Given a connected, undirected graph, two players (black and red) play the following game:
Note: Initially, all vertices do not contain any tokens.
- To begin the game, each player (black or red) receives a designated an amount (equal) of tokens.
- Players alternate turns. For each turn, a player puts his colored token (black or red) on any vertex on the graph.
- The player that has the majority of tokens and reaches the threshold (vertex degree) of that vertex wins all of the tokens on this specific vertex and the vertex topples ("Explodes"). The player then donates one token of the wining color to each of its neighbors. This may, in turn, cause some of the neighbors to topple, and so on. Note: In the case of a tie in an even degree vertex, the last colored token is the winner of this vertex.
- A player wins when he has managed to have a majority of his colored tokens in the graph: • when everyone finishes their initial tokens, and • when the game reaches the infinite topple—as can happen in the chip firing game.
I would be interested in any nontrivial statements for interesting tactics/heuristics to solve this game—how the classes of graphs (e.g., complete graphs, grid graphs, whatever) affect the game.