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.
Ty for reading. Run good!
- 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.
The following spoilered text is an explanation for math sharks.