Below Cant asked if I actually have read Chapter 13 and the answer is not for a few years until prompted. I thank Cant for prompting me to do that because almost all the material in Chapter 13 validates the proof and the corollary. Throughout this entire sequence of Cant saying one thing and then being confronted and backing away to another tack, he has had one consistency. He keeps insisting the approximation techniques of early blackjack analysis necessitated by lack of computer capacity have relevance to the bow effect and the floating advantage. He has continued to insist the proof which unmistakingly used expected values which are tangible numbers and in no way approximate were in some way derived from approximation techniques like ones Griffin and others used when they did not have the access to computers we have today. He has continued to insist the approximation techniques make us not perceive things correctly without conceiving of times when approximation techniques have nothing to do with things as approximation techniques had nothing to do with my proof. That approximation techniques had nothing to do with the proof should have been apparent when I first defined terms.
People who have read the exchange with some thought and inquiry have had to understand expected values. An expected value is a comprehensive term and a definite term, not an approximation. To put the difference in perspective I will use an example Cant insists on using. Without any foundation at all, he continues to insist my term E(i,m) is not an expected value but, instead is a value (which he inaccurately terms a "mean" value) which is derived from the mean deck composition of deck possibilities of having a count of i at a depth of n-m. That is an approximation just as the Griffin UNNL and other devices are approximations.
However, in today's world the perfect analogue to expected values is sim data. No one disputes multitudinious combinations a specific count can arise and that each of these ways has a unique expectation value. Experience has found there is a strong similarity of value of expectation among all the subsets of 52 cards which are a subset of 412 cards and also have a zero true count or card counting would not work because that is the way card counters lump things without worrying about the inexactnesses. If the inexactnesses were not trivial, successful card counting would be impossible because the prediction it provides would be useless. But that is another story. The point is Cant correctly points out values are not identical.
When values of subsets are not identical, sampling of these values is a sampling of random variables. And when the sampling of a set of random variables approaches infinity (or all the variables are counted if possibilities are not openended because possible samples are finite), the mean of the samples is the expected value of the entire set.
Sim data approximates as closely as is humanly possible the expected value because it is a value derived by calculating the mean of the random variables which make up, for example E(i,m).
These asides are prepatory to quoting passages from Chapter 13 with commentary explaining how my proof is consistent with Griffin's writings. In fact, they are so consistent that if Cant had accused me from stealing from Griffin rather than stealing from Brett Harris as he did I would have to have pled guilty to that. In my defense to any claim of purposeful plaigarism, I hope people would have believed I had forgotten much of Chapter 13 and would certainly have used it to bolster my discussion if I had remembered it.
Chapter 13 begins on page 203 of the Elephant edition. I will summarize some of the beginning until it gets to the relevant portions on the floating advantage and the bow effect and then use more extensive quotes and comments.
Chapter 13 is entitled "Regression Implications for Blackjack and Baccarat" which tells us right off Griffin is talking about approximations. As Griffin says, "A regression function relates the value of a predicted value to the known value of some predicting value". That is, there is a true value of some item. Some mathematical device (the function) establishes an approximate value of the true value from other values which are known (predicting value). Since it is realized the function only produces an approximate value, a field of inquiry is to determine the amount of the inexactness of prediction because if the inexactness is too large, the prediction becomes of no value. Determining inexactnesses of prediction is what the chapter is about.
He then remarks the normal regression function used is a linear (straight line) approximation and that does not necessarily predict so well for expectations in blackjack because such expectations do not necessarily follow a straight line.
In the section entitled "The Problem" Griffin remarks strategy decisions approximate linear regression functions so the predictions used for play decisions is highly reliable and explains that, because of the shortage of computer time (now not a factor) he would use the shortcuts of analysing a "Woolworth" blackjack game which has 32 tens in the deck and 20 fives; sims of small subsets (he did not have the ability of simming larger subsets as we do), using the Thorp infinite deck approximation technique; and comparing blackjack to a simpler game, baccarat.
