Paper: arXiv 2603.13581

Authors: Christopher D. Long

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.

Complexity vs Empirical Score

  • Math Complexity: 4.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Philosophers — conceptual discussion, limited math and data

Why this score: This short theoretical note presents an intuitive re-interpretation and streamlined derivation of the classic multinomial Kelly criterion using state prices. While elegant and clear, it lacks empirical backtesting, datasets, or substantial novel mathematical techniques beyond existing literature.

Research Flowchart

  flowchart TD
    A["Research Goal: Simplify Support Selection in<br>Single-Event Full Kelly Optimization"] --> B["Inputs: Outcome Probabilities (p_i),<br>State Prices (q_i), & Nonnegativity Constraints"]
    B --> C["Core Insight: Reframe Cash Fraction (c)<br>as Implicit State Positions (cq_i)"]
    C --> D["Derive Formulas: Optimal Bets x_i = (p_i - cq_i)_+<br>& Active Threshold p_i/q_i > c"]
    D --> E["Algorithm: Sort Outcomes by Ratio p_i/q_i<br>& Run One-Pass Greedy Search for c"]
    E --> F["Outcome: Compact Proof & Fast Support Selection<br>via Implicit-Position Viewpoint"]