BR management and probability

[NYplayer]

let's say i play Limit L1 and have a winrate of 1.55bb/100 and standard deviation of 25bb/100. now i can calculate that i need a bankroll of X to stay within 3 stdev rusk of ruin.

if i play limit L5 which is 5 times bigger and everything else is the same i need a bankroll of 5X.

However, what if i play limit L5 only 1% of the time. I need a bankroll bigger than x but surely less than 5X. how do i figure out what it should be? what if i play L5 10% of the time? 50% of the time?

[pzhon]

[ QUOTE ]

However, what if i play limit L5 only 1% of the time. I need a bankroll bigger than x but surely less than 5X. how do i figure out what it should be? what if i play L5 10% of the time? 50% of the time?

[/ QUOTE ]
There are multiple ways you could mix L5 and L1. One is to make sure that you are playing one hand at L5 for every 9 hands of L1. This is relatively easy to analyze (see below). Another would be to choose which level to play at randomly. This would require a larger bankroll. However, what you do in practice is to play a session of L5, then a few sessions of L1, etc. In that case, the bankroll you need is even larger, and depends on your session lengths.

Assuming perfect mixing, your win rate is the weighted average of the win rates, and your variance is the weighted average of the variances. That's because the variance of the sum of independent random variables is the sum of their variances.

The bankroll you need is c * variance/(win rate).
Let the win rate at L1 be W, so the win rate at L5 is 5W.
Let the variance at L1 be V, so the variance at L5 is 25V.
The bankroll for L1 is c*V/W. The bankroll for L5 is 5c*V/W.

With a 50-50 mixture, the variance is (V+25V)/2 = 13V, and the win rate is (W+5W)/2 = 3W, so the bankroll is 13/3 c*V/W.

With a 90-10 mixture, the variance is (9V+25V)/10 = 17/5 V, and the win rate is (9W+5W)/10 = 7/5 W, so the bankroll is 17/7 c*V/W.

Again, since you don't mix the levels evenly, your variance is greater in practice, so these are just lower bounds.