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

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.

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

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 where and are the active probability and price masses of event , is the budget multiplier, and is the continuation factor seen by inactive outcomes of that event. Consequently, after sorting each event by the edge ratio , 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.
Paper Structure (4 sections, 4 theorems, 46 equations)

This paper contains 4 sections, 4 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Model and admissible utilities
  3. Fixed-support stationarity and reduced costs
  4. The threshold identity

Key Result

Proposition 2

Fix a support family $A=(A_1,\dots,A_m)$, and let $(c,g)$ be an optimal portfolio for the restriction of eq:mainproblem to portfolios supported on $A$. Assume $c>0$. Then there exists a multiplier $\lambda>0$ such that and, for every event $\ell$ and every active outcome $i\in A_\ell$, If $j\notin A_\ell$, define Then the one-sided directional derivative in the feasible direction $g_{\ell j}\ma

Theorems & Definitions (14)

  • Definition 1
  • Proposition 2: Fixed-support first-order and reduced-cost conditions
  • proof
  • Theorem 3: Utility-invariant threshold identity
  • proof
  • Corollary 4: Eventwise prefix support and exact decoupling
  • proof
  • Corollary 5: Single-event utility-invariant support
  • proof
  • Remark 6: One-dimensional reduction in the single-event problem
  • ...and 4 more