About a Ball Removal Process on Bins

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 nonempty bin chosen uniformly at random. The process ends when a single nonempty 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 nonuniform choice among the nonempty bins. Funding: Financial support by the Israel Science Foundation [Grant 211/22] and by Agencia Nacional de Investigación y Desarrollo Chile through the Programa de Investigación Asociativa (Project FB210005) and Fondo Nacional de Desarrollo Científico y Tecnológico [Grant 1260036] is gratefully acknowledged.