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

TL;DR

This paper establishes a utility-invariant support theorem for optimal betting strategies on independent events, revealing that the support structure is independent of the specific utility function and derived from state-price geometry.

Contribution

It introduces a novel support theorem for simultaneous independent events, showing support independence from utility functions using state-price geometry.

Findings

Support is utility-invariant and independent of the utility function.

The exact active support can be determined by sorting event ratios.

The support theorem extends previous results to simultaneous independent events.

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 \[ \frac{\lambda}{K_{\ell}^{(U)}}=\frac{1-P_{\ell,A}}{1-Q_{\ell,A}}, \] where $P_{\ell,A}$ and $Q_{\ell,A}$ are the active probability and price masses of event $\ell$, $\lambda$ 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 $p_{\ell i}/\pi_{\ell i}$, 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.