  The dollar game is a chip-firing game introduced by Baker and Norine (2007)
as a context in which to formulate and prove the Riemann-Roch theorem for
graphs. A divisor on a graph is a formal integer sum of vertices. Each
determines a dollar game, the goal of which is to transform the given divisor
into one that is effective (nonnegative) using chip-firing moves. We use Duval,
Klivans, and Martin's theory of chip-firing on simplicial complexes to
generalize the dollar game and results related to the Riemann-Roch theorem for
graphs to higher dimensions. In particular, we extend the notion of the degree
of a divisor on a graph to a (multi)degree of a chain on a simplicial complex
and use it to establish two main results. The first of these is Theorem 18,
generalizing the fact that if a divisor on a graph has large enough degree (at
least as large as the genus of the graph), it is winnable; and the second is
Corollary 34, generalizing the fact that trees (graphs of genus 0) are exactly
the graphs on which every divisor of degree 0, interpreted as an instance of
the dollar game, is winnable.
