Optimal betting of multiple hands

  • Eric

    Optimal betting of multiple hands

    I’ve been trying to figure out the optimal bet for blackjack, given a certain number of hands and my advantage. I think my math is wrong, because I’m getting some funky results. Does anyone know the correct math for this? Read on for my attempt.

    I’m using the kelly function:
    bet = advantage * bankroll / variance

    Assume my bankroll is 1000, my advantage is 0.05, and the variance of 1 hand is 1.32.

    bet = 0.05 * 1000 / 1.32 = 38

    Now, you would assume that it would be be better to spread those 38 bets to 5 hands, rather than have them all on the same hand.

    Computing the bet again, for 5 hands this time, I get some weird results. The variance of 5 hands of blackjack is 16.20. (using the function 1.32*n + 0.48*n*(n-1) )

    bet = 0.05 * 1000 / 16.20 = 3.09

    3.09*5 hands = 15. Shouldn’t this number be greater than 38?

    Should I be dividing by standard deviation instead of variance?

  • KenSmith


    The variance you calculated for all 5 hands is fine, but you then need to divide it by 5 before using it in your bet-sizing formula, which expects the variance per bet.

    So, the average variance per bet for each of your five bets is 16.20 / 5 = 3.24.

    Your optimal bet on each hand is 0.05 * 1000 / 3.24 = 15.43.

    Your overall action of 15.43 * 5 = 77.15, which is as expected, quite a bit larger than the 38 units you should bet on a single hand.

    P.S. For anyone following along, the magic numbers in the equation are variance=1.32, covariance=0.48.

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