30+ buyins, mathematical explanation required

[Maulik]

earlier this evening, I was trying to explain to a friend why having 30 buyins is the widely accepted number for avoiding the phenomen known as 'risk of ruin' (at lower buyins). Kelly Criterion (google search) (http://www.google.com/search?q=kelly+criterion&sourceid=mozilla-search&start=0&start=0&ie=utf-8&oe=utf-8&client=firefox&rls=org.mozilla:en-GB:official) for more information. KC is especially useful for blackjack players enter Edward Thorp if you're curious.

Assuming a 3% edge in limit poker yields a requirement of 300 buyins at the lower limits. however, here I am not sure what is assumed for the corresponding variance.

I'll search the archieves tomorrow and post anything relevant.

[curtains]

30 buyins is a joke honestly if the bankroll is not replenishible. Very easy to go on a 30 buyin downswing, and also if you start off losing 5-10, you are going to start to feel very uncomfortable with your remaining bankroll of 20-25.

[Mr_J]

Risk of ruin almost doesn't exist for a good player since you can just move down in stakes when needed. So in that sense 30 buyins at the 109s is fine, you won't go broke. BUT, who wants to have to jump back down??? Once you move up your wanna stay there, and it's not going to happen if you only keep 30 buyins in your bankroll.

[DJ Sensei]

From a strictly mathematical standpoint, theres no way to ever completely avoid risk of ruin. The best we can do is say "is 5% chance of ruin too much, or 2%, or 1%?". Next need to find the EV and variance and make a distribution of finishes.... etc, etc.

I'll bust out the calculator and work on it for a while and post again soon...

[Maulik]

DL - I should have made it obvious that I understand confidence intervals, "chance of ruin," that any engineer or computer science student would know with a formal degree. This would elimiate responses of BR being replinsable (not a flame).Merely interested in limiting the discussion to statistics.

[DJ Sensei]

true, true. I'm working out a little statistical analysis of a standard player, see if i can't figure something out.

Right now, i'm assuming a 15/12/10 finish distribution, which corresponds to a 19% ROI, to see what the ROR is over various run sizes (i.e., over 100 STTs, or 500, or 1000)

[DJ Sensei]

aight, heres what i have so far:

for the 19% ROI player at the $22's (15/12/10 finish distribution)

the chance he'll go broke with a bankroll B over a random sample of N STTs is as follows (not entirely perfect probably, but fairly accurate)

B $440 $660 $1100
N________________________________
20 <.0001 na na
30 .0008 <.0001 na
50 .003 <.0001 <.0001
100 .011 .002 <.0001
500 .001 .0005 <.0001
1000 <.0001 <.0001 <.0001


of course, the $440 corresponds to 20 buyins, $660 to 30, and $1100 to 50. I'm not sure about the math though... it seems to say that theres no more than about a 1.2% chance of ruin, even with only 20 buyins...

meh, its late.

[AleoMagus]

The actual calculations to determine a specific bankroll requirement or a specific ROR are:

B=-(SD^2/2W)LN(R)

r=EXP(-2WB/SD^2)

where,
W is your average profit per tourney ($)
SD is your standard deviation per tournament ($)
R is your desired risk of ruin
B is your bankroll ($)

These calculations assume that a player will continue to play at a certain level, and will not cash out profits. This is, of course, a foolish assumption. In reality, we will sometimes cash out profits, and we will sometimes move up or down in stakes.

Assuming we want a 1% ROR, and we have a SD of 1.7 buy-ins, this looks something like this:

ROI - Buy-ins required

5% - 133.1
10% - 66.5
15% - 44.4
20% - 33.3
25% - 26.6
30% - 22.2
35% - 19.0

Really, the old 30 buy-in rule comes from smaller buy-in players who can get 25%+ ROI. All the higher limit players then notoriously chime in that 30 is way too little. This is obviously just because at the limits they play they are far more likely to get 5-10% ROI and thus, require a lot more.

Keep in mind that if a player does decide to drop in stakes when they take a hit, they need much less. Also, as we are talking about a relatively small number of tourneys, very tilty players might need a lot more.

Oh, one more thing... if you do cash out profits, and keep a constant floating bankroll (say 30 buy-ins) then you will almost certainly go broke. In some way (even a small way), you should always keep your bankroll growing.

Regards
Brad S

[Kama45]

Is that taking rake into consideration?

[AleoMagus]

AAAAARRRRRGGGGHHHH.

I can take all this sim stuff anymore.

The actual statistical methods to just solve for any of this stuff has been covered (at least by me) on this forum a million times before.

But everyone just glazes over statistics, and writes simulators to tell them the same answer.

Sigh
Brad S

[Sabrazack]

You poor misunderstood math whiz /images/graemlins/smile.gif I think people have an easier time interpreting simulation results than naked statistics.

[USCSigma1097]

Aleo,

Thanks for putting up with all of us "non-math" guys and posting your response. I have saved it and will now flame anyone that asks this question again. And don't you worry, it will be a healthy flaming.

