This 0-person game is played on a graph (so that "neighbour" is defined). It is known under various names, like "chip firing" and "sandpile".
When sinks are present, every state will stabilize after finitely many steps: money disappears into the sinks until nobody has enough left to give one coin to each of his neighbours. (In the sandpile applet the "single step" button does one clock tick, the "go" button runs the time until the state has become stable.)
We have an operation * on Z: given two vectors u and v in Z, let u*v be the vector obtained by first taking the state u+v and then letting that stabilize. The set S together with the operation * is a monoid.
A state u in S is called recurrent if u*z = u for some strictly positive vector z. Let R be the set of recurrent states in S. Then (R,*) is an abelian group.
For every z in Z there is a unique r in R such that u*z = u*r for all u in R. This r is called the reduction of z. In particular, the identity of the group R is the reduction of the zero vector. The elements of R are their own reduction, so that "reduced" and "recurrent" are synonymous. (In the sandpile applet, the "reduce" button computes the reduction of a given state.)
Check with the sandpile applet that doubling the identity and reducing gives the identity again. Why? Show that this property characterizes the identity.
Show that every finitely generated commutative semigroup carries a group on its recurrent elements (provided there are any). How should recurrent be defined here?
Can something be said about the time needed to obtain a stable state?
The avalanche of a single point is simply connected. Why?