Since I was called on by name, I guess I have to respond to this thread.
I will respond in reverse chronological order.
Warning: math ahead
PM: "In other words, if you take 100 players with avarage win of 4$ net at the 11$ SNG, and give them each a bankroll of 88$, 5 of them will bust at one point, but 95 will continue to make $$ as long as they live (if they don't cash-out?)"
Yes. Except for corrections in his formula, detailed below.
The cashing out correction can be dealt with very simply: when your bankroll changes (because of a win, a loss or a cashout), your risk of ruin, from that point, changes. So, if you cash out any excess whenever bankroll is larger than B(r), you will have r probability of going broke after each cashout. Those times your BR falls below B(r) your risk of ruin, calculated at that time, will be larger.
While 1% risk of ruin doesn't sound like much, it can be significant over many opportunities for ruin, since if you play with this RoR always (never less), your risk of going broke sometime will increase linearly (~1%*NumberofSNG's/25). Playing 400 SNG's like this is more dangerous than playing with a bankroll 2/5 as big that you never touch.
Guy:
Your estimate is essentially the same one used by Mason in GTAOT. The problem is that it somewhat underestimates the RoR since it misses all the times where you go broke in < n = (std dev/avg win) ^ 2 trials but end up with finite money at n, as well as those times where you go broke after > n SnG's. Your approach yields BR=f(r)*SD^2/EV, where r is RoR, SD is one tourney standard deviation in $/sqrt(SnG), and EV is per tourney expectation (in $/SnG). The f(r) you get is Z(r)-1, where Z is the critical value: number of SD's away from the mean of a normal distribution for which a fraction r of the area is to the left, and 1-r to the right. Z(1%)=2.33, Z(5%)=1.64.
You made one mistake here, because you used the two sided Z instead of one sided. Obviously, you won't go broke if you run +2SD well. (You calculated the Z=2 result, which is 2.3% RoR, instead of the Z=1.64.)
A better result was derived by Patrick Sileo (
Risk and BR), which gives f(r)=-ln(r)/2 (for a total BR formula of B=-SD^2/(2*EV)*ln(r)). Not only is it a better estimate, but ln(r) is easier to calculate than Z(r). This gives BR numbers ~1.5-2.5 times as big, with the largest corrections coming for the largest RoR's. For 5% RoR, -ln(r)/2= 1.5, giving a 50% larger BR than Guy's erroneous estimate (the errors patially cancel).
This answer relies on the normal distribution, so there will be small errors since SnG results will vary somewhat from a normal distribution over the relatively short number required to break you with the relatively small SD of SnG's. So this eq. is better for ring games than SnG's, but errors are still probably small. One could correct for exact SnG results (%1st,2nd,3rd), but it doesn't look like it is worth the trouble.
As for your estimates, they are not too bad. Over my last 425 SnG's, I would have gone broke in one 25 SnG set if I started each with 8xBuyin:
1/17= 6%. However, I suspect that my game was suffering over some of the down times, and I know my table selection was. Just wanted to prove we are not off by orders of magnitude.
Aleo:
I think your SD and EV numbers are pretty good. At the $109 level, my SD is about $180 (2nd net - $11) since I am unfortunately not getting more 1st's than 2nd's (my EV is slightly less too).
For your #'s, B(5%)= $135, B(1%)=$208, B(25%)=$63.
For your general argument, as long as reloading to your bankroll is easy, physical bankroll requirements are not very meaningful. Psychological BR is probably more important. However, as you get to larger stakes, bankroll becomes less replaceable, so you should protect it more.
After spending almost all my bankroll on travel and expenses, I had less than $500 left. I figured there was some chance I'd go broke at $55 SnG's, but it was only a one time risk, since I was going to build my bankroll without touching it until it was much more comfortable. This worked out for me, but I probably wouldn't have done it if I didn't have any possible other sources to borrow replacement money from. In addition, the possibility, undesireable that it is, of dropping in limits will greatly reduce risk of ruin.
Craig