regarding the math of part II

I forgot to also add:

can anyone tell me if the increase in a given count using a specific counting method or all counting methods forms a linear increase in the players advantage or is it a curve so that each plus 1 always translates into a .01% increase in advantage (for example) or maybe its roughly an increase of .01% for the first 5 increments in count but then a greater or lesser increase after that. I know its a geeky question but I am looking for something in the math beyond just basic counting that represents a stone left unturned.

gern blanston,

If you click the link below, you will find the Blackjack Data Repository, which contains zillions of tables and charts that will allow you to find the answer to this query.

For the example I chose (6 Decks, 83% penetration, 1-8 spread, no cover, S17, DAS, High-Low), if you peruse the column labeled "Advantage", you can see how it changes as the TC (column 1) changes.

Note that when you get to extremely high (or extremely low) TC's, the sample size is very small. Thus, the results you see are most reliable near the center of the range.

To see the impact of different counting systems, choose a diffent system from the drop-down boxes at the top of the page.

Enjoy!

Dog Hand

The poster asked a clear question: Is the advantage linear to the count? I think he deserves a clear answer and actually you gave the answer below by posting the link to the effects of removal.
The logical consequence of EORs (or the condition for their validity)is basic linearity.
But now you post a link to a simulation where the linearity is somewhat distorted to say the least.A blatant contradiction. Now the question is: What is correct,the sim results or EORs?
I clearly favour the EORs. The general rule that any increase of TC by 1 is worth 0.5% is sufficiently precise,at least in the range of TC-3 to TC+3. Now the sim shows an increase of 0.72% between 0 and +1. There must be a flaw here!Furthermore, we know since Ed Thorp that the advantage increases slightly faster from TC+4 onwards because of the effect of insurance.But no sign of this in the sim.
Posting a sim makes only sense if it's accompanied with the necessary analysis.Here it doesn't make us anymore intelligent.

Francis Salmon

The logical consequence of EORs (or the condition for their validity)is basic linearity.

Sorry but you will have to do better than claim it's logical. I don't see the dots or connections.

Obviously advantage is not linear with TC in a practical use. How can it be when advantage is affected by indexes and you chose how many indexes to use? How can it be when opportunities present themselves at different points? The most obvious opportunity is Insurance, as you point out. This advantage play pokes its head in at a particular count. But that is only one index.

But forget the fact everyone plays differently. There is no reason to assume advantage is linear with TC to start with. EORs are a wonderful estimation. But, they are estimations.

You claim that a sim must be wrong because it doesn't match theory as you understand it. Consider that the theory does not match the fullness of real world situations or that you are not applying the theory correctly to those situations, ot that you are stretching the theory. In fact, I don't remember Griffin ever saying that Advantage is linear over TC and I believe you have been told this before. I think he did say that it is linear over the specific Insurance wager. That is a long way from general linearity.

Any of the math guys have a comment? In the meantime, here are 150,000 2,000,000,000 round sims that indicate non-linearity.

CVCX Online

A theory has to match reality, otherwise it's just wrong. Or would you say that Einstein's E=mc2 is only true in theory, but in reality it's different? Of course not.
The same goes with EORs which is a generally acceped theory.I don't understand why you cannot see the implications of it.To make it simple let's take the five-count. Griffin and Thorp before him found that with every five you remove from a deck the player's edge by 0.69%.If at the neutral deck the game is even the advantage would by 0.69% at +1,1.38% at +2,2.07% at +3 and so on.If you draw one of your beautiful graphs with these data you get a straight line.
Nobody claims perfect linearity because there are minor disturbances by some special subsets and situation but basically the whole thing is linear.
You shouldn't confuse the issue with insurance. This is a side bet and we can simply remove to get linearity again.
Your sim not only lacks linearity but also steadyness.Or how can you explain why the edge increases by 0.72% from 0 to +1 but only by 0.53% between +1 and +2.

