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 [ \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.

• The paper demonstrates that optimal support selection in multi-outcome betting is independent of the utility curvature, focusing solely on state-price geometry.
• It introduces an edge ratio mechanism and a precise threshold identity to decouple event supports, enabling modular and efficient portfolio optimization.
• The findings have practical implications for computational efficiency and algorithmic wagering, potentially benefiting statistical arbitrage and automated betting platforms.

Utility-Invariant Support Selection and Eventwise Decoupling in Multi-Outcome Betting Optimization

Problem Formulation and Structural Results

The paper "Utility-Invariant Support Selection and Eventwise Decoupling for Simultaneous Independent Multi-Outcome Bets" (2603.24064) investigates optimal portfolio allocation for simultaneous wagers across independent events with finitely many outcomes. The central object is the expected utility maximization over nonnegative wager allocations and an endogenous cash position, subject to a bankroll constraint. The utility function is assumed to be strictly increasing and strictly concave, encompassing classical log utility and CRRA forms as special cases.

A key structural property is revealed via stationarity and support analysis: in optimal portfolios with positive cash across simultaneous independent bets, the selection of which outcomes receive positive wager is entirely determined by state-price geometry and subjective probability measures, and is provably independent of the utility function. This utility-invariance resolves a long-standing structural ambiguity in multi-event Kelly-type optimization, previously addressed only for single-event problems.

The support selection mechanism is articulated through the "edge ratio" $r_{\ell i} = p_{\ell i} / \pi_{\ell i}$ for each outcome $i$ in event $\ell$, and an explicit threshold $\theta = (1-P)/(1-Q)$, where $P$ and $Q$ denote aggregate probability and price masses of the active support. The paper formally demonstrates that—once events are sorted by decreasing edge ratio—the optimal set of outcomes (active support) for each event is a prefix determined by this threshold, precisely as in the single-event case.

Technical Contributions and Theoretical Advances

A pivotal theorem asserts a utility-invariant threshold identity relating the Lagrange multiplier $\lambda$ and the continuation factor $K_\ell^{(U)}$ for each event. This identity is:

$$\frac{\lambda}{K_\ell^{(U)}} = \frac{1-P_{\ell,A}}{1-Q_{\ell,A}},$$

where $K_\ell^{(U)}$ is the expected marginal utility from the portfolio excluding event $\ell$, and $A$ is the active support. This result is derived by summing eventwise first-order KKT conditions and decomposing the cash stationarity condition within a fixed support. Crucially, the continuation factor is constant across all inactive outcomes within an event due to independence, enabling a self-contained proof.

Corollaries confirm the exact decoupling: the optimal simultaneous portfolio's support is the eventwise union of the single-event supports. This decoupling holds across all admissible utilities provided the cash position is positive, with support discovery governed by the identical prefix rule leveraged in single-event utility maximization.

Boundary remarks address cases with zero cash position, strict vigorish, and statistical dependence between events. The interior threshold identity collapses under constraints or dependence, establishing independence as the precise structural boundary for utility-invariant support.

For special utility cases (logarithmic, CRRA), explicit scalar reductions are derived for post-support numerical computation, confirming that risk aversion alters optimal weights but never the support.

Numerical and Formal Implications

The paper highlights several strong claims:

• Support selection is entirely independent of utility curvature within the admissible class: utility affects only wager size, not which outcomes are selected.
• Exact eventwise decoupling in the simultaneous expected utility maximization for independent events: optimal supports can be determined in isolation for each event, generalizing previous single-event results.
• Canonical uniqueness for strict vigorish markets: when all events have overround and no threshold ties, there is a unique optimal support, with natural tie-breaking toward cash positions.

These results clarify and sharpen foundational aspects of betting portfolio optimization, establish a rigorous demarcation for structural invariance, and unify disparate lines of work in both algorithmic and economic literature on multi-event allocation.

Practical and Theoretical Implications

Practically, the utility-invariant decoupling deeply streamlines computational approaches: support discovery is reduced to eventwise greedy prefix selection, independent of utility preference, allowing for modularized optimization pipelines. This separation enables more efficient algorithms for large-scale betting portfolios and may facilitate novel applications in statistical arbitrage and automated wagering platforms.

Theoretically, the paper situates independence as the precise boundary for utility-invariant support selection, suggesting that any model for dependent events cannot exploit decoupling and must reintroduce utility-specific support dependencies. The identification and rigorous formalization of the continuation factor and threshold identity provide new tools for analyzing state-price geometries in portfolio theory.

Future Directions

This analytic division between support selection and wager sizing recommends future work on computational efficiency in simulation and allocation for high-dimensional event spaces, exploration of threshold behavior under relaxed independence, and investigation of structural consequences for portfolios with exogenous exposure constraints. Extending these results to correlated events, fractional wagering, and other risk-related generalizations may further deepen theoretical understanding of utility-maximization in betting contexts.

Conclusion

The presented framework delivers a concise and comprehensive characterization of support selection in simultaneous independent multi-outcome bets under concave utility, formalizing utility-invariant eventwise decoupling and exact threshold criteria. The implications span algorithmic efficiency, portfolio theory, and foundational economic modeling, and delineate structural boundaries for expected utility maximization in uncertainty-driven allocation problems.