About a ball removal process on bins

Abstract

We consider a basic balls-and-bins question in which you have to distribute $n$ balls into $k$ bins. Then, round by round, a ball is removed from a non-empty bin chosen uniformly at random. The process ends when a single non-empty bin remains. The goal is to minimize the expected number of remaining balls. An open problem posed by Will Ma asks 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.