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...)
You are correct that you want to find a bet so that your return in every possible situation is positve. If you put amounts (a, b, c) on each outcome then your possible profits, as you correctly deduced, are 2a - b - c, 2b - a - c and 4c - a - b.
One way of proceeding is to look for a solution where you make the same amount of money in each scenario, which means setting the three quantities above equal to each other. In this case you discover that you require a = b and 3a = 5c. You can satisfy these constraints by picking a = b = 5 and c = 3, in which case the possibilities are
- A wins - you make 10 - 5 - 3 = 2
- B wins - you make 10 - 5 - 3 = 2
- C wins - you make 12 - 5 - 5 = 2
In each case you have a profit of 2, so this is your arbitrage. Indeed, the real-world probabilities are completely irrelevant.
The key is the idea of implied probabilities. Since you lose 1 if A doesn't happen, but win 2 if it does happen, then the implied probability $p$ (defined to be the probability that would give you zero expected return on this bet) satisfies
$$
(1-p)\times (-1) + p \times 2 = 0 \quad\Rightarrow\quad p = 1/3
$$
Similarly the implied probability of B is 1/3 and the implied probability of C is 1/5. These sum to less than one (in bookmaking this is called an under-round) so there is the possibility of arbitrage. The arbitrage exists because the odds given are incoherent (they don't sum to one).
To take advantage, you make bets which are proportional to the inverse of the implied probability.
In the real world, a book keeper will always set odds that sum to more than one, meaning that any combination of bets which guarantees a payout, will always guarantee a negative payout (so there are no arbitrages).
Best Answer
You are correct that there are three different optimal bet sizes. This is due to the requirement that the gambler commits to a specific outcome of the team game.
Three are three outcomes to the team game:
However, for any of those outcomes, there are only two outcomes to the gambler's bankroll:
The Kelly criterion is only concerned with the outcomes to the gambler's bankroll. Since the gambler is required to commit to a choice on the outcome of the team game, and this choice affects the outcomes to the gambler's bankroll, this choice must be determined before the evaluation of the Kelly criterion.
There are thus three optimal bet sizes, each dependent on the gambler's respective commitment to the outcome of the team game:
In the question "Kelly criterion with more than two outcomes" (where a colored jelly bean is grabbed at random from a bag of 10 colored jelly beans), there are three possible outcomes in the gamble:
Each of these outcomes affect the gambler's bankroll in a distinct way from the other outcomes. However, since no choice is involved in this gamble, there is only a single optimal bet size, that is determined by the three possible outcomes to the gambler's bankroll.
This is the difference between your team game example and the jelly bean bag example: choice. A choice must be committed before the Kelly criterion can be determined.