arXiv2026
In the cup game, an adversary distributes $1$ unit of water among $n$ initially empty cups during each time step. The player then selects a single cup from which to remove up to $1$ unit of water, with the goal of minimizing the backlog, i.e., the supremum of the height of the fullest cup over all time steps. In the cup flushing game, the player is additionally allowed to empty the chosen cup entirely. Past work has shown that the optimal backlog in both of these settings is $Θ(\log n)$. Furthermore, the \textbf{greedy} algorithm, which always removes water from the fullest cup, has been shown in previous work to be exactly optimal in both the cup game and the cup flushing game. We introduce a new model, the semi-oblivious cup game, in which the player is uncertain of the exact height of each cup. We analyze the performance of the \textbf{greedy} algorithm in this setting, which can be viewed as selecting an arbitrary cup within a constant multiplicative factor of the fullest cup. We prove matching upper and lower bounds showing that the \textbf{greedy} algorithm achieves a backlog of $Θ(n^{\frac{c-1}{c}})$ in the semi-oblivious cup game. We also establish matching upper and lower bounds of $2^{Θ(\sqrt{\log n})}$ in the semi-oblivious cup flushing game. Finally, we show that in an additive error setting, greedy is actually able to achieve backlog $Θ(\log n)$, via matching upper and lower bounds. All of our lower bounds apply for adaptive adversaries against any (even randomized) algorithm, proving that greedy is asymptotically optimal in the semi-oblivious model.