Two tokens are placed on vertices of a graph. At each time step, one token is chosen and is moved to a random neighboring point. In previous work, Tetali and Winkler studied the Angel strategy for bringing the tokens together as quickly as possible (on average), and the Demon strategy for delaying their collision as long as possible (on average). We build on these results by studying a game version of this process.
In our game, two players take turns choosing the token to move. The Angel player hopes to bring the tokens together while the Demon player tries to keep them apart. We present optimal strategies for both players on stars, different types of directed cycles, and paths. Our proofs employ couplings of random walks as well as strategy stealing arguments.
Bañuelos, Jorge, "A Simultaneous Random Walk Game" (2011). Mathematics, Statistics, and Computer Science Honors Projects. 22.
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