As I've been thinking much on my bankroll management recently, I've had an idea to write a 'deep' [nu]tty quartermilestone post on application of higher math to BRM, namely theory of random processes, though it was my toughest subject at uni (if you admit that str8 A students can have difficulties with some subjects). It will be a tl;dr series if you approve.
I'm considering this only from math positions with little understanding of poker itself, if you're experienced please feel free to advocate other BRM strategies, I'm interested.
Part 1. Minimal number of BIs for the lowest limit.
To make the long story short:
determine your base limit - the minimal one that can earn you a living (expenses aren't bigger than winrate);
for risk-of-ruin calculations, RB should be added to the winrate and life expenses subtracted from it, yielding the net winrate;
those who start a pro career should first check that their net winrate EV at the base limit is nonnegative;
to estimate your winrate EV even remotely accurately, you have to play a 6-digit number of hands;
net winnings (i.e. winnings+RB-expenses-taxes) with zero net winrate EV can be approximately modelled as a Wiener process;
BR size should be such that, with zero net winrate EV, the aggregate probability of quitting poker due to both wrong negative winrate EV estimation and ruin is small, e.g. 0.1.
Moving up/down limits may be considered in a forthcoming post, I'm thinking of a superlight 'shot-only' strategy (what are your thoughts on it?) when I move to a higher limit as soon as I earn the stoploss amount for a single session of that limit, if I lose that amount on that higher limit I immediately step back to the lower one. That means more hands will be played on the base limit and not higher, but even short shots can give info on higher limit gameplay and fruit for learning. Certainly, if moving to Stars is planned, PLO100 should be considered the new base limit, i.e. a lot of BIs for it should be earned before moving.
The following procedure of determining the BR size is proposed.
Estimate your std deviation per 100 hands s', I assume it's 160 and 120-200 is std for SSPLO (see the dedicated thread).
Estimate your RB minus life cost and income tax per 100 hands, e.g. with rake 17bb/100 at PLO€20, RB%=60%, 40K hs/month, expenses €400/month=5bb/100 (ascetic life in a rent flat in 50 km distance from the Kremlin), 13% income tax (though 99% of Russian players are lucky enough not to pay it) gross post-RB winrate should be at least (5/0.87)bb/100=5.7bb/100 and pre-RB winrate should be at least -4.5bb/100 - quite achievable for 2+2ers, I guess.
Set the probability p of mistaken end of poker career due to winrate underestimation you can stand, then the recommended number of test hands is n=(10*z(1-p)*s'/l')^2, where l' are your living expenses in bb/100, z is found from a normal distrubution quantile table. E.g. with std dev 160bb/100 and expenses 5.7bb/100 you can be 95% (z(0.95)=1.65) sure quitting poker if you lose post-RB after (10*160*1.65/5.7)^2=214515 hands.
Set the probability p' of mistaken quitting poker after ruin you can stand, then you'd better have a BR of sqrt(n)*s'*z(1-2p')/10=s'^2*z(1-2*p')*z(1-p)/l' (measured in bb; sqrt = square root). In our example it's 25600*1.96*1.65/5.7=14524 (bb), or 145 BI. Also, if after some k hands (a big sample!) you show a net loss of sqrt(k)*s'*z(1-2p')/10 bb, it's time to refrain from grinding and learn as you're 95% doing smth wrong.
Remember to have a liferoll for at least 6 months plus your test period separated from your BR (it may be better from tax perspective, e.g. in Russia funds uploaded to e-wallets are not deducted from income tax base).
Remember that if by the end of the test period you increase your bankroll but earn less than twice your expenses, this agrees with the hypothesis that you're a winning player but doesn't reject the opposite, you have to play more test hands and win enough to do so.
Ty for reading. Run good!
The following spoilered text is an explanation for math sharks.
Spoiler:
Ideally, results of poker hands including RB are viewed as independent identically distributed random variables X_k. Sums of such variables Y_n=X_1+X_2+...+X_n (aggregate winnings in our case) are known as random walks. Let's assume that our net winrate EV is zero (it's better if it's positive but we first need to estimate whether we're losing players or not so we can't assume we're winners), i.e. EX_k=l (measured in bb/1) where l*n is the life cost during these n hands divided by (100% - taxrate) (these are pretty fixed).
If we rescale and resample Y'_n=Y_n-l*n (net winnings) as S_n(k/n)=Y'_k/(sqrt(n)*s), where sqrt(n) is the square root of the number of hands, s is the std deviation of X_k in bb/1, and define S_n(t), 0<=t<=1, in other points so that the graph of S_n(t) is piecewise linear with fractures in k/n, then by Donsker's theorem S_n(t) weakly converges to W(t) (the std Wiener process on [0,1]). The 'running minimum' of W(t) - m(t)=min_{0<=tau<=t}W(tau) - has unilateral normal distribution with density p(m(t)>m)=sqrt(2/(pi*t))exp(-m^2/(2*t)), m<0; p(m(t)>m)=0, m>=0. E.g. if my S_n(1)<-z(0.975)=1.96, I can be 95% sure that I'm a loser.
