Effect of Standard Deviation
  • Blackjack basic strategy is based upon optimizing Expected Value. That is, for each decision conventional strategy recommends the action (doubling, hitting, surrendering, insuring, etc.) which has the highest EV. Intuitively, however, in "borderline cases" standard deviation ought to be looked at. That is, a decision such as doubling, splitting, or not-surrendering involves accepting higher risk, which should clearly be taken into account in making strategy decisions.
    Players who completely accept the Kelly principles will seek to optimize, not their EV but their Expected Logarithm (EL). Given this, they should select for each strategy decision, the choice which produces the greater EL.

    EL = [ m - f * SS / 2 ] * f
    where
    m = Mean Earnings
    SS = Sum of Squares
    f = Fraction of Bankroll bet on the Hand

    Now, when small amounts of the bankroll have been wagered, the second term is negligible and EL is just f * EV. So maximizing EL will be the same as maximizing EV when f is small. However as f increases, then the SS term becomes more important. Indeed, if the decision of highest EV also increases Variance (such as doubling or not-surrendering) then we may compare the EL vs. Ev for these decisions and obtain the correct play. The decision of lower-risk would be preferred.

    Examples:
    1. The Basic Strategy for 6 decks calls for not surrendering 87 vs. 10. However this a very close play. This means that a Kelly player who had only bet 1 unit on this play would not be justified in following the Basic Strategy unless the bankroll exceeded 1,470 units!
    2. BS calls for doubling Ace-2 vs. 5 in <=9 decks. However in 8 decks, more than 200,000 units are required to make this play. Even in 6 decks, the required bank for playing basic strategy is 576 units. <br />3. If you know that your first card will be an Ace, and if your bet is large enough, you should violate BS and not double. Even a lucrative double, like A7 v 6, is not justified if more than 13.7% of the bank has been bet.
  • Alex,

    EL = [ m - f * SS / 2 ] * f

    You need more parenthesis to clarify this otherwise this can be misconstrued many ways.


    SS = Sum of Squares

    Sum of what squares? Bets to date vs bankroll? Pit bosses to BJ tables in LV casinos?


    The Basic Strategy for 6 decks calls for not surrendering 87 vs. 10.

    BS recommends any (well definately the most common rules) 6D game with late surrender should surrender any 15 vs dealer 10.
  • slimeo said:
    Alex,

    EL = [ m - f * SS / 2 ] * f

    You need more parenthesis to clarify this otherwise this can be misconstrued many ways.


    [quote]SS = Sum of Squares

    Sum of what squares? Bets to date vs bankroll? Pit bosses to BJ tables in LV casinos?


    The Basic Strategy for 6 decks calls for not surrendering 87 vs. 10.

    BS recommends any (well definately the most common rules) 6D game with late surrender should surrender any 15 vs dealer 10.[/quote]

    Well, the SS = Sum of Squares of Pit bosses.
    You are absolutely right on the money!

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