Bankroll Formula?

[Patrick del Poker Grande]

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That's great. We've already established that. We're on to the last piece of the puzzle now:

What are the units on Risk of Ruin? Is it X% every hour or 100 hands, or is it X% lifetime? I don't see how it could be over your lifetime here, since all the other terms in the formula are expressed in terms of hours or 100 hands. Some sort of sense suggests that it should be over lifetime, though, as you'd have to run it out to about 6+ decimal places if you were to do it for your lifetime in a per 100 hands basis.

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risk of ruin is unitless. it's the probablity that your bankroll will ever be depleted given the win rate and SD.

in the bankroll calculation it's just a number picked that you feel comfortable with. if you're okay with a 10% risk of ruin, then you can probably play on a 100BB bankroll.

you can calculate your risk of ruin by plugging in your actual bankroll, win rate, SD, and solving for RoR.

obviously things change...your bankroll is going to increase and if you stay at the same stakes with the same win rate and SD, then your risk of ruin will continuously get smaller.

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Yeah, but a 1% risk over 100 hands is different than 1% risk over an hour and it's certainly different than 1% over your lifetime. I realize it's actually unitless - my wording was chosen poorly. What is the... time frame we're talking about? If you have a 1% risk of going broke with every 100 hands you play, then wouldn't you be 'certain' to go broke after 100 such sessions, or 10,000 hands? This can't be right. So... essentially what you're saying is Risk of Ruin is over your lifetime, no?

[BruceZ]

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Yeah, but a 1% risk over 100 hands is different than 1% risk over an hour and it's certainly different than 1% over your lifetime. I realize it's actually unitless - my wording was chosen poorly. What is the... time frame we're talking about? If you have a 1% risk of going broke with every 100 hands you play, then wouldn't you be 'certain' to go broke after 100 such sessions, or 10,000 hands? This can't be right. So... essentially what you're saying is Risk of Ruin is over your lifetime, no?

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Hi Patrick,

You are using the same risk of ruin formula that I introduced here. It is for playing forever. If you plotted risk of ruin as a function of time for a typical winning player, it would approach this value asymptotically with almost all of the risk taking place within the first several hundred hours. After that time, if you have not already gone broke, your bankroll will typically have grown to the point that your risk of ruin has been reduced to almost nothing. For example, if you start with a 1% risk of ruin and then double your bankroll, your new risk of ruin will be 1% * 1% = 0.01% since to go broke you must now lose the equivalent of your 1% bankroll twice.

Also note that these formulas give the risk of ruin assuming that you allow all winnings to accumulate in your bankroll, and that you do not remove any money from your bankroll. If you were to remove all winnings from your bankroll, then your risk of ruin would be 100% (for infinite time) since you are simply waiting for a losing streak large enough to wipe out your bankroll, and it is guaranteed that such a steak will eventually come if you play long enough. You must subtract any money you plan to spend from your win rate before you compute the risk of ruin.

Also, this formula is not limited to b&m play or single table play. You need the same bankroll to play multiple tables as you do to play a single table at the same limit, as long as your win rate and SD per table do not degrade. If you play multiple tables at a lower limit, you can have the same or higher EV as with a single table, but with a lower overall SD, and in that case your bankroll requirements would actually be less for the same risk of ruin.

-Bruce

[IsaacW]

Thanks for explaining this.

[Patrick del Poker Grande]

Thank you, Bruce. I'm still not sure I'm 100% satisfied, but I can see that this is as close as it's going to get.

[BruceZ]

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Thank you, Bruce. I'm still not sure I'm 100% satisfied, but I can see that this is as close as it's going to get.

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Why not satisfied? Here is a derivation, with an explanation of assumptions, and links to a couple other papers.

[Rudbaeck]

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Actually multitabling will cause a need for a bigger bankroll. Your standard deviation will go up slightly, and your win rate down slightly, due to mistakes and shallow reads. It's nothing near double though.

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why do people always say this? even if your SINGLE TABLE win rate goes down and your SINGLE TABLE SD goes up, then, assuming you don't turn into a losing player, your win rate for the session (the time spent playing 8 tables) will be 8 times your single table win rate and the session SD will be sqrt(SD_1^2*8), which you'll find is significantly less than if you single tabled a game that had a winrate equal to your 8 table rate.

to use the numbers from the example given:
WR = 2 BB/hour, SD = 16 BB/hour

your 8 table win rate will be 16BB/hr and your 8 table SD = sqrt(16^2 * 8) = 45.2 BB/hr

the SD isn't 8x your win rate any more, it's less than 3x.

so now, calculate the 8 table bankroll using these numbers:
BR = -(SD^2 * Ln R)/(2 * WR)
= -(2048 * -4.605)/(2*16) = 294BB

A LOWER BANKROLL REQUIRED

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Could someone who isn't running a fever check the math? There should be no difference if the conversion is done correctly, and if you use a BB/100 unit for both winnings and SD then multitabling will definitely require a higher bankroll as win rate will go down and SD will go up.

If correctly converting SD and win rate to BB/hour and doing the calculations again leaves us with another bankroll something is fundamentally fishy.

I'm 99% an online player, and so think naturally in BB/100 units, and do all my bankroll calculations in those. If you do the conclusion that multitabling requires a bigger bankroll.

I can't think straight, but this seems weird.

[Rudbaeck]

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Also note that these formulas give the risk of ruin assuming that you allow all winnings to accumulate in your bankroll, and that you do not remove any money from your bankroll. If you were to remove all winnings from your bankroll, then your risk of ruin would be 100% (for infinite time) since you are simply waiting for a losing streak large enough to wipe out your bankroll...

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This bears repeating over and over again. I didn't understand this until pretty recently, my thanks to BruceZ for clearing it up for me earlier.

Many, if not most, poker authors say that a fixed bankroll of 300BB, from which you withdraw all your winnings at the end of the month is enough for playing forever with a very small risk of ruin.

It's no wonder that 'everyone goes broke' with that glaring error in fundamental bankroll management.