$k$ -一致超图上的服务员-顾客和顾客-服务员着色对策及$k$ - SAT对策
Waiter-Client and Client-Waiter colourability games on a $k$-uniform hypergraph and the $k$-SAT game
摘要：
服务员-客户和客户-服务员博弈是两个玩家的完全信息博弈，没有机会移动，在有限集（板）上进行，具有称为获胜集的特殊子集。每一轮有偏见的$(1:q)$游戏开始时，服务员提供$q+1$以前无人认领的棋盘元素给客户，客户认领一个。剩余的$q$元素然后由Waiter声明。如果客户在所有棋盘元素都被提供时完全声称获胜，他在客户-服务员游戏中获胜，在服务员-客户游戏中失败。我们给出了一个估计的阈值偏差的$(1:q)$服务员-客户端和客户端-服务员版本的两个不同的游戏：非2-colourability游戏，发挥完整的$k$ -均匀超图，和$k$ -SAT游戏。特别是，我们表明，唯一的价值$q$的赢家的客户端-服务员版本的非2-colorability游戏的变化是$\frac{1}{n}\binom{n}{k}2^{-k(1+o_k(1))}$和，对于服务员-客户端版本，相应的价值$q$是$\frac{1}{n}\binom{n}{k}2^{Θ_k(k)}$。此外，我们表明，阈值偏差的服务员-客户端和客户端-服务员版本的$k$ -SAT游戏是$\frac{1}{n}\binom{n}{k}$的一个因素，是指数和多项式分别在$k$。这表明，这些游戏表现出“概率直觉”。
英文摘要：
Waiter-Client and Client-Waiter games are two-player, perfect information games, with no chance moves, played on a finite set (board) with special subsets known as the winning sets. Each round of the biased $(1:q)$ game begins with Waiter offering $q+1$ previously unclaimed elements of the board to Client, who claims one. The $q$ elements remaining are then claimed by Waiter. If Client fully claims a winning set by the time all board elements have been offered, he wins in the Client-Waiter game and loses in the Waiter-Client game. We give an estimate for the threshold bias of the $(1:q)$ Waiter-Client and Client-Waiter versions of two different games: the non-2-colourability game, played on the complete $k$-uniform hypergraph, and the $k$-SAT game. In particular, we show that the unique value of $q$ at which the winner of the Client-Waiter version of the non-2-colourability game changes is $\frac{1}{n}\binom{n}{k}2^{-k(1+o_k(1))}$ and, for the Waiter-Client version, the corresponding value of $q$ is $\frac{1}{n}\binom{n}{k}2^{Θ_k(k)}$. Additionally, we show that the threshold bias for the Waiter-Client and Client-Waiter versions of the $k$-SAT game is $\frac{1}{n}\binom{n}{k}$ up to a factor that is exponential and polynomial in $k$ respectively. This shows that these games exhibit the "probabilistic intuition".
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。
