Exact Finite-Horizon Quantile Kelly for Repeated Multi-Outcome Events

Abstract: We formulate and prove an exact finite-horizon quantile theorem for repeated identical multi-outcome Kelly wagering in wealth-profile / Arrow--Debreu coordinates. For a fixed $m$-outcome event repeated independently over a horizon $n$, the terminal wealth induced by a one-period wealth profile $W$ is a monomial $W^N$ in the multinomial count vector $N$. We show that every fixed upper quantile of terminal wealth is a positively homogeneous piecewise-monomial function on the closed Arrow--Debreu wealth simplex, equivalently piecewise linear in log-wealth coordinates on the positive interior. The pieces are indexed by the chambers of the multinomial count arrangement, and on each chamber the quantile objective is exactly a one-period Kelly objective for a count-based \emph{shadow law} $k/n$. Consequently the finite-horizon quantile problem decomposes into finitely many shadow-Kelly subproblems. We then refine the interior chamber picture to a finite stratification of the full closed simplex by support faces and arrangement faces, and we prove a weak exact recursive boundary algorithm. We also prove a natural first-order asymptotic collapse to ordinary Kelly, showing that the optimal scaled log-quantile converges to the ordinary Kelly value and that exact finite-horizon maximizers converge to the Kelly wealth profile. For illustration, we include worked binary and ternary examples in the main text and expanded versions in the appendices. We conclude with further remarks and conjectures concerning stronger pruning, higher-order finite-horizon corrections, and extensions to simultaneous wagers.

• The paper introduces an exact finite-horizon quantile optimization theorem using a chamber-based decomposition to isolate shadow-Kelly subproblems.
• It rigorously decomposes the compound wealth maximization into piecewise monomial optimization within defined chambers on the Arrow–Debreu simplex.
• The approach demonstrates first-order asymptotic convergence to the classic Kelly criterion while highlighting finite-horizon deviations and boundary effects.

Exact Finite-Horizon Quantile Kelly for Repeated Multi-Outcome Events

Problem Formulation and Main Structural Results

This paper establishes a precise finite-horizon quantile optimization theorem for repeated, i.i.d. multi-outcome events under the Kelly criterion, shifting the perspective to the state-contingent wealth-profile (Arrow--Debreu) framework. The central object is the optimal upper quantile (not only expectation) of the compound wealth produced by applying the same one-period wealth profile $W$ over $n$ independent repetitions of an $m$-outcome event. The key result is an exact algebraic decomposition of the finite-horizon quantile optimization into finitely many deterministic subproblems—each corresponding to a "chamber" in log-wealth space—such that within each chamber, the quantile is equivalent to a one-period Kelly optimization for a "shadow" empirical measure $k/n$ determined by the chamber.

Terminal wealth in this context is given by $X_n(W) = \prod_{i=1}^m W_i^{N_i}$, where $N = (N_1,\ldots,N_m)$ is a multinomial count vector. The upper $\alpha$-quantile of $X_n(W)$ (for arbitrary $\alpha$) thus reduces to maximizing piecewise monomial functions over the Arrow--Debreu simplex: each piece (chamber) is aligned with a strict order of the monomials $W^k$, with subproblems corresponding to maximizing $n$0 for shadow count vectors $n$1.

The paper rigorously proves that the finite-horizon quantile problem

$n$2

can be decomposed as

$n$3

where chambers $n$4 and their closures $n$5 partition the feasible space, and for each chamber the optimand is monomial with respect to a shadow count. The Arrow--Debreu simplex supports a finite, tractable combinatorial structure for this decomposition.

Chamber Geometry and Shadow-Kelly Reduction

A central innovation is the geometric organization of the quantile problem via the multinomial count-induced hyperplane arrangement $n$6 in log-wealth space. These hyperplanes partition the Arrow--Debreu simplex into combinatorial "chambers" (open regions with fixed monomial rankings). Each chamber is indexed by a unique ordering of monomials $n$7, stemming from the linear functionals $n$8 with $n$9. The upper quantile within a chamber is given by $m$0, where $m$1 is determined by summing multinomial probabilities in the chamber's fixed order until exceeding $m$2.

Inside each chamber, the problem reduces to maximizing $m$3 subject to the Arrow--Debreu constraint. The paper proves that, for any $m$4, this is solved by the "shadow-Kelly" profile:

$m$5

which corresponds to the one-period Kelly solution for the empirical law $m$6. Uniqueness of the optimizer holds unless it lies on the boundary, in which case supremal values lie on lower-dimensional faces. Thus, finite-horizon quantile Kelly is "piecewise shadow-Kelly".

