  We study zero-sum games, a variant of the classical combinatorial Subtraction
games (studied for example in the monumental work "Winning Ways", by Berlekamp,
Conway and Guy), called Cumulative Subtraction (CS). Two players alternate in
moving, and get points for taking pebbles out of a joint pile. We prove that
the outcome in optimal play (game value) of a CS with a finite number of
possible actions is eventually periodic, with period $2s$, where $s$ is the
size of the largest available action. This settles a conjecture by Stewart in
his Ph.D. thesis (2011). Specifically, we find a quadratic bound, in the size
of $s$, on when the outcome function must have become periodic. In case of two
possible actions, we give an explicit description of optimal play. We
generalize the periodicity result to games with a so-called reward function,
where at each stage of game, the change of `score' does not necessarily equal
the number of pebbles you collect.
