  The problem of scheduling tasks on $p$ processors so that no task ever gets
too far behind is often described as a game with cups and water. In the
$p$-processor cup game on $n$ cups, there are two players, a filler and an
emptier, that take turns adding and removing water from a set of $n$ cups. In
each turn, the filler adds $p$ units of water to the cups, placing at most $1$
unit of water in each cup, and then the emptier selects $p$ cups to remove up
to $1$ unit of water from. The emptier's goal is to minimize the backlog, which
is the height of the fullest cup.
  The $p$-processor cup game has been studied in many different settings,
dating back to the late 1960's. All of the past work shares one common
assumption: that $p$ is fixed. This paper initiates the study of what happens
when the number of available processors $p$ varies over time, resulting in what
we call the \emph{variable-processor cup game}.
  Remarkably, the optimal bounds for the variable-processor cup game differ
dramatically from its classical counterpart. Whereas the $p$-processor cup has
optimal backlog $\Theta(\log n)$, the variable-processor game has optimal
backlog $\Theta(n)$. Moreover, there is an efficient filling strategy that
yields backlog $\Omega(n^{1 - \epsilon})$ in quasi-polynomial time against any
deterministic emptying strategy.
  We additionally show that straightforward uses of randomization cannot be
used to help the emptier. In particular, for any positive constant $\Delta$,
and any $\Delta$-greedy-like randomized emptying algorithm $\mathcal{A}$, there
is a filling strategy that achieves backlog $\Omega(n^{1 - \epsilon})$ against
$\mathcal{A}$ in quasi-polynomial time.
