[BillC]

For those people in this thread sporting a conundrum withering in a cloud of mental haze (or are 6 years old), let me say that the way out is to do the math. The infinite and infinitesimal often lead to paradoxes. Figure out the risk of ruin in 1 bet, n bets, and then ask what happens as n gets big. That is the way to balance your expected increase in bank vs the risk of a downturn. That is assuming you have an advantage; for a pure coinflip, you will always go broke. (Aside: in a one-dimensional random walk, you will return to zero with probability one. This also true for a 2-dim random walk; but it fails for higher dimensions. (The probability of return to origin is called the Polya number) So maybe it is better to keep >2 independent bankrolls...).

Just to recap: In games of great skew, you need to use a more precise calculation to determine ROR and bankroll. It would seem apt to ask what BR would be needed to enter a big poker tournament, e.g WSOP, assuming you are a somewhat above average player. It has to be huge.

The question of applying the Central Limit Theorem always dogs these sort of applied random walk models. It seems that the way out are error bounds involving higher moments as in the article by Stu Ethier (et. al) (thanks for the link BruceZ). Unfortunely the math gets way beyond the ken of most. They just have to trust us. The normal approximation is OK for poker.

Sileo's derivation was just an explanation for the gambling conference of math that is really old, almost as old as stochastic analysis itself. The gambling conference proceedings are not refereed and are below the standards of academic math journals. You can say anything. They have published pure bunk. That is not to say that Sileo is wrong, he just reivented the wheel, and explained it for non-math people like Don Schlesinger and the BlackJack community.

Final note: if you bet more than 2 times Kelly, your win rate will be negative. You will never go broke with fractional betting. But you will asymptotically go to zero.