Wait, you meant to say B = variance / ROI? The formula as written doesn't make sense to me, as it says the higher our ROI is, the more BI's we need. Thanks for your great post, but thought this might confuse other players as well..

b = percentage of your bankroll not the number of buyins

Standard deviation squared. If HEM says your standard deviation is for example 40ptbb/100, then the number you're looking for is 40^2=1600

yeah that looks about right

I can tell you from my own example. Few years ago I accidentally found forgotten $80 on some room and since I had 2k-ish BR on my main room I decided to have some fun and gamb00l: the plan was to use super aggressive BR managemant and play nl10 till $100 than nl25 till $200 then nl50 till $400 and then nl100 till $1000. Started hot, moved up and up, the peak was like $600 then cooler hit, dropped to $200 then moved down, dropped to $100 then moved down ... ending at $80 where I started before deciding to give up and just withdraw the money. After the roller coaster was over I imported the data in my tracker and to my surprise I was in green ... weird, I thought, did the tracker miss some hands or something? Then I noticed I was actually looking at bb graph, when I switched to $ graph everything was fine, me being at zero ... wtf? WTF? So I was switching between bb (I was 3 BI up there) and $ graph trying to figure out what the **** was going on when it hit me:

It doesn't matter how much bb really is! Proportion is the same, 100bb stack is 100 bb stack whether the blind is $0.02 or $10. Essentially I was winning 2 buyins at lower stakes only to break even $-wise after losing 1 buyin at higher stakes (not exactly like that but you get the idea). That being said: having to move down is a disaster.

this is completely not true, you overall expectation if you are a winner is still positive as is your expected growth, think about the theory behind gambler's ruin; its a short term theory rather than a long term one

its the second part of your argument which is the reason why brm is so important, that is the oppurtunity cost of going bust and/or playing much lower than you should.

wow thanks for the post jspazz. definitely appreciate it!

I haven't gotten around to calculating my risk of ruin and kelly criterion yet (since I'm reading the Mathematics of Poker) and haven't gotten to that part. But you have a lot of information that's sort of hard to digest.

If I play deepstack 20 hu sngs, what would be the optimal br I should have before I choose to move up to 30 hu sngs? I know this might be an annoying question but I'd appreciate it if you could help me answer

P.S. My roi is 12.5% as of now but I've only played 151 games at the 20 level... so it's pretty small sample

What is missing here is how we feel so much better, less stressed and less prone to tilt when we're way overrolled.

Which is why I feel stressed out playing NL1000 and agonise over every bet and raise and if I play it will only play that one table, while at NL200 I can lose 7 BI and barely tilt anymore.

good post Jspazz--thanks.

while counting hourly you should also take into account time spent waiting for action imo.

Nice post jspazz, brm is a topic that's brought up often, but rarely discussed in mathematical detail.

Oh and I forgot to give credits to jspazz for clearing some things. Nice posts.

Man I must be rly dumb, cuz Im having trouble calculating that B= ROI/variance.

Say I have a 10 ROI at the 50's

B=10/1
b=10

wtf?
PS: I suck at math

Proper bankroll management gives you the best shot to have a higher mean net worth after n games. Improper bankroll management stunts this curve. It has nothing to do with your immediate edge... it has to do with the fact that any strategy you pick now you're assumed to stick with, so bankrolling too short means you're losing half your bankroll way more often than is optimal for long-term growth.

That's exactly right. You can risk a portion of your bankroll that is equal to your ROI. If your ROI is 10%, you can risk 10% of your bankroll. If your ROI is 5%, you can risk 5% of your bankroll (also known as 10 and 20 buyins, respectively).

Hey man, it seems we were confused for the very same reason. I think it works both ways though; if you stick to B=ROI/var, your b=10% and thus you only need 10 BI. On the other hand, if you use B=var/ROI, your b=10BI which speaks itself. I think what matters here is the unit (% or BI).

Yeah, I didn't realised that B = % of roll

You stated earlier that kelly mathematically beats all other methods in terms of bankroll growth, however, does 1/2 kelly still beat bi rules or at least can I see some mathematical explanation for why 1/2 kelly is better than a bi rule? An intuitive explanation behind the math i.e. 1/2 kelly grows according to one function whereas a bi-rule grows according to another function would be fine I don't need a million equations.

i think ur reasoning for the guy who wants to withdraw on a regular basis is a bit flawed, since your formular might account for the increased chance to go bust if you take money out of your roll, but it still leaves a big risk that you will have to move down, which can rly mess with your plan on withdrawals for a certain $amount.
it obviously depends on how flexible u are with ur spending habits, but if i.e. u need 2k a month, that is well doable on the 100s and maybe also on the 50s if u grind a large volume, but having to move down to the 30s will deplenish ur monthly income in a way that u might to have cut down drasticly on ur spending. so for people who play at least semiprofessionally and rely on a relatively high $amount/month, its very rational to have a very tight (like 1/8 kelly or 100bi whatever) brm in order to always have enough $$$ to play a limit where they can make a living.

