Bankroll formulas with hourly rate and standard deviation

[HighStakesPro]

After over a year of playing poker online for stakes that were too high for my bankroll, I think I've finally learned my lesson, and I'm trying to play with a proper strategy that will insure that I gradually increase my bankroll and insure that i have virtually no chance of going broke.

I am following Mark Blade's reccomendation of four hundred big bets. I am four-tabling $0.50/1 Limit Hold'em (full table) on PokerStars, and I started with $400. I have forced myself to play absolutely nothing else. I am keeping track of my results with CardPlayer's Poker Analyst, and so far after 76:36 of total playing time, I have made $116.50 for a total of 1.52 BB/hour. I am adding up the time for each individual table I play, so for every one hour of real time, it amounts to approximately four hours of total playing time, so the figure is really 1.52 BB per table per hour. Poker Analyst also tells me that my standard deviation, through 19 sessions of around one hour each (12 wins 7 losses), is 22.62. Here is what I want to know:

1. Does multi-tabling increase my fluctuation/volatility or increase my risk of going broke wiht 400 BB?
2. Is it accurate for me to record four hours of time if I play for one hour at four tables?
3. How many hours and/or sessions do I need to play for my hourly rate and standard deviation to be stable enough to be truthfully reflected by my results?
4. What is a "typical" standard deviation, and if my standard deviation is low enough, can I jump up to $1/2, or whatever the next level is, with fewer big bets in my bankroll? How low would my standard deviation have to be to do this?
5. Once I have acccurate enough figures for hourly rate and standard deviation, how do I then calculate what my bankroll should be? What is the formula?
6. Is there any literature about bankroll strategy and formulas (other than Blade, who I have read) that would be helpful to me?

Whoever has gotten to this point, thank you for reading the entire query and hopefully answering some of my questions.

[AlanBostick]

Read Gambling Theory and Other Topics by Mason Malmuth, particularly the section titled "How Much Do You Need?"

Mason doesn't give this formula outright but if you know a bit of algebra you can pull it out of what he wrote. If S is your hourly standard deviation -- the square root of the variance your play records show you experience in one hour's play -- and W is how much you expect to win in one hour's play, then the minimum bankroll you need to play with negligible risk of going broke is:

BR = 9/4 * S*S/W

Typical values for W are in the range of 1-2BB and a typical value for S would be 10. This results with a bankroll requirement of but 225 BB for a win rate of 1 BB/hour and only 113 BB for a win rate of 2 BB/hour.

Mason warns of "non-self-weighting" effects of running bad (when your cards are cold, he says, your opponents notice you losing and take shots at you, so your win rate ) and reccomends increasing this by a margin of safety 20 to 30%. The 300 BB number I often see presumably includes a 50% margin of safety, and Mark Blade's 400 BB number is a 100% margin of safety.

1. Multitabling should increase your win rate by a factor of the number of tables you play, and it should increase your standard deviation by the square root of the number of tables. If your hourly win at one table is W and your hourly standard deviation is S, then your hourly win playing four tables should be 4*W and your hourly standard deviation should be 2*W. Interestingly enough, in the bankroll formula I give above, those factors cancel, and the bankroll requirement stays the same. (This makes sense, because playing four tables at once for one hour is the equivalent of playing one table for four hours.)

2. It's not unreasonable; but I personally would track each table as a separate session.

3. The uncertainty of your win rate -- the likely difference between your actual results and your "true" win rate, is given by U = S/sqrt(T), where T is the total number of hours of play that you've recorded. To trust your results, you want U to be small compared to W. With W ~ 1 and S ~ 10, after a hundred hours, U ~ 1, the same size as W, and there's about 1 chance in 6 that we're actually a losing player. After 900 hours, U ~ 1/3, and so we are rather more confident about the value of W.

