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，数学才提供了任何真实的见解。大多数数学都涉及“自然”，或者描述正在发生的情况。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捕者的入侵者的问题。如果入侵者比追赶者快得多，这可以用作净化通道的化学或生物泄漏的模型。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。博弈论起源于经济学、生物学和心理学领域，在博弈论中有许多参与者，同时行动和隐藏信息。只有在过去的50年里，“最后一个移动的玩家获胜”游戏的数学理论才有了基础（例如，国际象棋、象棋、围棋和十六进制）已奠定。一个重要的结果是，一个游戏可以被一个等效的，唯一的最简单的游戏，从而减少了识别好的策略的复杂性。即使确定一个“无穷小”（即，一个非常，非常小的优势）在游戏中可以获得一个举动，可以赚取额外的钱，为专业球员或扭转选举在紧张的比赛。在过去的5年里，该理论已被大大扩展，以涵盖赢家有更大的分数。这项研究将识别和利用这些类别的游戏中隐藏的结构，以获得良好的战略。这项研究还将扩展的理论，以类游戏的球员同时移动，但其中一个球员有先进的知识，对手的举动。这是“作弊机器人”模型，其中“机器人”足够快，可以识别“人类”正在进行的移动并做出反应。博弈是确定性的，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.