Modeling billiards games

Two-player games of billiards, of the sort seen in recent Computer Olympiads held by the International Computer Games Association, are an emerging area with unique challenges for A.I. research. Drawing on the insight gained from our victory in the 2008 ICGA billiards tournament, we define a game-theoretic model of these types of billiards games. The modeling is surprisingly subtle. While sharing features with existing models (including stochastic games, games on a square, recursive games, and extensive form games), our model is distinct, and consequently requires novel analysis. We focus on the basic question of whether the game has an equilibrium. For finite versions of the game it is not hard to show the existence of a pure strategy Markov perfect Nash equilibrium. In the infinite case, it can be shown that under certain conditions a stationary pure strategy Markov perfect Nash equilibrium is guaranteed to exist.