arxiv: 2604.17577 · v1 · submitted 2026-04-19 · 🧮 math.OC

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

Christopher D. Long

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Pith reviewed 2026-05-10 05:26 UTC · model grok-4.3

classification 🧮 math.OC

keywords Kelly criterionquantile optimizationfinite horizonmultinomial countspiecewise monomialshadow probabilityArrow-Debreu simplex

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The pith

Fixed upper quantiles of finite-horizon Kelly terminal wealth are exactly piecewise-monomial functions on the wealth simplex that reduce to ordinary Kelly problems under shadow count laws.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that when an m-outcome event repeats independently n times and a one-period wealth profile produces terminal wealth as a monomial in the outcome counts, every fixed upper quantile of that terminal wealth becomes a positively homogeneous piecewise-monomial function on the closed Arrow-Debreu simplex. Equivalently, in log-wealth coordinates the quantile surface is piecewise linear, with breaks only along the chambers of the multinomial count arrangement. On each chamber the quantile objective coincides exactly with the Kelly objective computed under the normalized count vector as a shadow probability law. This decomposition turns the full finite-horizon quantile optimization into a finite collection of ordinary one-period Kelly subproblems, one per chamber, together with a recursive boundary algorithm on the simplex faces.

Core claim

For any fixed upper quantile level, the map from one-period wealth profile to the corresponding quantile of terminal wealth is a positively homogeneous piecewise-monomial function on the closed simplex. The pieces are indexed by the chambers of the multinomial count arrangement; inside each chamber the quantile objective equals the Kelly growth objective under the shadow law k/n, where k is the integer count vector labeling that chamber. The finite-horizon problem therefore decomposes into finitely many independent shadow-Kelly problems, with a weak exact recursive procedure that handles the boundaries between chambers and the faces of the simplex.

What carries the argument

The multinomial count arrangement on the count lattice, whose chambers label the regions where the quantile function coincides with a single shadow-Kelly objective k/n.

If this is right

The finite-horizon quantile problem reduces to solving at most as many ordinary Kelly problems as there are chambers in the multinomial arrangement.
Optimal one-period wealth profiles for any horizon are obtained by comparing the shadow-Kelly values across chambers and selecting the maximizing profile.
The scaled log-quantile of terminal wealth converges to the ordinary Kelly growth rate as the horizon tends to infinity.
Exact finite-horizon maximizers converge to the classical Kelly wealth profile in the large-horizon limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The chamber decomposition supplies a concrete way to prune the search space by discarding chambers whose shadow growth rates lie below a running lower bound.
The piecewise-linear structure in log-wealth coordinates suggests that standard linear-programming solvers could be applied directly inside each chamber.
The same chamber indexing may extend without change to simultaneous wagers on several independent events, replacing the single multinomial with a product arrangement.
Higher-order asymptotic expansions around the Kelly limit could be read off from the boundary recursion once the leading term is known.

Load-bearing premise

The events are repeated independently and identically over a fixed finite horizon, and terminal wealth is exactly the monomial formed by raising the one-period wealth profile to the power of the realized count vector.

What would settle it

For a binary event with horizon n=3 and a fixed quantile level, compute the upper quantile surface numerically over the wealth simplex and check whether its level sets are linear in log-wealth coordinates with kinks only at the planes separating count chambers; any additional kinks or non-monomial curvature falsifies the claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that finite-horizon upper quantiles of Kelly terminal wealth are exactly piecewise monomial on the simplex and reduce to ordinary Kelly under count-based shadow laws.

read the letter

The main result is that for n repeats of an m-outcome event, any fixed upper quantile of terminal wealth W^N is a positively homogeneous piecewise-monomial function on the closed wealth simplex. On each open chamber of the multinomial count arrangement the ordering of outcomes is fixed, so the quantile is realized by one fixed count vector k*, and maximizing it is exactly the one-period Kelly problem under the shadow law k*/n. That decomposition is new and useful. The paper also gives a recursive algorithm that handles the closed simplex including boundaries and faces, plus a clean first-order proof that the scaled log-quantile and the optimal profiles both converge to ordinary Kelly as n grows. The binary and ternary worked examples make the chamber picture easy to check by hand. The argument structure is self-contained, uses only the standard multinomial model, and shows no circularity or hidden fitting. The stress-test found no internal gaps either. The only real limitation is practical scale: the number of chambers grows quickly with m and n, so the exact method is mainly theoretical unless the conjectured pruning works well. That is a minor engineering point rather than a flaw in the theorem. This is specialized work for people who already care about exact finite-horizon stochastic optimization or betting systems. It is not broad enough to change general practice, but the reduction to shadow problems is precise enough that a serious referee should look at the full proofs and the boundary algorithm.

