his version of step 4(ML)/Postulate III(myself)
that is based upon the Graham-Stokes result for the topomorphic
properties of the true count prediction of edge and the actual edge.
His version of error, in claiming this need to modify Theorem III, is
different than ML's. His derivation misses how what is set here as
Postulate II is a boundary condition for the Graham-Stokes result to
be applied and is not modified.
Those who accept simulation reports rather blindly usually make Case
III errors.
This started an an alternative proof of the True Count Theorem, that
was requested by Rob McCarvey, but the alternative proof of the TCT,
allowed a new simple proof of ML's step 4, that did not involve his
Case I and Case II errors, and proceded as a necessity from that. I
appologise for making a new post on this topic, after everyone has
been justifiable bored by it, in past posts by myself and ML, but a
friend's request and a need for completion compelled me. Woggy's
request to, "avoid the science fiction" that results from including
partition theory, led to adding Axiom V, to the proof that more edge
variance is involved with deeper penetration, but this also led to
the proof of Postulate IIb, that should forever bury the idea that
penetration modifies basic strategy in any way. I hope there are no
missunderstandings for doing this.