The “why” of the floating advantage

[Three]

I just said he started a proof with an assumption. If that was the only way he proved it I find it sloppy at best. But if he had proved it another way it is a different story. I never said he was wrong. Anytime I said anything Don being right or wrong I said I believed he was right. Obviously from a basic strategists point of view there is no difference between early and late in the shoe. But you can get an advantage without doing anything to get it or even knowing you got it. Eric work in the thread at least showed the BS HE never changed as depths got deeper until a bias for rounds played at a low TC kicked in due to high TC having already used up the cards. I didn't expect to see anything different.

It would have been interesting to see the depth study I challenged DogHand to do. The Hilo players would have learned a lot from that sim but I never thought it would show anything different. Only that it would prove with starting with an assumption that Don was right by proving the assumption fact. Don, I know you are confident it isn't an assumption and why. I agree with the logic but don't see it as proof rather than assumption. That has been my point from my first post. I just thought a depth based sim would be really interesting from a counting standpoint. I have never seen one published.


This is what I said about Don's explanation in my first post in the original thread:

I think Don said he explained it is his "Eureka!" paragraph at the bottom of page 70. He may be right but I am not sure the basic strategists advantage must be constant on average throughout the shoe is a given[/B]. If I am right that he feels this is the explanation, then the reason is that the advantage above TC +4 and below TC -4 falls at deep pen so the advantage in-between must increase. If you accept that BS advantage must be constant on average at different penetration levels throughout the shoe this must be true. I am not sure that you can jump to that conclusion.

After that I just pushed for a sim that wasn't based on an assumption that BS advantage is the same at all pen to prove Don right. The sims Don proposed all started with that assumption.

I said this about what I expected to be shown if DogHand simmed what I suggested:

In the process of actually proving Don right you get all this bonus data to learn from.

Everything in between was lobbying for the sim to be run. Unfortunately the two that took the challenge chose not to do it in a way that gave us the advantage by depth study for Hilo. I appreciate their effort but I was so sure Don was right that I didn't want people to waste their time unless very valuable depth based counting sim data was collected in the process. Prove a mundane aside we all are sure is right and get lots of useful data for counters at the same time.

Somehow my lobbying for the sim by finding oddities that may support the very small chance that the assumption is not true Don decided I didn't understand. I am surprised Don didn't understand my point was you can't prove anything that is based on an assumption. And until there is proof the BS players advantage never changes throughout the shoe it is an assumption that is almost certainly right. Don decided that despite me saying many times I thought he was right but his proof was flawed he decided I disagreed with him. I based everything on my view that the Basic Strategist must have the same advantage throughout the shoe was not a proven fact. I never believed or stated I thought it was incorrect. Again if Don had answered that his assertion that the BS advantage never changes was proven in some other way than it makes sense so it must be true we would not have had me in the discussion at all after my first post.

If you accept that BS advantage must be constant on average at different penetration levels throughout the shoe this must be true. I am not sure that you can jump to that conclusion.

But he never said that, which implied its truth is based on logic rather than sim data. I always logically believed Don was right but would never think using a logical assumption in a proof is something anyone with a strong math background would do. But Don's silence on the requested info on any other basis for the assumption being fact left it an assumption in my book and not eligible to be used to prove anything. Without proving this first Don's balloon logic is a mathematical nonstarter, because it is based on an assumption rather than fact.

[DSchles]

To clarify: I am not a theoretical mathematician. But the statement by Three that, "And until there is proof the BS players advantage never changes throughout the shoe it is an assumption that is almost certainly right" is disingenuous and enraging. THE PROOF EXISTS. I'm sorry if I can't furnish it for you. But then, as you're railing against my failure to understand your convoluted logic, answer this: since the proof exists, but I can't furnish it for you, does that mean it doesn't exist? Oh!

GO FIND IT SOMEWHERE! I don't have to do your work for you. Don't act so entitled. Go do some legwork and find it. But this much is NOT an assumption: there is a proof. Period. And, if you think the logic is that, because you don't have it in front of your eyes, then it is just an assumption, then you're a bigger ass than I imagined.

As for Angle, Optimus wrote this to you: "Your hangup appears to be confusing HE with advantage at TC 0. HE doesn't change with how many cards remain. Advantage at TC 0 does. That's it. I've said all I can say." Your response to him, which was patently false, shows your continuing misunderstanding of the entire discussion. You answered: "For the BS player they are interchangeable." NO, THEY ARE NOT!!!!!! After five of six decks are dealt, the basic strategist has precisely the same GLOBAL edge as he had at the start of the shoe. But his edge specifically at TC = 0 is greater than at the start, while his edge at extreme positive and negative counts is less. Can you understand that?

Next, you ask, "but WHY?" And that was answered as well. And the balloon analogy answers it FOR ALL COUNTS, not just the extreme ones. You claim to understand the extremes (more pushes, failed doubles). But then you want to understand the mechanism at zero. It's a CLOSED SYSTEM. If the extremes are less and the total is the same, then the middle is MORE! This isn't even mathematics; it's physics.

Anyway, as was the case in the first thread that you and Three hijacked, this will be my last post in this thread, as well. There is a limit to my patience, and sometimes, I have to be satisfied that I've explained something satisfactorily to only 98% of the people. That I haven't succeeded for 100% is much less a commentary on my skills as an educator as it is on those of the 2% to understand. It might have been nice for me to get 100%, but I'm going to try not to lose any sleep over it.

