Equal probabilities maximize the expected deficit in the siblings of the coupon collector

Abstract: In the siblings (or brotherhood) variant of the coupon collector's problem, a main collector draws coupons until her own album is complete and passes every duplicate down a chain of siblings; the $j$th collector is then left with $U_j^N$ empty places, $j\ge 2$. It has been conjectured [stated as an open problem in the work that introduced the model] that, for every fixed number of coupon types $N$ and every $j\ge 2$, the expected deficit $\E[U_j^N]$ is maximized by the equiprobable coupon distribution. We prove this in a sharp, finite-$N$ form: $\E[U_j^N]$ is strictly larger at the uniform vector than at any other probability vector, and indeed strictly increases along every ray running from an arbitrary distribution toward the uniform one. The proof is exact and elementary in its ingredients. An inclusion--exclusion step turns the governing Poissonized integral into a one-dimensional integral with a separable integrand; a single integration by parts then rewrites the radial derivative of $\E[U_j^N]$ as a positively weighted covariance of an increasing function, whose sign is settled by Chebyshev's correlation inequality. We show that $\E[U_j^N]$ is \emph{not} Schur-concave, so that no majorization or pairwise-smoothing argument can yield the result, and we explain why the recent variance-extremality method of Long~[Long, arXiv:2604.25108, 2026] does not transfer. As by-products we obtain a finite closed form for $\E[U_j^N]$ over subsets of the coupon set and the exact Hessian of $\E[U_j^N]$ at the uniform vector. The argument extends without change to all real $j>1$.

• The paper proves that a uniform coupon distribution uniquely maximizes the expected deficit in each sibling’s album.
• It uses dual analytical methods—a finite subset sum representation and a Poissonized integral—to rigorously establish monotonicity and Hessian negativity.
• The findings have practical implications for optimizing sequential allocation systems and advance understanding of extremal behavior in non-Schur-concave settings.

Maximization of Expected Deficits in the Siblings Variant of the Coupon Collector’s Problem

Problem Setting and Main Results

The paper addresses the siblings (brotherhood) extension of the classical coupon collector’s problem, wherein a sequence of collectors, each referred to as a sibling, successively inherits duplicate coupons collected by the predecessor. The primary quantity of interest is the expected deficit $E[U_j^N]$—the expected number of empty slots remaining in the $j$th sibling’s album when the first sibling completes her collection—under a general coupon probability vector $p = (p_1, ..., p_N)$.

The principal result rigorously establishes the conjecture that, for all $N \geq 2$ and $j \geq 2$, the expected deficit $E[U_j^N]$ is maximized uniquely at the uniform distribution $p_k = 1/N$ for all $k$. Moreover, the paper demonstrates strict monotonicity of $E[U_j^N]$ along any line segment (ray) from an arbitrary non-uniform distribution toward the uniform vector in the probability simplex.

Analytical Framework and Structural Insights

The authors utilize two complementary exact representations for $E[U_j^N]$. The first is a closed form involving finite sums over all non-empty subsets of coupon types; the second—a one-dimensional Poissonized separable integral—enables effective differentiation with respect to changes in the probability vector.

A key technical achievement is the derivation, via inclusion-exclusion and Poissonization, of

$j$0

where $j$1 and $j$2 are explicitly defined, enabling clear separation of coordinate dependencies.

Through this integral form, the main argument leverages a radial differentiation scheme: the derivative of $j$3 along a segment from any $j$4 to the uniform vector is reduced—following an integration by parts—to a covariance between monotonic functions under positive weights. This pivotal realization allows application of the Chebyshev correlation inequality, guaranteeing non-negativity and, on nontrivial rays, strict negativity of the derivative, thus ensuring strict maximality at the uniform vector.

Numerical Values and Structural Properties

The Hessian of $j$5 at the uniform vector is determined explicitly and is shown to be negative definite (on the subspace of perturbations preserving the sum constraint), confirming the strictness and nondegeneracy of the maximum. For instance, for $j$6 and $j$7 the tangent Hessian has eigenvalue $j$8, and for $j$9, $p = (p_1, ..., p_N)$0.

The function $p = (p_1, ..., p_N)$1 is not Schur-concave; that is, majorization or pairwise smoothing of coordinates does not necessarily increase expected deficits. This is analytically justified by sign changes in the Schur–Ostrowski quantity and confirmed by explicit computation. Consequently, the maximization principle cannot simply be deduced from standard majorization theory, distinguishing this result from other works in the coupon collector literature.

Comparison With Related Extremal Laws

In contrast to recent work on variance minimization for completion times in coupon-collecting models—where majorization and log-concavity arguments are applicable—the present analysis shows such methods do not transfer: (1) the deficit is evaluated at a random stopping time, not as a function of terminal waiting times; (2) the extremality pertains to a maximization, rather than minimization, of expectation. The present proof, reliant on elementary inequalities and explicit integration, sidesteps log-likelihood or variance decompositions.

Additionally, the critical structure of the result holds for all real $p = (p_1, ..., p_N)$2, not just integer values, by leveraging the monotonicity properties embedded in the integral expressions.

Implications and Future Directions

These findings have nontrivial implications for optimal design of sequential allocation and collection systems where maximizing expected deficits downstream is of interest, as well as for the theory of extremal expectations over probability simplices where neither convexity nor Schur-concavity is available. The radial monotonicity framework and covariance-based sign determination may prove valuable for analyzing other expectation-driven functionals lacking Schur properties.

On a theoretical level, the result highlights scenarios where equalizing stochastic input delays progress for agents in multi-stage inheritance/transfer models, providing sharp finite-$p = (p_1, ..., p_N)$3 guarantees. From a probabilistic combinatorics perspective, the methods presented reinforce the utility of elementary covariance inequalities even in non-classical extremal problems.

Future research may consider generalizations to more complex transfer rules, to random networked sibling structures, or to moments of higher order, perhaps examining the precise behavior of $p = (p_1, ..., p_N)$4 under various inhomogeneity regimes, and seeking other functionals on probability simplices exhibiting non-Schur extremality.

Conclusion

This work resolves a standing conjecture in the siblings variant of the coupon collector's problem by proving that the equiprobable coupon distribution uniquely maximizes the expected deficit in every sibling's album for all $p = (p_1, ..., p_N)$5 and all $p = (p_1, ..., p_N)$6. The proof, analytic and non-asymptotic, utilizes exact Poissonized representations and a reduction by integration by parts to a positive covariance term, explicit calculation of the Hessian, and an analysis of the lack of Schur-concavity. The arguments supplied are both elementary and robust, suggesting a paradigm for further extremal analysis in combinatorial stochastic processes.