Kelly vs ROR math

Discussion in 'Skilled Play - Card Counting, Advanced Strategies' started by Midwestern, May 3, 2011.

  1. Midwestern

    Midwestern Well-Known Member

    so i just posted advice on ROR in another thread and then i got to thinking about Kelly Bet sizing.

    If im doing my math right, Kelly bet for a 1000 unit bankroll, a 1% edge, and 10$ units should be

    (( Payoff x probability of win) - probability of loss) / ( payoff)

    for blackjack, we use a 1-to-1 payoff. If we assume a 1% long term edge because of AP, then our formula should be:

    ((1)(.51) - .49) / 1

    which gives us an optimal kelly bet of .02 of our bankroll or = 200 for a 10,000 (or 1,000 unit bankroll.) now this number is certainly not equal to 1 unit, because the kelly criterion assumes infinite trials at the same odds.

    my questions
    1) at what point in the bet ramp should this be placed so that our ROR is not affected? TC3? TC4? is this maxbet? certainly not TC1 or TC2.
    2) if we change bankroll size, this kelly bet size changes. Is it proper to have bankroll "X" amount to figure ROR and then bankroll "Y" ?

    ive read that alot of people use half kelly or .3 kelly.
    but that could mean anything given people's bet-ramps.

    basically, has anyone done research to find optimal kelly betsizing for a given Bankroll size and target ROR? ideally the optimal kelly bet would be the bet which grows the BR fastest, without increasing the ROR.
  2. Midwestern

    Midwestern Well-Known Member

    im gonna do some searching on and other websites but i figured id ask my homies on this website if any of you have a good 'quick and dirty"
  3. Sonny

    Sonny Well-Known Member

    You don't want to use that formula for BJ because the payoffs are not even money. With BJs, splits, doubles and surrenders the payouts can vary quite a bit. Instead you can use a simplified version of the Kelly formula which is Edge/Odds. For BJ that translates roughly to Advantage/Variance. The variance is usually around 1.33 units per hand while the advantage fluctuates. You can use the simplified formula for each TC to set up your bet ramp, although you'll have to fudge it a bit since you have min/max limitations. For example, you might have a max bet that you don't want to exceed (for bankroll or heat reasons) and/or a minimum bet that you must make when you don't have an advantage.

    The basics are explained starting in lesson 8 of the Blackjack School and there are several good (and math heavy) examples on bjmath. Blackjack Attack also has great chapters on bet spreads and RoR for different games and playing styles.

  4. Midwestern

    Midwestern Well-Known Member

    thanks sonny this is great. The advice which really helped was figuring out each kelly bet for my estimated advantage at each TC

    i eyeballed the chart above and i now see that TC 1 id be using advantage 0.15, TC 2 advantage 0.75%, TC 3 advantage 1.5%, TC 4 advantage 2%

    then using edge/1.33 variance, kelly would tell me to bet the following betramp for a 1000 unit bankroll

    TC1= 1 unit
    TC2= 5.5 units
    TC 3 = 11.27 units
    TC 4 = 15 Units

    The kelly amounts i get for my bet ramp are much larger than my original bet ramp of 1,4,8,12 So this means that now my BR has a higher ROR, but at least i'm growing it at an Optimal Rate instead of underbetting my advantage.
  5. tthree

    tthree Banned

    What am I missing?

    This is where my knowledge is really weak. Wouldnt you want your optimal bet ramp to have a low risk of ruin? It seems like you would be resizing your bets in dollars relative to your bankroll fluctuations in order to keep the same unit to bankroll ratio and a RoR of for all practical purposes zero.

    If you up your RoR to even 1% wouldnt you guarantee ruin if you played for a long enough time? Midwestern's original post seemed to indicate he understood this but then he seemed excited by the prospect of abandoning it.

    Am I missing something or is this just a gamblers mentality? If you start with a bankroll that demands a higher RoR isnt the game to grow your BR enough to reduce your RoR to all but zero before increasing either your ramp or your unit size? I would think strict adherence to this concept should all but guarantee reaching the long term results desired without the long term effect insuring ruin be an event you experience.

    Maybe Midwestern is still in the process of reaching the all but zero risk of ruin and thinks the increased in variance will help him get there. I have no choice but to have an undesirable RoR but look forward to hopefully reducing it to all but zero. I figure at that point, if Im lucky enough not to experience ruin before getting there, I should be able to follow this model and optimize from that point.

    When I reach a desired return rate i can just grow my bankroll on average and hope the variance doesnt cause a setback large enough to have me revert to a prior step in the process where RoR must be increased.

    Any comments from those who are experts on the subject who have been there and done that? I know it sounds easier than it is.
  6. Midwestern

    Midwestern Well-Known Member

    Yes. I calculated a 2.5% risk of ruin with my previous betramp and bankroll, which was low enough for my tolerance. However, i felt that my bet ramp of 1 4 8 12 was sub-optimal and this post was me exploring ways to make my bet ramp optimal. I.e. i was underbetting higher-advantage situations. Finding this out allows me to adjust my bet ramp in order to to capture more EV in those high-TC situations

    Yes! the kelly criterion is exactly this: for an advantageous gambling situation, the "kelly bet" is the proportion of your bankroll you should bet in those situations so that you MAXIMIZE growth of the bankroll and minimize risk of ruin. and because the bet size is "proportional" to bankroll, theoretically you should never go bust. The challenge here is to re-size the bets according to a fluctuating bankroll (if you are losing) to keep the zero risk of ruin. However, i figure if i DIDNT re-size and i started losing, at worst i'd still have the 2.5 % ROR (or slightly higher) as i calculated on my previous betramp.

