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Thread: Myooligan: bet sizing question

  1. #1
    Myooligan
    Guest

    Myooligan: bet sizing question

    I don't suppose there's a right answer to this, I'm just interested in how people would approach this:

    I currently have a bankroll of $2k, and will be able to add $1k per month from non-BJ sources. I'm aiming to play 20hrs/month, minimum.

    Do I bet full Kelly for my current bankroll? If I drop to $1000, do I half my unit size? Or keep it the same, knowing that my bank will be up and going in another two months?

    Now that I think about it, I'm sure there's a mathematical solution to this. Anybody?

  2. #2
    Don Schlesinger
    Guest

    Don Schlesinger: Re: bet sizing question

    > I don't suppose there's a right answer to this, I'm
    > just interested in how people would approach this:

    Actually, there is a sort of right answer, but, as usual, there's an "it depends" involved. :-)

    > I currently have a bankroll of $2k, and will be able
    > to add $1k per month from non-BJ sources. I'm aiming
    > to play 20hrs/month, minimum.

    And, what will be your starting unit size and spread? I ask, because, frankly, with $2K, there isn't a whole lot you can do right now.

    > Do I bet full Kelly for my current bankroll?

    If you had, say, $20K, instead of $2K, would you be betting full Kelly, if no more funds were available?

    > If I drop to $1000,

    Which has a 50% chance, if you bet full Kelly right off the bat, with your $2K.

    > do I half my unit size? Or keep it the same,
    > knowing that my bank will be up and going in another
    > two months?

    Depends on whether you want to be playing BJ for two months or sitting on the sidelines. I'd cut back, but with $1K, you're effectively out of business. You can't cut back from $2K and make enough to make it worthwhile for your time.

    > Now that I think about it, I'm sure there's a
    > mathematical solution to this. Anybody?

    Some people would tell you to take the present value of your income stream and to play as if you had all of that money right now. That's fine, except that, you don't have all of that money right now, and you will eventually hit a losing streak, which might put you out of business for a while.

    So, there really isn't one "right" answer. I think I'd play as if I had a several-thousand-dollar bankroll, but then I'd play half or third Kelly, rather than full Kelly.

    Don

  3. #3
    Myooligan
    Guest

    Myooligan: Re: bet sizing question

    Don! Good to hear from you. And thanks for answering. . .

    > And, what will be your starting unit size and spread?

    Spreading $10-$50.

    > If you had, say, $20K, instead of $2K, would you be
    > betting full Kelly, if no more funds were available?

    Yes, although. . . This is a separate question, but related: When people ask me "at what point do you throw in the towel." For instance if I were to reach a 4 sd loss without ever having been in the black, would I quit then? When discussing it with non-ap'ers, I simply avoid the question, because I know they'll think I'm a lunatic. But my real answer is, "After thoroughly reviewing and analyzing the results, as long as I'm convinced I have an advantage, I'll keep putting new money into it." And I do this under the assumption that as one's bankroll and play time approach infinity (no comments about the difference between my current bank and infinity, please), ROR also approaches 0.

    I suppose I should make sure I'm correct on this before committing my income for the next 30 years to gambling.

    But to answer your above question, I would bet full Kelly with a non-renewing $20k bank, but I'd scale my bet sizes back (or forward) continuously so that I'm never overbetting kelly for the current bankroll.

    > Which has a 50% chance, if you bet full Kelly right
    > off the bat, with your $2K.

    Good, although unpleasant, to know.

    > Some people would tell you to take the present value
    > of your income stream and to play as if you had all of
    > that money right now. That's fine, except that, you
    > don't have all of that money right now, and you will
    > eventually hit a losing streak, which might put you
    > out of business for a while.

    Present value, that's the answer I was looking for. Maybe one day I'll actually be able to put the concept to use.

  4. #4
    MJ
    Guest

    MJ: Re: bet sizing question

    > Which has a 50% chance, if you bet full Kelly right
    > off the bat, with your $2K.

    So, whenever you are playing at full Kelly there is a 50% chance of losing 1/2 your starting BR assuming you never resize your unit. OK, but what are the chances of losing 1/2 your starting BR if you are playing at either .8 or .4 kelly? Is there a mathematical formula that can be used to figure this out like the one I use below? If so, please provide an example.

