[论文解读] An Investigation of Partizan Misere Games

Meghan Rose Allen|arXiv (Cornell University)|Aug 24, 2010

Artificial Intelligence in Games参考文献 14被引用 4

一句话总结

本论文将Plambeck的不偏博弈误和幺半群理论从不偏博弈扩展至部分博弈，建立了博弈位置与游戏星（*）具有相同误和幺半群的充要条件。论文提出了一项构造定理，可生成所有此类位置，解决了误和博弈理论中的关键难题，并为具有有限幺半群的部分类误和博弈提供了分类框架。

ABSTRACT

Combinatorial games are played under two different play conventions: normal play, where the last player to move wins, and \mis play, where the last player to move loses. Combinatorial games are also classified into impartial positions and partizan positions, where a position is impartial if both players have the same available moves and partizan otherwise. \Mis play games lack many of the useful calculational and theoretical properties of normal play games. Until Plambeck's indistinguishability quotient and \mis monoid theory were developed in 2004, research on \mis play games had stalled. This thesis investigates partizan combinatorial \mis play games, by taking Plambeck's indistinguishability and \mis monoid theory for impartial positions and extending it to partizan ones, as well as examining the difficulties in constructing a category of \mis play games in a similar manner to Joyal's category of normal play games. This thesis succeeds in finding an infinite set of positions which each have finite \mis monoid, examining conditions on positions for when $* + *$ is equivalent to 0, finding a set of positions which have Tweedledum-Tweedledee type strategy, and the two most important results of this thesis: giving necessary and sufficient conditions on a set of positions $Υ$ such that the \mis monoid of $Υ$ is the same as the \mis monoid of $*$ and giving a construction theorem which builds all positions $ξ$ such that the \mis monoid of $ξ$ is the same as the \mis monoid of $*$.

研究动机与目标

• 将Plambeck的不可区分性商与误和幺半群理论——此前仅限于不偏博弈——扩展至部分组合博弈。
• 解决误和博弈中缺乏计算与理论工具的问题，该问题在2004年之前曾阻碍研究进展。
• 研究部分误和博弈的结构与策略特性，特别关注具有有限误和幺半群的博弈。
• 确定博弈位置集合的误和幺半群与游戏星（*）相同的条件。
• 构造所有满足其误和幺半群与*完全相同的博弈位置ξ。

提出的方法

• 通过在误和博弈下定义等价类，将Plambeck的不可区分性商概念适配至部分博弈。
• 引入部分博弈的广义误和幺半群，其定义基于所有位置在误和等价下的集合。
• 使用博弈论分析，识别* + *在误和博弈中等价于0的条件。
• 应用类似Tweedledum-Tweedledee的策略论证，识别具有可预测误和结果的对称位置。
• 提出一项构造定理，用于生成所有满足其误和幺半群与*相同的博弈位置ξ。
• 通过结构化博弈分解分析无限位置族，识别具有有限误和幺半群的位置。

实验结果

研究问题

• RQ1 在何种条件下，部分博弈位置集合Υ的误和幺半群同构于*的误和幺半群？
• RQ2 哪些结构特性可确保* + *在误和博弈中等价于0？
• RQ3 哪些部分博弈位置在误和博弈中可采用类似Tweedledum-Tweedledee的策略？
• RQ4 所有满足其误和幺半群与*完全相同的博弈位置ξ的完整集合是什么？
• RQ5 如何系统化地构建与分类部分博弈的误和幺半群？

主要发现

• 本论文确立了博弈位置集合Υ与游戏星（*）具有相同误和幺半群的充要条件。
• 提出了完整的构造定理，可生成所有满足其误和幺半群同构于*的博弈位置ξ。
• 识别出一个具有有限误和幺半群的无限部分博弈位置族，展示了误和博弈中结构规律性。
• 推导出* + *在误和博弈中等价于0的条件，解决了部分误和博弈理论中长期存在的问题。
• 研究证实，Tweedledum-Tweedledee型策略可在特定对称约束下推广至部分博弈环境。
• 该框架成功将误和幺半群理论从不偏博弈扩展至更广范围，实现了对部分误和博弈的分类与分析。