Single-Event Multinomial Full Kelly via Implicit State Positions

Battle Count: 69

The Kelly criterion is foundational to position sizing and capital growth in quantitative trading. This paper provides a cleaner derivation and a practical greedy algorithm for support selection in discrete-outcome scenarios. However, it addresses a single-event multinomial setting rather than continuous portfolio allocation, making it more directly relevant to betting/discrete-event trading than to typical multi-asset quantitative strategies. The implicit state-position viewpoint offers pedagogical value for understanding cash as a state-contingent claim in portfolio construction.

The algorithm is extremely simple: sort outcomes by edge ratio r_i = p_i/q_i in descending order, then greedily add outcomes while r_{k+1} > c_k, updating c_k = (1 - P_k)/(1 - Q_k) at each step. The final formula x_i = (p_i - c* q_i)+ is a single vectorized operation. No iterative solvers, no Lagrange multipliers, no numerical optimization libraries needed. The entire algorithm is O(n log n) due to sorting.

The paper is a self-contained mathematical derivation with complete proofs. All formulas, lemmas, propositions, and the greedy algorithm are fully specified. No external data or code is required. The result is standard in substance, so verification against classical literature (Kelly 1956, Cover & Thomas, Rosner 1975, Smoczynski & Tomkins 2010) is straightforward.