In the section "Woolworth Blackjack", Griffin analyses EOR's and then looks at subsets of different sizes and how actual expectations correlate with predicted expectations. He finds them close but further finds both a floating advantage for the actual expectations near the middle of the possible distributions (zero count) and a bow effect for expectations at high plus and minus counts. A linear function does not work very well, in other words.
The next section "Digression-the Count of Zero" is the section most relevant to my proof and I will quote at length from portions.
Griffin first stated he had not believed in a floating advantage in the range of zero for perfect decks and discussed how play strategy variations could not account for such a thing so he had not believed the advantage of 0/52 taken from eight decks should be higher than 0/416. He then goes on to say,
"Having publicly, privately, and righteously condemned for many years this heresy that a count of zero indicates an increase in player advantage, I was stunned by the 2.33% expectation for a normal 13 card subset of 5 fives and 8 tens in Woolworth blackjack, this being almost 3% above the full deck figure. Suddenly I knew I had been very wrong, misled by my erroneous presumption that the distribution of player advantage was symmetric as the distribution of a point count. What to do? Feverish recalculations only confirmed the error in my judgement. Could I conceal the finding perhaps until after my death? Unlikely. I decided upon the route of confession:: at least then I would be the one to prove myself wrong!" (ML comment: This graceful admission of an "error in my judgement" by Griffin is the precise claim Cant keeps making. Cant keeps saying the floating advantage does not exist for the whole count of zero in a diminished pack or any other count because variant distributions pull the expectation down to the expectation for the starting pack. Cant says the is true for only the mean subset. Griffin says he (Griffin) was wrong for saying that).
"The correct explanation of this count of zero phenomonon goes like this: the player's *gain* in expectation for unusually high counts will be of a smaller magnitude than his *loss* in expectation for correspondingly negative counts. In the former situation more pushes will begin to occur due to extra tens in the deck and double down opportunities will become less frequent. On the other hand, with outrageously negative counts, the basic strategist will often be doubling, splitting, and standing when the dealer is very unlikely to bust. For this unbalance to occur yet result, as is provable, in no change in the basic strategy expectation, there must be a small rise in the expectation somewhere in the middle of the distribution, quite possibly at counts close to and including zero." (ML Comment. See step 4 of my proof. That step shows formally what Griffin put into words. I had averaged the expected value of all i-j and all i+j counts and replaced the values of both with an average value. I then showed by induction, each value of each expectation as replaced was smaller than the value of i. If Griffins informal observations are accurate, my formal presentation is also accurate.)
"To illustrate this, again using the Hi-Lo count, one can calculate a basic strategy advantage of 18% for a +13 with 13 cards remaining from a single deck. Note that this is below the 26% we would presume using .5% per true count. But for a -13% count with 13 cards remaining, the basic strategy expectation is a whopping -135% because of the many hopeless doubles and splits. This is far below the extimated -26%. It therefore follows that for at least one of the running counts between -12 and +12 the actual expectation must be higher than the .5% per true count figure would indicate. This is because the overall expectation with 13 cards left must be precisely the full deck 0%." (ML Comment. Please look at the last sentence and my Theorems One and Two. In those theorems I put the expectation of the original deck E(i,n) equal to all the subsets. Griffin remarks here that is proper because "the overall expectation with ... cards left must equal the full deck 0% (ML: expectation)". Notice Griffin is discussing exact expectations here and expected values are exact expectations. He is not approximating by using mean composition expectations. The next section dealing with sims makes this even clearer.)
In the next section "Actual Blackjack, 10, 13, and 16 card subsets" Griffin uses the sims then available to further explain the phenomona he was exploring. Sims results approximate expected values because they calculate the means of samples of random variables. They are certainly not approximations or calculations of expectations for mean distributions.
When the Chapter was written it was not feasible to sim large subsets but it was reasonably easy to extract small subsets from larger numbers of cards and sim those small subsets and that was what Griffin did. He pulled 1000 subsets of ten, 13, and 16 cards and simmed to get what amount to expected values and, since the sets were so small, doubled the results on the first card of a split to keep from using up all the cards but felt that did not create a significant bias. A figure is included which shows the data points, and the linear approximation line, and the curved best fit line for the data. The best fit curve is parabolic.
