Analytical solution for Kelly's criterion for multiple outcomes

Jan Vecer

Quantitative Finance, 2026, vol. 26, issue 5, 671-684

Abstract: We present an elementary analytical representation of the optimal portfolio in multi-outcome prediction markets, extending the well-known Kelly criterion from the binary model. Specifically, the solution is to invest a fraction $ \mathbb {P}^M(\omega ) $ PM(ω) of one's wealth in each market outcome ω, where $ \mathbb {P}^M $ PM represents the agent's subjective (physical) measure and $ \mathbb {P}^Y $ PY the quoted (risk-neutral) measure. The resulting final portfolio payoff is represented by the likelihood ratio $ \frac {\mathbb {P}^M}{\mathbb {P}^Y} $ PMPY. This ratio is shown to be optimal for logarithmic utility, so once identified, no further calculations are required. Moreover, its expected logarithmic return corresponds to the relative entropy (Kullback–Leibler divergence), offering an intriguing connection to information theory. This framework also accommodates trading constraints: in such cases, the agent must find the replicable portfolio V, associated with a state price density $ \mathbb {P}^V $ PV, that is closest to $ \mathbb {P}^M $ PM in the sense of relative entropy.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:taf:quantf:v:26:y:2026:i:5:p:671-684

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DOI: 10.1080/14697688.2026.2629952

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