Proof of the Strong and Weak means---without contradiction.

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!

Clarke is pleasant, knowledgeable and helpful. He kindly shared with me some multi-parameter information at his own volition. The subject matter he discusses is often obtuse, but that's hardly his fault. And he patiently explains and re-explains and does so without remuneration. We should be thankful.

If you check over my Blackjack Therapy manuscript, on bjrnet.com, you will find some of the information you seek on the questions that caused you to reconsider the Floating Advantage.

If you had a computer doing your shuffle tracking you could avoid entirely the question of how come one can treat shuffle tracking subdecks as subdecks with some running count offset in them. You would not have anything like BE and PE to consider, and thus would have no bow-effect, but in this case the subdecks would be subject to something that appears very like the Strong Floating Advantage. Actually this is not the case:

(1) In the BJT ms. is a formula that appears to work in showing how BE can be used to estimate how well a count covers the type of skews that have comeup in regard to the floating advantage.

(2) In shuffle tracking you also have that running count offset for that subdeck.

(3) The formula mentioned in (1) gives you an estimate for how such skewing causes your edge in playing that subdeck to be less than playing a perfect subdeck of this sort, that is averaged. Actually for most subdeck sizes, when you base them on the actual number of shuffle passes and the actual grab sizes, there is no such thing as a perfect mean subdeck. But the formula given does show an estimate for how well a given count performs when, for example, the exspected number of 5s is traded for double the expected number of 4s, and all similar possible skews.

Combined, the shuffle tracking situation and the situation of perfectly mean decks, and their effect on normal penetration can be seperated:

In the shuffle tracking example the weakness of the weaker means is clearly seen---there are enough changes in composition with all the shuffles and washes in between the initial card order and the final results that we don't expect the perfect means to have any effect on the overall mean edge. It is just muddied when normal penetration is involved. Because such means work for estimating indexes and the like we assume they have more influence than they do. Think of shuffle tracking therefore as removing some of the uncertainty, in part, by using your count to track different segments, that parentage, from a larger initial pack, causes too be added to each subdeck size, in normal play, with penetration, by only counting the overall count for all cards removed.

So overall shuffle tracking is a different animal altogether in that simply shuffle tracking removes some of the uncertainty that draws your edge away from the edge that perfect mean compositions would indicate, and back to the mean edge for the entire initial shoe overall, with normal counting tactics.

With the ace and ten variations for a given true count you also have a decrease in the edge overall for that given true count. That is why, when applied to all such variations, even as the Strong Floating Advantage says you should go by the perfect means of the various subpacks, to predict the basic strategy edge, the basic strategy edge, because there are more such variations coming from a large shoe being the original pack and parent to all such subpacks, is pulled by such variations to match the basic strategy edge for the entire original pack. And the same applies to any subdeck composed for a given true count; the edge, at each TC, is the edge for the largest subpack you can compose for any given TC, from the original pack, except the true count is not a perfect predictor of your edge and has the bow-effect as a result of such errors accumulating with penetration.

I know you are a smart and an accomplished mathematician, but Clark, this is a place where people are trying to learn how not to lose money playing this game. What floating advantage got to do with the real game out there? Have you ever used the FA during real play?

Me, personally I don�t really care about the FA.
If you want to help, please talk about the basic counting, index, betting and playing strategy. That�s all they will ever need.
Oh! .., and some heavy experience in real games will be helpful. But not the FA, Clarke!

Puiu

I am not going to bother with this Floating Advantage argument more and am leaving for vacation.

But I will point out once again how Cant uses the words of mathamatics without substance.

A mathamatical "Proof" is simply that: a rigorous step by step process, using mathamatical operators which demonstrates unequivocably a proposition is accurate. If one has taken geometry one remembers the form. Good examples of mathamatical proofs occur in blackjack literature and can be found on bjmath. Examples are the Janacek and Brh papers on optimal betting.

Contrast real proofs with the post above which Cant calls a proof. Admittedly I do not understand a thing which was written above so I cannot vouch for the accuracy or inaccuracy of any statement but I can vouch whatever was being said is not a proof because none of the formal logic required for a proof exists. Calling those words a proof is a fraud.

