utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes expected utility, so there is no conflict; moreover, Kelly's original paper clearly states the need for a utility function in the case of gambling games which are played finitely many times). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations. Advanced mathematics For a rigorous and general proof, see Kelly's original paper or some of the other references listed below. Some corrections have been published. We give the following non-rigorous argument for the case with (a 50:50 "even money" bet) to show the general idea and provide some insights. When , a Kelly bettor bets times their initial wealth , as shown above. If they win, they have after one bet. If they lose, they have . Suppose they make bets like this, and win times out of this series of bets. The resulting wealth will be: Note that the ordering of the wins and losses does not affect the resulting wealth. Suppose another bettor bets a different amount, for some value of (where may be positive or negative). They will have after a win and after a loss. After the same series of wins and losses as the Kelly bettor, they will have: Take the derivative of this with respect to and get: The function is maximized when this derivative is equal to zero, which occurs at: which implies that but the proportion of winning bets will eventually converge to: according to the weak law of large numbers. So in the long run, final wealth is maximized by setting to zero, which means following the Kelly strategy. This illustrates that Kelly has both a deterministic and a stochastic component. If one knows K and N and wishes to pick a constant fraction of wealth to bet each time (otherwise one could cheat and, for example, bet zero after the Kth win knowing that the rest of the bets will lose), one will end up with the most money if one bets: each time. This is true whether is small or large. The "long run" part of Kelly is necessary because K is not known in advance, just that as gets large, will approach .