Chip-firing game

A move starts with selecting a vertex w which has at least as many chips as its degree: s(w) ≥ deg(w).

Starting from an initial configuration, the game proceeds with the following results (on a connected graph).

For finite chip-firing games, the possible orders of firing events can be described by an antimatroid.

Two variants of dollar game are prominent in the literature: In this dollar game, negative integer values (representing debt) are assigned to some of the vertices of the finite graph G. A game is called winnable if there exists a state where all the vertices can be made positive.

[2][3] In a variant form of chip-firing closely related to the sandpile model, also known as the dollar game, a single special vertex q is designated as the bank, and is allowed to go into debt, taking a negative integer value s(q) < 0.

Eventually, q will accumulate so many chips that no other vertex can fire: only in such a state, vertex q can fire chips to neighbouring vertices to "jump start the economy".

The set of states which are stable (i.e. for which only q can fire) and recurrent for this game can be given the structure of an abelian group which is isomorphic to the direct product of