Utility-Invariant Support Selection and Eventwise Decoupling for Simultaneous Independent Multi-Outcome Bets

Paper: arXiv 2603.24064

Authors: Christopher D. Long

Abstract

For simultaneous independent events with finitely many outcomes, consider the expected-utility problem with nonnegative wagers and an endogenous cash position. We prove a short support theorem for a broad class of strictly increasing strictly concave utilities. On any fixed support family and at any optimal portfolio with positive cash, summing the active first-order conditions and comparing that sum with cash stationarity yields the exact identity [ \fracλ{K_{\ell}^{(U)}}=\frac{1-P_{\ell,A}}{1-Q_{\ell,A}}, ] where $P_{\ell,A}$ and $Q_{\ell,A}$ are the active probability and price masses of event $\ell$, $λ$ is the budget multiplier, and $K_{\ell}^{(U)}$ is the continuation factor seen by inactive outcomes of that event. Consequently, after sorting each event by the edge ratio $p_{\ell i}/π_{\ell i}$, the exact active support is the eventwise union of the single-event supports, and this support is independent of the utility function. The single-event utility-invariant support theorem is already explicit in the free-exposure pari-mutuel setting in Smoczynski and Miles; the point of the present note is that the simultaneous independent-events analogue follows from the same state-price geometry once the right continuation factor is identified.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical derivation proving utility-invariance for optimal support selection across simultaneous independent events. While mathematically strong and insightful, it is purely theoretical with no empirical backtesting or real-world dataset validation included in the excerpt.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze Optimal Support & Decoupling for Simultaneous Independent Bets] --> B[Inputs: Edge Ratios p_i/π_i, Independent Events, Utility U, Cash Position]
    B --> C[Methodology: State-Price Geometry & Portfolio Expected Utility Maximization]
    C --> D[Derive First-Order Conditions & Continuation Factor K_l^U]
    D --> E[Establish Active Probability-to-Price Mass Identity]
    E --> F[Key Outcome 1: Utility-Invariant Support Selection via Eventwise Union]
    E --> G[Key Outcome 2: Exact Decoupling into Independent Single-Event Problems]