Math: Need help regarding the Kelly criterion

[LordMushroom2]

I wanna use the Kelly criterion to determine if I am bankrolled to play certain HU STT buy-ins. But it doesn´t take into account that I take money out of the bankroll to pay for bills. How can I take that into account?

My current method is to divide the dollars I take out of the bankroll per year by the number of HU STTs I play per year (I play these exclusively), and add that to the buy-in in the Kelly criterion formula. What do you think of this approach.

[Drag]

When you take money out of your bankroll you 'effectively' decrease your winrate. This would be the right approach.

Effectively increasing your buy-in doesn't work the same way.

[LordMushroom2]

Thanks.

[jason1990]

When you take money out to pay bills, do you generally take out the same dollar amount each unit of time, or do you take out the same percentage of your bankroll each time? If it is the latter, then you can easily incorporate that into the Kelly system. If Kelly tells you to wager a fraction f, then you should wager f(1 - r), where r is the fraction you are removing with each wager.

If, however, you are removing the same dollar amount each time, then it is quite complicated. Suppose you start with a bankroll of B_0. Let B_n be your bankroll after n wagers. Without removing anything, your bankroll changes as

B_n = X_n B_{n-1},

where X_n is the proportional change in your bankroll due to the n-th wager. Solving this recursion,

B_n = B_0 e^{S_n},

where S_n = Y_1 + ... + Y_n, and Y_j = log(X_j). For large n, S_n ~ un, where u = E[log(X_j)]. This is why the Kelly formula tells you to maximize the expected logarithm. In the long run, your bankroll will look like B_n ~ B_0 e^{un}, so hopefully u is positive. (It will be if you have an edge, and if you follow the Kelly system.)

Now, if you are taking out C dollars with each wager, then the recursion changes to

B_n = X_n B_{n-1} - C.

Solving this,

B_n = (B_0 - CZ_n)e^{S_n},

where Z_n = \sum_1^n e^{-S_j}. If u > 0, then this sum will converge as n goes to infinity to some random limit Z. In the long run, your bankroll will look like

B_n ~ (B_0 - CZ)e^{un}.

Two things to notice. First, if Z > B_0/C, then you will go broke. So there is a risk of ruin. Second, if you survive, and Z < B_0/C, then your bankroll will grow at the same rate as before, and you do not need to adjust the original Kelly recommendation.

As for risk of ruin, it is P(Z > B_0/C), and this looks like a very difficult probability to compute. But here is a (possibly crude) approximation.

Approximate Z by the integral from 0 to infinity of e^{-S(t)}, where S(t) is some continuous interpolation of S_n. Now, S_n is a random walk whose steps have mean u and variance s^2, where u and s are the mean and standard deviation of log(X_j). (The parameters u and s must be estimated from your tournament results.) Therefore, we might try using

S(t) = sB(t) + ut,

where B is a Brownian motion. In that case, it is known that the integral from 0 to infinity of e^{-S(t)} has the same distribution as 2/(Gs^2), where G has the gamma distribution with scale parameter 1 and shape parameter a = 2u/s^2. Using this, we can estimate that your risk of ruin is less than

[(2C/B_0)^a]/(a*Gamma(a)*s^{2a}).

By the way, Gamma(a) is the Gamma function.

[LordMushroom2]

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.

[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.