The Kelly Criterion
The neat thing about the Kelly Criterion that I did not know is that it is based on resizing your bet and compound interest. Not only that, but it is based on resizing your bet based on your CHANGING bankroll - your bankroll getting bigger as a result of winning or your bankroll getting smaller as a result of losing. So, technically, using the Kelly Criterion perfectly will lead one to never exhausting his/her entire bankroll because, for example, even if your bankroll shrinks to $20, the Kelly Criterion math (based on a 1% player advantage) dictates your currently resized bet should now be about $2.00, not, say, $75 - which, at the outset of your gambling trip is what the Kelly Criterion math might have said it should be. Does this sound accurate? Do you guys recalculate your Kelly Criterion every hour?, every bet?, every day?
The Half Kelly sounds interesting - a more conservative approach. Simply take 1/2 of what the Kelly Criterion says your bet should be. Interestingly, apparently the Half Kelly has a bigger negative impact on your chances of increasing your bankroll at the benefit of not as frequently experiencing the phenomenon of losing half your bankroll about 1/3? of the time as you would when using the "Full" Kelly Criterion.
Quote: StevenBlackif your bankroll shrinks to $20, the Kelly Criterion math (based on a 1% player advantage) dictates your currently resized bet should now be about $2.00
no, something around 1% for 1% advantage, and less if there is large variance. So, less than 20 cents actually. Not that I am the guy to ask.
Quote: StevenBlackfor example, even if your bankroll shrinks to $20, the Kelly Criterion math (based on a 1% player advantage) dictates your currently resized bet should now be about $2.00, not, say, $75 -
The KC is ((payout*probability of win)-probability of loss)/payout, so for a 1% advantage on an even money game, you should wager ((1*.505)-.495)/1=1% of your bankroll.
If your bankroll is $20, your wager should be $.20. Given the next line about a $75 bet, did you mean you'd have a $200 remaining bankroll?
The Kelly Criterion is great for things like betting the horses or sports or prop bets with Son-of-Soopoo where you can think you have a static advantage, but it's hard to use in games where the probability of winning changes rapidly like in blackjack.
The problem with applying it to a game like BJ is that your advantage is changing based on the cards played in each hand. Unless you really think you have a static advantage all the time, you'd need/want to recalculate it every bet. But there's not enough time to do that, plus you should also be counting cards and it's really hard to do both the Kelly math and the card counting at the same time.
Quote: StevenBlack
Not only that, but it is based on resizing your bet based on your CHANGING bankroll - your bankroll getting bigger as a result of winning or your bankroll getting smaller as a result of losing. So, technically, using the Kelly Criterion perfectly will lead one to never exhausting his/her entire bankroll
While this is true for the Kelly Criterion, it is not a general consequence of any betting system resizing your bet with your bankroll.
Think of an even-paid game with massive 10% advantage. If you always bet half your bankroll you will get broke in the long run. Although you never bet your full bankroll.
Don't believe ? Play such a game on your kitchen table.
Quote: MangoJWhile this is true for the Kelly Criterion, it is not a general consequence of any betting system resizing your bet with your bankroll.
Think of an even-paid game with massive 10% advantage. If you always bet half your bankroll you will get broke in the long run. Although you never bet your full bankroll.
Don't believe ? Play such a game on your kitchen table.
I think it depends on the definition of broke. Say I sit at my table with $100 and play your theoretical game. Say I lose 19 hands in a row right from the start. I now have $.000191 in front of me. Am I broke? I'm not out of money, and I can never have 0 or negative money. Over the long term, I still expect to considerably increase my bankroll if I can keep playing. But if I have to be more practical and bet in $1, then I'm definitely "broke" when/if my bankroll gets so low that the rounding leaves me with less than 1 bet remaining (after 7 losses in my example).
Quote: rdw4potusI think it depends on the definition of broke. Say I sit at my table with $100 and play your theoretical game. Say I lose 19 hands in a row right from the start. I now have $.000191 in front of me. Am I broke? I'm not out of money, and I can never have 0 or negative money. Over the long term, I still expect to considerably increase my bankroll if I can keep playing. But if I have to be more practical and bet in $1, then I'm definitely "broke" when/if my bankroll gets so low that the rounding leaves me with less than 1 bet remaining (after 7 losses in my example).
Yes it does depend on definitions. It does not only depend on the definition of "broke", but it does also depend on the definition of "long run".
The long run - usually understood as the mathematical limit of your bankroll towards infinite time - will be zero "almost surely" in the given scenario.
So what do mathematicians mean with "almost surely" ? There is exactly zero probability of happening differently (which doesn't mean it's impossible).
Hence,what could we agree on "broke" ? I think the idea of having zero bankroll almost surely is quite close to being broke.
Quote: rdw4potusThe KC is ((payout*probability of win)-probability of loss)/payout, so for a 1% advantage on an even money game, you should wager ((1*.505)-.495)/1=1% of your bankroll.
your formula doesnt seem to factor variance, or am I missing that? I am interested in this although not sanguine about ever looking at any real player advantage looming for me.
quoting the wizard mid-page at his WoO page on this:
Quote:Most gamblers use advantage/variance as an approximation, which is a very good estimator. For example, if a bet had a 2% advantage, and a variance of 4, the gambler using "full Kelly" would bet 0.02/4 = 0.5% of his bankroll on that event.
Quote: MangoJYes it does depend on definitions. It does not only depend on the definition of "broke", but it does also depend on the definition of "long run".
The long run - usually understood as the mathematical limit of your bankroll towards infinite time - will be zero "almost surely" in the given scenario.
So what do mathematicians mean with "almost surely" ? There is exactly zero probability of happening differently (which doesn't mean it's impossible).
Hence,what could we agree on "broke" ? I think the idea of having zero bankroll almost surely is quite close to being broke.
If I reasonably expect to win 110% back on average (the reasonable part is important), and I always bet 50% of what I have in front of me (including fractional amounts), I literally can never hit 0 and the expectation as time approaches infinity is infinity.
With a negative expectation, maybe winning back 90% on average, the limit of the bankroll as time approaches infinity is 0.
In either case, the bankroll never actually touches either bound.
Quote: odiousgambityour formula doesnt seem to factor variance, or am I missing that? I am interested in this although not sanguine about ever looking at any real player advantage looming for me.
quoting the wizard mid-page at his WoO page on this:
I won't challenge the Wiz. I just pulled out an old finance textbook and pulled the formula. I'm sure he's more accurate on the gambling implications of the math.
Either way, I think it's too much to keep up in a live BJ game while also counting.
Quote: odiousgambityour formula doesnt seem to factor variance, or am I missing that?
The formula does include variance, and in fact is all about variance. In a 2 outcome game (win or loss) variance (as well as EV) is described by the probability of win and probability of loss, together with the payout.
In a multi-outcome game (as blackjack) things are different for an exact Kelly-equivalent criterion. However it is still useful approximation - since doubles and split are not the majority of hands.
