Single-Event Multinomial Full Kelly via Implicit State Positions

TL;DR

This paper presents a simplified derivation of the full Kelly betting strategy for a single event with multiple outcomes, using an implicit state-position perspective that streamlines the solution process.

Contribution

It introduces an implicit position framework that simplifies the derivation and understanding of the Kelly optimal stakes for finitely many outcomes.

Findings

Provides a compact proof of the Kelly formula using implicit state positions.

Introduces a greedy algorithm for support selection based on outcome ratios.

Clarifies the role of baseline cash positions in Kelly betting strategies.

Abstract

For a single event with finitely many mutually exclusive outcomes, the full Kelly problem is to maximize expected log wealth over nonnegative stakes together with an optional cash position. The optimal formula is classical, but the support-selection step is often presented via Lagrange multipliers. This note gives a shorter state-price derivation. A cash fraction $c$ acts as an implicit position in every outcome: in terminal-wealth terms, it is equivalent to a baseline stake $cq_i$ on outcome $i$, where $q_i$ is the state price. On any active support, explicit bets therefore only top up favorable outcomes from this baseline $cq_i$ to the optimal total stake $p_i$. This yields the formula $x_i = (p_i - c q_i)_+$, the threshold rule $p_i/q_i > c$, and, after sorting outcomes by $p_i/q_i$, a one-pass greedy algorithm for support selection. The result is standard in substance, but the implicit-position viewpoint gives a compact proof and a convenient way to remember the solution.