Suppose we have $n$ independent events.
We know their probabilities $p_i,\dotsc,p_n$.
We are given betting odds $c_1,\dotsc,c_n$.
We can make bets to any of the events, and also any combination of events.
For every bet (be it a single or combined-event), the bookkeepers take a certain percentage $t$ from the winnings amount.
It is obvious that we can make $N=\sum_{i=1}^{n}\binom{n}{i}$ different bets.
We bet a fraction $b_i$ of our "bankroll" for the $i$-th of the N possible bets. If $b_i=0$, it means we don't make the $i$-th bet at all.
Is there a way of determining the optimal set of $b_1,\dotsc,b_N$ so that, in the long run, the "bankroll" increases maximally? (Something like the Kelly criterion)
Best Answer
Using the strategy described in the Wikipedia article on the Kelly Criterion, there absolutely are situations where you should place different bets on multiple different outcomes, including outcomes with a less-than-maximum Expected Value. There are even situations where you should bet on outcomes with a negative Expected Value.
Strategy:
To summarize the strategy described in the article (for multiple, exclusive, "Many horses" event situations):
1) Order all of the possible events from most to least profitable (highest to lowest Expected Value).
2) For each event, see if the Expected Value for that event exceeds the "Reserve Rate" for your existing set of bets. (Your "Reserve Rate" is initially "1" when your set of planned bets is empty.) If the Expected Value is higher, then add that event to your set of bets.
3) Once you have discovered your optimal set of outcomes to bet on, compute the optimal bet size for each outcome using the "Fraction to Bet" equation below (where "Fraction to Bet" is a fraction of your current bankroll.)
Reserve Rate = (1 – (sum of each probability bet on)) / (1 – (sum of each 1/payoff))
Fraction to Bet = Probability – Reserve Rate/Payoff
Example:
Here is a gambling article which gives a horse-racing example in which your optimal betting set would include a horse with a negative Expected Value. In this example situation, betting on sub-optimal horses allow you to safely wager a much larger percentage of your overall bankroll, which enables you to maximize your long-term expected profit as defined by the Kelly Criterion.
"Independent" vs. "Exclusive" Events:
Your question deals with Independent events, and the Kelly article deals with Exclusive events. But you can still use the strategy by considering all of the Exclusive possibilities. First compute your Reserve Rate for not betting. Then compute your Reserve Rate for the highest EV bet. Then compute the Reserve Rate for the top two highest EV bets where you separately compute the EV for each exclusive outcome (A and B, A and Not B, B and not A, etc...)