Games and Graphs

玩游戏是人类最古老的智力活动之一，但直到最近100年，数学才提供了一些真正的见解。大多数数学都与“自然”有关，或者描述正在进行的情况。游戏的数学必须考虑到聪明的对手。例如，考虑捕获在通道系统中逃离追捕者的入侵者的问题。如果入侵者比追赶者快得多，这可以作为清除化学或生物泄漏通道的模型。最坏的情况是，污染物/入侵者是一个聪明的对手！第二个重要的例子是在投票前发挥影响力。主要目标是确定人类可以理解和实施的良好策略。起源于经济学、生物学和心理学领域的博弈论已经取得了巨大的成功，在这些领域中有许多参与者，同时移动和隐藏信息。直到最近50年，“最后走一步赢”游戏（如象棋、跳棋、围棋和Hex）的数学理论基础才被建立起来。一个重要的结果是，一个游戏可以被一个等价的、独特的、最简单的游戏所取代，从而减少识别好策略的复杂性。即使识别出游戏中的“无穷小”（游戏邦注：即非常非常小的优势）也能够为职业玩家带来额外收益，或者在激烈的竞争中扭转局面。在过去的5年里，这一理论得到了显著的扩展，涵盖了得分较高的赢家。该研究将识别和利用这些游戏类别中的隐藏结构，以获得良好的策略。这项研究还将把这一理论扩展到玩家同时移动，但其中一个玩家对对手的移动有更深入的了解的游戏类别。这是“作弊机器人”模型，其中“机器人”足够快地识别出“人类”的动作并做出反应。游戏是确定性的，不涉及任何概率。

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! A second important example is playing for influence before a vote. 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. Only in the last 50 years has the foundations of a mathematical theory for "last-player-to-move-wins" games (e.g., Chess, Checkers, Go, and Hex) been laid. An important result is that a game can be replaced by an equivalent, unique simplest game thereby reducing the complexity in identifying good strategies. 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 or turn an election in tight races. In the last 5 years, the theory has been extended significantly to cover where the winner has the greater score. The proposed research will identify and exploit the hidden structures in these classes of games to obtain good strategies. This research will also extend the theory to classes of games where the players move simultaneously but where one player has advanced knowledge of the opponent's move. This is the `Cheating Robot' model where the `robot' is fast enough to recognize what move the `human' is making and respond. The games are deterministic and do not involve any probabilities.