I will leave this topic with one observation which is easily demonstrated by a combinatorial analysis program or a sim program like SBA where the deck can be stacked. Take any 52 card sample one wishes and look at the expectation off the top. Then take the same by adding one identical card for each card in the 52 card example making a 104 card sample of identical compostition. Look at the expectation off the top. The expectation for the 104 card sample will be smaller. The difference will be close to the difference in expectation for one deck and two decks. This is an invariable and easily testable.

I have done such sims and have reported the results. If Cant did such sims and accurately reported the results he would report them the same way but we will not observe that because he will not do this very simple test which will disprove his speculations.

What is written above would never be taken by anyone even slightly familar with mathamatical proofs as a proof in any sense of the word.

have you ever been to a strip joint?if so,when a girl is giving you a lap dance does she have to talk about the FLOATING ADVANTAGE to get you excited.???

"Calling this a proof is a fraud."

There is nothing wrong with being informal.

ML's second paragraph:

I have already clearly given how this example of "doubling up" a composed deck is a fraud. Once again, for the billionth time, and really this is insulting to people who still read ML's supposed rebutals: the lesser expectation he claims is not universal, exceptions are fairly common; read Griffin's Theory of Blackjack discussion of the deck of 8s and 7s; all such larger packs, expanding upon a given density pack of smaller size are less likely to come from the common parent.

ML's next to last paragraph:

I have already stated how ML's request for specific decks is a fraudulent request. See above. To answer this request is to simply spam in that ML has already stated his intention is to take whatever deck composition I post and double it and thus claim that his opinion on mean compositions is valid.That just feeds further trolling by him, on this. It is your call readers. We can cycle through 8*52! different decks (multiplication by 8 is just to limit the expansions ML is intent on doing up to 8 decks I presume) and we will find that something like 8*27! will exhibit higher expectation in the larger packs of the same density (estimate based upon sims posted on Grinder's posted sims on why expectation is greater with smaller initial numbers of decks and where exceptions occur). Big deal that more are likely to show higher edges with the smaller composed packs because the plain fact is that the probability of the mean occuring shrinks. For each 1/4 deck the probability of a perfect mean pack occuring from another perfect mean pack, is .034, if you are looking for mean packs that match the initial full pack. The probability of this occuring from any other pack is at least as low as (.034)^2.

It is actually the minority of packs that exhibit the expectation that ML claims even though they make up about 60% of the probable packs.

Now for a more formal proof:

The invariance of the basic strategy edge with penetration is absolute. Proof: the rounds that start anywhere in a shuffled pack can be brought into play by simply cutting the pack to bring that penetration point into play next.

The true count is not linear in its prediction of expectation. It overestimates the player's edge at the extremes and underestimates the edge in middle ranges. Proof: all sims dating back to E.O. Thorp's first 10 count sim rounds as found in Beat the Dealer, for the ten count, to any done today.

The true count grows more extreme with penetration. Proof: you divide the running count by the ever decreasing number of cards left with penetration.

The bow-effect increases with penetration. Proof: the true count and the actual edge are 2 different distributions whose correlation is generally estimated by the correlation coefficient between the best obtained effects of removal and the point count used. The distribution of hands and of the player edge is invariant, with basic strategy, but the true count is more extreme. By topomorphic process, the true count has to have the same limits while its overall amplitude in most ranges increases, the true count bows more with increased penetration.

You have the bow-effect increasing, as penetration increases, but no general increase in edge as penetration increases and no other posibility exists with an imperfect true count measuring edge and an invariant overall mean basic strategy edge.

The Proof of the Strong and Weak means is informal and need not be repeated or be restated more formaly in that it is simply the general case of Abdul Jalib's True Count Theorem. Whatever the mean of a parent pack, that is the mean of all the rounds that can be drawn from that pack, in exactly the same way that a true count is the true count of any portion of the unseen portion of a pack. Simply subsitute a perfect analysis of edge for true count and you have this proof, in formal form, where Jalib's formal analysis need not be repeated!

The perfect means never even overcome the skewing that results from coming from the larger initial pack.

Since the the power of the various means is true, the alternative way to derive the bow-effect, from analysis of where skewing accumulates with penetration, must be considered, and you must consider what the general effect is of packs that match the true count of larger packs, but have more skews in their densities of cards. The larger pack has less skewing of densities compared to smaller packs of the same true count, that come from the same still larger parent pack. It is possible for 8 aces to come from 2 decks, with one deck left, but simply impossible to have 16 aces coming from 2 decks with 2 decks to be played. ML is not playing with a full deck if he really believes his doubling of pack size claims.