4. Typical hourly standard deviations, as I've said, are in the neighborhood of 10 BB.

5. See the start of this message.

6. As I've said, Gambling Theory and Other Topics by Mason Malmuth.

[ohnonotthat]

Please explain why multi-tabling will [sic] "increase my SD by the square root of the number of tables played" (sq rt. of 4 = 2 so a 4-tabler will have a SD that is twice as high ?)

I'm NOT saying this is wrong - just asking the formula you used to arrive at this conclusion.

*

P.S. I may well not be, as you have alluded to in two prior answers, "as good of a player as I think" (actually I am probably better - I work hard to keep a tight reign on ego but I digress . . .) nonetheless, I and I suspect the others would love a direct anwser as opposed to another, "if you don't know you're not 'all that' ".

The gauntlet has been thrown down - care to retrieve it ?

*

P.S. Vitriol notwithstanding your responses are typicall excellent, albeit often laced with opinions that neither you nor anyone else can verify the validity thereof.

Scratch the second part of that sentence; in the spirit of the holidays I will alter it to, "your responses are typically excellent".

I feel the warmth of human kindness washing over me.

HEY, this stuff is NOT human kindness - would someone PLEASE get me a towel ? [img]/images/graemlins/shocked.gif[/img]

[borges]

Well, if S is your hourly std. deviation (when playing one table). Then
S^2 (S times S or S squared) is your variace when playing one table. If
you are multitabling N tables. Then since the variance scales linearly....(since
the variance of the sum of independent random variables is the sum of
the variances) your variance per hour would be N*S*S...or the standard deviation would
be S*sqrt{N} (S times the square root of N). So yes, if you are multitabling
4 tables your hourly standard deviation would double (but your win rate
would quadruple).

Hope this helps.

[BruceZ]

[ QUOTE ]
Read Gambling Theory and Other Topics by Mason Malmuth, particularly the section titled "How Much Do You Need?"

Mason doesn't give this formula outright but if you know a bit of algebra you can pull it out of what he wrote. If S is your hourly standard deviation -- the square root of the variance your play records show you experience in one hour's play -- and W is how much you expect to win in one hour's play, then the minimum bankroll you need to play with negligible risk of going broke is:

BR = 9/4 * S*S/W

[/ QUOTE ]

This formula will give a 1.1% risk of going broke due to an issue with the GTAOT derivation (see footnote in new version), and the 9/4 needs to be replaced by -ln(ror)/2 for a general risk of ruin ror. See Bankroll formulas for an explanation and the correct formulas. Also,

Derivation of bankroll formulas

Tool for bankroll and risk of ruin

[borges]

Interesting links. Thanks Bruce. I just spent a few moments looking at
Stu Ethier's paper.....He is basically saying that this naive approach of just
matching up the win rate and variance is not justified. The bounds provided
in the paper rest upon matching the first four moments.

Did you by any chance ever compute/compare the bounds that Ethier et al provide with the simpler formulas given in the bankroll derivations? Everything here is very tail dependent right? Heavy tails would significantly skew a ruin probability calculation.....something that would be brought out by computing and matching higher order moments......am I wrong on this?

take care,

Jim

PS: I'm also sure that none of this will make any real practical difference to
a poker player looking for guidance on bankroll requirements. [img]/images/graemlins/smile.gif[/img]

[pzhon]

[ QUOTE ]

He is basically saying that this naive approach of just matching up the win rate and variance is not justified.
...
I'm also sure that none of this will make any real practical difference to a poker player looking for guidance on bankroll requirements. [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]
I don't think that was the main point of the Ethier-Khoshnevisan paper.

The most relevant applications might be to MTT players and deep-stacked NLHE players. (In one live game, players sometimes buy in for 3000 times the big blind. The question is how many such buy-ins you need. It should be lower than the number of 100 BB buyins, but is 5 enough? 3?) In both cases, though, it is tough to get good estimates of the win rate and variance.

IIRC, the standard formulas overestimate the bankroll needed to play MTTs. Has anyone looked at this in particular?