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About a Ball Removal Process on Bins

Jose Correa, Marcos Kiwi, Vasilis Livanos, Eilon Solan, Ron Solan

TL;DR

The problem asks to maximize the total number of removed balls—equivalently minimize the expected number of remaining balls—when repeatedly removing from non-empty bins chosen at random. The authors prove that, for any fixed total $n$ and number of bins $k$, the initial allocation minimizing the expected remaining balls $f(\vec{n})$ is balanced in the sense that $|n_i-n_j|\le 1$ for all $i,j$, with a coupling argument establishing the $k=2$ case and a discussion of an independent Erlang-embedding approach that yields the general $k$ result. They show that $\mathbb{E}[X_{\vec{n}}]$ strictly decreases when shifting one unit from a larger bin to a smaller one whenever the imbalance exceeds 1, and they provide a detailed partition of the sample space to support the coupling. The paper also analyzes non-uniform removal probabilities, giving a counterexample where the proportional initial allocation is not optimal and presenting exact recurrences for the non-uniform case, along with a conjecture that the deviation from balance grows like $O(\sqrt{n})$. Together, these results resolve Ma's conjecture in the uniform case and illuminate when proportional stocking is suboptimal, with implications for related assortment optimization models.

Abstract

Consider the following process whereby $n$ balls are distributed into $k$ bins. Repeatedly, a ball is removed from a non-empty bin chosen uniformly at random. The process ends when a single non-empty bin remains. Will Ma (see~\cite[Sec.~1.1]{GS24}) asked whether the initial assignment that minimizes the expected number of remaining balls is one that is as balanced as possible. Using a coupling argument we answer this conjecture positively, and we discuss the case of non-uniform choice among the non-empty bins.

About a Ball Removal Process on Bins

TL;DR

The problem asks to maximize the total number of removed balls—equivalently minimize the expected number of remaining balls—when repeatedly removing from non-empty bins chosen at random. The authors prove that, for any fixed total and number of bins , the initial allocation minimizing the expected remaining balls is balanced in the sense that for all , with a coupling argument establishing the case and a discussion of an independent Erlang-embedding approach that yields the general result. They show that strictly decreases when shifting one unit from a larger bin to a smaller one whenever the imbalance exceeds 1, and they provide a detailed partition of the sample space to support the coupling. The paper also analyzes non-uniform removal probabilities, giving a counterexample where the proportional initial allocation is not optimal and presenting exact recurrences for the non-uniform case, along with a conjecture that the deviation from balance grows like . Together, these results resolve Ma's conjecture in the uniform case and illuminate when proportional stocking is suboptimal, with implications for related assortment optimization models.

Abstract

Consider the following process whereby balls are distributed into bins. Repeatedly, a ball is removed from a non-empty bin chosen uniformly at random. The process ends when a single non-empty bin remains. Will Ma (see~\cite[Sec.~1.1]{GS24}) asked whether the initial assignment that minimizes the expected number of remaining balls is one that is as balanced as possible. Using a coupling argument we answer this conjecture positively, and we discuss the case of non-uniform choice among the non-empty bins.
Paper Structure (2 sections, 6 theorems, 15 equations)

This paper contains 2 sections, 6 theorems, 15 equations.

Table of Contents

  1. Proof of the main result
  2. Non-uniform selection of bins

Key Result

Theorem 1

Fix $n \in \mathbb{N}$. The quantity $\min\{ f(\vec{n}) \colon \vec{n}\in\mathbb{N}^k, \sum_{i\in [k]} n_i = n \}$ is attained only by assignments $\vec{n} =(n_1,\dots,n_k) \in\mathbb{N}^k$ satisfying $\sum_{i\in [k]}n_i=n$ and $|n_i - n_j| \leq 1$ for every $i,j\in [k]$.

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6