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What I meant was that the usual ROR formula assumes small bet sizes and a relatively unskewed payoff distribution.
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The initial bankroll can be a single bet. In my
derivation of the ROR formula, which is essentially Sileo’s derivation, the ROR for a bankroll of size B is derived as the ROR for a 1 bet bankroll raised to the B power. This assumes that the winnings are reinvested in the bankroll, and that your win rate and standard deviation do not change. My comments above assumed that we are maintining a constant win rate.
If your increase your betting limits as your bankroll grows, then you will go broke with probability 1 if you continue to bet more than twice the Kelly fraction of your bankroll, where the Kelly fraction is approximately EV/sigma^2.
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For highly skewed games such as video poker (and i think most poker tournaments) the ROR calculation is not so nice
see
this article
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The ROR formula that I linked to is derived by assuming that the game with skewed payoffs can be modeled as a coin flip game with the same win rate and variance via the central limit theorem. There may be games for which this model breaks down, but it works well for blackjack and poker, and any game for which the risk of ruin depends primarily on the win rate and standard deviation, and very little on the higher moments.