[jason1990]

[ QUOTE ]
Thanks for all the work you have done, jason1990, but the math is too complicated for me. So I wondered if you could show how a neanderthal like me should go about calculating how big my bankroll has to be to play a certain buy-in by doing it for this example:

Buy-in: $50+2,50
Winrate: 58%
I take $1 out of my bankroll per STT regardless of how big my bankroll is.

[/ QUOTE ]
Since you are taking out a constant dollar amount with each tourney, you can calculate your bankroll normally.

p = 0.58
q = 1 - p = 0.42
b = (100 - 52.50)/52.50 = 0.905
f = (bp - q)/b = 0.116
fB = 52.50
B = 52.50/f = 453

So there is your bankroll requirement. Remember, since you are using Kelly, you must move up and down in stakes as your bankroll changes.

Now we must account for removing $1 each tourney. This adds a risk of ruin.

u = p*ln(1 + bf) + q*ln(1 - f)
= (0.58)(0.0996) + (0.42)(-0.123)
= 0.0061

s^2 = (0.58)(0.0996)^2 + (0.42)(-0.123)^2 - (0.0061)^2
= 0.0121

a = 2u/s^2 = 1.00992

risk or ruin = r = [(2C/B)^a]/[Gamma(a+1)*(s^2)^a]

The factor Gamma(a+1) can be computed in Excel with EXP(GAMMALN(a+1)). This gives

r = [(C/227)^(1.00992)]/0.0116

Taking C = 1 gives r = 36%. You have a very high risk of ruin. In plain English, the idea is this. That $1 may not seem like much. But it is going to slow down the growth of your bankroll. That means you will not move up as quickly. In fact, you will need to move down much sooner than you normally would. When you move down, that $1 becomes more significant, since it is a higher percentage of your bankroll.

Now suppose you want a risk of ruin of 5%. Then we need to solve

0.05 = [(C/227)^(1.00992)]/0.0116

for C. This gives

C = 227(0.05*0.0116)^(1/1.00992)
= 0.1416.

So you should only be taking out 14 cents per tourney on average. Alternatively, if you really need $1 per tourney, you could increase your bankroll. That will be more complicated to compute. Increasing your bankroll amounts to decreasing f. That means u will change, s^2 will change, and a will change. So we would need to go back to the beginning of all this and recalculate with a smaller f. It is easy to redo the calculations for different f values if you use an Excel spreadsheet to keep track of everything. For example, a starting bankroll of $1000 corresponds to f = 0.0525. With C = 1, this gives a 4.35% risk of ruin. In that case, though, remember that you need to move up and down in stakes so that your buy-in is always about 5.25% of your bankroll.