DavidC
11-24-2005, 03:14 AM
(The post in Probabilities (http://forumserver.twoplustwo.com/showflat.php?Cat=0&Number=4017925&page=0&vc=#Post4 017925) will be shorter, if you want.)
Over the past few months, I've become less of a "poker player" and more of an "advantage gambler". This has lead me to questions that I never had to seriously think about as a poker player.
I'm asking this question of the probabilities forum and I'm cross-posting it here, because I believe that this question is pertinant to micro-limits and the mid-high forums, and that the mid-high guys should know this already anyways (or if not, they should read the probabilities forum to find out).
The reason I think it's important for us is that we can move up limits very quickly. The guys in mid-high have selection of a wide variety of games that are within their rolls, for the most part (100-200 all the way to 1000-2000++). With small stakes players, the increments between levels tend to be smaller and players tend to settle at their current limits for longer.
---
To avoid issues of sample size, I'll give you a scenario:
You're coming out of the shower one morning (or afternoon), and lo and behold, The Supreme Being is in front of you... God, Jesus, Jehova, Allah, Genesh, or maybe just Einstein if you're an atheist. Regardless, there he is. He passes you a towel. After drying your face, you ask, "Is it really you?" He gives you some sort of wise smile and nods. He then says, "I know you have questions, and I am here to help. You may ask only two questions."
You've been waiting for this a long time, so you ask, "What's my BB per 100 hands winrate?", and he tells you it's 2. You're sure he means for the number of tables that you currently play, so you ask next what your STD DEV per 100 hands is, and he tells you that it's 15 bets. He vanishes and you give a sigh of contentment at the final resolution of the greatest mysteries of your life.
---
ROR = exp(-2*BR*EV/VAR)
VAR = STD DEV ^ 2
= 15 ^ 2
= 225
EV = 2
ROR = exp(-2*BR*EV/VAR)
= exp(-4*BR/225)
Generally I like to have ROR = 0.1% = 0.001
Isolating BR for ROR:
ROR = exp(-4*BR/225)
ln ROR = -4/255 * BR
-4/255 * BR = ln ROR
BR = -255/4 * ln (0.001)
= -63.75 * ln (0.001)
= -63.75 * -6.907755
BR = 440.36 BB
So, when Miller says that you need 500bb + health insurance + cash for emergencies + three months' living expenses, he's being pretty conservative, but he's not being much more conservative than a 1/1000 chance of losing your roll, and he's preventing us from having to dig into our roll to pay for all the stuff he just mentioned (thus maintaining our ROR instead of accidentally exceeding our ROR target, which may be 1/1000).
So this basically boils down to: If I'm going to move up in limits, I like the idea of having 440BB when I do so, unless you don't mind the possibility of losing your roll, which may not be so bad depending on your circumstances. Needless to say, we can never TOTALLY eliminate the possibility of losing our roll, because variance exists and has effects on any sized roll, but 1/1000 is a risk I'm willing to take.
---
Anyways, that's all fine and dandy, but now I'm curious:
If I've got my "threshold" of 1/1000 chance of bust, how do I manage my roll within that threshold?
Here's a few options:
A) I start playing 2/4 LHE with 440 bb. I play an orbit, lose 2bb, and go back to 1/2 because now I'm playing outside my roll.
B) I start playing 2/4 LHE with 440 bb. I play until I lose 220 bb of that, then I drop to 1/2, where I have 440 bb, and I can re-take my 1/1000 chance of bust, going back to 2/4 when I have 440bb at 2/4.
C) Play 2/4 until I bust. Just to be a nit, I'm going to say that busting means I've got 12bb left in my stack, so that my EV never changes, because if it did, that would have thrown off my initial calculation on my required BR.
Looking at option C a little more closely. Although I originally had taken a 1/1000 ROR shot, when I got down to 30bb in my stack, there's some chance that I won't bust, but it ain't great. /images/graemlins/smile.gif
Specifically:
ROR = exp(-4*BR/225)
= exp(-4*30/225) | I've got some change in my pocket or whatever to make sure I always have 12bb in my stack.
