关于非对称单架洗牌后的猜牌问题
On card guessing after an asymmetric single-shelf shuffle
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中文总结 AI 辅助
研究非对称单架洗牌后猜牌游戏，明确最优策略并运用解析组合学方法研究正确猜测数量，找出其分布、均值和方差，证明对数凹性及极限行为、相变等，此前\(p = 1/2\)特殊情况研究较局限。
中文摘要 AI 辅助
我们对参数为\(p\in(0,1)\)的非对称单架洗牌后完全反馈猜牌游戏中正确猜测的数量进行了确定性分析。明确描述了使正确猜测期望数量最大化的最优策略。运用解析组合学方法研究了最优策略下正确猜测的数量，找出了其明确分布及均值和方差，证明其分布是对数凹的。还研究了牌数趋于无穷时正确猜测数量的极限行为，证明了（局部）中心极限定理和大偏差原理，以及在\(p = 0\)和\(p = 1\)附近正确猜测数量的相变。此前仅知道\(p = 1/2\)特殊情况下的最优策略及最优策略下正确猜测的期望数量。
英文摘要
We provide a definitive analysis of the number of correct guesses in the complete-feedback card guessing game after an asymmetric single-shelf shuffle with parameter $p\in (0, 1)$. We explicitly describe the optimal strategy that maximizes the expected number of correct guesses. We study the number of correct guesses, under an optimal strategy, using methods from analytic combinatorics. In addition, we find the explicit distribution for the number of correct guesses, and thus find the mean and the variance for the number of correct guesses. We show that the distribution of the number of correct guesses is log-concave. We also study the limiting behaviour of the number of correct guesses as the number of cards goes to infinity. In particular, we prove a (local) central limit theorem and a large deviation principle with an explicit rate function. We also prove phase transitions for the number of correct guesses near $p=0$ and $p=1$. Prior to this work, only the optimal strategy and the expected number of correct guesses under the optimal strategy were known for the special case of $p=1/2$.