For the bow-effect to occur when examining these skewed packs you must hypothesize that such skewing generally causes packs that exhibit the bow effect. This can and is demonstrated by taking any perfect mean pack at any true count, and, for hi-lo as an example (it is the best combination of simplicity and basic strategy prediction), trading aces for tens and tens for aces. Trade for 10 cards and lower expectation for all true counts--less blackjacks than normal--and increase the player edge in middle ranges---more 20 hands-- and have less edge at the extreme counts---less blackjacks. The same type of edge changes occur with trading ten cards for aces. It is trivial to demonstrate (how come ML does not try this?).

Thus what is necessary to derive the bow-effect from examination of the increasingly skewed density packs that result with increased penetration is demonstrated.

Now the bow-effect has been demonstrated by both Griffin's original argument of the basic strategy edge being invariant for any given starting pack size and has been demonstrated by the general effects of how packs actually skew, for given TCs, with penetration.

The ultimate and predictable insult is for someone like ML to now give the excuse that he was just trying to help and drawout these points in the required fashion. That old altruistic intent excuse burns me. All of this spamming and abuse just to....

What remains is to follow up on 2 different claims that appear to be valid coming from ETF and T-Hopper. ETF claims quite well---but again without formal proof---that a perfect playing analysis of a deck composition would not exhibit any bow-effect. T-hopper has found, in simulations he posted on bjrnet.com, that adding more indexes, or more playing index gains in general, results in more gains than can be explained in the way playing gains are normally isolated by examining flat betting. Somewhere in between you must have a general flattening of the bow effect, and a large somewhat unexpected increase in overall betting efficiency, as you add playing gains to just using basic strategy.

that is about it! Tough if ansering these trolls has made for retaliatory posting!

Noun. The property of an imperfect measure of a finite population to exhibit distortion with bilateral symmetry, such that the integral of such distortions, from the general curve of its accuracy, sums to zero. Analysis of this property was important in the developement of the Graham-Stokes equation and theory.

I referred the board to bjmath and the formal Janecek and Brh proofs. The Abdul proof has similar rigor.

Nothing in the Cant post has any relation to a proof. Most of us here took geometry and that contained proofs. No steps are left out. No steps are assumed. No propositions are asserted unless they are supported by theorems are earlier steps in the logical process.

Nothing in the Cant post above has any relation to a formal proof even if he calls the verbage a formal proof. Does what was said bear any relation to Euclid, as an example?

As to the only proposition I understood in the verbage, Cant finds fault with my approach of doubling the cards, card for card, in any arbitrary sample of 52 cards and comparing the expectation of 104 cards with the expectation of 52 cards showing identical proportions of each denomination as artificial because there are circumstances where it is impossible to construct 104 cards the same way if starting with an six or eight deck stack. That limitation is an artificial red herring. If we start with sixty-four decks, which is certainly permissable for theoretical studies, that objection goes away. The sainted Griffin has a composition dependent change in basic strategy somewhere around eighteen decks. That such can theoretically occur is a theoretical thing but still real in the realm of math.

Again, what you saw above was not a proof. It was not even close.

This is more the tactics you are responsible to use to defend an otherwise doomed client in a court of law than legitimate discussion.

Look up topomorophic. Reread the Jalib's, True Count Theorem. You accept that style when he uses it but not me?

The basic strategy edge is absolutely invariant with penetration, even when, on very very rare occasions, there is a smaller perfect mean subpack burried within. That is just part of the overall possible spectrum.

When will you accept, or not disemble, that your example of the means is a flawed example of trying to extend the mean to a distribution of outcomes where it does not apply? In your latest example it is perfectly reasonable to even go to a 128 deck shoe, but it is not reasonable to go to a larger parent when the analysis calls for subpacks that come from the same parent.

invariant basic strategy edge+topomorphic process bending of the true count's ability to predict your edge=the bow-effect. This topomorphic process is present with any combination of a finite population and any flawed measure of that population, as sample size increases. It is the fundamental part of the derivation of Stokes-Graham theory, and the the S-G equation. Didn't you study this?

That subpacks from a larger pack exhibit the skewing that can only come from the larger parent pack, and includes the easily demonstrated fact that skewed subdecks, rather than showing a reversion to the means of such subdecks, which is weak and only strong for the initial pack, show bow-effects within themselves. That is why I took up consideration of how subdecks show bow-effects within themselves, even though cutting the deck well proves the invariance of the basic strategy edge overall.

