In an earlier post, I described the *dollar game* played on a finite graph , and mentioned (for the “borrowing binge variant”) that the total number of borrowing moves required to win the game is independent of which borrowing moves you do in which order. A similar phenomenon occurs for the *pentagon game* described in this post.

In this post, I’ll first present a simple general theorem due to Mikkel Thorup which implies both of these facts (and also applies to many other ‘chip firing games’ in the literature). Then, following Anders Björner, Laszlo Lovasz, and Peter Shor, I’ll explain how to place such results into the context of *greedoid languages*, which have interesting connections to matroids, Coxeter groups, and other much-studied mathematical objects.

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