TBH I think naymlis' point needs to be considered a little more carefully. People generally use bankroll guidelines to decide when to move up, and Kelly is only useful for that insofar as you know your ROI at the next level before you actually play there.

Telling a 20$ player that he will have about the same ROI at the 30s is probably close enough, but I only know that since I've been there already. Same thing for 30s->50s, imo, but the move to 100s will probably be a little harder for most people. And at some point on the way up, you start getting sat by spamz/sss/livb when they're bored. Mr Kelly didn't think of that, did he?

Yes we agree, just pointing out that bad brm doesn't make you a loser if you are a winning player, just less of a winner.

just to check: this includes the consideration that losing means playing smaller thus taking longer to make it back right?

also, how does it differ for those of us playing nl2-nl1k?

Kelly does not guarantee maximum expected value, but maximum expected growth (expected utility), which is the growth of the logarithm of bankroll size, not linear growth of the bankroll size. Since log(x) grows slower than x itself, winning some amount will always be worth less than losing that same amount. At some point, the impact of the logarithmic loss will be so big and the logarithmic win so small, compared to the relationship of real money win/loss amounts, that the expected utility of the bet will be negative, resulting in a constant shrinking of the roll over an infinite amount of identical ROI bets. If you bet your whole bankroll on every bet (unless it's an arbitrage), your expected growth will always be -100%, as a single loss, no matter how infrequent, will doom you. After a finite series of bets, you may find yourself immensely rich occasionally, but in the long run, you will bust every time. Similarly, if a bet is 0EV, you should not bet any amount on it, as the expected growth will always be negative (not zero!)

You can go w 10% ROI to err on the smaller side b4 you get a real sample, altho 12% is def very sustainable at these stakes in deepstax. I've found villains at 20s and 30s pretty similar, altho I've seen some midstakes regs sitting 30s occasionally. If you assume 10% ROI and variance = 1, then B = 0.10 / 1 / 2 (half Kelly) = 0.05. Thus you'd need 20 buyins. If you assume 12.5% ROI, B = 0.125 / 1 / 2 = 0.0625 and you'd need 16 buyins.

I dunno how big of a sample you have at 1k NL, but even if your ptbb winrate is 1/5 as much as your 200 NL winrate, you're better off playing 1k NL money-wise. in my personal opinion poker is more about cold hard cash than keeping your head straight, but if you value your mental health so much, 1k NL is obv a mentally (not monetarily) non-profitable game for you.

tbh I can't offer a mathematical proof right now altho I'm gonna play a bit w the formulas and see what comes up. Half Kelly is typically used coz ppl are scared of swings that full Kelly has, and you don't wanna be moving up and down a ton if you're living off the money.

yeah I think I ****ed up that part somewhat. Tho using 1/4 Kelly BRM should square the risk of moving down (i.e. if it was 10%, it's now 10%^2=1%), which should be enough for most ppl. I'm gonna think about this some more but I believe this is the necessary correction for most purposes.

haha obv I didn't advise him to go sit 500s on a whim coz he can, since he has no way of knowing his real ROI at 500s ATM, so playing them would be a bad idea. However, if he were to have 10% ROI at 500s, playing them w 5k would be max EG. He does seem confident in his sample at 20s, so ime he will be a similarly profitable player at 30s. I obv don't draw the conclusion from Kelly, but from experience that low stakes levels are fairly close together in skill level. Right now he seems to be overrolled, but since he has no way of knowing his ROI at high stakes, there's nothing he can do :\

In the beginning Kelly criterion was mostly used in betting and investment. iirc it was first put in wide usage by Edward O. Thorp who used it as BRM in his blackjack card counting. In such situations the ROI of the bet does not depend on the bet's size, so you want to bet as much as is correct to give yourself the best EG. In poker obv different "bet" sizes give different ROIs, so the correct situation might be investing 1% of your roll in a game where you have 20% ROI and variance=1 simply because you aren't profitable enough at higher buyins

very good thread.

btw, a few years ago i read a terrific book called Fortune's Formula. i highly recommend it if any of you are bored and want a captivating read on the origins of the Kelly Criterion. Lots of blackjack and Wall Street stuff in the book too.

So we don't actually know that 1/2 kelly provides greater expected growth than any of the other common br rules then?

I'm pretty sure it's still superior to something like muad'dib, but yes. However, if you don't like the swings of full Kelly, you prob won't like any non-Kelly strategies that give higher yield than 1/2 Kelly

where does variance 1 and 2 numbers come from ? what number should i use for super turbos ?