About a Ball Removal Process on Bins

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.