= exp(-120/225)
= 0.5866
= about a 60% chance of busting.
---
Option A represents what I will call a "Continuous" risk threshold. This means that you're constantly looking at your BR to determine what you should be doing. This is going into super-conservative territory, but it may be the correct approach. I'm just not sure yet if I should be using this approach or not. I suspect that this approach makes your actualized ROR much much lower than 1/1000, since you'll be reducing bet sizes in a downswing.
Option C represents what I'll call a "Static" risk threshold. This is actually pretty funky: if you had something like a million dollar roll, but for some reason you could only accept swings of a maximum of $1760, you'd go ahead and set this threshold, and there's only a 1/1000 chance that you'd ever see an absolute downswing (not relative to EV) this violent. On the other hand, if you play your bankroll like a "system", accepting one shot at risk, in a sense, this is it. You take your 1/1000 chance, no matter what happens, you'll never exceed that chance.
FWIW, the reason that a 300bb roll is acceptable is because of people's willingness to drop down in limits if they hit a bad swing. Otherwise, you may be best to keep a 440 roll just to protect you from the really bad swings. They DO happen.
---
Option B, which is what most people take, is what I'll call the "step-wise" approach to bankroll management.
Here's how it might work in practice.
You have a roll of X. You decide to bet at threshold ROR=0.1% until your roll is either 1.5X or 0.5X, at which point, you will re-evaluate your bet size (stakes) to be consistent with ROR = 0.1% at new bankroll X.
You might say that you want to play until X = 0X or 2X, in which case, you'd up your bets to double what you were betting (going from 1/2 to 2/4 in this case, or 2/4 to 4/8).
---
I've shown some different bankroll management practices, and I've shown you guys why Miller says what he says (in the 2p2 mag) about bankroll management, but I can't tell you guys what is best. On one hand, it's subjective, and I can't tell you what an acceptable level of risk is for YOU, and on the other, I don't have a clue which of these three approaches is the most suited to an advantage gambler in general.
Any input on this would be appreciated.
I'm going to post a trimmed-down version of this in Probabilities (http://forumserver.twoplustwo.com/showflat.php?Cat=0&Number=4017925&page=0&vc=#Post4 017925).
Over the past few months, I've become less of a "poker player" and more of an "advantage gambler". This has lead me to questions that I never had to seriously think about as a poker player.
I'm asking this question of the probabilities forum and I'm cross-posting it here, because I believe that this question is pertinant to micro-limits and the mid-high forums, and that the mid-high guys should know this already anyways (or if not, they should read the probabilities forum to find out).
The reason I think it's important for us is that we can move up limits very quickly. The guys in mid-high have selection of a wide variety of games that are within their rolls, for the most part (100-200 all the way to 1000-2000++). With small stakes players, the increments between levels tend to be smaller and players tend to settle at their current limits for longer.
---
To avoid issues of sample size, I'll give you a scenario:
You're coming out of the shower one morning (or afternoon), and lo and behold, The Supreme Being is in front of you... God, Jesus, Jehova, Allah, Genesh, or maybe just Einstein if you're an atheist. Regardless, there he is. He passes you a towel. After drying your face, you ask, "Is it really you?" He gives you some sort of wise smile and nods. He then says, "I know you have questions, and I am here to help. You may ask only two questions."
You've been waiting for this a long time, so you ask, "What's my BB per 100 hands winrate?", and he tells you it's 2. You're sure he means for the number of tables that you currently play, so you ask next what your STD DEV per 100 hands is, and he tells you that it's 15 bets. He vanishes and you give a sigh of contentment at the final resolution of the greatest mysteries of your life.
---
ROR = exp(-2*BR*EV/VAR)
VAR = STD DEV ^ 2
= 15 ^ 2
= 225
EV = 2
ROR = exp(-2*BR*EV/VAR)
= exp(-4*BR/225)
Generally I like to have ROR = 0.1% = 0.001
Isolating BR for ROR:
ROR = exp(-4*BR/225)
ln ROR = -4/255 * BR
-4/255 * BR = ln ROR
BR = -255/4 * ln (0.001)
= -63.75 * ln (0.001)
= -63.75 * -6.907755
BR = 440.36 BB
So, when Miller says that you need 500bb + health insurance + cash for emergencies + three months' living expenses, he's being pretty conservative, but he's not being much more conservative than a 1/1000 chance of losing your roll, and he's preventing us from having to dig into our roll to pay for all the stuff he just mentioned (thus maintaining our ROR instead of accidentally exceeding our ROR target, which may be 1/1000).