Flamethrower.
aka Sigma

[Nicholasp27]

how do u calculate the floating bankroll required for a buy-in at a certain roi%?

that is, say u are 30% on 11s/22s...u wanna have a br of x buyins and then at the end of each week/month you wanna cash out all money above x buyins...what is needed then?


i guess what that is asking is what is the largest dip you expect to see (not streak of ootm, but lowest possible point u would get to) in, say, 1000 tournaments


for instance, u could go 10ootm then get 3rd in 2 tournies then 10 more ootm, so ur actual drop isn't just 10 buy ins, but 18...



say x for 30%roi is 50 buy-ins...then i could just deposit 550 into pp and play the 10s and each friday i can pay myself any amount that is over 550 in my account...if it's 550 or less i don't get paid...over 1000 tournies, i expect to cash out $3300 total, so if i then go broke on the 550 at some point after 1000 tournies, then that's fine, i'll just reload

is there a formula or would a bunch of sims just have to be ran?

[SirKraggen]

</font><blockquote><font class="small">Svar på:</font><hr />
You poor misunderstood math whiz /images/graemlins/smile.gif I think people have an easier time interpreting simulation results than naked statistics.

[/ QUOTE ]

That might be true, but in the case of Senseis simulations (try saying that a few times!), the simulated results are incorrect. For example, if you have 1.1% chance of going broke in 100 games with 20 buyins, then you have at least 1.1% of going broke in 500 or 1000 games. If you're broke after 100 games, you're broke, period. No more games for you!

I'm not trying to flame or put anyone down, I have very likely made the same mistake in the past (maybe that's why I recognize it /images/graemlins/laugh.gif) ).

[Ryendal]

[ QUOTE ]
SD is your standard deviation per tournament ($)

[/ QUOTE ]

How do you find your standard deviation ? It's hard for me to believe we can get a good number for that.

[AleoMagus]

[ QUOTE ]
How do you find your standard deviation ? It's hard for me to believe we can get a good number for that.




[/ QUOTE ]

Standard deviation per tournamnet in SNG poker is equal to

SQRT((F1)(p1^2)+(F2)(p2^2)+(F3)(p3^2)+(Fn)(pn^2)+( W^2))

where,
F1= probability of finishing 1st
F2= probability of finishing 2nd
F3= proabability of finishing 3rd
Fn= probability of finishing nth

(note: all OTM finishes may be combined as one probability for ease of calculation)

p1= Net profit for 1st
p2= net profit for 2nd
p3= Net profit for 3rd
pn= Net profit(loss) for nth

W= win(loss) rate in net $/tourney

Or, you can just use one of the spreadsheets out there which calculates this for you.

www.aleomagus.freeservers.com/spreadsheet (http://www.aleomagus.freeservers.com/spreadsheet)

My 'confidence calculator' on this site has a pretty simple spreadsheet which will do this.

As for how 'good' these numbers are, I can't be exactly sure what you mean, but I will say that I think SD is a very reliable stat. This is mostly just because your SD will change so little so long as you are a winning player. Bigger winners will have a slightly higher SD, but almost all players will have an SD of 1.6-1.8 buy-ins (or 1,6-1,8 for those in chile).

Regards
Brad S

[johnnybeef]

[ QUOTE ]
AAAAARRRRRGGGGHHHH.

I can take all this sim stuff anymore.

The actual statistical methods to just solve for any of this stuff has been covered (at least by me) on this forum a million times before.

But everyone just glazes over statistics, and writes simulators to tell them the same answer.

Sigh
Brad S

[/ QUOTE ]

This is an incredibly close minded post. Many people (myself included) learn by doing. Just because you have ran and published all of the simulations doesn't necesarily mean that someone else will learn it best by reading your publishings. Take for instance a difficult calculus problem. I can look at a problem step by step, and see what to do and why to do it and then not remember why ten minutes later. If instead, I get a blank piece of paper in front of me and am given a problem with someone looking over my shoulder, and figure it out with a little guidance, I will remember how and why something is done much better. People do the same thing differently for a multitude of reasons. As a poker player, it should be your goal to understand these reasons, as that is where your profit comes from. I have a lot of respect for what you have done for this forum Aleo, but many of these sims that you have done are being redone by people in an effort to understand the results as well as you do.

Sigh
Johnny

[gumpzilla]

[ QUOTE ]
I have a lot of respect for what you have done for this forum Aleo, but many of these sims that you have done are being redone by people in an effort to understand the results as well as you do.

[/ QUOTE ]

a) His point is that what he was doing was not simulations, but statistical calculations.

b) You have a good point that people learn in different ways, but if your argument is that people should play with the simulations themselves, how does posting to 2+2 with simulation results any different than posting with statistical analysis? In neither case are readers doing it for themselves, and the person posting a simulation isn't offering any kind of theoretical underpinning to why the results are as they are, usually.

[johnnybeef]

[ QUOTE ]
b) You have a good point that people learn in different ways, but if your argument is that people should play with the simulations themselves, how does posting to 2+2 with simulation results any different than posting with statistical analysis? In neither case are readers doing it for themselves, and the person posting a simulation isn't offering any kind of theoretical underpinning to why the results are as they are, usually.

[/ QUOTE ]

fair enough, i guess i missed the point.