Graphs and Games

玩游戏是人类最古老的智力活动之一，但直到最近100年，数学才提供了真正的见解。大多数数学都是关于“自然”的，或者说它描述的是正在进行的情况。游戏的数学必须考虑到一个聪明的对手。例如，考虑在通道系统中捕获正在逃离追踪者的入侵者的问题。如果入侵者比追赶者快得多，这可以作为清除通道中化学或生物泄漏的模型。最糟糕的情况是，污染物/入侵者是一个聪明的对手！主要目标是确定人类可以理解和实施的良好战略。起源于经济学、生物学和心理学领域的博弈论在参与者多、同时行动和隐藏信息等领域取得了巨大的成功。这在纯策略游戏中很有用，如国际象棋、跳棋和十六进制。只有在过去的40年里，才奠定了“最后一步获胜”游戏(例如国际象棋、跳棋和围棋)的数学理论基础。在这种情况下，即使在一场比赛中识别出一个“无穷小”(即一个非常、非常小的优势)也可以获得一步棋，这可能会为职业球员赚取额外的钱。本研究将识别和挖掘这些游戏中的隐藏结构，以获得良好的策略。这项研究还将把技术从最后一个玩家移动的博弈论和经济博弈论扩展到纯粹的战略，得分游戏(例如，点和盒和反转)。

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).