    A while back you said to determine the %ROR for any given Kelly factor simply raise .135 to the reciprocal of the Kelly factor. IE for .4 Kelly, this correlates with a .135 ^ (5/2) = .66% ROR.

    MJ

  5. #5
    Don Schlesinger
    Guest

    Don Schlesinger: Re: bet sizing question

    > So, whenever you are playing at full Kelly there is a
    > 50% chance of losing 1/2 your starting BR assuming you
    > never resize your unit.

    Yes. And the chance and the bankroll percentage always add to 100%. For example, if you want the chance of losing, say, 30% of your bank at any time, it's 70%.

    > OK, but what are the chances
    > of losing 1/2 your starting BR if you are playing at
    > either .8 or .4 kelly? Is there a mathematical formula
    > that can be used to figure this out like the one I use
    > below? If so, please provide an example.

    I'm sure there must be, but it doesn't come to mind immediately. Let me think about it. Related chart below, taken from Red Taylor's classic Kelly FAQ on Richard Reid's bjmath site, considers a related topic, when you don't resize and have a goal of doubling.

    > A while back you said to determine the %ROR for any
    > given Kelly factor simply raise .135 to the reciprocal
    > of the Kelly factor. IE for .4 Kelly, this correlates
    > with a .135 ^ (5/2) = .66% ROR.

    Right.

    here's that post I referred to above:

    "Suppose you make optimal small bets (that is, with advantage less than 10%) and keep playing until you lose half your bankroll or double your bankroll, whichever comes first. In the following chart, "Failure" is given by 1/(1+2^(2/k - 1)) and is the chance that you will end up with only half your original bankroll if you play until you either halve or double your bankroll, whichever comes first."

    Kelly Frac. Failure

    100% 33.33%

    80% 26.12%

    60% 16.56%

    50% 11.11%

    40% 5.88%

    30% 1.93%

    20% 0.19%

    10% 0.0002%

    Note that, for pure Kelly, when you're constantly resizing, the values are quite different from not resizing and not stopping when you hit a goal. Both of these make risk of losing half much more than in the baove chart.

    Don

  6. #6
    MJ
    Guest

    MJ: Re: bet sizing question

    > Yes. And the chance and the bankroll percentage always
    > add to 100%. For example, if you want the chance of
    > losing, say, 30% of your bank at any time, it's 70%.

    Thanks for the information. I was not aware of this relationship you describe. For the above to hold true, does that assume that you stop playing when you double the starting BR?

    > I'm sure there must be, but it doesn't come to mind
    > immediately. Let me think about it. Related chart
    > below, taken from Red Taylor's classic Kelly FAQ on
    > Richard Reid's bjmath site, considers a related topic,
    > when you don't resize and have a goal of doubling.

    > "Suppose you make optimal small bets (that is,
    > with advantage less than 10%) and keep playing until
    > you lose half your bankroll or double your bankroll,
    > whichever comes first. In the following chart,
    > "Failure" is given by 1/(1+2^(2/k - 1)) and
    > is the chance that you will end up with only half your
    > original bankroll if you play until you either halve
    > or double your bankroll, whichever comes first."

    > Kelly Frac. Failure

    > 100% 33.33%

    > 80% 26.12%

    > 60% 16.56%

    > 50% 11.11%

    > 40% 5.88%

    > 30% 1.93%

    > 20% 0.19%

    > 10% 0.0002%

    > Note that, for pure Kelly, when you're constantly
    > resizing, the values are quite different from not
    > resizing and not stopping when you hit a goal. Both of
    > these make risk of losing half much more than in the
    > baove chart.

    Im a bit confused. Is there resizing of the unit or isn't there in the above chart? You wrote toward the top "when you don't resize and have a goal of doubling". In the above paragraph you seem to be implying there was resizing of the unit. Which is it?

    BTW, I plugged in the Kelly values for the formula and get the corresponding % failures. It seems like you answered exactly what I asked; what is the % of losing 1/2 the starting BR given a certain Kelly-factor. Your formula answers that question quite nicely. :-)

    MJ

  7. #7
    Don Schlesinger
    Guest

    Don Schlesinger: I forgot about this

    Go here. It answers every question you could possibly have on the topic.

    Don

    www.bjmath.com/bjmath/proport/riskpaper1.pdf

  8. #8
    MJ
    Guest

    MJ: Kelly Question for Don

    > Yes. And the chance and the bankroll percentage always
    > add to 100%. For example, if you want the chance of
    > losing, say, 30% of your bank at any time, it's 70%.

