About a Ball Removal Process on Bins

TL;DR

This paper proves that in a process removing balls from bins, the initial balanced distribution minimizes the expected remaining balls, confirming a conjecture and exploring non-uniform removal probabilities.

Contribution

It provides a coupling-based proof that the most balanced initial distribution minimizes expected remaining balls, confirming a conjecture and extending to non-uniform cases.

Findings

Balanced initial distributions minimize expected remaining balls.

Coupling argument confirms the conjecture positively.

Analysis includes non-uniform bin removal probabilities.

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.