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Given a connected, undirected graph, two players (black and red) play the following game:

Note: Initially, all vertices do not contain any tokens.

  1. To begin the game, each player (black or red) receives a designated an amount (equal) of tokens.
  2. Players alternate turns. For each turn, a player puts his colored token (black or red) on any vertex on the graph.
  3. 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.
  4. 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.

I built a demo game application [4MB] so you can test the game

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closed as too broad by Ramiro, bengoesboom, Jonathan Hobbs, Pat Ludwig Jul 20 at 17:44

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

1  
This is a very open-ended question. –  Joe Jun 12 at 16:32
    
What happens when a topple causes two neighbors to topple at once? In particular, if chain toppling causes a vertex to overload, reaching a token count greater than its degree before its topple can be resolved, what happens to the extra tokens? –  user2357112 Jun 13 at 22:42
    
I'm firing by vertex degree(High -> Low). Extra tokens stay in the vertex. –  Shimon Jun 14 at 12:00

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