    Don, you wrote above that when you bet full-kelly, then the percentage chance and BR percentage always add up to 100%. So there is a 10% chance of losing 90% of your BR if betting at full-kelly. Similarly, there should be a 1% chance of losing 99% of your BR if betting full-kelly.

    NOW, if there is only a 1% chance of losing 99% of your BR, then how can there be a 13.5% chance of losing 100% of your BR? If the relationship you described at the top is correct, then losing 100% of your BR is impossible. What am I missing here? Thanks.

    MJ

  9. #9
    Don Schlesinger
    Guest

    Don Schlesinger: Re: Kelly Question for Don

    > Don, you wrote above that when you bet full-kelly,
    > then the percentage chance and BR percentage always
    > add up to 100%. So there is a 10% chance of losing 90%
    > of your BR if betting at full-kelly. Similarly, there
    > should be a 1% chance of losing 99% of your BR if
    > betting full-kelly.

    If I said it, then oit must be true! :-) And, it is.

    > NOW, if there is only a 1% chance of losing 99% of
    > your BR, then how can there be a 13.5% chance of
    > losing 100% of your BR? If the relationship you
    > described at the top is correct, then losing 100% of
    > your BR is impossible. What am I missing here? Thanks.

    Losing 100% of your BR is, indeed, theoretically impossible with pure Kelly betting, although table minimums and chips that are not infinitely divisible (you can't bet $5.67), make the theory impossible to apply in practice.

    The 13.5% you quote is not for pure Kelly betting, where bet sizes are continually resized as bankroll fluctuates. The 13.5% refers to sizing your initial bets according to Kelly and then fixing those wagers, despite the inevitable BR fluctuations that will then ensue. The result is a 13.5% chance of ruin, if you never resize your bets from their initial levels.

    Clear?

    Don

  10. #10
    MJ
    Guest

    MJ: Re: Kelly Question for Don

    > Losing 100% of your BR is, indeed, theoretically
    > impossible with pure Kelly betting, although table
    > minimums and chips that are not infinitely divisible
    > (you can't bet $5.67), make the theory impossible to
    > apply in practice.

    Another problem with playing pure kelly is trying to calculate the new unit size after every wager. It simply is not possible.

    > The 13.5% you quote is not for pure Kelly betting,
    > where bet sizes are continually resized as bankroll
    > fluctuates. The 13.5% refers to sizing your initial
    > bets according to Kelly and then fixing those
    > wagers, despite the inevitable BR fluctuations that
    > will then ensue. The result is a 13.5% chance of ruin,
    > if you never resize your bets from their initial
    > levels.

    Ok I got it. Basically, we are discussing two ends of the spectrum. On one end, there is betting pure kelly. On the other end, there is betting full-kelly on the intial wager without ever resizing the unit, regardless of BR fluctuations. The former is not practical and the latter is reckless. Let us try to find a happy medium between the two. What if you played full-kelly (or any kelly fraction) and then resize the unit after a BR fluctuation of +/- 10%? This way, you resize the unit but not after every hand. Now, if you lose 9% of the initial starting BR, the ROR will go above 13.5% (or the ROR pertaining to the Kelly fraction)
    but not too far above 13.5%. If the BR does dip the full 10%, then you resize the unit and the ROR resets to 13.5%.

    If you play full kelly and never resize the unit, what is the probability of losing half the bank? I guess it is much greater than 50%, which you said was for pure kelly.

    Also, if pure kelly (with full kelly factor) is associated with a 50% chance of losing 1/2 the starting BR, then why does the chart you provided from BJmath say it is 33.33%?

    MJ

    "Suppose you make optimal small bets (that is, with advantage less than 10%) and keep playing until you lose half your bankroll or double your bankroll, whichever comes first. In the following chart, "Failure" is given by 1/(1+2^(2/k - 1)) and is the chance that you will end up with only half your original bankroll if you play until you either halve or double your bankroll, whichever comes first."

    Kelly Frac. Failure

    100% 33.33%

    80% 26.12%

    60% 16.56%

    50% 11.11%

    40% 5.88%

    30% 1.93%

    20% 0.19%

    10% 0.0002%

    Note that, for pure Kelly, when you're constantly resizing, the values are quite different from not resizing and not stopping when you hit a goal. Both of these make risk of losing half much more than in the baove chart.

