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Thread: Kelly formula for multiple outcomes with different payouts?

  1. #1


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    Question Kelly formula for multiple outcomes with different payouts?

    Per the title how do you calculate a Kelly fraction of your bankroll when there are outcomes with different payouts?

    Note - to create a positive expectation I have removed all 8 rank cards from the shoe.

    For example the bac Dragon 7 side bet probability is now .. 0.0276273404125582 .. payout 40 to 1 thus using ..

    f = p(b + 1) - 1 / b =

    0.00276..(40 + 1) - 1 / 40 = 0.3318%


    Now on the other hand the Dragon Bonus (excuse the ambiguity) side bet has the following payouts ..


    Win by 9 30.00
    Win by 8 10.00
    Win by 7 6.00
    Win by 6 4.00
    Win by 5 2.00
    Win by 4 1.00
    Natural win 1.00
    Natural tie 0.00
    Loss -1.00


    How would I change the simple formula above to handle all of these payouts and their probabilities?

    I found an app online which displays this bets (player side) Kelly as one value. I would like to calculate that.

    Thanks!
    Last edited by Super Natural; 03-22-2021 at 07:59 PM.

  2. #2


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    I'm not understanding. What does a Kelly fraction have to do with this? If you have no edge for any of the bets, then Kelly doesn't apply. What am I missing?

    Don

  3. #3


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    Don - that is a fair Q .. I am showing Kelly values at a full 8 deck shoe.

    I am writing a simulator and before each round I am calculating the Kelly value to decide whether and how much to bet.

    Thanks for the time.

  4. #4


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    Quote Originally Posted by Super Natural View Post
    Don - that is a fair Q .. I am showing Kelly values at a full 8 deck shoe.

    I am writing a simulator and before each round I am calculating the Kelly value to decide whether and how much to bet.

    Thanks for the time.
    But you're using the term "Kelly value" incorrectly. If the expectation is negative, the proper Kelly bet is zero. So, your example didn't make any sense.

    Don

  5. #5


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    Quote Originally Posted by DSchles View Post
    If the expectation is negative, the proper Kelly bet is zero.


    Don

    Sounds good .. I have updated my post to make sense.

  6. #6


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    Quote Originally Posted by Super Natural View Post
    0.00276..(40 + 1) - 1 / 40 = 0.3318%
    So, since the payout is 40 to 1, the correct Kelly wager for this bet is 0.3318%/40 = 0.0083% of your bank. Hardly worth playing.

    The answer to your second question involves extremely complicated math. I refer you to the following link:

    https://math.stackexchange.com/quest...n-two-outcomes

    Don

  7. #7


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  8. #8


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    Thanks Don I had visited that link before and came running to BJTF after reading log and sigma which I haven't used since pre-calc.

    I will give it a better read.

    Quote Originally Posted by DSchles View Post
    So, since the payout is 40 to 1, the correct Kelly wager for this bet is 0.3318%/40 = 0.0083% of your bank.
    I don't follow. I am already dividing by 40 in the formula ...

    f = p(b + 1) - 1 / b =

    f = 0.00276..(40 + 1) - 1 / 40 = 0.3318%

    Quote Originally Posted by DSchles View Post
    Nice .. it has a calculator I can compare results with.

  9. #9


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    Quote Originally Posted by Super Natural View Post
    I don't follow. I am already dividing by 40 in the formula ...

    f = p(b + 1) - 1 / b =

    f = 0.00276..(40 + 1) - 1 / 40 = 0.3318%
    Ah. OK. Sorry about that. Thought the formula you were using was just to calculate the edge. But it's the Kelly fraction. My bad.

    Don

  10. #10


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    Using the formula from the math stack exchange, I tried the simpler baccarat main wagers .. (Don, ride with me on these values being negative )

    Player
    f(x) = 0.45860log(1-x)+0.44625log(1+x)+0.09516log(1+0x) , optimum x -0.013649

    Banker
    f(x) = 0.44625log(1-x)+0.45860log(1+0.95x)+0.09516log(1+0x) , optimum x -0.12308

    Tie
    f(x) = 0.44625log(1-x)+0.45860log(1-x)+0.09516log(1+8x) , optimum x -0.017946


    These match the bac calculator app I am comparing results with, all good. (app Kelly value is multiplied by 100). Now for the dragon bonuses ..

    Dragon Bonus Player
    f(x) = 0.69224log(1-x)+0.00368log(1+30x)+0.00682log(1+10x)+0.01792log( 1+6x)+0.02826log(1+4x)+0.03324log(1+2x)+0.03737log (1+x)+0.16259log(1+x)+0.01787log(1+0x) , optimum x -0.004023

    Dragon Bonus Banker
    f(x) = 0.69933log(1-x)+0.00307log(1+30x)+0.00566log(1+10x)+0.01590log( 1+6x)+0.02384log(1+4x)+0.03146log(1+2x)+0.04024log (1+x)+0.16258log(1+x)+0.01787log(1+0x) , optimum x -0.013016


    Player bonus is close but not quite. Banker bonus I am way off. Did I miss something or is the app wrong?

    Functions can be viewed here .. https://www.desmos.com/calculator/jose0b9gtj
    Attached Images Attached Images
    Last edited by Super Natural; 04-03-2021 at 08:24 AM.

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