BR formula when stepping down

[Angel]

I was discussing the formula for determining one's needed bankroll [-(SD^2/2*hourly win rate)ln(risk of ruin)] recently and was asked if there was a formula if one was willing to step down to a smaller game if one lost half their bankroll. I have seen a couple of estimates for this - most notably I believe Malmuth said you could cut your needed bankroll to 2/3's of the formulaic total if you were willing to step down if you lost a significant portion of your bankroll. But playing with the numbers I came up with [-(SD^2/2*hourly win rate)ln(SQRT risk of ruin)] which calculates a bankroll size of 1/2 the original.

Could someone explain where this formula fails - if in fact it does?

[BruceZ]

If you step down to half the level after losing half your bankroll, and never step down again, then you need 2/3 as much bankroll as if you never step down for the same overall risk of ruin. To go broke when stepping down, you must first lose half your bankroll, and then lose as many total bets as would have had to lose if you never stepped down, so it is like having a bankroll 3/2 times greater.

Doubling your bankroll squares your risk of ruin, making it smaller since it is less than one. This is because you must lose the original bankroll twice. Reducing your bankroll by 1/2 has the effect of taking the square root of the risk of ruin, making it larger. Multiplying your bankroll by any factor raises your risk of ruin by the power of that factor. If your risk of ruin if you never step down is r, then the risk of losing half your bankroll is r^(1/2). The risk of losing half your bankroll, stepping down, and then losing the rest of your bankroll is
r^(1/2)*r = r^(3/2). So stepping down raises your risk of ruin to the 3/2 power, meaning it gets smaller, and it is the same as if you increased your bankroll by a factor of 3/2, so you only need 2/3 of the bankroll for the same risk of ruin r. This is all in agreement with the bankroll formula you have given.

[Angel]

Bruce,

Just wanted to thank you for the feedback.

[Bozeman]

If you are a winning player (at all relevant limits), and there is always a lower limit to step down to, your risk of ruin becomes zero for any bankroll. If you have m limits you can play in, each half as big as the higher one, and you step down every time you get to 1/2 your previous stepdown bankroll, your risk of ruin will be (r^1/2)^(m+1)=r^[(m+1)/2] where r is the risk of ruin you would have with this bankroll staying at the big limit. (This assumes you have the same win rate at all limits.) Obviously, RoR decreases rapidly as m goes up. For example, if you had a 10% risk of ruin, you could reduce it to 1% by having 2 more levels you could drop into.

Craig