  11. #11
    Don Schlesinger
    Guest

    Don Schlesinger: Re: Kelly Question for Don

    > Ok I got it. Basically, we are discussing two ends of
    > the spectrum. On one end, there is betting pure kelly.
    > On the other end, there is betting full-kelly on the
    > intial wager without ever resizing the unit,
    > regardless of BR fluctuations.

    Right.

    > The former is not
    > practical and the latter is reckless.

    "Reckless" is a relative term, not a mathematical one. Let's just say that, for most high-stakes players, or professionals, or teams, full Kelly is considered too risky. Most opt for one-half to one-third. Some, even less.

    > Let us try to
    > find a happy medium between the two. What if you
    > played full-kelly (or any kelly fraction) and then
    > resize the unit after a BR fluctuation of +/- 10%?

    Are you asking my opinion? I wouldn't resize until I'd lost half my bank. But, that's just me. If you were supposed to bet $90, instead of $100, are you really going to fiddle around with all those chips? You'll lose even more with the time you and the dealer waste making and paying your bets. And, that isn't a joke; it's a fact. Paying $45 blackjacks is a pain in the ass. Wouldn't be me.

    > This way, you resize the unit but not after every
    > hand. Now, if you lose 9% of the initial starting BR,
    > the ROR will go above 13.5% (or the ROR pertaining to
    > the Kelly fraction)
    > but not too far above 13.5%. If the BR does dip the
    > full 10%, then you resize the unit and the ROR resets
    > to 13.5%.

    Whatever makes you happy. :-)

    > If you play full kelly and never resize the unit, what
    > is the probability of losing half the bank? I guess it
    > is much greater than 50%, which you said was for pure
    > kelly.

    Yes, correct. I'll try to give you a value in a little while.

    > Also, if pure kelly (with full kelly factor) is
    > associated with a 50% chance of losing 1/2 the
    > starting BR, then why does the chart you provided from
    > BJmath say it is 33.33%?

    Just read it! The explanation is in the text that you provided, no?? The values given are for doubling the bank before halving. So, every time you double, that's one fewer time that you can possibly halve, instead. The last paragraph explains that, no?

    Apparently, I wrote:

    > "Suppose you make optimal small bets (that is,
    > with advantage less than 10%) and keep playing until
    > you lose half your bankroll or double your bankroll,
    > whichever comes first. In the following chart,
    > "Failure" is given by 1/(1+2^(2/k - 1)) and
    > is the chance that you will end up with only half your
    > original bankroll if you play until you either halve
    > or double your bankroll, whichever comes first."

    > Kelly Frac. Failure

    > 100% 33.33%

    > 80% 26.12%

    > 60% 16.56%

    > 50% 11.11%

    > 40% 5.88%

    > 30% 1.93%

    > 20% 0.19%

    > 10% 0.0002%

    > Note that, for pure Kelly, when you're constantly
    > resizing, the values are quite different from not
    > resizing and not stopping when you hit a goal. Both of
    > these make risk of losing half much more than in the
    > above chart.

    Don

  12. #12
    Don Schlesinger
    Guest

    Don Schlesinger: Best guess

    I don't have a formula for the question you asked above, but I think the answer is probably around 75%, instead of 50%.

    Don

  13. #13
    MJ
    Guest

    MJ: Re: Kelly Question for Don

    Thanks for the response. But I'm still confused on the last question.

    > Just read it!
    I did.

    > The explanation is in the text that you
    > provided, no?? The values given are for doubling the
    > bank before halving.

    Not quite. The chart says you "keep playing until
    you lose half your bankroll or double your bankroll,
    whichever comes first."

    > So, every time you double, that's
    > one fewer time that you can possibly halve, instead.

    I don't understand your point here. The assumption of the chart is that the life of the BR ends after you either half or double it, period.

    The only thing I can think of to reconcile the percentage difference you quoted with that of the chart are the respective assumptions. The chart uses pure kelly which is what you were discussing. When you say the chance of losing 1/2 the starting BR is 50% (pure kelly), do you stop playing if the bank has been doubled or is the upward growth of the bank infinite?

    Other then that, I do not see any other difference in the assumptions between what you said and that of the chart.

    MJ


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