Boundary Structure and Recursive Optimization

To precisely account for boundary behavior, the paper develops a finite stratification of the closed simplex by support faces and chamber arrangements within each support face, yielding a recursive boundary algorithm. For any face specified by support $m$7, with $m$8 for $m$9, the problem further stratifies via an arrangement on ratio coordinates. On each stratum, the upper quantile is again either identically zero (if the stratum cannot cover the quantile's required mass) or a fixed monomial.

The main recursive result shows that the maximization over the entire simplex reduces to strictly concave (or lower-dimensional) subproblems on each stratum/chamber, passing to "child" strata when the optimum does not lie in the interior. This recursive procedure is finite, exact, and comprehensive but may not be computationally efficient for very large $k/n$0 or $k/n$1 due to exponential chamber proliferation.

Asymptotics and Collapse to Ordinary Kelly

A critical theoretical contribution is the proof that as $k/n$2, the finite-horizon quantile Kelly problem collapses, at first order, to the classic one-period Kelly criterion. That is, the scaled log-quantile function converges uniformly on compact subsets of the positive interior of the simplex to the expected log-utility, and all exact finite-horizon maximizers converge to the asymptotic Kelly portfolio:

$k/n$3

where $k/n$4 is the true underlying distribution. The result holds for all fixed quantile levels $k/n$5.

This first-order asymptotic convergence does not address second-order effects or rates; the possibility and characterization of higher-order corrections near the Kelly point are raised as open questions.

Illustrative Binary and Ternary Cases

The binary ($k/n$6) specialization yields transparent expressions: the chambers correspond to $k/n$7 and $k/n$8 for wealth shares $k/n$9, and the quantile optimization over three or more periods can be computed explicitly. For the median ($X_n(W) = \prod_{i=1}^m W_i^{N_i}$0), the optimal solution coincides with prior known results in the literature on median-optimal fortune, now unified under the chamber framework.

In the ternary case, even for small horizons, the recursion naturally gives rise to nontrivial chamber and support-face behavior, with optimal monomials and shadow laws shifting as the support face is traversed. The explicit worked examples demonstrate these geometric and combinatorial aspects in detail and highlight departures from naive or asymptotic Kelly solutions at finite horizons.

Practical and Theoretical Implications

This decomposition concretely delineates the finite-horizon structure of quantile-optimal Kelly wagering under repeated, identical, multi-outcome events. In practical terms, this allows for exact numerical and analytic analysis of downside risk-sensitive objectives (such as the median or upper quantiles of fortune), rather than only expected log growth, in repeated betting or investment contexts with known or assessed outcome laws.

Theoretically, the result precisely demarcates how and why the Kelly solution emerges as the limiting quantile maximizer in the large-$X_n(W) = \prod_{i=1}^m W_i^{N_i}$1 regime, but also provides the machinery for characterizing and computing finite-horizon deviations. It is immediately relevant for risk-constrained Kelly frameworks, structured wager menus, and situations requiring exact finite-horizon guarantees. The framework is extensible to simultaneous wagers and certain forms of path-dependent optimization, under partitioned state spaces.

Open Directions and Future Developments

The paper notes that while the recursive stratification algorithm is exact, it is not necessarily computationally efficient for large-scale problems. Stronger pruning and dual formulations (potentially via Bregman projections or entropic duality) are identified as key theoretical next steps. Higher-order asymptotic corrections near the Kelly optimizer (beyond first-order convergence) remain open and potentially tractable via local combinatorial-geometric expansions.

The methods clearly make possible further generalizations: to risk-constrained one-period families, simultaneous event spaces, and perhaps to non-identical or dependent repetition settings by lifting to path space and adjusting the chamber construction.

Conclusion

This paper rigorously characterizes the exact finite-horizon quantile-optimal policy for repeated multi-outcome events, demonstrating that the Arrow--Debreu wealth-profile formulation enables a piecewise-mononomial, chamber-based decomposition into shadow-Kelly subproblems. The explicit recursive structure provides both theoretical clarity and a concrete pathway for quantile-based portfolio or betting optimization at finite horizons, robustly explaining the emergence and limitations of the ordinary Kelly solution and highlighting non-asymptotic behavior. The framework lays the foundation for future dual, algorithmic, and asymptotic refinements.