There is no strong floating advantage.

There is a bow-effect that is cut by PE gains.

Combining PE gains in deep games results in unexpected gains overall. That is what T-hopper is working on.

Shifting the betting ramp to try to exploit the bow-effects is a weak tactic and maximizing your count's PE brings far better results. ETF has been clear in explaining how PE, if you had a PE=1, would wipeout the bow-effect.(His disembling tactic is to just claim I am disagreeing with him. It is an interesting tactic, as interesting as his not using an recent edition of Theory of Blackjack to deny that "bow-effect" is used.).

Moving your betting ramp to match the bow-effect distortions IMHO will not be as profitable as what T-hopper is working on.

So who the hell is your client in that your tactics are those of an attorney who is formerly hired and has a definite client. See your state's legal cannons. Such deception is highly frowned upon when it is taken outside of such a formal hiring, and representation, in purely personal matters outside of the courts, even though it is very propper and called for in the courtroom. I don't care to mention the non-blackjack possibilities, but I know of one disbarred attorney who actually went down for doing such without representing a bonifide client. I am sure you have a client. But it sure ain't involving the points being discussed IMHO. It sure ain't truth or beauty either.

of yours next time you claim that you never lose your cool and that you're always polite with everyone and what not.

Yet another humanly impossible claim.

I have always said that I am a mean and vindictive SOB. Really you can ask my ex-wives and my exinlaws. Really you can ask the families of a few who have not had family members around because of self defense incidents. Really I have had a rough life and have had to be on occasion a very nasty guy.

Really though an attorney is required to disclose when he is working on behalf of a client. ML keeps engaging in a claim that is known in (real) math circles as "a redactive appeal to the reversion of the means." He is quite aware of how the bow-effect is topomorphic--he has posted before on the Graham-Stokes equation, very well, and even came up with a unique derivation of the proof for why overbetting, past a 2XKelly fraction leads to a crash. So he should know that he is engaging in such an appeal, as given above, when the means of such reversions are altered by coming from a larger parent pack than the current size pack left to be dealt. Bigger initial pack =less reversion!

Everything he has posted has been a reset of that.

ML is a damn good attorney in real life and is very skilled at what is pushing the limits, and how an attorney is required to push the limits when actually representing a client. But the same conduct, when not on behalf of a client is highly frowned upon.

As for my own experience, it is not revealing anything that is not known about me already, that few people appearing pro-per are ever given listing, in any record as an, "attorney of record," when they are not attorneys. An attorney appearing representing himself is appearing "pro-se." The only other in the entire 9th USCCA is the late George Franklin, in his first term as a Clark County Nevada commissioner; just before he passed the bar exams he had to represent the county in 1941. I will not disclose what case that minute order of the court was issued!

That was to permit me chambers access, as a pro-per otherwise, trying to get me to negotiate a settlement issue. I am fully opposed to their being any such settlement.

When anyone who is an attorney seems to be evasive it is considered quite fair tactics to ask who their client is? That is not considered rude at all. It is considered to be required etiquet to ask when someone who is an attorney seems to be evasive. You simply don't have the knowledge or the context to which I asked the question.

I am not an attorney; I have been ordered to take certain powers reserved for attorneys by the courts in question though, and to stay within those rules. That is why the form listing change was made in the courts involved. But if there is any other non-blackjack reason, or non-blackjack client or issue, related to ML's points, which seem to be invalid, if the issue is what is the truth about the FA, but very valid if the issue is representation of some other person, who might be threatened by my prevailing, then I have every right and a definite responsibility--if I want to maintain access to certain of my own court records or pleadings-- to ******ing ask?

That is just the way the game of law is played, with regard to attorneys.

your new-found proclivity to post in HTML colors and various font types! But this too will get irrelevant soon.

How did we get from topomorphic effects to the bar exams of the Clark County Commissioner?

You plotted the route!

I tried to find a link to the photos of surgery Dennis Rodman had to get when a lapdance resulted--he claimed in his second auto-bio that it was due to hitting bottom during regular intercourse--in a fracture of the penis. Real story came out later when his sometimes wife deflated some of his claims in an interview with Penthouse.

I should be so lucky in that while I preffer short redhaired petites what I attract is the usual Vegas tall blonde types. That has very little risk of such fractures for me....

multiple personalities