[Reef]

note: authored by "izverg" on the BW forums

Definitions and assumptions:
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D=deposit (doesn't enter in this discussion)
B=bonus
X=your target (you aim for $X total or $0=bust)
BR=your whole gambling bankroll
Bets have no house advantage (good approximation for Playtech with BJ switch)

X is the function of
1)size of the bonus B
2)your risk tolerance which should be determined by your bankroll

X does not depend on D.

Since there is no house advantage, for any X, you can extract more EV out of the bonus. Let's say you've reached a value of X, and you set your goal to reach X+dX or bust trying. Simple calculation shows that for this bet:

EV=B/X * dX
SD=(X-B)/sqrt(X) * sqrt(dX)

EV decreases with X, and SD grows with X -- obvious results.

Your bankroll limits the max Var/EV=SD^2/EV of the bets that you are making. When you get to X, you should continue if your bankroll can support bets of

Var/EV = (X-B)^2/B

Using arguments similar to the ones justifying 300BB bankroll requirement in poker, my personal criterion for bankroll is: (I am conservative when it comes to bankroll)

BR = 4*Var/EV

I think this is equivalent to requiring ~500BB bankroll for limit hold'em.

Finally, target X of the sticky bonus B is determined by solving the following equation:

BR = 4*(X-B)^2/B

When I reach X, I reach the risk threshold that my bankroll allows. Time to cashout.

X = B + sqrt(BR*B)/2

Some examples:
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Bonus=200
BR=5,000: X=700
BR=10,000: X=900
BR=20,000: X=1200