As for winrate EV estimation, assume that EX_k=a and the std dev of X_k is s (in bb/1), then by the central limit theoremsqrt(n)(Y_n/n-a)/s converges in distribution to N~N(0,1) - a normally distributed variable with zero EV and std deviation 1. Thus for big n we have P(Y_n>0)=P(sqrt(n)(Y_n/n-a)/s>-sqrt(n)a/s)~P(N>-sqrt(n)a/s). So if e.g. P(N>-sqrt(n)l/s)>0.95 (we need a>=l) we can say that if we quit poker after not increasing our bankroll in n hands, the probability of us underestimating ourselves is less than 5%.
Last edited by coon74; 06-01-2012 at 06:50 AM.
Reason: Typo in the running minimum formula
I don't know where this new bank roll management hype has come from but to be honest I don't think it is that hard.
You can easily play plo100 and plo1000 with same bank roll as long as you know when to stop playing games you only have 15bins and when to move down to plo50.
People should have some common knowledge how to deal with money. Unfortunately many people don't have any clue how they should handle their money (it is even more common in poker world which is kinda sad because our main tool is money.. especially "propokerkidslivinginmothersbasemetsplayingPLO5kfo rliving" suck at it). Amount of dollars in your roll does not have anything to do with the ability to handle your monies. I've withnessed my friends busting out their 6digit rolls few times simply because they have tilted and played too high and stopped giving a **** about the sums they play...
if you're playing for fun but taking it seriously 50-75 buyins for your normal game
if you're a pro at least 100 buy-ins + 3 months expenses in a checking account (not your life savings).
take shots. realize when you're shots aren't going well. stop. back to basics. learn. repeat. move up. having the perfect brm strategy ain't all that important. they are just guidelines to not go busto if you don't have a good feel for the variance of the game.
if you're playing for fun but taking it seriously 50-75 buyins for your normal game
if you're a pro at least 100 buy-ins + 3 months expenses in a checking account (not your life savings).
take shots. realize when you're shots aren't going well. stop. back to basics. learn. repeat. move up. having the perfect brm strategy ain't all that important. they are just guidelines to not go busto if you don't have a good feel for the variance of the game.
I've written this because I was curious if there is math underlying all this 50-75-100 BI advice - it seems based on pure experience which is insufficient to my unready mind whose transformation is just starting.
As you've seen I've arrived at the 45 BI figure (100 BIs out of those 145 are in fact life expenses for 5 test months) for a 10% risk of failure, which quite surprised me though it was just an example and the risk is probably too high (what percentage do you find acceptable?).
I agree that the model is poor as it doesn't account constant learning process and tilt and thus variable winrate, but it has allowed me to understand BRM better.
Also I meant to question the existing hierarchical view of limits and perception of going down as a grave failure. Imo people stick to these models because they see them irl - it's hardly possible to be an ordinary vendor today, a top manager tomorrow and demoted back to a vendor the day after tomorrow, people are just fired if they do smth wrong but usually find an analogous position elsewhere.
I have no idea whether loose smallish shots are good (they can't be possible without stoploss software btw as it's so hard to stop manually) - they may cut time spent at higher limits with higher hourly as backsteps will occur very often, but the entry price is smaller and if one is really talented, there will be fewer backsteps anyway. I want posters to sort this question out.
Oh btw thanks for reminding, napsus. In fact I came to these math considerations after I played a lot with this soft. It just draws a graph where I can see peak points but it's very hard to count the EV of the BR size in the end - not only peaks contribute to it, tails are also important and I'm too lazy to count this for every sim. Is there a way to easily find the average expected BR or the average time needed to move up?
i don't think there is. "moving up" is kind of misunderstood concept i think, since you really shouldn't be focusing on playing just one level but rather to play the best games. let's say that you have 80bi for plo50 and thus 40bi for plo100. your rule is that you only play higher games when you have 60bi for the level. then you see 5 megafish sitting at plo100 table where you can just wait for hands and try to bink. do you or do you not do go for it? so instead of "written in stone" brm guideline you should stay flexible in your approach.
for me the most important things to take out from that simulation are the risk of ruin and median expected bankroll at the end of period with the assumed conditions. i adjust the brm so that my risk of ruin is less than 1% of go ahead with that.
Two Plus Two
BRM higher math
quartermilestone post on application of higher math to BRM, namely theory of random processes, though it was my toughest subject at uni (if you admit that str8 A students 
can have difficulties with some subjects). It will be a tl;dr series if you approve.
Linear Mode
