Graphs and Games

Playing games is one of the oldest human intellectual activities but is only within the last 100 years that mathematics has offered any real insights. Most of mathematics deals with `nature' or it describes ongoing situations. The mathematics of games has to take in to account an intelligent opponent. For example, consider the problems of capturing an intruder who is fleeing pursuers in a system of passageways. If the intruder is much faster than the pursuers this can be used as a model for decontaminating the passageways from chemical or biological spills. The worst-case scenario is that the contaminant/intruder is an intelligent adversary! The main goal is to identify good strategies that humans can understand and implement. There has been great success in the game theory originating in the fields of economics, biology and psychology where there are many players, simultaneous moves and hidden information. This has been useful in pure strategy games such as Chess, Checkers and Hex. Only in the last 40 years has the foundations of a mathematical theory for "last-player-to-move-wins" games (e.g., Chess, Checkers and Go) been laid. In these cases, even identifying an `infinitesimal' (i.e. a very, very small edge) in a game can gain a move that could earn extra money for professional players. This research will identify and exploit the hidden structures in these games to obtain good strategies. This research will also extend the techniques from both the last-player-to-move and economic game theories to consider pure strategy, scoring games (e.g., Dots-and-Boxes and Reversi).