Francis Salmon

Page 24. "Insurance is interesting for another reason; it is the one situation in Blackjack which is truly 'linear'...."

We have been through all this before on advantageplayer.com. You are incorrect in saying that Blackjack is linear under accepted theory. It is not according to Griffin and the sims, from ALL simulators, bear this out.

Modern shoe blackjack is basically linear. Modern penetration means you never see a count which varies enough from neutral to produce significant non-linearity. The exception is the insurance issue.

Francis wants to treat insurance as a side-wager. That is not accepted practice I believe in the literature, but it is a reasonable perspective nonetheless, probably as valid as the conventional one.

If you can find a deeply penetrated game linearity does go completely out of the window. The gain from highly accurate end-deck subset analysis can be extraordinary. Sadly, this is just a hypothetical theoretical curiosity unless blackjack conditions radically improve in the near future.

Of course insurance is perfectly linear.It only depends on the number of tens remaining per deck.
I think you misinterpret the quotation from Griffin: The emphasis there is on 'truly', meaning that the rest is still practically linear.We can stop counting if this isn't true.
You still haven't answered my question on your sim.

Francis Salmon

I am not going to repeat this long thread - it has all been said before. You believe that every simulator is wrong. So be it.

I think your country the U.K. doesn't even allow insurance. So at least all the Brits should accept this practice of separating insurance from the whole issue.
Penetration is yet another issue and I'm perfectly aware that the edge varies according to the deck levels.Curously enough, Norm's sim doesn't even distinguish different penetration levels. He throws everything into the same basket. May be that's why he got such messy results.

Francis Salmon

Saying something is almost linear is like saying someone is almost pregnant. Linearity is an absolute and cannot be modified. It is certainly true that advantage increases with TC if you ignore both end of deck subsets and very high magnitude TCs. However, the effect cannot be linear for two reasons. First, Insurance, DDs and Splits add to advantage at varying points. Second, no counting strategy attains 100% efficiency. Besides, as a purely practical matter, players don't use every index and don't calculate depth to the card. And useful sims are based on the way humans actually play.

You're gonna be surprised but I have no doubt that your simulation results are correct.However your interpretation thereof is bogus.When a simulation seemingly contradicts an accepted theorie, the first step would be to analyse the whole thing and not just think that the theory has been proven wrong.
You are unable or unwilling to explain the extradinary increase in edge of 0.72% between 0 and +1 according to your sim.So,I'll try to do it in your place:
I think it has a lot to do with the way you define TC.When you label a TC as +1 you actually mean all the TCs ranging from +1 to +1.9 so there you get an average of say TC+1.4 (lower counts are more frequent). Likewise a TC of 0 includes all TCs from 0 to +0.9. Here you would think that the average is again about +0.4.But this is wrong.We know that every shoe starts with a count of 0 and together with all the other frequent occurrences of a neutral count we clearly get an average of considerably less than +0.4, I would estimate it at +0.2. So now we see that the difference between your TC 0 and TC +1 is actually not 1 but 1.2 which explains your exaggerated value in this range. Note that the increase between your TC+1 and TC+2 and TC+3 shows a normal and steady 0.53%.
So your sim actually confirms the linearity if we analyse it properly.

Francis Salmon

First, Insurance, DDs and Splits add to advantage at varying points.
Second, no counting strategy attains 100% efficiency.

I don't disagree. It is just that in terms of % edge at least, this isn't of much import.
Now, if you are trying to grow your bankroll from a small base, such apparently minor details take on some importance as you get a kind of compound interest effect if you know how to exploit such aberrations from linearity. Sadly, this is probably something very few players consider or exploit probably. Moreover, this is true of the many fine details of blackjack, and you have to draw a line under the practicalities of applying your blackjack strategy at some point.

Regarding 100% efficiency-you are right, the complexities of non-linear advantage have an important theoretical property. It is essential to understand this stuff if you are going to do research into the mechanics of blackjack. I think Francis is focusing on the implications for regular counters though.

