The Kelly criterion maximizes the expected logarithmic growth rate of a portfolio. For a single binary bet, the answer is simple: bet a fraction of your bankroll equal to your edge divided by the odds. For a single event with multiple outcomes — a horse race, a multi-way election — the criterion extends to multinomial Kelly, where the optimal allocation across multiple mutually exclusive outcomes maximizes expected log growth.
The paper derives the full multinomial Kelly solution for a single event with implicit state positions. “Implicit state” means the bet payoffs define a complete set of states — each combination of outcomes maps to a specific portfolio return — and the optimization accounts for the portfolio's behavior across all states simultaneously.
The solution is closed-form. Previous treatments either used numerical optimization or handled special cases (binary bets, equal payoffs). The general multinomial solution involves a system of equations that can be solved analytically, yielding optimal fractions that depend on the full matrix of payoff-probability relationships, not just individual bet edges.
The key practical insight: in a multi-outcome event, the optimal bet sizes interact. Adding a bet on one outcome changes the optimal bet on every other outcome, because the portfolio's logarithmic growth depends on the joint distribution of returns across all states. Optimizing each bet independently — treating each as a separate binary Kelly problem — is suboptimal and can be significantly wrong when outcomes are correlated through the event structure. The multinomial solution captures the interaction; the binary approximation misses it.