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Single-Event Multinomial Full Kelly via Implicit State Positions

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.

Single-Event Multinomial Full Kelly via Implicit State Positions

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 acts as an implicit position in every outcome: in terminal-wealth terms, it is equivalent to a baseline stake on outcome , where is the state price. On any active support, explicit bets therefore only top up favorable outcomes from this baseline to the optimal total stake . This yields the formula , the threshold rule , and, after sorting outcomes by , 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.
Paper Structure (6 sections, 3 theorems, 39 equations)

This paper contains 6 sections, 3 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Model
  3. A weighted full-investment lemma
  4. Optimization on a fixed support
  5. Greedy support selection
  6. Concluding comment

Key Result

Lemma 1

Let $a_i>0$ with $A:=\sum_{i=1}^m a_i$, and let $S>0$. Then is uniquely maximized over $z_i>0$ subject to $\sum_{i=1}^m z_i=S$ at

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Proposition 2: Fixed-support optimizer
  • proof
  • Theorem 3: Greedy characterization of the canonical Kelly strategy
  • proof
  • Remark 4: Binary reduction
  • Remark 5: Interpretation