Btw I wrote an entire book essentially based on the principle of non-linearity for baccarat card counters, so I'm certainly not down on the whole concept. It is fascinating what happens with blackjack (and baccarat as Griffin notes, and indeed all similar card games) when the deck structure breaks down at close to 100% penetration, and the EOR's go out the window.

An additional point: is advantage &risk/ TC linear? I would think not.

I have never seen such a stupid comparison but here is a better one: The relation between TC and advantage is like the one between altitude and temperature.The higher you get the cooler it is.That doesn't mean that all the places of same altitude must have the same temperature.Of course you have shady places and sunny places or windy ones and quiet ones.But no meteorologist would say that because of these other influences the relationship between altitude and temperature is not linear.
Norm, you have a problem of constantly confusing issues and being unable to separate one issue from others. I have said repeatedly that insurance is a different matter,yet you come back with it every time.
What you said about doubles and splits is totally wrong.The profitability of any double situation is perfectly proportional to the count!
We all know that counting systems are not 100%efficient yet they are all based on linearity.
Bringing in human incapacity and lack of precision in order to prove the non-linearity of Blackjack is absolutely ridiculous.So for an expert counter it's linear but for a beginner it's not?Bloody hell!

Francis Salmon

If that's the reason for the "bulge" in advantage between a TC of zero and a TC of one, then why is there an increase in advantage of 0.652% when going from -1 to 0?

If what you're saying is true, then the advantage figure at a TC of 0 is actually "out of position" since it's "dragged downward" about 40% from where it would "normally" be if not for the existence of zero true counts at top-of-deck. Yet, in spite of this lower figure, the decrease in advantage when going from TC of zero to TC of -1 is still bigger than we'd expect. Wouldn't you think the figure should actually be smaller than 0.652?

Just curious. I suspect the bulge around zero has more to do with indices than anything else, in the same way that insurance kicks in at higher counts.

I saw this contradiction myself and frankly I don't know the answer,after all it's not me who conceived,ran and published this simulation. Unfortunately,the author refuses to debate. So,I can only tell you my suspicion.
I don't think it has anything to do with indices, those are spread all over the place without affecting linearity in any way (insurance excepted).
It looks very much that in the sim the TCs were actually truncated and not floored. This would result in a TC of 0 regrouping all counts from -0.9 to +0.9 with an average of exactly 0.But the TC labelled -1 would include all TCs from -1 to -1.9 with an average of -1.4. A difference in edge of 0.65 looks quite plausible in that respect.Note that the edge between TC labelled -1 and TC labelled -2 decreases only by 0.45%.That's also plausible under this assumption because the average of the TCs labelled -2 should be roughly -2.3 with a resulting difference in TC of 0.9.
I admit these are only conjectures but what shall I do else if Norm refuses to debate.

Francis Salmon

Sure, linearity is an absolute, but there is nothing wrong with saying almost linear.

Vertical is an absolute. There is nothing wrong with saying that something is almost vertical.

Immediately is an absolute. What's wrong with almost immediately?

Yes, a girl is either pregnant or not, and there is no such thing as almost pregnant, but it's a bad analogy in this case.

I admit these are only conjectures but what shall I do else if Norm refuses to debate.

Francis, we have not "refused to debate." We all had the debate on advantageplayer.com. I am trying to get a release out and simply do not have time to keep going over the same arguments again and again and again. You can go right on calling us all silly and claiming that sims don't work. But, please stop trying to pull me back into a dead argument. We have said what we have to say and frankly it's become boring.

The fact that you're using the 'pluralis majestatis' shows that you think you stand over the matter.You're not open to arguments that contradict your ideas even if they reconcile your sim with the longestablished theory of linearity.
I'm not rehashing old arguments. The points I made above about your sim are new and might explain why its results show erratic fluctuations.So,it would be helpful if you took position.

Francis Salmon