    I was most excited on the prospect finding a Bet ramp of maximizing growth of my BR. Much like a spread of 1-4 is a losing spread for most 6D shoe games, my previous betramp, although adequate, may not have been optimal.

    here is where you nailed it on the head. I am comfortable with my risk of ruin at 2.5% and the increased variance and full-kelly betramp should help me maximize the growth of my BR. i'm just excited that the math worked out as i needed it to for my BR.
    Basically, i realized i can afford this hobby and i have enough to play properly :grin:
  7. kewljason

    kewljason Well-Known Member

    That "theoretically, you should never go bust" wording is where the problems come in. Kind of like how the Martingale looks good in theory, but in real life you run up against betting limitations. With Kelly if you keep resizing downword, you will not be resizing your entire spread as most likely the minimum wager is set by table minimums, so you are resizing your top wager and reducing your spread. this makes it more difficult to recoup earlier losses. Imagine trying to win back losses acheived with a $10-$100 spread when kelly resizing has now reduced your spread to $10-$50. :eek:

    Another observation about complete optimal betting is that you have to round off those amounts which by nature immediately makes it not 100% optimal. Let's say you were a quarter player. Your previous 1,4,8,12 betting ramp would have you wagering nice even amounts of $25, $100, $200, $300. Not great for cover purposes, but extremely efficent as far as time. If you get a nice juicy heads up game with dealer, you can bang out 100's of hands in an hour. But even rounding optimal spread, you are wagering $25, $150, $275, $375. That actually doesn't seem to bad but still would slow the game down slightly, with blackjack and insurance payouts at those high wager amounts, so your optimal spread can actually cost you money. :rolleyes:
  8. The Chaperone

    The Chaperone Well-Known Member

    Actually the Martingale looks terrible in theory.
  9. Midwestern

    Midwestern Well-Known Member

    haha chap i dont theory, martingale works as long as you have infinite funds and no capped max bet
  10. tthree

    tthree Banned


    Isnt the martingale the US governments favored way of money management.
  11. Midwestern

    Midwestern Well-Known Member

    I like this idea alot though for long-term bankroll growth. By figuring out whatever kelly bet we would do at different TC (i.e. the betramp i figured out earlier)

    As the bankroll grows, the size of your bets grow when you have advantage, but at lower TC situations, Kelly still calls for minimum bets.

    Basically this means you increase/decrease your spread proportionally according to your bankroll growth.

    KJ, that means that even though you play 25-400, you should try playing 5-400, beacuse you are probably overbetting in some low-edge scenarios. I can see you getting away with this by betting 25 off the top of a shoe (so a PB eyes you as a $25 player and still thinks you spread 1-16), but scale it down little by little so that you're betting 5 if the count isnt fully TC+1. This increases your true spread to 1-80 which sounds to me like a very powerful game.
  12. blackjack avenger

    blackjack avenger Well-Known Member

    Conservative Kelly Resizing Wins

    Betting .5 kelly has the same growth rate as 1.5 kelly but the former has far less variance.

    Betting double kelly the bank meanders up and down.

    Betting over double kelly your bank shrinks.

    Given the above it's probably best to be sure to always bet less then kelly.

    Probably way less then kelly if you have a large bank.
  13. tthree

    tthree Banned

    The big rolls I talk to like 1/4 kelly.
  14. blackjack avenger

    blackjack avenger Well-Known Member


    Fixed ror is just that. A 1% ror means you have a 1 in 100 chance of losing all.

    In theory Kelly continuous resizing does not lose all, but in the real world one can lose so much they cannot play due to table minimums. Also, we are not sure of our advantage and Kelly punishes for overbetting. With smaller banks half Kelly may be reasonable with larger banks using third to fourth Kelly. The more important bank preservation the more conservative the bets.
  15. To predict the magnitude of fluctuation,
    I recommend this book,

    “An introduction to stochastic modeling Third Edition ”
    by Samuel Karlin, Howard M. Taylor

    It's better than Don's BJA3.
  16. blackjack avenger

    blackjack avenger Well-Known Member

    theory vs reality

    .992 for 1/4
    .998 for 1/5
    Chance of never losing half with continuous resizing. So in theory 1/5 does not give much more safety. In the real world risk of drawdown is probably higher due to human error.

    A problem with 1/4 is you still may have to resize down on losses which raises N0. At 1/8 + Kelly, resizing is less of an issue, which some pros use.

  17. "Chance of losing half with continuous resizing" is
    0.78125% for 1/4 Kelly
    0.1953125% for 1/5 Kelly
  18. blackjack avenger

    blackjack avenger Well-Known Member


    For catching my error in text. I believe I fixed it. On your numbers, well yeah if you want to get precise!:)

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