Stack Exchange has many posts regarding bet sizing on multiple, mutually exclusive/independent events/gambles.
I'm curious if anyone is familiar with using Kelly to bet on 1+ gambles that are not mutually exclusive
For example, in blackjack, some casinos offer Side Bets.
Side Bet 1 (Perfect Pair): Pays off if both player cards are identical (same rank/suit) Side Bet 2 (Suited Trips): Pays off if both player cards and dealer up card are identical (same rank/suit)
You can place a bet on Side Bet 1, Side Bet 2, or both together. In addition, each bet can be a different size.
As these outcomes are not mutually exclusive, the standard Kelly formula cannot be used.
Here is my thinking so far:
For each bet combination (Side Bet 1, Side Bet 2, or both), calculate the bankroll fraction that will maximize r. When solving for 'both', each bet will have a separate bankroll fraction variable to enable each bet to have a different bet size.
Plug the fractional bet sizes back into the geometric growth rate formula and find the bet combination with the greatest R.
Wager on the bet combination with the greatest R (from step 2) and bet bankroll fraction on each bet as calculated in step 1.
If my logic above is correct, how do you set up step 1? Would it look something like this:
$R = \sum_{i=1}^{TotalBets} \left(1 + f_i \cdot payoff_i\right)^{prwin_i} \cdot \left(1 - f_i \cdot loose_i\right)^{1 - prloose_i}$ where $f_i$ is the bankroll fraction wagered for gamble i.
Thank you in advance!