Don

[DSchles]

Follow-up to the above. After writing that I couldn't find a proof about the invariability of the BS edge throughout the pack, I remembered something: my correspondence with Ed Thorp on this very topic!

In any event, the June 1993 issue of Snyder's Blackjack Forum, entitled "Thorp Speaks," contains an enlightening article on ... "Does Basic Strategy Have the Same Expectation for Each Round?" The article is 18 pages long! I'll summarize the findings: YES, it does!

He also answers other questions, such as, "Does it matter if others are at the table?" Answer: no. "Does it matter if the others use a different (not basic) strategy to try to affect the odds?" Answer: No.

Case closed.

Don

[k_c]

Ah, that's VERY different than saying there are 27 subsets with RC = 0! 26, 0, 26 might comprise 24 deuces and two 3s or it might comprise 24 5s and 2 6s. You're talking about VERY different subsets there. In reality, there are so many subsets with RC = 0, that I don't even know how to express the number that describes them.

Don

There would be a rather large number of subsets if each rank is considered separately. As an example this program has computed the total number of 26 card subsets with running count = 0 dealt from a single deck considering each rank independently:

Number of decks: 1
Cards in subset: 26

Number of subsets: 112695
Subsets are entered in subsets_26_0.txt
List subsets = 112695
Prob RC = 0: 0.124165

Press any key to exit

This program has computed the number of subsets by considering the 3 groups of cards that comprise the HiLo count (There are only 7 HiLo subsets of 26 cards from a single deck with RC=0 when only the groups are considered):

Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: 0
Subgroup removals: None
Specific removals (1 - 10): {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

....computing, please wait

Number of subsets for above conditions: 7
Prob of running count 0 with above removals from 1 deck: 0.124165

p[1] 0.0769231 p[2] 0.0769231 p[3] 0.0769231 p[4] 0.0769231 p[5] 0.0769231
p[6] 0.0769231 p[7] 0.0769231 p[8] 0.0769231 p[9] 0.0769231 p[10] 0.307692

Press x or X to exit program (it may take some time to close,)
any other key to enter more data for same count tags/decks:

Notice that the probability of a 26 card subset with running count = 0 is the same using either approach. The probability of a given rank could also have been computed using the first approach. The second approach allows for dealing with 8 decks whereas with the first approach even 2 decks takes a long time. 6 decks would be virtually untenable using the first approach. The maximum number of total subsets occurs at mid-shoe. There are 1868755 26 card subsets and 375268773 2 deck 52 card subsets. The maximum number of 6 deck subsets would be extremely large.

k_c

[BoSox]

He also answers other questions, such as, "Does it matter if others are at the table? Answer: no. Does it matter if the others use a different (not basic) strategy to try to affect the odds? Answer: No.

Case closed.

Don, shouldn't there be another " at some point?

[Freightman]

Don, shouldn't there be another " at some point?

Bosox, I agree with you. Don’s flaunting of a missing - “ - may be characterized as an egregious affront to one’ s grammatical sensibilities. My righteous indignation knows no bounds. Don may seek redemption through judicious use of those associated traits - grovelling and sucking up - by which, execution at their highest levels, may as be considered as an art form.

Further, I must point out, correction of my “auto correct”, which, in this post, had previously identified you as Bison. Mind you, Bison is better than Bisexual, which my auto correct has also previously identified you as.

Welcome back.

[DSchles]

Don, shouldn't there be another " at some point?

Yup. Thanks. Fixed.

Don

[BoSox]

Don may seek redemption through judicious use of those associated traits - grovelling and sucking up - by which, execution at their highest levels, may as be considered as an art form.

Freighter, are you not doing the same exact same thing "groveling"? Hoping that your ball scratching side count will appear in the BJ ATTACK 4 EDITION.

[SiMi]

http://www.edwardothorp.com/wp-conte...rEachRound.pdf

[Midwest Player]

I have to admit I thought basic strategy would not have the same expectation through out the shoe. However, I really didn't follow this thread very closely because "Three's" posts are so long and difficult to understand. However, if Don and Thorp say it is so I will believe them.

I spent a few minutes and did a google search. I found this. I don't know if this is what Don meant is the proof by Thorp.
http://www.edwardothorp.com/wp-conte...rEachRound.pdf

[Midwest Player]

http://www.edwardothorp.com/wp-conte...rEachRound.pdf

You beat me by 7 minutes. I'm slow at typing.

[DSchles]

Nice that these things have found their way to the Internet. In any event, maybe next time when I mention that something has already been proven, the naysayers will listen.

Don

[ericfarmer]

Ah, that's VERY different than saying there are 27 subsets with RC = 0! 26, 0, 26 might comprise 24 deuces and two 3s or it might comprise 24 5s and 2 6s. You're talking about VERY different subsets there. In reality, there are so many subsets with RC = 0, that I don't even know how to express the number that describes them.

Don

There are "only" 233,715,029 distinct subsets of 52 cards from a 6-deck shoe with RC=0. A lot, but not necessarily intractable. But as k_c says, more than I'm interested in evaluating .

(This is the coefficient of x^52 in the generating function in x and y as shown below.)

SeriesCoefficient[
    Sum[(x/y)^k, {k, 0, 24}]^5
    Sum[x^k, {k, 0, 24}]^3
    Sum[(x y)^k, {k, 0, 96}]
    Sum[(x y)^k, {k, 0, 24}],
  {x, 0, 52}, {y, 0, 0}]