“帽子”博弈中的集团与构造子
Cliques and constructors in "Hats'"game
摘要：
下面分析确定性帽子博弈的一般变体。几个戴着彩色帽子的圣人占据了一个图的顶点，第$k$个圣人可以有$h(k)$颜色之一的帽子。每个圣人都试图猜测自己的帽子的颜色，只是根据观察他的邻居的帽子，而不交换任何信息。如果一个预先确定的猜测策略保证每个颜色分配至少有一个正确的个人猜测，那么它就是获胜的。对于完全图和循环，我们解决了描述函数$h(k)$的圣人获胜的问题。在这里，我们证明了完全图上的圣人的获胜策略，并分析了几乎完全图上的帽子游戏。我们开发了“构造器理论”，这是一个定理的集合，展示了如何构建新的图形，其中圣人获胜。我们还定义了一个新的与4圈上的Hats对策等价的对策“车来检查”，并对该对策进行了全面的分析。
英文摘要：
The following general variant of deterministic Hats game is analyzed. Several sages wearing colored hats occupy the vertices of a graph, the $k$-th sage can have hats of one of $h(k)$ colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. For complete graphs and for cycles we solve the problem of describing functions $h(k)$ for which the sages win. We demonstrate here winning strategies for the sages on complete graphs, and analyze the Hats game on almost complete graphs. We develop "theory of constructors", that is a collection of theorems demonstrating how one can construct new graphs for which the sages win. We define also new game "Check by rook" which is equivalent to Hats game on 4-cycle and give complete analysis of this game.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
