Let the result of a one-unit bet as a proportion applied to a bankroll be . Let be the size of the bet and be the bankroll. The Kelly Criterion sets to maximize
Using the Taylor's series approximation and simplifying the expectation, we are to maximize
Setting the derivative with respect to equal to zero yields the Kelly Criterion
when is small.
Let be the result of simultaneous, independent, one-unit bets with the same exprectation and variance as . Then and because of the independence . Applying the Kelly Criterion here
The optimal size of each of the individual simultaneous bets is unchanged from the non-simultaneous case.
This result is approximate because of the use of the Taylor's series approximation that is only accurate when is small. So it should not be used to justify betting a large portion of the bankroll on a number of simultaneous bets.
In the case of blackjack, simultaneous bets are not independent and the variance is where is the single hand variance and is the covariance between hands. Then
The optimal size of the individual simultaneous bets decreases with the number of simultaneous hands.