Bulletin of Economic Research, 2025
The Kelly criterion (described in many “Kelly calculators” online) is built for a simple world: you place a bet that either wins or loses. But many real betting problems don’t look like that. You can have multiple mutually exclusive outcomes—for example, several teams that could win a league title, where only one can.
This paper works out the optimal betting strategy in that setting for standard concave utility functions. The big message is that if you solve the true problem, you don’t just apply a two-outcome Kelly rule separately to each bet. You take into account that bets can hedge each other across states of the world. That logic makes the optimal strategy more aggressive than the usual two-outcome rules, and it can even recommend placing some bets with negative expected returns because they reduce risk in other states.
There’s a sting in the tail. If the odds are “fair” in the sense that they correctly reflect underlying probabilities but include a bookmaker margin, then this more aggressive strategy loses more money and delivers lower realised utility than simpler two-outcome rules.
Key research finding: With multiple mutually exclusive outcomes, the true utility-maximising strategy is often more aggressive than Kelly-style two-outcome rules, and can involve negative-EV bets as hedges.
Practical advice: Be careful applying Kelly calculators to multi-outcome situations. If you don’t genuinely have an edge, “optimal” multi-bet strategies can simply lose more, faster.