So this basically boils down to: If I'm going to move up in limits, I like the idea of having 440BB when I do so, unless you don't mind the possibility of losing your roll, which may not be so bad depending on your circumstances. Needless to say, we can never TOTALLY eliminate the possibility of losing our roll, because variance exists and has effects on any sized roll, but 1/1000 is a risk I'm willing to take.
---
Anyways, that's all fine and dandy, but now I'm curious:
If I've got my "threshold" of 1/1000 chance of bust, how do I manage my roll within that threshold?
Here's a few options:
A) I start playing 2/4 LHE with 440 bb. I play an orbit, lose 2bb, and go back to 1/2 because now I'm playing outside my roll.
B) I start playing 2/4 LHE with 440 bb. I play until I lose 220 bb of that, then I drop to 1/2, where I have 440 bb, and I can re-take my 1/1000 chance of bust, going back to 2/4 when I have 440bb at 2/4.
C) Play 2/4 until I bust. Just to be a nit, I'm going to say that busting means I've got 12bb left in my stack, so that my EV never changes, because if it did, that would have thrown off my initial calculation on my required BR.
Looking at option C a little more closely. Although I originally had taken a 1/1000 ROR shot, when I got down to 30bb in my stack, there's some chance that I won't bust, but it ain't great. /images/graemlins/smile.gif
Specifically:
ROR = exp(-4*BR/225)
= exp(-4*30/225) | I've got some change in my pocket or whatever to make sure I always have 12bb in my stack.
= exp(-120/225)
= 0.5866
= about a 60% chance of busting.
---
Option A represents what I will call a "Continuous" risk threshold. This means that you're constantly looking at your BR to determine what you should be doing. This is going into super-conservative territory, but it may be the correct approach. I'm just not sure yet if I should be using this approach or not. I suspect that this approach makes your actualized ROR much much lower than 1/1000, since you'll be reducing bet sizes in a downswing.
Option C represents what I'll call a "Static" risk threshold. This is actually pretty funky: if you had something like a million dollar roll, but for some reason you could only accept swings of a maximum of $1760, you'd go ahead and set this threshold, and there's only a 1/1000 chance that you'd ever see an absolute downswing (not relative to EV) this violent. On the other hand, if you play your bankroll like a "system", accepting one shot at risk, in a sense, this is it. You take your 1/1000 chance, no matter what happens, you'll never exceed that chance.
FWIW, the reason that a 300bb roll is acceptable is because of people's willingness to drop down in limits if they hit a bad swing. Otherwise, you may be best to keep a 440 roll just to protect you from the really bad swings. They DO happen.
---
Option B, which is what most people take, is what I'll call the "step-wise" approach to bankroll management.
Here's how it might work in practice.
You have a roll of X. You decide to bet at threshold ROR=0.1% until your roll is either 1.5X or 0.5X, at which point, you will re-evaluate your bet size (stakes) to be consistent with ROR = 0.1% at new bankroll X.
You might say that you want to play until X = 0X or 2X, in which case, you'd up your bets to double what you were betting (going from 1/2 to 2/4 in this case, or 2/4 to 4/8).
---
I've shown some different bankroll management practices, and I've shown you guys why Miller says what he says (in the 2p2 mag) about bankroll management, but I can't tell you guys what is best. On one hand, it's subjective, and I can't tell you what an acceptable level of risk is for YOU, and on the other, I don't have a clue which of these three approaches is the most suited to an advantage gambler in general.
Any input on this would be appreciated.
I'm going to post a trimmed-down version of this in Probabilities (http://forumserver.twoplustwo.com/showflat.php?Cat=0&Number=4017925&page=0&vc=#Post4 017925).