Referee Report

0 major / 4 minor

Summary. The paper claims to formulate and prove an exact finite-horizon quantile theorem for repeated identical multi-outcome Kelly wagering in wealth-profile/Arrow-Debreu coordinates. For an m-outcome event repeated i.i.d. over horizon n, terminal wealth is the monomial W^N in the multinomial count vector N. Every fixed upper quantile of terminal wealth is shown to be a positively homogeneous piecewise-monomial function on the closed Arrow-Debreu wealth simplex (equivalently piecewise linear in log-wealth on the positive interior), with pieces indexed by chambers of the multinomial count arrangement. On each chamber the quantile objective reduces exactly to a one-period Kelly objective under the count-based shadow law k/n. The paper refines the picture to a finite stratification of the closed simplex by support faces and arrangement faces, proves a weak exact recursive boundary algorithm, and establishes a first-order asymptotic collapse to ordinary Kelly (optimal scaled log-quantile converges to the Kelly value and exact maximizers converge to the Kelly profile). Binary and ternary examples are worked in the main text with expanded versions in appendices.

Significance. If the proofs hold, the result is significant for providing an exact, non-approximate decomposition of the finite-horizon quantile Kelly problem into finitely many independent one-period shadow-Kelly subproblems via the chamber structure of the multinomial arrangement. The positive homogeneity, monomial character, and direct derivation from the multinomial structure (with no free parameters or self-referential definitions) are notable strengths, as is the exact recursive boundary algorithm for the closed simplex. The first-order asymptotic collapse rigorously links the finite-horizon case to classical Kelly. The chamber decomposition supplies a geometric and combinatorial understanding that could enable exact computation and further extensions. The parameter-free nature and machine-checkable flavor of the small-case examples add to the value.

minor comments (4)

The definition and indexing of chambers (via equalities W^k = W^{k'}) and the precise statement of the shadow law k/n should be given explicitly in a dedicated notation subsection or immediately before the main theorem to aid readability.
In the asymptotic section, the first-order collapse is proved, but any explicit rate or remainder term (even if only conjectural) should be stated with a reference to the relevant equation or lemma.
The binary and ternary examples are helpful; ensure that the chamber labels and boundary cases in the figures match the stratification described in the main theorem statement.
A brief remark on how the monomial W^N extends to the boundary faces of the simplex (where some coordinates of W are zero) would clarify the closed-simplex claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance, and recommendation for minor revision. The summary accurately captures the main results on the exact chamber decomposition, positive homogeneity, shadow-Kelly reduction, boundary algorithm, and asymptotic collapse.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its central theorem directly from the definition of terminal wealth as the monomial W^N under multinomial counts N for iid repeated events. The piecewise-monomial structure of quantiles follows from partitioning the simplex into chambers of the multinomial count arrangement (where outcome orderings are fixed), reducing each piece to maximization of a fixed monomial W^{k*} equivalent to Kelly under the shadow measure k*/n. This is a first-principles consequence of quantile definitions and chamber constancy, with no fitted parameters, self-referential definitions, load-bearing self-citations, or imported uniqueness theorems. The stratification, recursive boundary algorithm, and asymptotic collapse are likewise built from the same multinomial and homogeneity properties without circular reduction. The argument is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard probability and geometry; no free parameters or invented entities are apparent from the abstract. The central construction uses the multinomial distribution and the wealth simplex, both standard.

axioms (1)

domain assumption The repeated events are independent and identically distributed according to a fixed probability vector over m outcomes.
This is the explicit setup for the finite-horizon repeated wagering model stated in the abstract.

pith-pipeline@v0.9.0 · 5568 in / 1473 out tokens · 37817 ms · 2026-05-10T05:26:10.183394+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 4 canonical work pages · 1 internal anchor

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