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Thread: soft21: Theory of Blackjack, Chapter 15

  1. #1
    soft21
    Guest

    soft21: Theory of Blackjack, Chapter 15

    Hi,

    I've been struggling to fully understand the 'Interactive Approximation' mechanism described in chapter 15 of TOBJ, and I'm hoping someone might be able to assist me.

    There are a few things I am unsure about, but my first bit of confusion comes right at the start, in the 6-card example; I don't understand how the figures for P12, P13 and P23 are arrived at.

    Given the formula -
    Pkl = (N/2)M + ((N-2)/2)Mkl - ((N-1)/2)(Mk+Ml)
    and the various M values, I get different results to the ones given.
    E.g.,
    M=516, M1=768, M2=636, M12=1080
    therefore,
    P12 = 3*516 + 2*1080 - 2.5 * (768+636) = 198
    [but the given figure is 190]

    I've been putting together some software to generate approximate dealer probabilities in the way that is described in this chapter. I get reasonable answers when doing it the simple, 'Method A', way; but my attempt at the two-card-removal, 'Method B' approach is way off.

    The fact that I don't really understand the theoretical basis behind this approach, and can't quite make sense of the introductory example leaves me rather clueless at this point. :-)

  2. #2
    Don Schlesinger
    Guest

    Don Schlesinger: Re: Theory of Blackjack, Chapter 15

    What edition is this?? I have no Chapter 15 in my copy.

    Don

  3. #3
    soft21
    Guest

    soft21: Re: Theory of Blackjack, Chapter 15

    It's the sixth edition. Chapter 15 is entitled 'Interactive Approximations To Facilitate Rapid Blackjack Computations'.

  4. #4
    Don Schlesinger
    Guest

    Don Schlesinger: Re: Theory of Blackjack, Chapter 15

    > It's the sixth edition. Chapter 15 is entitled
    > 'Interactive Approximations To Facilitate Rapid
    > Blackjack Computations'.

    Sorry; I have only the fifth edition and, frankly, I didn't even know that Peter had added that chapter to the sixth edition.

    Don

  5. #5
    soft21
    Guest

    soft21: Re: Theory of Blackjack, Chapter 15

    Ah, I see. I hadn't realised it was a new section, but I see from the RGE catalog that it was the only change in the sixth edition -

    [This newest edition of Griffin's masterpiece is identical to the 5th Edition, except for the addition of an 8-page Chapter 15, "Interactive Approximations to Facilitate Rapid Blackjack Computations." This chapter would be of interest to programmers who are looking to write faster game analysis programs by taking shortcuts, but without losing any significant amount of accuracy. If you already have the 5th Edition, and this topic is not useful to you, then you don't need this book. With the passing of Peter Griffin in October of last year, this will be the final version of his life's work. Ppb; 270 pages.]

    I'm a programmer, with a fairly basic grasp of mathematics, trying to do exactly what is described in the above quote.

    Without reproducing the whole thing, perhaps I can explain some of the difficulty I am having.

    To demonstrate a simple case of an 'interactive model', Prof. Griffin introduces a simple game dealt from a deck of 6 cards, numbered 1 to 6. The result of the game is given by dealing exactly 3 cards :- mutiply the two low cards, subtract the high card if it's odd, or add it if it's even, mutiply the result by 60.

    He lists the 20 possible 3-card subsets, together with their exact expectations (as defined above), their simple linear estimates, and their interactive estimates.

    So, for the subset 123, he gives -
    Expectation: -60, Simple Linear Estimate: -144, Interactive Estimate: 90

    He also gives the overall mean M=516, and the means after single-card removals M1=768, M2=636, M3=540. I'm able to see how these figures are computed - For M, average all 20 subsets; for each Mk, average all the subsets without card k present.

    We then have the single-card payoffs, given by the formula Pk = NM-(N-1)Mk [i.e. 6 * 516 - 5 * Mk]. P1, P2, and P3 are given and the values are what I would expect -
    P1=-744, P2=-84, P3=396. The above estimate, -144, is thus (-744 + -84 + 396)/3.

    So far, so good, and I've put together a spreadsheet which generates all the same simple linear estimates as are printed in the book, save for one; for the subset 456, the book figure is 1184, but I get 1176.

    He also gives the following for decks with 2 cards removed - M12=1080, M13=885, M23=690. Again, I get the same values in my spreadsheet, and have generated the rest of the Mkl values too.

    The formula given for a 'two-card payoff' is Pkl=(N/2)M + ((N-2)/2)Mkl - ((N-1)/2)(Mk+Ml) [i.e. 3*516 + 2*Mkl - 2.5*(Mk+Ml)]

    And yet, after listing the above values, the book says "from these we get P12=180, P13=-30, and P23=120, whose average value is 90, the interactive estimate for subset {1,2,3}"

    Applying the above formula, the figures which I get are P12=198, P13=48, and P23=-12.

    Almost by accident, I discovered that summing these values with the linear estimate gives the interactive estimate quoted in the book, not just in this case but in all 20 cases. E.g., -144 + 198 + 48 + -12 = 90.

    I wish I could understand the significance of this. I had assumed, as is implied by the "from these we get..." quote, that the interactive estimate is computed by averaging all the two-card payoffs, just as the simple linear estimate is calculated by averaging the single-card payoffs, but it looks like maybe that is not the case.

    So, to summarise, my initial stumbling block is to understand where the statement "from these we get P12=180, P13=-30, and P23=120 [...]" comes from, and also precisely how the calculated two-card payoffs are supposed to be used to generate the interactive estimate.

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