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) 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