Bankroll (again)

The risk of ruin formula is understandably flawed, since it assumes you play at a given level until you go broke.
Since we have so many levels available to us as poker players, this is obviously incorrect.

1: Assuming
a)I am playing 10/20 HE now
b)I will move down to 5/10 when my bankroll reaches X
where X= -(sigma^2/2u)ln(r) or 300BBs
how would I derive a bankroll equation for R
where R is the risk of moving down in stakes?

2: How can we apply the Kelly Criterion to the stakes we are playing in such a way as to maximize growth of our bankroll, assuming we will move up or down in stakes when our bankroll reaches X for the next higher or lower limit?

[BruceZ]

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The risk of ruin formula is understandably flawed, since it assumes you play at a given level until you go broke.
Since we have so many levels available to us as poker players, this is obviously incorrect.

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That's not a flaw. That is the condition for which that risk of ruin formula was derived. It is useful for players who would rather not step down in level because they prefer to make a certain hourly rate, which is pretty much all players. These players compute the amount of money they will win per hour by playing at a certain level, and they want to know the bankroll that will enable them to do so forever with a certain probability of success.


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1: Assuming
a)I am playing 10/20 HE now
b)I will move down to 5/10 when my bankroll reaches X
where X= -(sigma^2/2u)ln(r) or 300BBs
how would I derive a bankroll equation for R
where R is the risk of moving down in stakes?

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In this case, you would just use the same risk of ruin formula substituting R for the risk of ruin, and the formula will then compute the amount you need over and above X in order for the probability of having to step down to be R.


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2: How can we apply the Kelly Criterion to the stakes we are playing in such a way as to maximize growth of our bankroll, assuming we will move up or down in stakes when our bankroll reaches X for the next higher or lower limit?

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The Kelly bankroll is sigma^2/u. If you never adjusted this bankroll, the above formula shows that it would produce a risk of ruin of 13.5%. Most everyone agrees that maintaining only the Kelly bankroll is much too risky, even if you step up and down as required, because the fluctuations will be too great. Most people play some fraction of Kelly, like 1/2 or 1/4 Kelly, meaning that you maintain 2 or 4 times the Kelly bankroll. Just remember that in poker, unlike blackjack, stepping up usually means that the game gets tougher, so you need to wait longer before stepping up.