After commenting on correlation Griffin said in commenting on the figure: "Figure A contains a plot of 200 pieces of data from ten card subsets of a single deck and provides insight into the behavior of least square estimates. Note the bowed, parabolic nature suggested for the regression function, similar to the shape we would observe if we plotted the 13 card regression function for the Woolworth game. [B]". Footnote B says the best fit is cubic. In the footnote Griffin further comments, "the characteristic of overestimating (ML: using linear estimation) advantage at the positive and negative extremes and underestimating in the middle of the distribution undoubtedly occurs regardless of how many cards remain." He then goes on to say the curve flattens as more cards remain. (ML Comment. Clearly the bow effect exists for the expected values (sim results) of true counts. I would suggest this statement by Griffin adds much credibility to the corollary contained in my proof and therefore adds much credibility to the proof.)
"Other data gathered in these experiments shed light on the 'Count of Zero' phenomonon. The player's actual advantage, as a function of two well known card counting systems, the Ten Count and the Hi-Lo, also displayed the same parbolic shape. (ML Comment. That is, when sorted, the expected values (sim results) of specific true counts using both the Ten Count and Hi-Lo displayed the bow effect.) Consequently counts near zero, reflecting normal proportions of unplayed cards, had expectations higher than linear theory would predict. (ML Comment. Remember what linear theory would predict. At a zero count linear theory would predict full deck expectation, -.02, because there would be no floating advantage under linear theory. The Cant formulation would predict the larger deck expectation at zero) For example, with 13 cards remaining, a Hi-Lo count of zero was associated with an expectation of +1.6% while 13 card subset with precisely four tens had a player advantage of +1.87."
"The most probable 13 card subset, one card of each denomination, had a 2.05% expectation for single deck basic strategy, 2.07% above the linear estimate of -.02%." (ML Comment: These figures shed some light on the Cant quibble not all combinations of cards making up a specific true count give expectations as large as "perfect" combinations. Translated to full deck expectancies, the value of all subsets of zero count have an expectancy .11 lower than the "perfect" combination. I had suspected that but thought it irrelevant because my proof contained expected values which did take diminished expectations of nonperfect subsets into account. The Griffin findings demonstrate both thoughts accurate. Overall expectations are diminished from "perfect" subsets but not diminished enough to be relevant to the proof. Expectation remains 1.62% for thirteen cards or normalized to a full deck, .4% better than expected. My original post on RGE which prompted Cant to post a "Proof" here which was not a proof at all and pissed me off for his abuse of mathematics and prompted me to post a real mathematical proof on the topic and all the silliness which has followed remarked one could expect about a .4% increased advantage. Griffin's sims coming from a different direction ended up the same place. Cant's formulation would say the expectation for all zero combinations should have been -.02 when they are, in fact, 1.6.)
The remainder of the chapter discusses the Thorp use of infinite decks and correction factors and Baccaret. The baccaret section is not relevant to my published proof. Sims were done of subsets taken from infinite decks of numbers of cards from 26 to 416 in the infinite deck section and Griffin points out the discrepency between predicted expectation (infinite deck expectation) increases with fewer cards in proportion to the expectation of starting decks of the same number, another situation where the floating advantage is demonstrated by sims (expected value). He attributes these differences mainly to the fact nonreplacement helps the expectation of double downs because the small cards cannot reappear with nonreplacement.
In short, Chapter 13 totally validates my proof and my postulate and conclusively proves the floating advantage exists not just for mean compositions but for expected values.
That Cant pointed out a writing by the premier blackjack theorist which completely destroys Cant's theories rather than supporting them is instructive. Cant simply is incapable of reading anything mathematical and understanding it. Like many ignorant people he thinks his views are the only truth and the more learned are bound to have the same beliefs even if he does not have the mathematical capabilities to grasp what the more learned are saying. In this situation he certainly presumed wrong. He failed to understand the sim data showing a 1.6% advantage for a true count of 0 in the 13 card
subset absolutely destroys his position.
This thing is finished now that Cant has pointed out how Griffin destroys his position but there are a few ends left unravelled and after discussing these ends I will be done.
It is almost impossible to understand what Cant drives at but he has now been pinned down. I could not understand his "proofs" which were just assertions and had no mathematical content but now I see what he was driving at. I also now see what he has been driving at in accusing me of switching means and/or using the values of expectancy of the mean distribution of cards of specific true counts rather than the expected value of specific true counts which I unmistakably did.
To help with the explanation I will return to expected values and restate the definition one more time. It is the mean of a set of random variables as the samples taken of the random variables approach infinity. If something happens with a probability of p, the expected value of p is n/N where wins are n and total plays are N. Samples taken of n/N might not exactly equal .5 when fair coin tosses are recorded but that does not change the expected value which is .5. In blackjack most certainly the value of various combinations of 52 cards which by happenstance have a Hi-Lo true count of 0 which are aalso subsets of larger stacks of cards will not have identical expectations. We see in Griffin one number for all sampled 0 sets, another number for all sets containing four tens, and another number for the "perfect" set of 13 card sets. Griffin did not have the capability of simming large numbers of outcomes and there is bound to be some error existing but he did derive what amounted to an expected value by looking at the outcomes for all 0 sets.
This aside should aid insight into Cant's completely inaccurate proofs and mathematical reasoning. He starts from two presumptions, only one of which is right. The first which Griffin demonstrated is wrong is that if the overall expectation of all subsets or a certain number is the same as the expectation of the set (and that is true for blackjack) it is necessary the expectation of all subsets of a specific size and true count be the same as the expectation of the true count of the set. That is simply not true. Griffin's sim data showing an expected value of 1.6% rather than -.02 puts that at rest forever. The accurate statement Cant makes is the expectation of the mean combination of cards of the subsets of a specific true count exhibit a floating advantage. He then goes on to conclude that since the overall expectation (expected value) of all subsets of a specific true count must be equal in value to the expectation of the same true count for the set, the expectation of combinations away from the mean composition must be enough smaller than the floating advantage to make up the difference. Then getting things totally backasswards because he did not understand the notation which would have been understandable to anyone with any mathematical background, he claims I was using the expectation of the mean composition rather than the expected value because the expected value does not behave the way I said it did but only the expectation of the mean composition does.
The import of my proof and Griffin's findings demonstrate both the expected value of the subsets of a specific true count and the expectation of the mean composition both exhibit a floating advantage and a bow effect. Perhaps values are different but the values share the same relations. And as Griffin pointed out, and as the proof points out, the expected value of the expectation of the 0 (or in my proof i) is larger but overall expectation remains the same as the starting set because of the bow effect.
Cant says the overall expectation for m cards cannot be preserved unless the true count of all the subsets of size m and true count zero have an expected value identical to the expectation of the original m cards. Griffin (and the proof) say overall expectation for m cards cannot be preserved unless the expected value of the subsets of n size and 0 or i count are larger. Whom should be trusted?
And that is why the multidimensional stuff, talking about Gauss and LaGrange operators (there are no such things applicable in any way to blackjack problems) and the other stuff is a crock. All that is needed to deal with blackjack statistics are expected values and these days expected values are easily found by sims. But that is Cant's methodology. He thinks dropping names lends credibility to his arguments and makes him appear expert. But since he drops references and names while not understanding the import of what these people actually said (witness his using Chapter 13 which totally disproves his assertions) he unmistakably demonstrates how ignorant he really is.
The ignorance has shown throughout this exchange, first beginning when he mistook expected values for something else. It was shown more clearly when, rather than stating expected values did not demonstrate the bow effect (because if they do, a floating advantage must exist as Griffin points out in Chapter 13) he asserted I was not using expected values but substituting something else when the clear statement of the postulate to the proof said expected values demonstrated the bow effect.
I can only say I am glad this is over because it is abundantly clear Cant is hoisted by his own petard, the citing of Chapter 13.
